Full review

Paper 1: OverviewBy Terezinha Nunes, Peter Bryant and Anne Watson, University of Oxford

Key understandings in

mathematics learning

A review commissioned by the Nuffield Foundation

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2 SUMMARY – PAPER 2: Understanding whole numbers2 Paper 1: Overview

In 2007, the Nuffield Foundation commissioned ateam from the University of Oxford to review theavailable research literature on how children learnmathematics. The resulting review is presented in aseries of eight paper s:

Paper 1: OverviewPaper 2: Understanding extensive quantities and

whole numbersPaper 3: Understanding rational numbers and

intensive quantitiesPaper 4: Understanding relations and their graphical

representationPaper 5: Understanding space and its representation

in mathematicsPaper 6: Algebraic reasoningPaper 7: Modelling, problem-solving and integrating

conceptsPaper 8: Methodological appendix

Papers 2 to 5 focus mainly on mathematics relevantto primary schools (pupils to age 11 y ears), whilepapers 6 and 7 consider aspects of mathematics in secondary schools.

Paper 1 includes a summar y of the review, which has been published separately as Introduction andsummary of findings.

Summaries of papers 1-7 have been publishedtogether as Summary papers.

All publications are available to download from our website, www.nuffieldfoundation.org

ContentsSummary of findings 3

Overview 10

References 36

About the authorsTerezinha Nunes is Professor of EducationalStudies at the University of Oxford. Peter Bryant is Senior Research Fellow in theDepartment of Education, University of Oxford.Anne Watson is Professor of MathematicsEducation at the University of Oxford.

About the Nuffield FoundationThe Nuffield Foundation is an endowedcharitable trust established in 1943 by WilliamMorris (Lord Nuffield), the founder of MorrisMotors, with the aim of advancing social w ellbeing. We fund research and pr acticalexperiment and the development of capacity to under take them; working across education,science, social science and social policy. Whilemost of the Foundation’s expenditure is onresponsive grant programmes we also undertake our own initiatives.

About this review

3 Key understandings in mathematics lear ning

AimsOur aim in the review is to present a synthesis of research on mathematics lear ning by childrenfrom the age of f ive to the age of sixteen years and to identify the issues that are fundamental tounderstanding children’s mathematics learning. Indoing so, we concentrated on three main questionsregarding key understandings in mathematics.

• What insights must students have in order tounderstand basic mathematical concepts?

• What are the sources of these insights and howdoes informal mathematics knowledge relate to school learning of mathematics?

• What understandings must students have inorder to build new mathematical ideas usingbasic concepts?

Theoretical framework

While writing the review, we concluded that thereare two distinct types of theor y about how childrenlearn mathematics.

Explanatory theories set out to explain how children’s mathematical thinking and knowledgechange. These theories are based on empir icalresearch on children’s solutions to mathematicalproblems as well as on exper imental and longitudinalstudies. Successful theories of this sor t shouldprovide insight into the causes of children’smathematical development and worthwhilesuggestions about teaching and lear ning mathematics.

Pragmatic theories set out to investigate what childrenought to learn and understand and also identifyobstacles to learning in formal educational settings.

Pragmatic theories are usually not tested for theirconsistency with empirical evidence, nor examinedfor the parsimony of their explanations vis-à-vis otherexisting theories; instead they are assessed in m ultiplecontexts for their descriptive power, their credibilityand their effectiveness in practice.

Our star ting point in the review is that children needto learn about quantities and the relations betweenthem and about mathematical symbols and theirmeanings. These meanings are based on sets ofrelations. Mathematics teaching should aim to ensurethat students’ understanding of quantities, relationsand symbols go together.

Conclusions

This theoretical approach underlies the six mainsections of the review. We now summarise the mainconclusions of each of these sections.

Whole numbers• Whole numbers represent both quantities and

relations between quantities, such as differencesand ratio. Primary school children must establishclear connections between numbers, quantitiesand relations.

• Children’s initial understanding of quantitativerelations is largely based on correspondence.One-to-one correspondence underlies theirunderstanding of cardinality, and one-to-manycorrespondence gives them their first insightsinto multiplicative relations. Children should be

Summary of findings

4 SUMMARY – PAPER 2: Understanding whole numbers

encouraged to think of number in terms of theserelations.

• Children star t school with varying levels of ability in using different action schemes to solvearithmetic problems in the context of stor ies.They do not need to know arithmetic facts tosolve these problems: they count in differentways depending on whether the prob lems they are solving involve the ideas of addition,subtraction, multiplication or division.

• Individual differences in the use of actionschemes to solve problems predict children’sprogress in learning mathematics in school.

• Interventions that help children lear n to use theiraction schemes to solve problems lead to betterlearning of mathematics in school.

• It is more difficult for children to use numbers torepresent relations than to represent quantities.

Implications for the classroomTeaching should make it possible for children to:

• connect their knowledge of counting with theirknowledge of quantities

• understand additive composition and one-to-many correspondence

• understand the inverse relation between additionand subtraction

• solve problems that involve these keyunderstandings

• develop their multiplicative understandingalongside additive reasoning.

Implications for further researchLong-term longitudinal and intervention studies with large samples are needed to suppor t curriculumdevelopment and policy changes aimed atimplementing these objectives. There is also a need for studies designed to promote children’scompetence in solving problems about relations.

Fractions• Fractions are used in pr imary school to represent

quantities that cannot be represented by a singlewhole number. As with whole numbers, childrenneed to make connections between quantitiesand their representations in fr actions in order tobe able to use fractions meaningfully.

• Two types of quantities that are taught inprimary school must be represented by fractions.The first involves measurement: if you want torepresent a quantity by means of a number and the quantity is smaller than the unit ofmeasurement, you need a fraction; for example, a half cup or a quar ter inch. The second involvesdivision: if the dividend is smaller than the divisor,the result of the division is represented b y afraction; for example, three chocolates sharedamong four children.

• Children use different schemes of action in thesetwo different situations. In division situations, theyuse correspondences between the units in thenumerator and the units in the denominator. Inmeasurement situations, they use par titioning.

• Children are more successful in under standingequivalence of fractions and in order ing fractionsby magnitude in situations that in volve divisionthan in measurement situations.

• It is crucial for children’s understanding of fractionsthat they learn about fractions in both types ofsituation: most do not spontaneously transfer whatthey learned in one situation to the other.

• When a fraction is used to represent a quantity,children need to learn to think about how thenumerator and the denominator relate to thevalue represented by the fraction. They must thinkabout direct and inverse relations: the larger thenumerator, the larger the quantity, but the largerthe denominator, the smaller the quantity.

• Like whole numbers, fractions can be used torepresent quantities and relations betweenquantities, but they are rarely used to representrelations in primary school. Older students oftenfind it difficult to use fractions to represent relations.


• use their understanding of quantities in divisionsituations to understand equivalence and orderof fractions

• make links between different types of reasoningin division and measurement situations

• make links between understanding fractionalquantities and procedures

• learn to use fractions to represent relationsbetween quantities, as well as quantities.

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Implications for further researchEvidence from experimental studies with largersamples and long-term interventions in the classroomare needed to establish how division situations relateto learning fractions. Investigations on how linksbetween situations can be built are needed to suppor tcurriculum development and classroom teaching.

There is also a need f or longitudinal studies designedto clarify whether separation between proceduresand meaning in fractions has consequences forfurther mathematics learning.

Given the impor tance of understanding andrepresenting relations numerically, studies thatinvestigate under what circumstances pr imary schoolstudents can use fractions to represent relationsbetween quantities, such as in propor tionalreasoning, are urgently needed.

Relations and their mathematicalrepresentation

• Children have greater difficulty in understandingrelations than in understanding quantities. This is true in the context of both additiv e andmultiplicative reasoning problems.

• Primary and secondar y school students oftenapply additive procedures to solve multiplicativeproblems and multiplicative procedures to solveadditive problems.

• Teaching designed to help students become awareof relations in the context of additive reasoningproblems can lead to significant improvement.

• The use of diagrams, tables and graphs torepresent relations in multiplicative reasoningproblems facilitates children’s thinking about thenature of the relations between quantities.

• Excellent curriculum development work hasbeen carried out to design progr ammes thathelp students develop awareness of their implicitknowledge of multiplicative relations. This workhas not been systematically assessed so far.

• An alternative view is that students’ implicitknowledge should not be the star ting point forstudents to learn about propor tional relations;teaching should focus on formalisations ratherthan informal knowledge and only later seek toconnect mathematical formalisations with applied

situations.This alternative approach has also notbeen systematically assessed yet.

• There is no research that compares the results of these diametrically opposed ideas.


• distinguish between quantities and relations• become explicitly aware of the different types

of relations in different situations• use different mathematical representations to

focus on the relevant relations in specif ic problems• relate informal knowledge and formal learning.

Implications for further researchEvidence from experimental and long-termlongitudinal studies is needed on which approaches to making students aware of relations in problemsituations improve problem solving. A study comparingthe alternative approaches – star ting from informalknowledge versus starting from formalisations –would make a significant contribution to the literature.

Space and its mathematicalrepresentation

• Children come to school with a great deal ofinformal and often implicit knowledge aboutspatial relations. One challenge in mathematicaleducation is how best to harness this knowledgein lessons about space .

• This pre-school knowledge of space is mainlyrelational. For example, children use a stablebackground to remember the position andorientation of objects and lines.

• Measuring length and area poses par ticularproblems for children, even though they are able to understand the underlying logic ofmeasurement. Their difficulties concern iteration ofstandard units and the need to appl y multiplicativereasoning to the measurement of area.

• From an ear ly age children are ab le toextrapolate imaginary straight lines, which allows them to learn how to use Car tesian co-ordinatesto plot specific positions in space with littledifficulty. However, they need help from teacher son how to use co-ordinates to work out therelation between different positions.



• Learning how to represent angle mathematicall yis a hard task for young children, even thoughangles are an impor tant par t of their everydaylife. Initially children are more aware of angle inthe context of movement (turns) than in othercontexts. They need help from to teacher s to beable to relate angles across diff erent contexts.

• An important aspect of learning about geometryis to recognise the relation between transformedshapes (rotation, reflection, enlargement). This canbe difficult, since children’s preschool experienceslead them to recognise the same shapes asequivalent across such transformations, rather thanto be aware of the nature of the tr ansformation.

• Another aspect of the under standing of shape is the fact that one shape can be tr ansformedinto another by addition and subtraction of itssubcomponents. For example, a parallelogramcan be transformed into a rectangle of the samebase and height by the addition and subtractionof equivalent tr iangles. Research demonstrates adanger that children learn these transformationsas procedures without under standing theirconceptual basis.


• build on spatial relational knowledge fromoutside school

• relate their knowledge of relations andcorrespondence to the conceptual basis ofmeasurement

• iterate with standard and non-standard units• understand the difference between

measurements which are/are not multiplicative• relate co-ordinates to extrapolating imaginary

straight lines• distinguish between scale enlargements and area

enlargements.

Implications for further researchThere is a ser ious need for longitudinal research on the possible connections between children’s pre-school spatial abilities and how well they learnabout geometry at school.

Psychological research is needed on: children’s abilityto make and understand transformations and theadditive relations in compound shapes; the exactcause of children’s difficulties with iteration; how

transitive inference, inversion and one-to-onecorrespondence relate to problems with geometr y,such as measurement of length and area.

There is a need for intervention studies on methodsof teaching children to work out the relationbetween different positions, using co-ordinates.

Algebra• Algebra is the way we express generalisations

about numbers, quantities, relations and functions.For this reason, good understanding of connectionsbetween numbers, quantities and relations isrelated to success in using algebr a. In par ticular,understanding that addition and subtr action areinverses, and so are multiplication and division,helps students understand expressions and solveequations.

• To understand algebraic symbolisation, studentshave to (a) under stand the under lying operationsand (b) become fluent with the notational r ules.These two kinds of learning, the meaning and thesymbol, seem to be most successful whenstudents know what is being expressed and havetime to become fluent at using the notation.

• Students have to learn to recognise the differentnature and roles of letter s as: unknowns,variables, constants and parameters, and also themeanings of equality and equivalence . Thesemeanings are not always distinct in algebra anddo not relate unambiguously to arithmeticalunderstandings.

• Students often get confused, misapply, ormisremember rules for transforming expressionsand solving equations. They often try to applyarithmetical meanings inappropriately toalgebraic expressions. This is associated withover-emphasis on notational manipulation, or on ‘generalised arithmetic’, in which they may try to get concise answers.


• read numerical and algebraic expressionsrelationally, rather than as instr uctions to calculate (as in substitution)

• describe generalisations based on proper ties(arithmetical rules, logical relations, structures) as well as inductive reasoning from sequences

6 Paper 1: Overview

• use symbolism to represent relations• understand that letters and ‘=’ have a range of

meanings• use hands-on ICT to relate representations• use algebra purposefully in multiple experiences

over time• explore and use algebraic manipulation software.

Implications for further researchWe need to know how explicit work onunderstanding relations between quantities enablesstudents to move successfully between arithmeticalto algebraic thinking.

Research on how expressing generality enablesstudents to use algebra is mainly in small-scaleteaching interventions, and the problems of large-scale implementation are not so well repor ted. Wedo not know the longer-term comparative effects ofdifferent teaching approaches to ear ly algebra onstudents’ later use of algebr aic notation and thinking.

There is little research on higher algebr a, except for teaching experiments involving functions. Howlearners synthesise their knowledge of elementar yalgebra to understand polynomial functions, theirfactorisation and roots, simultaneous equations,inequalities and other algebraic objects beyondelementary expressions and equations is not kno wn.

There is some research about the use of symbolicmanipulators but more needs to be lear ned aboutthe kinds of algebraic exper tise that developsthrough their use.

Modelling, solving problems andlearning new concepts in secondarymathematicsStudents have to be fluent in under standing methodsand confident about using them to know why and when to apply them, but such application does notautomatically follow the learning of procedures. Studentshave to understand the situation as well as to be able tocall on a familiar reper toire of facts, ideas and methods.

Students have to know some elementar y concepts well enough to apply them and combine them to form new concepts in secondar ymathematics. For example, knowing a range offunctions and/or their representations seems to benecessary to understand the modelling process, andis cer tainly necessary to engage in modelling.

Understanding relations is necessar y to solveequations meaningfully.

Students have to learn when and how to useinformal, experiential reasoning and when to use formal, conventional, mathematical reasoning.Without special attention to meanings, manystudents tend to apply visual reasoning, or betriggered by verbal cues, rather than analysesituations to identify var iables and relations.

In many mathematical situations in secondar ymathematics, students have to look for relationsbetween numbers, and variables, and relationsbetween relations, and proper ties of objects, andknow how to represent them.


• learn new abstract understandings, which isneither achieved through learning procedures,nor through problem-solving activities, withoutfurther intervention

• use their obvious reactions to perceptions andbuild on them, or understand conflicts with them

• adapt to new meanings and develop from ear liermethods and conceptualizations over time

• understand the meaning of new concepts ‘knowabout’, ‘know how to’, and ‘know how to use’

• control switching between, and compar ing,representations of functions in order tounderstand them

• use spreadsheets, graphing tools, and othersoftware to suppor t application and authenticuse of mathematics.

Implications for further researchExisting research suggests that where contextual andexploratory mathematics, integrated through thecurriculum, do lead to fur ther conceptual learning it isrelated to conceptual learning being a r igorous focusfor curriculum and textbook design, and in teacherpreparation, or in specifically designed projects basedaround such aims. There is therefore an urgent need forresearch to identify the key conceptual understandingsfor success in secondar y mathematics. There is noevidence to convince us that the new U.K. curricula willnecessarily lead to better conceptual understanding ofmathematics, either at the elementar y level which isnecessary to learn higher mathematics, or at higherlevels which provide the confidence and foundation for further mathematical study.



We need to under stand the ways in which studentslearn new ideas in mathematics that depend oncombinations of ear lier concepts, in secondar yschool contexts, and the characteristics ofmathematics teaching at higher secondar y levelwhich contribute both to successful conceptuallearning and application of mathematics.

Common themes

We reviewed different areas of mathematical activity,and noted that many of them involve commonthemes, which are fundamental to lear ningmathematics: number, logical reasoning, reflection onknowledge and tools, understanding symbol systemsand mathematical modes of enquir y.

NumberNumber is not a unitar y idea, which children learn in a linear fashion. Number develops incomplementary strands, sometimes withdiscontinuities and changes of meaning. Emphasis onprocedures and manipulation with numbers, ratherthan on understanding the under lying relations andmathematical meanings, can lead to over-relianceand misapplication of methods in ar ithmetic, algebra,and problem-solving. For example, if children formthe idea that quantities are onl y equal if they arerepresented by the same number, a principle thatthey could deduce from lear ning to count, they willhave difficulty understanding the equivalence offractions. Learning to count and to under standquantities are separate strands of development.Teaching can play a major role in helping childrenco-ordinate these two forms of knowledge withoutmaking counting the only procedure that can beused to think about quantities.

Successful learning of mathematics includesunderstanding that number describes quantity; beingable to make and use distinctions between different,but related, meanings of number; being able to userelations and meanings to inform application andcalculation; being able to use number relations tomove away from images of quantity and usenumber as a structured, abstract, concept.

Logical reasoningThe evidence demonstrates beyond doubt thatchildren must rely on logic to lear n mathematics and

that many of their difficulties are due to failures tomake the correct logical move that would have ledthem to the correct solution. Four different aspectsof logic have a crucial role in learning aboutmathematics.

The logic of cor respondence (one-to-one and one-to-many correspondence) The extension of the use ofone-to-one correspondence from sharing to workingout the numerical equivalence or non-equivalence oftwo or more spatial ar rays is a vastly important stepin early mathematical learning. Teaching multiplicationin terms of one-to-many correspondence is moreeffective than teaching children about multiplicationas repeated addition.

The logic of inversion Longitudinal evidence shows thatunderstanding the inverse relation between additionand subtraction is a strong predictor of children’smathematical progress. A flexible understanding ofinversion is an essential element in children’sgeometrical reasoning as well. The concept ofinversion needs a great deal more prominence than it has now in the school curriculum.

The logic of class inclusion and additive compositionClass inclusion is the basis of the under standing ofordinal number and the number system. Children’sability to use this form of inclusion in lear ning aboutnumber and in solving mathematical prob lems is atfirst rather weak, and needs some support.

The logic of transitivity All ordered series, includingnumber, and also forms of measurement involvetransitivity (a > c if a > b and b > c: a = c if a = band b = c). Learning how to use transitive relationsin numerical measurements (for example, of area) isdifficult. One reason is that children often do not grasp the impor tance ofiteration (repeated units of measurement).

The results of longitudinal research (although there isnot an exhaustive body of such work) support theidea that children’s logic plays a critical par t in theirmathematical learning.

Reflection on knowledge and toolsChildren need to re-conceptualise their intuitiv emodels about the world in order to access themathematical models that have been developed inthe discipline. Some of the intuitive models used bychildren lead them to appropr iate mathematicalproblem solving, and yet they may not know why

8 Paper 1: Overview

they succeeded. Implicit models can interfere withproblem solving when students rely on assumptionsthat lead them astray.

The fact that students use intuitiv e models when learning mathematics, whether the teacherrecognises the models or not, is a reason forhelping them to develop an awareness of theirmodels. Students can explore their intuitive modelsand extend them to concepts that are less intuitiv e,more abstract. This pragmatic theory has beenshown to have an impact in pr actice.

Understanding symbol systemsSystems of symbols are human in ventions and thus are cultural tools that have to be taught.Mathematical symbols are human-made tools thatimprove our ability to control and adapt to theenvironment. Each system makes specific cognitivedemands on the learner, who has to understand thesystems of representation and relations that arebeing represented; for example place-value notationis based on additive composition, functions depictcovariance. Students can behave as if theyunderstand how the symbols work while they donot understand them completely: they can learnroutines for symbol manipulation that remaindisconnected from meaning. This is true of rationalnumbers, for example.

Students acquire informal knowledge in theireveryday lives, which can be used to giv e meaningto mathematical symbols learned in the classroom.Curriculum development work that takes thisknowledge into account is not as widespread as onewould expect given discoveries from past research.

Mathematical modes of enquirySome important mathematical modes of enquiryarise in the topics covered in this synthesis.

Comparison helps us make new distinctions and createnew objects and relations Comparisons are related tomaking distinctions, sor ting and classifying; studentsneed to learn to make these distinctions based onmathematical relations and properties, rather thanperceptual similarities.

Reasoning about properties and relations rather thanperceptions Throughout mathematics, students haveto learn to interpret representations before theythink about how to respond. They need to think

about the relations between different objects in thesystems and schemes that are being represented.

Making and using representations Conventionalnumber symbols, algebraic syntax, coordinategeometry, and graphing methods, all affordmanipulations which might otherwise be impossib le.Coordinating different representations to explore andextend meaning is a fundamental mathematical skill.

Action and reflection-on-action In mathematics, actions may be physical manipulation, or symbolicrearrangement, or our obser vations of a dynamicimage, or use of a tool. In all these contexts, weobserve what changes and what stays the same as a result of actions, and make inferences about theconnections between action and effect.

Direct and inverse relations It is impor tant in allaspects of mathematics to be ab le to construct anduse inverse reasoning. As well as enabling moreunderstanding of relations between quantities, thisalso establishes the importance of reverse chains ofreasoning throughout mathematical problem-solving,algebraic and geometrical reasoning.

Informal and formal reasoning At first young childrenbring everyday understandings into school andmathematics can allow them to formalise these andmake them more precise . Mathematics also providesformal tools, which do not descr ibe everydayexperience, but enable students to solve problemsin mathematics and in the world which would beunnoticed without a mathematical per spective.

Epilogue

We have made recommendations about teaching andlearning, and hope to have made the reasoning behindthese recommendations clear to educationalists (inthe extended review). We have also recognised thatthere are weaknesses in research and gaps in currentknowledge, some of which can be easil y solved byresearch enabled by significant contributions of pastresearch. Other gaps may not be so easily solved, andwe have described some pragmatic theories that areor can be used by teachers when they plan theirteaching. Classroom research stemming from theexploration of these theor ies can provide new insights for further research in the future , alongsidelongitudinal studies which focus on learningmathematics from a psychological per spective.



AimsOur aim in this review is to present a synthesis ofresearch on key aspects of mathematics learning bychildren from the age of 5 to the age of 16 years:these are the ages that compr ise compulsoryeducation in the United Kingdom In prepar ing thereview, we have considered the results of a lar gebody of research car ried out by psychologists and bymathematics educators over approximately the lastsix decades. Our aim has been to dev elop atheoretical analysis of these results in order to attaina big picture of how children learn, and sometimesfail to learn, mathematics and how they could learn itbetter. Our main target is not to provide an answerto any specific question, but to identify issues thatare fundamental to under standing children’smathematics learning. In our view theories ofmathematics learning should deal with three mainquestions regarding key understandings inmathematics: • What insights must students have in order to

understand basic mathematical concepts?• What are the sources of these insights and how

does informal mathematics knowledge relate toschool learning of mathematics?

• What understandings must students have in orderto build new mathematical ideas using basicconcepts?

Theoretical analysis played a major role in thissynthesis. Many theoretical ideas were alreadyavailable in the literature and we sought to examinethem critically for coherence and for consistencywith the empir ical evidence. Cooper (1998) suggeststhat there may be occasions when new theoreticalschemes must be developed to provide anoverarching understanding of the higher-orderrelations in the research domain; this was cer tainly

true of some of our theoretical anal ysis of theevidence that we read for this review.

The answers to our questions should allow us totrace students’ learning trajectories. Confrey (2008)defined a learning trajectory as ‘a researcher-conjectured, empirically-supported description of the ordered network of experiences a studentencounters through instruction (i.e. activities, tasks,tools, forms of interaction and methods ofevaluation), in order to move from informal ideas,through successive refinements of representation,articulation, and reflection, towards increasinglycomplex concepts over time.’ If students’ learningtrajectories towards understanding specific conceptsare generally understood, teachers will be muchbetter placed to promote their advancement.

Finally, one of our aims has been to identify a set of research questions that stem from our cur rentknowledge about children’s mathematics learning andmethods that can provide relevant evidence aboutimportant, outstanding issues.

Scope of the reviewAs we reviewed existing research and existingtheories about mathematics learning, it soon becameclear to us that there are tw o types of theoriesabout how children learn mathematics. The first areexplanatory theories. These theories seek to explainhow children’s thinking and knowledge change.Explanatory theories are based on empiricalresearch on the strategies that children adopt insolving mathematical problems, on the difficulties and misconceptions that affect their solutions to

10 Paper 1: Overview

Overview

such problems, and on their successes and theirexplanations of their own solutions. They also drawon quantitative methods to describe age or schoolgrade levels when cer tain forms of knowledge areattained and to make inferences about the nature ofrelationships observed during learning (for example,to help understand the relation between informalknowledge and school learning of mathematics).

We have called the second type of theor y pragmatic.A pragmatic theory is rather like a road map forteachers: its aims are to set out what children m ustlearn and understand, usually in a clear sequence ,about par ticular topics and to identify obstacles tolearning in formal educational settings and otherissues which teachers should keep in mind whendesigning teaching. Pragmatic theories are usually not tested for their consistency with empir icalevidence, nor examined for the parsimony of theirexplanations vis-à-vis other existing theor ies; insteadthey are assessed in multiple contexts for theirdescriptive power, their credibility and theireffectiveness in practice.

Explanatory theories are of great impor tance inmoving forward our understanding of phenomenaand have proven helpful, for example, in the domainof literacy teaching and learning. However, with someaspects of mathematics, which tend to be those thatolder children have to learn about, there simply isnot enough explanator y knowledge yet to guideteachers in many aspects of their mathematicsteaching, but students must still be taught even whenwe do not know much about how they think orhow their knowledge changes over time throughlearning. Mathematics educators have developedpragmatic theories to fill this gap and to takeaccount of the interplay of learning theory withsocial and cultural aspects of educational contexts.Pragmatic theories are designed to guide teachers indomains where there are no satisfactor y explanatorytheories, and where explanator y theory does notprovide enough information to design complexclassroom teaching. We have included both types oftheory in our review. We believe that both types arenecessary in mathematics education but that theyshould not be confused with each other.

We decided to concentrate in our review on issuesthat are specific to mathematics learning. Werecognise the significance of general pedagogicaltheories that stress, for example, the impor tance of giving learners an active role in developing theirthinking and conceptual under standing, the notion of

didactic transposition, the theory of situations, socialtheories regarding the impor tance of conflict andcooperation, the role of the teacher, the role and useof language, peer collaboration and argumentation inthe classroom. These are important ideas but theyapply to other domains of lear ning as well, and wedecided not to provide an analysis of such theor iesbut to mention them only in the context of specif icissues about mathematical learning.

Another decision that we made about the scope ofthe synthesis was about how to deal with cultur aldifferences in teaching and learning mathematics. Thefocus of the review is on mathematics learning by U.K.students during compulsory education. We recognisethat there are many differences between learners indifferent parts of the world; so, we decided to includemostly research about learners who can beconsidered as reasonably similar to U.K. students, i.e.those living in Western cultures with a relatively highstandard of living and plenty of oppor tunities toattend school. Thus the description of students whoparticipated in the studies is not presented in detailand will often be indicated only in terms of thecountry where the research was car ried out. In orderto offer readers a notion of the time in students’ liveswhen they might succeed or show difficulties withspecific problems, we used age levels or school gradelevels as references. These ages and years of schoolingare not to be gener alised to very differentcircumstances where, for example, children might begrowing up in cultures with different number systemsor largely without school par ticipation. Occasionalreference to research with other groups is used b utthis was purposefully limited, and it was included onl ywhen it was felt that the studies could shed light on aspecific issue.

We also decided to concentr ate on keyunderstandings that offer the foundation formathematics learning rather than on the differenttechnologies used in mathematics. Wartofsky (1979)conceives technology as any human made tool thatimproves our ability to control and adapt to theenvironment. Mathematics uses many such tools.Some representational tools, such as counting andwritten numbers, are par t of traditional mathematicslearning in primary school. They improve our abilitiesin amazing ways: for example, counting allows us torepresent precisely quantities which we could notdiscriminate perceptually and written numbers in theHindu-Arabic system create the possibility of columnarithmetic, which is not easil y implemented with oralnumber when quantities are lar ge or even with



written Roman numerals. We have argued elsewhere(Nunes, 2002) that systems of signs enhance,structure and empower their users but learnersmust still construct meanings that allow them to use these systems. Our choice in this review was to consider how learners construct meanings rather than explore in depth the enab ling role ofmathematical representations. We discuss in muchgreater detail how they learn to use whole andrational numbers meaningfully than how theycalculate with these numbers. Similarly, we discusshow they might learn the meaning and power ofalgebraic representation rather than how they might become fluent with algebraic manipulation.Psychological theories (Luria, 1973; Vygotsky, 1981)emphasise the empowering role of culturallydeveloped systems of signs in human reasoning b utstress that learners’ construction of meanings forthese signs undergoes a long development processin order for the signs to be tr uly empowering.Similarly, mathematics educators stress thattechnology is aimed not to replace , but to enhancemathematical reasoning (Noss and Hoyles, 1992).

Our reason for not focusing on technologies in thissynthesis is that there are so man y technologicalresources used today for doing mathematics that it is not possible to consider even those used orpotentially useful in pr imary school in the requireddetail in this synthesis. We recognise this gap andstrongly suggest that at least some of these issues be taken up for a synthesis at a later point, as someimportant comparative work already exists in thedomain of column ar ithmetic (e.g. Anghileri,Beishuizen and Putten, 2002; Treffers, 1987) and the use of calculator s (e.g. Ruthven, 2008).

We wish to emphasise , therefore, that this review isnot an exhaustive one. It considers a par t of today’sknowledge in mathematics education. There areother, more specific aspects of the subject which,usually for reasons of space , we decided to by-pass.We shall explain the reasons for these choices as we go along.

Methods of the reviewWe obtained the mater ial for the synthesis through a systematic search of peer reviewed journals, editedvolumes and refereed conference proceedings.1 Weselected the papers that we read by first screeningthe abstracts: our main cr iteria for selecting ar ticlesto read were that they should be on a relevant topic

and that they should repor t either the results ofempirical research or theoretical schemes f orunderstanding mathematics learning or both. We also consulted several books in order to readresearchers’ syntheses of their own empirical workand to access ear lier well-established reviews ofrelevant research; we chose books that provideuseful frameworks for research and theor ies inmathematics learning.

We hope that this review will become the object of discussion within the community of researchers,teachers and policy makers. We recognise that it is only one step towards making sense of the vastresearch on how students’ thinking and knowledge of mathematics develops, and that other steps must follow, including a thorough evaluation of this contribution.

Teaching and learningmathematics: What is the nature of this task?

Learning mathematics is in some ways similar (but of course not identical) to language lear ning: inmathematics as well as in language it is necessar y tolearn symbols and their meaning, and to know howto combine them meaningfully.

Learning meanings for symbols is often more diff icultthan one might think. Think of learning the meaningof the word ‘brother’. If Megan said to her f our-year-old friend Sally ‘That’s my brother’ and pointed toher brother, Sally might learn to say correctly andappropriately ‘That’s Megan’s brother’ but she wouldnot necessarily know the meaning of ‘brother’.‘Brother of ’ is a phrase that is based on a set ofrelationships, and in order to under stand its meaningwe need to under stand this set of relationships,which includes ‘mother of ’ and ‘father of ’. It is in thisway that learning mathematics is very like learning alanguage: we need to learn mathematical symbolsand their meanings, and the meaning of thesesymbols is based on sets of relations.

In the same way that Megan might point to herbrother, Megan could count a set of pens and sa y:‘There are 15 pens here’. Sally could learn to countand say ‘15 pens’ (or dogs, or stars). But ‘15’ inmathematics does not just refer to the result ofcounting a set: it also means that this set isequivalent to all other sets with 15 objects, has


fewer objects than any set with 16 or more , and has more objects than any set with 14 or f ewer.Learning about numbers involves more thanunderstanding the operations that are car ried out todetermine the word that represents the quantity. Inthe context of learning mathematics, we would likestudents to know, without having to count, that someoperations do and other s do not change a quantity.For example, we would like them to under stand thatthere would only be more pens if we added someto the set, and fewer if we subtracted some fromthe set, and that there would still be 15 pens if weadded and then subtracted (or vice versa) the same number of pens to andfrom the original set.

The basic numerical concepts that we want studentsto learn in primary school have these two sides to them: on the one hand, there are quantities,operations on quantities and relations betweenquantities, and on the other hand there are symbols,operations on symbols and relations betweensymbols. Mathematics teaching should aim to ensurethat students’ understanding of quantities, relationsand symbols go together. Anything we do with thesymbols has to be consistent with their under lyingmathematical meaning as well as logically consistentand we are not free to pla y with meaning inmathematics in quite the same ways we might playwith words.

This necessary connection is often neglected intheories about mathematics learning and in teachingpractices. Theories that appear to be contradictoryhave often focused either on students’ understandingof quantities or on their under standing of symbolsand their manipulations. Similarly, teaching is oftendesigned with one or the other of these two kindsof understanding in sight, and the result is that thereare different ways of teaching that have differentstrengths and weaknesses.

Language learners eventually reach a time when theycan learn the meaning of new words simply bydefinitions and connections with other words. Think ofwords like ‘gene’ and ‘theory’: we learn their meaningsfrom descriptions provided by means of other wordsand from the way they are used in the language.Mathematics beyond primary school often workssimilarly: new mathematical meanings are lear ned byusing previously learned mathematical meanings andways of combining these. There are also other ways inwhich mathematics and language learning are similar ;perhaps the most impor tant of these other similarities

is that we can use language to represent a largevariety of meanings, and mathematics has a similarpower. But, of course, mathematics learning differsfrom language learning: mathematics contains its owndistinct concepts and modes of enquir y whichdetermine the way that mathematics is used. Thisspecificity of mathematical concepts is reflected in thethemes that we chose to analyse in our synthesis.

The framework for this reviewAs we star t our review, there is a gener al point tobe made about the theoretical position that w e havereached from our review of research on children’smathematics. On the whole , the teaching of thevarious aspects of mathematics proceeds in a clearsequence, and with a cer tain amount of separationin the teaching of different aspects. Children aretaught first about the number sequence and thenabout written numbers and arithmetical operationsusing written numbers. The teaching of the fourarithmetical operations is done separately. At schoolchildren learn about addition and subtractionseparately and before they learn about multiplicationand division, which also tend to be taught quiteseparately from each other. Lessons about arithmeticstart years before lessons about propor tions and theuse of mathematical models.

This order of events in teaching has had a cleareffect on research and theor ies about mathematicallearning. For example, it is a commonplace thatresearch on multiplication and division is most often(though there are exceptions; see Paper 4) carriedout with children who are older than those whoparticipate in research on addition and subtraction.Consequently, in most theor ies additive reasoning ishypothesised (or assumed) to precede multiplicativereasoning. Until recently there have been very fewstudies of children’s understanding of the connectionbetween the different arithmetical operationsbecause they are assumed to be learned relativelyindependently of each other.

Our review of the relevant research has led us to us to a different position. The evidence quite clear lysuggests that there is no such sequence , at any ratein the onset of children’s understanding of some ofthese different aspects of mathematics. Much of thislearning begins, as our review will show, in informalcircumstances and before children go to school. Evenafter they begin to lear n about mathematics formally,there are clear signs that they can embar k on



genuinely multiplicative reasoning, for example, at atime when the instr uction they receive is all aboutaddition and subtraction. Similar obser vations can bemade about learning algebra; there are studies thatshow that quite young children are capable ofexpressing mathematical generalities in algebraicterms, but these are rare: the majority of studiesfocus on the ways in which learners fail to do so at the usual age at which this is taught.

Sequences do exist in children’s learning, but thesetend not to be about diff erent arithmetic operations(e.g. not about addition before multiplication). Instead,they take the form of children’s understanding of newquantitative relations as a result of w orking with andmanipulating relations that have been familiar forsome time. An example, which we describe in detailin Paper 2, is about the inverse relation betweenaddition and subtraction. Young children easilyunderstand that if you add some new items to a setof items and then subtr act exactly the same items,the number of items in the set is the same as it wasinitially (inversion of identity), but it takes some timefor them to extend their knowledge of this relationenough to understand that the number of items inthe set will also remain the same if you add somenew items and then subtr act an equal number itemsfrom the set, which are not the same ones you hadadded (inversion of quantity: a + b - b = a). Causalsequences of this kind play an impor tant par t in theconclusions that we reach in this review.

Through our review, we identified some keyunderstandings which we think children must achieveto be successful learners of mathematics and whichbecame the main topics for the review. In theparagraphs that follow, we present the argumentsthat led us to choose the six main topics.Subsequently, each topic is summar ised under aseparate heading. The research on which thesesummaries are based is analysed in Papers 2 to 7.

The main points that are discussed here , before weturn to the summar ies, guided the choice of paper sin the review.

Quantity and number

The first point is that there is a distinction to bemade between quantity and number and thatchildren must make connections as well asdistinctions between quantity and number in orderto succeed in learning mathematics.

Thompson (1993) suggested that ‘a personconstitutes a quantity by conceiving of a quality of an object in such a wa y that he or sheunderstands the possibility of measur ing it.Quantities, when measured, have numerical value,but we need not measure them or kno w theirmeasures to reason about them. You can think of your height, another per son’s height, and theamount by which one of you is taller than theother without having to know the actual values’(pp. 165–166). Children exper ience and learnabout quantities and the relations betw een themquite independently of learning to count. Similar ly,they can learn to count quite independentl y fromunderstanding quantities and relations betweenthem. It is cr ucial for children to lear n to makeboth connections and distinctions betweennumber and quantity. There are different theoriesin psychology regarding how children connectquantity and number ; these are discussed in Paper 2.

The review also showed that there are twodifferent types of quantities that primary schoolchildren have to understand and that these areconnected to different types of numbers. Ineveryday life, as well as in primary school, childrenlearn about quantities that can be counted. Someare discrete and each item can be counted as anatural unit; other quantities are continuous and weuse measurement systems, count the conventionalunits that are part of the system, and attributenumbers to these quantities. These quantities which are measured by the successive addition ofitems are termed extensive quantities. They arerepresented by whole numbers and give childrentheir first insights into number.

In everyday life children also learn about quantitiesthat cannot be counted like this. One reason whythe quantity might not be countable in this way isthat it may be smaller than the unit; for example, ifyou share three chocolate bars among four people,you cannot count how many chocolate bars eachone receives. Before being taught about fractions,some primary school students are aware that youcannot say that each person would be given onechocolate bar, because they realise that eachperson’s portion would be smaller than one: thesechildren conclude that they do not know a numberto say how much chocolate each person willreceive (Nunes and Bryant, 2008). Quantities that are smaller than the unit are represented byfractions, or more generally by rational numbers.


Rational numbers are also used to representquantities which we do not measure directly but only through a relation between two othermeasures. For example, if we want to say somethingabout the concentration of orange squash in a glass,we have to say something about the r atio ofconcentrate used to water. This type of quantity,measurable by ratio, is termed intensive quantity and is often represented by rational numbers.

In Papers 2 and 3 we discuss how children makeconnections between whole and rational numbersand the different types of quantities that theyrepresent.

Relations

Our second general point is about relations.Numbers are used to represent quantities as w ell as relations; this is why children must establish aconnection between quantity and number but alsodistinguish between them. Measures are numbersthat are connected to a quantity. Expressions such as 20 books, 3 centimetres, 4 kilos, and ½ achocolate are measures. Relations, like quantities, do not have to be quantified. For example, we cansimply say that two quantities are equivalent ordifferent. This is a qualitative statement about therelation between two quantities. But we can quantifyrelations and we use numbers to do so: for example,when we compare two measures, we are quantifyinga relation. If there are 20 children in a class and 17books, we can say that there are 3 more childrenthan books. The number 3 quantifies the relation. Wecan say 3 more children than books or 3 booksfewer than children; the meaning does not changewhen the wording changes because the number 3does not refer to children or to books, but to therelation between the two measures.

A major use of mathematics is to quantify relationsand manipulate these representations to expand our understanding of a situation. We came to theconclusion from our review that under standingrelations between quantities is at the root ofunderstanding mathematical models. Thompson(1993) suggested that ‘Quantitative reasoning is theanalysis of a situation into a quantitativ e structure – a network of quantities and quantitativerelationships… A prominent characteristic ofreasoning quantitatively is that numbers and numericrelationships are of secondar y importance, and donot enter into the pr imary analysis of a situation.

What is impor tant is relationships among quantities’(p. 165). Elsewhere, Thompson (1994) emphasisedthat ‘a quantitative operation is non-numerical; it hasto do with the comprehension (italics in the or iginal)of a situation.’ (p. 187). So relations, like quantities, are different from numbers but we use numbers toquantify them.

Paper 4 of this synthesis discusses the quantificationof relations in mathematics, with a focus on the sor tsof relations that are par t of learning mathematics inprimary school.

The coordination of basic conceptsand the development of higher orderconcepts

Students in secondar y school have the dual task ofrefining what they have learned in primary schooland understanding new concepts, which are basedon reflections about and combinations of previousconcepts. The challenge for students in secondar yschool is to learn to take a different perspective withrespect to their mathematics knowledge and, at thesame time, to learn about the power of this newperspective. Students can under stand much aboutusing mathematical representations (numbers,diagrams, graphs) for quantities and relations andhow this helps them solve problems. Students who have gone this far under stand the role ofmathematics in representing and helping usunderstand phenomena, and even generalisingbeyond what we know. But they may not haveunderstood a distinct and cr ucial aspect of theimportance of mathematics: that, above and beyondhelping represent and explore what you know, it canbe used to discover what you do not know. In thisreview, we consider two related themes of thissecond side of mathematics: algebraic reasoning andmodelling. Papers 6 and 7 summar ise the researchon these topics.

In the rest of this opening paper w e shall summariseour main conclusions from our review. In otherwords, Papers 2 to 7 contain our detailed reviews of research on mathematics lear ning; each of the six subsequent sections about a centr al topic inmathematics learning is a summary of Papers 2 to 7.



Key understandings inmathematics: A summary of the topics reviewed

Understanding extensive quantitiesand whole numbers

Natural numbers are a way of representingquantities that can be counted. When children learnnumbers, they must find out not just about thecounting sequence and how to count, but also abouthow the numbers in the counting system representquantities and relations between them. We found agreat deal of evidence that children are aware ofquantities such as the size of objects or the amountof items in groups of objects long before they learnto count or under stand anything about the numbersystem. This is quite clear in their ability todiscriminate objects by size and sets by number when these discr iminations can be made perceptually.

Our review also showed that children learn to countwith surprisingly little difficulty. Counting is an activityorganised by principles such as the order in varianceof number labels, one-to-one correspondencebetween items and counting labels, and the use ofthe last label to say how many items are in the set.There is no evidence of children being taught theseprinciples systematically before they go to schooland yet most children star ting school at the age offive years are already able to respect these principleswhen counting and identify other people’s errorswhen they violate counting principles.

However, research on children’s numericalunderstanding has consistently shown that at firstthey make very little connection between thenumber words that they learn and their existingknowledge about quantities such as siz e and the amounts of items. Our review showed thatThompson’s (1993) theoretical distinction betweenquantities and number is hugely relevant tounderstanding children’s mathematics. For example,many four-year-old children understand how to shareobjects equally between two or more people , on aone-for-A, one-for-B basis, but have some difficulty in understanding that the number of items in twoequally shared sets must be the same, i.e. that ifthere are six sweets in one set, there must be six in the other set as well. To make the connectionbetween number words and quantities, children haveto grasp two aspects of number, which are cardinal

number and ordinal number. By cardinal number, wemean that two sets with the same number of itemsin them are equal in amount. The term ordinalnumber refers to the fact that numbers are arrangedin an ordered ser ies of increasing magnitude:successive numbers in the counting sequence aregreater than the preceding number by 1. Thus, 2 is a greater quantity than 1 and 3 than 2 and it f ollowsthat 3 must also greater than 1.

There are three different theories about howchildren come to co-ordinate their knowledge of quantities with their knowledge of counting.

The first is Piaget’s theory, which maintains that thisdevelopment is based on children’s schemas ofaction and the coordination of the schemas witheach other. Three schemas of action are relevant to natural number: adding, taking away, and settingobjects in correspondence. Children must alsounderstand how these schemas relate to each other.They must, for example, understand that a quantityincreases by addition, decreases by subtraction, andthat if you add and take away the same amount toan original quantity, that quantity stays the same. Theymust also understand the additive composition ofnumber, which involves the coordination of one-to-one correspondence with addition and subtr action: if the elements of two sets are placed incorrespondence but one has more elements thanthe other, the larger set is the sum of the smaller setplus the number of elements for which there is nocorresponding item in the smaller set. Research hasshown that this insight is not attained b y youngchildren, who think that adding elements to thesmaller set will make it larger than the larger setwithout considering the number added.

A second view, in the form of a nativist theor y, has been suggested by Gelman and Butterworth(Gelman & Butterworth, 2005). They propose thatfrom bir th children have access to an innate , inexactbut powerful ‘analog’ system, whose magnitudeincreases directly with the number of objects in anarray, and they attach the number words to theproperties occasioning these magnitudes. Accordingto this view both the system f or knowing aboutquantities and the pr inciples of counting are innateand are naturally coordinated.

A third theoretical alternative, proposed by Carey(2004), star ts from a standpoint in agreement withGelman’s theory with respect to the innate analogsystem and counting pr inciples. However, Carey does


not think that these systems are coordinatednaturally: they become so through a ‘parallelindividuation’ system, which allows very youngchildren to make precise discriminations betweensets of one and two objects, and a little later,between two and three objects. During the sameperiod, these children also learn number words and,through their recognition of 1, 2 and 3 as distinctquantities, they manage to associate the r ight countwords (‘one’, ‘two’ and ‘three’) with the r ightquantities. This association between parallelindividuation and the count list eventually leads towhat Carey (2004) calls ‘bootstrapping’: the childrenlift themselves up by their own intellectualbootstraps by inducing a r ule that the next countword in the counting system is exactly one morethan the previous one . They do so, some timebetween the age of three – and f ive years and,therefore, before they go to school.

One important point to note about these threetheories is that they use diff erent definitions ofcardinal number, and therefore different criteria forassessing whether children under stand cardinality or not. Piaget’s criterion is the one that we havementioned already and which we ourselves think tobe right: it is the under standing that two or moresets are equal in quantity when the number of itemsin them is the same (and vice-v ersa). Gelman’s andCarey’s less demanding cr iterion for understandingcardinality is the knowledge that the last count wordfor the set represents the set’ s quantity: if I count‘one, two, three’ items and realise that means thatthat there are three in the set, I understand cardinalnumber. In our view, this second view of cardinality isinadequate for two reasons: first, it is actually basedon the position of the count w ord and is thus morerelated to ordinal than cardinal number; second, itdoes not include any consideration of the fact thatcardinal number involves inferences regarding theequivalence of sets. Piaget’s definition of cardinal andordinal number is much more str ingent and it hasnot been disputed by mathematics educators. Hewas sceptical of the idea that children w ouldunderstand cardinal and ordinal number conceptssimply from learning how to count and the evidencewe reviewed definitely shows that learning aboutquantities and numbers develop independently ofeach other in young children.

This conclusion has impor tant educationalimplications. Schools must not be satisfied withteaching children how to count: they must ensurethat children learn not only to count but also to

establish connections between counting and theirunderstanding of quantities.

Piaget’s studies concentrated on children’s ability to reason logically about quantitative relations. Heargued that children must understand the inverserelation between addition and subtraction and alsoadditive composition (which he termed class-inclusion and was later investigated under the labelof par t-whole relations) in order to tr uly understandnumber. The best way to test this sor t of causalhypothesis is through a combination of longitudinaland intervention studies. Longitudinal studies withthe appropriate controls can suggest that A iscausally related to B if it is a specif ic predictor of B at a later time . Intervention studies can test thesecausal ideas: if children are successfull y taught A and,as a consequence, their learning of B improves, it is safe to conclude that the natur al, longitudinalconnection between A and B is also a causal one .

It had been difficult in the past to use thiscombination of methods in the analysis of children’smathematics learning for a variety of reasons. First,researchers were not clear on what sor ts of logicalreasoning were vital to learning mathematics. Thereare now clearer hypothesis about this: the inverserelation between addition and subtraction andadditive composition of number appear as keyconcepts in the work of different researchers.Second, outcome measures of mathematics learningwere difficult to find. The current availability ofstandardised assessments, either developed forresearch or by policy makers for monitoring theperformance of educational systems, makes bothlongitudinal and inter vention studies possible, asthese can be seen as valid outcome measures. Ourown research has shown that researcher designedand government designed standardised assessmentsare highly correlated and, when used as outcomemeasures in longitudinal and inter vention studies,lead to convergent conclusions. Finally, in order tocarry out inter vention studies, it is necessar y todevelop ways of teaching children the key conceptson which mathematics learning is grounded.Fortunately, there are cur rently successfulinterventions that can be used f or fur ther researchto test the effect that learning about these keyconcepts has on children’s mathematics learning.

Our review identified two longitudinal studies thatshow that children’s understanding of logical aspectsof number is vital for their mathematics learning.One was carried out in the United Kingdom and



showed that children’s understanding of the inverserelation between addition and subtraction and ofadditive composition at the beginning of school are specific predictors of their results in NationalCurriculum maths tests (a government designed and administered measure of children’s mathematicslearning) about 14 months later, after controlling fortheir general cognitive ability, their knowledge ofnumber at school entr y, and their working memory.

The second study, carried out in Germany, showedthat a measure of children’s understanding of theinverse relation between addition and subtractionwhen they were eight years old was a predictor of their performance in an algebra test when theywere in university; controlling for the children’sperformance in an intelligence test giv en at age eighthad no effect on the strength of the connectionbetween their understanding of the inverse relationand their performance in the algebra measure.

Our review also showed that it is possib le toimprove children’s understanding of these logicalaspects of number knowledge. Children who wereweak in this under standing at the beginning of schooland improved this understanding through a shortintervention performed significantly better than acontrol group that did not receive this teaching.Together, these studies allow us to conclude that it iscrucial for children to coordinate their understandingof these logical aspects of quantities with theirlearning of numbers in order to make good progressin mathematics learning.

Our final step in this summar y of research on wholenumbers considered how children use additivereasoning to solve word problems. Additivereasoning is the logical analysis of problems thatinvolve addition and subtraction, and of cour se thekey concepts of additive composition and theinverse relation between addition and subtractionplay an essential role in this reasoning. The chief toolused to investigate additive reasoning is the wordproblem. In word problems a scene is set, usually inone or two sentences, and then a question is posed.We will give three examples.

A Bob has three marbles and Bill has four: howmany marbles do they have altogether? Combineproblem.

B Wendy had four pictures on her wall and herparents gave her three more: how many does shehave now? Change problem.

C Tom has seven books: Jane has five: how manymore books does Tom have than Jane? Compareproblem.

The main interest of these prob lems is that, althoughthey all involve very simple and similar additions andsubtractions, there are vast differences in the level oftheir difficulty. When the three kinds of prob lem aregiven in the form that we have just illustrated, theCompare problems are very much harder than theCombine and Change problems. This is not becauseit is too difficult for the children to subtr act 5 from 7,which is how to solve this par ticular Compareproblem, but because they find it hard to work outwhat to do so solve the problem. Compareproblems require reasoning about relations betweenquantities, which children find a lot more diff icultthan reasoning about quantities.

Thus the difficulty of these problems rests on howwell children manage to work out the ar ithmeticalrelations that they involve. This conclusion issupported by the fact that the relatively easyproblems become a great deal more diff icult if themathematical relations are less tr ansparent. Forexample, the usually easy Change problem is a lotharder if the result is giv en and the children have towork out the star ting point. For example, Wendyhad some pictures on her wall b ut then took 3 ofthem down: now she has 4 pictures left on the wall:how many were there in the f irst place? The reasonthat children find this problem a relatively hard oneis that the stor y is about subtraction, but the solutionis an addition. Pupils therefore have to call on theirunderstanding of the inverse relation betweenadding and subtracting to solve this problem.

One way of analysing children’s reactions to wordproblems is with the fr amework devised byVergnaud, who argued that these problems involvequantities, transformations and relations. A Changeproblem, for example, involves the initial quantity anda transformation (the addition or subtr action) whichleads to a new quantity, while Compare problemsinvolve two quantities and the relation betweenthem. On the whole , problems that involve relationsare harder than those involving transformations, butother factors, such as the stor y being about additionand the solution being a subtr action or vice versaalso have an effect.

The main impact of research on word problems hasbeen to reinforce the idea with which we began thissection. This idea is that in teaching children


arithmetic we must make a clear distinctionbetween numerical analysis and the children’sunderstanding of quantitative relations. We mustremember that there is a great deal more toarithmetical learning than knowing how to carry out numerical procedures. The children have tounderstand the quantitative relations in theproblems that they are asked to solve and how to analyse these relations with numbers.

Understanding rational numbers andintensive quantities

Rational numbers, like whole numbers, can be usedto represent quantities. There are some quantitiesthat cannot be represented by a whole number, andto represent these quantities, we must use rationalnumbers. Quantities that are represented by wholenumbers are formed by addition and subtraction: asargued in the previous section, as we add elementsto a set and count them (or conventional units, inthe case of continuous quantities), we find out whatnumber will be used to represent these quantities.Quantities that cannot be represented by wholenumbers are measured not by addition but bydivision: if we cut one chocolate, for example, inequal par ts, and want to have a number to representthe par ts, we cannot use a whole number.

We cannot use whole numbers when the quantitythat we want to represent numerically:• is smaller than the unit used for counting,

irrespective of whether this is a natural unit (e.g. we have less than one banana) or aconventional unit (e.g. a fish weighs less than a kilo)

• involves a ratio between two other quantities (e.g.the concentration of orange juice in a jar can bedescribed by the ratio of orange concentrate towater ; the probability of an event can be describedby the ratio between the number of favourablecases to the total number of cases). Thesequantities are called intensive quantities.

We have concluded from our review that there areserious problems in teaching children about fr actionsand that intensive quantities are not explicitl yconsidered in the cur riculum.

Children learn about quantities that are smaller than the unit through division. Two types of actionschemes are used by children in division situations:partitioning, which involves dividing a whole intoequal par ts, and correspondence situations, where

two quantities are involved, a quantity to be sharedand a number of recipients of the shares.

Partitioning is the scheme of action most often usedin primary schools in the United Kingdom tointroduce the concept of fr actions. Research showsthat children have quite a few problems to solvewhen they par tition continuous quantities: forexample, they need to anticipate the connectionbetween number of cuts and number of parts, andsome children find themselves with an even numberof par ts (e.g. 6) when they wanted to ha ve an oddnumber (e.g. 5) because they star t out bypartitioning the whole in half. Children also find itvery difficult to understand the equivalence betweenfractions when the par ts they are asked to comparedo not look the same . For example, if they areshown two identical rectangles, each cut in half but indifferent ways (e.g. horizontally and diagonally), many9- and 10-year-olds might say that the fractions arenot equivalent; in some studies, almost half of thechildren in these age levels did not recognize theequivalence of two halves that looked ratherdifferent due to being the result of diff erent cuts.Also, if students are asked to paint 2/3 of a f iguredivided into 9 par ts, many 11- to 12-year-olds maybe unable to do so, even though they can paint 2/3of a figure divided into 3 par ts; in a study in theUnited Kingdom, about 40% of the students did notsuccessfully paint 2/3 of f igures that had been dividedinto 6 or 9 sections.

Different studies that we reviewed showed thatstudents who learn about fractions through theengagement of the par titioning schema in divisiontend to reply on perception r ather than on thelogic of division when solving prob lems: they aremuch more successful with items that can besolved perceptually than with those that cannot.There is a clear lesson here f or education: numberunderstanding should be based on logic , not onperception alone, and teaching should be designedto guide children to think about the logic ofrational numbers.

The research that we reviewed shows that thepartitioning scheme develops over a long per iod oftime. This has led some researcher s to develop waysto avoid asking the children to par tition quantities byproviding them with pre-divided shapes or withcomputer tools that do the partitioning for thechildren. The use of these resources has positiv eeffects, but these positive effects seem to beobtained only after large amounts of instr uction.



In some studies, the students had diff iculties with theidea of improper fractions even after prolongedinstruction. For example, one student argued withthe researcher during instruction that you cannothave eight sevenths if you divided a whole into seven parts.

In contrast to the difficulties that children have withpartitioning, children as young as five or six years inage are quite good at using cor respondences indivision, and do so without having to carry out theactual par titioning. Some children seem tounderstand even before receiving any instruction on fractions that, for example, two chocolates sharedamong four children and four chocolates sharedamong eight children will give the children in the two groups equivalent shares of chocolate; theydemonstrate this equivalence in action by showingthat in both cases there is one chocolate to beshared by two children.

Children’s understanding of quantities smaller thanone is often ahead of their kno wledge of fractionalrepresentations when they solve problems using the correspondence scheme. This is true ofunderstanding equivalence and even more so ofunderstanding order. Most children at the age ofeight or so realise that dividing 1 chocolate amongthree children will give bigger pieces than dividingone chocolate among four children. This insight that they have about quantities is not necessar ilyconnected with their under standing of orderingfractions by magnitude: the same children might saythat 1/3 is less than 1/4 because three is less thanfour. So we find in the domain of rational numbersthe same distinction found in the domain of wholenumbers between what children know aboutquantities and what they know about the numbersused to represent quantities.

Research shows that it is possible to help childrenconnect their understanding of quantities with theirunderstanding of fractions and thus make progress in rational number knowledge. Schools could makeuse of children’s informal knowledge of fractionalquantities and work with problems about situations,without requiring them to use formalrepresentations, to help them consolidate thisreasoning and prepare them for formalization.

Reflecting about these two schemes of action and drawing insights from them places children indifferent paths for understanding rational number.When children use the correspondence scheme,

they can achieve some insight into the equivalenceof fractions by thinking that, if there are twice asmany things to be shared and twice as man yrecipients, then each one’s share is the same . Thisinvolves thinking about a direct relation betw een the quantities. The partitioning scheme leads tounderstanding equivalence in a different way: if awhole is cut into twice as man y par ts, the size ofeach par t will be halved. This involves thinking aboutan inverse relation between the quantities in theproblem. Research consistently shows that childrenunderstand direct relations better than inverserelations and this may also be tr ue of rationalnumber knowledge.

The arguments children use when stating thatfractional quantities resulting from shar ing are or arenot equivalent have been described in one study inthe United Kingdom. These arguments include theuse of correspondences (e.g. sharing four chocolatesamong eight children can be shown by a diagram tobe equivalent to shar ing two chocolates among fourchildren because each chocolate is shared amongtwo children), scalar arguments (twice the number ofchildren and twice the number of chocolates meansthat they all get the same), and an under standing ofthe inverse relation between the number of par tsand the size of the par ts (i.e. twice the number ofpieces means that each piece is halv ed in size). Itwould be impor tant to investigate whetherincreasing teachers’ awareness of children’s ownarguments would help teachers guide children’slearning in this domain of numbers more effectively.

Some researchers have argued that a better star tingpoint for teaching children about fractions is the useof situations where children can use cor respondencereasoning than the use of situations where thescheme of par titioning is the relevant one . Ourreview of children’s understanding of the equivalenceand order of fractions supports this claim. However,there are no inter vention studies comparing theoutcomes of these two ways of introducing childrento the use of fr actions, and inter vention studieswould be crucial to solve this issue: one thing ischildren’s informal knowledge but the outcomes ofits formalization through instruction might be quiteanother. There is now considerably more informationregarding children’s informal strategies to allow fornew teaching programmes to be designed andassessed. There is also considerable work oncurriculum development in the domain of teachingfractions in primary school. Research that comparesthe different forms of teaching (based on par titioning


or based on correspondences) and the introductionof different representations (decimal or ordinar y) is now much more feasible than in the past.Intervention research, which could be carried out in the classroom, is urgently needed. The availableevidence suggests that testing this h ypothesisappropriately could result in more successfulteaching and learning of rational numbers.

In the United Kingdom ordinar y fractions continueto play an impor tant role in pr imary schoolinstruction whereas in some countries greaterattention is given to decimal representation than toordinary fractions in primary school. Two reasons areproposed to justify the teaching of decimals bef oreordinary fractions . First, decimals are common inmetric measurement systems and thus theirunderstanding is cr itical for learning other topics,such as measurement, in mathematics and science.Second, decimals should be easier than ordinar yfractions to understand because decimals can betaught as an extension of place value representation;operations with decimals should also be easier andtaught as extensions of place value representation.

It is cer tainly true that decimals are used inmeasurement and thus learning decimals is necessar ybut ordinary fractions often appear in algebraicexpressions; so it is not clear a priori whether oneform of representation is more useful than the otherfor learning other aspects of mathematics. However,the second argument, that decimals are easier thanordinary fractions, is not suppor ted in sur veys ofstudents’ performance: students find it difficult tomake judgements of equivalence and order as m uchwith decimals as with ordinar y fractions. Studentsaged 9 to 11 years have limited success whencomparing decimals written with different numbersof digits after the decimal point (e.g. 0.5 and 0.36):the rate of correct responses varied between 36%and 52% in the three diff erent countries thatparticipated in the study, even though all the childrenhave been taught about decimals.

Some researchers (e.g. Nunes, 1997; Tall, 1992;Vergnaud, 1997) argue that different representationsshed light on the same concepts from differentperspectives. This would suggest that a way tostrengthen students’ learning of rational numbers isto help them connect both representations. Casestudies of students who received instruction thataimed at helping students connect the tw o forms ofrepresentation show encouraging results. However,the investigation did not include the appropr iate

controls and so it does not allo w for establishingfirmer conclusions.

Students can learn procedures for comparing, addingand subtracting fractions without connecting theseprocedures with their under standing of equivalenceand order of fractional quantities, independently ofwhether they are taught with ordinar y or decimalfractions representation. This is not a desiredoutcome of instruction, but seems to be a quitecommon one. Research that focuses on the use ofchildren’s informal knowledge suggests that it ispossible to help students make connections betweentheir informal knowledge and their learning ofprocedures but the evidence is limited and theconsequences of this teaching have not beeninvestigated systematically.

Research has also shown that students do notspontaneously connect their knowledge of fractionsdeveloped with extensive quantities smaller than theunit with their under standing of intensive quantities.Students who succeed in under standing that twochocolates divided among four children and fourchocolates divided among 8 children yield the samesize share do not necessar ily understand that a paintmixture made with two litres of white and two ofblue paint will be the same shade as one made withfour litres of white and f our of blue paint.

Researchers have for some time distinguishedbetween different situations where fr actions areused and argued that connections that seemobvious to an adult are not necessar ily obvious to children. There is now evidence that this is so .There is a clear educational implication of thisresult: if teaching children about fr actions in thedomain of extensive quantities smaller than theunit does not spontaneously transfer to theirunderstanding of intensive quantities, a completefractions curriculum should include intensivequantities in the progr amme.

Finally, this review opens the way for a fresh researchagenda in the teaching and lear ning of fractions. Thesource for the new research questions is the findingthat children achieve insights into relations betweenfractional quantities before knowing how torepresent them. It is possible to envisage a researchagenda that would not focus on children’smisconceptions about fractions, but on children’spossibilities of success when teaching star ts fromthinking about quantities rather than from learningfractional representations.



Understanding relations and theirgraphical representationChildren form concepts about quantities from theireveryday experiences and can use their schemas ofaction with diverse representations of the quantities(iconic, numerical) to solve problems. They oftendevelop sufficient awareness of quantities to discusstheir equivalence and order as well as how quantitiesare changed by operations. It is significantly moredifficult for them to become aware of the relationsbetween quantities and operate on relations.

The difficulty of understanding relations is clear bothwith additive and multiplicative relations betweenquantities. Children aged about eight to ten y earscan easily say, for example, how many marbles a boywill have in the end if he star ted a game with sixmarbles, won five in the first game, lost three in thesecond game, and won two in the third game.However, if they are not told ho w many marbles theboy had at the start and are asked how many moreor fewer marbles this boy will have after playing thethree games, they find this second problemconsiderable harder, par ticularly if the first gameinvolves a loss.

Even if the children are taught how to representrelations and recognise that winning five in the firstgame does not mean having five marbles, they ofteninterpret the results of oper ations on relations as ifthey were quantities. Children find both additive andmultiplicative relations significantly more difficult thanunderstanding quantities.

There is little evidence that the design ofmathematics curricula has so far taken into accountthe impor tance of helping students become a wareof the difference between quantities and relations.Some researchers have carried out exper imentalteaching studies which suggest that it is possib le topromote students’ awareness of additive relationsas different from quantities; this was not an easytask but the instruction seemed to have positiveresults (but note that there were no controlgroups). Fur ther research must be carried out toanalyse how this knowledge affects mathematicslearning: longitudinal and inter vention studieswould be crucial to clar ify this. If positive resultsare found, there will be imperative policyimplications.

The first teaching that children receive in schoolabout multiplicative relations is about propor tions.Initial studies on students’ understanding of

proportions previously led to the conclusion thatstudents’ problems with propor tional reasoningstemmed from their difficulties with multiplicativereasoning. However, there is presently muchevidence to show that, from a relatively early age(about five to six years in the United Kingdom),many children (our estimate is about tw o-thirds)already have informal knowledge that allows them to solve multiplicative reasoning problems.

Multiplicative reasoning problems are defined by thefact that they involve two (or more) measureslinked by a fixed ratio. Students’ informal knowledgeof multiplicative reasoning stems from the schemaof one-to-many correspondence, which they useboth in multiplication and division problems. Whenthe product is unknown, children set the elements in the two measures in cor respondence (e.g. onesweet costs 4p) and f igure out the product (howmuch five sweets will cost) by counting or adding.When the correspondence is unknown (e.g. if youpay 20p for five sweets, how much does each sweet cost), the children share out the elements(20p shared in five groups) to find what thecorrespondence is.

This informal knowledge is currently ignored in U.K. schools, probably due to the theor y thatmultiplication is essentially repeated addition anddivision is repeated subtraction. However, theconnections between addition and multiplication onthe one hand, and subtraction and division on theother hand, are procedural and not conceptual. So students’ informal knowledge of multiplicativereasoning could be developed in school from anearlier age.

Even after being taught other methods to solv eproportions problems in school, students continue to use one-to-many correspondences reasoning tosolve proportions problems; these solutions havebeen called building up methods. For example, if arecipe for four people is to be adapted to serve sixpeople, students figure out that six people is thesame as four people plus two people; so they figureout what half the ingredients will be and add this tothe quantity required for four people. Building upmethods have been documented in many differentcountries and also among people with lo w levels ofschooling. A careful analysis of the reasoning inbuilding-up methods suggests that the students focuson the quantities as they solve these problems, andfind it difficult to focus on the relations between the quantities.


Research carried out independently in differentcountries has shown that students sometimes useadditive reasoning about relations when theappropriate model is a multiplicative one. Somerecent research has shown that students also usemultiplicative reasoning in situations where theappropriate model is additive. These results suggestthat children use additive and multiplicative modelsimplicitly and do not make conscious decisionsregarding which model is appropr iate in a specificsituation. We concluded from our review thatstudents’ problems with propor tional reasoningstems from their difficulties in becoming explicitlyaware of relations between quantities. Greaterawareness of the models implicit in their solutionswould help them distinguish between situations thatinvolve different types of relations: additive,proportional or quadratic, for example.

The educational implication from these f indings is that schools should take up the task of helpingstudents become more aware of the models thatthey use implicitly and of ways of testing theirappropriateness to par ticular situations. Thedifferences between additive and multiplicativesituations rests on the relations between quantities;so it is likely that the cr itical move here is to helpstudents become aware of the relations betweenquantities implicit in the procedure they use to solve problems.

Two radically different approaches to teachingproportions and linear functions in schools can beidentified in the literature. These constitute pragmatictheories, which can guide teacher s, but have as yetnot been tested systematically. The first, described asfunctional and human in focus, is based on thenotion that students’ schemas of action should bethe star ting point for this teaching. Throughinstruction, they should become progressively moreaware of the relations between quantities that canbe identified in such problems. Diagrams, tables andgraphs are seen as tools that could help studentsunderstand the models of situations that they areusing and make them into models for othersituations later.

The second, described in the literature as algebraic,proposes that there should be a sharp separationbetween students’ intuitive knowledge, in whichphysical and mathematical knowledge areintertwined, and mathematical knowledge. Studentsshould be led to formalisations early on ininstruction and re-establish the connections between

mathematical structures and physical knowledge at a later point. Representations using ordinar y anddecimal fractions and the number line are seen asthe tools that can allow students to abstract early on from the physical situations. Students should learnearly on to represent equivalences between ordinaryfractions (e.g. 2/4 = 4/8), a representation that wouldprovide insight into propor tions, and alsoequivalences between ordinary and decimal fractions(2/4 = 0.5), which would provide insight into theordering and equivalence of fr actions marked on the number line.

Each of these approaches makes assumptions aboutthe significance of students’ informal knowledge atthe star t of the teaching progr amme. The functionalapproach assumes that students’ informal knowledgecan be formalised through instruction and that thiswill be beneficial to learning. The algebraic approachassumes that students’ informal knowledge is anobstacle to students’ mathematics learning. There isevidence from a combination of longitudinal andintervention methods, albeit with younger children,that shows that students’ knowledge of informalmultiplicative reasoning is a causal and positiv e factorin mathematics learning. Children who scored higherin multiplicative reasoning problems at the star t oftheir first year in school performed significantlybetter in the government designed and schooladministered mathematics achievement test thanthose whose scores were lower. This longitudinalrelationship remained significant after the appropr iatecontrols were taken into account. The interventionstudy provides results that are less clear because thechildren were taught not only about multiplicativereasoning but also about other concepts consideredkey to mathematics learning. Nevertheless, childrenwho were at r isk for mathematics learning andreceived teaching that included multiplicativereasoning, along with two other concepts, showedaverage achievement in the standardisedmathematics achievement tests whereas the control group remained in the bottom 20% of thedistribution, as predicted by their assessment at thestart of school. So, in terms of the assumptionsregarding the role of informal knowledge, thefunctional approach seems to have the edge overthe algebraic approach.

These two approaches to instr uction also differ inrespect to what students need to know to benefitfrom teaching and what they lear n during the courseof instruction. Within the functional approach, thetools used in teaching are diagr ams, tables, and



graphs so it is clear that students need to lear n toread graphs in order to be ab le to use them as toolsfor thinking about relations between quantities andfunctions. Research has shown that students haveideas about how to read graphs before instructionand these ideas should be taken into account whengraphs are used in the classroom. It is possible toteach students to read gr aphs and to use them inorder to think about relations in the course ofinstruction about propor tions, but much moreresearch is needed to show how students’ thinkingchanges if they do lear n to use graphs to analyse thetype of relation relevant in specific situations. Withinthe algebraic approach, it is assumed that studentsunderstand the equivalence of fr actions withoutreference to situations. Our review of students’understanding of fractions, summarised in theprevious section, shows that this is not tr ivial so it is necessar y to show that students can, in the course of this teaching, learn both about fractionequivalence and propor tional relations.

There is no evidence to show how either of theseapproaches to teaching works in promotingstudents’ progress nor that one of them is moresuccessful than the other. Research that can clar ifythis issue is urgently needed and could have a majorimpact in promoting better lear ning by U.K.students. This is par ticularly impor tant in view offindings from the international comparisons thatshow that U.K. students do relatively well in additivereasoning items but comparatively poorly inmultiplicative reasoning items.

Understanding space and itsrepresentation in mathematics

When children begin to be taught about geometr y,they already know a great deal about space , shape,size, distance and or ientation, which are the basicsubject matter of geometr y. They are also quitecapable of drawing logical inferences about spatialmatters. In fact, their spatial knowledge is soimpressive and so sophisticated that one mightexpect geometry to be an easy subject f or them.Why should they have any difficulty at all withgeometry if the subject just in volves learning how to express this spatial knowledge mathematically?

However, many children do find geometry hard andsome children continue to make basic mistakes rightthrough their time at school. There are two mainreasons for these well-documented difficulties. One

reason is that many of the spatial relations thatchildren must think about and lear n to analysemathematically in geometr y classes are different fromthe spatial relations that they lear n about in theirpre-school years. The second is that geometr y makesgreat demands on children’s spatial imagination. Inorder to measure length or area or angle , forexample, we have to imagine spaces divided intoequal units and this tur ns out to be quite hard f orchildren to learn to do systematically.

Nevertheless, pre-school children’s spatial knowledgeand spatial experiences are undoubtedly relevant tothe geometry that they must learn about later, and itis important for teachers and researchers alike torecognise this. From a very early age children areable to distinguish and remember diff erent shapes,including basic geometrical shapes. Children are ableto co-ordinate visual information about size anddistance to recognise objects by their actual size, and also to co-ordinate visual shape and or ientationinformation to recognise objects by their actualshapes. In social situations, children quite easily workout what someone else is looking at b y extrapolatingthat person’s line of sight often across quite lar gedistances, which is an impressive feat of spatialimagination. Finally, they are highly sensitive not justto the orientation of lines and of objects in theirenvironments, but also to the relation betweenorientations: for example, young children can,sometimes at least, recognise when a line in theforeground is parallel to a stable background feature.

These impressive spatial achievements must helpchildren in their efforts to understand the geometr ythat they are taught about at school, but there is littledirect research on the links between children’sexisting informal knowledge about space and theprogress that they make when they are eventuallytaught about geometry. This is a worrying gap,because research of this sor t would help teachers to make an effective connection between what theirpupils know already and what they have to learn intheir initial geometr y classes. It would also give us abetter understanding of the obstacles that childrenencounter when they are first taught about geometr y.

Some of these obstacles are immediatel y apparentwhen children learn about measurement, first oflength and then of area. In order to lear n how tomeasure length, children must grasp the underlyinglogic of measurement and also the role of iter ated(i.e. repeated) measurement units, e.g. the unit of 1cm repeated on a r uler. Using a r uler also involves an


active form of one-to-one correspondence, since the child must imagine and impose on the line beingmeasured the same units that are explicit andobvious on the ruler. Research suggests that childrendo have a reasonable understanding of theunderlying logic of measurement by the time thatthey begin to learn about geometr y, but that manyhave a great deal of diff iculty in grasping how toimagine one-to-one correspondence between theiterated units on the r uler and imagined equivalentunits on the line that they are measur ing. Onecommon mistake is to set the 1 cm r ather than the 0 cm point at one end of the line . The evidencesuggests that many children apply a poor lyunderstood procedure when they measure lengthand are not thinking, as they should, of one-to-onecorrespondence between the units on the r uler andthe length being measured. There is no doubt thatteachers should think about how to promotechildren’s reflection on measurement procedures.Nunes, Light and Mason (1993), for example,showed that using a broken ruler was one way to promote this.

Measurement of area presents additional prob lems.One is that area is often calculated from lengths,rather than measured. So, although the measurementis in one kind of unit, e.g. centimetres, the finalcalculation is in another, e.g. square centimetres. Thisis what Vergnaud calls a ‘product of measures’calculation. Another potential problem is that mostcalculations of area are multiplicative: with rectanglesand parallelograms, one has to multiply the figure’sbase by height, and with tr iangles one must calculatebase by height and then halve it. There is evidencethat many children attempt to calculate area b yadding par ts of the per imeter, rather than bymultiplying. One consequence of the multiplicativenature of area calculations is that doub ling a figure’sdimensions more than doubles its area. Think of arectangle with a base of 10 cm and a height of 4 cm,its area is 40 cm2: if you enlarge the figure bydoubling its base and height (20 cm x 8 cm), youquadruple its area (160 cm2). This set of relations is hard for pupils, and for many adults too, tounderstand.

The measurement of area also raises the questionof relations between shapes. For example the proofthat the same base by height rule for measuringrectangles applies to parallelograms as well rests on the demonstration that a rectangle can betransformed into a parallelogram with the sameheight and base without changing its area. In turn

the rule for finding the area of tr iangles, A = ½(base x height), is justified by the fact every trianglecan be transformed into a parallelogram with thesame base and height by doubling that tr iangle. Thus,rules for measuring area rest heavily on the relationsbetween geometric shapes. Although Wertheimerdid some ingenious studies on how children wereable to use of the relations betw een shapes to helpthem measure the area of some of these shapes,very little research has been done since then ontheir understanding of this centrally impor tantaspect of geometr y.

In contrast, there is a great deal of research onchildren’s understanding of angles. This researchshows that children have very little understanding of angles before they are taught about geometr y. The knowledge that they do have tends to bequite disconnected because children often fail tosee the connection between angles in dissimilarcontexts, like the steepness of a slope and ho wmuch a person has to turn at a corner. There isevidence that children begin to connect what theyknow about angles as they gro w older : theyacquire, in the end, a fairly abstract understandingof angle. There is also evidence , mostly fromstudies with the progr amming language Logo, thatchildren learn about angle relatively well in thecontext of movement.

Children’s initial uncer tainties with angles contrastsharply to the relative ease with which they adopt the Cartesian framework for plotting positions in any two-dimensional space. This framework requiresthem to be able to extrapolate imaginary pairs ofstraight lines, one of which is perpendicular to thevertical axis and the other to the hor izontal axis, andthen to work out where these imaginar y lines willmeet, in order to plot specif ic positions in space. At first sight this might seem an extraordinarilysophisticated achievement, but research suggests that it presents no intellectual obstacle at all to mostchildren. Their success in extrapolating imaginarystraight lines and working out their meeting point maystem for their ear ly experiences in social interactionsof extrapolating such lines when working out whatother people are looking at, but we need longitudinalresearch to establish whether this is so. Some fur therresearch suggests that, although children can usuallywork out specific spatial positions on the basis ofCartesian co-ordinates, they often find it hard to usethese co-ordinates to work out the relation betweentwo or more different positions in space .



We also need research on another possib leconnection between children’s early informal spatialknowledge and how well they learn about geometrylater on. We know that very young children tellshapes apart, even abstract geometric shapes,extremely well, but it is also clear that when childrenbegin to learn about geometr y they often find ithard to decompose complicated shapes into sev eralsimpler component shapes. This is a worryingdifficulty because the decomposition of shapes pla ysan important par t in learning about measurement ofarea and also of angles. More research is needed onhow children learn that par ticular shapes can bebroken down into other component shapes.

Overall, research suggests that the relation betweenthe informal knowledge that children build up beforethey go to school and the progress that they mak eat school in geometr y is a crucial one. Yet, it is arelation on which there is very little research indeedand there are few theories about this possible link aswell. The theoretical frameworks that do exist tendto be pragmatic ones. For example, the InstituteFreudenthal group assume a strong link betw eenchildren’s preschool spatial knowledge and theprogress that they make in learning about geometr ylater on, and argue that improving children’s earlyunderstanding of space will have a beneficial effecton their learning about geometr y. Yet, there is nogood empirical evidence for either of these twoimportant claims.

Algebraic reasoning

Research on learning algebra has considered a r angeof new ideas that have to be under stood in schoolmathematics: the use and meaning of letters andexpressions to represent numbers and variables;operations and their proper ties; relations, functions,equations and inequalities; manipulation andtransformation of symbolic statements. Youngchildren are capable of understanding the use of aletter to take the place of an unknown number, andare also able to construct statements aboutcomparisons between unknown quantities, butalgebra is much more than the substitution of letter sfor numbers and numbers for letters. Letters areused in mathematics in var ying ways. They are usedas labels for objects that have no numerical value,such as vertices of shapes or for objects that dohave numerical value, such as lengths of sides ofshapes. They denote fixed constants such as g, e or πand also non-numerical constants such as I and they

represent unknowns and variables. Distinguishingbetween these meanings is usuall y not taughtexplicitly, and this lack of instr uction might causechildren some difficulty: g, for example, can indicategrams, acceleration due to gravity, an unknown in anequation, or a var iable in an expression.

Within common algebraic usage, Küchemann (1981)identified six different ways adolescents used letter sin the Chelsea diagnostic test instr ument (Har t et al.,1984). Letters could be evaluated in some wa y,ignored, used as shor thand for objects or treated asobjects used as a specif ic unknown, as a generalisednumber, or as a var iable. These interpretationsappear to be task-dependent, so learners haddeveloped a sense of what sor ts of question weretreated in what kinds of ways, i.e. generalising(sometimes idiosyncratically) about question-typesthrough familiarity and prior experience.

The early experiences students have in algebra aretherefore very impor tant, and if algebra is presentedas ‘arithmetic with letters’ there are many possibleconfusions. Algebraic statements are aboutrelationships between variables, constructed usingoperations; they cannot be calculated to find ananswer until numbers are substituted, and the samerelationship can often be represented in manydifferent ways. The concept of equivalentexpressions is at the heart of algebraic manipulation,simplification, and expansion, but this is not alwaysapparent to students. Students who do notunderstand this try to act on algebr aic expressionsand equations in ways which have worked inarithmetical contexts, such as trial-and-error, ortrying to calculate when they see the equals sign, or rely on learnt rules such as ‘BODMAS’ which can be misapplied.

Students’ prior experience of equations is oftenassociated with finding hidden numbers usingarithmetical facts, such as ‘what number, times by 4,gives 24?’ being expressed as 4p = 24. An algebraicapproach depends on under standing operations orfunctions and their inverses, so that addition andsubtraction are understood as a pair, andmultiplication and division are under stood as a pair.This was discussed in an ear lier section. Later on,roots, exponents and logar ithms also need to beseen as related along with other functions and theirinverses. Algebraic understanding also depends onunderstanding an equation as equating twoexpressions, and solving them as f inding out for what values of the var iable they are equal. New


technologies such as graph-plotters and spreadsheetshave made multiple representations available andthere is substantial evidence that students who ha vethese tools available over time develop a strongerunderstanding of the meaning of expressions, andequations, and their solutions, than equivalentstudents who have used only formal pencil-and-paper techniques.

Students have to learn that whereas themathematical objects they have understood inprimary school can often be modelled with material objects they now have to deal with objectsthat cannot always be easily related to theirunderstanding of the mater ial world, or to their out-of-school language use. The use of concrete modelssuch as rods of ‘unknown’ related lengths, tiles of‘unknown’ related areas, equations seen as balances,and other diagrammatical methods can providebridges between students’ past experience andabstract relationships and can enable them to makethe shift to seeing relations r ather than number asthe main focus of mathematics. All these metaphorshave limitations and eventually, par ticularly with theintroduction of negative numbers, the metaphorsthey provide break down. Indeed it was thisrealisation that led to the in vention of algebraic notation.

Students have many perceptions and cognitivetendencies that can be har nessed to help them lear nalgebra. They naturally try to relate what they areoffered to what they already know. While this can be a problem if students refer to computationalarithmetic, or alphabetic meaning of letters (e.g. a = apples), it can also be useful if they ref er totheir understanding of relations between quantitiesand operations and inverses. For example, whenstudents devise their own methods for mentalcalculation they often use relations between numbersand the concepts of distr ibutivity and associativity.

Students naturally try to generalise when they seerepeated behaviour, and this ability has been usedsuccessfully in approaches to algebr a that focus onexpressing generalities which emerge inmathematical exploration. When learners need toexpress generality, the use of letter s to do so makessense to them, although they still have to learn theprecise syntax of their use in order to comm unicateunambiguously. Students also respond to the visualimpact of mathematics, and make inferences basedon layout, graphical interpretation and patterns intext; their own mathematical jottings can be

structured in ways that relate to under lyingmathematical structure. Algebraic relationshipsrepresented by graphs, spreadsheets anddiagrammatic forms are often easier to understandthan when they are expressed in symbols. Forexample, students who use function machines aremore likely to understand the order of oper ations in inverse functions.

The difficulties learners have with algebra insecondary school are near ly all due to their inabilityto shift from ear lier understandings of arithmetic tothe new possibilities afforded by algebraic notation.

• They make intuitive assumptions and applypragmatic reasoning to a symbol system they donot yet understand.

• They need to grasp the idea that an algebraicexpression is a statement about relationshipsbetween numbers and operations.

• They may confuse equality with equivalence and tryto get answers rather than transform expressions.

• They get confused between using a letter to standfor something they know, and using it to stand forsomething they do not know, and using it to standfor a variable.

• They may not have a purpose for using algebra,such as expressing a generality or relationship, socannot see the meaning of what they are doing.

New technologies offer immense possibilities forimbuing algebraic tasks with meaning, and forgenerating a need for algebraic expression.

The research synthesis sets these obser vations out in detail and focuses on detailed aspects of algebr aicactivity that manifest themselves in schoolmathematics. It also formulates recommendations for practice and research.

Modelling, problem-solving andintegrating concepts

Older students’ mathematical learning involvessituations in which it is not immediately apparentwhat mathematics needs to be done or applied, nor how this new situation relates to previousknowledge. Learning mathematics includes learningwhen and how to adapt symbols and meanings to



apply them in unfamiliar situations and also knowing when and how to adapt situations andrepresentations so that familiar tools can be brought to bear on them. Students need to lear nhow to analyse complex situations in a var iety ofrepresentations, identify var iables and relationships,represent these and develop predictions orconclusions from working with representations of variables and relationships. These might bepresented graphically, symbolically, diagrammaticallyor numerically.

In secondary mathematics, students possess not onlyintuitive knowledge from outside mathematics andoutside school, but also a range of quasi-intuitiveunderstandings within mathematics, derived fromearlier teaching and generalisations, metaphors,images and strategies that have served them well inthe past. In Tall and Vinner’s pragmatic theory, (1981)these are called ‘concept images’, which are a r agbagof personal conceptual, quasi-conceptual, perceptualand other associations that relate to the language ofthe concept and are loosely connected by thelanguage and observable artefacts associated withthe concept. The difference between students’concept images and conventional definitions causesproblems when they come to lear n new conceptsthat combine different earlier concepts. They have to expand elementar y meanings to under stand newabstract concepts, and sometimes these concepts do not fit with the images and models that studentsknow. For example, rules for combining quantities donot easily extend to negative numbers; multiplicationas repeated addition does not easily extend tomultiplying decimals.

There is little research and theoretical explor ationregarding how combinations of concepts areunderstood by students in general. For example, it would be helpful to know if students whounderstand the use of letters, ratio, angle, functions,and geometrical facts well have the same difficultiesin learning early trigonometry as those whoseunderstanding is more tenuous. Similarly, it would be helpful to know if students whose algebraicmanipulation skills are fluent understand quadraticfunctions more easily, or differently, from studentswho do not have this, but do understandtransformation of graphs.

There is research about how students learn to useand apply their knowledge of functions, par ticularly in the context of modelling and prob lem-solving.Students not only have to learn to think about

relationships (beyond linear relationships with whichthey are already familiar), but they also need to thinkabout relations between relations. Our analysis (see Paper 4) suggested that cur ricula presently donot consider the impor tant task of helping studentsbecome aware of the distinctions between quantitiesand relations; this task is left to the studentsthemselves. It is possible that helping students makethis distinction at an ear lier age could have a positiveimpact on their later lear ning of algebra.

In the absence of specif ic instructions, students tendto repeat patterns of learning that have enabledthem to succeed in other situations o ver time.Students tend to star t on new problems withqualitative judgements based on a particular context,or the visual appearance of symbolic representations,then tend to use additive reasoning, then formrelationships by pattern recognition or repeatedaddition, and then shift to proportional and relationalthinking if necessar y. The tendency to use addition asa first resor t persists as an obstacle into secondar ymathematics. Students also tend to check theirarithmetic if answers conflict rather than adaptingtheir reasoning by seeing if answers make sense ornot, or by analysing what sorts of relations areimportant in the problem. Pedagogic interventionover time is needed to enab le learners to look forunderlying structure and, where multiplerepresentations are available (graphs, data, formulae,spreadsheets), students can, over time, develop newhabits that focus on covariation of variables.However, they need knowledge and experience of arange of functions to dr aw on. Students are unlikelyto detect an exponential relationship unless theyhave seen one before, but they can descr ibe changesbetween nearby values in additive terms. A shift todescribing changes in multiplicative terms does nothappen naturally.

We hoped to find evidence about how studentslearn to use mathematics to solve problems when itis not immediately clear what mathematics theyshould be using. Some evidence in elementar ysituations has been descr ibed in an ear lier section,but at secondar y level there is only evidence ofsuccessful strategies, and not about how studentscome to have these strategies. In modelling andsome other problem-solving situations successfulstudents know how to identify var iables and how toform an image of simultaneous variation. Successfulstudents know how to hold one var iable still whilethe change in another is obser ved. They are also ableto draw on a reper toire of known function-types to


say more about how the changes in var iables are related. It is more common to f ind secondarystudents treating each situation as ad hoc and using trial-and-adjustment methods which arearithmetically-based. Pedagogic intervention overtime is needed to enable them to shift towardsseeing relationships, and relations between relations,algebraically and using a range of representationaltools to help them do so .

The tendencies described above are specificinstances of a more general issue. ‘Outside’experiential knowledge is seldom appropr iate as a source for meaning in higher mathematics, andstudents need to learn how to distinguish betweensituations where ear lier and ‘outside’ understandingsare, and are not, going to be helpful. For example, isit helpful to use your ‘outside’ knowledge aboutcooking when solving a r atio problem about the sizeof cakes? In abstract mathematics the same is tr ue:the word ‘similar’ means something rather vague in everyday speech, but has specific meaning inmathematics. Even within mathematics there areambiguities. We have to understand, for example,that -40 is greater in magnitude than -4, but a smaller number.

All students generalise inductively from the examplesthey are given. Research evidence of secondar ymathematics reveals many typical problems that arise because of generalising irrelevant features of examples, or over-generalising the domain ofapplicability of a method, but we found littlesystematic research to show instances where theability to generalise contributes positively to learningdifficult concepts, except to generate a need to lear nthe syntax of algebra.

Finally, we found considerable evidence thatstudents do, given appropriate experiences overtime, change the ways in which they approachunfamiliar mathematical situations and newconcepts. We only found anecdotal evidence thatthese new ways to view situations are extendedoutside the mathematics classroom. There isconsiderable evidence from long-term curriculumstudies that the procedures students have to learnin secondary mathematics are learnt more easily ifthey relate to less formal explorations they havealready undertaken. There is evidence thatdiscussion, verbalisation, and explicitness aboutlearning can help students make these changes.

Five common themes across the topics reviewedIn our view, a set of coherent themes cuts across therich, and at first sight heterogeneous, topics aroundwhich we have organised our outline. These themesrise naturally from the mater ial that we havementioned, and they do not include recent attemptsto link brain studies with mathematical education. Inour view, knowledge of brain functions is not yetsophisticated enough to account for assigning meaning,forming mathematical relationships or manipulatingsymbols, which we have concluded are the significanttopics in studies of mathematical lear ning.

In this section, we summarise five themes thatemerged as significant across the research on thedifferent topics, summarised in the previous sections.

Number

Number is not a unitar y idea that developsconceptually in a linear fashion. In learning, and inmathematical meaning, understanding of numberdevelops in complementar y strands, sometimes withdiscontinuities and changes of meaning. Emphasis oncalculation and manipulation with numbers ratherthan on understanding the under lying relations andmathematical meanings can lead to over-reliance andmisapplication of methods.

Most children star t school with everydayunderstandings that can contr ibute to their ear lylearning of number. They understand ‘more’ and ‘less’without knowing actual quantities, and can comparediscrete and continuous quantities of familiar objects.Whole number is the tool which enab les them to beprecise about comparisons and relations betweenquantities, once they under stand cardinality.

Learning to count and under standing quantities areseparate strands of development which have to beexperienced alongside each other. This allowscomparisons and combinations to be made that areexpressed as relations. Counting on its own does notprovide for these. Counting on its own also meansthat the shift from discrete to contin uous number isa conceptual discontinuity rather than an extensionof meaning.

Rational numbers (we have used ‘fraction’ and‘rational number’ interchangeably in order to focuson their meaning for learners, rather than on their



mathematical definitions) arise naturally for childrenfrom understanding division in sharing situations,rather than from par titioning wholes. Understandingrational numbers as a way of comparing quantities isfundamental to the development of multiplicativeand proportional reasoning, and to applications ingeometry, science, and everyday life. This is not thesame as saying that children should do ar ithmeticwith rational numbers. The decimal representationdoes not afford this connection (although it isrelatively easy to do additive arithmetic with decimalfractions, as long as the same n umber of digitsappears after the decimal point).

The connection between number and quantitybecomes less obvious in higher mathematics, e.g. onthe co-ordinate plane the numbers indicate scaledlengths from the axes, but are more usefullyunderstood as values of the var iables in a function.Students also have to extend the meaning ofnumber to include negative numbers, infinitesimals,irrationals, and possibly complex numbers. Numberhas to be abstracted from images of quantity andused as a set of related, continuous, values whichcannot all be expressed or depicted precisel y.Students also have to be able to handle number-likeentities in the form of algebraic terms, expressionsand functions. In these contexts, the idea of numberas a systematically related set (and subsets) is centr alto manipulation and transformation; they behave likenumbers in relations, but are not defined quantitiesthat can be enumerated. Ordinality of number alsohas a place in mathematics, in the domain offunctions that generate sequences, and also inseveral statistical techniques.

Successful learning of mathematics includesunderstanding that number describes quantity; beingable to make and use distinctions between different,but related, meanings of number; being able to userelations and meanings to inform application andcalculation; being able to use number relations tomove away from images of quantity and use n umberas a structured, abstract, concept.

Logical reasoning plays a crucialpart in every branch of mathematicallearning

The importance of logic in children’s understandingand learning of mathematics is a centr al theme in ourreview. This idea is not a new one , since it was alsothe main claim that Piaget made about children’ s

understanding of mathematics. However, Piaget’stheory has fallen out of favour in recent years, andmany leading researchers on mathematics learningeither ignore or actively dismiss his and his colleagues’contribution to the subject. So, our conclusion aboutthe importance of logic may seem a surprising onebut, in our view, it is absolutely inescapable. Weconclude that the evidence demonstr ates beyonddoubt that children rely on logic in lear ningmathematics and that many of their difficulties insolving mathematical problems are due to failures ontheir par t to make the correct logical move whichwould have led them to the cor rect solution.

We have reviewed evidence that four differentaspects of logic have a crucial role in learning aboutmathematics. Within each of these aspects we havebeen able to identify definite changes over time inchildren’s understanding and use of the logic inquestion. The four aspects follow.

The logic of correspondence (one-to-oneand one-to-many correspondence) Children must understand one-to-onecorrespondence in order to lear n about cardinalnumber. Initially they are much more adept atapplying this kind of cor respondence when theyshare than when they compare spatial arrays ofitems. The extension of the use of one-to-onecorrespondence from sharing to working out thenumerical equivalence or non-equivalence of two ormore spatial arrays is a vastly important step in ear lymathematical learning.

One–to-many correspondence, which itself is anextension of children’s existing knowledge of one-to-one correspondences, plays an essential, but untilrecently largely ignored, par t in children’s learningabout multiplication. Researchers and teachers havefailed to consider that one-to-many correspondenceis a possible basis for children’s initial multiplicativereasoning because of a wide-spread assumption thatthis reasoning is based on children’s additiveknowledge. However, recent evidence on how tointroduce children to multiplication shows thatteaching them multiplication in terms of one-to-manycorrespondence is more effective than teaching themabout multiplication as repeated addition.

The logic of inversion The subject of inversion was also neglected untilfairly recently, but it is now clear that under standingthat the addition and subtr action of the samequantity leaves the quantity of a set unchanged is of


great importance in children’s additive reasoning.Longitudinal evidence also shows that thisunderstanding is a strong predictor of children’smathematical progress. Experimental researchdemonstrates that a flexible understanding ofinversion is an essential element in children’ sgeometrical reasoning as well. It is highly likely thatchildren’s learning about the inverse relationbetween multiplication and division is an equallyimportant a par t of mathematical learning, but theright research still has to be done on this question.Despite this gap, there is a clear case for giving theconcept of inversion a great deal more prominencethan it has now in the school cur riculum.

The logic of class inclusion and additivecomposition Numbers consist of other numbers. One cannotunderstand what 6 means unless one also kno wsthat sets of 6 are composed of 5 + 1 items, or 4 +2 items etc . The logic that allows children to workout that every number is a set of combination ofother numbers is known as class inclusion. This formof inclusion, which is also referred to as additivecomposition of number, is the basis of theunderstanding of ordinal number: every number in the number series is the same as the one thatprecedes it plus one . It is also the basis f or learningabout the decade str ucture: the number 4321consists of four thousands, three hundreds two tensand one unit, and this can only be proper lyunderstood by a child who has thoroughl y graspedthe additive composition of number. This form ofunderstanding also allows children to comparenumbers (7 is 4 more than 3) and thus tounderstand numbers as a way of expressingrelations as well as quantities. The evidence clear lyshows that children’s ability to use this form ofinclusion in learning about number and in solvingmathematical problems is at first rather weak, andneeds some suppor t.

The logic of transitivity All ordered series, including number, and also formsof measurement involve transitivity (a > c if a > band b > c: a = c if a = b and b = c). Empiricalevidence shows that children as young as 5-years of age do to some extent gr asp this set of relations,at any rate with continuous quantities like length.However, learning how to use transitive relations in numerical measurements (for example, of area) isan intricate and to some extent a diff icult business.Research, including Piaget’s initial research onmeasurement, shows that one powerful reason

for children finding it difficult to apply transitivereasoning to measurement successfully is that theyoften do not grasp the impor tance of iteration(repeated units of measurement). These difficultiespersist through primary school.

One of the reasons why Piaget’s ideas about theimportance of logic in children’s mathematicalunderstanding have been ignored recently isprobably the nature of evidence that he off ered forthem. Although Piaget’s main idea was a positiveone (children’s logical abilities determine theirlearning about mathematics), his empir ical evidencefor this idea was mainly negative: it was aboutchildren’s difficulties with the four aspects of logicthat we have just discussed. A constant theme inour review is that this is not the best wa y to test a causal theor y about mathematical learning. Weadvocate instead a combination of longitudinalresearch with inter vention studies. The results of thiskind of research do strongly suppor t the idea thatchildren’s logic plays a critical par t in theirmathematical learning.

Children should be encouraged toreflect on their implicit models andthe nature of the mathematical tools

Children need to re-conceptualise their intuitiv emodels about the world in order to access themathematical models that have been developed inthe discipline. Some of the intuitive models used bychildren lead them to appropr iate mathematicalproblem solving, and yet they may not know whythey succeeded. This was exemplified by students’use of one-to-many correspondence in the solutionof proportions problems: this schema of action leadsto success but students may not be aware of theinvariance of the ratio between the variables whenthe scheme is used to solve problems. Increasingstudents’ awareness of this invariant should improvetheir mathematical understanding of proportions.

Another example of implicit models that lead tosuccess is the use of distr ibutivity in oral calculationof multiplication and division. Students who knowthat they can, instead of multiplying a number by 15,multiply it by 10 and then add half of this to theproduct, can be credited with implicit kno wledge ofdistributivity. It is possible that they would benefitlater on, when learning algebra, from the awarenessof their use of distr ibitivity in this context. Thisunderstanding of distr ibutivity developed in a



context where they could justify it could be used for later learning.

Other implicit models may lead students astray.Fischbein, Deri, Nello and Mar ino (1985) and Greer (1988) have shown that some implicit modelsinterfere with students’ problem solving. If, forexample, they make the implicit assumption that in a division the dividend must always be larger thanthe divisor, they might shift the numbers around inimplementing the division operation when thedividend is actually smaller than the divisor. So, whenstudents have developed implicit models that leadthem astray, they would also benefit from greaterawareness of these implicit models.

The simple fact that students do use intuitiv e modelswhen they are learning mathematics, whether theteacher recognises the models or not, is a reason forwanting to help students develop an awareness ofthe models they use . Instruction could and shouldplay a crucial role in this process.

Finally, reflecting on implicit models can help studentsunderstand mathematics better and also linkmathematics with reality and with other disciplinesthat they learn in school. Freudenthal (1971) arguedthat it would be difficult for teachers of otherdisciplines to tie the bonds of mathematics to realityif these have been cut by the mathematics teacher. In order to tie these bonds, mathematics lessons canexplore models that students use intuitively andextend these models to scientific concepts that havebeen shown to be challenging for students. One ofthe examples explored in a mathematics lessondesigned by Treffers (1991) focuses on themathematics behind the concept of density. He tellsstudents the number of bicycles owned by people in United Kingdom and in the Nether lands. He alsotells them the population of these tw o countries. He then asks them in which countr y there are morebicycles. On the basis of their intuitiv e knowledge,students can easily engage in a discussion that leadsto the concept of density: the number of bicyclesshould be considered in relation to the n umber of people. A similar discussion might help studentsunderstand the idea of population density and ofdensity in physics, a concept that has been sho wn tobe very difficult for students. The discussion of howone should decide which countr y has more bicyclesdraws on students’ intuitive models; the concept ofdensity in physics extends this model. Streefland andVan den Heuvel-Panhuizen (see Paper 4) suggestedthat a model of a situation that is under stood

intuitively can become a model f or other situations,which might not be so accessib le to intuition.Students’ reflection about the mathematicsencapsulated in one concept is termed by Treffershorizontal mathematising; looking across conceptsand thinking about the mathematics tools themselv esleads to vertical mathematising, i.e. a re-constructionof the mathematical ideas at a higher lev el ofabstraction. This pragmatic theory about howstudents’ implicit models develop can be easily put to test and could have an impact on mathematics aswell as science education.

Mathematical learning depends onchildren understanding systems ofsymbols

One of the most powerful contributions of recentresearch on mathematical learning has come fromwork on the relation of logic , which is universal, tomathematical symbols and systems of symbols, whichare human inventions, and thus are cultur al tools thathave to be taught. This distinction plays a role in allbranches of mathematical learning and has ser iousimplications for teaching mathematics.

Children encounter mathematical symbolsthroughout their lives, outside school as well as inthe classroom. They first encounter them in lear ningto count. Counting systems with a base pro videchildren with a powerful way of representingnumbers. These systems require the cognitive skillsinvolved in generative learning. As it is impossible tomemorise a very long sequence of words in a fixedorder, counting systems with a base solv e thisproblem: we learn only a few symbols (the labels forunits, decades, hundred, thousand, million etc.) bymemory and generate the other ones in a r ule-based manner. The same is tr ue for the Hindu-Arabicplace value system for writing numbers: when weunderstand how it works, we do not need tomemorise how each number is written.

Mathematical symbols are technologies in the sensethat they are human-made tools that impro ve ourability to control and adapt to the environment. Eachof these systems makes specific cognitive demandsfrom the learner. In order to under stand place-valuerepresentation, for example, students’ mustunderstand additive composition. If students haveexplicit knowledge of additive composition and howit works in place-value representation, they arebetter placed to learn column arithmetic, which


should then enable students to calculate with verylarge numbers; this task is very taxing withoutwritten numbers. So the costs of lear ning to usethese tools are worth paying: the tools enablestudents to do more than they can do without thetools. However, research shows that students shouldbe helped to make connections between symbolsand meanings: they can behave as if they under standhow the symbols work while they do notunderstand them completely: they can learn routinesfor symbol manipulation that remain disconnectedfrom meaning.

This is also tr ue of rational numbers. Children can learn to use wr itten fractions by counting thenumber of par ts into which a whole was cut andwriting this below a dash, and counting the numberof par ts painted and wr iting this above the dash.However, these symbols can remain disconnectedfrom their logical thinking about division. Thesedisconnections between symbols and meaning arenot restricted to writing fractions: they are alsoobserved when students learn to add and subtractfractions and also later when students lear nalgebraic symbols.

Plotting variables in the Cartesian plane is anotheruse of symbol systems that can empo wer students:they can, for example, more easily analyse change by looking at graphs than they can by intuitivecomparisons. Here, again, research has shown howreading graphs also depends on the inter pretationsthat students assign to this system of symbols.

A recurrent theme in the review of research acrossthe different topics was that the disconnectionbetween symbols and meanings seems to explainmany of the difficulties faced by primary schoolstudents in learning mathematics. The inevitableeducational implication is that teaching aims shouldinclude promoting connections between symbolsand meaning when symbols are introduced and usedin the classroom.

This point is, of course, not new, but it is wellworth reinforcing and, in par ticular, it is well worthremembering in the light of cur rent findings. Thehistory of mathematics education includes thedevelopment of pedagogical resources that weredeveloped to help students attr ibute meaning tomathematical symbols. But some of these resources,like Dienes’ blocks and Cuisinaire’s rods, are onlyencountered by students in the classroom; thepoint we are making here is that students acquire

informal knowledge in their everyday lives, whichcan be used to give meaning to mathematicalsymbols learned in the classroom. Research inmathematics education over the last f ive decades or so has helped descr ibe the situations in whichthese meanings are learned and the way in whichthey are structured. Curriculum development workthat takes this knowledge into account has alreadystarted (a major example is the research b ymembers of the Freudenthal Institute) b ut it is not as widespread as one would expect given thediscoveries from past research.

Children need to learn modes of enquiryassociated with mathematicsWe identify some impor tant mathematical modes of enquiry that arise in the topics covered in thissynthesis.

Comparison helps us make newdistinctions and create new objects and relationsA cycle of creating and naming new objects throughacting on simple objects per vades mathematics, andthe new objects can then be related and compared to create higher-level objects. Making additive andmultiplicative comparisons is an aspect ofunderstanding relations between quantities andarithmetic. These comparisons are manifested preciselyas difference and ratio. Thus difference and ratio ariseas two new mathematical ideas, which become newmathematical objects of study and can be representedand manipulated. Comparisons are related to makingdistinctions, sorting and classifying based onperceptions, and students need to lear n to makethese distinctions based on mathematical relations and properties, rather than perceptual similar ities.

Reasoning about properties and relationsrather than perceptionsMany of the problems in mathematics that studentsfind hard occur when immediate perceptions lead tomisapplication of learnt methods or informalreasoning. Throughout mathematics, students have tolearn to interpret representations before they thinkabout how to respond. They need to think about therelations between different objects in the systemsand schemes that are being represented.

Making and using representationsConventional number symbols, algebraic syntax,coordinate geometry, and graphing methods, all affordmanipulations that might otherwise be impossib le.Coordinating different representations to explore and



extend meaning is a fundamental mathematical skillthat is implicit in the use of the n umber line torepresent quantities, for example, the use of gr aphs toexpress functions. Equivalent representations, such asfor number, algebraic relationships and functions, canprovide new insights through compar ison andisomorphic analogical reasoning.

Action and reflection-on-actionLearning takes place when we reflect on the effects of actions. In mathematics, actions may be physicalmanipulation, or symbolic rearrangement, or ourobservations of a dynamic image, or use of a tool. In all these contexts, we observe what changes andwhat stays the same as a result of actions, and makeinferences about the connections between action andeffect. In ear ly mathematics such reflection is usuall yembedded in children’s classroom activity, such aswhen using manipulatives to model changes inquantity. In later mathematics changes and invariancemay be less obvious, particularly when change isimplicit (as in a situation to be modelled) or usefulvariation is hard to identify (as in a quadr atic function).

Direct and inverse relations Direct and inverse relations are discussed in severalof our papers. While it may sometimes be easier toreason in a direct manner that accords with action, itis important in all aspects of mathematics to be ab leto construct and use inverse reasoning. Addition and subtraction must be understood as a pair, andmultiplication and division as a pair, rather than as aset of four binary operations. As well as enablingmore understanding of relations between quantities,this also establishes the impor tance of reverse chainsof reasoning throughout mathematical problem-solving, algebraic and geometrical reasoning. Forexample, using reverse reasoning makes it morelikely that students will lear n the dualism embeddedin Car tesian representations; that all points on thegraph fulfil the function, and the function gener atesall points on the gr aph.

Informal and formal reasoningAt first young children bring everyday understandingsinto school, and mathematics can allow them toformalise these and make them more precise . On theother hand, intuitions about continuity, approximation,dynamic actions and three-dimensional space mightbe over-ridden by early school mathematics – yet areneeded later on. Mathematics also provides formaltools which do not descr ibe everyday outsideexperience, but enable students to solve problems inmathematics and in the world which would be

unnoticed without a mathematical per spective. In the area of word problems and realistic problemslearning when and how to apply informal and formalreasoning is impor tant. Later on, counter-intuitiveideas have to take the place of ear ly beliefs, such as‘multiplication makes things bigger’ and students haveto be wary of informal, visual and immediateresponses to mathematical stimuli.

A recurring issue in the papers is that students find it hard to coordinate attention on local and globalchanges. For example, young children confusequantifying ‘relations between relations’ with theoriginal quantities; older children who cannot identifycovariation of functions might be able to talk aboutseparate variation of variables; students readily seeterm-to-term patterns in sequences rather than thegenerating function; changes in areas are confusedwith changes in length.

EpilogueOur aim has been to wr ite a review that summar isesour findings from the detailed analysis of a largeamount of research. We sought to make it possiblefor educators and policy makers to take a fresh lookat mathematics teaching and lear ning, star ting fromthe results of research on key understandings, ratherthan from previous traditions in the organisation ofthe curriculum. We found it necessar y to organiseour review around ideas that are already core ideasin the curriculum, such as whole and r ationalnumber, algebra and problem solving, but also tofocus on ideas that might not be identif ied so easilyin the current curriculum organisation, such asstudents’ understanding of relations betweenquantities and their under standing of space.

We have tried to make cogent and convincingrecommendations about teaching and lear ning, and to make the reasoning behind theserecommendations clear to educationalists. We have also recognised that there are w eaknesses inresearch and gaps in cur rent knowledge, some ofwhich can be easily solved by research enabled bysignificant contributions of past research. Other gapsmay not be so easil y solved, and we have describedsome pragmatic theories that are, or can be , used byteachers when they design instr uction. Classroomresearch, stemming from the explor ation of thesepragmatic theories, can provide new insights forfurther research in the future .


Endnotes1 Details of the search process is pro vided in Appendix 1. This

contains the list of data bases and jour nals consulted and thetotal number of papers read although not all of these can becited in the six paper s that comprise this review.



Anghileri, J., Beishuizen, M., and Putten, K. V. (2002).From informal strategies to structured procedures:mind the gap! Educational Studies in Mathematics ,49, 149–170.

Carey, S. (2004). Bootstrapping and the origins ofconcepts. Daedalus, 133, 59–68.

Cooper, H. M. (1998). Synthesizing research: a guidefor literature reviews. Thousand Oaks, CA: SagePublications.

Confrey, J. (2008). A synthesis of research on r ationalnumber reasoning. Paper presented at theInternational Conference on MathematicsEducation – ICME 11, Monterrey, Mexico.

Fischbein, E., Deri, M., Nello, M., & Marino, M. (1985).The role of implicit models in solving verbalproblems in multiplication and division. Journal forResearch in Mathematics Education, 16, 3–17.

Freudenthal, H. (1971). Geometry between the Deviland the Deep Sea. Educational Studies inMathematics, 3, 413–435.

Greer, B. (1988). Nonconservation of multiplicationand division: Analysis of a symptom. Journal ofMathematical Behavior, 7, 281–298.

Hart. K. (1981) Children’s Understanding ofMathematics 11–16. London (U.K.): Murray.

Küchemann, D. (1981) Algebra, in K. Har t (Ed.)Children’s Understanding of Mathematics 11–16 ,102–119 London (U.K.); Murray.

Luria, A. R. (1973). The working brain. An introduction toneuropsychology. Harmondsworth (U.K.): Penguin.

Noss, R., and Hoyles, C. (1992). Looking back andlooking forward. In C . Hoyles and R. Noss (Eds.),Learning mathematics and LOGO (pp. 431–470).Cambridge, MA: The MIT Press.

Nunes, T. (1997). Systems of signs and mathematicalreasoning. In T. Nunes amd P. Bryant (Eds.),Learning and Teaching Mathematics. AnInternational Perspective (pp. 29–44). Hove (UK):Psychology Press.

Nunes, T. (2002). The role of systems of signs inreasoning. In T. Brown and L. Smith (Eds.),Reductionism and the Development of Knowledge(pp. 133–158). Mawah, NJ: Lawrence Erlbaum.

Nunes, T., and Br yant, P. (2008). Rational Numbersand Intensive Quantities: Challenges and Insightsto Pupils’ Implicit Knowledge. Anales de Psicologia,in press.

Ruthven, K. (2008). Creating a calculator-awarenumber curriculum. Canadian Journal of Science,Mathematics and Technology Education, 3, 437–450.

Tall, D. (1992). The transition to advancedmathematical thinking: Functions, limits, infinity, andproof. In D. A. Grouws (Ed.), Handbook ofResearch on Mathematics Teaching and Learning(pp. 495–511). New York, N.Y.: Macmillan.

Thompson, P. W. (1993). Quantitative Reasoning,Complexity, and Additive Structures. EducationalStudies in Mathematics , 25 , 165–208.

Thompson, P. (1994). The Development of theConcept of Speed and Its Relationship toConcepts of Rate. In G. Harel and J. Confrey(Eds.), The Development of Multiplicative Reasoningin the Learning of Mathematics (pp. 181–236).Albany, N.Y.: State University of New York Press.

Treffers, A. (1987). Integrated column ar ithmeticaccording to progressive schematisation.Educational Studies in Mathematics , 18 125–145.

Treffers, A. (1991). Didactical background of amathematics programme for primary education.In L. Streefland (Ed.), Realistic MathematicsEducation in Pr imary School (pp. 21–56). Utrecht:Freudenthal Institute, Utrecht University.

Vergnaud, G. (1997). The nature of mathematicalconcepts. In T. Nunes and P. Bryant (Eds.), Learningand Teaching Mathematics. An InternationalPerspective (pp. 1–28). Hove (UK): Psychology Press.

Vygotsky, L. S. (1981). The genesis of higher mentalfunctions. In J. V. Wertsch (Ed.), The concept ofactivity in Soviet Psychology (pp. 57–93). Amonk,N.Y.: Sharpe.

Wartofsky, M. W. (1979). Models: representation andthe scientific understanding. Dordrecht: Reidel.


References

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Paper 2: Understanding whole numbersBy Terezinha Nunes and Peter Bryant, University of Oxford




2

2 SUMMARY – PAPER 2: Understanding whole numbers2 Paper 2: Understanding whole numbers












ContentsSummary of paper 2 3

Understanding extensive quantities and whole numbers 7

References 34

About the authorsTerezinha Nunes is Professor of EducationalStudies at the University of Oxford. Peter Bryant is Senior Research Fellow in theDepartment of Education, University of Oxford.


About this review


Headlines• Whole numbers are used in primary school to

represent quantities and relations. It is crucial forchildren’s success in learning mathematics inprimary school to establish clear connectionsbetween numbers, quantities and relations.

• Using different schemes of action, such as settingobjects in correspondence, children can judgewhether two quantities are equivalent, and if they arenot, make judgements about their order of magnitude.These insights are used in understanding the numbersystem beyond simply producing a string of numberwords in a fixed order. It takes children some time tomake links between their understanding of quantitiesand their knowledge of number.

• Children start school with varying levels of ability in using different action schemes to solvearithmetic problems in the context of stories. Theydo not need to know arithmetic facts to solvethese problems: they count in different waysdepending on whether the problems they aresolving involve the ideas of addition, subtraction,multiplication or division.

• Individual differences in the use of action schemesto solve problems predict children’s progress inlearning mathematics in school.

• Interventions that help children learn to use theiraction schemes to solve problems lead to betterlearning of mathematics in school.

• It is considerably more difficult for children to usenumbers to represent relations than to representquantities. Understanding relations is crucial for theirfurther development in mathematics in school.

In children’s everyday lives and before they startschool, they have experiences of manipulating andcomparing quantities. For example, even at age four,many children can share sweets fairly between tworecipients by using correspondences: they share givingone-for-you, one-for-me, until there are no sweets left.They do sometimes make mistakes but they knowthat, when the sharing is done fairly, the two peoplewill have the same amount of sweets at the end. Evenyounger children know some things about quantities:they know that if you add sweets to a group ofsweets, there will be more sweets there, and if youtake some away, there will be fewer. However, theymight not know that if you add a certain number andtake away the same number, there will be just as manysweets as there were before.

At the same time that young children are developingthese ideas about quantities, they are often learningto count. They learn to say the sequence of numberwords in the right order, they know that each objectthat they are counting must be counted once andonly once, and that it does not matter if you count arow of sweets from left to right or from right to left,you should get to the same number.

Four-year-olds are thus amazing learners ofmathematics. But they lack one thing which is cruciallyimportant: they do not at first make connectionsbetween their understanding of quantities and theirknowledge of numbers. So if you ask a four-year-old, who just shared some sweetsfairly between two dolls, to count the sweets thatone doll has and then tell you, without counting, howmany sweets the other doll has, the majority (about60%) will tell you that they do not know. Knowingthat the dolls have the same quantity is not sufficient

Summary of paper 2: Understanding whole numbers


to know that if one has 8 sweets, the other one has 8 sweets also, i.e. has the same number.

Quantities and numbers are not the same thing. Wecan use numbers as measures of quantities, but wecan think about quantities without actually having ameasure for them. Until children can understand theconnections between numbers and quantities, theycannot use their knowledge of quantities to supporttheir understanding of numbers and vice versa.Because the connections between quantities andnumbers are many and varied, learning about theseconnections could take three to four years inprimary school.

An important link that children must make betweennumber and quantity is the link between the order of number words in the counting sequence and themagnitude of the quantity represented. How dochildren come to understand that the any number in the counting sequence is equal to the precedingnumber plus 1?

Different explanations have been proposed in theliterature. One is that they simply see that magnitudeincreases as they count. But this explanation doesnot work well: our perception of magnitude isapproximate and knowing that any number is equalto its predecessor plus 1 is a very precise piece ofknowledge. A second explanation is that children use perception, language and inferences together toreach this understanding. Young children discriminatewell, for example, one puppet from two puppets andtwo puppets from three puppets. Because theyknow these differences precisely, they put these two pieces of information together, and learn thattwo is one more than one, and three is one morethan two. They then make the inference that allnumbers in the counting sequence are equal to thepredecessor plus one. But this sort of generalisationcould not be stretched into helping childrenunderstand that any number is also equal to the last-but-one in the sequence plus 2. This process ofputting together perception with language and thengeneralising is an explanation for only the n + 1 idea;it would be much better if we could have a moregeneral explanation of how children understand the connection between quantities and the number sequence.

The third explanation for how children connect heir knowledge of quantities with the magnitude ofnumbers in the counting sequence is that children’sschemes of action play the most important part in this

development. The actions of adding and taking awayhelp them understand part–whole relations. Whenthey can link their understanding of part–wholerelations with counting, they will understand manythings about relations between numbers. A criticalchange in young children’s behaviour when they addtwo sets is from ‘count all’ to ‘count on’. If they knowthat they have 5 sweets, and you add 4 to the 5, theycould either start from 1 and count all the sweets(count all) or they could point to the 5, and count onfrom there. ‘Count on’ is a sign that the children havelinked their knowledge of part–whole relations withthe counting sequence: they have understood theadditive composition number. This explanation worksfor the relation between a number and its immediatepredecessor and any of its predecessors. It issupported by much research that shows that countingon is a sign of abstraction in part–whole relations,which opens the way for children to solve many otherproblems: they can add a quantity to an invisible set,count coins of different denominations to form a singletotal, and are ready to learn to use place value torepresent numbers in writing.

Adding and subtracting elements to sets also givechildren the opportunity to understand the inverserelation between addition and subtraction. This insightis not gained in an all-or-nothing fashion: children firstapply it only to quantities and later on to number also.The majority of five-year-olds realises that if you add 3sweets to a set of sweets and then take the samesweets away, the number of sweets in the set remainsthe same. However, many of these children will notrealise that if you add 3 sweets to the set and thentake 3 other sweets away, the number of sweets is stillthe same. They see that adding and taking away thesame quantity leaves the original quantity the same butthis does not immediately mean to them that addingand taking away the same number also leaves theoriginal number the same. Research shows that thestep from understanding the inverse relation betweenaddition and subtraction of quantities is a useful start ifone wants to teach children about the inverse relationbetween addition and subtraction of number.

Adding, taking away and understanding part–wholerelations form one part of the story of whatchildren know about quantities and numbers in theearly years of primary school. They relate to howadditive reasoning develops. The other part of thestory is surprising to many people: children alsoknow quite a lot about multiplicative reasoningwhen they start school.

4 Paper 2: Understanding whole numbers

Children use two different schemes of action to solvemultiplication and division problems before they aretaught about these operations in school: they useone-to-many correspondence and sharing. If five- and six-year-olds are shown, for example, four littlehouses in a row, told that they should imagine that ineach live three dogs, and asked how many dogs livein the street, the majority can say the correct number.Many children will point three times to each houseand count in this way until they complete thecounting at the fourth house. They are notmultiplying: they are solving the problem using one-to-many correspondence. Children can also shareobjects to recipients and answer problems aboutdivision. They do not know the arithmetic operations,but they can use their reasoning to count in differentways and solve the problem. So children manipulatequantities using multiplicative reasoning and solveproblems before they learn about multiplication anddivision in school.

If children are assessed in their understanding of theinverse relation between addition and subtraction, of additive composition, and of one-to-manycorrespondence in their first year of school, thisprovides us with a good way of anticipating whetherthey will have difficulties in learning mathematics inschool. Children who do well in these assessments go on to attain better results in mathematicsassessments in school. Those who do not do well canimprove their prospects through early intervention.Children who received specific instruction on theserelations between quantities and how to use them tosolve problems did significantly better than a similargroup who did not receive such instruction.

Finally, many studies have used story problems toinvestigate which uses of additive reasoning areeasier and which are more difficult for children ofprimary school age. Two sorts of difficulties havebeen identified.

The first relates to the need to understand thataddition and subtraction are the inverse of eachother. One story that requires this understanding is:Ali had some Chinese stamps in his collection and hisgrandfather gave him 2; now he has 8; how manystamps did he have before his grandfather gave himthe 2 stamps? This problem exemplifies a situation inwhich a quantity increases (the grandfather gave him2 stamps) but, because the information about theoriginal number in his collection is missing, theproblem is not solved by an addition but rather by a subtraction. The problem would also be an inverse

problem if Ali had some Chinese stamps in hiscollection and gave 2 to his grandfather, leaving hiscollection with 6. In this second problem, there is adecrease in the quantity but the problem has to besolved by an increase in the number, in order to getus back to Ali’s collection before he gave 2 stampsaway. There is no controversy in the literature: inverseproblems are more difficult than direct problems,irrespective of whether the arithmetic operation thatis used to solve it is addition or subtraction.

The second difficulty depends on whether thenumbers in the problem are all about quantities orwhether there is a need to consider a relationbetween quantities. In the two problems about Ali’sstamps, all the numbers refer to quantities. An exampleof a problem involving relations would be: In Ali’s classthere are 8 boys and 6 girls; how many more boysthan girls in Ali’s class? (Or how many fewer girls thanboys in Ali’s class?). The number 2 here refers neitherto the number of boys nor to the number of girls: itrefers to the relation (the difference) between numberof boys and girls. A difference is not a quantity: it is arelation. Problems that involve relations are moredifficult than those that involve quantities. It should notbe surprising that relations are more difficult to dealwith in numerical contexts than quantities: the majority,if not all, the experiences that children have withcounting have to do with finding a number torepresent a quantity, because we count things and notrelations between things. We can re-phrase problemsthat involve relations so that all the numbers refer toquantities. For example, we could say that the boysand girls need to find a partner for a dance; how manyboys won’t be able to find a girl to dance with? Thereare no relations in this latter problem, all the numbersrefer to quantities. This type of problem is significantlyeasier. So it is difficult for children to use numbers torepresent relations. This could be one step thatteachers in primary school want to help their childrentake, because it is a difficult move for every child.



Recommendations


Research about mathematicallearning

Children’s pre-school knowledge ofquantities and counting developsseparately.

When children star t school, they can solvemany different problems using schemes ofaction in coordination with counting,including multiplication and divisionproblems.

Three logical-mathematical reasoningprinciples have been identified in research,which seem to be causally related tochildren’s later attainment in mathematics in primary school. Individual differences inknowledge of these pr inciples predict laterachievement and interventions reducelearning difficulties.

Children’s ability to solve word problemsshows that two types of problem causedifficulties for children: those that involvethe inverse relation between addition andsubtraction and those that involve thinkingabout relations.

Recommendations for teaching and research

Teaching Teachers should be aware of the impor tance of helping children make connections between theirunderstanding of quantities and their knowledge of counting.

Teaching The linear view of development, according to which understanding addition precedes multiplication, is notsupported by research. Teachers should be aware of children’smathematical reasoning, including their ability to solvemultiplication and division problems, and use their abilities for fur ther learning.

Teaching A greater emphasis should be given in the curriculum to promoting children’s understanding of theinverse relation between addition and subtraction, additivecomposition, and one-to-many correspondence. This would helpchildren who star t school at r isk for difficulties in learningmathematics to make good progress in the f irst years.Research Long-term longitudinal and inter vention studieswith large samples are needed before curriculum and policychanges can be proposed. The move from the laboratory tothe classroom must be based on research that identif iespotential difficulties in scaling up successful inter ventions.

Teaching Systematic use of problems involving thesedifficulties followed by discussions in the classroom would givechildren more oppor tunities for making progress in usingmathematics in contexts with which they ha ve difficulty.Research There is a need for intervention studies designedto promote children’s competence in solving problems aboutrelations. Brief experimental interventions have paved the wayfor classroom-based research but large-scale studies areneeded.

Counting and reasoningAt school, children’s formal learning aboutmathematics begins with natural numbers (1, 2,…17…103 …525…). Numbers are symbols forquantities: they make it possible for the child tospecify single values precisely and also to work out the relations between different quantities. Bycounting, the child can tell you that there are 20books in the pile on the teacher’s desk (a singlequantity), and eventually should be able to work out that there is 1 book for every child in the class if there are 20 children there, or that there are 5 more books than children (a relation between twoquantities) if there are 15 children in the class.

Quantities and numbers are not the same. Thompson(1993) suggested that ‘a person constitutes a quantityby conceiving of a quality of an object in such a waythat he or she understands the possibility of measuringit. Quantities, when measured, have numerical value,but we need not measure them or know theirmeasures to reason about them. You can think of yourheight, another person's height, and the amount bywhich one of you is taller than the other withouthaving to know the actual values’ (pp. 165–166).1

Children experience and learn about quantities andthe relations between them quite independently oflearning to count. Similarly, they can learn to countquite independently from understanding quantitiesand relations between them. We shall argue in thissection that the most important task for a child whois learning about natural numbers is to connect thesenumbers to a good understanding of quantities andrelations. The connection should work at two levels.

First, children must realise that their knowledge ofquantities and numbers should agree with one

another. If Sean has 15 books and Patrick 17, Patrick has more books than Sean. Unless childrenunderstand that numbers are a precise way ofexpressing quantities, the number system will haveno meaning for them.

Second, they must realise eventually that the numbersystem enhances their knowledge of quantities in anincreasingly powerful way. They may not be able tolook at a pile of books and tell without counting thatthe one with 17 has more books than the one with15; indeed, the thickness of books varies and the pileof 15 books could well be taller than the pile with 17.By counting they can know which pile has morebooks. When they know how to count, we can alsoadd and subtract numbers, and work out the exactrelations between them. If we understand lots ofthings about quantities, e.g. how to create equivalentquantities and how their equivalence is changed, butwe don’t have numbers to represent them, wecannot add and subtract.

In this section, therefore, we will focus on theconnections that children make, and sometimes fail to make, between their growing knowledge aboutquantities and the number system. In many ways thisis an unusual thing to do. Most existing accounts ofhow children learn about number are morerestricted. Either they leave out the number systemaltogether and concentrate instead on children’sability to reason about quantities, or they are strictlyconfined to how well children count sets of objects.

Piaget’s theory (Piaget, 1952) is an example of thefirst kind of theory. His view that children have to beable to reason logically about quantity in order tounderstand number and the number system is


Understanding extensivequantities and wholenumbers


almost certainly right, but it left out the possibilitythat learning to count eventually transforms thisreasoning in children by making it more powerfuland more precise.

In the opposite corner, Gelman’s influential theory(Gelman and Gallistel, 1978), which focuses on howchildren count single sets of objects and has little tosay about children’s quantitative reasoning, has theserious disadvantage that it by-passes children’sreasoning about relations between quantities. In theend, numbers are only important because they allowus to represent quantities and make sense ofquantitative relations.

The first part of this section is an account of how children connect numbers with quantity. We will start this account with a detailed list of theconnections that they need to make. We argue thatchildren need to make three types of connectionsbetween number words and quantities in order tomake the most of what they learn when they beginto count: they need to understand cardinality; theyneed to understand ordinal numbers, and they needto understand the relation between cardinality andaddition and subtraction. The second part of thissection is an account of how children learn to usenumbers to solve problems. We argue that numbersare used to represent quantities but that childrenmust also learn to use them to representtransformations and relations, and that the differentmeanings that numbers can have affect how easilychildren solve problems.

Giving meaning to numbersYoung children’s dissociation ofquantities and numbers

Children may know that two quantities are the sameand still not make the inference that the number ofobjects in one is the same as the number of objectsin the other. Conversely, they may know how tocount and yet not make use of counting when askedto create two equal sets. We review here brieflyresearch within two different traditions, inspired byPiaget’s and Gelman’s theories, that shows that youngchildren do not necessarily make a connectionbetween what they know about quantities and what they know about counting.

Equivalence of sets in one-to-onecorrespondence and its connection tonumber wordsNumbers have both cardinal and ordinal properties.Two sets have the same cardinal value when theitems in one set are in one-to-one correspondencewith those in the other. There are as many eggcups ina box of six egg-cups as there are eggs in a carton ofsix eggs, and if there are six people at the breakfasttable each will have one of those eggs on its owneggcup to eat. Thus, the eggcups, eggs and people areall in one-to-one correspondence since there is oneegg and eggcup for each one person, which meansthat each of these three sets has the same number.

We shall deal with the ordinal properties of numberin a later section. At the moment, all that we need tosay is that numbers are arranged in an ordered series.

To return to cardinality, Piaget argued quitereasonably that no one can understand the meaningof ‘six’ unless he or she also understands thenumber’s cardinal properties, and by this he meantunderstanding not only that any set of six containsthe same number of items as any other set of six butalso that that the items in a set of six are in one-to-one correspondence with any other set of six items.So, if we are to pursue the approach of studying thelinks between children’s quantitative reasoning andhow they learn about natural numbers, we need tofind out how well children understand the principlethat sets which are in one-to-one correspondencewith each other are equal in quantity, and also howclearly they apply what they understand about one-to-one correspondence to actual numbers like ‘six’.

Piaget based his claim that young children have a verypoor understanding of one-to-one correspondenceon the mistakes that they make when they are shownone set of items (e.g. a row of eggs) and are asked toform another set (e.g. of eggcups) of the samenumber. Four- and five-year-olds often match the newset with the old one on irrelevant criteria such astwo rows’ lengths and make no effort to put therows into one-to-one correspondence. Their abilityto establish one-to-one correspondence betweensets grows over time: it cannot be taken for granted.

However, even when children do establish a one-to-one correspondence between two sets, they do notnecessarily infer that counting the elements in one settells them how many elements there are in the otherset. Piaget (1952) established this in an experiment in


which he proposed to buy sweets from the children,using a one-to-one exchange between pence andsweets. For each sweet that the child gave to Piaget,he gave the child a penny. As they exchanged penceand sweets, the child was asked to count how manypence he/she had. Piaget ensured that he stopped thisexchange procedure without going over the child’scounting range. When he stopped the exchange, heasked the child how many pence the child had. Thechildren were able to answer this without difficulty asthey had been counting their coins. He then asked thechild how many sweets he had. Piaget reports thatsome children were unable to make the inferencethat the number of sweets Piaget had was the sameas the number of pence that the child himself/herselfhad. Unfortunately, Piaget gave no detailed descriptionof how the ability to make this inference related tothe children’s age.

More recent research, which offers quantitativeinformation, shows that many four-year-olds who dounderstand one-to-one correspondence well enoughto share fairly do not make the inference thatequivalent sets have the same number of elements.Frydman and Bryant (1988) asked four-year-oldchildren to share a set of ‘chocolates’ to tworecipients. At this age, children often share thingsbetween themselves, and they typically do so on aone-for-A, one-for-B, one-for-A, one-for-B basis. In thisstudy, the children established the correspondencethemselves; this contrasts with Piaget’s study, wherePiaget controlled the exchange of sweets and pence.When the child had done the sharing, theexperimenters counted out the number of items thathad been given to one recipient, which was six.Having done this, they asked the child how manychocolates had been given to the other recipient.

None of the children immediately made theinference that there were the same number ofchocolates in one set as in the other, and thereforethat there were also six items in the second set.Instead, every single child began to count the secondset. In each case, the experimenter then interruptedthe child’s counting, and asked him or her if there wasany other way of working out the number of items inthe second recipient’s share. Only 40% of the groupof four-year-olds made the correct inference that thesecond recipient had also been given six chocolates.The failure of more than half of the children is aninteresting one. The particular pre-school childrenwho made it knew that the two recipients’ shareswere equal, and they also knew the number of itemsin one of the shares. Yet, they did not connect what

they knew about the relative quantities to thenumber symbols. Other children, however, did makethis connection, which we think is the first significantstep in understanding cardinality. Whether all childrenwill have made this connection by the time that theystart learning about numbers and arithmetic atschool depends on many factors: for example, theage they start school and their previous experienceswith number are related to whether they have takenthis important step by then (e.g. socio-economicstatus related to maths ability at school entry: seeGinsburg, Klein, and Starkey, 1998; Jordan,Huttenlocher, and Levine, 1992; Secada, 1992).

Counting and understanding relationsbetween quantitiesPiaget’s theory of how children develop anunderstanding of cardinality was confronted by analternative theory, by Gelman’s nativist view ofchildren’s counting and its connection to cardinalnumber knowledge (Gelman and Gallistel, 1978).Gelman claimed that children are born with agenuine understanding of natural number, and thatthis makes it possible for them to learn and use thebasic principles of counting as soon as they begin tolearn the names for numbers. She outlined five basiccounting principles. Anyone counting a set of objectsshould understand that:• you should count every object once and only once

(one-to-one correspondence principle)• the order in which you count the actual objects

(from left-to right, from right to left or from themiddle outwards) makes no difference (orderirrelevance principle)

• you should produce the number words in aconstant order when counting: you cannot count 1-2-3 at one time and 1-3-2 at another (fixedorder principle)

• whether the objects in a set are all identical to eachother or all quite different has no effect on theirnumber (the abstraction principle)

• the last number that you count is the number ofitems in the set (cardinal principle).

Each of these principles is justified in the sense thatanyone who does not respect them will end upcounting incorrectly. A child who produces countwords in different orders at different times is boundto make incorrect judgements about the number ofitems in a set. So will anyone who does not obey theone-to-one principle.

Gelman’s original observations of children countingsets of objects, and the results of some subsequent



experiments in which children had to spot errors inother protagonists’ counting (e.g. Gelman and Meck,1983), all supported her idea that children obey andapparently understand all five of these principleswith small sets of items long before they go toschool. The young children’s success in countingsmaller sets allowed her to dismiss their morefrequent mistakes with large sets of items asexecutive errors rather than failures inunderstanding. She agued that the children knewthe principles of counting and therefore of number,but lacked some of the skills needed to carry themout. This view became known as the ‘principles-before-skills hypothesis’.

These observations of Gelman’s provoked a greatdeal of useful further research on children’s counting,most of which has confirmed her original results,though with some modifications. For example, five-year-old children do generally count objects in a one-to-one fashion (one number word for each object)but not all of the time (Fuson, 1988). They tend eitherto miss objects or count some more than once indisorganised arrays. It is now clear that gestures playan important part in helping children keep trackduring counting (Albilali and DiRusso, 1999) butsometimes they point at some of the objects in atarget set without counting them.

Many of the criticisms of Gelman’s hypothesis were against her claims that children understandcardinality. Ironically, even critics of Gelman (e.g.Carey, 2004; LeCorre and Carey, 2007)) have intheir own research accepted her all too limiteddefinition of understanding cardinality (that it is therealisation that the last number counted representsthe number of objects). However, severalresearchers have criticised her empirical test ofcardinality. Gelman had argued that children, whocount a set of objects and emphasise the lastnumber (‘one-two-three-FOUR’) or repeat it (‘one-two-three-four- there are four’), understandthat this last number represents the quantity of thecounted set. However, Fuson (Fuson, and Hall,1983; Fuson, Richards and Briars, 1982) andSophian (Sophian, Wood, and Vong, 1995) bothmade the reasonable argument that emphasising orrepeating the last number could just be part of anill-understood procedure.

Although Gelman’s five principles cover someessential aspects of counting, they leave others out. The five principles, and the tools that Gelmandevised to study children’s understanding of these

principles, only apply to what someone must knowand do in order to count a single set of objects.They tell us nothing about children’s understandingof numerical relations between sets. Piaget’sresearch on number, on the other hand, was almostentirely concerned with comparisons betweendifferent quantities, and this has the confusingconsequence that when Gelman and Piaget usedthe same terms, they gave them quite differentmeanings. For Piaget, understanding cardinality was about grasping that all and only equivalent sets are equal in number : for Gelman it meantunderstanding that the last number countedrepresents the number of items in a single set.When Piaget studied one-to-one correspondence,he looked at children’s comparisons between twoquantities (eggs and egg cups, for example):Gelman’s concern with one-to-onecorrespondence was about children assigning one count word to each item in a set.

Since two sets are equal in quantity if they containthe same number of items and unequal if they donot, one way to compare two sets quantitatively is to count each of them and to compare the twonumbers. Another, for much the same reason, is touse one-to-one correspondence: if the sets are incorrespondence they are equal; if not, they areunequal. This prompts a question: how soon and howwell do children realise that counting sets is a validway, and sometimes the only feasible valid way, ofcomparing them quantitatively? Another way ofputting the same question is to ask: how soon andhow well do children realise that numbers are ameasure by which they can compare the quantities of two or more different sets.

Most of the research on this topic suggests that ittakes children some time to realise that they can, andoften should, count to compare. Certainly many pre-school children seem not to have grasped theconnection between counting and comparing even ifthey have been able to count for more than one year.

One source of evidence comes from the work bySophian (1988), who asked children to judge whethersomeone else (a puppet) was counting the right waywhen asked to do two things. The puppet was facedwith two sets of objects, and was asked in some trialsto say whether the two sets were equal or not and inothers to work out how many items there were onthe table altogether. Sometimes the puppet did theright thing, which was to count the two setsseparately when comparing them and to count all the


items together when working out the grand total. At other times he got it wrong, e.g. counted all theobjects as one set when asked to compare the twosets. The main result of Sophian’s study was that thepre-school children found it very hard to make thisjudgement. Most 3-year-olds judged counting each setwas the right way to count in both tasks while 4-year-olds judged counting both sets together was the rightway to count in both tasks. Neither age group couldidentify the right way to count reliably.

A second type of study shows that even at schoolage many children seem not to understand fully the significance of numbers when they makequantitative comparisons. There is, for example, thestriking demonstration by Pierre Gréco (1962), acolleague of Piaget’s, that children will count tworows of counters, one of which is more spread outand longer than the other, and correctly say thatthey both have the same number (this one has six,and so does the other) but then will go on to saythat there are more counters in the longer rowthan in the other. A child who makes this mistakeunderstands cardinality in Gelman’s sense (i.e. isable to say how many items in the set) but doesnot know what the word ‘six’ means in Piaget’ssense. Barbara Sarnecka and Susan Gelman (2004)recently replicated this observation. They reportthat children three- and four-year-olds know that ifa set had five objects and you add some to it, it nolonger has five objects; however they did not knowthat equal sets must have the same number word.

Another source of evidence is the observation,repeated in many studies, that children, who cancount quite well, nevertheless fail to count theitems in two sets that they have been asked tocompare numerically (Cowan, 1987; Cowan andDaniels, 1989; Michie, 1984; Saxe, Guberman andGearhart, 1987); instead they rely on perceptualcues, like length, which of course are unreliable.Children who understand the cardinality of number should understand that they can make thecomparison only by counting or using one-to-onecorrespondence, and yet at the age of five and sixyears most of them do neither, even when, as in theCowan and Daniels study, the one-to-one cues areemphasised by lines drawn between items in thetwo sets that the children were asked to compare.

Finally, the criterion for the cardinality principle hasitself been criticised as insufficient to show thatchildren understand cardinality. The criticism is boththeoretical and also based on empirical results. From

a theoretical standpoint, Vergnaud (2008) pointed out that Gelman’s cardinality criterion should actuallybe viewed as showing that children have someunderstanding of ordinal, not of cardinal, number:Gelman’s criterion is indeed based on the position ofthe number word in the counting sequence, becausethe children use the last number word to representthe set. Vergnaud argues that ordinal numbers cannotbe added whereas cardinal numbers can. He predictsthat children whose knowledge of cardinal number isrestricted to Gelman’s cardinality principle will not beable to continue counting to answer how manyobjects are in a set if you add some objects to theset that they have just counted: they will need tocount again from one. Research by Siegler andRobinson (1982) and Starkey and Gelman (1982)produced results in line with this prediction: 3-year-olds do not spontaneously count to solve additionproblems after counting the first set. Ginsburg, Kleinand Starkey (1998) also interpreted such results asindicative of an insufficient development of theconcept of cardinality in young children. We return to the definition of cardinality later on, after we havediscussed alternative explanations to Gelman’s theoryof an innate counting principle as the basis forlearning about cardinality.

Three further studies will be used here to illustratethat some children who are able to use Gelman’scardinality principle do not seem to have a full graspof when this principle should be applied; so meetingthe criterion for the cardinality principle does notmean understanding cardinality.

Fuson (1988) showed that three-year-old children whoseem to understand the cardinality principle continueto use the last number word in the counting sequenceto say how many items are in a set even if the countingstarted from two, rather than from one. Counting inthis unusual way should at least lead the children toreject the last word as the cardinal for the set.

Using a similar experimental manoeuvre, Freeman,Antonuccia and Lewis (2000) assessed three- andfive-year-olds’ rejection of the last word after countingif there had been a mistake in counting. The childrenparticipated in a few different tasks, one of which wasa task where a puppet counted an array with either 3 or 5 items, but the puppet miscounted, either bycounting an item twice or by skipping an item. Thechildren were asked whether the puppet had countedright, and if they said that the puppet had not, theywere asked: How many does the puppet think thereare? How many are there really? All children had



shown that they could count 5 items accurately (2 of 22 could count accurately to 6, another 4 could do so to 7, and the remaining 18 could count itemsaccurately to 10). However, their competence incounting was no assurance that they realised that thepuppet’s answer was wrong after miscounting: onlyabout one third of the children were able to say thatthe answer was not right after they had detected theerror. The children’s rejection of the puppet’s wronganswer increased with age: 82% of the five-year-oldscorrectly rejected the puppet’s answer in all threetrials when a mistake had been made. However, themajority of the children could not say what thecardinal for the set was without recounting: themajority counted the set again in order to answer thequestion ‘how many are there really?’ They neither saidimmediately the next number when the puppet hadskipped one nor used the previous number when thepuppet double-counted an item. So, quite a few of theyounger children passed Gelman’s cardinality principlebut did not necessarily see that the cardinalityprinciple should not be applied when the countingprinciples are violated. Most of the older children, whorejected the use of the cardinality principle, did notuse it to deduce what the correct cardinal should be;instead, all they demonstrated was that they couldreplace the wrong routine with the correct one, andthen they could say what the number really was.Understanding that the next number is the cardinalfor the set if the puppet skipped one item, withouthaving to count again, would have demonstrated thatthe children have a relatively good grasp of cardinality.Freeman and his colleagues reported that only aboutone third of the children who detected the puppet’serror were able to say what the correct number ofitems was without recounting. In the subsequentsection we return to the importance of knowing whatthe next number is for the concept of cardinality.

The third study we consider here was by Bermejo,Morales and deOsuna (2004), who argued that ifchildren really understand cardinality, and not just theGelman’s cardinality principle, they should do betterthan just re-implement the counting in a correct way.For example, they should be able to know how manyobjects are in a set even if the counting sequence isimplemented backwards. If you count a set by saying‘three, two, one’, and you reach the last item when yousay ‘one’, you know that there are three objects in theset. If you count backwards from three and the label‘one’ does not coincide with the last object, you knowthat the set does not have three objects. Just likestarting to count from two, counting backwards isanother way of separating out Gelman’s cardinality

principle from understanding cardinality: when youcount backwards, the first number label is the cardinalfor the set if there is a one-to-one correspondencebetween number labels and objects. Bermejo andcolleagues showed that four- and six-year-old childrenwho can say that there are three objects in a set whenyou count forward cannot necessarily say that if youcount backwards from four and the last number label is‘two’, this does not mean that there are two objects inthe set. In fact, many children did not realize that therewas a contradiction between the two answers: forthem, the set could have three objects if you count oneway and two if you count in another way. They alsoshowed that children who were given the opportunityto discuss what the cardinal for the set was when thecounting was done backwards showed markedprogress in other tasks of understanding cardinality,which included starting to count from other numbersin the counting sequence than the number one, as inFuson’s task. They concluded that reflecting about theuse of counting and the different actions involved inachieving a correct counting created opportunities forchildren to understanding cardinality better.

The evidence that we have presented so farsuggests very strongly and remarkably consistentlythat learning to count and understanding relationsbetween quantities are two different achievements.On the whole, children can use the procedures forcounting long before they realise how countingallows them to measure and compare differentquantities, and thus to work out the relationsbetween them. We think that it is only whenchildren establish a connection between what they know about relations between quantities and counting that they can be said to know themeaning of natural numbers.2

Summary1 Natural numbers are a way of representing

particular quantities and relations betweenquantities.

2 When children learn numbers, they must find outnot just about the counting sequence and ho w tocount, but also about how the numbers in thecounting system represent quantities and relationsbetween them.

3 One basic aspect of this representation is thecardinality of number: all sets with the samenumber have the same quantity of items in them.


4 Another way of expressing cardinality is to say that all sets with the number are in one-to-onecorrespondence with each other.

5 There is evidence that young children’s firstsuccessful experiences with one-to-onecorrespondence come through shar ing; however,even if they succeed in shar ing fairly and know thenumber of items in one set, many do not make theinference that the number of items in the otherset is the same.

6 Because of its cardinal proper ties, number is ameasure: one can compare the quantity of items in two different sets by counting each set.

7 Several studies have shown that many children asold as six years are reluctant to count, althoughthey know how to count, when asked to comparesets. They resor t to perceptual compar isonsinstead.

8 This evidence suggests that lear ning aboutquantities and learning about numbers developindependently of each other in young children. Butin order to under stand natural numbers, childrenmust establish connections between quantities andnumbers. Thus schools must ensure that childrenlearn not only to count but also learn to establishconnections between counting and theirunderstanding of quantities.

Current theories about the origin ofchildren’s understanding of themeaning of cardinal numberWe have seen that Piaget’s theory defines children’sunderstanding of number on the basis of theirunderstanding of relations between quantities; forhim, cardinality is not just saying how many items are in sets but grasping that sets in one-to-onecorrespondence are equivalent in number and vice-versa. He argued that children could only be said tounderstand numbers if they made a connectionbetween numbers and the relations betweenquantities that are implied by numbers. He alsoargued that this connection was established bychildren as they reflected about the effect of their actions on quantities: setting items incorrespondence, adding and taking items away areschemes of action which form the basis for children’sunderstanding of how to compare and to changequantities. Piaget acknowledges that learning to count

can accelerate this process of reflection on actions,and so can other forms of social interaction, becausethey may help the children realise the contradictionsthat they fall into when they say, for example, thattwo quantities are different and yet they are labelledby the same number. However, the process thateventually leads to their understanding of themeanings of natural numbers and the implications of these representations is the child’s growingunderstanding of relations between quantities.

Piaget’s studies of children’s understanding of therelations between quantities involved three differentideas that he considered central to understandingnumber: understanding equivalence, order, and class-inclusion (which refers to the idea that the whole isthe sum of the parts, or that a set with 6 itemscomprises a set with 5 items plus 1). The methodsused in these studies have been extensivelycriticised, as has the idea that children developthrough a sequence of stages that can be easilytraced and are closely associated with age. However,to our knowledge the core idea that children cometo understand relations between quantities byreflecting upon the results of their actions is still avery important hypothesis in the study of howchildren learn about numbers. We do not review thisvast literature here as there are several collections ofpapers that do so (see, for example, Steffe, Cobband Glaserfeld, 1988; Steffe and Thompson, 2000).Later sections of this paper will revisit Piaget’s theoryand discuss related research.

This is not the only theory about how children cometo understand the meaning of cardinal numbers. Thereare at least two alternative theories which are widelydiscussed in the literature. One is a nativist theory,which proposes that children have from birth accessto an innate, inexact but powerful ‘analog’ system,whose magnitude increases directly with the numberof objects in an array, and they attach the numberwords to the properties occasioning thesemagnitudes (Dehaene 1992; 1997; Gallistel andGelman, 1992; Gelman and Butterworth, 2005; Xuand Spelke, 2000; Wynn, 1992; 1998). This gives all of us from birth the ability to make approximatejudgements about numerical quantities and wecontinue through life to use this capacity. Thediscriminations that this system allows us to make aremuch like our discriminations along other continua,such as loudness, brightness and length. One featureof all these discriminations is that the greater thequantities (the louder, the brighter or the longer theyare) the harder they are to discriminate (known, after



the great 19th century psycho-physicist whometiculously studied perceptual sensitivity, as the‘Weber function’). To quote Carey (2004): ‘Tap out asfast as you can without counting (you can preventyourself from counting by thinking 'the' with each tap)the following numbers of taps: 4, 15, 7, and 28. If youcarried this out several times, you'd find the meannumber of taps to be 4, 15, 7, and 28, with the rangeof variation very tight around 4 (usually 4, occasionally3 or 5) and very great around 28 (from 14 to 40 taps,for example). Discriminability is a function of theabsolute numerical value, as dictated by Weber's law’(p. 63). The evidence for this analog system being aninnate one comes largely from studies of infants (Xuand Spelke, 2000; McCrink and Wynn, 2004) and to a certain extent studies of animals as well, and isbeyond the scope of this review. The evidence for itsimportance for learning about number and arithmeticcomes from studies of developmental or acquireddyscalculia (e.g. Butterworth, Cipolotti andWarrington, 1996; Landerl, Bevana and Butterworth,2004). However important this basic system may beas a neurological basis for number processing, it is notclear how the link between an analog and imprecisesystem and a precise system based on counting canbe forged: ‘ninety’ does not mean ‘approximatelyninety’ any more than ‘eight’ could mean‘approximately eight’. In fact, as reported in theprevious section, three- and four-year-olds know thatif a set has 6 items and you add one item to it, it nolonger has 5 objects: they know that ‘six’ is not thesame as ‘approximately six’.

A third well-known theoretical alternative, whichstarts from a standpoint in agreement with Gelman’stheory, is Susan Carey’s (2004) hypothesis aboutthree ways of learning about number. Carey acceptsGelman and Gallistel’s (1978) limited definition of thecardinality principle but rejects their conclusionsabout how children first come to understand thisprinciple. Carey argues that initially (by which shemeans in the first three years of life), very youngchildren can represent number in three differentways (Le Corre and Carey, 2007). The first is theanalog system, described in the previous paragraphs.However, although Carey thinks that this system playsa part in people’s informal experiences of quantitythroughout their lives, she does not seem to assign it a role in children’s learning about the countingsystem, or in any other part of the mathematics thatthey learn about at school.

In her theory, the second of Carey’s three systems,which she calls the ‘parallel individuation’ system, plays

the crucial part in making it possible for children tolearn how to connect number with the countingsystem. This system makes it possible for infants torecognise and represent very small numbers exactly(not approximately like the analog system). Thesystem only operates for sets of 1, 2 and 3 objectsand even within this restricted scope there is markeddevelopment over children’s first three years.

Initially the system allows very young children torecognise sets of 1 as having a distinct quantity. Thechild understands 1 as a quantity, though he or shedoes not at first know that the word ‘one’ applies tothis quantity. Later on the child is able to discriminateand recognise – in Carey’s words ‘to individuate’ –sets of 1 and 2 objects, and still later, around the ageof three- to four-years, sets of 1, 2 and 3 objects asdistinct quantities. In Carey’s terms young childrenprogress from being ‘one-knowers’ to becoming ‘two-knowers’ and then ‘three-knowers’.

During the same period, these children also learnnumber words and, though their recognition of 1, 2and 3 as distinct quantities does not in any waydepend on this verbal learning, they do manage toassociate the right count words (‘one’, ‘two’ and ‘three’)with the right quantities. This association betweenparallel individuation and the count list eventually leadsto what Carey (2004) calls ‘bootstrapping’: the childrenlift themselves up by their own intellectual bootstraps.They do so, some time in their fourth or fifth year, andtherefore well before they go to school.

This bootstrapping takes two forms. First, with the helpof the constant order of number words in the countlist, the children begin to learn about the ordinalproperties of numbers: 2 always comes after 1 in thecount list and is always more numerous than 1, and 3 ismore numerous than 2 and always follows 2. Second,since the fact that the count list that the children learngoes well beyond 3, they eventually infer that thenumber words represent a continuum of distinctquantities which also stretches beyond ‘three’. They alsobegin to understand that the numbers above three areharder to discriminate from each other at a glance thansets of 1, 2 and 3 are, but that they can identify bycounting. In Carey’s words ‘The child ascertains themeaning of 'two' from the resources that underlienatural language quantifiers, and from the system ofparallel individuation, whereas she comes to know themeaning of 'five' through the bootstrapping process –i.e., that 'five' means one more than four, which is onemore than three – by integrating representations ofnatural language quantifiers with the external serial


ordered count list’. Carey called this new understanding‘enriched parallel individuation’ (Carey, 2004; p. 65).

Carey’s main evidence for parallel individuation andenriched parallel individuation came from studies inwhich she used a task, originally developed by Wynn,called ‘Give – a number’. In this, an experimenter asksthe child to give her a certain number of objectsfrom a set of objects in front of them: ‘Could youtake two elephants out of the bowl and place themon the table?’ Children sometimes put out thenumber asked for and sometimes just grab objectsapparently randomly. Using this task Carey showedthat different three-, four- and five-year-old childrencan be classified quite convincingly as ‘one-’ ‘two-’ or‘three-knowers’ or as ‘counting-principle-knowers’.The one-knowers do well when asked to provideone object but not when asked the other numberswhile the two- and three-knowers can respectivelyprovide up to two and three objects successfully. The ‘counting-principle-knowers’ in contrast countquantities above three or four.

The evidence for the existence of these threegroups certainly supports Carey’s interesting idea of a radical developmental change from ‘knowing’some small quantities to understanding that thenumber system can be extended to other numbersin the count list. The value of her work is that itshows developmental changes in children’s learningabout the counting system. These had been by-passed both by Piaget and his colleagues becausetheir theory was about the underlying logic neededfor this learning and not about counting itself, andalso by Gelman, because her theory about countingprinciples was about innate or rapidly acquiredstructures and not about development. However,Carey’s explanation of children’s counting in termsof enriched parallel individuation suffers thelimitation that we have mentioned already: it has no proper measure of children’s understanding ofcardinality in its full sense. Just knowing that the lastnumber that you counted is the number of the setis not enough.

The third way in which children learn number,according to Carey’s theory, is through a systemwhich she called ‘set-based quantification’: this isheavily dependent on language and particularly onwords like ‘a’ and ‘some’ that are called ‘quantifiers’.Thus far the implications of this third hypothesisedsystem for education are not fully worked out, andwe shall not discuss it further.

Carey’s theory has been subjected to much criticismfor the role that it attributes to induction or analogyin the use of the ‘next’ principle and to language.Gelman and Butterworth (2005), for example, arguethat groups that have very restricted numberlanguage still show understanding of larger quantities;their number knowledge is not restricted to smallnumerosities as suggested in Carey’s theory. Rips,Asmuth and Bloomfield (2006; 2008) address itmore from a theoretical standpoint and argue thatthe bootstrapping hypothesis presupposes the veryknowledge of number that it attempts to explain.They suggest that, in order to apply the ‘nextnumber’ principle, children would have to knowalready that 1 is a set included in 2, 2 in 3, and 3 in 4.If they already know this, then they do not need touse the ‘next number’ principle to learn about whatnumber words mean.

Which of these approaches is right? We do not thinkthat there is a simple answer. If you hold, as we do,that understanding number is about understanding an ordered set of symbols that represent quantitativerelations, Piaget’s approach definitely has the edge.Both Gelman’s and Carey’s theory only address thequestion of how children give meaning to numberwords: neither entertains the idea that numbersrepresent quantities and relations between quantities,and that it is necessary for children to understandthis system of relations as well as the fact that theword ‘five’ represents a set with 5 items in order tolearn mathematics. Their research did nothing to dentPiaget’s view that children of five years and six yearsare still learning about very basic relations betweenquantities, sometimes quite slowly.

Summary

1 Piaget’s studies of learning about numberconcentrated on children’s ability to reasonlogically about quantitative relations, and bypassedtheir acquisition of the counting system.

2 In contrast many current theories concentrate on children learning to count, and omit children’sreasoning about quantitative relations. The mostnotable omission in these theor ies is the questionof children’s understanding of cardinality.

3 Gelman’s studies of children’s counting,nevertheless, did establish that even very youngchildren systematically obey some basic countingprinciples when they do count.



Ordinal number

Numbers, as we have noted, come in a fixed order,and this order represents a quantitative series.Numbers are arranged in an ascending scale: 2 ismore than 1 and 3 more than 2 and so on. Also thenext number in the scale is always 1 more than thenumber that precedes it. Ordinal numbers indicatethe position of a quantity in a series.

Piaget developed much the same argument aboutordinal number as about cardinal number. He claimedthat children learn to count, and therefore toproduce numbers in the right fixed order, long beforethey understand that this order represents an ordinalseries. This claim about children’s difficulties withordinality was based on his experiments on ‘seriation’and also on ‘transitivity’.

In his ‘seriation’ experiments, Piaget and his colleagues(Piaget, 1952) showed children a set of sticks alldifferent in length and arranged in order fromsmallest to largest, and then jumbled them up andasked the child to re-order them in the same way.However, the children were asked to do so not byconstructing the visual display all at once, which theywould be able to do perceptually and by trial-and-error, but by giving the sticks to the experimenterone by one, in the order that they think they shouldbe placed.

This is a surprisingly difficult task for young childrenand, at the age range that we are considering here(five- to six-years), children tend to form groups ofordered sticks instead of creating a single orderedseries. Even when they do manage to put the sticksinto a proper series, they tend then to fail anadditional test, which Piaget considered to be theacid test of ordinal understanding: this was to insertanother stick which he then gave them into itscorrect place in the already created series, which wasnow visible. These difficulties, which are highly reliableand have never been refuted or explained away, arecertainly important, but they may not be true ofnumber. Children’s reactions to number series maywell be different precisely because of the extensivepractice that they have with producing numbers in a fixed order.

Recently, however, Brannon (2002) made the strikingclaim that even one-year-old-children understandordinal number relations. The most direct evidencethat Brannon offered for this claim was a study inwhich she showed the infant sequences of three

cards, each of which depicted a different number ofsquares. Each three-card sequence constituted eitheran increasing or a decreasing series. In somesequences the number of squares increased fromcard to card e.g. 2, 4, 8 and 3, 6, 12: in others thenumbers decreased e.g. 16, 8, 4.

Brannon’s results suggested that 11-month-old infantscould discriminate the two kinds of sequence (afterseeing several increasing sequences they were moreinterested in looking at a decreasing than at yetanother increasing sequence, and vice versa), and she concluded that even at this young age childrenhave some understanding of seriation.

However, her task was a very weak test of theunderstanding of ordinality. It probably shows thatchildren of this age are to some extent aware of therelations ‘more’ and ‘less’, but it does not establish thatthe children were acting on the relation between allthree numbers in each sequence.

The point here is that in order to understandordinality the child must be able to co-ordinate a set of ‘more’ and ‘less’ relations. This meansunderstanding that b is smaller than a and at thesame time larger than c in an a >b > c series. Piagetwas happy to accept that even very young childrencan see quite clearly that b is smaller than a and atanother time that it is larger than c, but he claimedthat in order to form a series children have tounderstand that intermediary quantities like b aresimultaneously larger than some values and smallerthan others. Of course, Brannon did not showwhether the young children in her study could orcould not grasp these two-way relations.

Piaget’s (1921) most direct evidence for children’sdifficulties with two-way relations came from anotherkind of task – the transitivity task. The relationsbetween quantities in any ordinal series are transitive.If A > B and B > C, then it follows that A > C, andone can draw this logical conclusion without everdirectly comparing A with C. This applies to numberas well: since 8 is more than 4 and 4 more than 2, 8 ismore than 2.

Piaget claimed that children below the age of roughlyeight years are unable to make these inferencesbecause they find it difficult to understand that B can be simultaneously smaller than one quantity (A) and larger than another (C). In his experiments ontransitivity Piaget did find that children very rarelymade the indirect inference between A and C on the


basis of being shown that A > C and B > C, but thiswas not very strong evidence for his hypothesisbecause he failed to check the possibility that thechildren failed to make the inference because theyhad forgotten the premises – a reason which hasnothing directly to do with logic or with reasoning.

Subsequent studies, in which care was taken tocheck how well the children remembered thepremises at the time that they were required tomake the A > C inference (Bryant and Trabasso,1971; Bryant and Kopytynska, 1976 ; Trabasso, 1977)consistently showed that children of five years orolder do make the inference successfully, providedthat they remember the relevant premises correctly.Young children’s success in these tasks throws somedoubt on Piaget’s claim that they do not understandordinal quantitative relations, but by and large thereis still a host of unanswered questions aboutchildren’s grasp of ordinality. We shall return to the issue of transitivity in the section on Space and Geometry.

Above all we need a comprehensive set of seriationand transitivity experiments in which the quantitiesare numbers (discontinuous quantities), and notcontinuous quantities like the rods of different lengthsthat have been the staple diet of previous work onthese subjects.

Summary

1 The count list is ar ranged in order of the magnitudeof the quantities represented by the numbers. Therelations between numbers in this ser ies aretransitive: if A > B and B > C , then A > C.

2 Piaget argued that young children find ordinalrelations, as well as cardinal relations, difficult tounderstand. He attr ibuted these difficulties to an inability, on the par t of young children, tounderstand that, in an A > B > C ser ies, B issimultaneously smaller than A and larger than B.

3 Piaget’s evidence for this claim came from studiesof seriation and transitivity. The difficulties thatchildren have in the ser iation experiment, in whichthey have to construct an ordered ser ies of sticks,are surprising and very striking.

4 However, the criterion for constructing the seriesin the seriation experiment (different lengths ofsome sticks) cannot be applied by counting.

Therefore, seriation studies do not deal directl ywith children’s understanding of natural number.The question of the ser iation of number is still an open one.

Cardinality, additive reasoning andextensive quantities

So far we have discussed how children give meaningto number and how easy or difficult it is for them tomake connections between what they understandabout quantity and the numbers that they learnwhen they begin to count. Now we turn to anotheraspect of cardinal number, its connection withaddition and subtraction – or, more generally, withadditive reasoning. There are undeniable connectionsbetween the concept of cardinality and additivereasoning and we shall explore them in this section,which is about the additive composition of number,and in the subsequent section, which is about theinverse relations between addition and subtraction.

Piaget (1952) included in his definition of children’sunderstanding of number their realisation that aquantity (and its numerical representation) is onlychanged by addition or subtraction, not by otheroperations such as spreading the elements orbunching them together. This definition, he indicated,is valid for the domain of extensive quantities, whichare measured by the addition of units because thewhole is the sum of the parts. If the quantity is madeof discrete elements (e.g. a set of coins), the task ofmeasuring it and assigning a number to it is easy: allthe children have to do is to count. If the quantity iscontinuous (e.g. a ribbon), the task of measuring it ismore difficult: normally a conventional unit would beapplied to the quantity and the number thatrepresents the quantity is the number of iterations ofthese units. Extensive quantities differ from intensivequantities, which are measured by the ratio betweentwo other quantities. For example, the concentrationof a juice is measured by the ratio between amountof concentrate and amount of water used to makethe juice. These quantities are considered in Paper 4.

His studies of children’s understanding of theconservation of quantities have been criticised onmethodological grounds (e.g. Donaldson, 1978;Light, Buckingham and Robbins, 1979; Samuel andBryant, 1984) but, so far as we know, his idea thatchildren must realise that extensive quantitieschange either by addition or by subtraction has not been challenged.



Piaget (1952) also made the reasonable suggestionthat you cannot understand what ‘five’ is unless youalso know that it is composed of numbers smallerthan it. Any set of five items contains a sub-set of 4items and another sub-set of 1, or one sub-set of 3and another of 2. A combination of or, in otherwords, an addition of each of these pairs of setsproduces a set of five. This is called the additivecomposition of number, which is an important aspectof the understanding of relations between numbers.

Piaget used the idea of class-inclusion to describe thisaspect of number; others (e.g. Resnick and Ford,1981) have called it part-whole relations. Piaget’sstudies consisted in asking children about thequantitative relations between classes, one of whichwas included in the other. For example, in some taskschildren were asked to compare the number of dogswith the number of animals in sets which includedother animals, such as cats. For an adult, there is noneed to know the actual number of dogs, cats, andanimals in such a task: there will be always moreanimals than dogs because the class of dogs isincluded in the class of animals. However, somechildren aged four- to six-years do not necessarilythink like adults: if the number of dogs is quite a bitlarger than the number of cats, the children mightanswer that there are more dogs than animals.According to Piaget, this answer which to an adultseems entirely illogical, was the result of the children’sdifficulties with thinking of the class of dogs assimultaneously included in the class of animals andexcluded from it for comparison purposes. Once theymentally excluded the dogs from the set of animals,they could no longer think of the dogs as part of theset of animals: they would then be unable to focus onthe fact that the whole (the overall class, animals) isalways larger than one part (the included class, dogs).

Piaget and his colleagues (Piaget, 1952; Inhelder,Sinclair and Bovet, 1974) did use a number ofconditions to try to eliminate alternative hypothesesfor children’s difficulties. For example, they asked thechildren whether in the whole world there would bemore dogs or more animals. This question used thesame linguistic format but could be answered withoutan understanding of the necessary relation between apart and a whole: the children could think that thereare many types of animals in the world and thereforethere say that there are more animals than dogs.Children are indeed more successful in answering thisquestion than the class-inclusion one. Anothermanipulation Piaget and his colleagues used was toask the children to circle with a string the dogs and

the animals: this had no effect on the children’sperformance, and they continued to exclude the dogs mentally from the class of animals. The onlymanipulation that helped the children was to ask thechildren to first think of the set of animals withoutseparating out the dogs, then replace the dogs withvisual representations that marked their inclusion inthe class of animals, while the dogs were set in aseparate class: the children were then able to create asimultaneous representation of the dogs included inthe whole and as a separate part and answer thequestion correctly. After having answered the questionin this situation, some children went on to answer itcorrectly when other class-inclusion problems werepresented (for example, about flowers and roses)without the support of the extra visual signs.

The Piagetian experiments on class-inclusion havebeen criticised on many grounds: for example, it hasbeen argued that the question the children are askedis an anomalous question because it uses disjunction(dogs or animals) when something can besimultaneously a dog and an animal (Donaldson,1978; Markman, 1979). However, Piaget’s hypothesisthat part-whole relations are an important aspect ofnumber understanding has not been challenged. Asdiscussed in the previous sections, it has been argued(e.g. by Rips, Asmuth and Bloomfield, 2008) that it ismost unlikely that a child will understand theordinality of number until she has grasped theconnection between the next number and the plus-one compositions: i.e. that a set of 5 items contains aset of 4 items plus a set of 1, and a set of 4 items iscomposed of a set of 3 plus 1, and so on.

For exactly the same reasons, the understanding ofadditive composition of number is essential in anycomparison between two numbers. To judge thedifference, for example, between 7 and 4, somethingwhich as we shall see is not always easy for youngchildren, you need to know that 7 is composed of 4and 3, which means that 7 is 3 greater than 4.

Of course, even very young children have a great dealof informal experience of quantities increasing ordecreasing as a result of additions and subtractions.There is good evidence that pre-schoolers dounderstand that additions increase and subtractionsdecrease quantities (Brush, 1978; Cooper, 1984; Klein,1984) but this does not mean that they realise thatthe only changes that affect quantity are addition andsubtraction. It is possible that their understanding ofthese changes is qualitative in the sense that it lacksprecision. We can take as an example what happens


when young children are shown two sets that areunequal and are arranged in one-to-onecorrespondence, as in Figure 2.1, so that it is possiblefor the children to see the size of the difference (sayone set has 10 objects and the other 7). Theexperimenter proposes to add to the smaller setfewer items than the difference (i.e. she proposes toadd 2 to the set with 7). Some preschoolers judgethat the set to which elements are added will becomelarger than the other. Others think that it is now thesame as the other set. It is only at about six- or seven-years that children actually take into account theprecise difference between the sets in order to knowwhether they will be the same or not after theaddition of items to the smaller set (Klein, 1984;Blevins-Knabe, Cooper, Mace and Starkey, 1987).

The basic importance of the additive composition ofnumber means that learning to count and learning toadd and subtract are not necessarily two successiveand separate intellectual steps, as common sensemight suggest. At first glance, it seems quite aplausible suggestion that children must understandnumber and know about the counting system inorder to do any arithmetic, like adding andsubtracting. It seems simply impossible that theycould add 6 and 4 or subtract 4 from 6 withoutknowing what 6 and 4 mean. However, we have nowseen that this link between counting and arithmeticmust work in the opposite direction as well, becauseit is also impossible that children could know what 6or 4 or any other number mean, or anything aboutthe relations between these numbers, without alsounderstanding something about the additivecomposition of number.

Given its obvious importance, there is remarkably littleresearch on children’s grasp of additive composition ofnumber. The most relevant information, though it issomewhat indirect, comes from the well-knowndevelopmental change from ‘counting-all’ to ‘counting-on’. As we have seen, five-year-old children generallyknow how to count the number of items in a setwithin the constraints of one-to-one counting.

However, their counting is not always economic. If, forexample, they are given a set of 7 items which theyduly count and then 6 further items are added to thisset and the children are asked about the total numberin the newly increased set, they tend to count all the13 items in front of them including the subset thatthey counted before. Such observations have beenreplicated many times (e.g. Fuson, 1983; Nunes andBryant, 1996; Wright, 1994) and have given origin to a widely used analysis of children’s progress inunderstanding cardinality (e.g. Steffe, Cobb and vonGlaserfeld, 1988; Steffe, Thompson and Richards, 1982;Steffe, von Glasersfeld, Richards and Cobb, 1983). Thiscounting of all the items is not wrong, of course, butthe repeated counting of the initial items isunnecessary. The children could just as well and muchmore efficiently have counted on from the initial set(not ‘1, 2, 3,…..13’ but ‘8, 9, 10…..13’).

According to Vergnaud (2008), the explanation forchildren’s uneconomical behaviour is conceptual: asreferred in the previous section, he argues that theirunderstanding of number may be simply ordinal(i.e. what they know is that the last number wordrepresents the set) and so they cannot add becauseordinal numbers cannot be added. They can, however,count a new, larger set, and give to it the label of thelast number word used in counting.

Studies of young children’s reactions to the kind of situation that we have just described haveconsistently produced two clear results. The first isthat young children of around the age of five yearsconsistently count all the items. The second is thatbetween the ages of five and seven years, there is adefinite developmental shift from counting-all tocounting-on: as children grow older they begin toadopt the more economic strategy of counting-onfrom the previously counted subset. This new strategyis a definite sign of children’s eventual recognition ofthe additive composition of the new set: they appearto understand that the total number of the new setwill contain the original 7 items plus the newly added6 items. The fact that younger children stick to


Figure 2.1: Two sets in correspondence; the difference between sets is easily seen.


counting-all does not establish that they cannotunderstand the additive composition of the new set (as is often the case, it is a great deal easier toestablish that children do understand some principlethan that they do not). However, the developmentalchange that we have just described does suggest animprovement in children’s understanding of additiverelations between numbers during their first twoyears at school.

The study of the connections that children make, orfail to make, between understanding number, additivecomposition and additive reasoning plainly supportsthe Piagetian thesis that children give meaning tonumbers by establishing relations between quantitiesthough their schemes of action. They do need tounderstand that addition increases and subtractiondecreases the number of items in a set. This forms afoundation for their understanding of the precise wayin which the number changes: adding 1 to set acreates a number that is equal to a + 1. This numbercan be seen as a whole that includes the parts a and1. Instead of relying on the ‘next number’ induction oranalogy, children use addition and the logic of part-whole to understand numbers.

Summary

1 In order to under stand number as an ordinalseries, children have to realise that numbers arecomposed of combinations of smaller numbers.

2 This realisation stems from their progressiv eunderstanding of how addition affects number: atfirst they understand that addition increases thenumber of items in a set without being preciseabout the extent of this increase b ut, as they co-ordinate their knowledge of addition with theirunderstanding of par t-whole relations, they canalso become more precise about additiv ecomposition.

3 Young children’s tendency to count-all r ather thanto count-on suggests either that they do notunderstand the additive composition of number or that their grasp of additive composition is tooweak for them to take advantage of it.

4 Their difficulties suggest that children should be taught about additive composition, andtherefore about addition, as they learn about the counting system.

The decade structure and additivecomposition

Additive composition and the understanding ofnumber and counting are linked for anotherimportant reason. The power and the effectiveness ofcounting rest largely on the invention of base systems,and these systems depend on additive composition.The base-10 system, which is now widespread, freesus from having to remember long strings of numbers,as indeed any base system does. In English, once weknow the simple rules for the decimal system andremember the number words for 1 to 20, for thedecades, and then for a hundred, a thousand and amillion, we can generate most of the natural numbersthat we will ever need to produce with very littleeffort or difficulty. The link between understandingadditive composition and adopting a base system isquite obvious. Base systems rest on the additivecomposition of number and the decade structure is ineffect a clear reminder that ‘fourteen’ is a combinationof 10 and 4, and ‘thirty-five’ of three 10s and 5.

Additive composition is the basic concept thatunderlies any counting system with a base, oral orwritten. This includes of course the Hindu-Arabicplace-value system that we use to write numbers. Forexample, the decimal system explicitly represents thefact that all the numbers between 10 and a 100 mustbe a combination of one or more decades and anumber less than 10: 17 is a combination of 10 and 7and 23 a combination of two 10s and 3. The digitsexpress the additive composition of any numberfrom 10 on: e.g. in 23, 2 represents two 10s whichare added to 3, which represents three units.

Additive composition is also at the root of our abilityto count money using coins and notes of differentdenominations. When we have, for example, one 10pand five 1p coins, we can only count the 10p and the1p coins together if we understand about additivecomposition.

The data from the ‘shop task’, a test devised by Nunesand Schliemann (1990), suggest that initially childrenfind it hard to combine denominations in this way. Inthe shop task children are shown a set of toys in a‘shop’, are given some (real or artificial) money and areasked to choose a toy that they would like to buy.Then the experimenter asks them to pay a certainsum for their choice. Sometimes the child can pay forthis with coins of one denomination only: for example,the experimenter charges a child 15p for a toy car andthe child has that number of pence to make the


purchase or the charge is 30p and the child can paywith three 10p coins. In other trials, the child can payonly by combining denominations: the car costs 15pand the child, having fewer 1p coins than that, must pay with the combination of a 10p and five 1p coins.Although the values that the children are asked to paywhen they use only 10p coins are larger than thosethey pay when using combinations of different values,children can count in tens (ten, twenty, thirty etc.)using simple correspondences between the countinglabels and the coins. This task does not requireunderstanding additive composition. So Nunes andSchliemann predicted that the mixed denominationtrials would be significantly more difficult than theother trials in the task. They found that the mixeddenominations trials were indeed much harder for thechildren than the single denomination trials and thatthere was a marked improvement between the agesof five and seven years in children’s performance in thecombined denomination trials. This work was originallycarried out in Brazil and the results have beenconfirmed in other research in the United Kingdom(Krebs, Squire and Bryant, 2003; Nunes et al., 2007).

A fascinating observation in this task is that childrendon’t change from being unable to carry out theadditive composition to counting on from ten as theyadd 1p coins to the money they are counting. Theshow the same count-all behaviour that they showwhen they have a set of objects and more object areadded to the set. However, as there are no visible 1pcoins within the 10p, they point to the 10p ten timesas they count, or they lift up 10 fingers and say ‘ten’,and only then go on to count ‘eleven, twelve, thirteenetc.’ This repeated pointing to count invisible objectshas been documented also by Steffe and his colleagues(e.g. Steffe, von Glasersfeld, Richards and Cobb, 1983),who interpreted it, as we do, as a significant step incoordinating counting with a more matureunderstanding of cardinality.

In a recent training study (Nunes et al., 2007), weencouraged children who did not succeed in theshop-task to use the transition behaviour we hadobserved, and asked them to show us ten with theirfingers; we then pointed to their fingers and the 10pcoin and asked the children to say how much therewas in each display; finally, we encouraged them to goon and count the money. Our study showed thatsome children seemed to be able to grasp the ideaof additive composition quite quickly after thisdemonstration and others took some time to do so, but all children benefited significantly from brieftraining sessions using this procedure.

Since children appear to be finding out about theadditive nature of the base-10 system and at the sametime (their first two years at school) about the additivecomposition of number in general, one can reasonablyask what the connection between these two is. Onepossibility is that children must gain a full understandingof the additive composition of number before they canunderstand the decade structure. Another is thatinstruction about the decade structure is children’s firstentrée to additive composition. First they learn that 12is a combination of 10 and 2 and then they extend thisknowledge to other combinations (e.g. 12 is also acombination of 8 and 4). The results of a recent studyby Krebs, Squire and Bryant (2003), in which the samechildren were given the shop task and countingall/counting on tasks, favour the second hypothesis. Allthe children who consistently counted on (the moreeconomic strategy) also did well on the shop task, butthere were some children who scored well in theshop task but nevertheless tended not to count on.However, no child scored well in the counting all/ontask but poorly in the shop task. This pattern suggeststhat the cues present in the language help childrenlearn about the decimal system first and then extendtheir new understanding of additive composition tocombinations that do not involve decades.

Summary

1 The decimal system is a good example of aninvented and culturally transmitted mathematicaltool. It enhances our power to calculate and freesus from having to remember extended sequencesof number.

2 Once we know the rules for the decade systemand the names of the diff erent classes and order s(tens, hundreds, thousands etc.), we can use thesystem to count by generating numbers ourselves.

3 However, the system also makes some quitedifficult intellectual demands. Children find it hardat first to combine different denominations, such astens and ones.

4 Teachers should be aware that the ability tocombine denominations rests on a thorough gr aspof additive composition.

5 There is some evidence that exper ience with thestructure of the decimal system may enhanceschildren’s understanding of additive composition.There is also evidence that it is possib le to use



money to provoke children’s progress inunderstanding additive composition.

The inverse relation betweenaddition and subtraction

The research we have considered so far suggeststhat by the age of six or seven children understandquite a lot about number : they understandequivalence well enough to know that if two setsare equivalent they can infer the number thatdescribes one by counting the other ; theyunderstand that addition and subtraction are theoperations that change the number in a set; theyunderstand additive composition and what must be added to one set to make it equivalent to theother ; they understand that they can count on ifyou add more elements to a set; and theyunderstand about ordinal number and can maketransitive inferences. However, there is an insightabout how addition and subtraction affect thenumber of elements in a set that we still need toconsider. This is the insight that addition is theinverse of subtraction and vice versa, and thus thatequal additions and subtractions cancel each otherout: 27 + 19 – 19 = 27 and 27 – 19 + 19 = 27.

It is easy to see that one cannot understand eitheraddition or subtraction or even number fully withoutalso knowing about the inverse relation of each ofthese operations to the other. It is absolutely essentialwhen adding and subtracting to understand thatthese are reversible actions. Otherwise one will notunderstand that one can move along the numberscale in two opposite directions – up and down.

The understanding of any inverse relation should,according to Piaget (1952), be particularly hard foryoung children, since in his theory young children arenot able to carry out ‘reversible’ thought processes.Children in the five- to eight-year range do not seethat if 4 + 8 = 12, therefore 12 – 8 = 4 becausethey do not realise that the original addition (+8) iscancelled out by the inverse subtraction (–8). Thisclaim is a central part of Piaget’s theory aboutchildren’s arithmetical learning, but he never tested itdirectly, even though it would have been quite easyto do so.

In one of his last publications, Piaget and Moreau(2001) did report an ingenious, but rather toocomplicated, study of the inverse relations betweenaddition and subtraction and also between

multiplication and division. They asked children,aged from six- to ten-years, to choose a numberbut not to tell them what this was. Then they askedthe child first to add 3 to this number, next todouble the sum and then to add 5 to the result ofthe multiplication. Next, they asked the child whatthe result was, and went on to tell him or her whatwas the number that s/he chose to start with.Finally the experimenters asked each child toexplain how they had managed to work out whatthis initial number was.

Piaget and Moreau reported that this was a difficulttask. The youngest children in the sample did notunderstand that the experimenters had performedthe inverse operations, subtracting where the childhad added and dividing where s/he had multiplied.The older children did show some understandingthat this was how the experimenters reached theright number, but did not understand that the orderof the inverse operations was important. Theexperimenters accounted for the younger children’sdifficulties by arguing that these children had failed tounderstand the adult’s use of inversion (equaladditions and subtractions and equal multiplicationsand divisions) because they did not understand theprinciple of inversion.

This was a highly original study but Piaget andMoreau’s conclusions from it can be questioned. Onealternative explanation for the children’s difficulties isthat they may perfectly have understood the inverserelations between the different operations, but theymay still not have been able to work out how theadult used them to solve the problem. The children,also, had to deal with two kinds of inversion(addition-subtraction and multiplication-division) inorder to explain the adult’s correct solution, and so their frequent failures to produce a coherentexplanation may have been due to their not knowingabout one of the inverse relations, e.g. betweenmultiplication and division, even though they werecompletely at home with the other, e.g. betweenaddition and subtraction.

Nevertheless, some following studies seemed toconfirm that young school-children are oftenunaware aware that inverse transformations canceleach other out in a + b – b sums. In two studies(Bisanz and Lefevre, 1990; Stern, 1992), the vastmajority of the younger children did no better withinverse a + b – b sums in which they could takeadvantage of the inversion principle than with controla + b – c sums where this was not possible. For


example, Stern reported that only 13% of the seven-year-old children and 48% of the nine-year-olds in herstudy used the inversion principle consistently, whensome of the problems that they were given were a + b – b sums and others a + b – c sums.

This overall difficulty was confirmed in a further studyby Siegler and Stern (1998), who gave eight-year-oldGerman children inversion problems in eightsuccessive sessions. Their aim was to see whether the children improved in their use of the inversionprinciple to solve problems. The children were alsoexposed to other traditional scholastic problems (e.g. a + b – c), which could not be solved by usingthe inverse principle. In the last of the eight sessions,Siegler and Stern also gave the children controlproblems, which involved sequences such as a + b +b, so that the inversion principle was not appropriatefor solution. The experimenters recorded how wellchildren distinguished the problems that could besolved through the inversion principle from thosethat had to be solved in some other way.

The study showed that the children who were givenlots of inversion problems in the first seven sessionstended to get better at solving these problems overthese sessions, but in the final session in which thechildren were given control as well as inversionproblems they often, quite inappropriately,overgeneralised the inversion strategy to the controlsums: they would give a as the answer in a + b + bcontrol problems as well as in inverse a + b – bproblems. Their relatively good performance with the inversion problems in the previous sessions,therefore, was probably not the result of anincreasing understanding of inversion. They seem tohave learned some lower-level and totally inadequatestrategy, such as ‘if the first number (a) is followed byanother number (b) which is then repeated, theanswer must be a’.

The pervasive failures of the younger school childrenin these studies to take advantage of the inversionprinciple certainly suggest that it is extremely difficultfor them to understand and to learn how to use thisprinciple, as Piaget first suggested. However, in allthese tasks the problems were presented eitherverbally or in written form. Other studies, whichemployed sets of physical objects, paint a differentpicture. (Bryant, Christie and Rendu, 1999;Rasmussen, Ho and Bisanz, 2003). For example,Bryant et al. used sets of bricks to present five- and six-year-old children with a + b – b inversionproblems and a + a – b control problems. In this

particular task young children did a great deal betterwith the inversion problems than with the controlproblems, which is good evidence that they wereusing the inversion principle when they could. In thesame study the children were also given equivalentinversion and control problems as verbal sums (27 +14 – 14): they used the inversion principle much lessoften in this task than in the task with bricks, a resultwhich resonates well with Hughes’ (1981) discoverythat pre-school children are much more successful atworking out the results of additions and subtractionsin problems that involve concrete objects than inabstract, verbal sums.

The fact that young children are readier to use theinversion principle in concrete than in abstractproblems suggests that they may learn aboutinversion initially through their actions with concretematerial. Bryant et al. raised this possibility, and theyalso made a distinction between two levels in theunderstanding of the inverse relation betweenaddition and subtraction. One is the level of identity:when identical stuff is added to and then subtractedfrom an object, the final state of this object is thesame as the initial state. Young children have manyinformal experiences of inverse transformations atthis level. A child gets his shirt dirty (mud is added toit) and then it is cleaned (mud is subtracted) and theshirt is as it was before. At meal-times various objects(knives, forks etc.) are put on the dining room tableand then subtracted when the meal is over; the tabletop is as empty after the meal, as it was before.

Note that understanding the inversion of identitymay not involve quantity. The child can understandthat, if the same (or identical) stuff is added and thenremoved, the status quo is restored without having toknow anything about the quantity of the stuff.

The other possible level is the understanding of theinversion of quantity. If I have 10 sweets and someonegives me 3 more and then I eat 3, I have the samenumber left as at the start, and it doesn’t matterwhether the 3 sweets that I ate are the same 3sweets as were given to me or different ones.Provided that I eat the same number as I was given,the quantitative status quo is now restored.

In a second study, again using toy bricks, Bryant et al.established that five- and six-year-old children foundproblems, called identity problems, in which exactlythe same bricks were added to and then subtractedfrom the initial set (or vice versa), easier than otherproblems, called quantity problems, in which the



same number of bricks was added and thensubtracted (or vice versa), but the actual brickssubtracted were quite different from the bricks thathad been added before. Bryant et al. also found agreater improvement with age in children’sperformance in quantity inversion problems than inidentity inversion problems. These results point to adevelopmental hypothesis: children’s understandingof the inversion of identity precedes, and mayprovide the basis for, their understanding of theinversion of quantity. First they understand thatadding and subtracting the same stuff restores thephysical status quo. Then they extend this knowledgeto quantity, realising now that adding and subtractingthe same quantity restores the quantitative statusquo, whether the addend and subtrahend are thesame stuff or not.

However, the causal determinants of learning aboutinversion might vary between children. Certainlythere are many reports of substantial individualdifferences within the same age groups in theunderstanding of the inversion principle. Many of theseven- and nine-year-olds in Bisanz and LeFevre'sstudy (1990) used the inversion principle to solveappropriate problems but over half of them did not.Over half of the ten-year-olds tested in Stern’s(1992) original study did take advantage of theprinciple, but around 40% seemed unable to do so.

Recent work by Gilmore (Gilmore and Bryant, 2006;Gilmore and Papadatou-Pastou, 2008) suggests thatthe underlying pattern of these individual differencesmight take a more complex and also a moreinteresting form than just a dichotomy betweenthose who do and those who do not understandthe inversion principle. She used cluster analysis with samples of six- to eight-year-olds who had been given inversion and control problems (againthe control problems had to be solved throughcalculation), and consistently found three groups of children. One group appeared to have a clearunderstanding of inversion and good calculation skillsas well; these children did better in the inversionthan in the control problems, but their scores in thecontrol problems were also relatively high. Anothergroup consisted of children who seemed to havelittle understanding of inversion and whosecalculation skills were weak as well. The remaininggroup of children had a good understanding ofinversion, but their calculation skills were weak: inother words, these children did better in theinversion than in the control problems, but theirscores in the control problems were particularly low.

Thus, the discrepancy between knowing aboutinversion and knowing how to calculate went oneway but not the other. Gilmore identified a group ofchildren who could use the inversion principle andyet did not calculate well, but she found no evidenceat all for the existence a group of children whocould calculate well but were unable to use theinversion principle. Children, therefore, do not haveto be good at adding and subtracting in order tounderstand the relation between these twooperations. On the contrary, they may need tounderstand the inverse relation before they canlearn to add and subtract efficiently.

How can knowledge of inversion facilitate children’sability to calculate? Our answer to this is onlyhypothetical at this stage, but it is worth examininghere. If children understand well the principle ofinversion, they may use their knowledge of numberfacts more flexibly, and thus succeed in moreproblems where calculation is required than childrenwho cannot use their knowledge flexibly. Forexample, if they know that 9 + 7 = 16 andunderstand inversion, they can use this knowledge to answer two more questions: 16 – 9 = ? and 16 – 7 = ? Similarly, the use of ‘indirect addition’ tosolve difficult subtraction problems depends onknowing and using the inverse relation betweenaddition and subtraction. One must understandinversion to be able to see, for example, that an easyway to solve the subtraction 42 – 39 is to convert itinto an addition: the child can count up from 39 to42, find that this is 3, and will then know that 42 – 39must equal 3. In our view, no one could reason thisway without also understanding the inverse relationbetween addition and subtraction.

If this hypothesis is correct, it has fascinatingeducational implications. Children spend much timeat home and in school practising number facts,perhaps trying to memorise them as if they wereindependent of each other. However, a mixture oflearning about number facts and about mathematicalprinciples that help them relate one number fact toothers, such as inversion, could provide them withmore flexible knowledge as well as more interestinglearning experiences. So far as we know, there is nodirect evidence of how instruction that focuses bothon number facts and principles works in comparisonwith instruction that focuses only on number facts.However, there is some preliminary evidence on the role of inversion in facilitating children’sunderstanding of the relation between the sum a + b = c and c – b (or – a) = ?


Some researchers have called this ‘the complement’question and analysed its difficulty in a quite directway by telling children first that a + b = c and thenimmediately asking them the c – a = ? question(Baroody, Ginsburg and Waxman, 1983; Baroody,1999; Baroody and Tiilikainen, 2003; Resnick, 1983;Putnam, de Bettencourt and Leinhardt, 1990). Thesestudies established that the step from the first to thesecond sum is extremely difficult for children in theirfirst two or three years at school, and most of themfail to take it. Only by the age of about eight years doa majority of children use the information from theaddition to solve the subtraction, and even at this agemany children continue to make mistakes. Wouldthey be able to do better if their understanding ofthe inverse relation improved?

The study by Siegler and Stern (1998) describedearlier on, with eight-year-olds, seems to suggest thatit is not that easy to improve children’s understandingof the inverse relation between addition andsubtraction: after solving over 100 inversionproblems, distributed over 7 days, the children didvery poorly in using it selectively; i.e. using it when itwas appropriate, and not using it when it was notappropriate. However, the method that they usedhad several characteristics, which may not havefacilitated learning. First, the problems were allpresented simply as numbers written on cards, withno support of concrete materials or stories. Second,the children were encouraged to answer correctlyand also quickly, if possible, but they did not receiveany feedback on whether they were correct. Finally,they were asked to explain how they had solved theproblem, but if they indicated that they had used theinverse relation to solve it, they were neither toldthat this was a good idea nor asked to think moreabout it if they had used it inappropriately. In brief, itwas not a teaching study.

Recently we completed two studies on teachingchildren about the inverse relation between additionand subtraction (Nunes, Bryant, Hallett, Bell andEvans, 2008). Our aims were to test whether it ispossible to improve children’s understanding of theinverse relation and to see whether they wouldimprove in solving the complement problem afterreceiving instruction on inversion.

One of the studies was with eight-year-olds, i.e.children of the same age as those who participated in the Siegler and Stern study. Our study wasconsiderably briefer, as it involved a pre-test, twoteaching sessions, and a post-test. In the pre-test the

children answered inverse problems (a + b – b),control problems (a + b – c) and complementproblems (a + b = c; c – a/b = ?). During the training,they only worked on inversion problems. So if thetaught groups improved significantly on thecomplement problems, this would have to be aconsequence of realising the relevance of the inverse principle to this type of problem.

For the teaching phase, the children were randomlyassigned to one of three groups: a Control group, who only received practice in calculation; a VisualDemonstration group and a Verbal Calculator group,both receiving instruction on the inverse relation. Theform of the instruction varied between the two groups.

The Visual Demonstration group was taught with thesupport of concrete materials, and started with aseries of trials that took advantage of the identityinversion. First the children counted the number of bricks in a row of Unifix bricks, which wassubsequently hidden under a cloth so that nocounting was possible after that. Next, theexperimenter added some bricks to the row andsubtracted others. The child was then asked howmany bricks were left under the cloth. The number ofbricks added and subtracted was either the same ordiffered by one; this required the children to attendduring all trials, as the answer was not in all examplesthe same number as before the additions andsubtractions, but they could still use the inverseprinciple easily because the difference of one did notmake the task too different from an exact inversiontrial. When they had given their answer, they receivedfeedback and explained how they had found theanswer. If they had not used the inversion principle,they were encouraged to think about it (e.g. Howmany were added? How many were taken away?Would the number be the same or different?).

The Verbal Calculator group received the samenumber of trials but no visual demonstration. Afterthey had provided their answer, they were encouragedto repeat the trial verbally as they entered theoperations into a calculator and checked the answer.Thus they would be saying, for example, ‘fourteen pluseight minus eight is’ and looking at the answer.

As explained, we had three types of problems in the pre- and post-test: inversion, control andcomplement problems, which were transfer problemsfor our intervention group, as they had not learnedabout these directly during the training. We did notexpect the groups to differ in the control trials, as the



amount of experience they had between pre- and post-test was limited, but we expected theexperimental teaching groups to perform significantlybetter than the Control group in the inversion andtransfer problems.

The results were clear :

• Both taught groups made more progress than theControl group from the pre-test to the post-test inthe inversion problems.

• The Visual Demonstration group made moreprogress than the Control group in the transferproblems; the Verbal Calculator group’simprovement did not differ from the improvementin the Control group in the transfer problems.

• The children’s performance did not improvesignificantly in the control problems in any of the groups.

Thus with eight-year-olds both Visual and Verbalmethods can be used to promote children’s reflectionabout the inverse relation between addition andsubtraction. Although the two methods did not differwhen directly compared to each other, they differedwhen compared with the fixed-standard provided bythe control group: the Visual Demonstration methodwas effective in promoting transfer from the types ofitems used in the training to new types of items, of aformat not presented during the training, and theVerbal method did not.

In our second teaching study, we worked with muchyounger children, whose mean age was just five years.We carried out the study using the same methods,with a pre-test, two teaching sessions, and a post-test,but this time all the children were taught using theVisual Demonstration method. Because the childrenwere so young, we did not use complement problemsto assess transfer, but we included a delayed post-test,given to the children about three weeks after theyhad completed the training in order to see whetherthe effects of the intervention, if any, would remainsignificant at a later date without further instruction.

The intervention showed significant effects for thechildren in one school but not for the children in theother school; the effects persisted until the delayedpost-test was given. Although we cannot be certain,we think that the difference between the schools was due to the fact that in the school where theintervention did not have a significant effect we were

unable to find a quite room to work with thechildren without interruptions and the children haddifficulty in concentrating.

The main lesson from this second study was that it ispossible for this intervention to work with such youngchildren and for the effects to last without furtherinstruction, but it is not certain that it will do so.

Finally, we need to consider whether knowing aboutinversion is really as important as we have claimedhere. Two studies support this claim. The first was byStern (2005). She established in a longitudinal studythat German children’s performance in inversiontasks, which they solved in their second year atschool, significantly predicted their performance in analgebra assessment given about 15 years later, whenthey were 23 years old and studying in university. Infact, the brief inversion task that she gave to thechildren had a higher correlation with theirperformance in the algebra test than the IQ testgiven at about the same time as the inversion task.Partialling out the effect of IQ from the correlationbetween the inversion and the algebra tests did notaffect this predictive relation between the inversiontask and the algebra test.

The second study was by our own team (Nunes etal., 2007). It combined longitudinal and interventionmethods to test whether the relation betweenreasoning principles and mathematics learning is acausal one. The participants in the longitudinal studywere tested in their first year in school. In the secondyear, they completed the mathematics achievementtests administered by the teachers and designedcentrally in the United Kingdom. The gap betweenour assessment and the mathematics achievementtest was about 14 months. One of the componentsof our reasoning test was an assessment of children’sunderstanding of the inverse relation betweenaddition and subtraction; the others were additivecomposition (assessed by the shop task) andcorrespondence (in particular, one-to-manycorrespondence). We found that children’sperformance in the reasoning test significantlypredicted their mathematics achievement even aftercontrolling for age, working memory, knowledge ofmathematics at school entry, and general cognitiveability. We did not report the specific connectionbetween the inversion problems and the children’smathematics achievement in the original paper, so wereport it here. We used a fixed-order regressionanalysis so that the connection between the inversiontask and mathematics achievement could be


considered after controlling for the children’s age,general cognitive ability and working memory. Theinversion task remained a significant predictor of thechildren’s mathematics achievement, and explained12% extra variance. This is a really remarkable result:6 inversion problems given about 14 months beforethe mathematics achievement test made a significantcontribution to predicting children’s achievementafter such stringent controls.

Our study also included an intervention component.We identified children who were underperforming inthe logical assessment for their age at the beginningof their first year in school and created a control andan intervention group. The control children receivedno intervention and the intervention group receivedinstruction on the reasoning principles for one hour aweek for 12 weeks during the time their peers wereparticipating in mathematics lessons. So they had noextra time on maths but specialised instruction onreasoning principles. We then compared theirperformance in the state-designed mathematicsachievement tests with that of the control group. Theintervention group significantly outperformed thecontrol group. The mean for the control group in themathematics assessment was at the 28th percentileusing English norms; the intervention group’s meanwas just above the 50th percentile, i.e. above themean. So a group of children who seemed at risk fordifficulties with mathematics caught up through thisintervention. In the intervention study it is notpossible to separate out the effect of inversion; thechildren received instruction on three reasoningprinciples that we considered of great importance asa basis for their learning. It would be possible to carryout separate studies of how each of the threereasoning principles that we taught the childrenaffects their mathematics performance but we didnot consider this a desirable approach, as our view isthat each one of them is central to children’smathematics learning.

The combination of longitudinal and interventionmethods in the analysis of the causes of success anddifficulties in learning to read is an approach that wasextremely successful (Bradley and Bryant, 1983). The study by Nunes et al., (2007) shows that thiscombination of methods can also be usedsuccessfully in the analysis of how children learnmathematics. However, three caveats are called forhere. First, the study involved relatively small samples:a replication with a larger sample is highlyrecommended. Second, it is our view that it is alsonecessary to attempt to replicate the results of the

intervention in studies carried out in the classroom.Experimental studies, such as ours, provide a proof ofexistence: they show that it is possible to accomplishsomething under controlled conditions. But they donot show that it is possible to accomplish the sameresults in the classroom. The step from the laboratoryto the classroom must be carefully considered (seeNunes and Bryant, 2006, for a discussion of thisissue). Finally, it is clear to us that developmentalprocesses that describe children’s development whenthey do not have any special educational needs (theydo not have brain deficiencies, for example, and havehearing and sight within levels that grant them accessto information normally accessed by children) mayneed further analysis when we want to understandthe development of children who do have specialeducational needs. We exemplify here briefly thesituation of children with severe or profound hearingloss. The vast majority of deaf children are born tohearing parents (about 90%), who may not knowhow to communicate with their children withoutmuch additional learning. Mathematics learninginvolves logical reasoning, as we have argued, and alsoinvolves learning conventional representations fornumbers. Knowledge of numbers can be used toaccelerate and promote children’s reflections abouttheir schemes of action, and this takes place throughsocial interaction. Parents teach children a lot aboutcounting before they go to school (Schaeffer,Eggleston and Scott, 1974; Young-Loveridge, 1989)but the opportunities for these informal learningexperiences may be restricted for deaf children. Theywould enter school with less knowledge of countingand less understanding of the relations betweenaddition, subtraction, and number. This does not meanthat they have to develop their understanding ofnumbers in a different way from hearing children, butit does mean that they may need to learn in a muchmore carefully planned environment so that theirlearning opportunities are increased and appropriatefor their visual and language skills. In brief, there maybe special children whose mathematical developmentrequires special attention. Understanding theirdevelopment may or may not shed light on a moregeneral theory of mathematics learning.



Summary

1 The inversion principle is an essential par t ofadditive reasoning: one cannot understand eitheraddition or subtraction unless one alsounderstands their relation to each other.

2 Children probably first recognise the inverserelation between adding and subtracting theidentical stuff. We call this the inversion of identity.

3 The understanding of the inversion of quantity is a step-up. It means under standing that a quantitystays the same if the same n umber of items isadded to it and subtr acted from it, even thoughthe added and subtracted items are different fromeach other.

4 The inversion of quantity is more diff icult for youngchildren to understand, but in tasks that involveconcrete objects, many children in the f ive- toseven-year age range do grasp this form ofinversion to some extent.

5 There are however strong individual differencesamong children in this form of understanding.Children in the five- to eight-year range fall intothree main groups. Those who are good acalculating and also good at using the in versionprinciple, those who are weak in both things, and those who are good at using the in versionprinciple, but weak in calculating.

6 The evidence suggests that children’sunderstanding of the inversion principle plays animportant causal role in their progress in learningabout mathematics. Children’s understanding ofinversion is a good predictor of their mathematicalsuccess, and improving this understanding has theresult of improving children’s mathematicalknowledge in general.

Additive reasoning andproblem solvingIn this section we continue to analyse children’s ability to solve additive reasoning problems. Additivereasoning refers to reasoning used to solve problemswhere addition or subtraction are the operationsused to find a solutions. We prefer to use thisexpression, rather than addition and subtractionproblems, because it is often possible to solve thesame problem either by addition or by subtraction.

For example, if you buy something that costs £35,you may pay with two £20-notes. You can calculateyour change by subtraction (40 – 35) or by addition(35 + 5). So, problems are not addition orsubtraction problems in themselves, but they can bedefined by the type of reasoning that they require,additive reasoning.

Although preschoolers’ knowledge of addition andsubtraction is limited, as we argued in the previoussection, it is clear that their initial thinking about thesetwo arithmetical operations is rooted in theireveryday experiences of seeing quantities beingcombined with, or taken away from, other quantities.They find purely numerical problems like ‘what is 2and 1 more?’ a great deal more difficult thanproblems that involve concrete situations, even whenthese situations are described in words and leftentirely to the imagination (Ginsburg, 1977; Hughes,1981; 1986; Levine, Jordan and Huttenlocher, 1992).

The type of knowledge that children develop initiallyseems to be related to two types of action: puttingmore elements in a set (or joining two sets) andtaking out elements from one set (or separating twosets). These schemes of action are used by childrento solve arithmetic problems when they arepresented in the context of stories.

By and large, three main kinds of story problem havebeen used to investigate children’s additive reasoning:

• the Change problem (‘Bill had eight apples and thenhe gave three of them away. How many did hehave left?’).

• the Combine problem (‘Jane has three dolls and Maryhas four. How many do they have altogether?’).

• the Compare problem (‘Sam has five books andSarah has eight. How many more books does Sarahhave than Sam?’).

A great deal of research (e.g. Brown, 1981; Carpenter,Hiebert and Moser, 1981; Carpenter and Moser, 1982;De Corte and Verschaffel, 1987; Kintsch and Greeno,1985; Fayol, 1992; Ginsburg, 1977; Riley, Greeno andHeller, 1983; Vergnaud, 1982) has shown that ingeneral, the Change and Combine problems are much easier than the Compare problems. The mostinteresting aspect of this consistent pattern of resultsis that problems that are solved by the samearithmetic operation – or in other words, by the samesum – can differ radically in how difficult they are.


Usually pre-school children do make the appropriatemoves in the easiest Change and Combine problems:they put together and count up (counting on orcounting all) and separate and count the relevant set to find the answer. Very few pre-school children seemto know addition and subtraction facts, and so theysucceed considerably more if they have physical objects(or use their fingers) in order to count. Research byCarpenter and Moser (1982) gives an indication ofhow pre-school children perform in the simplerproblems. These researchers interviewed children (agedabout four to five years) twice before they had hadbeen given any instruction about arithmetic in school;we give here the results of each of these interviews, asthere is always some improvement worth notingbetween the testing occasions.

For Combine problems (given two parts, find thewhole), 75% and 82% of the answers were correctwhen the numbers were small and 50% and 71%when the numbers were larger; only 13% of theresponses with small numbers were obtained throughthe recall of number facts and this was the largestpercentage of recall of number facts observed in theirstudy. For Change problems (Tim had 11 candies; hegave 7 to Martha; how many did he have left?), thepre-schoolers were correct 42% and 61% with largernumbers (Carpenter and Moser do not report thefigures for smaller numbers) at each of the twointerviews; only 1% of recall of number facts isreported. So, pre-school children can do relatively wellon simple Change and Combine problems beforethey know arithmetic facts; they do so by putting setstogether or by separating them and counting.

This classification of problems into three types –Combine, Change and Compare – is not sufficientto describe story problem-solving. In a Changeproblem, for example, the story might provide theinformation about the initial state and the change(Tim had 11 candies; he gave 7 to Martha); the childis asked to say what the final state is. But it is alsopossible to provide information, for example, aboutthe transformation and the end state (Tim hadsome candies; he gave Martha 7 and he has 4 left)and ask the child to say what the initial state was(how many did he have before he gave candies toMartha?). This sort of analysis has resulted in morecomplex classifications, which consider whichinformation is given and which information must besupplied by the children in the answer. Stories thatdescribe a situation where the quantity decreases, asin the example above, but have a missing initial statecan most easily be solved by an addition. The conflict

between the decrease in quantity and the operationof addition can be solved if the children understandthe inverse relation between subtraction andaddition: by adding the number that Tim still has and the number he gave away, one can find out how many candies he had before.

Different analyses of word problems have beenproposed (e.g. Briars and Larkin, 1984; Carpenter andMoser, 1982; Fuson, 1992; Nesher, 1982; Riley, Greenoand Heller, 1983; Vergnaud, 1982). We focus here onsome aspects of the analysis provided by GérardVergnaud, which allows for the comparison of manydifferent types of problems and can also be used tohelp understand the level of difficulty of further typesof additive reasoning problems, involving directednumbers (i.e. positive and negative numbers).

First, Vergnaud distinguishes between numerical andrelational calculation. Numerical calculation refers to thearithmetic operations that the children carry out tofind the answer to a problem: in the case of additivereasoning, addition and subtraction are the relevantoperations. Relational calculation refers to theoperations of thought that the child must carry out inorder to handle the relations involved in the problem.For example, in the problem ‘Bertrand played a gameof marbles and lost 7 marbles. After the game he had3 marbles left. How many marbles did he have beforethe game?’, the relational calculation is the realisationthat the solution requires using the inverse ofsubtraction to go from the final state to the initialstate and the numerical calculation would be 7 + 3.

Vergnaud proposes that children perform theserelational calculations in an implicit manner: to use hisexpression, they rely on ‘theorems in action’. Thechildren may say that they ‘just know’ that they haveto add when they solve the problem, and may beunable to say that the reason for this is that additionis the inverse of subtraction. Vergnaud reportsapproximately twice as many correct responses byFrench pre-school children (aged about five years) toa problem that involves no relational calculation(about 50% correct in the problem: Pierre had 6marbles. He played a game and lost 4; how many didhe have after the game?) than to the problem above(about 26% correct responses), where we are toldhow many marbles Bertrand lost and asked howmany he had before the game.

Vergnaud also distinguished three types ofmeanings that can be represented by naturalnumbers: quantities (which he calls measures),



transformations and relations. This distinction has an effect on the types of problems that can becreated starting from the simple classification inthree types (change, combine and compare) andtheir level of difficulty.

First, consider the two problems below, the firstabout combining a quantity and a transformation andthe second about combining two transformations.

• Pierre had 6 marbles. He played one game and lost 4 marbles. How many marbles did he haveafter the game?

• Paul played two games of marbles. He won 6 in the first game and lost 4 in the second game. What happened, counting the two games together?

French children, who were between pre-school andtheir fourth year in school, consistently performedbetter on the first than on the second type ofproblem, even though the same arithmetic calculation(6-4) is required in both problems. By the second yearin school, when the children are about seven years old,they achieve about 80% correct responses in the firstproblem, and they only achieve a comparable level ofsuccess two years later in the second problem. So,combining transformations is more difficult thancombining a quantity and a transformation.

Brown (1981) confirmed these results with Englishstudents in the age range 11 to 16. In her task,students are shown a sign-post that indicates thatGrange is 29 miles to the west and Barton is 58miles to the east; they are asked how do they workout how far they need to drive to go from Grangeto Barton. There were eight choices of operationsconnecting these two numbers for the students toindicate the correct one. The rate of correctresponses to this problem was 73%, which contrastswith 95% correct responses when the problemreferred to a union of sets (a combine problem).

The children found problems even more difficultwhen they needed to de-combine transformationsthan when they had to combine them. Here is anexample of a problem with which they needed tode-combine two transformations, because the storyprovides the result of combining operations and thequestion that must be answered is about the state of affairs before the combination took place.

• Bruno played two games of marbles. He played thefirst and the second game. In the second game he

lost 7 marbles. His final result, with the two gamestogether, was that he had won 3 marbles. Whathappened in the first game?

This de-combination of transformations was still verydifficult for French children in the fourth year inschool (age about nine years): they attained less than50% correct responses.

Vergnaud’s hypothesis is that when children combinetransformations, rather than quantities, they have togo beyond natural numbers: they are now operatingin the domain of whole numbers. Natural numbersare counting numbers. You can certainly count thenumber of marbles that Pierre had before he startedthe game, count and take away the marbles that helost in the second game, and say how many he hadleft at the end. In the case of Paul’s problem, if youcount the marbles that he won in the first game, youneed to count them as ‘one more, two more, threemore etc.’: you are actually not counting marbles butthe relation between the number that he now has tothe number he had to begin with. So if the startingpoint in a problem that involves transformations isnot known, the transformations are now relations. Ofcourse, children who do solve the problem aboutPaul’s marbles may not be fully aware of thedifference between a transformation and a relation,and may succeed exactly because they overlook thisdifference. This point is discussed in Paper 4, when weconsider in detail how children think about relations.

Finally, problems where children are asked to quantifyrelations are usually difficult as well:

• Peter has 8 marbles. John has 3 marbles. How manymore marbles does Peter have than John?

The question in this problem is neither about aquantity (i.e. John’s or Peter’s marbles) nor about atransformation (no-one lost or got more marbles): itis about the relation between the two quantities.Although most pre-school children can say that Peterhas more marbles, the majority cannot quantify therelation (or the difference) between the two. Thebest known experiments that demonstrate thisdifficulty were carried out by Hudson (1983) in theUnited States. In a series of three experiments, heshowed the children some pictures and asked themtwo types of question:

• Here are some birds and some worms. How manymore birds than worms?


• Here are some birds and some worms. The birdsare racing to get a worm. How many birds won’tget worms?

The first question asks the children to quantify therelation between the two sets, of worms and birds;the second question asks the children to imagine thatthe sets were matched and quantify the set that hasno matching elements. The children in the first year ofschool (mean age seven years) attained 64% correctresponses to the first question and 100% to thesecond question; in nursery school (mean age fouryears nine months) and kindergarten (mean age 6years 3 months), the rate of correct responses was,respectively, 17% and 25% to the first question and83% and 96% to the second question.

It is, of course, difficult to be completely certain thatthe second question is easier because the childrenare asked a question about quantity whereas the firstquestion is about a relation. The reason for thisambiguity is that two things have to change at thesame time for the story to be different: in order tochange the target of the question, so that it is either a quantity or a relation, the language used in theproblem also varies: in the first problem, the word‘more’ is used, and in the second it is not.

Hudson included in one of his experiments a pre-test of children’s understanding of the word ‘more’(e.g. Are there more red chips or more whitechips?’) and found that they could answer thisquestion appropriately. He concluded that it was thelinguistic difficulty of the ‘How many more…?’question that made the problem difficult, not simplythe difficulty of the word ‘more’. We are notconvinced by his conclusion and think that moreresearch about children’s understanding of how toquantify relations is required. Stern (2005), on theother hand, suggests that both explanations arerelevant: the linguistic form is more difficult andquantifying relations is also more difficult than usingnumbers to describe quantities.

In the domain of directed numbers (i.e. positive and negative numbers), it is relatively easier to studythe difference between attributing numbers toquantities and to relations without asking the ‘howmany more’ question. Unfortunately, studies withlarger sample, which would allow for a quantitativecomparison in the level of difficulty of theseproblems, are scarce. However, some indication that quantifying relations is more difficult forstudents is available in the literature.

Vergnaud (1982) pointed out that relationshipsbetween people could be used to create problemsthat do not contain the question ‘how many more’.Among others, he suggested the following example.

• Peter owes 8 marbles to Henry but Henry owes6 marbles to Peter. What do they have to do toget even?

According to his analysis, this problem involves acomposition of relations.

Marthe (1979) compared the performance of Frenchstudents in the age range 11 to 15 years in twoproblems involving such composition of relations with their performance in two problems involving a change situation (i.e. quantity, transformation,quantity). In order to control for problem format, allfour problems had the structure a + x = b, in which xshows the place of the unknown. The problems usedlarge numbers so that students had to go throughthe relational calculation in order to determine thenumerical calculation (with small numbers, it ispossible to work in an intuitive manner, sometimesstarting from a hypothetical amount and adjusting thestarting point later to make it fit). An example of aproblem type using a composition of relations isshown below.

• Mr Dupont owes 684 francs to Mr. Henry. But MrHenry also owes money to Mr Dupont. Takingeverything into account, Mr Dupont must give back327 francs to Mr Henry. What amount did MrHenry owe to Mr Dupont?

Marthe did find that problems about relations werequite a bit more difficult than those about quantitiesand transformations; there was a difference of 20%between the rates of correct responses for theyounger children and 10% for the older children.However, the most important effect in theseproblems seemed to be whether the students hadto deal with numbers that had the same or differentsigns: problems with same signs were consistentlyeasier than those with different signs.

In summary, different researchers have argued that itis one thing to learn to use numbers to representquantities and a quite different one to use numbersto quantify relations. Relations are more abstract andmore challenging for students. Thompson (1993)hypothesises that learning to quantify and thinkabout numbers as measures of relations is a crucialstep that students must take in order to understand



algebra. We are completely sympathetic to thishypothesis, but we think that the available evidenceis a bit thin.

More than two decades ago, Dickson, Brown andGibson (1984) reviewed research on additive reasoningand problem solving, and pointed out how difficult it isto come to firm conclusions when no single study hascovered the variety of problems that any theoreticalmodel would aim to compare. We have to piece theevidence together from diverse studies, and of coursesamples vary across different locations and cohorts. Inthe last decade research on additive reasoning hasreceived less attention in research on mathematicseducation than before. Unfortunately, this has left somequestions with answers that are, at best, based on singlestudies with limited numbers of students. It is time touse a new synthesis to re-visit these questions and seek for unambiguous answers within a single researchprogramme.

Summary

1 In word problems children are told a br ief storywhich ends in an ar ithmetical question. Theseproblems are widely used in school textbooks and also as a research tool.

2 There are three main kinds of w ord problem:Combine, Change and Compare .

3 Vergnaud argued that the crucial elements in these problems were Quantities (measures),Transformations and Relations. On the whole ,problems that involve Relations are harder thanthose involving Transformations.

4 However, other factors also affect the level ofdifficulty in word problems. Any change in sign isoften hard for children to handle: when the stor yis about an addition but the solution is to subtr act,as in missing addend problems, children often failto use the inverse operation.

5 Overall the extreme var iability in the level ofdifficulty of different problems, even when thesedemand exactly the same mathematical solution(the same simple additions or subtr actions)confirms the view that there is a great deal moreto arithmetical learning than knowing how to carryout par ticular operations.

6 Research on word problems supports a differentapproach, which is that ar ithmetical learning

depends on children making a coherentconnection between quantitative relations and the appropriate numerical analysis.

Overall conclusions andeducational implications• Learning about quantities and numbers are two

different matters: children can understand relationsbetween quantities and not know how to makeinferences about the numbers that are used torepresent the quantities; they might also learn tocount without making a connection betweencounting and what it implies for the relationsbetween quantities.

• Some ideas about quantities are essential forunderstanding number: equivalence betweenquantities, their order of magnitude, and the part-whole relations implicit in determining the numberof elements in a set.

• These core ideas, in turn, require that childrencome to understand yet other logical principles:transitive relations in equivalence and order,which operations change quantities and which donot, and the inverse relation between additionand subtraction. These notions are central tounderstanding numbers and how they representquantities; children who have a good grasp ofthem learn mathematics better in school.Children who have difficulties with these ideasand do not receive support to come to gripswith them are at risk for difficulties in learningmathematics, but these difficulties can beprevented to a large extent if they receiveappropriate instruction.

• There is no question that word problems give us avaluable insight into children’s reasoning aboutaddition and subtraction. They demonstrate thatthere is a great deal more to understanding theseoperations than just learning how to add andsubtract. Children’s solutions do depend on theirability to reason about the relations betweenquantities in a logical manner. There is no doubtabout these conclusions, even if there is need forfurther research to pin down some of the details.

• Learning to count and to use numbers torepresent quantities is an important element inthis developmental process. Children can moreeasily reason about the relation between


addition, subtraction, and number when theyknow how to represent quantities by counting.But this is not a one-way relation: it is by adding,subtracting, and understanding the inverserelation between these operations that childrenunderstand additive composition and learn tosolve additive reasoning problems.

• The major implication from this review is thatschools should take very seriously the need toinclude in the curriculum instruction that promotesreflection about relations between quantities,operations, and the quantification of relations.

• These reflections should not be seen asappropriate only for very young children: whennatural numbers start to be used to representrelations, directed numbers become a new domain of activity for children to re-construct their understanding of additive relations. Theconstruction of a solid understanding of additiverelations is not completed in the first years ofprimary school: some problems are still difficult forstudents at the age of 15.

Endnotes

1 Gelman and Butterworth (2005) make a similar distinctionbetween numerosity and the representation of number: ‘weneed to distinguish possession of the concept of n umerosityitself (knowing that any set has a numerosity that can bedetermined by enumeration) from the possession ofrerepresentations (in language) of par ticular numerosities’ (pp. 6). However, we adopt here the term ‘quantities’ because it has an established definition and use in the context ofchildren’s learning of mathematics.

2 It is noted here that evidence from cases studies of acquireddyscalculia (a cognitive disorder affecting the ability to solvemathematics problems observed in patients after neurologicaldamage) is consistent with the idea that under standingquantities and number knowledge can be dissociated:calculation may be impaired and conser vation of quantities may be intact in some patients whereas in other s calculation is intact and conceptual knowledge impaired (Mittmair-Delazer,Sailer and Benke, 1995). Dissociations between arithmetic skillsand the meaning of numbers were extensively described byMcCloskey (1992) in a detailed review of cases of acquireddyscalculia.



Alibali, M. W., & DiRusso, A. A. (1999). The function ofgesture in learning to count: More than keepingtrack. Cognitive Development, 14, 37-56.

Baroody, A. J., Ginsburg, H.P., & Waxman, B. (1983).Children’s use of mathematical structure. Journal forResearch in Mathematics Education, 14, 156-168.

Baroody, A. J., & Tiilikainen, S. H. (2003). Twoperspectives on addition development. In A. J.Baroody & A. Dowker (Eds.), The development of arithmetic concepts and skills (pp. 75-126).Erlbaum: Mahwah, New Jersey.

Bermejo, V., Morales, S., & deOsuna, J. G. (2004).Supporting children’s development of cardinalityunderstanding. Learning and Instruction, 14 381-398.

Bisanz, J., & Lefevre, J.-A. (1990). Strategic and non-strategic processing in the development ofmathematical cognition. In D. J. Bjorklund (Ed.),Children’s strategies: Contemporary views ofcognitive development (pp. 213-244).

Blevins-Knabe, B., Cooper, R. G., Mace, P. G., &Starkey, P. (1987). Preschoolers sometimes knowless than we think: The use of quantifiers to solveaddition and subtraction tasks. Bulletin of thePsychometric Society, 25, 31-34.

Bradley, L., & Br yant, P. E. (1983). Categorising soundsand learning to read -a causal connection. Nature,301, 419-521.

Brannon, E. M. (2002). The development of ordinalnumerical knowledge in infancy. Cognition, 83, 223-240.

Briars, D. J., & Larkin, J. H. (1984). An integrated modelof skills in solving elementary arithmetic wordproblems. Cognition and Instruction, 1, 245-296.

Brown, M. (1981). Number operations. In K. Hart(Ed.), Children’s Understanding of Mathematics: 11-16 (pp. 23-47). Windsor, UK: NFER-Nelson.

Brush, L. R. (1978). Preschool children’s knowledge ofaddition and subtraction. Journal for Research inMathematics Education, 9, 44-54.

Bryant, P. E., & Kopytynska, H. (1976). Spontaneousmeasurement by young children. Nature, 260, 773.

Bryant, P. E., & Trabasso, T. (1971). Transitive inferences andmemory in young children. Nature, 232, 456-458.

Bryant, P., Christie, C., & Rendu, A. (1999). Children’sunderstanding of the relation between additionand subtraction: Inversion, identity anddecomposition. Journal of Experimental ChildPsychology, 74, 194-212.

Butterworth, B., Cipolotti, L., & Warrington, E. K.(1996). Shor t-term memory impairments andarithmetical ability. Quarterly Journal ofExperimental Psychology, 49A, 251-262.

Carey, S. (2004). Bootstrapping and the or igin of concepts. Daedalus, 133(1), 59-69.

Carpenter, T. P., & Moser, J. M. (1982). Thedevelopment of addition and subtr actionproblem solving. In T. P. Carpenter, J. M. Moser & T. A. Romberg (Eds.), Addition and subtraction: Acognitive perspective (pp. 10-24). Hillsdale (NJ):Lawrence Erlbaum.

Carpenter, T. P., Hieber t, J., & Moser, J. M. (1981).Problem structure and first grade children’s initialsolution processes for simple addition andsubtraction problems. Journal for Research inMathematics Education, 12, 27-39.

Cooper, R. G. (1984). Early number development:discovering number space with addition andsubtraction. In C . Sophian (Ed.), The origins ofcognitive skill (pp. 157-192). Hillsdale, NJ:Lawrence Erlbaum Ass.

Cowan, R. (1987). When do children tr ust countingas a basis for relative number judgements? Journalof Experimental Child Psychology, 43, 328-345.

Cowan, R., & Daniels, H. (1989). Children’s use ofcounting and guidelines in judging relativ enumber. British Journal of Educational Psychology,59, 200-210.


References

De Corte, E., & Verschaffel, L. (1987). The effect ofsemantic structure on first graders’ solutionstrategies of elementar y addition and subtractionword problems. Journal for Research inMathematics Education, 18, 363-381.

Dehaene, S. (1992). Varieties of numerical abilities.Cognition, 44 1-42.

Dehaene, S. (1997). The Number Sense . London:Penguin.

Dickson, L., Brown, M., & Gibson, O. (1984). Childrenlearning mathematics: A teacher’s guide to recentresearch. Oxford: The Alden Press.

Donaldson, M. (1978). Children’s minds. London:Fontana.

Fayol, M. (1992). From number to numbers in use:solving arithmetic problems. In J. Bideaud, C.Meljac & J.-P. Fischer (Eds.), Pathways to Number(pp. 209-218). Hillsdale, NJ: LEA.

Freeman, N. H., Antonuccia, C., & Lewis, C. (2000).Representation of the cardinality pr inciple: earlyconception of error in a counterfactual test.Cognition, 74, 71-89.

Frydman, O., & Br yant, P. E. (1988). Sharing and theunderstanding of number equivalence by youngchildren. Cognitive Development, 3, 323-339.

Fuson, K. (1992). Research on whole numberaddition and subtraction. In D. A. Grouws (Ed.),Handbook of Research on Mathematics Teachingand learning (pp. 243-275). New York: MacmillaPublishing Company.

Fuson, K. C. (1988). Children’s Counting and Conceptsof Number. New York: Springer Verlag.

Fuson, K. C., Richards, J., & Briars, D. J. (1982). Theacquisition and elaboration of the number wordsequence. In C . J. Brainerd (Ed.), Children’s logicaland mathematical cognition (pp. 33-92). New York:Springer Verlag.

Fuson, K., & Hall, J. W. (1983). The Acquisition of Ear lyNumber Word Meanings: A Conceptual Analysisand Review. In H. P. Ginsburg (Ed.), TheDevelopment of Mathematical Thinking (pp. 50-109). New York: Academic Press.

Gallistel, C. R., & Gelman, R. (1992). Preverbal and verbalcounting and computation. Cognition, 44, 43-74.

Gelman, R., & Butterworth, B. (2005). Number andlanguage: how are they related? Trends in CognitiveSciences, 9, 6-10.

Gelman, R., & Gallistel, C. R. (1978). The child’sunderstanding of number. Cambridge, Mass:Harvard University Press.

Gelman, R., & Meck, E. (1983). Preschoolers’ counting:Principles before skill. Cognition, 13, 343-359.

Gilmore, C., & Br yant, P. (2006). Individual differencesin children’s understanding of inversion andarithmetical skill. British Journal of EducationalPsychology, 76, 309-331.

Gilmore, C., & Papadatou-Pastou, M. (2009).Patterns of Individual Differences in ConceptualUnderstanding and Arithmetical Skill: A Meta-Analysis. In special issue on “Young Children’sUnderstanding and Application of the Addition-Subtraction inverse Principle”. MathematicalThinking and Learning, in press .

Ginsburg, H. (1977). Children’s Arithmetic: The LearningProcess. New York: Van Nostrand.

Ginsburg, H. P., Klein, A., & Starkey, P. (1998). TheDevelopment of Children’s MathematicalThinking: Connecting Research with Pr actice. InW. Damon, I. E. Siegel & A. A. Renninger (Eds.),Handbook of Child Psychology. Child Psychology inPractice (Vol. 4, pp. 401-476). New York: JohnWiley & Sons.

Gréco, P. (1962). Quantité et quotité: nouvellesrecherches sur la correspondance terme-a-termeet la conservation des ensembles. In P. Gréco & A.Morf (Eds.), Structures numeriques elementaires:Etudes d’Epistemologie Genetique Vol 13(pp. 35-52). Paris.: Presses Universitaires de France.

Hudson, T. (1983). Correspondences and numericaldifferences between sets. Child Development, 54,84-90.

Hughes, M. (1981). Can preschool children add andsubtract? Educational Psychology, 3, 207-219.

Hughes, M. (1986). Children and number. Oxford:Blackwell.

Inhelder, B., Sinclair, H., & Bovet, M. (1974). Learningand the development of cognition. London:Routledge and Kegan Paul.

Jordan, N., Huttenlocher, J., & Levine, S. (1992).Differential calculation abilities in young childrenfrom middle- and low-income families.Developmental Psychology, 28, 644-653.

Kintsch, W., & Greeno, J. G. (1985). Understandingand solving word arithmetic problems.Psychological Review, 92, 109-129.

Klein, A. (1984). The early development of ar ithmeticreasoning: Numerative activities and logicaloperations. Dissertation Abstracts International, 45, 131-153.

Krebs, G., Squire, S., & Bryant, P. (2003). Children’sunderstanding of the additive composition ofnumber and of the decimal structure: what is therelationship? International Journal of EducationalResearch, 39, 677-694.



Landerl, K., Bevana, A., & Butterworth, B. (2004).Developmental dyscalculia and basic numericalcapacities: A study of 8-9 year old students.Cognition 93, 99-125.

Le Corre, M., & Carey, S. (2007). One, two, three,nothing more: an investigation of the conceptualsources of verbal number principles. Cognition,105, 395-438.

Light, P. H., Buckingham, N., & Robbins, A. H. (1979).The conservation task as an interactional setting.British Journal of Educational Psychology, 49, 304-310.

Markman, E. M. (1979). Classes and collections:conceptual organisation and numerical abilities.Cognitive Psychology, 11, 395-411.

Marthe, P. (1979). Additive problems and directednumbers. Paper presented at the Proceedings ofthe Annual Meeting of the Inter national Groupfor the Psychology of Mathematics Education,Warwick, UK.

McCrink, K., & Wynn, K. (2004). Large numberaddition and subtraction by 9-month-old infants.Psychological Science, 15, 776-781.

Michie, S. (1984). Why preschoolers are reluctant tocount spontaneously. British Journal ofDevelopmental Psychology, 2, 347-358.

Mittmair-Delazer, M., Sailer, U., & Benke, T. (1995).Impaired arithmetic facts but intact conceptualknowledge - A single-case study of dyscalculia.Cortex, 31, 139-147.

Nesher, P. (1982). Levels of description in the analysisof addition and subtraction word problems. In T. P.Carpenter, J. M. Moser & T. A. Romberg (Eds.),Addition and subtraction. Hillsdale,NJ: LawrenceErlbaum Ass.

Nunes, T., & Br yant, P. (2006). Improving Literacythrough Teaching Morphemes: Routledge.

Nunes, T., & Schliemann, A. D. (1990). Knowledge ofthe numeration system among pre-schooler s. InL. S. T. Wood (Ed.), Transforming early childhoodeducation: International perspectives (pp. pp.135-141). Hillsdale, NJ: Lawrence Erlbaum.

Nunes, T., Bryant, P., Evans, D., Bell, D., Gardner, S.,Gardner, A., & Carraher, J. N. (2007). TheContribution of Logical Reasoning to theLearning of Mathematics in Pr imary School. BritishJournal of Developmental Psychology, 25, 147-166.

Nunes, T., Bryant, P., Hallett, D., Bell, D., & Evans, D.(2008). Teaching Children about the InverseRelation between Addition and Subtraction.Mathematical Thinking and Learning, in press.

Piaget, J. (1921). Une forme verbale de lacomparaison chez l’enfant. Archives de Psychologie,18, 141-172.

Piaget, J. (1952). The Child’s Conception of Number.London: Routledge & Kegan Paul.

Piaget, J., & Moreau, A. (2001). The inversion ofarithmetic operations (R. L. Campbell, Trans.). In J. Piaget (Ed.), Studies in Reflecting Abstraction(pp. 69-86). Hove: Psychology Press.

Putnam, R., deBettencour t, L. U., & Leinhardt, G.(1990). Understanding of derived fact strategiesin addition and subtraction. Cognition andInstruction, 7, 245-285.

Rasmussen, C., Ho, E., & Bisanz, J. (2003). Use of themathematical principle of inversion in youngchildren. Journal of Experimental Child Psychology,85, 89-102.

Resnick, L., & Ford, W. W. (1981). The Psychology ofMathematics for Instruction. Hillsdale, NJ: LawrenceErlbaum Ass.

Resnick, L. B. (1983). A Developmental theory ofNumber Understanding. In H. P. Ginsburg (Ed.),The Development of Mathematical Thinking (pp.110-152). New York: Academic Press.

Riley, M., Greeno, J. G., & Heller, J. I. (1983).Development of children’s problem solving abilityin arithmetic. In H. Ginsburg (Ed.), Thedevelopment of mathematical thinking(pp. 153-196). New York: Academic Press.

Rips, L. J., Asmuth, J., & Bloomfield, A. (2006). Givingthe boot to the bootstr ap: How not to learn thenatural numbers. Cognition, 101, B51-B60.

Rips, L. J., Asmuth, J., & Bloomfield, A. (2008).Discussion. Do children learn the integers byinduction? Cognition, 106 940-951.

Samuel, J., & Br yant, P. E. (1984). Asking only onequestion in the conser vation experiment. Journalof Child Psychology and Psychiatr y, 25, 315-318.

Sarnecka, B. W., & Gelman, S. A. (2004). Six does notjust mean a lot: preschoolers see number wordsas specific. Cognition, 92 329-352.

Saxe, G., Guberman, S. R., & Gearhar t, M. (1987). Social and developmental processes in children’sunderstanding of number. Monographs of the Societyfor Research in Child Development, 52, 100-200.

Schaeffer, B., Eggleston, V. H., & Scott, J. L. (1974).Number development in young children. CognitivePsychology, 6, 357-379.

Secada, W. G. (1992). Race, ethnicity, social class,language, and achievement in mathematics. In D.A. Grouws (Ed.), Handbook of research onmathematics teaching and learning (pp. 623-660).New York: Macmillan.

Siegler, R. S., & Robinson, M. (1982). The developmentof numerical understanding. In H. W. Reese & L. P.Lipsitt (Eds.), Advances in child development andbehavior. New York: Academic Press.


Siegler, R. S., & Stern, E. (1998). Conscious andunconscious strategy discoveries: a microgeneticanalysis. Journal of Experimental Psychology-General, 127, 377-397.

Sophian, C. (1988). Limitations on preschoolchildren’s knowledge about counting: usingcounting to compare two sets. DevelopmentalPsychology, 24, 634-640.

Sophian, C., Wood, A. M., & Vong, C. I. (1995). Makingnumbers count: the ear ly development ofnumerical inferences. Developmental Psychology,31, 263-273.

Spelke, E. S. (2000). Core knowledge. AmericanPsychologist, 55, 1233-1243.

Starkey, P., & Gelman, R. (1982). The development ofaddition and subtraction abilities prior to formalschooling in arithmetic. In T. P. Carpenter, J. M.Moser & T. A. Romberg (Eds.), Addition andsubtraction: a cognitive per spective (pp. 99-115).Hillsdale,NJ: Elrbaum.

Steffe, L. P., & Thompson, P. W. (2000). Radicalconstructivism in action : building on the pioneer ingwork of Ernst von Glasersfeld. New York Falmer.

Steffe, L. P., Cobb, P., & Glaserfeld, E. v. (1988).Construction of arithmetical meanings andstrategies. London Springer-Verlag.

Steffe, L. P., Thompson, P. W., & Richards, J. (1982).Children’s Counting in Arithmetical ProblemSolving. In T. P. Carpenter, J. M. Moser & T. A.Romberg (Eds.), Addition and Subtraction: ACognitive Perspective (pp. 83-96). Hillsdale, NJ:Lawrence Erlbaum Associates.

Steffe, L. P., von Glasersfeld, E., Richards, J., & Cobb, P.(1983). Children’s Counting Types: Philosophy, Theoryand Application. New York: Praeger.

Stern, E. (1992). Spontaneous used of conceptualmathematical knowledge in elementar y schoolchildren. Contemporary Educational Psychology, 17,266-277.

Stern, E. (2005). Pedagogy - Learning for Teaching.British Journal of Educational Psychology, MonographSeries, II, 3, 155-170.

Thompson, P. W. (1993). Quantitative Reasoning,Complexity, and Additive Structures. EducationalStudies in Mathematics , 3, 165-208.

Trabasso, T. R. (1977). The role of memory as asystem in making transitive inferences. In R. V. Kail& J. W. Hagen (Eds.), Perspectives on thedevelopment of memory and cognition. Hillsdale, NJ:Lawrence Erlbaum Ass.

Vergnaud, G. (1982). A classification of cognitive tasksand operations of thought involved in additionand subtraction problems. In T. P. Carpenter, J. M.Moser & R. T. A (Eds.), Addition and subtraction: Acognitive perspective (pp. 60-67). Hillsdale (NJ):Lawrence Erlbaum.

Vergnaud, G. (2008). The theory of conceptual fields.Human Development, in press.

Wynn, K. (1992). Evidence against empir icist accountsof the origins of numerical knowledge. Mind andLanguage, 7, 315-332.

Wynn, K. (1998). Psychological foundations ofnumber: numerical competence in human infants.Trends in Cognitive Science , 2, 296-303.

Xu, F., & Spelke, E. (2000). Large number discriminationin 6-month-old infants. Cognition, 74, B1-B11.

Young-Loveridge, J. M. (1989). The relationshipbetween children’s home experiences and theirmathematical skills on entr y to school. Early ChildDevelopment and Care, 1989 (43), 43 - 59.




ISBN 978-0-904956-69-6





3

Paper 3: Understanding rational numbers and intensive quantitiesBy Terezinha Nunes and Peter Bryant, University of Oxford

2 SUMMARY – PAPER 2: Understanding whole numbers2 Paper 3: Understanding rational numbers and intensive quantities












ContentsSummary of Paper 3 3

Understanding rational numbers and intensive quantities 7

References 28

About the authorsTerezinha Nunes is Professor of EducationalStudies at the University of Oxford. Peter Bryant is Senior Research Fellow in theDepartment of Education, University of Oxford.


About this review


Headlines• Fractions are used in primary school to represent

quantities that cannot be represented by a singlewhole number. As with whole numbers, childrenneed to make connections between quantities andtheir representations in fractions in order to beable to use fractions meaningfully.

• There are two types of situation in which fractionsare used in primary school. The first involvesmeasurement: if you want to represent a quantityby means of a number and the quantity is smallerthan the unit of measurement, you need a fraction– for example, a half cup or a quarter inch. Thesecond involves division: if the dividend is smallerthan the divisor, the result of the division isrepresented by a fraction. For example, when youshare 3 cakes among 4 children, each child receives¾ of a cake.

• Children use different schemes of action in thesetwo different situations. In division situations, theyuse correspondences between the units in thenumerator and the units in the denominator. Inmeasurement situations, they use partitioning.

• Children are more successful in understandingequivalence of fractions and in ordering fractions by magnitude in situations that involve division thanin measurement situations.

• It is crucial for children’s understanding of fractionsthat they learn about fractions in both types ofsituation: most do not spontaneously transfer whatthey learned in one situation to the other.

• When a fraction is used to represent a quantity,children need to learn to think about how the

numerator and the denominator relate to the value represented by the fraction. They must thinkabout direct and inverse relations: the larger thenumerator, the larger the quantity but the largerthe denominator, the smaller the quantity.

• Like whole numbers, fractions can be used torepresent quantities and relations betweenquantities, but in primary school they are rarelyused to represent relations. Older students often find it difficult to use fractions to represent relations.

There is little doubt that students f ind fractions achallenge in mathematics. Teachers often say that itis difficult to teach fr actions and some think that itwould be better for everyone if children were nottaught about fractions in pr imary school. In orderto understand fractions as numbers, students mustbe able to know whether two fractions areequivalent or not, and if they are not, which one isthe bigger number. This is similar to under standingthat 8 sweets is the same number as 8 marbles andthat 8 is more than 7 and less than 9, for example.These are undoubtedly key understandings aboutwhole numbers and fractions. But even after theage of 11 many students have difficulty in knowingwhether two fractions are equivalent and do notknow how to order some fractions. For example , ina study carried out in London, students were askedto paint 2/3 of f igures divided in 3, 6 and 9 equalparts. The majority solved the task cor rectly whenthe figure was divided into 3 par ts but 40% of the11- to 12-year-old students could not solve itwhen the figure was divided into 6 or 9 par ts,which meant painting an equivalent fraction (4/6 and 6/9, respectively).

Summary of paper 3: Understanding rational numbers andintensive quantities


Fractions are used in pr imary school to representquantities that cannot be represented by a singlewhole number. If the teaching of fr actions were to beomitted from the pr imary school curriculum, childrenwould not have the suppor t of school learning torepresent these quantities. We do not believe that itwould be best to just forget about teaching fractionsin primary school because research shows thatchildren have some informal knowledge that could beused as a basis for learning fractions. Thus the questionis not whether to teach fractions in primary schoolbut what do we know about their informal knowledgeand how can teachers draw on this knowledge.

There are two types of situation in which fr actionsare used in pr imary school: measurement anddivision situations.

When we measure anything, we use a unit ofmeasurement. Often the object we are measuringcannot be described only with whole units, and weneed fractions to represent a par t of the unit. In thekitchen we might need to use a ½ cup of milk andwhen setting the margins for a page in a documentwe often need to be precise and def ine the marginas, for example, as 3.17 cm. These two examplesshow that, when it comes to measurement, we use two types of notation, ordinary and decimalnotation. But regardless of the notation used, wecould not accurately describe the quantities in thesesituations without using fractions. When we speak of ¾ of a chocolate bar, we are using fractions in ameasurement situation: we have less than one unit,so we need to descr ibe the quantity using a fr action.

In division situations, we need a fraction to represent aquantity when the dividend is smaller than the divisor .For example, if 3 cakes are shared among 4 children, itis not possible for each one to have a whole cake, butit is still possible to carry out the division and torepresent the amount that each child receiv es using a number, ¾. It would be possible to use decimalnotation in division situations too, but this is rarely thecase. The reason for preferring ordinary fractions inthese situations is that there are two quantities indivision situations: in the example , the number of cakes and the number of children. An ordinary fraction represents each of these quantities b y a wholenumber: the dividend is represented by the numerator,the divisor by the denominator, and the operation ofdivision by the dash between the two numbers.

Although these situations are so similar f or adults, wecould conclude that it is not necessar y to distinguish

between them, however, research shows thatchildren think about the situations diff erently.Children use different schemes of action in each of these situations.

In measurement situations, they use par titioning. If a child is asked to show ¾ of a chocolate, the childwill try to cut the chocolate in 4 equal par ts andmark 3 parts. If a child is asked to compare ¾ and6/8, for example, the child will par tition one unit in 4 par ts, the other in 8 par ts, and tr y to compare thetwo. This is a difficult task because the par titioningscheme develops over a long per iod of time andchildren have to solve many problems to succeed in obtaining equal par ts when par titioning. Althoughpartitioning and comparing the par ts is not the onlyway to solve this problem, this is the most lik elysolution path tr ied out by children, because theydraw on their relevant scheme of action.

In division situations, children use a different schemeof action, correspondences. A problem analogous tothe one above in a division situation is: there are 4children sharing 3 cakes and 8 children shar ing 6identical cakes; if the two groups share the cakesfairly, will the children in one group get the sameamount to eat as the children in the other group?Primary school pupils often approach this problemby establishing correspondences between cakes andchildren. In this way they soon realise that in bothgroups 3 cakes will be shared by 4 children; thedifference is that in the second group there are tw olots of 3 cakes and two lots of 4 children, but thisdifference does not affect how much each child gets.

From the beginning of primary school, many childrenhave some informal knowledge about division thatcould be used to under stand fractional quantities.Between the ages of five and seven years, they arevery bad at par titioning wholes into equal par ts but can be relatively good at thinking about theconsequences of sharing. For example, in one studyin London 31% of the f ive-year-olds, 50% of the six-year-olds and 81% of the seven-year-olds understoodthe inverse relation between the divisor and theshares resulting from the division: they knew that themore recipients are shar ing a cake, the less each onewill receive. They were even able to ar ticulate thisinverse relation when asked to justify their answers.It is unlikely that they had at this time made aconnection between their understanding ofquantities and fractional representation; actually, it isunlikely that they would know how to represent thequantities using fractions.

4 Paper 3: Understanding rational numbers and intensive quantities

The lack of connection between students’understanding of quantities in division situations andtheir knowledge about the magnitude of fr actions isvery clearly documented in research. Students whohave no doubt that recipients of a cak e sharedbetween 3 people will fare better than those of acake shared between 5 people may, nevertheless, saythat 1/5 is a bigger fr action than 1/3 because 5 is abigger number than 3. Although they understand theinverse relation in the magnitude of quantities in adivision situation, they do not seem to connect thiswith the magnitude of fr actions. The link betweentheir understanding of fractional quantities andfractions as numbers has to be developed throughteaching in school.

There is only one well-controlled experiment whichcompared directly young children’s understanding ofquantities in measurement and division situations. Inthis study, carried out in Portugal, the children weresix- to seven-years-old. The context of the problemsin both situations was very similar : it was aboutchildren eating cakes, chocolates or pizzas. In themeasurement problems, there was no shar ing, only par titioning. For example, in one of themeasurement problems, one gir l had a chocolate barwhich was too large to eat in one go . So she cut herchocolate in 3 equal pieces and ate 1. A boy had anidentical bar of chocolate and decided to cut his into6 equal par ts, and eat 2. The children were askedwhether the boy and the gir l ate the same amountof chocolate. The analogous division problem wasabout 3 gir ls sharing one chocolate bar and 6 bo yssharing 2 identical chocolate bar s. The rate of correctresponses in the par titioning situation was 10% forboth six- and seven-year-olds and 35% and 49%,respectively, for six- and seven-year-olds in thedivision situation.

These results are relevant to the assessment ofvariations in mathematics cur ricula. Differentcountries use different approaches in the initialteaching of fraction, some star ting from division and others from measurement situations. There is no direct evidence from classroom studies to show whether one star ting point results in higherachievement in fractions than the other. The scarceevidence from controlled studies supports the ideathat division situations provide children with moreinsight into the equivalence and order of quantitiesrepresented by fractions and that they can lear n howto connect these insights about quantities withfractional representation. The studies also indicatethat there is little tr ansfer across situations: children

who succeed in compar ing fractional quantities andfractions after instruction in division situations do no better in a post-test when the questions areabout measurement situations than other children in a control group who received no teaching. Theconverse is also tr ue: children taught in measurementsituations do no better than a control group indivision situations.

A major debate in mathematics teaching is the relative weight to be given to conceptualunderstanding and procedural knowledge in teaching.The difference between conceptual understandingand procedural knowledge in the teaching offractions has been explored in many studies. Thesestudies show that students can lear n procedureswithout understanding their conceptual significance.Studies with adults show that knowledge ofprocedures can remain isolated from under standingfor a long time: some adults who are ab le toimplement the procedure they lear ned for dividingone fraction by another admit that they have no ideawhy the numerator and the denominator exchangeplaces in this procedure . Learners who are able toco-ordinate their knowledge of procedures withtheir conceptual understanding are better at solvingproblems that involve fractions than other learnerswho seem to be good at procedures b ut show lessunderstanding than expected from their knowledgeof procedures. These results reinforce the idea that itis very important to tr y to make links betweenchildren’s knowledge of fractions and theirunderstanding of fractional quantities.

Finally, there is little , if any, use of fr actions torepresent relations between quantities in pr imaryschool. Secondary school students do not easil yquantify relations that involve fractions. Perhaps this difficulty could be attenuated if some teachingabout fractions in pr imary school involvedquantifying relations that cannot be descr ibed by a single whole number.


6 SUMMARY – PAPER 2: Understanding whole numbers6 Paper 3: Understanding rational numbers and intensive quantities

Recommendations


Children’s knowledge of fractionalquantities starts to develop before theyare taught about fractions in school.

There are two types of situation relevantto primary school teaching in whichquantities cannot be represented by asingle whole number: measurement anddivision.

Children do not easily transfer theirunderstanding of fractions from division tomeasurement situations and vice-versa.

Many students do no make links betweentheir conceptual understanding of fractionsand the procedures that they are taughtto compare and operate on fractions inschool.

Fractions are taught in pr imary school onlyas representations of quantities.


Teaching Teachers should be aware of children’s insightsregarding quantities that are represented by fractions andmake connections between their understanding of thesequantities and fractions.

Teaching The primary school curriculum should include thestudy of both types of situation in the teaching of fr actions.Teachers should be aware of the different types of reasoningused by children in each of these situations.Research Evidence from experimental studies with largersamples and long-term interventions in the classroom areneeded to establish whether division situations are indeed a better star ting point for teaching fractions.

Teaching Teachers should consider how to establish linksbetween children’s understanding of fractions in division andmeasurement situations.Research Investigations on how links between situations canbe built are needed to suppor t curriculum development andclassroom teaching.

Teaching Greater attention may be required in the teachingof fractions to creating links between procedures andconceptual understanding.Research There is a need for longitudinal studies designed to clarify whether this separation between procedural andconceptual knowledge does have important consequences forfurther mathematics learning.

Teaching Consideration should be given to the inclusion ofsituations in which fractions are used to represent relations.Research Given the impor tance of understanding andrepresenting relations numerically, studies that investigateunder what circumstances pr imary school students can usefractions to represent relations between quantities areurgently needed.

IntroductionRational numbers, like natural numbers, can be usedto represent quantities. There are some quantitiesthat cannot be represented by a natural number, andto represent these quantities, we must use rationalnumbers. We cannot use natural numbers when thequantity that we want to represent numerically:• is smaller than the unit used for counting,

irrespective of whether this is a natural unit (e.g. we have less than one banana) or aconventional unit (e.g. a fish weighs less than a kilo)

• involves a ratio between two other quantities (e.g.the concentration of orange juice in a jar can bedescribed by the ratio of orange concentrate towater ; the probability of an event can be describedby the ratio between the number of favourablecases to the total number of cases).1

The term ‘fraction’ is often identified with situationswhere we want to represent a quantity smaller thanthe unit. The expression ‘rational number’ usuallycovers both sor ts of examples. In this paper, we willuse the expressions ‘fraction’ and ‘rational number’interchangeably. Fractions are considered a basicconcept in mathematics learning and one of thefoundations required for learning algebra (Fennell,Faulkner, Ma, Schmid, Stotsky, Wu et al. (2008); sothey are impor tant for representing quantities andalso for later success in mathematics in school.

In the domain of whole numbers, it has been knownfor some time (e .g. Carpenter and Moser, 1982;Ginsburg, 1977; Riley, Greeno and Heller, 1983) thatit is impor tant for the development of children’smathematics knowledge that they establishconnections between the numbers and thequantities that they represent. There is littlecomparable research about rational numbers

(but see Mack, 1990), but it is reasonable to expectthat the same hypothesis holds: children should learnto connect quantities that must be represented byrational numbers with their mathematical notation.However, the difficulty of learning to use rationalnumbers is much greater than the difficulty oflearning to use natural numbers. This paper discusseswhy this is so and presents research that showswhen and how children have significant insights intothe complexities of rational numbers.

In the first section of this paper, we discuss whatchildren must learn about rational numbers and whythese might be difficult for children once they havelearned about natural numbers. In the second sectionwe describe research which shows that these areindeed difficult ideas for students even at the end ofprimary school. The third section compares children’sreasoning across two types of situations that havebeen used in different countries to teach childrenabout fractions. The fourth section presents a br iefoverview of research about children’s understandingof intensive quantities. The fifth section considerswhether children develop sound understanding ofequivalence and order of magnitude of fr actionswhen they learn procedures to compare fr actions.The final section summarises our conclusions anddiscusses their educational implications.

What children must know inorder to understand rationalnumbers

Piaget’s (1952) studies of children’s understanding of number analysed the crucial question of whether


Understanding rationalnumbers and intensivequantities


young children can under stand the ideas ofequivalence (cardinal number) and order (ordinalnumber) in the domain of natur al numbers. He alsopointed out that learning to count may help thechildren to understand both equivalence and order.All sets that are represented by the same numberare equivalent; those that are represented by adifferent number are not equivalent. Their order ofmagnitude is the same as the order of the numberlabels we use in counting, because each number labelrepresents one more than the previous one in thecounting string.

The understanding of equivalence in the domain of fractions is also crucial, but it is not as simplebecause language does not help the children in the same way. Two fractional quantities that havedifferent labels can be equivalent, and in fact there is an infinite number of equivalent fractions: 1/3, 2/6,6/9, 8/12 etc. are different number labels but theyrepresent equivalent quantities. Because rationalnumbers refer, although often implicitly, to a whole , itis also possible for two fractions that have the samenumber label to represent different quantities: 1/3 of12 and 1/3 of 18 are not representations ofequivalent quantities.

In an analogous way, it is not possible simply totransfer knowledge of order from natural to rationalnumbers. If the common fr action notation is used,there are two numbers, the numerator and thedenominator, and both affect the order of magnitudeof fractions, but they do so in diff erent ways. If thedenominator is constant, the larger the numerator,the larger is the magnitude of the fr action; if thenumerator is constant, the larger the denominator,the smaller is the fr action. If both var y, then moreknowledge is required to order the fr actions, and it is not possible to tell which quantity is more b ysimply looking at the fr action labels.

Rational numbers differ from whole numbers also inthe use of two numerical signs to represent a singlequantity: it is the relation between the numbers, nottheir independent values, that represents thequantity. Stafylidou and Vosniadou (2004) analysedGreek students’ understanding of this form ofnumerical representation and obser ved that moststudents in the age r ange 11 to 13 years did notseem to interpret the written representation offractions as involving a multiplicative relationbetween the numerator and the denominator : 20% of the 11-year-olds, 37% of the 12-year-olds and 48% of the 13-year-olds provided this type of

interpretation for fractions. Many younger students(about 38% of the 10-year-olds in grade 5) seemedto treat the numerator and denominator asindependent numbers whereas others (about 20%)were able to conceive fractions as indicating a par t-whole relation but many (22%) are unable to offer a clear explanation for how to interpret thenumerator and the denominator.

Rational numbers are also different from naturalnumbers in their density (see, for example,Brousseau, Brousseau and Warfield, 2007;Vamvakoussi and Vosniadou, 2004): there are nonatural numbers between 1 and 2, for example, but there is an inf inite number of fractions between1 and 2. This may seem unimpor tant but it is thisdifference that allows us to use r ational numbers torepresent quantities that are smaller than the units.This may be another source of diff iculty for students.

Rational numbers have another proper ty which isnot shared by natural numbers: every non-zerorational number has a multiplicative inverse (e.g. theinverse of 2/3 is 3/2). This proper ty may seemunimportant when children are taught aboutfractions in primary school, but it is impor tant forthe understanding of the division algor ithm (i.e. wemultiply the fraction which is the dividend by theinverse of the fraction that is the divisor) and will berequired later in school, when students learn aboutalgebra. Booth (1981) suggested that students oftenhave a limited under standing of inverse relations,particularly in the domain of fr actions, and thisbecomes an obstacle to their under standing ofalgebra. For example, when students think offractions as representing the number of par ts intowhich a whole was cut (denominator) and thenumber of par ts taken (numerator), they find it verydifficult to think that fr actions indicate a division andthat it has, therefore, an inverse.

Finally, rational numbers have two common writtennotations, which students should lear n to connect: 1/2and 0.5 are conceptually the same number with twodifferent notations. There isn’t a similar variation innatural number notation (Roman numerals aresometimes used in specific contexts, such as clocksand indices, but they probably play little role in thedevelopment of children’s mathematical knowledge).Vergnaud (1997) hypothesized that differentnotations afford the understanding of differentaspects of the same concept; this would imply thatstudents should learn to use both notations f orrational numbers. On the one hand, the common


fractional notation 1/2 can be used to help studentsunderstand that fractions are related to the oper ationof division, because this notation can be inter pretedas ‘1 divided by 2’. The connection between fractionsand division is cer tainly less explicit when the decimalnotation 0.5 is used. It is reasonable to expect thatstudents will find it more difficult to understand whatthe multiplicative inverse of 0.5 is than the in verse of1/2, but unfortunately there seems to be no evidenceyet to clarify this.

On the other hand, adding 1/2 and 3/10 is acumbersome process, whereas adding the samenumbers in their decimal representation, 0.5 and 0.3,is a simpler matter. There are disagreements regardingthe order in which these notations should be taughtand the need for students to learn both notations inprimary school (see, for example, Brousseau,Brousseau and Warfield, 2004; 2007), but, to ourknowledge, no one has proposed that one notationshould be the only one used and that the other oneshould be banned from mathematics classes. There isno evidence on whether children f ind it easier tounderstand the concepts related to r ational numberswhen one notation is used r ather than the other.

Students’ difficulties with rational numbersMany studies have documented students’ difficultiesboth with understanding equivalence and order ofmagnitude in the domain of r ational numbers (e.g.Behr, Harel, Post and Lesh, 1992; Behr, Wachsmuth,Post and Lesh, 1984; Har t, 1986; Har t, Brown,Kerslake, Küchermann and Ruddock, 1985; Kamiiand Clark, 1995; Kerslake, 1986). We illustrate herethese difficulties with research car ried out in theUnited Kingdom.

The difficulty of equivalence questions varies acrosstypes of tasks. Kerslake (1986) noted that whenstudents are given diagrams in which the same shapesare divided into different numbers of sections andasked to compare two fractions, this task is relativelysimple because it is possib le to use a perceptualcomparison. However, if students are given a diagramwith six or nine divisions and asked to mark 2/3 ofthe shape, a large proportion of them fail to mar kthe equivalent fractions, 4/6 and 6/9. Hart, Brown,Kerslake, Küchermann and Ruddock (1985), workingwith a sample of students (N =55) in the age r ange11 to 13 years, found that about 60% of the 11- to12-year-olds and about 65% of the 12- to 13-y ear-

olds were able to solve this task. We (Nunes, Bryant,Pretzlik and Hurry, 2006) gave the same item morerecently to a sample of 130 pr imary school studentsin Years 4 and 5 (mean ages, respectively, 8.6 and 9.6years). The rate of correct responses across theseitems was 28% for the children in Year 4 and 49% forthe children in Year 5. This low percentage of correctanswers could not be explained by a lack ofknowledge of the fraction 2/3: when the diagram wasdivided into three sections, 93% of the students inthe study by Hart el al. (1985) gave a correct answer;in our study, 78% of the Year 4 and 91% of the Year 5students’ correctly shaded 2/3 of the f igure.

This quantitative information is presented here toillustrate the level of difficulty of these questions. A different approach to the analysis of how the levelof difficulty can var y is presented later, in the thirdsection of this paper.

Students often have difficulty in ordering fractionsaccording to their magnitude . Hart et al. (1985) askedstudents to compare two fractions with the samedenominator (3/7 and 5/7) and two with the samenumerator (3/5 and 3/4). When the fractions have thesame denominator, students can respond cor rectly byconsidering the numerators only and ordering them asif they were natural numbers. The rate of correctresponses in this case is relatively high but it does noteffectively test students’ understanding of rationalnumbers. Hart et al. (1985) observed approximately90% correct responses among their students in theage range 11 to 13 years and we (Nunes et al., 2006)found that 94% of the students in Year 4 and 87% ofthe students in Year 5 gave correct responses. Incontrast, when the numerator was the same and thedenominator varied (comparing 3/5 and 3/4), and thestudents had to consider the value of the fractions in away that is not in agreement with the order of natur alnumbers, the rate of correct responses wasconsiderably lower: in the study by Hart et al.,approximately 70% of the answers were correct,whereas in our study the percent of cor rect responseswere 25% in Year 4 and 70% among in Year 5.

These difficulties are not par ticular to U.K. students:they have been widely reported in the literature onequivalence and order of fr actions (for examples inthe United States see Behr, Lesh, Post and Silver, 1983;Behr, Wachsmuth, Post and Lesh 1984; Kouba, Brown,Carpenter, Lindquist, Silver and Swafford, 1988).

Difficulties in comparing rational numbers are notconfined to fractions. Resnick, Nesher, Leonard,



Magone, Omanson and Peled (1989) have shown that students have difficulties in comparing decimalfractions when the number of places after the decimalpoint differs. The samples in their study were relativelysmall (varying from 17 to 38) but included studentsfrom three different countries, the United States, Israeland France, and in three gr ade levels (4th to 6th). Thechildren were asked to compare pair s of decimalssuch as 0.5 and 0.36, 2.35 and 2.350, and 4.8 and 4.63.The rate of correct responses varied between 36%and 52% correct, even though all students hadreceived instruction on decimals. A more recent study(Lachance and Confrey, 2002) of 5th grade students(estimated age approximately 10 years) who hadreceived an introduction to decimal fr actions in theprevious year showed that only about 43% were ableto compare decimal fractions correctly.2 Rittle-Johnson,Siegler and Alibali (2001) confirmed students’difficulties when comparing the magnitude of decimals: the rate of correct responses by thestudents (N = 73; 5th grade; mean age 11 years 8months) in their study was 19%.

In conclusion, the very basic ideas about equivalenceand order of fractions by magnitude, without which wecould hardly say that the students have a good sensefor what fractions represent, seems to elude manystudents for considerable periods of time. In thesection that follows, we will contrast two situationsthat have been used to introduce the concept offractions in primary school in order to examine thequestion of whether children’s learning may differ as a function of these differences between situations.

Children’s schemas of action in division situationsMathematics educators and researchers may notagree on many things, but there is a clear consensusamong them on the idea that r ational numbers arenumbers in the domain of quotients (Brousseau,Brousseau and Warfield, 2007; Kieren, 1988; 1993;1994; Ohlsson, 1988): that is, numbers defined by theoperation of division. So, it seems reasonable to seekthe origin of children’s understanding of rationalnumbers in their under standing of division.3

Our hypothesis is that in division situations childrencan develop some insight into the equivalence andorder of quantities in fr actions; we will use the ter mfractional quantities to refer to these quantities.These insights can be developed even in theabsence of knowledge of representations for fractions,

either in written or in oral form. Two schemes of action that children use in division ha ve beenanalysed in the liter ature: par titioning andcorrespondences (or dealing).

Behr, Harel, Post and Lesh (1992; 1993) pointed outthat fractions represent quantitites in a diff erent wayacross two types of situation. The first type is the part-whole situation. Here one star ts with a single quantity,the whole, which is divided into a cer tain number ofparts (y), out of which a specified number is taken (x);the symbol x/y represents this quantity in terms ofpart-whole relations. Partitioning is the scheme ofaction that children use in par t-whole tasks. The mostcommon type of fraction problem that teachers giveto children is to ask them to par tition a whole into afixed number of parts (the denominator) and show acertain fraction with this denominator. For example,the children have to show what 3/5 of a pizza is. 4

The second way in which fractions representquantities is in quotient situations. Here one startswith two quantities, x and y, and treats x as thedividend and y as the divisor, and by the operation ofdivision obtains a single quantity x/y. For example, thequantities could be 3 chocolates (x) to be sharedamong 5 children (y).The fractional symbol x/yrepresents both the division (3 divided by 5) and thequantity that each one will receive (3/5). A quotientsituation calls for the use of correspondences as thescheme of action: the children establishcorrespondences between por tions and recipients.The por tions may be imagined by the children, notactually drawn, as they must be when the childrenare asked to par tition a whole and show 3/5.5

When children use the scheme of par titioning inpart-whole situations, they can gain insights aboutquantities that could help them under stand someprinciples relevant to the domain of rational numbers.They can, for example, reason that, the more par tsthey cut the whole into, the smaller the par ts will be.This could help them under stand how fractions areordered. If they can achieve a higher level of precisionin reasoning about par titioning, they could develop some understanding of theequivalence of fractions: they could come tounderstand that, if they have twice as many par ts,each par t would be halved in size. For example, youwould eat the same amount of chocolate aftercutting one chocolate bar into two parts and eatingone par t as after cutting it into four par ts and eatingtwo, because the number of par ts and the size of theparts compensate for each other precisely. It is an


empirical question whether children attain theseunderstandings in the domain of whole n umbers andextend them to rational numbers.

Partitioning is the scheme that is most often used to introduce children to fractions in the UnitedKingdom, but it is not the onl y scheme of actionrelevant to division. Children use correspondencesin quotient situations when the dividend is onequantity (or measure) and the divisor is anotherquantity. For example, when children share outchocolate bars to a number of recipients, thedividend is in one domain of measures – thenumber of chocolate bar s – and the divisor is inanother domain – the number of children. Thedifference between par titioning and correspondencedivision is that in par titioning there is a single whole(i.e. quantity or measure) and in cor respondencethere are two quantities (or measures).

Fischbein, Deri, Nello and Marino (1985) hypothesisedthat children develop implicit models of divisionsituations that are related to their exper iences. Weuse their hypothesis here to explore what sorts ofimplicit models of fractions children may develop fromusing the par titioning or the correspondence schemein fractions situations. Fischbein and colleaguessuggested, for example, that children form an implicitmodel of division that has a specific constraint: thedividend must be larger than the divisor. We ourselveshypothesise that this implicit model is developed onlyin the context of par titioning. When children use thecorrespondence scheme, precisely because there aretwo domains of measures, young children readilyaccept that the dividend can be smaller than thedivisor: they are ready to agree that it is perfectlypossible to share one chocolate bar among three children.

At first glance, the difference between these twoschemes of action, par titioning and correspondence,may seem too subtle to be of interest when we arethinking of children’s understanding of fractions.Certainly, research on children’s understanding offractions has not focused on this distinction so far.However, our review shows that it is a cr ucialdistinction for children’s learning, both in terms of what insights each scheme of action aff ords and in terms of the empir ical research results.

There are at least four differences between whatchildren might learn from using the par titioningscheme or the scheme of correspondences.

• The first is the one just pointed out: that, whenchildren set two measures in correspondence, thereis no necessary relation between the size of thedividend and of the divisor. In contrast, in partitioningchildren form the implicit model that the sum of theparts must not be larger than the whole. Therefore,it may be easier for children to develop anunderstanding of improper fractions when they formcorrespondences between two fields of measuresthan when they partition a single whole. They mighthave no difficulty in understanding that 3 chocolatesshared between 2 children means that each childcould get one chocolate plus a half. In contrast, inpartitioning situations children might be puzzled ifthey are told that someone ate 3 parts of achocolate divided in 2 parts.

• A second possible difference between the twoschemes of action may be that, when usingcorrespondences, children can reach the conclusionthat the way in which partitioning is carried out doesnot matter, as long as the correspondences betweenthe two measures are ‘fair’. They can reason, forexample, that if 3 chocolates are to be shared by 2children, it is not necessary to divide all 3 chocolatesin half, and then distribute the halves; giving a wholechocolate plus a half to each child would accomplishthe same fairness in sharing. It was argued in the firstsection of this paper that this an important insight inthe domain of rational numbers: different fractionscan represent the same quantity.

• A third possible insight about quantities that canbe obtained from correspondences more easilythan from partitioning is related to ordering ofquantities. When forming correspondences,children may realize that there is an inverserelation between the divisor and the quotient: themore people there are to share a cake, the lesseach person will get: Children might achieve thecorresponding insight about this inverse relationusing the scheme of partitioning: the more partsyou cut the whole into, the smaller the parts.However, there is a difference between theprinciples that children would need to abstractfrom each of the schemes. In partitioning, theyneed to establish a within-quantity relation (themore parts, the smaller the parts) whereas incorrespondence they need to establish abetween-quantity relation (the more children, the less cake). It is an empirical matter to find outwhether or not it is easier to achieve one ofthese insights than the other.



• Finally, both partitioning and correspondencescould help children to understand somethingabout the equivalence between quantities, but thereasoning required to achieve this understandingdiffers across the two schemes of action. Whensetting chocolate bars in correspondence withrecipients, the children might be able to reasonthat, if there were twice as many chocolates andtwice as many children, the shares would beequivalent, even though the dividend and thedivisor are different. This may be easier than thecomparable reasoning in partitioning. Inpartitioning, understanding equivalence is based on inverse proportional reasoning (twice as manypieces means that each piece is half the size)whereas in contexts where children use thecorrespondence scheme, the reasoning is based ona direct proportion (twice as many chocolates andtwice as many children means that everyone stillgets the same).

This exploratory and hypothetical analysis of howchildren can reach an understanding of equivalenceand order of fractions when using par titioning orcorrespondences in division situations suggests thatthe distinction between the two schemas is worthinvestigating empirically. It is possible that the schemeof correspondences affords a smoother transitionfrom natural to rational numbers, at least as far asunderstanding equivalence and order of fr actionalquantities is concerned.

We turn now to an empir ical analysis of this question.The literature about these schemes of action is vastbut this paper focuses on research that sheds light on whether it is possib le to findcontinuities between children’s understanding ofquantities that are represented by natural numbersand fractional quantities. We review research oncorrespondences first and then research onpartitioning.

Children’s use of thecorrespondence scheme injudgements about quantitiesPiaget (1952) pioneered the study of ho w and whenchildren use the correspondence scheme to drawconclusions about quantities. In one of his studies,there were three steps in the method.

• First, Piaget asked the children to place one pinkflower into each one of a set of vases;

• next, he removed the pink flowers and asked thechildren to place a blue flower into each one of thesame vases;

• then, he set all the flowers aside, leaving on thetable only the vases, and asked the children to takefrom a box the exact number of straws required ifthey wanted to put one flower into each straw.

Without counting, and only using correspondences,five- and six-year old children were able to makeinferences about the equivalence between straws andflowers: by setting two straws in correspondence witheach vase, they constructed a set of str aws equivalentto the set of vases. Piaget concluded that the children’sjudgements were based on ‘multiplicative equivalences’(p. 219) established by the use of the cor respondencescheme: the children reasoned that, if there is a 2-to-1correspondence between flowers and vases and a 2-to-1 correspondence between straws and vases, thenumber of flowers and straws must be the same.

In Piaget’s study, the scheme of cor respondence was used in a situation that in volved ratio but notdivision. Frydman and Br yant (1988) carried out a series of studies where children establishedcorrespondences between sets in a division situationwhich we have described in more detail in Paper 2,Understanding whole numbers. The studies showedthat children aged four often shared pretend sweetsfairly, using a one-for-you one-for-me type ofprocedure. After the children had distr ibuted thesweets, Frydman and Br yant asked them to countthe number of sweets that one doll had and thendeduce the number of sweets that the other dollhad. About 40% of the four-year-olds were able tomake the necessar y inference and say the exactnumber of sweets that the second doll had; thisproportion increased with age . This result extendsPiaget’s observations that children can makeequivalence judgements not only in multiplication butalso in division problems by using correspondence.

Frydman and Br yant’s results were replicated in anumber of studies by Davis and his colleagues (Davisand Hunting, 1990; Davis and Pepper, 1992; Pitkethlyand Hunting, 1996), who refer to this scheme ofaction as ‘dealing’. They used a var iety of situations,including redistribution when a new recipient comes,to study children’s ability to use cor respondences indivision situations and to make inferences aboutequality and order of magnitude of quantities. Theyalso argue that this scheme is basic to children’ sunderstanding of fractions (Davis and Pepper, 1992).


Correa, Nunes and Br yant (1998) extended thesestudies by showing that children can make inferencesabout quantities resulting from a division not onl ywhen the divisors are the same but also when theyare different. In order to circumvent the possibilitythat children feel the need to count the sets afterdivision because they think that they could ha vemade a mistake in sharing, Bryant and his colleaguesdid not ask the children to do the shar ing: thesweets were shared by the experimenter, outside thechildren’s view, after the children had seen that thenumber of sweets to be shared was the same.

There were two conditions in this study: samedividend and same divisor versus same dividend and different divisors. In the same dividend and samedivisor condition, the children should be ab le toconclude for the equivalence between the sets thatresult from the division; in the same dividend anddifferent divisor condition, the children shouldconclude that the more recipients there are , thefewer sweets they receive; i.e. in order to answercorrectly, they would need to use the inverserelation between the divisor and the result as aprinciple, even if implicitly.

About two-thirds of the five-year-olds, the vastmajority of the six-year-olds, and all the seven-year-olds concluded that the recipients had equivalentshares when the dividend and the divisor were thesame. Equivalence was easier than the in verserelation between divisor and quotient: 34%, 53% and 81% of the children in these three age lev els,respectively, were able to conclude that the morerecipients there are, the smaller each one’s share willbe. Correa (1994) also found that children’s successin making these inferences improved if they solvedthese problems after practising sharing sweetsbetween dolls; this indicates that thinking about ho wto establish correspondences improves their abilityto make inferences about the relations between thequantities resulting from shar ing.

In all the previous studies, the dividend wascomposed of discrete quantities and was lar ger than the divisor. The next question to consider iswhether children can make similar judgements aboutequivalence when the situations involve continuousquantities and the dividend is smaller than thedivisor : that is, when children have to think aboutfractional quantities.

Kornilaki and Nunes (2005) investigated thispossibility by comparing children’s inferences in

division situations in which the quantities w erediscrete and the dividends were larger than thedivisors to their inferences in situations in which thequantities were continuous and dividends smallerthan the divisors. In the discrete quantities tasks, thechildren were shown one set of small toy fishes tobe distributed fairly among a group of white cats andanother set of fishes to be distr ibuted to a group ofbrown cats; the number of fish was always greaterthan the number of cats. In the continuous quantitiestasks, the dividend was made up of f ish-cakes, to bedistributed fairly among the cats: the number ofcakes was always smaller than the number of cats,and varied between 1 and 3 cakes, whereas thenumber of cats to receive a por tion in each groupvaried between 2 and 9. Following the paradigmdevised by Correa, Nunes and Br yant (1998), thechildren were neither asked to distr ibute the fish norto par tition the fish cakes. They were asked whether,after a fair distr ibution in each group, each cat in onegroup would receive the same amount to eat aseach cat in the other group .6

In some tr ials, the number of fish (dividend) and cats(divisor) was the same; in other tr ials, the dividendwas the same but the divisor was different. So in thefirst type of tr ials the children were asked aboutequivalence after sharing and in the second type thechildren were asked to order the quantities obtainedafter sharing.

The majority of the children succeeded in all theitems where the dividend and the divisor w ere thesame: 62% of the f ive-year-olds, 84% of the six-year-olds and all the seven-year-olds answered all thequestions correctly. When the dividend was the sameand the divisors differed, the rate of success was31%, 50% and 81%, respectively, for the three agelevels. There was no difference in the level of successattained by the children with discrete versuscontinuous quantities.

In almost all the items, the children explained theiranswers by referring to the type of relation betw eenthe dividends and the divisor s: same divisor, sameshare or, with different divisors, the more catsreceiving a share, the smaller their share . The use of numbers as an explanation for the relative size ofthe recipients’ shares was obser ved in 6% of answersby the seven-year-olds when the quantities werediscrete and less often than this b y the youngerchildren. Attempts to use numbers to speak aboutthe shares in the continuous quantities tr ials werepractically non-existent (3% of the seven-year-olds’



explanations). Thus, the analysis of justificationssupports the idea that the children w ere reasoningabout relations between quantities rather than usingcounting when they made their judgments ofequivalence or ordered the quantities that w ould beobtained after division.

This study replicated the previous f indings, which we have mentioned already, that young children canuse correspondences to make inferences aboutequivalences and also added new evidence relevantto children’s understanding of fractional quantities:many young children who have never been taughtabout fractions used correspondences to orderfractional quantities. They did so successfully whenthe division would have resulted in unitar y fractionsand also when the dividend was greater than 1 andthe result would not be a unitar y fraction (e.g. 2 fishcakes to be shared by 3, 4 or 5 cats).

A study by knowledge Mamede, Nunes and Br yant(2005) confirmed that children can make inferencesabout the order of magnitude of fractions in sharingsituations where the dividend is smaller than thedivisor (e.g. 1 cake shared by 3 children compared to 1 cake shared by 5 children). She worked withPortuguese children in their f irst year in school, whohad received no school instr uction about fractions.Their performance was only slightly weaker than thatof British children: 55% of the six-year-olds and 71%of the seven-year-olds were able to make theinference that the larger the divisor, the smaller theshare that each recipient would receive.

These studies strongly suggest that children canlearn principles about the relationship betweendividend and divisor from exper iences with shar ingwhen they establish correspondences between thetwo domains of measures, the shared quantitiesand the recipients. They also suggest that childrencan make a relatively smooth transition fromnatural numbers to rational numbers when theyuse correspondences to understand the relationsbetween quantities. This argument is central toStreefland’s (1987; 1993; 1997) hypothesis aboutwhat is the best star ting point for teachingfractions to children and has been advanced b yothers also (Davis and Pepper, 1992; Kieren, 1993;Vergnaud, 1983).

This research tell an encour aging story aboutchildren’s understanding of the logic of division ev enwhen the dividend is smaller than the divisor, butthere is one fur ther point that should be considered

in the transition between natural and rationalnumbers. In the domain of r ational numbers there isan infinite set of equivalences (e .g. 1/2 = 2/4 = 3/6etc) and in the studies that we have described so farthe children were only asked to make equivalencejudgements when the dividend and the divisor in theequivalent fractions were the same. Can they stillmake the inference of equivalence in shar ingsituations when the dividend and the divisor aredifferent across situations, but the dividend-divisorratio is the same?

Nunes, Bryant, Pretzlik, Bell, Evans and Wade (2007)asked British children aged between 7.5 and 10 years,who were in Years 4 and 5 in school, to makecomparisons between the shares that would bereceived by children in shar ing situations where thedividend and divisor were different but their ratio wasthe same. Previous research (see , for example, Behr,Harel, Post and Lesh, 1992; Kerslake, 1986) shows thatchildren in these age levels have difficulty with theequivalence of fractions. The children in this study hadreceived some instruction on fractions: they had beentaught about halves and quar ters in problems aboutpartitioning. They had only been taught about one pairof equivalent fractions: they were taught that one halfis the same as two quarters. In the correspondenceitem in this study, the children were presented withtwo pictures: in the first, a group of 4 gir ls was going toshare fairly 1 pie; in the second, a group of 8 boys wasgoing to share fair ly 2 pies that were exactly the sameas the pie that the gir ls had. The question was whethereach girl would receive the same share as each boy.The overall rate of correct responses was 73% (78%in Year 4 and 70% in Year 5; this difference was notsignificant). This is an encouraging result: the childrenhad only been taught about halves and quar ters;nevertheless, they were able to attain a high r ate ofcorrect responses for fractional quantities that couldbe represented as 1/4 and 2/8.

In the studies reviewed so far the children wereasked about quantities that resulted from divisionand always included two domains of measures; thusthe children’s correspondence reasoning wasengaged in these studies. However, they did notinvolve asking the children to represent thesequantities through fractions. The final study reviewedhere is a br ief teaching study (Nunes, Bryant, Pretzlik,Evans, Wade and Bell, 2008), where the childrenwere taught to represent fr actions in the context oftwo domains of measures, shared quantities andrecipients, and were asked about the equivalencebetween fractions. The types of arguments that the


children produced to justify the equivalence of thefractions were then analyzed and compared to theinsights that we hypothesized would emerge in thecontext of sharing from the use of thecorrespondence scheme.

Brief teaching studies are of great value in researchbecause they allow the researchers to know whatunderstandings children can construct if they aregiven a specific type of guidance in the inter actionwith an adult (Cooney, Grouws and Jones, 1988;Steffe and Tzur, 1994; Tzur, 1999; Yackel, Cobb, Wood, Wheatley and Merkel, 1990). They also havecompelling ecological validity: children spend much of their time in school tr ying to use what they havebeen taught to solve new mathematics problems.Because this study has only been published in asummary form (Nunes, Bryant, Pretzlik and Hurry,2006), some detail is presented here .

The children (N = 62) were in the age r ange from7.5 to 10 years, in Years 4 and 5 in school. Children inYear 4 had only been taught about half and quar tersand the equivalence between half and two quarters;children in Year 5 had been taught also about thirds.They worked with a researcher outside theclassroom in small groups (12 groups of betw een 4and 6 children, depending on the class size) and wereasked to solve each problem first individually, andthen to discuss their answers in the group. Thesessions were audio- and video-recorded. Thechildren’s arguments were transcribed verbatim; theinformation from the video-tapes was latercoordinated with the transcripts in order to help theresearchers understand the children’s arguments.

In this study the researcher s used problemsdeveloped by Streefland (1990). The children solvedtwo of his shar ing tasks on the f irst day and anequivalence task on the second day of the teachingstudy. The tasks were presented in booklets withpictures, where the children also wrote theiranswers. The tasks used in the first day were:

• Six girls are going to share a packet of biscuits. Thepacket is closed; we don’t know how many biscuitsare in the packet. (a) If each girl received onebiscuit and there were no biscuits left, how manybiscuits were in the packet? (b) If each girl receiveda half biscuit and there were no biscuits left, howmany biscuits were in the packet? (c) If some moregirls join the group, what will happen when thebiscuits are shared? Do the girls now receive more or less each than the six girls did?

• Four children will be sharing three chocolates. (a)Will each child be able to get one bar of chocolate?(b) Will each child be able to get at least a half barof chocolate? (c) How would you share thechocolate? (The booklets contained a picture withthree chocolate bars and four children and thechildren were asked to show how they would sharethe chocolates) Write what fraction each one gets.

After the children had completed these tasks, the researcher told them that they were going topractice writing fractions which they had not yetlearned in school. The children were asked to write‘half ’ with numerical symbols, which they knewalready. The researcher taught the children to wr itefractions that they had not yet learned in school inorder to help the children re-inter pret the meaningof fractions. The numerator was to be used torepresent the number of items to be divided, thedenominator should represent the number ofrecipients, and the dash between them two numbersshould represent the sign for division (for a discussionof children’s interpretation of fraction symbols in thissituation, see Char les and Nason, 2000, and Empson,Junk, Dominguez and Turner, 2005).

The equivalence task, presented on the second day, was:

• Six children went to a pizzeria and ordered twopizzas to share between them. The waiter broughtone first and said they could start on it because itwould take time for the next one to come. (a)How much will each child get from the first pizzathat the waiter brought? Write the fraction thatshows this. (b) How much will each child get fromthe second pizza? Write your answer. (c) If you addthe two pieces together, what fraction of a pizzawill each child get? You can write a plus signbetween the first fraction and the second fraction,and write the answer for the share each child getsin the end. (d) If the two pizzas came at the sametime, how could they share it differently? (e) Arethese fractions (the ones that the children wrotefor answers c and d equivalent?

According to the hypotheses presented in theprevious section, we would expect children todevelop some insights into r ational numbers bythinking about different ways of sharing the sameamount. It was expected that they w ould realize that:



1 it is possible to divide a smaller number by a largernumber

2 different fractions might represent the sameamount

3 twice as many things to be divided and twice asmany recipients would result in equivalent amounts

4 the larger the divisor, the smaller the quotient.

The children’s explanations for why they thoughtthat the fractions were equivalent provided evidencefor all these insights, and more, as described below.

It is possible to divide a smallernumber by a larger number There was no difficulty among the students inattempting to divide 1 pizza among 6 children. Inresponse to par t a of the equivalence prob lem, allchildren wrote at least one fr action correctly (somechildren wrote more than one fr action for the sameanswer, always correctly).

In response to par t c, when the children were askedhow they could share the 2 pizzas if both pizzascame at the same time and what fr action wouldeach one receive, some children answered 1/3 andothers answered 2/12 from each pizza, giving a totalshare of 4/12. The latter children, instead of shar ing 1 pizza among 3 gir ls, decided to cut each pizza in12 par ts: i.e. they cut the sixths in half.

Different fractions can represent thesame amountThe insight that different fractions can represent the same amount was expressed in all groups. Forexample, one child said that, ‘They’re the sameamount of people, the same amount of pizzas, andthat means the same amount of fr actions. It doesn’tmatter how you cut it.’ Another child said, ‘Because itwouldn’t really matter when they shared it, they’d getthat [3 gir ls would get 1 pizza], and then they’d getthat [3 gir ls would get the other pizza], and then itwould be the same .’ Another child said, ‘It’s the sameamount of pizza. They might be different fractionsbut the same amount [this child had offered 4/12 asan alternative to 2/6].’ Another child said: ‘Erm, wellbasically just the time doesn’t make much difference,the main thing is the number of things.’

When the dividend is twice as largeand the divisor is also twice as large,the result is an equivalent amount

The principle that when the dividend is twice aslarge and the divisor is also twice as large, the resultis an equivalent amount was expressed in 11 of the12 groups. For example, one child said, ‘It’s half thegirls and half the pizzas; three is a half of six and oneis a half of two.’ Another child said, ‘If they have twopizzas, then they could give the first pizza to threegirls and then the next one to another three gir ls.(…) If they all get one piece of that each, and theyget the same amount, they all get the same amount’.

So all three ideas we thought that could appear inthis context were expressed by the children. But two other principles, which we did not expect toobserve in this cor respondence problem, were alsomade explicit by the children.

The number of parts and size ofparts are inversely proportionalThe principle that the number of par ts and size ofparts are inversely proportional was enunciated in 8 ofthe 12 groups. For example, one child who cut thepizzas the second time around in 12 par ts each said,‘Because it’s double the one of that [total number ofpieces] and it’s double the one of that [number ofpieces for each], they cut it twice and each is half thesize; they will be the same’. Another child said, ‘Becauseone sixth and one sixth is actuall y a different way infractions [from 1 third] and it doub led [the number ofpieces] to make it [the size of the piece] littler, andhalving [the number of pieces] makes it [the size of thepiece] bigger, so I halved it and it became one third’.

The fractions show the same part-whole relationThe reasoning that the fr actions show the samepart-whole relation, which we had not expected toemerge from the use of the cor respondencescheme, was enunciated in only one group (out of12), initially by one child, and was then reiter ated bya second child in her own terms. The first child said,‘You need three two sixths to make six [6/6 – heshows the 6 pieces mar ked on one pizza], and youneed three one thirds to make three (3/3 – showsthe 3 pieces marked on one pizza). [He wrote thecomputation and continued] There’s two sixths, add


two sixths three times to make six sixths. With onethird, you need to add one third three times tomake three thirds.’

To summarize: this br ief teaching exper iment wascarried out to elicit discussions betw een thechildren in situations where they could use thecorrespondence scheme in division. The first set ofproblems, in which they are ask ed about shar ingdiscrete quantities, created a background for thechildren to use this scheme of action. Theresearchers then helped them to constr uct aninterpretation for written fractions where thenumerator is the dividend, the denominator is thedivisor, and the line indicates the operation ofdivision. This interpretation did not replace theiroriginal interpretation of number of par ts takenfrom the whole; the two meanings co-existed andappeared in the children’s arguments as theyexplained their answers. In the subsequentproblems, where the quantity to be shared wascontinuous and the dividend was smaller than thedivisor, the children had the oppor tunity toexplore the different ways in which continuousquantities can be shared. They were not asked toactually par tition the pizzas, and some made mar kson the pizzas whereas other s did not. The mostsalient feature of the children’s drawings was thatthey were not concerned with par titioning per se ,even when the par ts were marked, but with thecorrespondences between pizzas and recipients.Sometimes the correspondences were carried outmentally and expressed verbally and sometimesthe children used dr awings and gestures whichindicated the correspondences.

Other researchers have identified children’s use ofcorrespondences to solve problems that involvefractions, although they did not necessar ily use thislabel in describing the children’s answers. Empson(1999), for example, presented the followingproblem to children aged about six to sev en years(first graders in the USA): 4 children got 3 pancakesto share; how many pancakes are needed for 12children in order for the children to have the sameamount of pancake as the first group? She repor tedthat 3 children solved this problem by par titioningand 3 solved it by placing 3 pancakes incorrespondence to each group of 4 children. Similarstrategies were repor ted when children solvedanother problem that involved 2 candy bar s sharedamong 3 children.

Kieren (1993) also documented children’s use ofcorrespondences to compare fractions. In hisproblem, the fractions were not equivalent: therewere 7 recipients and 4 items in Group A and 4recipients and 2 items in Group B . The childrenwere asked how much each recipient would get in each group and whether the recipients in bothgroups would get the same amount. Kierenpresents a drawing by an eight-year-old, where the items are par titioned in half and thecorrespondences between the halves and therecipients are shown; in Group A, a line without arecipient shows that there is an extr a half in thatgroup and the child ar gues that there should beone more person in Group A for the amounts tobe the same. Kieren termed this solution‘corresponding or ‘ratiolike’ thinking’ (p. 54).

Conclusion

The scheme of correspondences develops relativelyearly: about one-third of the five-year-olds, half of six-year-olds and most seven-year-olds can usecorrespondences to make inferences aboutequivalence and order in tasks that in volve fractionalquantities. Children can use the scheme ofcorrespondences to:• establish equivalences between sets that have the

same ratio to a reference set (Piaget, 1952)• re-distribute things after having carried out one

distribution (Davis and Hunting, 1990; Davis andPepper, 1992; Davis and Pitkethly, 1990; Pitkethlyand Hunting, 1996);

• reason about equivalences resulting from divisionboth when the dividend is larger or smaller thanthe divisor (Bryant and colleagues: Correa, Nunesand Bryant, 1994; Frydman and Bryant, 1988; 1994;Empson, 1999; Nunes, Bryant, Pretzlik and Hurry,2006; Nunes, Bryant, Pretzlik, Bell, Evans and Wade,2007; Mamede, Nunes and Bryant, 2005);

• order fractional quantities (Kieren, 1993; Kornilakiand Nunes, 2005; Mamede, 2007).

These studies were carried out with children up tothe age of ten years and all of them producedpositive results. This stands in clear contr ast with theliterature on children’s difficulties with fractions andprompts the question whether the difficulties mightstem from the use of par titioning as the star tingpoint for the teaching of fr actions (see also Lamon,1996; Streefland, 1987). The next section examinesthe development of children’s par titioning action andits connection with children’s concepts of fractions.



Children’s use of the scheme of partitioning in makingjudgements about quantitiesThe scheme of par titioning has been also namedsubdivision and dissection (Pothier and Sawada,1983), and is consistently defined as the process of dividing a whole into par ts. This process isunderstood not as the activity of cutting somethinginto par ts in any way, but as a process that must beguided from the outset by the aim of obtaining apre-determined number of equal par ts.

Piaget, Inhelder and Szeminska (1960) pioneered thestudy of the connection between partitioning andfractions. They spelled out a number of ideas thatthey thought were necessary for children to developan understanding of fractions, and analysed them inpartitioning tasks. The motivation for par titioning wassharing a cake between a number of recipients, butthe task was one of par titioning. They suggested that‘the notion of fraction depends on two fundamentalrelations: the relation of par t to whole (…) and therelation of par t to par t’ (p. 309). Piaget and colleaguesidentified a number of insights that children need toachieve in order to under stand fractions:

1 the whole must be conceived as divisible, an ideathat children under the age of about tw o do notseem to attain

2 the number of par ts to be achieved is determinedfrom the outset

3 the par ts must exhaust the whole (i.e . thereshould be no second round of par titioning and noremainders)

4 the number of cuts and the number of par ts arerelated (e.g. if you want to divide something in 2parts, you should use only 1 cut)

5 all the par ts should be equal

6 each par t can be seen as a whole in itself, nestedinto the whole but also susceptible of fur therdivision

7 the whole remains invariant and is equal to thesum of the par ts.

Piaget and colleagues observed that children rarelyachieved correct partitioning (sharing a cake) beforethe age of about six years. There is variation in the

level of success depending on the shape of the whole(circular areas are more diff icult to par tition thanrectangles) and on the number of par ts. A majorstrategy in carrying out successful par titioning was theuse of successive divisions in two: so children are ableto succeed in dividing a whole into f ourths beforethey can succeed with thirds. Successive halvinghelped the children with some fr actions: dividingsomething into eighths is easier this way. However, itinterfered with success with other fr actions: somechildren, attempting to divide a whole into f ifths,ended up with sixths by dividing the whole f irst inhalves and then subdividing each half in three parts.

Piaget and colleagues also investigated children’sunderstanding of their seventh criterion for a trueconcept of fraction, i.e. the conser vation of thewhole. This conservation, they argued, would requirethe children to under stand that each piece could not be counted simply as one piece , but had to beunderstood in its relation to the whole. Somechildren aged six and even seven years failed tounderstand this, and argued that if someone ate acake cut into 1/2 + 2/4 and a second per son ate acake cut into 4/4, the second one would eat morebecause he had four par ts and the first one only hadthree. Although these children would recognise thatif the pieces were put together in each case theywould form one whole cake, they still maintainedthat 4/4 was more than 1/2 + 2/4. Finally, Piaget andcolleagues also obser ved that children did not haveto achieve the highest level of development in thescheme of par titioning in order to under stand theconservation of the whole.

Children’s difficulties with par titioning continuouswholes into equal par ts have been confirmed manytimes in studies with pre-schooler s and children intheir first years in school (e .g. Hieber t andTonnessen, 1978; Hunting and Sharpley, 1988 b,)observed that children often do not anticipate thenumber of cuts and fail to cut the whole extensiv ely,leaving a par t of the whole un-cut. These studies alsoextended our knowledge of how children’s exper tisein par titioning develops. For example, Pothier andSawada (1983) and Lamon (1996) proposed moredetailed schemes for the analysis of the developmentof par titioning schemes and other researcher s(Hiebert and Tonnessen, 1978; Hunting and Sharpley,1988 a and b; Miller, 1984; Novillis, 1976) found thatthe difficulty of par titioning discrete and continuousquantities is not the same , as hypothesized by Piaget.Children can use a procedure f or par titioningdiscrete quantities that is not applicable to


continuous quantities: they can ‘deal out’ the discretequantities but not the continuous ones. Thus theyperform significantly better with the former than thelatter. This means that the tr ansition from discrete tocontinuous quantities in the use of par titioning is notdifficult, in contrast to the smooth tr ansition noted inthe case of the cor respondence scheme.

These studies showed that the scheme ofpartitioning continuous quantities develops slowly,over a longer per iod of time. The next question toconsider is whether par titioning can promote theunderstanding of equivalence and order ing offractions once the scheme has developed.

Many studies investigated children’s understanding ofequivalence of fractions in par titioning contexts (e.g.Behr, Lesh, Post and Silver, 1983; Behr, Wachsmuth,Post and Lesh, 1984; Larson, 1980; Kerslake, 1986),but differences in the methods used in these studiesrender the comparisons between par titioning andcorrespondence studies ambiguous. For example, ifthe studies star t with a representation of thefractions, rather than with a problem aboutquantities, they cannot be compared to the studiesreviewed in the previous section, in which childrenwere asked to think about quantities withoutnecessarily using fractional representation. We shallnot review all studies but only those that usecomparable methods.

Kamii and Clark (1995) presented children withidentical rectangles and cut them into fr actions usingdifferent cuts. For example, one rectangle was cuthorizontally in half and the second was cut across adiagonal. The children had the oppor tunity to verifythat the rectangles were the same size and that thetwo parts from each rectangle were the same in size.They asked the children: if these were chocolatecakes, and the researcher ate a par t cut from the first rectangle and the child ate a par t cut from thesecond, would they eat the same amount? Thismethod is highly comparable to the studies byKornilaki and Nunes (2005) and by Mamede (2007),where the children do not have to carry out theactions, so their difficulty with par titioning does notinfluence their judgements. They also use similar lymotivated contexts, ending in the question of whetherrecipients would eat the same amount. However, thequestion posed by Kamii and Clark draws on thechild’s understanding of par titioning and the relationsbetween the par ts of the two wholes because eachwhole corresponds to a single recipient.

The children in Kamii’s study were considerably olderthan those in the cor respondence studies: they werein the fifth or sixth year in school (approximately 11and 12 years). Both groups of children had beentaught about equivalent fractions. In spite of havingreceived instruction, the children’s rate of successwas rather low: only 44% of the fifth graders and51% of the sixth gr aders reasoned that they wouldeat the same amount of chocolate cak e becausethese were halves of identical wholes.

Kamii and Clark then showed the children twoidentical wholes, cut one in fourths using a hor izontaland a vertical cut, and the other in eighths, using onlyhorizontal cuts. They discarded one fourth from thefirst ‘chocolate cake’, leaving three fourths be eaten,and asked the children to take the same amountfrom the other cake, which had been cut intoeighths, for themselves. The percentage of cor rectanswers was this time even lower: 13% of the f ifthgraders and 32% of the sixth gr aders correctlyidentified the number of eighths required to take thesame amount as three fourths.

Recently, we (Nunes and Br yant, 2004) included asimilar question about halves in a sur vey of Englishchildren’s knowledge of fractions. The children in ourstudy were in their fourth and fifth year (mean ageseight and a half and nine and a half, respectively) inschool. The children were shown pictures of a boyand a gir l and two identical rectangular areas, the‘chocolate cakes’. The boy cut his cake along thediagonal and the gir l cut hers horizontally. Thechildren were asked to indicate whether they ate thesame amount of cake and, if not, to mark the childwho ate more. Our results were more positive thanKamii and Clark’s: 55% of the children in y ear four(eight and a half year olds) in our study answeredcorrectly. However, these results are weak bycomparison to children’s rate of correct responseswhen the problem draws on their under standing ofcorrespondences. In the Kornilaki and Nunes study,100% of the seven-year-olds (third graders) realizedthat two divisions that have the same dividend andthe same divisor result in equivalent shares. Ourresults with fourth graders, when both the dividendand the divisor were different, still shows a higherrate of correct responses when correspondences areused: 78% of the fourth graders gave correct answerswhen comparing one fourth and two eighths.

In the preceding studies, the students had to thinkabout the quantities ignoring their perceptualappearance. Har t et al. (1985) and Nunes et al.



(2004) presented students with verbal questions,which did not contain drawings that could lead toincorrect conclusions based on perception. In bothstudies, the children were told that two boys hadidentical chocolate bars; one cut his into 8 par ts andate 4 and the other cut his into 4 par ts and ate 2.Combining the results of these two studies, it ispossible to see how the rate of correct responseschanged across age: 40% at ages 8 to 9 y ears, 74% at10 to11 years, 60% at 11 to 12 y ears, and 64% at 12to 13 years. The students aged 8 to 10 wereassessed by Nunes et al. and the older ones by Har tet al. These results show modest progress on theunderstanding of equivalence questions presented inthe context of par titioning even though thequantities eaten were all equivalent to half.

Mamede (2007) carried out a direct compar isonbetween children’s use of the cor respondence andthe par titioning scheme in solving equivalence andorder problems with fractional quantities. In this well-controlled study, she used stor y problems involvingchocolates and children, similar pictures andmathematically identical questions; the divisionscheme relevant to the situation was the onl yvariable distinguishing the problems. Incorrespondence problems, for example, she askedthe children: in one par ty, three gir ls are going toshare fairly one chocolate cake; in another par ty, sixboys are going to share fairly two chocolate cakes.The children were asked to decide whether eachboy would eat more than each girl, each gir l wouldeat more than each boy, or whether they wouldhave the same amount to eat. In the par titioningproblems, she set the following scenario: This girl andthis boy have identical chocolate cakes; the cakes aretoo big to eat at once so the gir l cuts her cake in 3 identical par ts and eats one and the boy cuts hiscake in 6 identical par ts and eats 2. The childrenwere asked whether the girl and the boy ate thesame amount or whether one ate more than theother. The children (age range six to seven) werePortuguese and in their f irst year in school; they hadreceived no instruction about fractions.

In the correspondence questions, the responses of35% of the six-year-olds and 49% of the seven-year-olds were correct; in the par titioning questions, 10%of the answers of children in both age lev els werecorrect. These highly significant differences suggestthat the use of cor respondence reasoning suppor tschildren’s understanding of equivalence betweenfractions whereas par titioning did not seem to affordthe same insights.

Finally, it is impor tant to compare students’ argumentsfor the equivalence and order of quantitiesrepresented by fractions in teaching studies wherepartitioning is used as the basis for teaching. Manyteaching studies that aim at promoting students’understanding of fractions through par titioning havebeen reported in the literature (e.g., Behr, Wachsmuth,Post and Lesh, 1984; Brousseau, Brousseau andWarfield, 2004; 2007; Empson, 1999; Kerslake, 1986;Olive and Steffe, 2002; Olive and Vomvoridi, 2006;Saenz-Ludlow, 1994; Steffe, 2002). In most of thesestudies, students’ difficulties with par titioning arecircumvented either by using pre-divided mater ials(e.g. Behr, Wachsmuth, Post and Lesh, 1984) or byusing computer tools where the computer carries out the division as instructed by the student (e.g. Olive and Steffe, 2002; Olive and Vomvoridi, 2006).

Many studies combine par titioning withcorrespondence during instruction, either becausethe researchers do not use this distinction (e .g.Saenz-Ludlow, 1994) or because they wish toconstruct instruction that combines both schemes in order to achieve a better instr uctional program(e.g. Brousseau, Brousseau and Warfield, 2004; 2007).These studies will not be discussed here. Two studiesthat analysed student’s arguments focus theinstruction on par titioning and are presented here .

The first study was car ried out by Berhr, Wachsmuth,Post and Lesh (1984). The researchers used objectsof different types that could be manipulated dur inginstruction (e.g. counters, rectangles of the same siz eand in different colours, pre-divided into fractionssuch as halves, quar ters, thirds, eighths) but alsotaught the students how to use algor ithms (divisionof the denominator by the numerator to find aratio) to check on the equivalence of fr actions. Thestudents were in fourth grade (age about 9) andreceived instruction over 18 weeks. Behr et al.provided a detailed analysis of children’s argumentsregarding the ordering of fractions. In summary, theyreport the following insights after instr uction.

• When ordering fractions with the same numeratorand different denominators, students seem to be ableto argue that there is an inverse relation between thenumber of parts into which the whole was cut andthe size of the parts. This argument appears eitherwith explicit reference to the numerator (‘there aretwo pieces in each, but the pieces in two fifths aresmaller.’ p. 328) or without it (‘the bigger the numberis, the smaller the pieces get.’ p. 328).


• A third fraction can be used as a reference pointwhen two fractions are compared: three ninths isless than three sixths because ‘three ninths is …less than half and three sixths is one half ’ (p. 328). It is not clear how the students had learned that3/6 and 1/2 are equivalent but they can use thisknowledge to solve another comparison.

• Students used the ratio algorithm to verify whetherthe fractions were equivalent: 3/5 is not equivalentto 6/8 because ‘if they were equal, three goes intosix, but five doesn’t go into eight.’ (p. 331).

• Students learned to use the manipulative materialsin order to carry out perceptual comparisons: 6/8equals 3/4 because ‘I started with four parts. Then Ididn’t have to change the size of the paper at all. Ijust folded it, and then I got eight.’ (p. 331).

Behr et al. report that, after 18 weeks of instruction,a large proportion of the students (27%) continuedto use the manipulatives in order to car ry ourperceptual comparisons; the same propor tion (27%)used a third fraction as a reference point and asimilar proportion (23%) used the r atio algorithmthat they had been taught to compare fr actions.

Finally, there is no evidence that the students w ereable to understand that the number of par ts and sizeof par ts could compensate for each other precisely in a propor tional manner. For example, in thecomparison between 6/8 and 3/4 the students couldhave argued that there were twice as many par tswhen the whole was cut into 8 par ts in comparisonwith cutting in 4 par ts, so you need to take twice asmany (6) in order to have the same amount.

In conclusion, students seemed to develop someinsight into the inverse relation between the divisorand the quantity but this only helped them when thedividend was kept constant: they could not extendthis understanding to other situations where thenumerator and the denominator differed.

The second set of studies that f ocused onpartitioning was carried out by Steffe and hiscolleagues (Olive and Steffe, 2002; Olive andVomvoridi, 2006; Steffe, 2002). Because the aim ofmuch of the instruction was to help the childrenlearn to label fractions or compose fractions thatwould be appropriate for the label, it is not possib leto extract from their repor ts the children’sarguments for equivalence of fractions.

However, one of the protocols (Olive and Steffe,2002) provides evidence for the student’s difficultywith improper fractions, which, we hypothesise, couldbe a consequence of using par titioning as the basisfor the concept of fr actions. The researcher asked Joeto make a stick 6/5 long. Joe said that he could notbecause ‘there are only five of them’. After prompting,Joe physically adds one more f ifth to the five alreadyused, but it is not clear whether this ph ysical actionconvinces him that 6/5 is mathematicall y appropriate.In a subsequent example , Joe labels a stick made with9 sticks, which had been def ined as ‘one seventh’ ofan original stick, 9/7, but according to the researcher s‘an important perturbation’ remains. Joe later counts8 of a stick that had been designated as ‘one seventh’but doesn’t use the label ‘eight sevenths’. When theresearcher proposes this label, he questions it: ‘Howcan it be EIGHT sevenths?’ (Olive and Steffe, 2002, p. 426). Joe later refused to make a stick that is 10/7,even though the procedure is physically possible.Subsequently, on another day, Joe’s reaction toanother improper fraction is: ‘I still don’t under standhow you could do it. How can a fraction be biggerthan itself? (Olive and Steffe, 2002, p. 428; emphasis in the original).

According to the researchers, Joe only sees thatimproper fractions are acceptable when theypresented a problem where pizzas were to be sharedby people. When 12 friends ordered 2 slices each ofpizzas cut into 8 slices, Joe realized immediately thatmore than one pizza would be required; the traditionalpartitioning situation, where one whole is divided intoequal parts, was transformed into a less usual one ,where two wholes are required but the size of thepart remains fixed.

This example illustrates the difficulty that studentshave with improper fractions in the context ofpartitioning but which they can overcome bythinking of more than one whole .

Conclusion

Partitioning, defined as the action of cutting a whole into a pre-determined number of equal par ts,shows a slower developmental process thancorrespondence. In order for children to succeed,they need to anticipate the solution so that the r ightnumber of cuts produces the right number of equalparts and exhausts the whole . Its accomplishment,however, does not seem to produce immediateinsights into equivalence and order of fr actional



quantities. Apparently, many children do not see it asnecessary that halves from two identical wholes areequivalent, even if they have been taught about theequivalence of fractions in school.

In order to use this scheme of action as the basis for learning about fractions, teaching schemes andresearchers rely on pre-cut wholes or computertools to avoid the difficulties of accurate par titioning.Students can develop insight into the inverse relationbetween the number of par ts and the size of theparts through the par titioning scheme but there isno evidence that they realize that if you cut a wholein twice as many par ts each one will be half in siz e.Finally, improper fractions seem to cause uneasinessto students who have developed their conception offractions in the context of par titioning; it is impor tantto be aware of this uneasiness if this is the schemechosen in order to teach fr actions.

Rational numbers and children’sunderstanding of intensivequantitiesIn the introduction, we suggested that rationalnumbers are necessar y to represent quantities thatare measured by a relation between two otherquantities. These are called intensive quantities andthere are many examples of such quantities both in everyday life and in science . In everyday life, weoften mix liquids to obtain a cer tain taste. If you mix fruit concentrate with water to make juice, theconcentration of this mixture is descr ibed by arational number: for example, 1/3 concentrate and2/3 water. Probability is an intensive quantity that isimportant both in mathematics and science and ismeasured as the number of favourable cases dividedby the number of total cases.7

The conceptual difficulties involved in understandingintensive quantities are largely similar to thoseinvolved in understanding the representation ofquantities that are smaller than the whole. In order to understand intensive quantities, students must form a concept that takes two variablessimultaneously into account and realise that there isan inverse relation between the denominator andthe quantity represented.

Piaget and Inhelder descr ibed children’s thinkingabout intensive quantities as one of the manyexamples of the development of the scheme ofproportionality, which they saw as one of the

hallmarks of adolescent thinking and f ormaloperations. They devoted a book to the anal ysis ofchildren’s understanding of probabilities (Piaget andInhelder, 1975) and descr ibed in great detail thesteps that children take in order to understand thequantification of probabilities. In the mostcomprehensive of their studies, the children wereshown pairs of decks of cards with diff erent numbersof cards, some marked with a cross and other sunmarked. The children were asked to judge whichdeck they would choose to draw from if theywanted to have a better chance of drawing a card marked with a cross.

Piaget and Inhelder obser ved that many of theyoung children treated the number of marked andunmarked cards as if they were independent:sometimes they chose one deck because it hadmore marked cards than the other and sometimesthey chose a deck because it had f ewer blank cardsthan the other. This approach can lead to cor rectresponses when either the number of marked cardsor the number of unmarked cards is the same inboth decks, and children aged about seven yearswere able to make correct choices in such problems.This is rather similar to the obser vations of children’ssuccesses and difficulties in comparing fractionsreported earlier on: they can reach the cor rectanswer when the denominator is constant or whenthe numerator is constant, as this allow them tofocus on the other value . When they must think ofdifferent denominators and numerators, thequestions become more difficult.

Around the age of nine , children star ted makingcorrespondences between marked and unmarkedcards within each deck and were able to identifyequivalences using this type of procedure . Forexample, if asked to compare a deck with onemarked and two unmarked cards (1/3 probability)with another deck with two marked and fourunmarked cards (2/6 probability), the children wouldre-organise the second deck in two lots, setting onemarked card in correspondence with two unmarked,and conclude that it did not mak e any differencewhich deck they picked a card from. Piaget andInhelder saw these as empir ical proportionalsolutions, which were a step towards the abstractionthat characterises proportional reasoning.

Noelting (1980 a and b) replicated these results withanother intensive quantity, the taste of or ange juicemade from a mixture of concentr ate and water. Inbroad terms, he described children’s thinking and its


development in the same way as Piaget and Inhelderhad done. This is an impor tant replication of Piaget’sresults results considering that the content of theproblems differed marked across the studies,probability and concentration of juice.

Nunes, Desli and Bell (2003) compared students’ability to solve problems about extensive andintensive quantities that involved the same type ofreasoning. Extensive quantities can be representedby a single whole number (e.g. 5 kilos, 7 cows, 4days) whereas intensive quantities are representedby a ratio between two numbers. In spite of thesedifferences, it is possible to create problems whichare comparable in other aspects but differ withrespect to whether the quantities are extensiv e orintensive. Intensive quantities problems alwaysinvolve three var iables. For example , three var iablesmight be amount of or ange concentrate, amount ofwater, and the taste of the or ange juice, which isthe intensive quantity. The amount of orangeconcentrate is directly related to how orangey thejuice tastes whereas the amount of water isinversely related to how orangey the juice tastes. A comparable extensive quantities problem wouldinvolve three extensive quantities, with the oneunder scrutiny being inversely propor tional to oneof the var iables and directly propor tional to theother. For example , the number of days that thefood bought by a farmer lasts is directlyproportional to the amount of f ood purchased andinversely propor tional to the number of animalsshe has to feed. In our study, we analysed students’performance in compar ison problems where theyhad to consider either intensive quantities (e .g. howorangey a juice would taste) or extensive quantities(e.g. the number of days the farmer’s food supplywould last). Students performed significantly betterin the extensive quantities problems even thoughboth types of problem involved propor tionalreasoning and the same number of var iables. So,although the difficulties shown by children acrossthe two types of problem are similar, their level ofsuccess was higher with extensive than intensivequantities. This indicates that students f ind it difficultto form a concept where two variables must becoordinated into a single constr uct,8 and thereforeit may be impor tant for schools and teachers toconsider how they might promote thisdevelopment in the classroom.

We shall not review the lar ge literature on intensivequantities here (see, for example, Erickson, 1979;Kaput, 1985; Schwar tz, 1988; Stavy, Strauss, Orpaz

and Carmi, 1982; Stavy and Tirosh, 2000), but there islittle doubt that students’ difficulties in understandingintensive quantities are very similar to those thatthey have when thinking about fr actions whichrepresent quantities smaller than the unit. They treatthe values independently, they find it difficult to thinkabout inverse relations, and they might think of therelations between the numbers as additive instead ofmultiplicative.

There is presently little information to indicatewhether students can transfer what they havelearned about fractions in the context ofrepresenting quantities smaller than the unit to therepresentation and understanding of intensivequantities. Brousseau, Brousseau, and Warfield (2004)suggest both that teacher s believe that students willeasily go from one use of fr actions to another, andthat nonetheless the differences between these twotypes of situation could actuall y result in interferencerather than in easy tr ansfer of insights acrosssituations. In contrast, Lachance and Confrey (2002)developed a curriculum for teaching third gradestudents (estimated age about 8 years) about ratiosin a variety of problems, including intensive quantitiesproblems, and then taught the same students infourth grade (estimated age about 9 years) aboutdecimals. Their hypothesis is that students wouldshow positive transfer from learning about ratios tolearning about decimals. They claimed that theirstudents learned significantly more about decimalsthan students who had not par ticipated in a similarcurriculum and whose performance in the samequestions had been descr ibed in other studies.

We believe that it is not possible at the moment toform clear conclusions on whether knowledge offractions developed in one type of situation tr ansferseasily to the other, shows no transfer, or actuallyinterferes with learning about the other type ofsituation. In order to settle this issue, we must carry out the appropr iate teaching studies andcomparisons.

However, there is good reason to conclude that theuse of rational numbers to represent intensivequantities should be explicitly included in thecurriculum. This is an impor tant concept in everydaylife and science, and causes difficulties for students.



Learning to use mathematicalprocedures to determine theequivalence and order of rational numbersPiaget’s (1952) research on children’s understandingof natural numbers shows that young children, agedabout four, might be able to count two sets ofobjects, establish that they have the same number,and still not conclude that they are equivalent if thesets are displayed in very different perceptualarrangements. Conversely, they might establish theequivalence between two sets by placing theirelements in correspondence and, after counting theelements in one set, be unable to infer what thenumber in the other set is (Piaget, 1952; Frydmanand Bryant, 1988). As we noted in Paper 2,Understanding whole numbers, counting is aprocedure for creating equivalent sets and placingsets in order but many young children who knowhow to count do not use counting when ask ed tocompare or create equivalent sets (see, for example,Michie, 1984, Cowan and Daniels, 1989; Cowan,1987; Cowan, Foster and Al-Zubaidi, 1993; Saxe,Guberman and Gearhar t, 1987).

Procedures to establish the equivalence and order of fractional quantities are much more complex thancounting, par ticularly when both the denominatorand the numerator differ. Students are taughtdifferent procedures in different countries. Theprocedure that seems most commonly taught inEngland is to check the equivalence b y analysing the multiplicative relation between or within thefractions. For example, when comparing 1/3 with4/12, students are taught to f ind the factor thatconnects the numerators (1 and 4) and then applythe same factor to the denominators. If thenumerator and the denominator of the secondfraction are the product of the n umerator and thedenominator of the first fraction by the samenumber, 4 in this case , they are equivalent. Analternative approach is to f ind whether themultiplicative relation between the numerator and the denominator of each fraction is the same (3 in this case): if it is, the fractions are equivalent.

If students learned this procedure and applied itconsistently, it should not matter whether the factoris, for example, 2, 3 or 5, because these are well-known multiplication associations. It should also notmatter whether the fraction with larger numeratorand denominator is the f irst or the second. However,research shows that these var iations affect students’

performance. Har t et al. (1985) presented studentswith the task of identifying the missing values inequivalent fractions. The children were presentedwith the item below and asked which numbersshould replace the square and the tr iangle:

2/7 = �/14 = 10/�

The rate of correct responses by 11 to 12 and 12- to13-year-olds for the second question was about halfthat for the first one: about 56% for the first questionand 24% for the second. The within-fraction methodcannot be easily applied in these cases but the factorsare 2 and 5, and these multiplication tables should bequite easy for students at this age level.

We recently replicated these different levels ofdifficulty in a study with 8- to 10-year-olds. Theeasiest questions were those where the commonfactor was 2; the rates of correct responses for 1/3 = 2/� and 6/8 = 3/� were 52% and 45%,respectively. The most difficult question was 4/12 = 1/� this was only answered correctly by 16% of the students. It is unlikely that the difficulty ofcomputation could explain the differences inperformance: even weak students in this age rangeshould be able to identify 3 as the factor connecting4 and 12, if they had been taught the within-fr actionmethod, or 4 as the factor connecting 1 and 4, ifthey were taught the between-fractions method.

A noteworthy aspect of our results was the lo wcorrelations between the different items: althoughmost were significant (due to the lar ge sample size;N = 188), only two of the nine cor relations wereabove .4. This suggests that the students were notable to use the procedure that they learnedconsistently to solve five items that had the sameformat and could be solved by the same procedure .

Our assessment, like the one by Har t et al. (1985),also included an equivalence question set in thecontext of a story: two boys have identical chocolatebars, one cuts his into 8 equal par ts and eats 4 andthe other cuts his into 4 equal par ts and eats 2; thechildren are asked to indicate whether the boys eatthe same amount of chocolate and, if not, who eatsmore. This item is usually seen as assessing children’sunderstanding of quantities as it is not expressed infraction terms. In our sample , no student wrote thefractions 4/8 and 2/4 and compared them by meansof a procedure. We analysed the correlationsbetween this item and the f ive items described inthe previous paragraph. If the students used the


same reasoning or the same procedure to solv e theitems, there should be a high cor relation betweenthem. This was not so: the highest of the cor relationsbetween this item and each of the f ive previous oneswas 0.32, which is low. This result exemplifies theseparation between understanding fractionalquantities and knowledge of procedures in thedomain of rational numbers. This is much the sameas observed in the domain of natur al numbers.

It is possible that understanding the relations betweenquantities gives students an advantage in lear ning theprocedures to establish the equivalence of fractions,but it may not guarantee that they will actually learn itif teachers do connect their under standing with theprocedure. When we separated the students into twogroups, one that answered the question about theboys and the chocolates correctly and the other thatdid not, there was a highly significant differencebetween the two groups in the r ate of correctresponses in the procedural items: the group whosucceeded in the chocolate question showed 38%correct responses to the procedur al items whereasthe group who failed only answered 18% of theprocedural questions correctly.

A combination of longitudinal and inter ventionstudies is required to clar ify whether students whounderstand fractional quantities benefit more whentaught how to represent and compare fr actions.There are presently no studies to clarify this matter.

Research that analyses students’ knowledge ofprocedures used to find equivalent fractions and itsconnection with conceptual knowledge of fractionshas shown that there can be discrepancies betw eenthese two forms of knowledge. Rittle-Johnson, Sieglerand Alibali (2001) argued that procedural andconceptual knowledge develop in tandem butKerslake (1986) and Byrnes and colleagues (Byrnes,1992; Byrnes and Wasik, 1991), among others,identified clear discrepancies between students’conceptual and procedural knowledge of fractions.Recently we (Hallett, Nunes and Bryant, 2007)analysed a large data set (N = 318 children in Years4 and 5) and obser ved different profiles of relativeperformance in items that assess knowledge ofprocedures to compare fractions and understandingof fractional quantities. Some children show greatersuccess in procedural questions than would beexpected from their performance in conceptualitems, others show better performance in conceptualitems than expected from their perf ormance in theprocedural items, and still other s do not show any

discrepancy between the two. Thus, some studentsseem to learn procedures for finding equivalentfractions without an understanding of why theprocedures work, others base their approach tofractions on their under standing of quantities withoutmastering the relevant procedures, and yet othersseem able to co-ordinate the two forms ofknowledge. Our results show that the third group ismore successful not only in a test about fr actions butalso in a test about intensive quantities, which didnot require the use of fractions in the representationof the quantities.

Finally, we ask whether students are better at usingprocedures to compare decimals than to compareordinary fractions. The students in some of the gr adelevels studied by Resnick and colleagues (1989)would have been taught how to add and subtractdecimals: they were in grades 5 and 6 (the estimatedage for U.S. students is about 10 and 11 years) andone of the ear ly uses of decimals in the cur riculumin the three par ticipating countries is addition andsubtraction of decimals. When students are taught toalign the decimal numbers by placing the decimalpoints one under the other before adding – forexample, when adding 0.8, 0.26 and 0.361 you needto align the decimal points before carrying out theaddition – they may not realise that they are using aprocedure that automatically converts the values tothe same denominator : in this case, x/1000. It ispossible that students may use this procedure ofaligning the decimal point without full y understandingthat this is a conversion to the same denominatorand thus that it should help them to compare thevalue of the fractions: after learning how to add andsubtract with decimals, they may still think that 0.8 isless than 0.36 but probably would not have said that0.80 is less than 0.36.

To conclude, we find in the domain of r ationalnumbers a similar separation betweenunderstanding quantities and learning to operatewith representations when judging the equivalenceand order of magnitude of quantities. Students aretaught procedures to test whether fr actions areequivalent but their knowledge of these proceduresis limited, and they do not appl y it across itemsconsistently. Similarly, students who solveequivalence problems in context are not necessar ilyexperts in solving problems when the fr actions arepresented without context.

The significance of children’s difficulties inunderstanding equivalence of fractions cannot be



overstressed: in the domain of r ational numbers,students cannot learn to add and subtr act withunderstanding if they do not realise that fractionsmust be equivalent in order to be added. Adding 1/3and 2/5 without transforming one of these into anequivalent fraction with the same denominator asthe other is like adding bananas and tins of soup: itmakes no sense. Above and beyond the fact thatone cannot be said to understand numbers withoutunderstanding their equivalence and order, in thedomain of rational numbers equivalence is a coreconcept for computing addition and subtr action.Kerslake (1986) has shown that students learn toimplement the procedures for adding andsubtracting fractions without having a glimpse at why they convert the fractions into commondenominators first. This separation between themeaning of fractions and the procedures cannotbode well for the future of these lear ners.

Conclusions and educationalimplications• Rational numbers are essential for the

representation of quantities that cannot berepresented by a single natural number. For thisreason, they are needed in everyday life as well asscience, and should be part of the curriculum in theage range 5 to 16.

• Children learn mathematical concepts by applyingschemes of action to problem solving and reflectingabout them. Two types of action schemes areavailable in division situations: partitioning, whichinvolves dividing a whole into equal parts, andcorrespondence situations, where two quantities(or measures) are involved, a quantity to be sharedand a number of recipients of the shares.

• Children as young as five or six years in age arequite good at establishing correspondences toproduce equal shares, whereas they experiencemuch difficulty in partitioning continuous quantities.Reflecting about these schemes and drawing insightsfrom them places children in different paths forunderstanding rational number. When they use thecorrespondence scheme, they can achieve someinsight into the equivalence of fractions by thinkingthat, if there are twice as many things to be sharedand twice as many recipients, then each one’s shareis the same. This involves thinking about a directrelation between the quantities. The partitioningscheme leads to understanding equivalence in a

different way: if a whole is cut into twice as manyparts, the size of each part will be halved. Thisinvolves thinking about an inverse relation betweenthe quantities in the problem. Research consistentlyshows that children understand direct relationsbetter than inverse relations.

• There are no systematic and controlledcomparisons to allow for unambiguous conclusionsabout the outcomes of instruction based oncorrespondences or partitioning. The availableevidence suggests that testing this hypothesisappropriately could result in more successfulteaching and learning of rational numbers.

• Children’s understanding of quantities is oftenahead of their knowledge of fractionalrepresentations when they solve problems usingthe correspondence scheme. Schools could makeuse of children’s informal knowledge of fractionalquantities and work with problems aboutsituations, without requiring them to use formalrepresentations, to help them consolidate thisreasoning and prepare them for formalization.

• Research has identified the arguments that childrenuse when comparing fractions and trying to seewhether they are equivalent or to order them bymagnitude. It would be important to investigatenext whether increasing teachers’ awareness ofchildren’s own arguments would help teachersguide children’s learning more effectively.

• In some countries, greater attention is given todecimal representation than to ordinary fractionsin primary school whereas in others ordinaryfractions continue to play an important role. Theargument that decimals are easier to understandthan ordinary fractions does not find support insurveys of students’ performance: students find itdifficult to make judgements of equivalence andorder both with decimals and with ordinaryfractions.

• Some researchers (e.g. Nunes, 1997; Tall, 1992;Vergnaud, 1997) argue that differentrepresentations shed light onto the same conceptsfrom different perspectives. This would suggest that a way to strengthen students’ learning ofrational numbers is to help them connect bothrepresentations. Moss and Case (1999) analysedthis possibility in the context of a curriculum basedon measurements, where ordinary fractions andpercentages were used to represent the same


information. Their results are encouraging, but thestudy does not include the appropriate controlsthat would allow for establishing firmer conclusions.

• Students can learn procedures for comparing,adding and subtracting fractions without connectingthese procedures with their understanding ofequivalence and order of fractional quantities,independently of whether they are taught withordinary or decimal representation. This is not adesired outcome of instruction, but seems to be aquite common one. Research that focuses on theuse of children’s informal knowledge suggests that itis possible to help students make connections (e.g.Mack, 1990), but the evidence is limited. There isnow considerably more information regardingchildren’s informal strategies to allow for newteaching programmes to be designed and assessed.

• Finally, this review opens the way for a freshresearch agenda in the teaching and learning offractions. The source for the new researchquestions is the finding that children achieve insightsinto relations between fractional quantities beforeknowing how to represent them. It is possible toenvisage a research agenda that would not focuson children’s misconceptions about fractions, buton children’s possibilities of success when teachingstarts from thinking about quantities rather thanfrom learning fractional representations.

Endnotes

1 Rational numbers can also be used to represent relations thatcannot be described by a single whole number but therepresentation of relations will not be discussed here .

2 The authors repor t a successful programme of instructionwhere they taught the students to estab lish connectionsbetween their understanding of ratios and decimals. Thestudents had received two years of instruction on ratios. A fulldiscussion of this very interesting work is not possible here asthe information provided in the paper is insufficient.

3 There are different hypotheses regarding what types ofsubconstructs or meanings for rational numbers should bedistinguished (see, for example, Behr, Harel, Post and Lesh,1992; Kieren, 1988) and how many distinctions are justifiable.Mathematicians and psychologists may well use differentcriteria and consequently reach different conclusions.Mathematicians might be looking for conceptual issues inmathematics and psychologists for distinctions that have animpact on children’s learning (i.e. show different levels ofdifficulty or no transfer of learning across situations). We havedecided not to pursue this in detail but will consider thisquestion in the final section of the paper.

4 Steffe and his colleagues have used a different type of problem,where the size of the par t is fixed and the children have toidentify how many times it fits into the whole .

5 This classification should not be confused with the classif icationof division problems in the mathematics education liter ature.Fischbein, Deri, Nello and Mar ino (1985) define par titivedivision (which they also term sharing division) as a model f orsituations in which ‘an object or collection of objects is dividedinto a number of equal fragments or sub-collections. Thedividend must be larger than the divisor ; the divisor (operator)must be a whole number; the quotient must be smaller thanthe dividend (operand)… In quotative division or measurementdivision, one seeks to determine how many times a givenquantity is contained in a lar ger quantity. In this case , the onlyconstraint is that the dividend must be larger than the divisor. Ifthe quotient is a whole number, the model can be seen asrepeated subtraction.’ (Fischbein, Deri, Nello and Mar ino, 1985,p.7). In both types of prob lems discussed by Fishbein et al., thescheme used in division is the same , par titioning, and thesituations are of the same type, par t-whole.

6 Empson, Junk, Dominguez and Turner (2005) have stressed that‘the depiction of equal shares of, for example, sevenths in apart–whole representation is not a necessar y step tounderstanding the fraction 1/7 (for contrasting views, seeCharles and Nason, 2000; Lamon, 1996; Pothier and Sawada,1983). What is necessar y, however, is understanding that 1/7 isthe amount one gets when 1 is divided into 7 same-siz ed par ts.’

7 Not all intensive quantities are represented by fractions;speed, for example, is represented by a ratio, such as in 70miles per hour.

8 Vergnaud (1983) proposed this hypothesis in his compar isonbetween isomorphism of measures and product of measuresproblems. This issue is discussed in greater detail in anotherpaper 4 of this review.



Behr, M., Harel, G., Post, T., and Lesh, R. (1992).Rational number, ratio, proportion. In D. A.Grouws (Ed.), Handbook of Research onMathematics Teaching and Learning (pp. 296-333).New York: Macmillan.

Behr, M. J., Harrel, G., Post, R., & Lesh, R. (1993).Toward a unified discipline of scientific enquiry. InT. Carpenter, P. E. Fennema & T. A. Romberg (Eds.),Rational Numbers: an integration of research (pp.13-48). Hillsdale, NJ: Lawrence Erlbaum Ass.

Behr, M., Lesh, R., Post, T., and Silver, E. A. (1983).Rational number concepts. In R. Lesh and M.Landau (Eds.), Acquisition of MathematicalConcepts and Processes (pp. 91-126). New York:Academic Press.

Behr, M. J., Wachsmuth, I., Post, T. R., and Lesh, R. (1984).Order and equivalence of r ational numbers: Aclinical teaching experiment. Journal for Research inMathematics Education, 15(5), 323-341.

Brousseau, G., Brousseau, N., and Warfield, V. (2004).Rationals and decimals as required in the schoolcurriculum Part 1: Rationals as measurement.Journal of Mathematical Behavior, 23, 1–20.

Brousseau, G., Brousseau, N., and Warfield, V.(2007). Rationals and decimals as required inthe school curriculum Part 2: From rationals to decimals. Journal of Mathematical Behavior 26 281–300.

Booth, L. R. (1981). Child-methods in secondarymathematics. Educational Studies in Mathematics ,12 29-41.

Byrnes, J. P. (1992). The conceptual basis ofprocedural learning. Cognitive Development, 7,235-237.

Byrnes, J. P. and Wasik, B. A. (1991). Role ofconceptual knowledge in mathematicalprocedural learning. Developmental Psychology,27(5), 777-786.

Carpenter, T. P., and Moser, J. M. (1982). Thedevelopment of addition and subtractionproblem solving. In T. P. Carpenter, J. M. Moser andT. A. Romberg (Eds.), Addition and subtraction: Acognitive perspective (pp. 10-24). Hillsdale (NJ):Lawrence Erlbaum.

Charles, K. and Nason, R. (2000). Young children’spartitioning strategies. Educational Studies inMathematics, 43, 191–221.

Cooney, T. J., Grouws, D. A., and Jones, D. (1988). Anagenda for research on teaching mathematics. InD. A. Grouws and T. J. Cooney (Eds.), Effectivemathematics teaching (pp. 253-261). Reston, VA:National Council of Teachers of Mathematics andHillsdale, NJ: Erlbaum.

Correa, J. (1994). Young children’s understanding of thedivision concept. University of Oxford, Oxford.

Correa, J., Nunes, T., and Br yant, P. (1998). Youngchildren’s understanding of division: Therelationship between division terms in anoncomputational task. Journal of EducationalPsychology, 90, 321-329.

Cowan, R., and Daniels, H. (1989). Children’s use ofcounting and guidelines in judging relative number.British Journal of Educational Psychology, 59, 200-210.

Cowan, R. (1987). When do children trust countingas a basis for relative number judgements? Journalof Experimental Child Psychology, 43, 328-345.

Cowan, R., Foster, C. M., and Al-Zubaidi, A. S. (1993).Encouraging children to count. British Journal ofDevelopmental Psychology, 11, 411-420.

Davis, G. E., and Hunting, R. P. (1990). Spontaneouspartitioning: Preschoolers and discrete items.Educational Studies in Mathematics, 21, 367-374.

Davis, G. E., and Pepper, K. L. (1992). Mathematicalproblem solving by pre-school children.Educational Studies in Mathematics, 23, 397-415.

Davis, G. E., and Pitkethly, A. (1990). Cognitive aspectsof sharing. Journal for Research in MathematicsEducation, 21, 145-153.


References

Empson, S. B. (1999). Equal sharing and sharedmeaning: The development of fraction concepts ina First-Grade classroom. Cognition and Instruction,17(3), 283-342.

Empson, S. B., Junk, D., Dominguez, H., and Turner, E.(2005). Fractions as the coordination ofmultiplicatively related quantities: a cross-sectionalstudy of children’s thinking. Educational Studies inMathematics, 63, 1-28.

Erickson, G. (1979). Children’s conceptions of heatand temperature. Science Education, 63, 221-230.

Fennell, F. S., Faulkner, L. R., Ma, L., Schmid, W., Stotsky,S., Wu, H.-H., et al. (2008). Report of the TaskGroup on Conceptual Knowledge and SkillsWashington DC: U.S. Depar tment of Education,The Mathematics Advisory Panel.

Fischbein, E., Deri, M., Nello, M., and Marino, M. (1985).The role of implicit models in solving v erbalproblems in multiplication and division. Journal forResearch in Mathematics Education, 16, 3-17.

Frydman, O., and Br yant, P. E. (1988). Sharing and theunderstanding of number equivalence by youngchildren. Cognitive Development, 3, 323-339.

Frydman, O., and Br yant, P. (1994). Children’sunderstanding of multiplicative relationships in theconstruction of quantitative equivalence. Journal ofExperimental Child Psychology, 58, 489-509.

Ginsburg, H. (1977). Children’s Arithmetic: The LearningProcess. New York: Van Nostrand.

Gréco, P. (1962). Quantité et quotité: nouvellesrecherches sur la cor respondance terme-a-termeet la conser vation des ensembles. In P. Gréco andA. Morf (Eds.), Structures numeriques elementaires:Etudes d’Epistemologie Genetique Vol 13 (pp. 35-52). Paris: Presses Universitaires de France.

Hart, K. (1986). The step to formalisation. Insititute ofEducation, University of London: Proceedings ofthe 10th International Conference of thePsychology of Mathematics Education, pp 159-164.

Hart, K., Brown, M., Kerslake, D., Kuchermann, D., &Ruddock, G. (1985). Chelsea DiagnosticMathematics Tests. Fractions 1. Windsor (UK):NFER-Nelson.

Hallett, D., Nunes, T., and Br yant, P. (2007). Individualdifferences in conceptual and procedur alknowledge when learning fractions. under review.

Hiebert, J., and Tonnessen, L. H. (1978). Developmentof the fraction concept in two physical contexts:An exploratory investigation. Journal for Researchin Mathematics Education, 9(5), 374-378.

Hunting, R. P. and Sharpley, C. F. (1988 a).Preschoolers’ cognition of fractional units. BritishJournal of Educational Psychology, 58, 172-183.

Hunting, R. P. and Sharpley, C. F. (1988 b). Fractionalknowledge in preschool children. Journal forResearch in Mathematics Education, 19(2), 175-180.

Kamii, C., and Clark, F. B. (1995). Equivalent Fractions:Their Difficulty and Educational Implications.Journal of Mathematical Behavior, 14, 365-378.

Kaput, J. (1985). Multiplicative word problems andintensive quantities: An integrated software response(Technical Report 85-19). Cambridge (MA):Harvard University, Educational Technology Center.

Kerslake, D. (1986). Fractions: Children’s Strategies andErrors: A Report of the Strategies and Errors inSecondary Mathematics Project. Windsor: NFER-Nelson.

Kieren, T. (1988). Personal knowledge of rationalnumbers: Its intuitive and formal development. InJ. Hiebert and M. Behr (Eds.), Number conceptsand operations in the middle-grades (pp. 53-92).Reston (VA): National Council of Teachers ofMathematics.

Kieren, T. E. (1993). Rational and Fractional Numbers:From Quotient Fields to RecursiveUnderstanding. In T. Carpenter, E. Fennema and T.A. Romberg (Eds.), Rational Numbers: AnIntegration of Research (pp. 49-84). Hillsdale, N.J.:Lawrence Erlbaum Associates.

Kieren, T. (1994). Multiple Views of MultiplicativeStructures. In G. Harel and J. Confrey (Eds.), TheDevelopment of Multiplicative Reasoning in theLearning of Mathematics (pp. 389-400). Albany,New York: State University of New York Press.

Kornilaki, K., and Nunes, T. (2005). GeneralisingPrinciples in spite of Procedur al Differences:Children’s Understanding of Division. CognitiveDevelopment, 20, 388-406.

Kouba, V. L., Brown, C. A., Carpenter, T. P., Lindquist, M.M., Silver, E. A., and Swafford, J. (1988). Results ofthe Four th NAEP Assessment of Mathematics:Number, operations, and word problems.Arithmetic Teacher, 35 (4), 14-19.

Lachance, A., and Confrey, J. (2002). Helping studentsbuild a path of under standing from ratio andproportion to decimal notation. Journal ofMathematical Behavior, 20 503–526.

Lamon, S. J. (1996). The Development of Unitizing: ItsRole in Children’s Partitioning Strategies. Journal forResearch in Mathematics Education, 27, 170-193.

Larson, C. N. (1980). Locating proper fractions onnumber lines: Effect of length and equivalence .School Science and Mathematics , 53, 423-428.

Mack, N. K. (1990). Learning fractions withunderstanding: Building on informal knowledge.Journal for Research in Mathematics Education, 21,16-32.



Mamede, E., Nunes, T., & Br yant, P. (2005). Theequivalence and order ing of fractions in part-wholeand quotient situations. Paper presented at the29th PME Conference, Melbourne; Volume 3,281-288.

Mamede, E. P. B. d. C. (2007). The effects ofsituations on children’s understanding of fractions.Unpublished PhD Thesis, Oxford BrookesUniversity, Oxford.

Michie, S. (1984). Why preschoolers are reluctant to count spontaneously. British Journal ofDevelopmental Psychology, 2, 347-358.

Miller, K. (1984). Measurement procedures and thedevelopment of quantitative concepts. In C .Sophian (Ed.), Origins of cognitive skills. TheEighteen Annual Carnegie Symposium onCognition (pp. 193-228). Hillsdale, New Jersey:Lawrence Erlbaum Associates.

Moss, J., and Case, R. (1999). Developing children’sunderstanding of rational numbers: A new modeland an experimental curriculum. Journal forResearch in Mathematics Education, 30, 122-147.

Noelting, G. (1980 a). The development ofproportional reasoning and the r atio conceptPart I - Differentiation of stages. EducationalStudies in Mathematics , 11, 217-253.

Noelting, G. (1980 b). The development ofproportional reasoning and the r atio conceptPart II - Problem-structure at sucessive stages:Problem-solving strategies and the mechanism ofadaptative restructuring. Educational Studies inMathematics, 11, 331-363.

Novillis, C. F. (1976). An Analysis of the FractionConcept into a Hierarchy of SelectedSubconcepts and the Testing of the HierarchicalDependencies. Journal for Research in MathematicsEducation, 7, 131-144.

Nunes, T. (1997). Systems of signs and mathematicalreasoning. In T. Nunes and P. Bryant (Eds.),Learning and Teaching Mathematics. AnInternational Perspective (pp. 29-44). Hove (UK):Psychology Press.

Nunes, T., and Br yant, P. (2004). Children’s insights andstrategies in solving fractions problems. Paperpresented at the Fir st International Conference ofthe Greek Association of Cognitive Psychology,Alexandropolis, Greece.

Nunes, T., Bryant, P., Pretzlik, U., and Hurry, J. (2006).Fractions: difficult but crucial in mathematicslearning. London Institute of Education, London:ESRC-Teaching and Learning ResearchProgramme.

Nunes, T., Bryant, P., Pretzlik, U., Bell, D., Evans, D., &Wade, J. (2007). La compréhension des fr actionschez les enfants. In M. Merri (Ed.), Activitéhumaine et conceptualisation. Questions à GérardVergnaud (pp. 255-262). Toulouse, France:Presses Universitaires du Mirail.

Nunes, T., Bryant, P., Pretzlik, U., Evans, D., Wade, J., andBell, D. (2008). Children’s understanding of theequivalence of fractions. An analysis of argumentsin quotient situations. submitted.

Nunes, T., Desli, D., and Bell, D. (2003). Thedevelopment of children’s understanding ofintensive quantities. International Journal ofEducational Research, 39, 652-675.

Piaget, J. (1952). The Child’s Conception of Number .London: Routledge and Kegan Paul.

Piaget, J., and Inhelder, B. (1975). The Origin of the Ideaof Chance in Children . London: Routledge andKegan Paul.

Piaget, J., Inhelder, B., and Szeminska, A. (1960). TheChild’s Conception of Geometr y. New York: Harperand Row.

Ohlsson, S. (1988). Mathematical meaning andapplicational meaning in the semantics offractions and related concepts. In J. Hieber t andM. Behr (Eds.), Number Concepts and Operationsin the Middle Grades (pp. 53-92). Reston (VA):National Council of Mathematics Teachers.

Olive, J., and Steffe, L. P. (2002). The construction ofan iterative fractional scheme: the case of Joe .Journal of Mathematical Behavior, 20 413-437.

Olive, J., and Vomvoridi, E. (2006). Making sense ofinstruction on fractions when a student lacksnecessary fractional schemes: The case of Tim.Journal of Mathematical Behavior, 25 18-45.

Piaget, J. (1952). The Child’s Conception of Number .London: Routledge and Kegan Paul.

Piaget, J., Inhelder, B., and Szeminska, A. (1960). TheChild’s Conception of Geometr y. New York: Harperand Row.

Pitkethly, A., and Hunting, R. (1996). A Review ofRecent Research in the Area of Initial FractionConcepts. Educational Studies in Mathematics , 30,5-38.

Pothier, Y., and Sawada, D. (1983). Partitioning: TheEmergence of Rational Number Ideas in YoungChildren. Journal for Research in MathematicsEducation, 14, 307-317.

Resnick, L. B., Nesher, P., Leonard, F., Magone, M.,Omanson, S., and Peled, I. (1989). Conceptualbases of ar ithmetic errors: The case of decimalfractions. Journal for Research in MathematicsEducation, 20, 8-27.


Riley, M., Greeno, J. G., and Heller, J. I. (1983).Development of children’s problem solving abilityin arithmetic. In H. Ginsburg (Ed.), Thedevelopment of mathematical thinking (pp. 153-196). New York: Academic Press.

Rittle-Johnson, B., Siegler, R. S., and Alibali, M. W.(2001). Developing Conceptual Understandingand Procedural Skill in Mathematics: An IterativeProcess. Journal of Educational Psychology, 93(2),46-36.

Saenz-Ludlow, A. (1994). Michael’s FractionSchemes. Journal for Research in MathematicsEducation, 25, 50-85.

Saxe, G., Guberman, S. R., and Gearhar t, M. (1987).Social and developmental processes in children’sunderstanding of number. Monographs of the Societyfor Research in Child Development, 52, 100-200.

Schwartz, J. (1988). Intensive quantity and referenttransforming arithmetic operations. In J. Hiebertand M. Behr (Eds.), Number concepts andoperations in the middle grades (pp. 41-52).Hillsdale,NJ: Erlbaum.

Stafylidou, S., and Vosniadou, S. (2004). Thedevelopment of students’ understanding of thenumerical value of fractions. Learning andInstruction 14, 503-518.

Stavy, R., Strauss, S., Orpaz, N., and Carmi, G. (1982).U-shaped behavioral growth in rationcomparisons, or that’s funny I would not havethought you were U-ish. In S. Strauss and R. Stavy(Eds.), U-shaped Behavioral Growth (pp. 11-36).New York: Academic Press.

Stavy, R., and Tirosh, D. (2000). How Students (Mis-)Understand Science and Mathematics. IntuitiveRules. New York: Teachers College Press.

Steffe, L. P. (2002). A new hypothesis concerningchildren’s fractional knowledge. Journal ofMathematical Behavior, 102, 1-41.

Steffe, L. P., and Tzur, R. (1994). Interaction andchildren’s mathematics. Journal of Research inChildhood Education, 8, 99- 91 16.

Streefland, L. (1987). How lo Teach Fractions So As toBe Useful. Utrecht, The Netherlands: The StateUniversity of Utrecht.

Streefland, L. (1990). Free productions in Teachingand Learning Mathematics. In K. Gravemeijer, M.van den Heuvel & L. Streefland (Eds.), ContextsFree Production Tests and Geometr y in RealisticMathematics Education (pp. 33-52). UtrechtNetherlands: Research group for MathematicalEducation and Educational Computer Centre -State University of Utrecht.

Streefland, L. (1993). Fractions: A Realistic Approach.In T. P. Carpenter, E. Fennema and T. A. Romberg(Eds.), Rational Numbers: An Integration of Research(pp. 289-326). Hillsdale, NJ: Lawrence ErlbaumAssociates.

Streefland, L. (1997). Charming fractions or fractionsbeing charmed? In T. Nunes and P. Bryant (Eds.),Learning and Teaching Mathematics. AnInternational Perspective (pp. 347-372). Hove (UK):Psychology Press.

Tall, D. (1992). The transition to advancedmathematical thinking: Functions, limits, infinity, andproof. In D. A. Grouws (Ed.), Handbook ofResearch on Mathematics Teaching and Learning(pp. 495-511). New York: Macmillan.

Tzur, R. (1999). An Integrated Study of Children’sConstruction of Improper Fractions and theTeacher’s Role in Promoting That Learning. Journalfor Research in Mathematics Education, 30, 390-416.

Vamvakoussi, X., and Vosniadou, S. (2004).Understanding the structure of the set of rationalnumbers: a conceptual change approach. Learningand Instruction, 14 453-467.

Vergnaud, G. (1997). The nature of mathematicalconcepts. In T. Nunes and P. Bryant (Eds.), Learningand Teaching Mathematics. An InternationalPerspective (pp. 1-28). Hove (UK): PsychologyPress.

Vergnaud, G. (1983). Multiplicative structures. In R. Lesh & M. Landau (Eds.), Acquisition ofmathematics concepts and processes(pp. 128-175). London: Academic Press.

Yackel, E., Cobb, P., Wood, T., Wheatley, G., andMerkel, G. (1990). The importance of socialinteraction in children’s construction ofmathematical knowledge. In T. J. Cooney and C . R.Hirsch (Eds.), Teaching and learning mathematics inthe 1990s. 1990 Yearbook of the National Councilof Teachers of Mathematics (pp. 12-21). Reston,VA: National Council of Teachers of Mathematics.




ISBN 978-0-904956-70-2





4

Paper 4: Understanding relations and their graphical representationBy Terezinha Nunes and Peter Bryant, University of Oxford

2 SUMMARY – PAPER 2: Understanding whole numbers2 Paper 4: Understanding relations and their graphical representation












ContentsSummary of paper 4 3

Understanding relations and their graphical representation 7

References 34

About the authorsTerezinha Nunes is Professor of Educational Studiesat the University of Oxford. Peter Bryant is Senior Research Fellow in theDepartment of Education, University of Oxford.


About this review


Headlines

• Children have greater difficulty in understandingrelations than in understanding quantities. This istrue in the context of both additive andmultiplicative reasoning problems.

• Primary and secondary school students often apply additive procedures to solve multiplicativereasoning problems and also apply multiplicativeprocedures to solve additive reasoning problems.

• Explicit instruction to help students become awareof relations in the context of additive reasoningproblems can lead to significant improvement inchildren’s performance.

• The use of diagrams, tables and graphs torepresent relations facilitates children’s thinkingabout and discussing the nature of the relationsbetween quantities in problems.

• Excellent curriculum development work has beencarried out to design instruction to help studentsdevelop awareness of their implicit knowledge ofmultiplicative relations. This programme has notbeen systematically assessed so far.

• An alternative view is that students’ implicitknowledge should not be the starting point forstudents to learn about proportional relations;teaching should focus on formalisations rather than informal knowledge and seek to connectmathematical formalisations with applied situationsonly later.

• There is no research comparing the results ofthese diametrically opposed ideas.

Children need to learn to co-ordinate theirknowledge of numbers with their under standing of quantities. This is cr itical for mathematics learning in pr imary school so that they can usetheir understanding of quantities to suppor t theirknowledge of numbers and vice versa. But this isnot all that students need to lear n to be able touse mathematics sensibly. Using mathematics alsoinvolves thinking about relations betweenquantities. Research shows quite unambiguouslythat it is more diff icult for children to solveproblems that involve relations than to solveproblems that involve only quantities.

A simple problem about quantities is: Paul had 5marbles. He played two games with his fr iend. In thefirst game, he won 6 marbles. In the second game helost 4 marbles. How many marbles does he havenow? The same numerical information can be useddifferently, making the problem into one which is allabout relations: Paul played three games of marbles.In the first game, he won 5 marbles. In the secondgame, he won 6. In the third game , he lost 4. Did heend up winning or losing marb les? How many?

The arithmetic that children need to use to solve isthe same in both problems: add 5 and 6 and subtr act4. But the second problem is significantly more difficultfor children because it is all about relations. They don’tknow how many marbles Paul actually had at any time,they only know that he had 5 more after the f irstgame than before, and 6 more after the second game ,and 4 fewer after the third game . Some children saythat this problem cannot be solved because we don’tknow how many marbles Paul had to begin with: theyrecognise that it is possible to operate on quantities,but do not recognise that it is possib le to operate onrelations. Why should this be so?

Summary of paper 4: Understanding relationsand their graphicalrepresentation


One possible explanation is the way in which weexpress relations. When we speak about quantities, wesay that Paul won marbles or lost marbles; these aretwo opposite statements. When we speak aboutrelations, statements that use opposite words maymean the same thing: after winning 5 marbles, we cansay that Paul now has 5 more marbles or that beforehe had 5 fewer. In order to gr asp the concept ofrelations fully, students must be able to view these two different statements as meaning the same thing.Research shows that some students are able to treatthese different statements as having the same meaningbut others find this difficult. Students who realise thatthe two statements mean the same thing are moresuccessful in solving problems about relations.

A second plausible explanation is that many childrendo not distinguish clear ly between quantities andrelations when they use numbers. When they aregiven a problem about relations, they interpret therelations as quantities. If they are given a problem like‘Tom, Fred, and Rhoda put their apples into a bag.Tom and Fred together had 17 more apples thanRhoda. Tom had 7 apples. Rhoda had 5 apples. Howmany apples did Fred have?’, they write down thatTom and Fred had 17 apples together (instead of 17 more than Rhoda). When they make thisinterpretation error, the problem seems very easy: if Tom had 7, Fred had 10. The information aboutRhoda seems irrelevant. But of course this is not thesolution. It is possible to teach children to representquantities and relations differently, and thus todistinguish the two: for example, they can be taughtto write ‘plus 17’ to show that this is not a quantitybut a relation. Children aged seven to nine years canadopt this notation and at the same time improvetheir ability to solve relational problems. However,even after this teaching, they still seem to be temptedto interpret relations as quantities. So, learning torepresent relations helps children take a step towardsdistinguishing relations and quantities but they needplenty of oppor tunity to think about this distinction.

A third difficulty is that relational thinking involvesbuilding a model of a prob lem situation in order totreat the relations in the problem mathematically. Inprimary school, children have little oppor tunity toexplore situations in their mathematics lessonsbefore solving a problem. If they make a mistake in solving a problem when their computation wascorrect, the error is explained as ‘choice of thewrong operation’, but the wrong choice ofoperation is a symptom, not an explanation forwhat went wrong during problem solving.

Models of situations are ways of thinking about them,and more than one way may be appropriate. It alldepends on the question that we want to answer.Suppose there are 12 gir ls and 18 boys in a class andthey are assigned to single-sex groups dur ing Frenchlessons. If there were not enough books for all ofthem and the Head Teacher decided to give 4 booksto the gir ls and 6 books to the bo ys, would this befair? If you give one book to each gir l, there are 8girls left without books; if you give one book to eachboy, there are 12 boys left without books. This seemsunfair. If you ask all the children to share , 3 gir ls willshare one book and 3 boys will share one book. Thisseems fair. The first model is additive: the questions itanswers are ‘How many more gir ls than books?’ and‘How many more boys than books?’ The secondmodel is multiplicative: it examines the r atio betweengirls and books and the r atio between boys andbooks. If the Head Teacher is planning to buy morebooks, she needs an additive model. If the HeadTeacher is not planning to buy more books, the ratio is more informative. A model of a situation isconstructed by the problem solver for a purpose;additive and multiplicative relations answer differentquestions about the same situation.

Children, but also adults, often make mistakes in thechoice of operation when solving problems: theysometimes use additive reasoning when they shouldhave used multiplicative reasoning but they can alsomake the converse mistake, and use multiplicativereasoning when additive reasoning would beappropriate. So, we need to examine research thatexplains how children can become more successful inchoosing the appropriate model to answer a question.

Experts often use diagrams, tables and graphs to helpthem analyse situations. These resources could suppor tchildren’s thinking about situations. But children seemto have difficulty in using these resources and have tolearn how to use them. They have to become literatein the use of these mathematical tools in order tointerpret them correctly. A question that has not beenaddressed in the literature is whether children canlearn about using these tools and about anal ysingsituations mathematically at the same time . Researchabout interpreting tables and graphs has been carriedout either to assess students’ previous knowledge (ormisconceptions) before they are taught or to testways of making them liter ate in the use of these tools.

A remarkable exception is found in the work ofresearchers in the Freudenthal Institute . One of theirexplicit aims for instruction in mathematics is to help

4 Paper 4: Understanding relations and their graphical representation

students mathematise situations: i.e. to help thembuild a model of a situation and later tr ansform it intoa model for other situations through their awarenessof the relations in the model. They argue that weneed to use diagrams, tables and graphs during theprocess of mathematising situations. These are built bystudents (with teacher guidance) as they explore thesituations rather than presented to the students readymade for interpretation. Students are encouraged touse their implicit knowledge of relations; by buildingthese representations, they can become aware ofwhich models they are using. The process of solutionis thus not to choose an operation and calculate butto analyse the relations in the prob lem and worktowards solution. This process allows the students tobecome aware of the relations that are conservedthroughout the different steps.

Streefland worked out in detail how this processwould work if students were asked to solve Hart’sfamous onion soup recipe problem. In this problem,students are presented with a recipe of onion soupfor 8 people and asked how much of each ingredientthey would need if they were preparing the soup for6 people. Many students use their everydayknowledge of relations in searching for a solution: theythink that you need half of the or iginal recipe (whichwould serve 4) plus half of this (which w ould serve 2people) in order to have a recipe for 6 people. Thisperfectly sound reasoning is actually a mixture ofadditive and multiplicative thinking: half of a recipe for8 serves 4 people (multiplicative reasoning) and half ofthe latter serves 2 (multiplicative reasoning); 6 peopleis 2 more than 4 (additive); a recipe for 6 is the sameas the recipe for 4 plus the recipe for 2 (additive).

Streefland and his colleagues suggested thatdiagrams and tables provide the sort ofrepresentation that helps students think about therelations in the problem. It is illustrated here by theratio table showing how much water should be usedin the soup. The table can be used to help studentsbecome aware that the first two steps in theirreasoning are multiplicative: they divide the number

of persons in half and also the amount of water in half. Additive reasoning does not work: thetransformation from 8 to 4 people would meansubtracting 4 whereas the par allel transformation in the amount of water would be to subtract 1. Sothe relation is not the same . If they can discover that multiplicative reasoning preserves the relation,whereas additive reasoning does not, they could beencouraged to test whether there is a multiplicativerelation that they can use to f ind the recipe for 6;they could come up with x3, trebling the recipe for2. Streefland’s ratio table can be used as a model fortesting if other situations f it this sor t of multiplicativereasoning. The table can be expanded to calculatethe amounts of the other ingredients.

An alternative approach in curriculum development isto start from formalisations and not to base teachingon students’ informal knowledge. The aim of thisapproach is to establish links between different formalrepresentations of the same relations. A programmeproposed by Adjiage and Pluvinage starts with linesdivided into segments: students learn how torepresent segments with the same fr action eventhough the lengths of the lines diff er (e.g. 3/5 of linesof different lengths). Next they move to using theseformal representations in other types of prob lems: forexample, mixtures of chocolate syr up and milk wherethe number of cups of each ingredient differs but theratio of chocolate to total number of cups is thesame. Finally, students are asked to write abstractionsthat they learned in these situations and formulaterules for solving the problems that they solved duringthe lessons. An example of generalisation expected is‘seven divided by four is equal to seven fourths’ or ‘7÷ 4 = 7/4’. An example of a rule used in problemsolving would be ‘Given an enlargement in which a 4cm length becomes a 7 cm length, then any length tobe enlarged has to be multiplied by 7/4.’

There is no systematic research that compares thesetwo very different approaches. Such research wouldprovide valuable insight into how children come tounderstand relations.


6 SUMMARY – PAPER 2: Understanding whole numbers6 Paper 4: Understanding relations and their graphical representation

Recommendations


Numbers are used to represent quantitiesand relations. Primary school childrenoften interpret statements about relationsas if they were about quantities and thusmake mistakes in solving problems.

Many problem situations involve bothadditive and multiplicative relations; whichone is used to solve a problem dependson the question being asked. Both childrenand adults can make mistakes in selectingadditive or multiplicative reasoning toanswer a question.

Experts use diagrams, tables and graphs toexplore the relations in a prob lemsituation before solving a problem.

Some researchers propose that informalknowledge interferes with students’learning. They propose that teachingshould star t from formalisations which areonly later applied to problem situations.


Teaching Teachers should be aware of children’s difficultiesin distinguishing between quantities and relations dur ingproblem solving.

Teaching The primary school curriculum should include thestudy of relations in situations in a more explicit wa y.Research Evidence from experimental studies is needed onwhich approaches to making students aware of relations inproblem situations improve problem solving.

Teaching The use of tables and graphs in the classroom mayhave been hampered by the assumption that students must firstbe literate in interpreting these representations before they canbe used as tools. Teachers should consider using these tools aspart of the learning process during problem solving.Research Systematic research on how students usediagrams, tables and graphs to represent relations dur ingproblem solving and how this impacts their later lear ning isurgently needed. Experimental and longitudinal methodsshould be combined.

Teaching Teachers who star t from formalisations should tr yto promote links across different types of mathematicalrepresentations through teaching.Research There is a need for experimental and longitudinalstudies designed to investigate the progress that studentsmake when teaching star ts from formalisations rather thanfrom students’ informal knowledge and the long-termconsequences of this approach to teaching students about relations.

Relations and their importancein mathematics

In our analysis of how children come to under standnatural and rational numbers, we examined theconnections that children need to make betweenquantities and numbers in order to under stand what numbers mean. Numbers are cer tainly used to represent quantities, but they are also used torepresent relations. The focus of this section is onthe use of numbers to represent relations. Relationsdo not have to be quantified: we can simply say, forexample, that two quantities are equivalent ordifferent. This is a qualitative statement about therelation between two quantities. But relations can be quantified also: if there are 20 children in theclass and 17 books, we can say that there are 3more children than books. The number 3 quantifiesthe additive relation between 17 and 20 and so wecan say that 3 quantifies a relation.

When we use numbers to represent quantities, thenumbers are the result of a measurement oper ation.Measures usually rely on culturally developedsystems of representation. In order to measurediscrete quantities, we count their units, and in orderto measure continuous quantities, we use systemsthat have been set up to allow us to represent themby a number of conventional units. Measures areusually described by a number followed by a noun,which indicates the unit of quantity the n umberrefers to: 5 children, 3 centimetres, 200 grams. Andwe can’t replace the noun with another nounwithout changing what we are talking about. Whenwe quantify a relation, the number does not refer toa quantity. We can say ‘3 more children than books’or ‘3 books fewer than children’: it makes nodifference which noun comes after the n umber

because the number refers to the relation betweenthe two quantities, how many more or fewer.

When we use qualitative statements about therelations between two quantities, the quantities mayor may not have been expressed numerically. Forexample, we can look at the children and the booksin the class and know that there are more childrenwithout counting them, especially if the difference isquite large. So we can say that there are morechildren than books without knowing how manychildren or how many books. But in order to quantifya relation between two quantities, the quantitiesneed to be measurable, even if, in the case ofdifferences, we can evaluate the relationship withoutactually measuring them. The ability to express therelationship quantitatively, without knowing the actualmeasures, is one of the roots of algebra (see Paper5). For this reason, we will often use the ter m‘measures’ in this section, instead of ‘quantities’, torefer to quantities that are represented n umerically.

It is perfectly possible that when children first appearto succeed in quantifying relations, they are actuallystill thinking about quantities: when they say ‘3children more than books’, they might be thinking ofthe poor little things who won’t have a book whenthe teacher shares the books out, not of the relationbetween the number of books and the number ofchildren. This hypothesis is consistent with results ofstudies by Hudson (1983), described in Paper 2:young children are quite able to answer thequestion ‘how many birds won’t get worms’ but theycan’t tell ‘how many more birds than worms’. We, asadults, may think that they understand something


Understanding relationsand their graphicalrepresentation


about relations when they answer the first question,but they may be talking about quantities, i.e. thenumber of birds that won’t get worms.

There is no doubt to us that children must grasphow numbers and quantities are connected in order to understand what numbers mean. Butmathematics is not only about representingquantities with numbers. A major use ofmathematics is to manipulate numbers thatrepresent relations and ar rive at conclusions without having to operate directly on the quantities.Attributing a number to a quantity is measur ing;quantifying relations and manipulating them isquantitative reasoning. To quote Thompson (1993):‘Quantitative reasoning is the analysis of a situationinto a quantitative structure – a network ofquantities and quantitative relationships… Aprominent characteristic of reasoning quantitativelyis that numbers and numeric relationships are ofsecondary importance, and do not enter into theprimary analysis of a situation. What is impor tant isrelationships among quantities’ (p. 165). Elsewhere,Thompson (1994) emphasised that ‘a quantitativeoperation is nonnumerical; it has to do with thecomprehension [italics in the original] of a situation.Numerical operations (which we have termedmeasurement operations) are used to evaluate aquantity’ (p. 187–188).

In order to reach the r ight conclusions inquantitative reasoning, one must use an appropr iaterepresentation of the relations between thequantities, and the representation depends on whatwe want to know about the relation between thequantities. Suppose you want to know whether youare paying more for your favourite chocolates atone shop than another, but the boxes of chocolatesin the two shops are of different sizes. Of course thebigger box costs more money, but are you payingmore for each chocolate? You don’t know unless youquantify the relation between price and chocolates.This relation, price per chocolate, is not quantified inthe same way as the relation ‘more children thanbooks’. When you want to know how many childrenwon’t have books, you subtract the number ofbooks from the number of children (or vice versa).When you want to know the price per chocolate,you shouldn’t subtract the number of chocolatesfrom the price (or vice versa); you should divide theprice by the number of chocolates. Quantifyingrelations depends on the nature of the question y ouare asking about the quantities. If you are asking howmany more, you use subtraction; if you are asking a

rate question, such as pr ice per chocolate, you usedivision. So quantifying relations can be done b yadditive or multiplicative reasoning. Additivereasoning tells us about the diff erence betweenquantities; multiplicative reasoning tells us about theratio between quantities. The focus of this section ison multiplicative reasoning but a brief discussion ofadditive relations will be included at the outset toillustrate the difficulties that children face when theyneed to quantify and operate on relations. However,before we turn to the issue of quantif ication ofrelations, we want to say why we use the termsadditive and multiplicative reasoning, instead ofspeaking about the four arithmetic operations.

Mathematics educators (e.g. Behr, Harel, Post andLesh, 1994; Steffe, 1994; Vergnaud, 1983) includeunder the term ‘additive reasoning’ those problemsthat are solved by addition and subtraction andunder the term ‘multiplicative reasoning’ those thatare solved by multiplication and division. This way ofthinking, focusing on the problem structure ratherthan on the ar ithmetic operations used to solveproblems, has become dominant in mathematicseducation research in the last three decades or so . It is based on some assumptions about ho w childrenlearn mathematics, three of which are made explicithere. First, it is assumed that in order to understandaddition and subtraction properly, children must alsounderstand the inverse relation between them;similarly, in order to under stand multiplication anddivision, children must understand that they also arethe inverse of each other. Thus a focus on specificand separate operations, which was more typical ofmathematics education thinking in the past, isjustified only when the focus of teaching is oncomputation skills. Second, it is assumed that thelinks between addition and subtraction, on onehand, and multiplication and division, on the other,are conceptual: they relate to the connectionsbetween quantities within each of these domains ofreasoning. The connections between addition andmultiplication and those between subtraction anddivision are procedural: you can multiply by carryingout repeated additions and divide by using repeatedsubtractions. Finally, it is assumed that, in spite of theprocedural links between addition and multiplication,these two forms of reasoning are distinct enough tobe considered as separate conceptual domains. Sowe will use the terms additive and multiplicativereasoning and relations rather than refer to thearithmetic operations.


Quantifying additive relations

The literature about additive reasoning consistentlyshows that compare problems, which involverelations between quantities, are more difficult than those that involve combining sets ortransformations. This literature was reviewed inPaper 1. Our aim in taking up this theme again hereis to show that there are three sources of diff icultiesfor students in quantifying additive relations: • to interpret relational statements as such, rather

than to interpret them as statements aboutquantities

• to transform relational statements into equivalentstatements which help them think about theproblem in a different way

• to combine two relational statements into a thirdrelational statement without falling prey to thetemptation of treating the result as a statementabout a quantity.

This discussion in the context of additiv e reasoningillustrates the role of relations in quantitativ ereasoning. The review is br ief and selective, becausethe main focus of this section is on m ultiplicativereasoning.

Interpreting relational statements asquantitative statementsCompare problems involve two quantities and arelation between them. Their general format is: A had x; B has y; the relation between A and B is z.This allows for creating a number of differentcompare problems. For example, the simplestcompare problems are of the form: Paul has 8marbles; Alex had 5 marbles; how many more doesPaul have than Alex? or How many fewer does Alexhave than Paul? In these problems, the quantities areknown and the relation is the unknown.

Carpenter, Hiebert and Moser (1981) observed that53% of the first grade (estimated age about 6 years)children that they assessed in compare prob lemsanswered the question ‘how many more does Ahave than B’ by saying the number that A has. This is the most common mistake reported in theliterature: the relational question is answered as aquantity mentioned in the problem. The explanationfor this error cannot be children’s lack of knowledgeof addition and subtraction, because about 85% ofthe same children used cor rect addition andsubtraction strategies when solving problems thatinvolved joining quantities or a tr ansformation of aninitial quantity. Carpenter and Moser repor t that

many of the children did not seem to know what todo when asked to solve a compare problem.

Transforming relational statements intoequivalent relational statementsCompare problems can also state how many itemsA has, then the value of the relation betw een A’sand B’s quantities, and then ask how much B has.Two problems used by Vershaffel (1994) will beused to illustrate this problem type. In the problem‘Chris has 32 books. Ralph has 13 more books thanChris. How many books does Ralph have?’, therelation is stated as ‘13 more books’ and the answeris obtained by addition; this problem type is referredto by Lewis and Mayer (1987) as involvingconsistent language. In the problem ‘Pete has 29nuts. Pete has 14 more nuts than Rita. How manynuts does Rita have?’, the relation is stated as ‘14more nuts’ but the answer is obtained bysubtraction; this problem types is referred to asinvolving inconsistent language. Verschaffel found thatBelgian students in sixth grade (aged about 12) gave82% correct responses to problems with consistentlanguage and 71% correct responses to problemswith inconsistent language. The operation itself,whether it was addition or subtr action, did not affect the rate of correct responses.

Lewis and Mayer (1987) have argued that the rateof correct responses to relational statements withconsistent or inconsistent language var ies becausethere is a higher cognitive load in processinginconsistent sentences. This higher cognitive load isdue to the fact that the subject of the sentence inthe question ‘how many nuts does Rita have?’ is theobject of the relational sentence ‘Pete has 14 morenuts than Rita’. It takes more effort to process thesetwo sentences than other two, in which the subjectof the question is also the subject of the relationalstatement. They provided some evidence for thishypothesis, later confirmed by Verschaffel (1994),who also asked the students in his study to retellthe problem after the students had alreadyanswered the question.

In the problems where the language was consistent,almost all the students who gave the r ight answersimply repeated what the researcher had said: therewas no need to rephr ase the problem. In theproblems where the language was inconsistent,about half of the students (54%) who gave correctanswers retold the problem by rephrasing itappropriately. Instead of saying that ‘Pete has 14 nutsmore than Rita’, they said that ‘Rita has 14 nuts less



than Pete’, and thus made Rita into the subject ofboth sentences. Verschaffel interviewed some of thestudents who had used this cor rect rephrasing byshowing them the wr itten problem that he had readand asking them whether they had said the samething. Some said that they changed the phr aseintentionally because it was easier to think about thequestion in this way; they stressed that the meaningof the two sentences was the same . Other studentsbecame confused, as if they had said somethingwrong, and were no longer cer tain of their answers.In conclusion, there is evidence that at least somestudents do reinterpret the sentences ashypothesised by Lewis and Mayer; some do thisexplicitly and others implicitly. However, almost asmany students reached correct answers withoutseeming to rephrase the problem, and may notexperience the extra cognitive load predicted.

It is likely that, under many conditions, we rephraserelational statements when solving problems. So twosignificant findings arise from these studies:

• rephrasing relational statements seems to be astrategy used by some people, which may placeextra cognitive demands on the problem solverbut nevertheless helps in the search for a solution

• rephrasing may be done intentionally and explicitly,as a strategy, but may also be carried out implicitlyand apparently unintentionally, producinguncertainty in the problem solvers’ minds if theyare asked about the rephrasing.

Combining relational statements into athird relational statementCompare problems typically involve two quantitiesand a relation between them but it is possible tohave problems that require children to work withmore quantities and relations than these simplerproblems. In these more complex prob lems, it maybe necessary to combine two relational statementsto identify a third one .

Thompson (1993) analysed students’ reasoning incomplex comparison problems which involved atleast three quantities and three relations. His aimwas to see how children interpreted complexrelational problems and how their reasoningchanged as they tackled more prob lems of the sametype. To exemplify his problems, we quote the firstone: ‘Tom, Fred, and Rhoda combined their applesfor a fruit stand. Fred and Rhoda together had 97more apples than Tom. Rhoda had 17 apples. Tom

had 25 apples. How many apples did Fred have?’ (p. 167). This problem includes three quantities(Tom’s, Fred’s and Rhoda’s apples) and threerelations (how many more Fred and Rhoda havethan Tom; how many fewer Rhoda has than Tom; acombination of these two relations). He asked sixchildren who had achieved different scores in a pre-test (three with higher and three with middle lev elscores) sampled from two grade levels, second(aged about seven) and fifth (aged about nine) todiscuss six problems presented over four differentdays. The children were asked to think about theproblems, represent them and discuss them.

On the first day the children went directly to tr yingout calculations and represented the relations asquantities: the statement ‘97 more apples than Tom’was interpreted as ‘97 apples’. They did not knowhow to represent ‘97 more’. This leads to theconclusion that Fred has 80 apples because Rhodahas 17. On the second day, working with problemsabout marbles won or lost dur ing the games, theresearcher taught the children to userepresentations by writing, for example, ‘plus 12’ to indicate that someone had won 12 marbles and‘minus 1’ to indicate that someone had lost 1marble. The children were able to work with theserepresentations with the researcher’s support, butwhen they combined two statements, for exampleminus 8 and plus 14, they thought that the answerwas 6 marbles (a quantity), instead of plus six (a relation). So at first they represented relationalstatements as statements about quantities,apparently because they did not know how torepresent relations. However, after having learnedhow to represent relational statements, theycontinued to have difficulties in thinking onlyrelationally, and unwittingly converted the result ofoperations on relations into statements aboutquantities. Yet, when asked whether it would alwaysbe true that someone who had won 2 marbles in agame would have 2 marbles, the children recognisedthat this would not necessarily be true. They didunderstand that relations and quantities are differentbut they interpreted the result of combining tworelations as a quantity.

Thompson describes this tension betweeninterpreting numbers as quantities or relations as themajor difficulty that the students faced throughouthis study. When they seemed to under stand‘difference’ as a relation between two quantitiesarrived at by subtraction, they found it difficult tointerpret the idea of ‘difference’ as a relation


between two relations. The children could correctlyanswer, when asked, that if someone has 2 marb lesmore than another per son, this does not mean thathe has two marbles; however, after combining tworelations (minus 8 and plus 14), instead of sayingthat this person ended up with plus 6 marb les, theysaid that he now had 6 marbles.

Summary

1 At first, children have difficulties in using additivereasoning to quantify relations; when asked abouta relation, they answer about a quantity.

2 Once they seem to conquer this, they continue tofind it difficult to combine relations and stay withinrelational reasoning: the combination of tworelations is often converted into a statementabout a quantity.

3 So children’s difficulties with relations are notconfined to multiplicative reasoning: they are alsoobserved in the domain of additiv e reasoning.

Quantifying multiplicative relations

Research on how children quantify multiplicativerelations has a long tr adition. Piaget and hiscolleagues (Inhelder and Piaget, 1958; Piaget andInhelder, 1975) originally assumed that children f irstthink of quantifying relations additively and can onlythink of relations multiplicatively at a later age . Thishypothesis led to the prediction of an ‘additivephase’ in children’s solution to multiplicativereasoning problems, before they would be able toconceive of two variables as linked by amultiplicative relation. This hypothesis led to muchresearch on the development of proportionalreasoning, which largely supported the claim thatmany younger students offer additive solutions toproportions problems (e.g. Har t, 1981 b; 1984;Karplus and Peterson, 1970; Karplus, Pulos and Stage,1983; Noelting, 1980 a and b). These results are notdisputed but their interpretation will be examined inthe next sections of this paper because cur rentstudies suggest an alternative interpretation.

Work carried out mostly by Lieven Verschaffel andhis colleagues (e.g. De Bock, Verschaffel et al., 2002;2003) shows that students also make the conversemistake, and multiply when they should be adding inorder to solve some relational problems. This type of

error is not confined to young students: pre-serviceelementary school teachers in the United States(Cramer, Post and Currier, 1993) made the samesort of mistake when asked to solve the problem:Sue and Julie were r unning equally fast around a track.Sue started first. When she had run 9 laps, Julie hadrun 3 laps. When Julie completed 15 laps , how manylaps had Sue run? The relation between Sue’s andJulie’s numbers of laps should be quantifiedadditively: because they were running at the samespeed, this difference would (in principle) beconstant. However, 32 of 33 pre-ser vice teachersanswered 45 (15 x 3), apparently using the ratiobetween the first two measures (9 and 3 laps) tocalculate Sue’s laps. This latter type of mistake wouldnot be predicted by Piaget’s theory.

The hypothesis that we will pursue in this chapter,following authors such as Thompson 1994) andVergnaud (1983), is that additive and multiplicativereasoning have different origins. Additive reasoningstems from the actions of joining, separating, andplacing sets in one-to-one cor respondence.Multiplicative reasoning stems from the action ofputting two variables in one-to-manycorrespondence (one-to-one is just a par ticularcase), an action that keeps the ratio between thevariables constant. Thompson (1994) made thispoint forcefully in his discussion of quantitativ eoperations: ‘Quantitative operations originate inactions: The quantitative operation of combining two quantities additively originates in the actions ofputting together to make a whole and separating awhole to make par ts; the quantitative operation ofcomparing two quantities additively originates in theaction of matching two quantities with the goal ofdetermining excess or deficits; the quantitativeoperation of comparing two quantitiesmultiplicatively originates in matching and subdividingwith the goal of shar ing. As one interiorizes actions,making mental operations, these operations in themaking imbue one with the ability to comprehendsituations representationally and enable one to drawinferences about numerical relationships that are notpresent in the situation itself ’ (pp. 185–186).

We suggest that, if students solve additive andmultiplicative reasoning problems successfully butthey are guided by implicit models, they will find itdifficult to distinguish between the two models.According to Fischbein (1987), implicit models and informal reasoning provide a star ting point for learning, but one of the aims of mathematicsteaching in primary school is to help students



formalize their informal knowledge (Treffers, 1987).In this process, the models will change and becomemore explicitly connected to the systems ofrepresentations used in mathematics.

In this section, we analyse how students establishand quantify relations between quantities inmultiplicative reasoning problems. We first discussthe nature of multiplicative reasoning and presentresearch results that descr ibe how children’sinformal knowledge of multiplicative relationsdevelops. In the subsequent section, we discuss therepresentation of multiplicative relations in tablesand graphs. Next we analyse how children establishother relations between measures, besides linearrelations. The final section sets out some hypothesesabout the nature of the diff iculty in dealing withrelations in mathematics and a research agenda f ortesting current hypotheses systematically.

The development of multiplicative reasoningMultiplicative reasoning is impor tant in many waysin mathematics learning. Its role in under standingnumeration systems with a base and place valuewas already discussed in Paper 2. In this section,we focus on a different role of multiplicativereasoning in mathematics learning, its role inunderstanding relations between measures orquantities, which has already been recognised by different researchers (e.g. Confrey, 1994;Thompson, 1994; Vergnaud, 1983; 1994).

Additive and multiplicative reasoning problems are essentially different: additive reasoning is used in one-variable problems, when quantities of the same kind are put together, separated or compared,whereas multiplicative reasoning involves twovariables in a fixed-ratio to each other. Even thesimplest multiplicative reasoning problems involvetwo variables in a fixed ratio. For example, in theproblem ‘Hannah bought 6 sweets; each sweet costs5 pence; how much did she spend?’ there are twovariables, number of sweets and price per sweet. Theproblem would be solved by a multiplication if, as in this example , the total cost is unknown. Thesame problem situation could be presented with a different unknown quantity, and would then besolved by means of a division: ‘Hannah bought somesweets; each sweet costs 5p; she spent 30p; howmany sweets did she buy?’

Even before being taught about multiplication anddivision in school, children can solve multiplicationand division problems such as the one about

Hannah. They use the schema of one-to-manycorrespondence.

Different researchers have investigated the use ofone-to-many correspondences by children to solvemultiplication and division problems before they aretaught about these operations in school. Piaget’swork (1952), described in Paper 3, showed thatchildren can understand multiplicative equivalences:they can construct a set A equivalent to a set B b yputting the elements in A in the same r atio that Bhas to a compar ison set.

Frydman and Br yant (1988; 1994) also showed that young children can use one-to-manycorrespondences to create equivalent sets. They used sharing in their study because youngchildren seem to have much experience withcorrespondence when sharing. In a shar ing situation,children typically use a one-for-you one-for-meprocedure, setting the shared elements (sweets) intoone-to-one correspondence with the recipients(dolls). Frydman and Br yant observed that childrenin the age range five to seven years becameprogressively more competent in dealing with one-to-many correspondences and equivalences in thissituation. In their task, the children were asked toconstruct equivalent sets but the units in the setswere of a different value. For example, one doll onlyliked her sweets in double units and the second dollliked his sweets in single units. The children wereable to use one-to-many correspondence to sharefairly in this situation: when they gave a double tothe first doll, they gave two singles to the second.This flexible use of correspondence to constructequivalent sets was interpreted by Frydman andBryant as an indication that the children’ s use of the procedure was not merely a copy of previouslyobserved and rehearsed actions: it reflected anunderstanding of how one-to-manycorrespondences can result in equivalent sets. Theyalso replicated one of Piaget’s previous findings:some children who succeed with the 2:1 r atio foundthe 3:1 ratio difficult. So the development of theone-to-many correspondence schema does nothappen in an all-or-nothing fashion.

Kouba (1989) presented young children in theUnited States, in first, second and third gr ade (aged about six to eight years), with multiplicativereasoning problems that are more typical of thoseused in school; for example: in a par ty, there were 6cups and 5 mar shmallows in each cup; how manymarshmallows were there?


Kouba analysed the children’s strategies in greatdetail, and classified them in terms of the types ofactions used and the level of abstraction. The level of abstraction varied from direct representation (i.e. all the information was represented by thechildren with concrete mater ials), through par tialrepresentation (i.e. numbers replaced concreterepresentations for the elements in a group and thechild counted in groups) up to the most abstractform of representation available to these children,i.e. multiplication facts.

For the children in f irst and second grade, who had not received instruction on multiplication anddivision, the most important factor in predicting the children’s solutions was which quantity wasunknown. For example, in the problem above, aboutthe 6 cups with 5 mar shmallows in each cup, whenthe size of the groups was known (i.e. the numberof marshmallows in each cup), the children usedcorrespondence strategies: they paired objects (ortallies to represent the objects) and counted oradded, creating one-to-many correspondencesbetween the cups and the mar shmallows. Forexample, if they needed to find the total number ofmarshmallows, they pointed 5 times to a cup (or itsrepresentation) and counted to 5, paused, and thencounted from 6 to 10 as they pointed to thesecond ‘cup’, until they reached the solution.Alternatively, the may have added as they pointed to the ‘cup’.

In contrast, when the number of elements in eachgroup was not known, the children used dealingstrategies: they shared out one mar shmallow (or itsrepresentation) to each cup, and then another, untilthey reached the end, and then counted the numberin each cup. Here they sometimes used tr ial-and-error: they shared more than one at a time andthen might have needed to adjust the number percup to get to the cor rect distribution.

Although the actions look quite diff erent, their aims are the same: to establish one-to-manycorrespondences between the marshmallows and the cups.

Kouba observed that 43% of the str ategies used by the children, including first, second, and thirdgraders, were appropriate. Among the first andsecond grade children, the overwhelming majority of the appropriate strategies was based oncorrespondences, either using direct representationor par tial representation (i.e . tallies for one variable

and counting or adding for the other); few usedrecall of multiplication facts. The recall of numberfacts was significantly higher after the children hadreceived instruction, when they were in third grade.

The level of success obser ved by Kouba amongchildren who had not yet received instruction ismodest, compared to that obser ved in twosubsequent studies, where the ratios were easier.Becker (1993) asked kindergarten children in theUnited States, aged four to five years, to solveproblems in which the cor respondences were 2:1 or 3:1. As reported by Piaget and by Frydman andBryant, the children were more successful with 2:1than 3:1 correspondences, and the level of successimproved with age. The overall level of correctresponses by the five-year-olds was 81%.

Carpenter, Ansell, Franke, Fennema and Weisbeck(1993) also gave multiplicative reasoning problemsto U.S. kindergarten children involvingcorrespondences of 2:1, 3:1 and 4:1. They observed71% correct responses to these problems.

The success rates leave no doubt that many youngchildren star t school with some under standing ofone-to-many correspondence, which they can useto learn to solve multiplicative reasoning problems in school. These results do not imply that childrenwho use one-to-many correspondence to solvemultiplicative reasoning problems consciouslyrecognise that in a multiplicative situation there is a fixed ratio linking the two variables. Their actionsmaintain the ratio fixed but it is most likely that thisinvariance remains, in Vergnaud’s (1997) terminology,as a ‘theorem in action’.

The importance of informal knowledgeBoth Fischbein (1987) and Treffers (1987) assumedthat children’s informal knowledge is a star ting point for learning mathematics in school but it isimportant to consider this assumption fur ther. Ifchildren star t school with some informal knowledgethat can be used for learning mathematics in school,it is necessar y to consider whether this knowledgefacilitates their learning or, quite the opposite , is anobstacle to learning. The action of establishing one-to-many correspondences is not the same as theconcept of ratio or as multiplicative reasoning: ratiomay be implicit in their actions b ut it is possible thatthe children are more aware of the methods thatthey used to figure out the numerical values of thequantities, i.e. they are aware of counting or adding.



Children’s methods for solving multiplicationproblems can be seen as a star ting point, if theyform a basis for fur ther learning, but also an obstacleto learning, if children stick to their counting andaddition procedures instead of lear ning about ratioand multiplicative reasoning in school. Resnick(1983) and Kaput and West (1994) argue that animportant lesson from psychological andmathematics education research is that, even afterpeople have been taught new concepts and ideas,they still resor t to their prior methods to solveproblems that differ from the textbook examples onwhich they have applied their new knowledge. Theimplementation of the one-to-many correspondenceschema to solve problems requires adding andcounting, and students have been repor ted to resor tto counting and adding even in secondar y school,when they should be multiplying (Booth, 1981). So is this informal knowledge an obstacle to betterunderstanding or does it provide a basis for learning?

It is possible that a precise answer to this questioncannot be found: whether informal knowledge helpsor hinders children’s learning might depend on thepedagogy used in their classroom. However, it ispossible to consider this question in pr inciple byexamining the results of longitudinal andintervention studies. If it is found in a longitudinalstudy that children who star t school with moreinformal mathematical knowledge achieve bettermathematics learning in school, then it can beconcluded that, at least in a gener al manner, informalknowledge does provide a basis for learning.Similarly, if inter vention studies show that increasingchildren’s informal knowledge when they are in theirfirst year school has a positive impact on theirschool learning of mathematics, there is fur thersupport for the idea that informal knowledge canoffer a foundation for learning. In the case of thecorrespondence schema, there is clear evidencefrom a longitudinal study but intervention studieswith the appropriate controls are still needed.

Nunes, Bryant, Evans, Bell, Gardner, Gardner andCarraher (2007) carried out the longitudinal study. In this study, British children were tested on theirunderstanding of four aspects of logical-mathematical reasoning at the star t of school; one of these was multiplicative reasoning. There were five items which were multiplicative reasoningproblems that could be solved by one-to-manycorrespondence. The children were also given theBritish Abilities Scale (BAS-II; Elliott, Smith andMcCulloch, 1997) as an assessment of their gener al

cognitive ability and a Working Memory Test,Counting Recall (Pickering and Gathercole, 2001), at school entr y. At the beginning of the study, thechildren’s age ranged from five years and one monthto six years and six months. About 14 months later,the children were given a state-designed andteacher-administered mathematics achievement test,which is entirely independent of the researcher s andan ecologically valid measure of how much theyhave learned in school. The children’s performance in the five items on correspondence at school entr ywas a significant predictor of their mathematicsachievement, after controlling for : (1) age at the timeof the achievement test; (2) performance on theBAS-II excluding the subtest of their kno wledge ofnumbers at school star t; (3) knowledge of numberat school entr y (a subtest of the B AS-II); (4)performance on the working memory measure; and (5) performance on the multiplicative reasoning,one-to-many correspondence items. Nunes et al.(2007) did not repor t the analysis of longitudinalprediction based separately on the items that assessmultiplicative reasoning; so these results arereported here. The results are presented visuall y inFigure 4.1 and descr ibed in words subsequently.

The total variance explained in the mathematicsachievement by these predictors was 66%; ageexplained 2% (non significant), the BAS generalscore (excluding the Number Skills subtest)explained a fur ther 49% (p < 0.001); the sub-test onnumber skills explained a fur ther 6% (p < 0.05);working memory explained a fur ther 4% (p < 0.05),and the children’s understanding of multiplicativereasoning at school entr y explained a fur ther 6% (p = 0.005). This result shows that children’sunderstanding of multiplicative reasoning at schoolentry is a specific predictor of mathematicsachievement in the first two years of school. Itsupports the hypothesis that, in a general way, thisinformal knowledge forms a basis for their schoollearning of mathematics: after 14 months and aftercontrolling for general cognitive factors at schoolentry, performance on an assessment ofmultiplicative reasoning still explained a signif icantamount of variance in the children’s mathematicsachievement in school.

It is therefore quite likely that instruction will be animportant factor in influencing whether studentscontinue to use the one-to-many schema of actionto solve such problems, even if replacing objectswith numbers but still counting or adding instead ofmultiplying, or whether they go on to adopt the use


of the operations of multiplication and division.Treffers (1987) and Gravemeijer (1997) argue thatstudents do and should use their inf ormalknowledge in the classroom when lear ning aboutmultiplication and division, but that it should be oneof the aims of teaching to help them f ormalise thisknowledge, and in the process develop a betterunderstanding of the ar ithmetic operationsthemselves. We do not review this work here butrecognise the impor tance of their argument,particularly in view of the strength of this inf ormalknowledge and students’ likelihood of using it evenafter having been taught other forms of knowledgein school. However, it must be pointed out thatthere is no evidence that teaching students aboutarithmetic operations makes them more aware ofthe invariance of the ratio when they use one-to-many correspondences to solve problems. Kaputand West (1994) also designed a teachingprogramme which aimed at using students’ informalknowledge of correspondences to promote theirunderstanding of multiplicative reasoning. In contrastto the programme designed by Treffers for theoperation of multiplication and by Gravemeijer forthe operation of division, Kaput and West’sprogramme used simple calculations and tried to

focus the students’ attention on the invariance ofratio in the correspondence situations. They useddifferent sor ts of diagrams which treated thequantities in correspondence as composite units: forexample, a plate and six pieces of tableware formeda single unit, a set-place for one person. The ideasproposed in these approaches to instruction arevery ingenious and merit further research with theappropriate controls and measures. The lack ofcontrol groups and appropr iate pre- and post-testassessments in these inter vention studies makes itdifficult to reach conclusions regarding the impact ofthe programmes.

Park and Nunes (2001) car ried out a br iefintervention study where they compared children’ssuccess in multiplicative reasoning problems after thechildren had par ticipated in one of two types ofintervention. In the first, they were taught aboutmultiplication as repeated addition, which is thetraditional approach used in Br itish schools and isbased on the procedural connection betweenmultiplication and addition. In the secondintervention group, the children were taught about multiplication by considering one-to-manycorrespondence situations, where these


Figure 4.1: A schematic representation of the degree to which individual differences in mathematics achievement are explainedby the first four factors and the additional amount of variance explained by children’s informal knowledge at school entry.


correspondences were represented explicitly. A thirdgroup of children, the control group, solved additionand subtraction problems, working with the sameexperimenter for a similar per iod of time. Thechildren in the one-to-many correspondence groupmade significantly more progress in solvingmultiplicative reasoning problems than those in therepeated addition and in the control group . Thisstudy does include the appropriate controls andprovides clear evidence for more successful learningof multiplicative reasoning when instruction drawson the children’s appropriate schema of action.However, this was a very brief intervention with asmall sample and in one-to-one teaching sessions. It would be necessar y to replicate it with largernumbers of children and to compare its level ofsuccess with other inter ventions, such as those usedby Treffers and Gravemeijer, where the children’sunderstanding of the ar ithmetic operations ofmultiplication and division was strengthened byworking with larger numbers.

Summary

1 Additive and multiplicative reasoning have theirorigins in different schemas of action. There doesnot seem to be an order of acquisition, withyoung children understanding at first only additivereasoning and only later multiplicative reasoning.Children can use schemas of action appropr iatelyboth in additive and multiplicative reasoningsituations from an ear ly age.

2 The schemas of one-to-many correspondence and sharing (or dealing) allow young children to succeed in solving multiplicative reasoningproblems before they are taught aboutmultiplication and division in school.

3 There is evidence that children’s knowledge ofcorrespondences is a specific predictor of theirmathematics achievement and, therefore, that theirinformal knowledge can provide a basis for furtherlearning. However, this does not mean that theyunderstand the concept of r atio: the invariance ofratio in these situations is lik ely to be known onlyas a theorem in action.

4 Two types of programmes have been proposedwith the aim of br idging students’ informal andformal knowledge. One type (Treffers, 1987;Gravemeijer, 1997) focuses on teaching thechildren more about the oper ations of

multiplication and division, making a transition from small to large numbers easier for thestudents. The second type (Kaput and West, 1994;Park and Nunes, 2001) focused on making thestudents more aware of the schema of one-to-many correspondences and the theorems inaction that it represents implicitl y. There isevidence that, with younger children solving smallnumber problems, an inter vention that focuses on the schema of cor respondences facilitates thedevelopment of multiplicative reasoning.

Finally, it is pointed out that all the examplespresented so far dealt with prob lems in which thechildren were asked questions about quantities.None of the problems focused on the relationbetween quantities. In the subsequent section, wepresent a classification of multiplicative reasoningproblems in order to aid the discussion of ho wquantities and relations are handled in the contextof multiplicative reasoning problems.

Different types of multiplicative reasoning problemsWe argued previously that many children solveproblems that involve additive relations, such ascompare problems, by thinking only about quantities.In this section, we examine different types ofmultiplicative reasoning problems and analysestudents’ problem solving methods with a view tounderstanding whether they are consider ing onlyquantities or relations in their reasoning. In order to achieve this, it is necessar y to think about thedifferent types of multiplicative reasoning problems.

Classifications of multiplicative reasoning situationsvary across authors (Brown, 1981; Schwartz, 1988;Tourniaire and Pulos, 1985; Vergnaud, 1983), butthere is undoubtedly agreement on whatcharacterises multiplicative situations: in thesesituations there are always two (or more) var iableswith a fixed ratio between them. Thus, it is arguedthat multiplicative reasoning forms the foundationfor children’s understanding of proportional relationsand linear functions (Kaput and West, 1994;Vergnaud, 1983).

The first classifications of problem situationsconsidered distinct possibilities: for example, rate andratio problem situations were distinguished initially.However, there seemed to be little agreementamongst researchers regarding which situationsshould be classified as rate and which as ratio. Lesh,


Post and Behr (1988) wrote some time ago: ‘there is disagreement about the essential char acteristicsthat distinguish, for example rates from ratios… Infact, it is common to f ind a given author changingterminology from one publication to another’ (p.108). Thompson (1994) and Kaput and West (1994)consider this distinction to apply not to situations,but to the mental oper ations that the problemsolver uses. These different mental operations couldbe used when thinking about the same situation:ratio refers to understanding a situation in terms ofthe par ticular values presented in the problem (e.g.travelling 150 miles over 3 hours) and rate refers tounderstanding the constant relation that applies toany of the pair s of values (in theor y, in any of the 3hours one would have travelled 50 miles). ‘Rate is areflectively abstracted constant ratio’ (Thompson,1994, p. 192).

In this research synthesis, we will work with theclassification offered by Vergnaud (1983), whodistinguished three types of problems.

• In isomorphism of measures problems, there is a simple proportional relation between twomeasures (i.e. quantities represented by numbers):for example, number of cakes and price paid forthe cakes, or amount of corn and amount of cornflour produced.

• In product of measures problems, there is aCartesian composition between two measures toform a third measure: for example, the number of T-shirts and number of shorts a girl has can becomposed in a Cartesian product to give thenumber of different outfits that she can wear; thenumber of different coloured cloths and thenumber of emblems determines the number of different flags that you can produce.

• In multiple proportions problems, a measure is in simple proportion to (at least) two othermeasures: for example, the consumption of cerealin a Scout camp is proportional to number ofpersons and the number of days.

Because this classification is based on measures, itoffers the oppor tunity to explore the differencebetween a quantity and its measure . Although thismay seem like a digression, exploring the differencebetween quantities and measures is helpful in thischapter, which focuses on the quantification ofrelations between measures. A quantity, as definedby Thompson (1993) is constituted when we think

of a quality of an object in such a wa y that weunderstand the possibility of measur ing it.‘Quantities, when measured, have numerical value,but we need not measure them or kno w theirmeasures to reason about them’ (p. 166). Twoquantities, area and volume, can be used here toillustrate the difference between quantities andmeasures.

Hart (1981 a) pointed out that the square unit canbe used to measure area by different measurementoperations. We can attr ibute a number to the areaof a rectangle, for example, by covering it withsquare units and counting them: this is a simplemeasurement operation, based on iteration of theunits. If we don’t have enough bricks to do this (seeNunes, Light and Mason, 1993), we can count thenumber of square units that make a row along thebase, and establish a one-to-many correspondencebetween the number of rows that fit along theheight and the number of square units in each row.We can calculate the area of the rectangle b yconceiving of it as an isomor phism of measuresproblem: 1 row corresponds to x units. If weattribute a number to the area of the rectangle b ymultiplying its base by its height, both measured withunits of length, we are conceiving this situation as aproduct of measures: two measures, the length ofthe base and that of its height, multiplied produce athird measure, the area in square units. Thus aquantity in itself is not the same as its measure , andthe way it is measured can change the complexity(i.e. the number of relations to be considered) ofthe situation.

Nunes, Light and Mason (1993) showed thatchildren aged 9 to 10 years were much moresuccessful when they compared the area of tw ofigures if they chose to use br icks to measure theareas than if they chose to use a r uler. Because thechildren did not have sufficient bricks to cover theareas, most used calculations. They had threequantities to consider – the number of rows thatcovered the height, the number of bricks in eachrow along the base , and the area, and the relationbetween number of rows and number of bricks inthe row. These children worked within anisomorphism of measures situation.

Children who used a r uler worked within a productof measures situation and had to consider threequantities – the value of the base, the value of theheight, and the area; and three relations to consider– the relation between the base and the height, the



relation between base and area, and the relationbetween the height and the area (the area isproportional to the base if the height is constantand proportional to the height if the base isconstant).

The students who developed an isomorphism of measures conception of area were able to use their conception to compare a rectangular with a triangular area, and thus expanded theirunderstanding of how area is measured. Thestudents who worked within a product of measuressituation did not succeed in expanding theirknowledge to think about the area of tr iangles.Nunes, Light and Mason speculated that, after this initial move, students who worked with anisomorphism of measures model might subsequentl ybe able to re-conceptualise area once again andmove on to a product of measures approach, butthey did not test this h ypothesis.

Hart (1981 a) and Vergnaud (1983) make a similarpoint with respect to the measurement of v olume: itcan be measured by iteration of a unit (how manylitres can fit into a container) or can be conceived asa problem situation involving the relations betweenbase, height and width, and described as product ofmeasures. Volume as a quantity is itself neither a uni-dimensional nor a three-dimensional measure andone measure might be useful f or some purposes(add 2 cups of milk to mak e the pancake batter)whereas a different one might be useful f or otherpurposes (the volume of a trailer in a lorry can beeasily calculated by multiplying the base, the heightand the width). Different systems of representationand different measurement operations allow us toattribute different numbers to the same quantity, andto do so consistently within each system.

Vergnaud’s classification of multiplicative reasoningsituations is used here to simplify the discussion in

this chapter. We will focus primarily on isomorphismof measures situations, because the analysis of howthis type of problem is solved by students ofdifferent ages and by schooled and unschooledgroups will help us under stand the operations ofthought used in solving them.

A diagram of isomorphism of measures situations,presented in Figure 4.2 and adapted from Vergnaud(1983), will be used to facilitate the discussion.

This simple schema shows that there are two sets of relations that can be quantif ied in this situation:• the relation between a and c is the same as that

between b and d; this is the scalar relation, whichlinks two values in the same measure space

• the relation between a and b is the same as thatbetween c and d; this is the functional relation, orthe ratio, which links the two measure spaces.

The psychological difference (i.e. the difference that it makes for the students) between scalar andfunctional relations is very important, and it hasbeen discussed in the literature by many authors(e.g. Kaput and West, 1994; Nunes, Schliemann andCarraher, 1993; Vergnaud, 1983). It had also beendiscussed previously by Noelting (1980 a and b) andTourniaire and Pulos (1985), who used the termswithin and between quantities relations. This paperwill use only the terms scalar and functional relationsor reasoning.

‘For a mathematician, a propor tion is a statement of equality of two ratios, i.e., a/b = c/d’ (Tourniaireand Pulos, 1985, p. 181). Given this definition, thereis no reason to distinguish between what has beentraditionally termed multiplication and divisionproblems and propor tions problems. We think thatthe distinction has been based, perhaps onlyimplicitly, on the use of a r atio with reference to theunit in multiplication and division problems. So one


Figure 4.2: Schema of an isomorphism of measures situation. Measures 1 and 2 are connected by a proportional relation.

Measure 1 Measure 2

should not be surprised to see that one-to-manycorrespondences reasoning is used in the beginningof primary school by children to solve simplemultiplicative reasoning problems and continues tobe used by older students to solve proportionsproblems in which the unit r atio is not given in theproblem description. Many researchers (e.g. Har t,1981 b; Kaput and West, 1994; Lamon, 1994; Nunesand Bryant, 1996; Nunes, Schliemann and Carraher,1993; Piaget, Grize, Szeminska and Bangh, 1977;Inhelder and Piaget, 1958; 1975; Ricco, 1982; Steffe,1994) have described students’ solutions based oncorrespondence procedures and many differentterms have been used to refer to these, such asbuilding up strategies, empirical strategies, halving or doubling, and replications of a composite unit. Inessence, these strategies consist of using the initialvalues provided in the problem and changing themin one or more steps to ar rive at the desired value.Hart’s (1981 b) well known example of the onionsoup recipe for 4 people, which has to beconverted into a recipe for 6 people, illustrates thisstrategy well. Four people plus half of 4 mak es 6people, so the children take each of the ingredientsin the recipe in tur n, half the amount, and add thisto the amount required for 4 people.

Students used yet another method in solvingproportions problems, still related to the idea ofcorrespondences: they first find the unit ratio andthen use it to calculate the desired value . Althoughthis method is taught in some countr ies (see Lave,1988; Nunes, Schliemann and Carraher, 1993; Ricco,1982), it is not necessar ily used by all students afterthey have been taught; many students rely onbuilding up strategies which change across differentproblems in terms of the calculations that are used,instead of using a single algor ithm that aims atfinding the unit ratio. Hart (1981 b) presented thefollowing problem to a large sample of students(2257) aged 11 to 16 years in 1976: 14 metres ofcalico cost 63p; find the price of 24 metres. Shereported that no child actuall y quoted the unitar ymethod in their explanation, even though somechildren did essentially seek a unitar y ratio. Ricco(1982), in contrast, found that some studentsexplicitly searched for the unit ratio (e.g. ‘First I needto know how much one notebook will cost andthen we will see’, p. 299, our translation) but othersseem to search for the unit ratio without makingexplicit the necessity of this step in their procedure .

Building up methods and f inding the unit ratio maybe essentially an extension of the use of the one-to-

many correspondence schema, which maintains theratio invariant without necessarily bringing with it anawareness of the fixed relation between thevariables. Unit ratio is a mathematical term but it isnot clear whether the children who w ere explicitlysearching for the price of one notebook in Ricco’sstudy were thinking of ratio as the quantification ofthe relation between notebooks and money. Whenthe child says ‘1 notebook costs 4 cents’, the child isspeaking about two quantities, not necessarily aboutthe relation between them. A statement about therelation between the quantities would be ‘thenumber of notebooks times 4 tells me the total cost’.

The use of these informal strategies by students in thesolution of propor tions problems is consistent withthe hypothesis that multiplicative reasoning developsfrom the schema of one-to-many correspondences:students may be simply using numbers instead ofobjects when reasoning about the quantities in theseproblems. In the same way that they build up thequantities with objects, they can build up the quantitieswith numbers. It is unlikely that students are thinking ofthe scalar relation and quantifying it when they solv eproblems by means of building up strategies. We thinkthat it can be concluded with some certainty thatstudents realise that whatever transformation theymake, for example, to the number of people in Har t’sonion soup problem, they must also make to thequantities of ingredients. It is even less likely that theyhave an awareness of the ratio between the twodomains of measures and have reached anunderstanding of a reflectively abstracted constantratio, in Thompson’s terms.

These results provoke the question of the role ofteaching in developing students’ understanding offunctional relations. Studies of high-school studentsand adults with limited schooling in Br azil throw somelight on this issue . They show that instruction aboutmultiplication and division or about propor tions perse is neither necessary for people to be able to solveproportions problems nor sufficient to promotestudents’ thinking about functional relations. Nunes,Schliemann and Carraher (1993) have shown thatfishermen and foremen in the construction industry,who have little formal school instruction, can solveproportions problems that are novel to them in threeways: (a) the problems use values that depart fromthe values they normally work with; (b) they areasked to calculate in a direction which they normallydo not have to think about; or (c) the content of theproblem is different from the problems with whichthey work in their everyday lives.



Foremen in the construction industry have to workwith blue-prints as representations of distances inthe buildings under construction. They haveexperience with a cer tain number of conventionallyused scales (e.g. 1:50, 1:100 and 1:1000). When theywere provided with a scale dr awing that did not f itthese specifications (e.g. 1:40) and did not indicatethe ratio used (e.g. they were shown a distance onthe blue-print and its value in the b uilding), mostforemen were able to use correspondences tofigure out what the scale would be and thencalculate the measure of a wall from its measure onthe blueprint. They were able to do so even whenfractions were involved in the calculations and thescale had an unexpected format (e.g. 3 cm:1 m usesdifferent units whereas scales typicall y use the sameunit) because they have extensive experience inmoving across units (metres, centimetres andmillimetres). Completely illiterate foremen (N = 4),who had never set foot in a school due to their lif ecircumstances, showed 75% correct responses tothese problems. In contrast, students who had beentaught the formal method known as the rule ofthree, which involves writing an equation of theform a/b = c/d and solving for the unknown value,performed significantly worse (60% correct). Thusschooling is not necessar y for multiplicativereasoning to develop and propor tions problems tobe solved correctly, and teaching students a gener alformula to solve the problem is not a guarantee thatthey will use it when the oppor tunity arises.

These studies also showed that both secondar yschool students and adults with relativ ely little

schooling were more successful when they coulduse building up strategies easily, as in problems ofthe type A in Figure 4.3. Problem B uses the samenumbers but arranged in a way that building upstrategies are not so easily implemented; the relationthat is easy to quantify in prob lem B is the functionalrelation.

The difference in students’ rate of success across thetwo types of problem was significant: they solvedabout 80% of type A problems correctly and only35% of type B problems. For the adults (fishermen),there was a difference between the rate of correctresponses (80% correct in type A and 75% correctin type B) but this was not statisticall y significant.Their success, however, was typically a result ofprowess with calculations when building up aquantity, and very few answers might have resultedfrom a quantification of the functional relation.

These results suggest three conclusions.

• Reasoning about quantities when solvingproportional problems seems to be an extensionof correspondence reasoning; schooling is notnecessary for this development.

• Most secondary school students seem to use thesame schema of reasoning as younger students;there is little evidence of an impact of instructionon their approach to proportional problems.

• Functional reasoning is more challenging and is notguaranteed by schooling; teaching students a


Figure 4.3: For someone who can easily think about scalar or functional relations, there should be no difference in the level of difficulty of

the two problems. For those who use building up strategies and can only work with quantities, problem A is significantly easier.

Measure 1

3

12

Measure 2

x 4

5

?

A

Measure 1

3

5

Measure 2

x 4

12

?

B

formal method, which can be used as easily forboth problem types, does not make functionalreasoning easier (see Paper 6 for furtherdiscussion).

The results obser ved with Brazilian students donot differ from those obser ved by Vergnaud(1983) in France, and Har t (1981 b; 1984) in theUnited Kingdom. The novelty of these studies isthe demonstration that the informal knowledgeof multiplicative reasoning and the ability to solv emultiplicative reasoning problems throughcorrespondences develop into more abstr actschemas that allow for calculating in the absenceof concrete forms of representation, such asmaniputatives and tallies. Both the students andthe adults with low levels of schooling were ableto calculate, for example, what should the actualdistances in a building be from their siz e inblueprint drawings. Relatively unschooled adultswho have to think about propor tions in thecourse of their occupations and secondar yschool students seem to rel y on these moreabstract schemas to solve propor tions problems.The similarity between these two groups, ratherthan the differences, in the forms of reasoningand rates of success is str iking. These resultssuggest that informal knowledge ofcorrespondences is a powerful thinking schemaand that schooling does not easily transform itinto a more powerful one by incorporatingfunctional understanding into the schema.

Different hypotheses have been considered in theexplanation of why this informal knowledge seemsso resistant to change . Har t (1981 b) considered thepossibility that this may rest on the diff iculty of thecalculations but the comparisons made by Nunes,Schliemann and Carraher (1993) rule out thishypothesis: the difficulty of the calculations was heldconstant across problems of type A and type B , andquantitative reasoning on the basis of the functionalrelation remained elusive.

An alternative explanation, explored by Vergnaud(1983) and Har t (personal communication), is thatinformal strategies are resistant to change becausethey are connected to reasoning about quantities,and not about relations. It makes sense to say that ifI buy half as much fish, I pay half as much money:these are manipulations of quantities and theirrepresentations. But what sense does it make todivide kilos of fish by money?

There has been some discussion of the diff erencebetween reasoning about quantities and relations inthe literature. However, we have not been able tofind studies that establish whether the difficulty ofthinking about relations might be at the root ofstudents’ difficulties in transforming their informal intoformal mathematics knowledge. The educationalimplications of these hypotheses are considerable butthere is, to our knowledge, no research thatexamines the issue systematically enough to providea firm ground for pedagogical developments. Theimportance of the issue must not be underestimated,particularly in the United Kingdom, where studentsseem do to well enough in the inter nationalcomparisons in additive reasoning but not inmultiplicative reasoning problems (Beaton, Mullis,Martin, Gonzalez, Kelly AND Smith, 1996, p. 94–95).

Summary

We draw some educational implications from thesestudies, which must be seen as hypotheses aboutwhat is impor tant for successful teaching ofmultiplicative reasoning about relations.

1 Before children are taught about multiplication and division in school, they already have schemasof action that they use to solv e multiplicativereasoning problems. These schemas of actioninvolve setting up correspondences between two variables and do not appear to dev elop from the idea of repeated addition. This informalknowledge is a predictor of their success inlearning mathematics and should be dr awn uponexplicitly in school.

2 Students’ schemas of multiplicative reasoningdevelop sufficiently for them to apply theseschemas to numbers, without the need to useobjects or tallies to represent quantities. But theyseem to be connected to quantities, and itappears that students do not f ocus on therelations between quantities in multiplicativereasoning problems. This informal knowledgeseems to be resistant to change under currentconditions of instruction.

3 In many previous studies, researchers drew theconclusion that students’ problems in understandingproportional relations were explained by theirdifficulties in thinking multiplicatively. Today, it seemsmore likely that students’ problems are based ontheir difficulty in thinking about relations, and not



about quantities, since even young children succeedin multiplicative reasoning problems.

4 Teaching approaches might be more successful inpromoting the formalisation of students’ informalknowledge if: (a) they draw on the students’informal knowledge rather than ignore it; (b) theyoffer the students a way of representing therelations between quantities and promote anawareness of these relations; and (c) they use avariety of situational contexts to help studentsextend their knowledge to new domains ofmultiplicative reasoning.

We examine now the conceptual underpinnings oftwo rather different teaching approaches to thedevelopment of multiplicative reasoning in search for more specific hypotheses regarding how greaterlevels of success can be achieved by U.K. students.

The challenge in attempting a synthesis of results isthat there are many ways of classifying teachingapproaches and there is little systematic research thatcan provide unambiguous evidence. The difficulty isincreased by the fact that by the time students aretaught about propor tions, some time between theirthird and their sixth year in school, they haveparticipated in a diversity of pedagogical approachesto mathematics and might already have distinctattitudes to mathematics learning. However, weconsider it plausible that systematic investigation ofdifferent teaching approaches would prove invaluablein the analysis of pathways to help childrenunderstand functional relations. In the subsequentsection, we explore two different pathways byconsidering the types of representations that areoffered to students in order to help them becomeaware of functional relations.

Representing functional relationsThe working hypothesis we will use in this section is that in order to become explicitl y aware ofsomething, we need to represent it. This hypothesisis commonplace in psychological theor ies: it is par tof general developmental theories, such as Piaget’stheory on reflective abstraction (Piaget, 1978; 2001;2008) and Karmiloff-Smith’s theory ofrepresentational re-descriptions in development(Karmiloff-Smith, 1992; Karmiloff-Smith and Inhelder,1977). It is also used to descr ibe development inspecific domains such as language and liter acy

(Gombert, 1992; Karmiloff-Smith, 1992), memory(Flavell, 1971) and the under standing of others(Flavell, Green and Flavell, 1990). It is beyond thescope of this work to review the liter ature onwhether representing something does help usbecome more aware of the represented meaning;we will treat this as an assumption.

The hypothesis concerning the impor tance ofrepresentations will be used in a diff erent form here.Duval (2006) pointed out that ‘the part played bysigns in mathematics, or more exactly by semioticsystems of representation, is not only to designatemathematical objects or to communicate but also towork on mathematical objects and with them.’ (p.107). We have so far discussed the quantif ication ofrelations, and in par ticular of functional relations, as ifthe representation of functional relations could onl y beattained through the use of numbers. Now we wish tomake explicit that this is not so . Relations, includingfunctional relations, can be represented by numbersbut there are many other ways in which relations canbe represented before a number is attributed to them;to put it more forcefully, one could say that relationscan be represented in different ways in order tofacilitate the attribution of a number to them.

When students are taught to wr ite an equation of the form a/b = c/x, for example, to represent aproportions problem in order to solve for x, thisformula can be used to help them quantify therelations in the problem. Har t (1981 b) repor ts thatthis formula was taught to 100 students in oneschool where she car ried out her investigations ofproportions problems but that it was only used by20 students, 15 of whom were amongst the highachievers in the school. This formula can be used to explore both scalar and functional relations in aproportions problem but it can also be taught as arule to solve the problem without any explorationof the scalar or functional relations that itsymbolises. In some sense , students can learn to usethe formula without developing an awareness of thenature of the relations between quantities that areassumed when the formula is applied.

Researchers in mathematics education have beenaware for at least two decades that one needs toexplore different forms of representation in order toseek the best ways to promote students’ awarenessof reasoning about the relations in a propor tionsproblem. It is likely that the large amount of researchon proportional reasoning, which exposed students’difficulties as well as their reliance on their o wn


methods even after teaching, played a crucial role inthis process. It did undoubtedly raise teachers’ andresearchers’ awareness that the representationthrough formulae (a/b = c/x) or algorithms did notwork all that well. In this section, we will seek toexamine the under lying assumptions of two verydifferent approaches to teaching students aboutproportions.

Two approaches to therepresentation of functional relationsKieren (1994) suggested that there are tw oapproaches to research about, and to the teachingof, multiplicative reasoning in school. The first isanalytic-functional: it is human in f ocus, andinvestigates actions, action schemes and oper ationsused in giving meaning to multiplicative situations.The second is algebraic: this focuses on mathematicalstructures, and investigates structures used in thisdomain of mathematics. Although the investigationof mathematics structures is not incompatible withthe analytic-functional approach, these arealternatives in the choice of star ting point forinstruction. They delineate radically distinct pathwaysfor guiding students’ learning trajectories.

Most of the research car ried out in the past aboutstudents’ difficulties did not describe what sort ofteaching students had par ticipated in; one of theexceptions is Har t’s (1981 b) descr iption of theteaching in one school, where students were taughtthe a/b = c/d, algebraic approach: the vast major ity ofthe students did not use this f ormula when they wereinterviewed about propor tions in her study, and itsuse was confined to the higher achievers in their tests.It is most urgent that a research programme thatsystematically compares these two approaches shouldbe carried out, so that U.K. students can benefit frombetter understanding of the consequences of howthese different pathways contribute to learning ofmultiplicative reasoning. In the two subsequentsections, we present one well developed programmeof teaching within each approach.

The analytic-functional approach:from schematic representations ofquantities in correspondence toquantifying relationsStreefland and his colleagues (Streefland, 1984; 1985a and b; van den Br ink and Streefland, 1979)

highlighted the role that dr awing and visualisationcan play in making children aware of relations. In aninitial paper, van den Br ink and Streefland (1979)analysed a boy’s reactions to propor tions indrawings and also pr imary school children’sreactions in the classroom when visual propor tionswere playfully manipulated by their teacher.

The boy’s reactions were taken from a discussionbetween the boy and his father. They saw a posterfor a film, where a man is br avely standing on awhale and tr ying to harpoon it. The whale’s size isexaggerated for the sake of sensation. The fatherasked what was wrong with the picture and the boyeventually said: ‘I know what you mean. That whaleshould be smaller. When we were in England wesaw an orca and it was onl y as tall as three men’(van den Brink and Streefland, 1979, p. 405). In linewith Bryant (1974), van den Br ink and Streeflandargued that visual propor tions are par t of the basicmechanisms of perception, which can be used inlearning in a var iety of situations, and suggested thatthis might be an excellent star t for making childrenaware of relations between quantities.

Van den Brink and Streefland then developedclassroom activities where six- to eight-year-oldchildren explored propor tional relations in drawings.Finally, the teacher showed the children a picture ofa house and asked them to mark their own heighton the door of the house . The children engaged inmeasurements of themselves and the door of theclassroom in order to tr anspose this size relation tothe drawing and mark their heights on the door. Thisactivity generated discussions relevant to thequestion of propor tions but it is not possib le toassess the effect of this activity on theirunderstanding of propor tions, as no assessmentswere used. The lesson ended with the teachershowing another par t of the same picture: a gir lstanding next to the house . The girl was much tallerthan the house and the children concluded that thiswas actually a doll house. Surprise and playfulnesswere considered by Streefland an impor tant factorin children’s engagement in mathematics lessons.

Van den Brink and Streefland suggested that childrencan use perceptual mechanisms to reflect aboutproportions when they judge something to be out ofproportion in a picture. They argued that it is not onlyof psychological interest but also of mathematical-didactical interest to discover why children can reasonin ratio and proportion terms in such situations,abstracting from perceptual mechanisms.



Streefland (1984) later developed fur ther activitiesin a lesson ser ies with the theme ‘with a giant’sregard’, which star ted with activities that exploredthe children’s informal sense of propor tions andprogressively included mathematical representationsin the lesson. The children were asked, for example,to imagine how many steps would a normal mantake to catch up with one of the giant’ s steps; later,they were asked to represent the man’s and thegiant’s steps on a number line and subsequently bymeans of a table. Figure 4.4 presents one exampleof the type of diagr am used for a visual comparison.

In a later paper, Streefland (1985 a and b) pur suedthis theme fur ther and illustrated how the diagramsused to represent visual meanings could be used ina progressively more abstract way, to representcorrespondences between values in other problemsthat did not have a visual basis. This was illustratedusing, among others, Hart’s (1981 b) onion soupproblem, where a recipe for onion soup for 8people is to be adapted f or 4 or 6 people . Thediagram proposed by Streefland, which the teachershould encourage the pupils to constr uct, showsboth (a) the correspondences between the values,which the children can find using their own, informalbuilding up strategies, and (b) the value of the scalartransformation. See Figure 4.5 for an example.

Streefland suggested that these schematicrepresentations could be used later in Har t’s onionsoup problem in a vertical orientation, morecommon for tables than the above diagram, andwith all ingredients listed on the same table indifferent rows. The top row would list the numberof people, and the subsequent rows would list eachingredient. This would help students realise that thesame scalar transformation is applied to all the

ingredients for the taste to be preserved when theamounts are adjusted. Streefland argued that ‘theratio table is a permanent record of propor tion asan equivalence relation, and in this way contributesto acquiring the correct concept. Applying the ratiotable contributes to the detachment from thecontext… In this quality the r atio table is, as it were,a unifying model for a variety of ratio contexts, aswell as for the various manifestations of ratio… The ratio table can contribute to discovering, making conscious and applying all proper ties thatcharacterise ratio-preserving mappings and to theiruse in numerical problems’ (Streefland, 1985, a, p.91). Ratio tables are then related to gr aphs, wherethe relation between two variables can be discussedin a new way.

Streefland emphasises that ‘mathematizing realityinvolves model building’ (Streefland, 1985a, p. 86); sostudents must use their intuitions to develop amodel and then learn how to represent it in orderto assess its appropriateness. He (Streefland, 1985 b;in van den Heuvel-Panhuizen, 2003) argued thatchildren’s use of such schematic models of situationsthat they understand well can become a model fornew situations that they would encounter in thefuture. The representation of their knowledge insuch schematic form helps them under stand what isimplied in the model, and make explicit a relationthat they had used only implicitly before.

This hypothesis is in agreement with psychologicaltheories that propose that reflection andrepresentation help make implicit knowledge explicit(e.g. Karmiloff-Smith, 1992; Piaget, 2001). However,the concept proposes a pedagogical str ategy inStreefland’s work: the model is chosen by theteacher, who guides the student to use it and adds


Figure 4.4: The giant’s steps and the man’s steps on a line; this drawing can be converted into a number line and a table which displays the

numerical correspondences between the giant’s and the man’s steps.

elements, such as the explicit representation of thescalar factor. The model is chosen because it can beeasily stripped of the specifics in the situation andbecause it can help the students mo ve from thinkingabout the context to discussing the mathematicalstructures (van den Heuvel-Panhuizen, 2003). Sochildren’s informal knowledge is to be transformedinto formal knowledge through changes inrepresentation that highlight the mathematicalrelations that remain implicit when students f ocuson quantities.

Finally, Streefland also suggested that teachingchildren about ratio and propor tions could star tmuch earlier in primary school and should be seenas a longer project than prescribed by currentpractice. Star ting from children’s informal knowledgeis a crucial aspect of his proposal, which is based on Freudenthal’s (1983) and Vergnaud’s (1979)argument that we need to know about children’simplicit mathematical models for problem situations,not just their ar ithmetic skills, when we want todevelop their problem solving ability. Streeflandsuggests that, besides the visual and spatial relationsthat he worked with, there are other conceptswhich children aged eight to ten years can grasp inprimary school, such as comparisons between thedensity (or crowdedness) of objects in space andprobabilities. Other concepts, such as percentagesand fractions, were seen by him as related toproportions, and he argued that connections shouldbe made across these concepts. However, Streeflandconsidered that they mer ited their own analyses inthe mathematics classroom. He argued, citingVergnaud (1979) that ‘different proper ties, almostequivalent to the mathematician, are not allequivalent for the child (Vergnaud, 1979, p. 264). Sohe also developed programmes for the teaching of

percentages (Streefland and van den Heuvel-Panhuizen, 1992; van den Heuvel-Panhuizen, 2003)and fractions (Streefland, 1993; 1997). Marja van denHeuvel-Panhuizen and her colleagues (Middletonand van den Heuvel-Panhuizen, 1995; Middleton, vanden Heuvel-Panhuizen and Shew, 1998) detailed theuse of the ratio table in teaching students in their3rd year in school about percentages andconnecting percentages, fractions and propor tions.

In all these studies, the use of the ratio table is seenas a tool for computation and also for discussion ofthe different relations that can be quantified in theproblem situations. Their advice is that teacher sshould allow students to use the table at their own level of understanding but always encouragestudents to make their reasoning explicit. In this way,students can compare their own reasoning withtheir peers’ approach, and seek to improve theirunderstanding through such compar isons.

Streefland’s proposal is consistent with many of theeducational implications that we drew from previousresearch. It star ts from the representation of thecorrespondences between quantities and moves tothe representation of relations. It uses schematicdrawings and tables that bring to the fore of eachstudent’s activity the explicit representation of thetwo (or more) measures that are in volved in theproblem. It is grounded on students’ informalknowledge because students use their building upsolutions in order to constr uct tables and schematicdrawings. It systematizes the students’ solutions intables and re-represents them by means of graphs.After exploring students’ work on quantities,students’ attention is focused on scalar relations,which they are asked to represent explicitly using thesame visual records. It draws on a var iety of contexts


Figure 4.5: A table showing the answers that the children can build up and the representation of the scalar transformations.


that have been previously investigated and whichstudents have been able to handle successfully. Finally,it uses graphs to explore the linear relations that areimplied in propor tional reasoning.

To our knowledge, there is no systematicinvestigation of how this proposal actually workswhen implemented either exper imentally or in the classroom. The work by Treffers (1987) andGravemeijer (1997) on the formalisation of students’understanding of multiplication and division focusedon the transition from computation with small tolarge numbers. The work by van den Heuvel-Panhuizen and colleagues focused on the use ofratio tables in the teaching of percentages andequivalence of fractions. In these paper s, the authorsoffer a clear descr iption of how teachers can guidestudents’ transition from their own intuitions to amore formal mathematical representation of thesituations. However, there is no assessment of howthe programmes work and limited systematicdescription of how students’ reasoning changes as the programmes develop.

The approach by researchers at the FreudenthalInstitute is described as developmental research andaims at constructing a curriculum that is designed and improved on the basis of students’ responses(Gravemeijer, 1994). This work is crucial to thedevelopment of mathematics education. However, itdoes not allow for the assessment of the eff ects ofspecific teaching approaches, as more exper imentalintervention research does. It leaves us with the sensethat the key to formalising students’ multiplicativereasoning may be already to hand but we do notknow this yet. Systematic research at this stage wouldoffer an invaluable contribution to the understandingof how students learn and to education.

Streefland was not the only researcher to proposethat teaching students about multiplicative relationsshould star t from their informal understanding ofthe relations between quantities and measures.Kaput and West (1994) developed an experimentalprogramme that took into account students’ buildingup methods and sought to formalise them throughconnecting them with tables. Their aim was to helpstudents create composite units of quantity, wherethe correspondences between the measures wererepresented iconically on a computer screen. Forexample, if in a problem the quantities are 3umbrellas for 2 animals, the computer screen woulddisplay cells with images for 3 umbrellas and 2animals in each cell, so that the group of umbrellas

and animals became a higher-level unit. The cells inthe computer screen were linked up with tables,which showed the values corresponding to the cellsthat had been filled with these composite units: forexample, if 9 cells had been filled in with the iconicrepresentations, the table displayed the values for 1through 9 of the composite units in columns headedby the icons for umbrellas and animals. Subsequently,students worked with non-integer values for theratios between the quantities: for example, theycould be asked to enlarge a shape and thecorresponding sides of the two figures had a non-integer ratio between them (e.g. one figure had aside 21 cm long and the other had thecorresponding side 35 cm long).

Kaput and West’s programme was delivered over 11lessons in two experimental classes, which included31 students. Two comparison classes, with a total of29 students, followed the instruction previously usedby their teachers and adopted from textbooks. Onecomparison class had 13 lessons: the first five lessonswere based on a textbook and covered exercisesinvolving ratio and propor tion; the last eightconsisted of computer-based activities using functionmachines with problems about rate and profit. Thesecond comparison class had only three lessons; thecontent of these is not descr ibed by the authors.The classes were not assigned randomly to thesetreatments and it is not clear ho w the teacherswere recruited to par ticipate in the study.

At pre-test, the students in the exper imental andcomparison classes did not differ in the percentageof correct solutions in a multiplicative reasoning test.At post-test, the students in the exper imental groupsignificantly out-performed those in both compar isonclasses. They also showed a larger increase in the use of multiplicative strategies than students in thecomparison classes. It is not possib le from Kaput andWest’s report to know whether these were buildingup, scalar or functional solutions, as they areconsidered together as multiplicative solutions.

In spite of the limitations pointed out, the studydoes provide evidence that students benef it fromteaching that develops their building up strategiesinto more formalised approaches to solution, bylinking the quantities represented by icons of objectsto tables that represent the same quantities. Thisresult goes against the view that inf ormal methodsare an obstacle to students’ learning in and ofthemselves; it is more likely that they are an obstacleif the teaching they are exposed to does not b uild


on the students’ informal strategies and does nothelp students connect what they know with thenew forms of mathematical representation that the teacher wants them to learn.

The algebraic approach:representing ratios and equivalencesIn contrast to the functional approach to theteaching of propor tions that was descr ibed in theprevious section, some researchers have proposedthat teaching should not star t from students’understanding of multiplicative reasoning, but from aformal mathematical definition of propor tions as theequality of two ratios. We found the most explicitjustification for this approach in a recent paper b yAdjiage and Pluvinage (2007). Adjiage and Pluvinage,citing several authors (Hart, 1981 b); Karplus, Pulosand Stage, 1983; Lesh, Post, and Behr, 1988), arguethat building up strategies are a weak indicator ofproportionality reasoning and that the link betw een‘interwoven physical and mathematicalconsiderations, present in the build-up strategy’(2007, p. 151) should be the representation ofproblems through rational numbers. For example, amixture that contains 3 par ts concentrate and 2parts water should be represented as 3/5, usingnumbers or marks on a number line. The level 1,which corresponds to an iconic representation of theparts used in the mixture , should be transformedinto a level 2, numerical representation, and studentsshould spend time working on such transformations.Similarly, a scale drawing of a figure where one side isreduced from a length of 5 cm to 3 cm should berepresented as 3/5, also allowing for the move froman iconic to a numerical representation. Finally, therepresentation by means of an equivalence of r atios,as in 3/5 = 6/10, should be introduced, to transformthe level 2 into a level 3 representation. The sameresults could be obtained by using decimals ratherthan ordinary fractions representations.

In brief, level 1 allows for an ar ticulation betweenphysical quantities: the students may realise that amixture with 3 par ts concentrate and 2 par ts watertastes the same as another with 6 par ts concentrateand 4 par ts water. Level 2 allows for articulationsbetween the physical quantities and a mathematicalrepresentation: students may realise that two differentsituations are represented by the same number. Level3 allows for articulations within the mathematicaldomain as well as conversions from one system ofrepresentation to another : 3/5 = 0.6 or 6/10.

Adjiage and Pluvinage (2007) ar gue that it isimportant to separate the physical from themathematical initially in order to ar ticulate themlater, and propose that three r ational registersshould be used to facilitate students’ attainment oflevel 3: linear scale (a number line with resourcessuch as subdividing, sliding along the line , zooming),fractional writing, and decimal wr iting should beused in the teaching of r atio and propor tions.

It seems quite clear to us that this proposal doesnot star t from students’ intuitions or strategies forsolving multiplicative reasoning problems, but ratheraims to formalise the representation of physicalsituations from the star t and to teach students howto work with these formalisations. The authorsindicate that their programme is inspired by Duval’s(1995) theory of the role of representations inmathematical thinking but we believe that there isno necessary link between the theor y and thisparticular approach to teaching students about ratio and propor tions.

In order to convey a sense for the programme,Adjiage and Pluvinage (2007) descr ibe five momentsexperienced by students. The researchers workedwith two conditions of implementation, which they termed the full exper iment and the par tialexperiment. Students in the par tial experiment did not par ticipate in the first moment using acomputer; they worked with pencil and paper tasksin moments 1 and 2.

• Moment 1The students are presented with threelines, divided into equal spaces. They are told thatthe lines are drawn in different scales. The lineshave different numbers of subdivisions – 5, 3 and 4,respectively. Points equivalent to 3/5, 2/3 and 1/4are marked on the line. The students are asked tocompare the segments from the origin to the pointon the line. This is seen as a purely mathematicalquestion, executed in the computer by students inthe full experiment condition. The computer hasresources such as dividing the lines into equalsegments, which the students can use to executethe task.

• Moment 2 A similar task is presented with paperand pencil.

• Moment 3The students are shown two picturesthat represent two mixtures: one is made with 3 cups of chocolate and 2 of milk (the cups areshown in the pictures in different shades) and the



other with 2 cups of chocolate and 1 of milk. Thestudents are asked which mixture tastes morechocolaty. This problem aims to link the physicaland the mathematical elements.

• Moment 4The students are asked in what way arethe problems in moments 2 and 3 similar. Studentsare expected to show on the segmented linewhich portion corresponds to the cups ofchocolate and which to the cups of milk.

• Moment 5This is described as institutionalization inDouady’s (1984) sense: the students are asked tomake abstractions and express rules. For instance,expressions such as these are expected: ‘7 dividedby 4 is equal to seven fourths (7 ÷ 4 = 7/4)’;‘Given an enlargement in which a 4 cm lengthbecomes a 7 cm length, then any length to beenlarged has to be multiplied by 7/4.’ (Adjiage andPluvinage, 2007, pp. 160–161).

The teaching programme was implemented overtwo school years, star ting when the students werein their 6th year (estimated age about 11 years) inschool. A pre-test was given to them before theystarted the programme; the post-test was carriedout at the end of the students’ 7th year (estimatedage about 12 years) in school.

Adjiage and Pluvinage (2007) worked with anexperienced French mathematics teacher, whotaught two classes using their exper imentalprogramme. In both classes, the students solved the same problems but in one class, referred to as a par tial experimental, the students did not use thecomputer-based set of activities whereas in theother one, referred to as full exper imental, they had access to the computer activities. The teachermodified only his approach to teaching r atio andproportions; other topics in the year were taught aspreviously, before his engagement in the exper iment.

The performance of students in these twoexperimental classes was compared to resultsobtained by French students in the same region (thebaseline group) in a national assessment and also tothe performance of non-specialist, prospectiveschool-teachers on a ratio and propor tions task. Thetasks given to the three groups were not the samebut the researchers considered them comparable.

Adjiage and Pluvinage repor ted positive resultsfrom their teaching programme. When the pupils inthe experimental classes were in grade 6 they had a

low rate of success in r atio and propor tionsproblems: about 13%. At the end of gr ade 7, theyattained 39% correct answers whereas the studentsin the sample from the same region (baselinegroup) attained 15% cor rect responses in thenational assessment. The students in the fullexperimental classes obtained significantly betterresults than those in the par tial experimentalclasses but the researchers did not provide separatepercentages for the two groups. Prospectiveteachers attained 83% on similar prob lems. Theresearchers were not satisfied with these resultsbecause, as they point out, the students performedsignificantly worse than the prospective teachers,who were taken to represent educated adults.

Although there are limitations to this study, itdocuments some progress among the students in the experimental classes. However, it is difficult to know from their repor t how much time wasdevoted to the teaching progr amme over the twoyears and how this compares to the instr uctionreceived by the baseline group.

In brief, this approach assumes that students’ maindifficulties in solving propor tions problems result fromtheir inability to co-ordinate different forms ofmathematical representations and to manipulate them.There is no discussion of the question of quantitiesand relations and there is no attempt to mak estudents aware of the relations between quantities in the problems. The aims of teaching are to:• develop students’ understanding of how to use

number line and numerical representationstogether in order to compare rational numbers

• promote students’ reflection on how the numericaland linear representations relate to problemsituations that involve physical elements (3 cups of chocolate and 2 of milk)

• promote students’ understanding of the relationsbetween the different mathematicalrepresentations and their use in solving problems.

A comparison between this example of thealgebraic approach and the functional approach asexemplified by Streefland’s work suggests that thisalgebraic approach does not offer students theopportunity to distinguish between quantities andrelations. The three forms of representation offeredin the Adjiage and Pluvinage programme focus onquantities; the relations between quantities are leftimplicit. Students are expected to recognise thatmixtures of concentrate that are numericallyrepresented as 3/5 and 6/10 are equivalent. In the


number line, they are expected to manipulate therepresentations of quantities in order to comparethem. We found no evidence in the descr iption oftheir teaching programme that students were askedto think about their implicit models of the situationsand explicitly discuss the transformations that wouldmaintain the equivalences.

Summary

1 It is possible to identify in the liter ature two ratherdifferent views of how students can best be taughtabout multiplicative reasoning. Kieren identifiedthese as the functional and the algebr aic approach.

2 The functional approach proposes that teachingshould star t from students’ understanding ofquantities and seek to make their implicit modelsof relations between quantities explicit.

3 The algebraic approach seeks to representquantities with mathematical symbols and leadstudents to work with symbols as soon aspossible, disentangling physical and mathematicalknowledge.

4 There is no systematic compar ison between thesetwo approaches. Because their explicit descriptionis relatively recent, this paper is the f irst detailedcomparison of their characteristics and provides a basis for future research.

Graphs and functional relationsThe previous sections focused on the visual and numerical representations of relations. Thissection will briefly consider the question of therepresentation of relations in the Car tesian plane.We believe that this is a f orm of representation thatmerits fur ther discussion because of the additionalpower that it can add to students’ reflections, ifproperly explored.

Much research on how students interpret graphshas shown that graph reading has to be learned, justas one must learn how to read words or numbers.Similarly to other aspects of mathematics lear ning,students have some ideas about reading gr aphsbefore they are taught, and researchers agree thatthese ideas should be considered when one designsinstruction about graph reading. Several papers can

be of interest in this context but this research is not reviewed here, as it does not contr ibute to thediscussion of how graphs can be used to helpstudents understand functional relations (forcomplementary reviews, see Friel, Curcio and Br ight,2001; Mevarech and Kramarsky, 1997). We focushere on the possibilities of using gr aphs to helpstudents understand functional relations.

As reported earlier in this chapter, Lieven Verschaffeland his colleagues have shown that students makemultiplicative reasoning errors in additive situationsas well as additive errors in multiplicative situations,and so there is a need for students to be offeredopportunities to reflect on the nature of the relationbetween quantities in problems. Van Dooren, Bock,Hessels, Janssens and Verschaffel (2004) go as far assuggesting that students fall prey to what they call anillusion of linearity, but we think that they haveoverstated their case in this respect. In fact, some ofthe examples that they use to illustrate the so-calledillusion of linearity are indeed examples of linearfunctions, but perhaps not as simple as the typicallinear functions used in school. In two examples oftheir ‘illusion of linearity’ discussed here, there is alinear function connecting the two variables but theproblem situation is more complex than many ofthe problems used in schools when students aretaught about ratio and propor tions. In our view,these problems demonstrate the impor tance ofworking with students to help them reflect aboutthe relations between the quantities in thesituations.

In one example, taken from Cramer, Post andCurrier (1993) and discussed ear lier on in thispaper, two girls, Sue and Julie , are supposed to berunning on a track at the same speed. Sue star tedfirst. When she had run 9 laps, Julie had run 3 laps.When Julie completed 15 laps, how many laps hadSue run? Although prospective teachers wronglyquantified the relation between the number of lapsin a multiplicative way, we do not think that they f ellfor the ‘illusion of linearity’, as argued by De Bock,Verschaffel et al., (2002; 2003). The function actuallyis linear, as illustrated in Figure 4.6. However, theintercept between Sue’s and Julie’s numbers of lapsis not at zero, because Sue must have run 6 lapsbefore Julie starts. So the prospective teachers’ erroris not an illusion of linearity but an inability to dealwith intercepts different from zero.

Figure 4.6 shows that three different curves wouldbe obtained: (1) if the girls were running at the



same speed but one star ted before the other, as in the Cramer, Post and Currier problem; (2) if one were running faster than the other and thisdifference in speed were constant; and (3) if theystarted out running at the same speed but one gir lbecame progressively more tired whereas the otherwas able to speed up as she war med up. Studentsmight hypothesise that this latter example is betterdescribed by a quadratic than a linear function, if thegirl who was getting tired went from jogging towalking, but they could find that the quadraticfunction would exaggerate the difference betweenthe girls: how could the strong gir l run 25 laps whilethe weak one ran 5?

The aim of this illustr ation is to show that relationsbetween quantities in the same context can var y andthat students can best investigate the nature of therelation between quantities is if they have a tool todo so. Streefland suggested that tables and graphscan be seen as tools that allo w students to explorerelations between quantities; even though they couldbe used to help students’ reasoning in this problem,we do not know of research where it has been used.

Van Dooren, et al. (2004) used graphs and tables inan intervention programme designed to help

student overcome the ‘illusion of linearity’ in asecond problem, which we argue also involvemislabelling of the phenomenon under study. Inseveral studies, De Bock, Van Dooren and theircolleagues (De Bock, Verschaffel and Janssens, 1998;De Bock, Van Dooren, Janssens and Verschaffel, 2002;De Bock, Verschaffel and Janssens, 2002; De Bock,Verschaffel, Janssens, Van Dooren and Claes, 2003)claim to have identified this illusion in questionsexemplified in this problem: ‘Farmer Carl needsapproximately 8 hours to manure a square piece ofland with a side of 200 m. How many hours wouldhe need to manure a square piece of land with aside of 600 m?’ De Bock, Van Dooren and colleaguesworked with relatively large numbers of Belgianstudents across their many studies, in the age r ange12 to 16 years. They summarise their findings byindicating that ‘the vast majority of students (even16-year-olds) failed on this type of prob lem becauseof their alarmingly strong tendency to apply linearmethods’ (Van Dooren, Bock, Hessels, Janssens andVerschaffel, 2004, p. 487) and that even withconsiderable support many students were not able to overcome this difficulty. Some students who did become more cautious about o ver-using alinear model, subsequently failed to use it when itwas appropriate.


Figure 4.6: Three graphs showing different relations between the number of laps run by two people over time

We emphasise here that in this problem, as in theprevious one, students were not falling prey to anillusion of linearity. The area of a rectangular f igure isindeed proportional to its side when the other side isheld constant; this is a case of m ultiple proportionsand thus the linear relation between the side andthe area can only be appreciated if the other sidedoes not change. Because the rectangle in theirproblem is the par ticular case of a square , if oneside changes, so does the other ; with both measureschanging at the same time, the area is not a simplelinear function of one of the measures.

Van Dooren et al. (2004) describe an inter ventionprogramme, in which students used gr aphs andtables to explore the relation between the measureof the side of a square , its area and its per imeter. Theintervention contains interesting examples in whichstudents have the oppor tunity to examine diagramsthat display squares progressively larger by 1 cm, inwhich the square units (1 cm2) are clear ly marked.Students thus can see that when the side of a squareincreases, for example, from 1 cm to 2 cm, its areaincreases from 1 cm2 to 4 cm2, and when theincrease is from 2 cm to 3 cm, the area increases to9 cm2. The graph associated with this table displays aquadratic function whereas the gr aph associate withthe perimeter displays a linear function.

Their programme was not successful in promotingstudents’ progress: the experimental group significantlydecreased the rate of responses using simpleproportional reasoning to the area prob lems but alsodecreased the rate of correct responses to perimeterproblems, although the per imeter of a square isconnected to its sides by a simple propor tion.

We believe that the lack of success of theirprogramme may be due not to a lack ofeffectiveness of the use of gr aphs and tables inpromoting students’ reasoning but from their use of an inadequate mathematical analysis of theproblems. Because the graphs and tables used onlytwo variables, measure of the side and measure ofthe area, the students had no opportunity toappreciate that in the area prob lem there is aproportional relation between area and each thetwo sides. The two sides var y at the same time inthe par ticular case of the square but in otherrectangular figures there isn’t a quadr atic relationbetween side and area. The relation between sidesand perimeter is additive, not multiplicative: ithappens to be multiplicative in the case of thesquare because all sides are equal; so to each

increase by 1 cm in one side corresponds a 4 cmincrease in the per imeter.

We think that it would be surprising if the studentshad made significant progress in under standing therelations between the quantities through theinstruction that they received in these problems:they were not guided to an appropr iate model ofthe situation, and worked with one measure , side,instead of two measures, base and height. One ofthe students remarked at the end of theintervention programme, after ten exper imentallessons over a two week period: ‘I really dounderstand now why the area of a square increases9 times if the sides are tr ipled in length, since theenlargement of the area goes in two dimensions. Butsuddenly I star t to wonder why this does not holdfor the perimeter. The perimeter also increases intwo directions, doesn’t it?’ (Van Dooren et al., 2004,p. 496). This student seems to have understood thatthe increase in one dimension of the square impliesa similar increase along the other dimension andthat these are multiplicatively related to the area butapparently missed the oppor tunity to understandthat sides are additively related to the perimeter.

In spite of the shor tcomings of this study, theintervention illustrates that it is possib le to relateproblem situations to tables and graphssystematically to stimulate students’ reflection aboutthe implicit models. It is a cur rent hypothesis bymany researchers (e.g. Carlson, Jacobs, Coe, Larsenand Hsu, 2002; Hamilton, Lesh, Lester and Yoon,2007; Lesh, Middleton, Caylor and Gupta, 2008) that modelling data, testing the adequacy of modelsthrough graphs, and comparing different model fitscan make an impor tant contribution to students’understanding of the relations between quantities. Itis consistently acknowledged that this process mustbe carefully designed: powerful situations must bechosen, clear means of hypothesis testing must beavailable, and appropriate teacher guidance shouldbe provided. Shortcomings in any of these aspectsof teaching experiments could easily result innegative results.

The hypothesis that modelling data, testing theadequacy of models through graphs, and comparingdifferent model fits can promote student’sunderstanding of different types of relationsbetween quantities seems entirely plausible but, toour knowledge, there is no research to pro vide clearsupport to it. We think that there are now manyideas in the literature that can be implemented to



assess systematically how effective the use of graphs and tables is as tools to support students’understanding of the different types of relation thatcan exist between measures. This research has thepotential to make a huge contribution to theimprovement of mathematics education in theUnited Kingdom.

Conclusions and implicationsThis review has identified results in the domain ofhow children learn mathematics that have significantimplications for education. The main points arehighlighted here.

1 Children form concepts about quantities fromtheir everyday experiences and can use theirschemas of action with diverse representations of the quantities (iconic, numerical) to solveproblems. They often develop sufficient awarenessof quantities to discuss their equivalence andorder as well as how they can be combined.

2 It is significantly more difficult for them to becomeaware of the relations between quantities andoperate on relations. Even after being taught howto represent relations, they often interpret theresults of operations on relations as if they were quantities. Children find both additive andmultiplicative relations significantly more difficultthan understanding quantities.

3 There is little evidence that the design ofinstruction has so far taken into account theimportance of helping students become aware of the difference between quantities and relations.Some researchers have carried out experimentalteaching studies that suggest that it is possib le topromote students’ awareness of relations. Fur therresearch must be carried out to analyse how thisknowledge affects mathematics learning. If positiveresults are found, there will be strong policyimplications.

4 Previous research had led to the conclusion thatstudents’ problems with propor tional reasoningstemmed from their difficulties with multiplicativereasoning. However, there is presently muchevidence to show that, from a relatively early age(about five to six years in the United Kingdom),children already have informal knowledge that

allows them to solve multiplicative reasoningproblems. We suggest that students’ problems withproportional reasoning stems from their difficultiesin becoming explicitly aware of relations betweenquantities. This awareness would help themdistinguish between situations that involve differenttypes of relations: additive, proportional orquadratic, for example.

5 Multiplicative reasoning problems are defined by the fact that they in volve two (or more)measures linked by a fixed ratio. Students’ informalknowledge of multiplicative reasoning stems fromthe schema of one-to-many correspondence,which they use both in multiplication and divisionproblems. When the product is unknown, childrenset the elements in the two measures incorrespondence (e.g. 1 sweet costs 4p) and f igureout the product (how much 5 sweets will cost).When the correspondence is unknown (e.g. if youpay 20p for 5 sweets, how much does each sweetcost), the children share out the elements (20pshared in 5 groups) to f ind what thecorrespondence is.

6 This informal knowledge is currently ignored inU.K. schools, probably due to the theor y thatmultiplication is essentially repeated addition anddivision is repeated subtraction. However, theconnections between addition and multiplication,on one hand, and subtraction and division, on theother hand, are procedural and not conceptual. So students’ informal knowledge of multiplicativereasoning could be developed in school from anearlier age.

7 A considerable amount of research car ried outindependently in different countries has shownthat students sometimes use additive reasoningabout relations when the appropr iate model is a multiplicative one. Some recent research hasshown that students also use multiplicativereasoning in situations where the appropr iatemodel is additive. These results suggest thatchildren use additive and multiplicative modelsimplicitly and do not make conscious decisionsregarding which model is appropr iate in a specificsituation. The educational implication from thesefindings is that schools should take up the task ofhelping students become more aware of themodels that they use implicitly and of ways oftesting their appropriateness to par ticularsituations.


8 Proportional reasoning stems from children’s useof the schema of one-to-many correspondences,which is expressed in calculations as b uilding-upstrategies. Evidence suggests that many studentswho use these strategies are not aware offunctional relations that characterises a linearfunction. This result reinforces the impor tance ofthe role that schools could pla y in helping studentsbecome aware of functional relations inproportions problems.

9 Two radically different approaches to teachingproportions and linear functions in schools can beidentified in the literature. One, identified asfunctional and human in focus, is based on thenotion that students’ schemas of action should bethe star ting point for this teaching. Throughinstruction, they should become progressivelymore aware of the scalar and functional relationsthat can be identified in such problems. Diagrams,tables and graphs are seen as tools that could helpstudents understand the models that they areusing of situations and make them into models forother situations later. The second approach,identified as algebraic, proposes that there shouldbe a sharp separation between students’ intuitiveknowledge, in which physical and mathematicalknowledge are inter twined, and mathematicalknowledge. Students should be led toformalisations early on in instruction and re-establish the connections between mathematicalstructures and physical knowledge at a later point.Representations using fractions, ordinary anddecimal, and the number line are seen as the toolsthat can allow students to abstract early on fromthe physical situations. There is no unambiguousevidence to show how either of these approachesto teaching succeeds in promoting students’progress, nor that either of them is moresuccessful than the less clear ly ar ticulated ideasthat are implicit in cur rent teaching in theclassroom. Research that can clar ify this issue isurgently needed and could have a major impact bypromoting better learning in U.K. students.

10 Students need to learn to read graphs in orderto be able to use them as tools f or thinkingabout functions. Research has shown thatstudents have ideas about how to read graphsbefore instruction and that these ideas should betaken into account when gr aphs are used in theclassroom. It is possible to teach students to readgraphs and to use them in order to think about

relations but much more research is needed toshow how students’ thinking changes if they dolearn to use graphs in order to analyse the typeof relation that is most relevant in specif icsituations.



Adjiage, R., and Pluvinage, F. (2007). An experimentin teaching ratio and propor tion. EducationalStudies in Mathematics , 65, 149-175.

Beaton, A. E., Mullis, I. V. S., Mar tin, M. O., Gonzalez, E.J., Kelly, D. L., & Smith, T. A. (1996). Mathematicsachievement in the middle school y ears: IEA’s ThirdInternational Mathematics and Science Study(TIMSS). Chestnut Hill, MA, USA: TIMSSInternational Study Center, Boston College.

Becker, J. (1993). Young children’s numerical use ofnumber words: counting in many-to-onesituations. Developmental Psychology, 19, 458-465.

Behr, M. J., Harel, G., Post, T., and Lesh, R. (1994). Unitsof quantity: A conceptual basis common to additiveand multiplicative structures. In G. Harel and J.Confrey (Eds.), The Development of MultiplicativeReasoning in the Learning of Mathematics (pp. 121-176). Albany (NY): SUNY Press.

Booth, L. R. (1981). Child-Methods in SecondaryMathematics. Educational Studies in Mathematics ,12, 29-41.

Brown, M. (1981). Number operations. In K. Hart(Ed.), Children’s Understanding of Mathematics: 11-16 (pp. 23-47). Windsor, UK: NFER-Nelson.

Bryant, P. (1974). Perception and Understanding inYoung Children. London and Co. Ltd.: Methuen.

Carlson, M., Jacobs, S., Coe, E., Larsen, S., and Hsu, E.(2002). Applying covariational reasoning whilemodeling dynamic events. Journal for Research inMathematics Education, 33, 352–378.

Carpenter, T. P., Ansell, E., Franke, M. L., Fennema, E.,and Weisbeck, L. (1993). Models of ProblemSolving: A Study of Kindergarten Children’sProblem-Solving Processes. Journal for Research inMathematics Education, 24, 428-441.

Carpenter, T. P., Hiebert, J., and Moser, J. M. (1981).Problem structure and first grade children’s initialsolution processes for simple addition andsubtraction problems. Journal for Research inMathematics Education, 12, 27-39.

Confrey, J. (1994). Splitting, Similarity and Rate ofChange: A New Approach to Multiplicationand Exponential Functions. In G. Harel & J.Confrey (Eds.), The Development ofMultiplicative Reasoning in the Lear ning ofMathematics (pp. 293-332). Albany, New York:State University of New York Press.

Cramer, K., Post, T., and Currier, S. (1993). Learningand teaching ratio and propor tion: researchimplications. In D. T. Owens (Ed.), Research ideasfor the classroom: Middle grades mathematics (pp.159-178). New York: Macmillan.

De Bock, D., Verschaffel, L., and Janssens, D. (1998).The predominance of the linear model insecondary school students’ solutions of wordproblems involving length and area of similarplane figures. Educational Studies in Mathematics,35, 65-83.

De Bock, D., Verschaffel, L., and Janssens, D. (2002).The effects of different problem presentationsand formulations on the illusion of linearity insecondary school students. Mathematical Thinkingand Learning, 4, 65-89.

De Bock, D., Van Dooren, W., Janssens, D., andVerschaffel, L. (2002). Improper use of linearreasoning: an in-depth study of the nature andthe irresistibility of secondar y school students’errors. Educational Studies in Mathematics , 50,311-334.

De Bock, D., Verschaffel, L., Janssens, D., Dooren, W.,and Claes, K. (2003). Do realistic contexts andgraphical representations always have a beneficialimpact on students’ performance? Negativeevidence from a study on modelling non-lineargeometry problems. Learning and Instruction, 13,441-463.

Douady, R. (1984). Jeux de cadres et dialectique outil-objet dans l’enseignement des mathématiques , uneréalisation de tout le cur sus primaire. Unpublished,Doctorat d’état, Paris.


References

Duval, R. (1995). Sémiosis et pensée humaine . Bern:Peter Lang.

Duval, R. (2006). A cognitive analysis of problems ofcomprehension in a learning of mathematics.Educational Studies in Mathematics, 61, 103-131.

Elliott, C. D., Smith, P., and McCulloch, K. (1997).British Ability Scale-II Technical Manual. London:nferNelson Publishing Company.

Flavell, J. H. (1971). What is memory developmentthe development of? Human Development, 14,272-278.

Flavell, J. H., Green, F. L., and Flavell, E. R. (1990).Developmental Changes in young children’sknowledge about the mind. CognitiveDevelopment, 5, 1-27.

Freudenthal, H. (1983). Didactical Phenomenology ofMathematical Structures. Dordrecht: D. Reidel.

Friel, S. N., Curcio, F. R., and Bright, G. W. (2001).Making Sense of Graphs: Critical FactorsInfluencing Comprehension and InstructionalImplications. Journal for Research in MathematicsEducation, 32, 124-158.

Frydman, O., & Br yant, P. E. (1988). Sharing and theunderstanding of number equivalence by youngchildren. Cognitive Development, 3, 323-339.

Gombert, J. E. (1992). Metalinguistic Development.Hemel Hempstead: Harverster Wheatsheaf.

Gravemeijer, K. (1994). Educational Developmentand Developmental Research in MathematicsEducation. Journal for Research in MathematicsEducation, 25, 443-471.

Gravemeijer, K. (1997). Mediating between concreteand abstract. In T. Nunes and P. Bryant (Eds.),Learning and Teaching Mathematics. AnInternational Perspective (pp. 315-346). Hove(UK): Psychology Press.

Hamilton, E., Lesh, R., Lester, F., and Yoon, C. (2007).The use of reflection tools in b uilding personalmodels of problem-solving. In R. Lesh, E.Hamilton and J. Kaput (Eds.), Foundations for thefuture in mathematics education (pp. 349–366).Mahwah, NJ: Lawrence Erlbaum Associates.

Hart, K. (1981 a). Measurement. In K. Har t (Ed.),Children’s Understanding of Mathematics: 11-16(pp. 9-22). London: John Murray.

Hart, K. (1981 b). Ratio and propor tion. In K. Har t(Ed.), Children’s Understanding of Mathematics: 11-16 (pp. 88-101). London: John Murray.

Hart, K. (1984). Ratio: children’s strategies and errors. Areport of the strategies and errors in secondar ymathematics project. Windsor: NFER-Nelson.

Hudson, T. (1983). Correspondences and numericaldifferences between sets. Child Development, 54,84-90.

Inhelder, B. and. Piaget, J. (1958). The Growth of LogicalThinking for Childhood to Adolescence. London:Routledge. and Kegan Paul.

Kaput, J., and West, M. M. (1994). Missing-Valueproportional reasoning problems: Factorsaffecting informal reasoning patterns. In G. Hareland J. Confrey (Eds.), The Development ofMultiplicative Reasoning in the Lear ning ofMathematics (pp. 237-292). Albany, New York:State University of New York Press.

Karmiloff-Smith, A. (1992). Beyond modularity: adevelopmental perspective on cognitive science .Cambridge, Mass: MIT Press.

Karmiloff-Smith, A., and Inhelder, B. (1977). ‘If youwant to get ahead, get a theor y’. In P. N. Johnson-Laird and P. C. Wason (Eds.), Thinking. Readings inCognitive Science (pp. 293-306). Cambridge:Cambridge University Press.

Karplus, R., and Peterson, R. W. (1970). Intellectualdevelopment beyond elementary school II: ratio,a survey. School-Science and Mathematics , 70, 813-820.

Karplus, R., Pulos, S., and Stage, E. K. (1983).Proportional reasoning of early adolescents. In R.Lesh and M. Landau (Eds.), Acquisition ofmathematics concepts and processes (pp. 45-90).London: Academic Press.

Kieren, T. (1994). Multiple Views of MultiplicativeStructures. In G. Harel and J. Confrey (Eds.), TheDevelopment of Multiplicative Reasoning in theLearning of Mathematics (pp. 389-400). Albany,New York: State University of New York press.

Kouba, V. (1989). Children’s solution strategies forequivalent set multiplication and division wordproblems. Journal for Research in MathematicsEducation, 20, 147-158.

Lamon, S. (1994). Ratio and Proportion: CognitiveFoundations in Unitizing and Norming. In G. Hareland J. Confrey (Eds.), The Development ofMultiplicative Reasoning in the Lear ning ofMathematics (pp. 89-122). Albany, New York:State University of New York Press.

Lave, J. (1988). Cognition in practice. Cambridge:Cambridge University Press.

Lesh, R., Middleton, J. A., Caylor, E., and Gupta, S.(2008). A science need: Designing tasks toengage students in modeling complex data.Educational Studies in Mathematics, in press.

Lesh, R., Post, T., and Behr, M. (1988). Proportionalreasoning. In J. Hieber t and M. Behr (Eds.),Number Concepts and Operations in the MiddleGrades (pp. 93-118). Reston, VA: National Councilof Teachers of Mathematics.



Lewis, A., and Mayer, R. (1987). Students’miscomprehension of relational statements inarithmetic word problems. Journal of EducationalPsychology, 79, 363-371.

Mevarech, Z. R., and Kramarsky, B. (1997). FromVerbal Descriptions to Graphic Representations:Stability and Change in Students’ AlternativeConceptions. Educational Studies in Mathematics ,32, 229-263.

Middleton, J. A., and Van den Heuvel-Panhuizen, M.(1995). The ratio table. Mathematics Teaching inthe Middle School, 1, 282-288.

Middleton, J. A., Van den Heuvel-Panhuizen, M., andShew, J. A. (1998). Using bar representations as amodel for connecting concepts of r ationalnumber. Mathematics Teaching in the MiddleSchool, 3, 302-311.

Noelting, G. (1980 a). The development ofproportional reasoning and the ratio conceptPart I - Differentiation of stages. EducationalStudies in Mathematics , 11, 217-253.

Noelting, G. (1980 b). The development ofproportional reasoning and the ratio conceptPart II - Problem-structure at successive stages:Problem-solving strategies and the mechanism ofadaptative restructuring. Educational Studies inMathematics, 11, 331-363.

Nunes, T., & Br yant, P. (1996). Children DoingMathematics. Oxford: Blackwell.

Nunes, T., Light, P., and Mason, J. (1993). Tools forthought: the measurement of lenght and area.Learning and Instruction, 3, 39-54.

Nunes, T., Schliemann, A. D., and Carraher, D. W.(1993). Street Mathematics and SchoolMathematics. New York: Cambridge UniversityPress.

Nunes, T., Bryant, P., Evans, D., Bell, D., Gardner, S.,Gardner, A., & Carraher, J. N. (2007). TheContribution of Logical Reasoning to the Lear ningof Mathematics in Pr imary School. British Journalof Developmental Psychology, 25, 147-166.

Park, J., & Nunes, T. (2001). The development of theconcept of multiplication. Cognitive Development,16, 1-11.

Piaget, J. (1952). The Child’s Conception of Number.London: Routledge & Kegan Paul.

Piaget, J. (1978). The Grasp of Consciousness. London:Routledge Kegan Paul.

Piaget, J. (2001). Studies in reflecting abstraction:translated by R. Campbell. Hove: Psychology Press.

Piaget, J. (2008). Intellectual Evolution fromAdolescence to Adulthood. Human Development,51(Reprint of Human Development 1972;15:1–12), 40-47.

Piaget, J., Grize, J., Szeminska, A., and Bangh, V. (1977).Epistemology and the Psychology of Functions .Dordrecth, Netherlands: D. Reidel.

Piaget, J., and Inhelder, B. (1975). The Origin of theIdea of Chance in Children . London: Routledge andKegan Paul.

Pickering, S., and Gathercole, S. (2001). Workingmemory test batter y for children (WMTB-C)Manual. London: The Psychological Corporation.

Resnick, L. B. (1983). Mathematics and ScienceLearning: A New Conception. Science, 220, 477-478.

Ricco, G. (1982). Les première acquisitions de lanotion de fonction linèaire chez l’enfant de 7 à11 ans. Educational Studies in Mathematics, 13,289-327.

Schwartz, J. (1988). Intensive quantity and referenttransforming arithmetic operations. In J. Hieber tand M. Behr (Eds.), Number concepts andoperations in the middle grades (pp. 41-52).Hillsdale, NJ: Erlbaum.

Steffe, L. (1994). Children’s multiplying schemes. In G.Harel and J. Confrey (Eds.), The Development ofMultiplicative Reasoning in the Lear ning ofMathematics (pp. 3-40). Albany, New York: StateUniversity of New York Press.

Streefland, L. (1984). Search for the Roots of Ratio:Some Thoughts on the Long Term LearningProcess (Towards...A Theory): Part I: Reflectionson a Teaching Experiment. Educational Studies inMathematics, 15, 327-348.

Streefland, L. (1985 a). Search for the Roots of Ratio:Some Thoughts on the Long Term LearningProcess (Towards...A Theory): Part II: The Outlineof the Long Term Learning Process. EducationalStudies in Mathematics , 16, 75-94.

Streefland, L. (1985 b). Wiskunde als activiteit en derealiteit als bron. Nieuwe Wiskrant, 5, 60-67.

Streefland, L. (1993). Fractions: A Realistic Approach.In T. P. Carpenter, E. Fennema and T. A. Romberg(Eds.), Rational Numbers: An Integration ofResearch (pp. 289-326). Hillsdale, NJ: LawrenceErlbaum Associates.

Streefland, L. (1997). Charming fractions or fractionsbeing charmed? In T. Nunes and P. Bryant (Eds.),Learning and Teaching Mathematics. AnInternational Perspective (pp. 347-372). Hove(UK): Psychology Press.

Streefland, L., and Van den Heuvel-Panhuizen, M.(1992). Evoking pupils’ informal knowledge onpercents. Paper presented at the Proceedings ofthe Sixteenth PME Conference, University ofNew Hampshire, Durham, NH.


Thompson, P. W. (1993). Quantitative Reasoning,Complexity, and Additive Structures. EducationalStudies in Mathematics , 25, 165-208.

Thompson, P. (1994). The Development of theConcept of Speed and Its Relationship toConcepts of Rate. In G. Harel and J. Confrey (Eds.),The Development of Multiplicative Reasoning in theLearning of Mathematics (pp. 181-236). Albany,New York: State University of New York Press.

Tourniaire, F., and Pulos, S. (1985). Proportionalreasoning: A review of the liter ature. EducationalStudies in Mathematics , 16 181-204.

Treffers, A. (1987). Integrated column arithmeticaccording to progressive schematisation.Educational Studies in Mathematics , 18 125-145.

Van den Brink, J. and Streefland, L. (1979). YoungChildren (6-8): Ratio and Propor tion. EducationalStudies in Mathematics , 10, 403-420.

Van de Heuvel-Panhuizen, M. (2003). The DidacticalUse of Models in Realistic Mathematics Education:An Example from a Longitudinal Trajectory onPercentage. Realistic Mathematics EducationResearch: Leen Streefland’s Work Continues.Educational Studies in Mathematics, 54, 9-35.

Van Dooren, W., Bock, D. D., Hessels, A., Janssens, D.,and Verschaffel, L. (2004). Remedying secondar yschool students’ illusion of linear ity: a teachingexperiment aiming at conceptual change .Learning and Instruction, 14, 485–501.

Vergnaud, G. (1979). The Acquisition of ArithmeticalConcepts. Educational Studies in Mathematics , 10,263-274.

Vergnaud, G. (1983). Multiplicative structures. In R.Lesh and M. Landau (Eds.), Acquisition ofMathematics Concepts and Processes (pp. 128-175). London: Academic Press.

Vergnaud, G. (1994). Multiplicative Conceptual Field:What and Why? In G. Harel & J. Confrey (Eds.), TheDevelopment of Multiplicative Reasoning in theLearning of Mathematics (pp. 41-60). Albany, NewYork: State University of New York press.

Vergnaud, G. (1997). The nature of mathematicalconcepts. In T. Nunes & P. Bryant (Eds.), Learningand Teaching Mathematics. An InternationalPerspective (pp. 1-28). Hove (UK): Psychology Press.

Verschaffel, L. (1994). Using Retelling Data to StudyElementary School Children’s Representationsand Solutions of Compare Problems. Journal forResearch in Mathematics Education, 25, 141-165.




ISBN 978-0-904956-71-9





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Paper 5: Understanding space and its representation in mathematicsBy Peter Bryant, University of Oxford

2 SUMMARY – PAPER 2: Understanding whole numbers2 Paper 5: Understanding space and its representation in mathematics













Understanding space and its representation in mathematics 7

References 37

About the authorPeter Bryant is Senior Research Fellow in theDepartment of Education, University of Oxford.


About this review


Headlines

• Children come to school with a great deal ofknowledge about spatial relations. One of the mostimportant challenges in mathematical education ishow best to harness this implicit knowledge inlessons about space.

• Children’s pre-school implicit knowledge of space ismainly relational. Teachers should be aware of kindsof relations that young children recognise and arefamiliar with, such as their use of stable backgroundto remember the position and orientation ofobjects and lines.

• Measuring of length and area poses particularproblems for children, even though they are able to understand the underlying logic ofmeasurement. Their difficulties concern iterationof standard units, which is a new idea for them,and also the need to apply multiplicative reasoningto the measurement of area.

• From an early age children are able to extrapolateimaginary straight lines, which allows them to learnhow to use Cartesian co-ordinates to plot specificpositions in space with no difficulty. However, theyneed instruction about how to use co-ordinates towork out the relation between different positions.

• Learning how to represent angle mathematically is ahard task for young children, even though angles arean important part of their everyday life. There isevidence that children are more aware of angle in thecontext of movement (turns) than in other contextsand learn about the mathematics of angle relativelyeasily in this context. However, children need a greatdeal of help from to teachers to understand how torelate angles across different contexts.

• An important aspect of learning about geometry is to recognise the relation between transformedshapes (rotation, reflection, enlargement). This also can be difficult, since children’s pre-schoolexperiences lead them to recognise the sameshapes as equivalent across such transformations,rather than to be aware of the nature of thetransformation. However, there is very littleresearch on this important question.

• Another aspect of the understanding of shape is thefact that one shape can be transformed into another,by addition and subtraction of its subcomponents.For example, a parallelogram can be transformedinto a rectangle of the same base and height by theaddition and subtraction of equivalent triangles andadding two equivalent triangles to a rectangle createsa parallelogram. Research demonstrates that there isa danger that children might learn about thesetransformations only as procedures withoutunderstanding their conceptual basis.

• There is a severe dearth of psychological researchon children’s geometrical learning. In particular weneed long-term studies of the effects of interventionand a great deal more research on children’sunderstanding of transformations of shape.

At school, children often learn formally about mattersthat they already know a great deal about in aninformal and often quite implicit way. Sometimes theirexisting, informal understanding, which for the mostpart is based on exper iences that they start to havelong before going to school, fits well with what theyare expected to learn in the classroom. At othertimes, what they know already, or what they thinkthey know, clashes with the formal systems that they

Summary of paper 5: Understanding space and its representation in mathematics


are taught at school and can ev en prevent them fromgrasping the significance of these formal systems.

Geometry is a good and an obvious example.Geometry lessons at school deal with the use ofmathematics and logic to analyse spatial relations andthe properties of shapes. The spatial relations and theshapes in question are cer tainly a common par t of any child’s environment, and psychological research has established that from a very early age children areaware of them and quite familiar with them. It has been shown that even very young babies not onlydiscriminate regular geometric shapes but can recognisethem when they see them at a tilt, thus co-ordinatinginformation about the or ientation of an object withinformation about the pattern of its contours.

Babies are also able to extrapolate imaginary straightlines (a key geometric skill) at any rate in socialsituations because they can work out what someoneelse is looking at and can thus constr uct that person’sline of sight. Another major ear ly achievement byyoung children is to master the logic that under liesmuch of the formal analysis of spatial relations thatgoes on in geometr y. By the time they f irst go toschool young children can make logical transitiveinferences (A > B , B > C , therefore A > C; A = B, B= C, therefore A = C), which are the logical basis ofall measurement. In their first few years at schoolthey also become adept at the logic of inversion (A+ B – B), which is a logical move that is an essentialpart of studying the relation between shapes.

Finally, there is strong evidence that most of theinformation about space that children use andremember in their everyday lives is relational innature. One good index of this is that children’ smemory of the or ientation of lines is lar gely basedon the relation between these lines and theorientation of stable features in the background. Forthis reason children find it much easier to rememberthe orientation of horizontal and vertical lines thanof diagonal lines, because horizontal and verticalfeatures are quite common in the child’s stablespatial environment. For the same reason, youngchildren remember and reproduce r ight angles(perpendicular lines) better than acute or obtuseangles. The relational nature of children’s spatialperception and memory is potentially a powerfulresource for learning about geometry, since spatialrelations are the basic subject matter of geometr y.

With so much relevant informal knowledge aboutspace and shape to dr aw on, one might think thatchildren would have little difficulty in translating this

knowledge into formal geometrical understanding.Yet, it is not always that easy. It is an unfortunate andwell-documented fact that many children havepersistent difficulties with many aspects of geometr y.

One evidently successful link between youngchildren’s early spatial knowledge and their moreformal experiences in the classroom is their lear ninghow to use Car tesian co-ordinates to plot positionsin two-dimensional space. This causes schoolchildrenlittle difficulty, although it takes some time for themto understand how to work out the relationbetween two positions plotted in this way.

Other links between informal and formal knowledgeare harder for young children. The apparently simple actof measuring a straight line, for example, causes themproblems even though they are usually perfectly able tomake the appropriate logical moves and understandthe importance of one-to-one correspondence, whichis an essential par t of relating the units on a r uler tothe line being measured. One problem here is thatthey find it hard to under stand the idea of iter ation:iteration is about repeated measurements, so that aruler consists of a set of iter ated (repeated) units likecentimetres. Iteration is necessary when a par ticularlength being measured is longer than the measur inginstrument. Another problem is that the one-to-onecorrespondence involved in measuring a line with aruler is asymmetrical. The units (centimetres, inches) arevisible and clear in the r uler but have to be imaginedon the line itself. It is less of a sur prise that it also takeschildren a great deal of time to come to ter ms withthe fact that measurement of area usuall y needs someform of multiplication, e.g. height x width withrectangles, rather than addition.

The formal concept of angle is another seriousstumbling block for children even though they arefamiliar enough with angles in their everyday spatialenvironments. The main problem is that they f ind ithard to grasp that two angles in very different contextsare the same, e.g. themselves turning 90o and thecorner of a page in a book. Abstraction is an essentialpart of geometry but it has very little to do withchildren’s ordinary spatial perception and knowledge.

For much the same reason, decomposing a relativelycomplex shape into several simpler componentshapes – again an essential activity in geometr y – issomething that many children find hard to do. In theirordinary lives it is usually more impor tant for them tosee shapes as unities, rather than to be able to breakthem up into other shapes. This difficulty makes it hardfor them to work out relationships between shapes.

4 Paper 5: Understanding space and its representation in mathematics

For example, children who easily grasp that a + b – b= a, nevertheless often fail to under stand completelythe demonstration that a rectangle and aparallelogram with the same base and height areequal in area because you can transform theparallelogram into the rectangle by subtracting atriangle from one end of the parallelogram and addingan exactly equivalent triangle to the other end.

We know little about children’s understanding oftransformations of shape or of any difficulties thatthey might have when they are taught about thesetransformations. This is a ser ious gap in research onchildren’s mathematical learning. It is well recognised,however, that children and some adults confuse scaleenlargements with enlargements of area. They think that doubling the length of the contour of ageometric shape such as a square or a rectangle alsodoubles its area, which is a serious misconception.Teachers should be aware of this potential difficultywhen they teach children about scale enlar gements.

Researchers have been more successful in identifyingthese obstacles than in showing us how to helpchildren to surmount them. There are some studiesof ways of preparing children for geometry in thepre-school period or in the ear ly years at school. This

research, however, concentrated on shor t-term gainsin children’s geometric understanding and did notanswer the question whether these ear ly teachingprogrammes would actually help children when theybegin to learn about geometr y in the classroom.

There has also been research on teaching childrenabout angle, mostly in the context of computer-based teaching programmes. One of the mostinteresting points to come out of this research is thatteaching children about angle in ter ms of movements(turning) is successful, and there is some evidencethat children taught this way are quite likely totransfer their new knowledge about angle to othercontexts that do not involve movement.

However, there has been no concer ted research on how teachers could take advantage of children’sconsiderable spatial knowledge when teaching themgeometry. We badly need long-term studies ofinterventions that take account of children’srelational approach to the spatial environment andencourage them to grasp other relations, such as therelation between shapes and the relation betweenshapes and their subcomponent par ts, which gobeyond their informal spatial knowledge.


Recommendations


Children’s pre-school knowledge of spaceis relational. They are skilled at using stab lefeatures of the spatial fr amework toperceive and remember the relativeorientation and position of objects in theenvironment. There is, however, noresearch on the relation between thisinformal knowledge and how well childrenlearn about geometr y.

Children already understand the logic ofmeasurement in their ear ly school years.They can make and understand transitiveinferences, they understand the inverserelation between addition and subtraction,and they can recognise and use one-to-one correspondence. These are threeessential aspects of measurement.


Teaching Teachers should be aware of the research onchildren’s considerable spatial knowledge and skills and shouldrelate their teaching of geometr ical concepts to thisknowledge.Research There is a ser ious need for longitudinal researchon the possible connections between children’s pre-schoolspatial abilities and how well they learn about geometr y at school.

Teaching The conceptual basis of measurement and not justthe procedures should be an impor tant par t of the teaching.Teachers should emphasise transitive inferences, inversion ofaddition and subtraction and also one-to-one cor respondenceand should show children their impor tance.Research Psychologists should extend their research on transitive inference, inversion and one-to-onecorrespondence to geometrical problems, such asmeasurement of length and area.

6 SUMMARY – PAPER 2: Understanding whole numbers6 Paper 5: Understanding space and its representation in mathematics


Many children have difficulties with theidea of iteration of standard units inmeasurement.

Many children wrongly apply additivereasoning, instead of multiplicativereasoning, to the task of measur ing area.Children understand this multiplicativereasoning better when they f irst think of itas the number of tiles in a row times thenumber of rows than when they tr y touse a base times height formula.

Even very young children can easilyextrapolate straight lines andschoolchildren have no difficulty in learninghow to plot positions using Car tesian co-ordinates, but it is difficult for them towork out the relation between differentpositions plotted in this way.

Research on pre-school inter ventionsuggests that it is possible to preparechildren for learning about geometr y byenhancing their understanding of spaceand shapes. However, this research has notincluded long-term testing and thereforethe suggestion is still tentative.

Children often learn about the relationbetween shapes (e.g. between aparallelogram and a rectangle) as aprocedure without understanding theconceptual basis for these transformations.

There is hardly any research on children’sunderstanding of the transformation ofshapes, but there is evidence of confusionin many children about the effects ofenlargement: they consider that doublingthe length of the per imeter of a square ,for example, doubles its area.


Teaching Teachers should recognise this diff iculty andconstruct exercises which involve iteration, not just withstandard units but with familiar objects like cups and hands.

Research Psychologists should study the exact cause ofchildren’s difficulties with iteration.

Teaching In lessons on area measurement, teachers canpromote children’s use of the reasoning ‘number in a rowtimes number of rows’ by giving children a number of tiles thatis insufficient to cover the area. They should also contrastmeasurements which do, and measurements which do not,rest on multiplication.

Teaching Teachers, using concrete mater ial, should relateteaching about spatial co-ordinates to children’s everydayexperiences of extrapolating imaginary straight lines.Research There is a need for intervention studies onmethods of teaching children to work out the relationbetween different positions, using co-ordinates.

Research There will have to be long-term predictive andlong-term intervention studies on this cr ucial, but neglected,question

Teaching Children should be taught the conceptual reasonsfor adding and subtracting shape components when studyingthe relation between shapes.Research Existing research on this topic was done a verylong time ago and was not v ery systematic. We need well-designed longitudinal and inter vention studies on children’sability to make and understand such transformations.

Teaching Teachers should be aware of the r isk that childrenmight confuse scale enlargements with area enlargements.Research Psychologists could easily study how childrenunderstand transformations like reflection and rotation butthey have not done so. We need this kind of research.

Recommendations (continued)

From informal understanding toformal misunderstanding of spaceThis paper is about children’s informal knowledge ofspace and spatial relations and about their f ormallearning of geometr y. It also deals with theconnection between these two kinds of knowledge.This connection is much the same as the onebetween knowledge about quantitative relations onthe one hand and about number on the other hand,which we described in Papers 1 and 2. We shall showhow young children build up a large and impressive,but often implicit, understanding of spatial relationsbefore they go to school and ho w this knowledgesometimes matches the relations that they lear n ingeometry very well and sometimes does not.

There is a r ich vein of research on children’s spatial knowledge – knowledge which they acquireinformally and, for the most par t, long before they goto school – and this research is ob viously relevant tothe successes and the diff iculties that they have whenthey are taught about geometry at school. Yet, with afew honourable exceptions, the most remarkable ofwhich is a recent thorough review b y Clements andSarama (2007 b), there have been very few attemptsindeed to link research on children’s informal, andoften implicit, knowledge about spatial relations totheir ability to car ry out the explicit analyses ofspace that are required in geometr y classes.

The reason for this gap is probably the str ikingimbalance in the contr ibution made by psychologistsand by maths educators to research on geometr icallearning. Although psychologists have studiedchildren’s informal understanding of space in detailand with great success, they have vir tually ignoredchildren’s learning about geometr y, at any rate inrecent years. Despite Wertheimer’s (1945) and

Piaget, Inhelder and Szeminska’s (1960) impressivepioneering work on children’s understanding ofgeometry, which we shall describe later, psychologistshave vir tually ignored this aspect of children’seducation since then. In contrast, mathematicseducators have made steady progress in studyingchildren’s geometry with measures of what childrenfind difficult and studies of the eff ects of differentkinds of teaching and classroom exper ience.

One effect of this imbalance in the contr ibution ofthe two disciplines to research on lear ning aboutgeometry is that the existing research tells us more about educational methods than about theunderlying difficulties that children have in learningabout geometry. Another result is that someexcellent ideas about enhancing children’sgeometrical understanding have been proposed byeducationalists but are still waiting for the kind ofempirical test that psychologists are good atdesigning and carrying out.

The central problem for anyone trying to make thelink between children’s informal spatial knowledgeand their understanding of geometr y is easy to state .It is the star k contrast between children’s impressiveeveryday understanding of their spatial environmentand the difficulties that they have in learning how to analyse space mathematically. We shall star t our review with an account of the basic spatialknowledge that children acquire informally longbefore they go to school.


Understanding space and its representation in mathematics


Early spatial knowledge:perception

Shape, size, position and extrapolationof imagined straight lines

Spatial achievements begin early. Over the last 30years, experimental work with young babies hasclearly shown that they are born with, or acquirevery early on in their life, many robust and effectiveperceptual abilities. They can discriminate objects bytheir shape, by their size and by their orientation andthey can perceive depth and distinguish differences indistance (Slater, 1999; Slater and Lewis 2002; Slater,Field and Hernadez-Reif, 2002; Bremner, Bryant andMareschal, 2006).

They can even co-ordinate information about size anddistance (Slater, Mattock and Brown, 1990), and theycan also co-ordinate information about an object’sshape and its orientation (Slater and Morrison, 1985).The first co-ordination makes it possible for them torecognise a par ticular object, which they first see closeup, as the same object when they see it again in thedistance, even though the size of the visual impressionthat it now makes is much smaller than it was before.With the help of the second kind of co-ordination,babies can recognise par ticular shapes even whenthey see them from completely different angles: theshape of the impression that these objects mak e onthe visual receptors varies, but babies can stillrecognise them as the same by taking the change inorientation into account.

We do not yet know how children so young arecapable of these impressive feats, but it is quite likelythat the answer lies in the relational nature of theway that they deal with siz e (and, as we shall seelater, with orientation), as Rock (1970) suggestedmany years ago. A person nearby makes a largervisual impression on your visual system than aperson in the distance but, if these two people areroughly the same size as each other, the relationbetween their size and that of familiar objects neareach of them, such as car s and bus-stops andwheelie-bins, will be much the same.

The idea that children judge an object’ s size in termsof its relation to the siz e of other objects at thesame distance receives some support from work on children’s learning about relations. When four-year-old children are asked to discriminate andremember a par ticular object on the basis of its siz e,

they do far better when it is possib le to solve theproblem on the basis of siz e relations (e.g. it isalways the smaller one) than when they ha ve toremember its absolute size (e.g. it is always exactlyso large) (Lawrenson and Br yant, 1972).

Another remarkable early spatial achievement byinfants, which is also relational and is highl y relevantto much of what they later have to learn ingeometry lessons, is their ability to extr apolateimaginary straight lines in three dimensional space(Butterworth, 2002). Extrapolation of imaginedstraight lines is, of course, essential for the use ofCartesian co-ordinates to plot positions in gr aphsand in maps, but it also is a basic ingredient of veryyoung children’s social communication (Butterworthand Cochrane, 1980; Butterworth and Grover, 1988).Butterworth and Jarrett (1991) showed this in astudy in which they asked a mother to sit oppositeher baby and then to stare at some predeter minedobject which was either in front and in full view ofthe child or was behind the child, so that he had toturn his head in order to see it. The question waswhether the baby would then look at the sameobject, and to do this he would have to extrapolatea straight line that represented his mother’ s line ofsight. Butterworth and Jarrett found that babiesyounger than 12 months manage to do this most ofthe time when the object in question was in front ofthem. They usually did not also tur n their heads tolook at objects behind them when these apparentlycaught their mothers’ attention. But 15-month-oldchildren did even that: they followed their mother’sline of sight whether it led them to objects already in full view or to ones behind them. A slightly laterdevelopment that also involves extrapolatingimaginary straight lines is the ability to point and tolook in the direction of an object that someone elseis pointing at, which infants manage do with greatproficiency (Butterworth and Morisette, 1996;Butterworth and Itakura, 2000).

Orientation and position

The orientation of objects and surfaces are asignificant and highly regular and predictable par t ofour everyday spatial environments. Walls usually are,and usually have to be, vertical: objects stay onhorizontal surfaces but tend to slide off slopingsurfaces. The surface of still liquid is hor izontal: theopposite edges of many familiar manufacturedobjects (doors, windows, television sets, pictures,book pages) are parallel: we ourselves are vertical


when we walk, horizontal when we swim. Yet,children seem to have more difficulty distinguishingand remembering information about orientation thaninformation about other familiar spatial variables.

Horizontals and verticals are not the problem. Five-year-old children take in and remember theorientation of horizontal and vertical lines extremelywell (Bryant 1969, 1974; Bryant and Squire, 2001). Incontrast, they have a lot of diff iculty in rememberingeither the direction or slope of ob liquely orientedlines. There is, however, an effective way of helpingthem over this difficulty with oblique lines. If there areother obliquely oriented lines in the background thatare parallel to an oblique line that they are asked toremember, their memory of the slope and directionfor this oblique line improves dramatically (see Figure5.1). The children use the par allel relation betweenthe line that they have to remember and stablefeatures in the background fr amework to store andrecognise information about the oblique line.

This result suggests a reason for the initial radicaldifference in how good their memor y is for verticaland horizontal features and how poor it is forobliquely oriented ones. The reason, again, is about

relations. It is that that there are usually ample stablehorizontal and vertical features in the background torelate these lines to. Stable, background features thatparallel par ticular lines which are not either v erticalor horizontal are much less common. If this idea isright, young children are already relying on spatialrelations that are at the hear t of Euclidean geometr yto store information about the spatial environmentby the time that they begin to be taught f ormallyabout geometry.

However, children do not always adopt this excellentstrategy of relating the orientation of lines topermanent features of the spatial environment.Piaget and Inhelder’s (1963) deser vedly famous andoften-repeated experiment about children drawingthe level of water in a tilted container is the bestexample. They showed the children tilted glasscontainers (glasses, bottles) with liquid in them(though the containers were tilted, the laws ofnature dictated that the level of the liquid in themwas horizontal). They also gave the children a pictureof an empty, tilted container depicted as just abo ve a table top which was an obvious horizontalbackground feature. The children’s task was to dr awin the level of the liquid in the dr awing so that it was


Figure 5.1: Children easily remember horizontal or vertical lines but not oblique lines unless they can relate oblique lies to astable background feature.

Child sees a line Short delay

Easy

Difficult

Easy

Child shown twolines and judges


exactly like the liquid in the exper imenter’s hand. Thequestion that Piaget and Inhelder asked was whetherthey would draw the liquid as par allel to the tabletop or, in other words, as horizontal.

Children below the age of roughly eight years didnot manage to do this. Many of them drew the liquidas perpendicular to the sides (when the sides werestraight) and parallel to the bottom of the container.It seems that the children could not tak e advantageof the parallel relation between the liquid and thetable top, probably because they were preoccupiedwith the glass itself and did not manage to shift theirattention to an external feature.

Piaget and Inhelder treated the young child’sdifficulties in this drawing task as a failure on thechild’s par t to notice and take advantage of a basicEuclidean relation, the parallel relation between two horizontal lines. They argued that a child whomakes this mistake does not have any idea abouthorizontality: he or she is unaware that horizontallines and surfaces are an impor tant par t of theenvironment and that some surfaces, such as stillliquid, are constantly horizontal.

Piaget and Inhelder then extended their ar gument toverticality. They asked children to copy pictures of

objects that are usually vertical, such as trees andchimneys. In the pictures that the children had tocopy, these objects were positioned on obliquelyoriented surfaces: the trees stood vertically on theside of a steeply sloping hill and vertical chimneyswere placed on sloping roofs. In their copies of these pictures, children younger than about eightyears usually drew the trees and chimneys asperpendicular to their baselines (the side of the hillor the sloping roof) and therefore with an obliqueorientation. Piaget and Inhelder concluded thatchildren of this age have not yet realised that thespace around them is full of stab le vertical andhorizontal features.

There is something of a conflict betw een the two setsof results that we have just presented. One (Bryant1969, 1974; Bryant and Squire, 2001) suggests thatyoung children detect, and indeed rely on, parallelrelations between objects in their immediateperception and stable background features. The other(Piaget and Inhelder, 1963) leads to the conclusionthat children completely disregard these relations.However, this is not a ser ious problem. In the first setof experiments the use that children made of parallelrelations was probably implicit. The second set ofexperiments involved drawing tasks, in which thechildren had to make explicit judgements about such


Figure 5.2: The perpendicular bias

When children see 2-line figure A and are asked to copy in the missing line on B either by placingor drawing a straight wire, they represent the line as nearer to the perpendicular than it is

relations. Children probably perceive and make use ofparallel relations without being aware of doing so. Theimplication for teaching children is an interesting one .It is that one impor tant task for the teacher ofgeometry is to transform their implicit knowledge into explicit knowledge.

There is another point to be made about thechildren’s mistakes in Piaget and Inhelder’s studies.One possible reason, or par tial reason, for thesemistakes might have been that in every case (theliquid in a tilted container, trees on the hillside ,chimneys on the sloping roofs) the task was to dr awthe crucial feature as non-perpendicular in relationto its baseline. There is plenty of evidence (Ibbotsonand Bryant, 1976) that, in copying, children find itquite difficult to draw one straight line that meetsanother straight line, the baseline, when the line thatthey have to draw is obliquely oriented to thatbaseline (see Figure 5.2).

They tend to misrepresent the line that they aredrawing either as perpendicular to the baseline or as closer to the per pendicular than it should be .There are various possible reasons for this‘perpendicular error’, but at the very least it showsthat children have some difficulty in representingnon-perpendicular lines. The work by Piaget et al.establishes that the presence of stab le, backgroundfeatures of the spatial environment, like the table top, does not help children surmount this bias.

Early spatial knowledge: logic and measurement

Inferences about space andmeasurement

The early spatial achievements that we havedescribed so far are, broadly speaking, perceptualones. Our next task is to consider ho w youngchildren reason about space . We must considerwhether young children are able to make logicalinferences about space and can under stand otherpeople’s inferential reasoning about space by the age when they first go to school.

We can star t with spatial measurements. Thesedepend on logical inferences about space.Measurement allows us to make comparisons betweenquantities that we cannot compare directly. We canwork out whether a washing line is long enough to

stretch between posts by measuring the line that wehave and the distance between the posts. We comparethe two lengths, the length of the line and the distancefrom one post to the other indirectl y, by comparingboth directly to the same measur ing instrument – atape measure or ruler. We combine two directcomparisons to make an indirect compar ison.

When we put two pieces of information together inthis way in order to produce a new conclusion, weare making a logical inference. Inferences aboutcontinua, like length, are called transitive inferences.We, adults, know that if A = B in length and B = C ,then A is necessarily the same length as C , eventhough we have never seen A and C together andtherefore have not been able to compare themdirectly. We also know, of course, that if A > B and B> C (in length), then A > C, again without making adirect comparison between A and C . In theseinferences B is the independent measure throughwhich A and C can be compared.

Piaget, Inhelder and Szeminska (1960) were the firstto discuss this link between understanding logic andbeing able to measure in their well-known book ongeometry. They argued that the main cause of thedifficulties that children have in learning aboutmeasurement is that they do not understandtransitive inferences. These authors’ claim about theimportance of transitive inferences in learning aboutmeasurement is indisputable and an extremelyimportant one. However, their idea that youngchildren cannot make or understand transitiveinferences has always been a controversial one, andit is now clear that we must make a fundamentaldistinction between being able to make the inferenceand knowing when this inference is needed and howto put it into effect.

There are usually two consecutive par ts to atransitive inference task. In the first, the child is giventwo premises (A = B , B = C) and in the second heor she has to tr y to draw an inference from thesepremises. For example, in Piaget’s first study oftransitive inferences, which was not about length b utabout the behaviour of some fictional people, he firsttold the children that ‘Mary is naughtier than Sar ah,and Sarah is naughtier than Jane’ and then asked then‘Who was the naughtier, Mary or Jane?’ Most childrenbelow the age of roughly nine years found, and stilldo find, this an extremely difficult question and oftensay that they cannot tell. The failure is a dr amatic one,but there are at least two possible reasons for it.



One, favoured by Piaget himself, that the failure is alogical one – that children of this age simpl y cannotput two premises about quantity together logicall y. It is worth noting that Piaget thought that the reasonthat young children did not make this logical movewas that they could not conceive that the middleterm (B when the premises are A > B and B > C)could simultaneously have one relation to A andanother, different, relation to C .

The second possible reason for children not makingthe transitive inference is about memor y. Thechildren may be unable to make the inference simplybecause they have forgotten, or because they didnot bother to commit to memory in the first place,one or both of the premises. The implication here isthat they would be able to make the inference ifthey could remember both premises at the time thatthey were given the inferential question.

One way to test the second hypothesis is to makesure that the children in the study do remember the premises, and also to take the precaution ofmeasuring how well they remember these premisesat the same time as testing their ability to draw atransitive inference. Bryant and Trabasso (1971) didthis by repeating the information about the premisesin the first par t of the task until the children hadlearned it thoroughly, and then in the second part checking how well they remembered thisinformation and testing how well they could answerthe inferential questions at the same time . In thisstudy even the four-year-olds were able rememberthe premises and they managed to put themtogether successfully to make the correct transitiveinference on 80% of the tr ials. The equivalent figurefor the five-year-olds was 89%.

Young children’s success in this inferential tasksuggests that they have the ability to make theinference that under lies measurement, but we stillhave to find out how well they apply this ability tomeasurement itself. Here, the research of Piaget etal. (1960) on measurement provides someinteresting suggestions. These researchers showedchildren a tower made of br icks of different sizes.The tower was placed on a small tab le and eachchild was asked to build another tower of the same height on another lower table that was usually, though not always, on the other side of the other side of apartition, so that the child had to create the replicawithout being able compare it directly to the originaltower. Piaget et al. also provided the child with

various possible measurement instruments, such asstrips of paper and a str aight stick, to help her withthe task, and the main question that they ask ed waswhether the child would use any of these asmeasures to compare the two towers.

Children under the age of (roughl y) eight years didnot take advantage of the measur ing instruments.Either they tr ied to do the task by remembering theoriginal while creating the replica, which did notwork at all well, or they used their hands or theirbody as a measur ing instrument. For example, somechildren put one hand at the bottom and the otherat the top of the or iginal tower and then walked tothe other tower trying at the same time to k eeptheir hands at a constant distance from each other .This strategy, which Piaget et al. called ‘manualtransfer’, tended not to be successful either, for thepractical reason that the children also had to use andmove their hands to add and subtr act bricks to theirown tower. Older children, in contrast, were happyto use the str ips of paper or the dowel rod as amakeshift ruler to compare the two towers. Piaget et al. claimed that the children who did not use themeasuring instruments failed the task because theywere unable to reason about it logicall y. They alsoargued that children’s initial use of their own bodywas a transitional step on the way to truemeasurement using an ‘independent middle term’.

This might be too pessimistic a conclusion. There isan alternative explanation for the reactions of thechildren who did not attempt to use a measure atall. It is that children not only have to be able tomake an inference to do well in any measuring task:they also have to realise that a direct compar isonwill not do, and thus that instead they should makean indirect, inferential, comparison with the help of areliable intervening measure.

There is some evidence to suppor t this idea. If it isright, children should be ready to measure in a taskin which it is made completel y obvious that directcomparisons would not work. Bryant and Kopytynska(1976) devised a task of this sor t. First, they gave agroup of five- and six-year-old children a version ofPiaget et al.’s two towers task, and all of them failed.Then, in a new task, they gave the children twoblocks of wood, each with a hole sunk in the middlein such a way that it was impossib le to see howdeep either hole was. They asked the children to f indout whether the two holes were as deep as eachother or not. The children were also given a rod withcoloured markings. The question was whether the


children, who did not measure in Piaget et al.’s task,would star t to use a measure in this new task inwhich it was clear that a direct compar ison would beuseless.

Nearly all the children used the rod to measureboth holes in the b locks of wood at least once (theywere each given four problems) and over half thechildren measured and produced the r ight answer inall four problems. It seems that children of f ive yearsor older are ready to use an inter vening measure tomake an indirect compar ison of two quantities. Theirdifficulty is in knowing when to distr ust directcomparisons enough to resor t to measurement.

Iteration and measurement

One interesting variation in the study of measurementby Piaget et al. (1960) was in the length of the str aightdowel rod, which was the main measur ing tool in thistask. The rod’s length equalled the height of theoriginal tower (R = T) in some problems but in othersthe rod was longer (R > T) and in others still it wasshorter (R < T) than the tower.

The older children who used the rod as a measurewere most successful when R = T. They were slightlyless successful when R > T and they had to mar k apoint on the rod which coincided with the summitof the tower. In contrast, the R < T problems wereparticularly difficult, even for the children who tr iedto use the rod as a measure . The solution to suchproblems is iteration which, in this case, is to applythe rod more than once to the to wer: the child hasto mark a point to represent the length of the r ulerand then to star t measuring again from this point.

It is worth noting that iteration also involves a greatdeal of care in its execution. You must cover all thesurface that you are measuring, all its length in theseexamples, but you must never overlap – nevermeasure any par t of the surface twice .

Iteration in measurement is interesting because the people who do it successfull y are actuallyconstructing their own measure and thereforecertainly have a strong and effective understandingof measurement. Piaget et al. (1960) also argued thatchildren’s eventual realisation that iteration is thesolution to some measur ing problems is the basis for their eventual understanding of the role ofstandardised units such as centimetres and metres.We use these units, they argued, in an iterative way:

1 metre is made up of 100 iter ations of 1centimetre, and one kilometre consists of 1000iterations of 1 metre . Children’s first insight into thisiterative system, according to Piaget et al., comesfrom their initial exper iences with R < T problems.This is an interesting causal h ypothesis that has someserious educational implications. It should be tested.

Conclusions about children’s earlyspatial knowledge• Children have a well-developed and effective

relational knowledge of shape, position, distance,spatial orientation and direction long before theygo to school. This knowledge may be implicit andnon-numerical for the most part, but it is certainlyknowledge that is related to geometry.

• The mistakes that children make in drawinghorizontal and vertical lines are probably due topreferring to concentrate on relations betweenlines close to each other (liquid in a glass isperpendicular to the sides of the glass) rather thanto separated lines (liquid in a glass is parallel tohorizontal surfaces like table tops). This is a mistakenot in relational perception, but in picking the rightrelation.

• Children are also able to understand and to maketransitive inferences, which are the basic logicalmove that underlies measurement, several yearsbefore being taught about geometry.

• We do not yet know how well they can cope withthe notion of iteration in the school years.

• There is no research on the possible causal linksbetween these impressive early perceptual andlogical abilities and the successes and difficulties thatchildren have when they first learn about geometry.This is a serious gap in our knowledge aboutgeometrical learning.



The connections betweenchildren’s knowledge of spacebefore being taught geometryand how well they learn whenthey are taught about geometry

To what extent does children’s early spatialdevelopment predict their success in geometr y later on?The question is simple , clear and overwhelminglyimportant. If we were dealing with some otherschool subject – say learning to read – we wouldhave no difficulty in finding an answer, perhaps morethan one answer, about the impor tance of ear ly,informal learning and experience, because of thevery large amount of work done on the subject.With geometry, however, it is different. Havingestablished that young children do have a r ich and in many ways sophisticated understanding of theirspatial environment, psychologists seem to havemade their excuses and left the room. Literallyhundreds of longitudinal and inter vention studiesexist on what children already know about languageand how they learn to read and spell. Yet, as far aswe know, no one has made a systematic attempt, inlongitudinal or inter vention research, to link whatchildren know about space to how they learn themathematics of spatial relations, even though thereare some extremely interesting and highly specificquestions to research.

To take one example, what connections are therebetween children’s knowledge of measurementbefore they learn about it and how well they learnto use and under stand the use of r ulers? To takeanother, we know that children have a bias towardsrepresenting angles as more per pendicular than theyare: what connection is there between the extent ofthis bias and the success that children ha ve inlearning about angles, and is the relation a positive ora negative one? These are practicable and immenselyinteresting questions that could easil y be answered inlongitudinal studies. It is no longer a matter of whatis to be done . The question that baffles us is: why arethe right longitudinal and inter vention studies notbeing done?

How can we inter vene to prepare young children in the pre-school per iod for geometry? If there is aconnection between the remarkable spatialknowledge that we find in quite young children and their successes and failures in lear ning aboutgeometry later on, it should be possib le to work

on these ear ly skills and enhance them in var iousways that will help them lear n about geometr y whenthe time comes.

Here the situation is r ather different. Educators haveproduced systematic programmes to preparechildren for formal instruction in geometr y. Some ofthese are ingenious and convincing, and they deser veattention. The problem in some cases is a lack ofempirical evaluation.

One notable programme comes from the highlyrespected Freudenthal Institute in the Nether lands. A team of educational researcher s there (van denHeuven-Panhuizen and Buys, 2008) have producedan ingenious and or iginal plan for enhancingchildren’s geometric skills before the age when theywould normally be taught in a f ormal way about thesubject. We shall concentrate here on therecommendations that van den Heuven-Panhuizenand Buys make for introducing kindergarten childrento some basic geometr ical concepts. However, weshall begin with the remar k that, though theirrecommendations deserve our serious attention, theFreudenthal team offer us no empir ical evidence atall that they really do work. Neither inter ventionstudies with pre-tests and post-tests and r andomlyselected treatment groups, nor longitudinal predictiveprojects, seem to have played any par t in thisparticular initiative.

The basic theoretical idea behind the Freudenthalteam’s programme for preparing children forgeometry is that children’s everyday life includesexperiences and activities which are relevant togeometry but that the geometr ic knowledge thatkindergarten children glean from these exper iencesis implicit and unsystematic . The solution that theteam offers is to give these young children asystematic set of enjoyable game-like activities withfamiliar material and after each activity is f inished todiscuss and to encourage the children to reflect onwhat they have just done.

Some of these activities are about measurement(Buys and Veltman, 2008). In one interesting example ,a teacher encourages the children to find out howmany cups of liquid would fill a par ticular bottle and,when they have done that, to work out how manycups of liquid the bottle would provide when thebottle is not completely full. This leads to the idea ofputting marks on the bottle to indicate when itcontains one or more cups’ worth of liquid. Thus, thechildren experience measurement units and also


iteration. In another measurement activity childrenuse conventional measures. They are given threerods each a metre long and are ask ed to measurethe width of the room. Typically the children star twell by forming the rods into a straight 3-metre line,but then hit the problem of measuring the remainingspace: their first reaction is to ask f or more rods, butthe teacher then provides the suggestion that insteadthey try moving the first rod ahead of the third andthen to move the second rod: Buys and Veltmanreport that the children readil y follow this suggestionand apparently understand the iteration involvedperfectly well.

Other exercises, equally ingenious, are aboutconstructing and operating on shapes (van den Heuvel-Panhuizen, Veltman, Janssen andHochstenbach, 2008). The Freudenthal team use the device of folding paper and then cutting outshapes to encourage children to think about therelationship between shapes: cutting an isoscelestriangle across the fold, for example, creates aregular parallelogram when the paper is unf olded.The children also play games that take the form offour children creating a four-part figure betweenthem with many symmetries: each child producesthe mirror-image of the figure that the previouschild had made (see Figure 5.3). The aim of suchgames is to give children systematic exper ience ofthe transformations, rotation and reflection, and toencourage them to reflect on these transformations.

If this group of researcher s is r ight, children’s earlyknowledge of geometric relationships andcomparisons, though implicit and unsystematic , playsan important par t in their eventual learning aboutgeometry. It is a resource that can be enhanced b ysensitive teaching of the kind that the Freudenthalgroup has pioneered. They may be r ight, but

someone has to establish, through empiricalresearch, how right they are.

There are a few empirical studies of ways ofimproving spatial skills in pre-school children. In thesethe children are given pre-tests which assess howwell they do in spatial tasks which are suitable forchildren of that age , then go through inter ventionsessions which are designed to increase some ofthese skills and finally, soon after the end of thisteaching, they are given post-tests to measureimprovement in the same skills.

Two well designed studies carried out by Casey andher colleagues take this form (Casey, Erkut, Cederand Young, 2008; Casey, Andrews, Schindler, Kersh andYoung, 2008). In both studies the researcher s wereinterested in how well five- and six-year-old childrencan learn to compose geometric shapes bycombining other geometric shapes and how well theydecompose shapes into component shapes, and alsowhether it is easier to improve this par ticular skillwhen it is couched in the context of a stor y thanwhen the context is a more f ormal and abstract one.

The results of these two studies showed that thespecial instruction did, on the whole , help children tocompose and decompose shapes and did have aneffect on related spatial skills in the children whowere taught in this way. They also showed that thenarrative context added to the eff ect of teachingchildren at this age . Recently, Clements and Sarama(2007 a) repor ted a very different study of slightlyyounger, nursery children. These researchers wereinterested in the effects of a pre-school progr amme,called Building Blocks, the aim of which is to preparechildren for mathematics in general includinggeometry. This programme is based on a theor yabout children’s mathematical development: as far as


Figure 5.3: An activity devised by van den Heuvel-Panhuizen, Veltman, Janssen and Hochstenbach: four children devise a four-partshape by forming mirror-images.


geometry is concerned Clement and Sarama’sstrongest interest is in children’s awareness of thecomposition of shapes and the relationship betw eendifferent shapes. They also believe that the actualteaching given to individual children should bedetermined by their developmental levels. Thus theday-to-day instruction in their programme dependson the children’s developmental trajectories.Clements and Sarama report that the young childrentaught in the Building Blocks progr ammes improvedfrom pre-test to post-test more r apidly than childrentaught in other ways in tasks that involvedconstructing or relating shapes.

These are interesting conclusions and a good star t.However, research on the question of the eff ects of intervention programmes designed to preparechildren for geometry need to go fur ther than this.We need studies of the eff ects of pre-schoolinterventions on the progress that children mak ewhen they are eventually taught geometr y at schoola few years later on. We cannot be sure that thechanges in the children’s skills that were detected inthese studies would have anything to do with theirsuccesses and failures later on in geometr y.

Summary

1 Children have a well-developed and effectiverelational knowledge of shape, position, distance,spatial orientation and direction long before theygo to school. This knowledge may be implicit andnon-numerical for the most par t, but it is cer tainlyknowledge that is related to geometry.

2 The mistakes that children make in drawinghorizontal and vertical lines are probably due tothem preferring to concentrate on relationsbetween lines close to each other (liquid in a glassis perpendicular to the sides of the glass) r atherthan to separated lines (liquid in a glass is par allelto horizontal surfaces like table tops). This is amistake not in relational perception, but in pickingthe right relation.

3 Children are also able to understand and to maketransitive inferences, which are the basic logicalmove that under lies measurement, several yearsbefore being taught about geometr y.

4 We do not yet know how well they can cope withthe notion of iteration in the school years.

5 There is little research on the possib le causal linksbetween these impressive early perceptual andlogical abilities and the successes and diff icultiesthat children have when they first learn aboutgeometry. This is a ser ious gap in our knowledgeabout geometrical learning.

Learning about geometryThe aim of teaching children geometry is to showthem how to reason logically and mathematicallyabout space, shapes and the relation betweenshapes, using as tools conventional mathematicalmeasures for size, angle, direction, orientation andposition. In geometr y classes children learn toanalyse familiar spatial exper iences in entirely newways, and the exper ience of this novel and explicitkind of analysis should allow them to perceive andunderstand spatial relationships that they knewnothing about before.

In our view the aspects of anal ysing spacegeometrically that are new to children coming to the subject for the first time are:• representing spatial relations which are already

familiar to them, like length, area and position, innumbers

• learning about relations that are new to them, atany rate in terms of explicit knowledge, such asangle

• forming new categories for shapes, such astriangles, and understanding that the properties ofa figure depends on its geometric shape

• understanding that there are systematic relationsbetween shapes, for instance between rectanglesand parallelograms

• understanding the relation between shapes acrosstransformations, such as rotation, enlargement andchanges in position.

Applying numbers to familiar spatial relations and forming relations between different shapes

Length measurement

Young children are clear ly aware of length. Theyknow that they grow taller as they grow older, andthat some people live closer to the school thanothers. However, putting numbers on these changes


and differences, which is one of their f irst geometricfeats, is something new to them.

Standard units of measurement are equal subdivisionsof the measuring instrument, and this means thatchildren have to understand that this instrument, aruler or tape measure or protr actor, is not just acontinuous quantity but is also subdivided into unitsthat are exactly the same as each other. The child hasto understand, for example, that by using a ruler, shecan represent an object’s length through an iter ationof measurement units, like the centimetre.

When children measure, for example, the length of astraight line, they must relate the units on the r uler tothe length that they are measur ing, which is a one-to-one correspondence, but of a relatively demandingform. In order to see that the measured length is, forexample, 10 cm long, they have to understand thatthe length that they are measur ing can also bedivided into the same unit and that ten of the unitson the ruler are in one-to-one cor respondence withten imaginary but exactly similar units on what isbeing measured. This is an active form of one-to-onecorrespondence, since it depends on the childrenunderstanding that they are converting a continuousinto a discontinuous quantity by dividing it intoimaginary units. Here, is a good example of ho w eventhe simplest of mathematical analysis of space makesdemands on children’s imagination: they must imagineand impose divisions on undivided quantities in orderto create the one-to-one cor respondence which isbasic to all measurement of length.

Measuring a straight line with a r uler is probably thesimplest form of measurement of all, but childreneven make mistakes with this task and their mistak essuggest that they do not at f irst grasp that measuringthe line takes the form of imposing one-to-onecorrespondence of the units on the measure withimagined units on the line . This was cer tainlysuggested by the answers that a large number ofchildren who were in their first three years ofsecondary school (11-, 12-, 13- and 14-year-olds)gave to a question about the length of a str aight line,which was par t of a test devised by Har t, Brown,Kerslake, Küchemann and Ruddock (1985). Thechildren were shown a picture of straight line besidea ruler that was marked in centimetres. One end ofthe line was aligned with the 1 cm mar k on the rulerand the other end with the 7 cm mar k. The childrenwere asked how long the line was and, in theyoungest group, almost as many of them (46%) gavethe answer 7 cm as gave the r ight answer (49%)

which was 6 cm. Thus even at the comparatively lateage of 11 years, after several years experience ofusing rulers, many children seemed not tounderstand, or at any rate not to under standperfectly, that it is the number of units on the r ulerthat the line corresponded to that decided its length.

A study by Nunes, Light and Mason (1993) gives ussome insight into this apparently persistent difficulty.These experimenters asked pairs of six- to eight-year-old children to work together in a measuringtask. They gave both children in each pair a piece ofpaper with a straight line on it, and the pair’s taskwas to find out whether their lines were the samelength or, if not whose was longer and whoseshorter. Neither child could see the other’ s linebecause the two children did the task in separaterooms and could only talk to each other over atelephone. Both children in each pair w ere given ameasure to help them compare these lines and theonly difference between the pairs was in themeasures that the experimenters gave them.

The pairs of children were assigned to three groups.In one group, both children in each pair were given astring with no markings: this therefore was a measurewithout units. In a second group, each child in everypair was given a standard ruler, marked in centimetres.In the third group the children w ere also given rulersmarked in centimetres, but, while one of the childrenhad a standard ruler, the other child in the pair wasgiven a ‘broken’ ruler: it star ted at four centimetres.The child with the broken ruler could not producethe right answer just by reading out the number fromthe ruler which coincided with the end of the line. Ifthe line was, for example, 7 cm long, that numberwould be 11 cm. The children in these pairs had topay particular attention to the units in the ruler.

The pairs in the second group did w ell. They cameup with the correct solution 84% of the time . Thefew mistakes that they made were mostly aboutplacing the ruler or counting the units. Some childrenaligned one endpoint with the 1 cm point on theruler rather than the 0 cm point and thusoverestimated the length by 1 cm. This suggests thatthey were wrongly concentrating on the boundar iesbetween units rather than the units themselves.Teachers should be aware that some children havethis misconception. Nevertheless the ruler did, onthe whole, help the children in this study since thosewho worked with complete rulers did much betterthan the children who were just given a str ing tomeasure with.



However, the broken ruler task was more diff icult.The children in the standard r uler group were right84%, and those in the broken ruler group 63%, ofthe time. Those who got it r ight despite having abroken ruler either counted the units or read off thelast number (e.g. 11 cm for a 7 cm line) and thensubtracted 4, and since they managed to do thismore often than not, their performance establishedthat these young children have a considerableamount of understanding of how to use the units ina measuring instrument and of what the units mean.However, on 30% of the tr ials the children in thisgroup seemed not to under stand the significance ofthe missing first four centimetres in the ruler. Eitherthey simply read off the number that matched theline’s endpoint (11 cm for 7 cm) or they did notsubtract the r ight amount from it.

A large-scale American study (Kloosterman, Warfield,Wearne, Koc, Martin and Strutchens, 2004; Sowder,Wearne, Martin and Strutchens, 2004) later confirmedthis striking difficulty. In this study, the children had tojudge the length of an object which was pictured justabove a ruler, though neither of its endpoints wasaligned with the zero endpoint on that r uler. Less than25% of the 4th graders (nine-year-olds) solved theproblem correctly and only about 60% of the 8 th

graders managed to find the correct answer. Thisstrong result, combined with those repor ted by Nuneset al., suggests that many children may know how touse a standard ruler, but do not fully understand thenature or structure of the measurement units that theyare dealing with when they do measure . Their mistake,we suggest, is not a misunder standing of the functionof a ruler: it is a failure in an activ e form of one-to-onecorrespondence – in imagining the same units on theline as on the r uler and then counting these units.

Summary

1 Measuring a straight line with a r uler is a procedureand it is also a consider able intellectual feat.

2 The procedure is to place the zero point of theruler at one end of the str aight line and to read offthe number of standardised units on the r uler thatcorresponds to the other end of this line . There isno evidence that this procedure causes y oungchildren any consistent difficulty.

3 The intellectual feat is to under stand that the ruleriterates a standardised unit (e.g. the centimetre)and that the length of the line being measured is

the number of units in the par t of the ruler that isin correspondence to the line. Thus measurementof length is a one-to-one correspondenceproblem, and the correspondence is between unitsthat are displayed on the ruler but have to beimagined on the line itself. This act of imaginationseem obvious and easy to adults but may not beso for young children.

4 Tests and experiments in which the line beingmeasured is not aligned with zero show thatinitially children do not completely understandhow measurement is based on imagining one-to-one correspondence of iterated units.

Measurement of area: learning aboutthe relationship between the areas ofdifferent shapesThere is a str iking contrast between young children’sapparently effortless informal discriminations of sizeand area and the diff iculties that they have in learninghow to analyse and measure area geometr ically.Earlier in this chapter we reported that babies areable to recognise objects by their size and can do soeven when they see these objects at diff erentdistance on different occasions. Yet, many children findit difficult at first to measure or to under stand thearea of even the simplest and most regular of shapes.

All the intellectual requirements for understandinghow to measure length, such as knowing abouttransitivity, iteration, and standardised units, apply aswell to measuring area. The differences are that:

• area is necessarily a more complex quantity tomeasure than length because now children have tolearn to consider and measure two dimensions andto co-ordinate these different measurements. Theco-ordination is always a multiplicative one (e.g.base x height for rectangles; πr2 for circles etc.).

• the standardised units of area – square centimetresand square metres or square inches etc. – are newto the children and need a great deal ofexplanation. This additional step is usually quite ahard one for children to take.

Rectangles

Youngsters are usually introduced to themeasurement of area by being told about the


base x height r ule for rectangles. Thus, rectanglesprovide them with their f irst experience of squarecentimetres. The large-scale study of 11- to 14-year-old children by Har t et al. (1985), which we havementioned already, demonstrates the difficulties thatmany children have even with this simplest of areameasurements. In one question the children w ereshown a rectangle, drawn on squared paper, whichmeasured 4 squares (base) by 2½ squares (height)and then were asked to draw another rectangle ofthe same area with a base of 5 squares. Only 44% ofthe 11-year-old children got this r ight: many judgedthat it was impossible to solve this problem.

We have to consider the reason for this difficulty.One reason might be that children find it hard tocome to terms with a new kind of measuring unit,the square. In order to explain these new unitsteachers often give children ‘covering’ exercises. Thechildren cover a rectangle with squares, usually 1 cmsquares, arranged in columns and rows and theteacher explains that the total number of squares isa measure of the rectangle’s area. The arrangementof columns and rows also provides a way ofintroducing children to the idea of m ultiplying heightby width to calculate a rectangle’ s area. If therectangle has five rows and four columns of squares,which means that its height is 5 cm and its width 4cm, it is covered by 20 squares.

This might seem like an easy transition, but it has itspitfalls. These two kinds of computation are based oncompletely different reasoning: counting is aboutfinding out the number that represents a quantityand involves additive reasoning whereas multiplyingthe base by the height involves understanding thatthere is a multiplicative relation between each ofthese measures and the area. Therefore, practice onone (counting) will not necessar ily encourage thechild to adopt the other f ormula (multiplying).Another radical difference is that the coveringexercise provides the unit, the square centimetre,from the star t but when the child uses a ruler tomeasure the sides and then to multiply height bywidth, she is measur ing with one unit, thecentimetre, but creating a new unit, the squarecentimetre (for fur ther discussion, see Paper 3).

This could be an obstacle . The French psychologist,Gerard Vergnaud (1983), rightly distinguishesproblems in which the question and the answ er areabout the same units (‘A plant is 5 cm high at thebeginning of the week and by the end of the week it is 2 cm higher. How high is it at the end of the

week?’) and those in which the question is couchedin one unit and the answer in another (‘The page onyour book is 15 cm high and 5 cm wide. What is the area of this page?’). The answer to the secondquestion must be in square centimetres even thoughthe question itself is couched onl y in terms ofcentimetres. Vergnaud categorised the first kind ofproblem as ‘isomorphism of measures’ and thesecond as ‘product of measures’. His point was thatproduct of measures problems are intr insically themore difficult of the two because, in order to solvesuch problems, the child has to under stand how onekind of unit can be used to create another .

At first, even covering tasks are difficult for manyyoung children. Outhred and Mitchelmore (2000)gave young children a rectangle to measure and justone 1 cm2 square tile to help them to do this. Thechildren also had pencils and were encouraged todraw on the rectangle itself. Since the children hadone tile only to work with, they could only ‘cover’ thearea by moving that tile about. Many of the youngerchildren adopted this strategy but carried it outrather unsuccessfully. They left gaps between theirdifferent placements of the tile and there were alsogaps between the squares in the dr awings that theymade to represent the different positions of the tiles.

These mistakes deserve attention, but they are hardto interpret because there are two quite differentways of accounting for them. One is that theseparticular children made a genuinely conceptualmistake about the iteration of the measuring unit.They may not have realised that gaps are not allo wed– that the whole area must be covered by thesestandardised units. The alternative account is that this was an executive, not a conceptual, failure. Thechildren may have known about the need forcomplete covering, and yet may have been unable to carry it out. Moving a tile around the rectangle , so that the tile covers every par t of it without anyoverlap, is a complicated task, and children need agreat deal of dexter ity and a highly organisedmemory to carry it out, even if they know exactlywhat they have to do. These ‘executive’ demands mayhave been the source of the children’s problems.Thus, we cannot say for sure what bear ing this studyhas on Vergnaud’s distinction between isomorphismand product of measures until we know whether themistakes that children made in applying the measurewere conceptual or executive ones.

Vergnaud’s analysis, however, fits other data that wehave on children’s measurement of area quite well.



Nunes et al. (1993) asked pairs of eight- and nine-year-old children to work out whether tworectangles had the same area or not. The dimensionsof the two rectangles were always different, evenwhen their areas were the same (e .g. 5 x 8 and 10 x 4 cm). The experimenters gave all the childrenstandard rulers, and also 1 cm3 bricks to help themsolve the problem.

The experimenters allowed the pairs of children tomake several attempts to solve each problem untilthey agreed with each other about the solution.Most pairs star ted by using their r ulers, as they hadbeen taught to at school, but many of them thendecided to use the br icks instead. Overall thechildren who measured with br icks were much moresuccessful than those who relied entirel y on theirrulers. This clear difference is a demonstration ofhow difficult it is, at first, for children to use onemeasurement unit (centimetres) to create another(square centimetres). At this age they are happierand more successful when working just with directrepresentation of the measurement units that theyhave to calculate than when they ha ve to use a r ulerto create these units.

The success of the children who used the br icks wasnot due to them just counting these bricks. Theyhardly ever covered the area and then labor iouslycounted all the br icks. Much more often, theycounted the rows and the columns of br icks andthen either multiplied the two figures or usedrepeated addition or a mixture of the tw o to comeup with the correct solution (A: ‘Eight bricks in a row.And 5 rows. What’s five eights?’ B: ‘Two eights is 16 and16 is 32. Four eights is 32. 32. 40’). In fact, the childrenwho used bricks multiplied in order to calculate thearea more than three times as often as the childrenwho used the ruler. Those who used rulers oftenconcentrated on the per imeter: they measured thelength of the sides and added lengths instead.

This confusion of area and per imeter is a ser iousobstacle. It can be tr aced back in time to asystematic bias in judgements that young childrenmake about area long before they are taught theprinciples of area measurement. This bias is towardsjudging the area of a f igure by its perimeter.

The bias was discovered independently in studies byWilkening (1979) in Germany and Anderson andCuneo (1978) and Cuneo (1983) in America. Bothgroups of researchers asked the same two questions(Wilkening and Anderson, 1982).

1 If you ask people to judge the area of differentrectangles that var y both in height and in width,will their judgement be affected by both thesedimensions? In other words, if you hold the widthof two rectangles constant will they judge thehigher of the two as larger, and if you hold theirheight constant will they judge the wider one asthe larger? It is quite possib le that young childrenmight attend to one dimension onl y, and indeedPiaget’s theory about spatial reasoning implies thatthis could happen.

2 If people take both dimensions into account, dothey do so in an additive or a multiplicative way?The correct approach is the multiplicative one,because the area of a rectangle is its heightmultiplied by its width. This means that thedifference that an increase in the rectangle’ s heightmakes to the area of the rectangle depends on itswidth, and vice versa. An increase of 3 cm in theheight of a 6 cm wide rectangle adds another 18cm2 to its area, but the same increase in height toan 8 cm wide rectangle adds another 24 cm 2. Theadditive approach, which is wrong, would be tojudge that an equal change in height to tw orectangles has exactly the same effect on theirareas, even if their widths differ. This is not tr ue ofarea, but it is tr ue of perimeter. To increase theheight of a rectangle by 3 cm has exactly the sameeffect on the per imeter of a 6 cm and an 8 cmwide rectangle (and increase of 6 cm) and thesame goes for increases to the width of rectangleswith different heights. Also, the same increase inwidth has exactly the same effect on the tworectangles’ perimeters, but very different effects ontheir area. It follows that anyone who persistentlymakes additive judgements about area is probablyconfusing area with per imeter.

The tasks that these two teams of exper imentersgave to children and adults in their studies wereremarkably similar, and so we will describe onlyWilkening’s (1979) experiment. He showed 5-, 8-and 11-year-old children and a group of adults aseries of rectangles that var ied both in height (6, 12and 18 cm) and in width (again 6, 12 and 18 cm). Hetold the par ticipants that these could be broken intopieces of a par ticular size, which he illustrated byshowing them also the size of one of these pieces.The children’s and adults’ task was to imagine whatwould happen if each rectangle was brok en up andthe pieces were arranged in a row. How long wouldthis row be?


The most str iking contrast in the pattern of thesejudgments was between the five-year-old childrenand the adults. To put it br iefly, five-year-old childrenmade additive judgements and adults mademultiplicative judgements.

The five-year-olds plainly did take both height andwidth into account, since they routinely judgedrectangles of the same height but different widths ashaving different areas and they did the same withrectangles of the same widths but different heights.This is an impor tant result, and it must be reassuringto anyone who has to teach young schoolchildrenabout how to measure area. They are apparentlyready to take both dimensions into account.

However, the results suggest that young childrenoften co-ordinate information about height andwidth in the wrong way. The typical five-year-oldjudged, for example, that a 6 cm diff erence in heightwould have the same effect on 12 cm and 18 cmwide rectangles. In contrast, the adults’ judgementsshowed that they recognised that the eff ect wouldbe far greater on the 18 cm than on the 12 cm widefigures. This is evidence that young children rely onthe figures’ perimeters, presumably implicitly, in orderto judge their area. As have already seen, whenchildren begin to use r ulers many of them fall intothe trap of measuring a figure’s perimeter in order towork out its area, (Nunes, Light and Mason, 1993).Their habit of concentrating on the per imeter whenmaking informal judgements about area may well bethe basis for this later mistake. The existence amongschoolchildren of serious confusion between areaand perimeter was confirmed in later research byDembo, Levin and Siegler (1997).

We can end this section with an interesting question.One obvious possible cause of the radical differencein the patterns of 5-year-olds’ and adults’ judgementsmight be that the 5-year-olds had not learned howto measure area while the adults had. In otherwords, mathematical learning could alter this aspectof people’s spatial cognition. The suggestion does notseem far-fetched, especially when one also consider sthe performance of the older children in Wilkening’sinteresting study. The 5-year-olds had not beentaught about measurement at all: the 8-year-olds hadhad some instruction, but not a great deal: the 12-year-olds were well-versed in measurement, butprobably still made mistakes. Wilkening found somesigns of a multiplicative pattern in the responses ofthe 8-year-olds, but this was slight: he found strongersigns of this pattern among the 12-year-olds, though

not as pronounced as in the adult group . Thesechanges do not prove that being taught how to measure and then becoming increasingl yexperienced with measuring led to this differencebetween the age groups, but they are cer tainlyconsistent with that idea. There is an alternativeexplanation, which is that adults and older childrenhave more informal experience than 8-year-olds doof judging and compar ing areas, as for example whenthey have to judge how much paint they need tocover different walls. Here is a signif icant andinteresting question for research: do teachers alonechange our spatial under standing of area or doesinformal experience play a par t as well?

Summary

1 Measuring area is a multiplicative process: weusually multiply two simple measurements (e.g.base by height for rectangles) to produce a totalmeasure of an area. The process also produces adifferent unit (i.e . product of measures): measuringthe base and height in centimetres and thenmultiplying them produces a measure in ter ms ofsquare centimetres.

2 Producing a new measure is a diff icult step forchildren to make. They find it easier to measure arectangle when they measure with units whichdirectly instantiate square centimetres than whenthey use a r uler to measure its base and height incentimetres.

3 The multiplicative aspect of area measurement isalso a problem for young children who show adefinite bias to judge the area of a rectangle byadding its base and height r ather than bymultiplying them. They confuse, therefore,perimeter and area.

Parallelograms: forming relationsbetween rectangles andparallelogramsThe measurement of parallelograms takes us into one of the most exciting aspects of lear ning aboutgeometry. The base-by-height rule applies to thesefigures as well as to rectangles. One way of justifyingthe base by height rule for parallelograms is that anyparallelogram can be transformed into a rectangle withthe same base and height measurements b y addingand subtracting congruent areas to the parallelogram.



Figure 5.4 presents this justif ication which is acommonplace in geometr y classes. It is based on theinversion principle (see Paper 2). Typically the teachershows children a parallelogram and then creates twocongruent triangles (A and S) by dropping verticallines from the top two corners of the parallelogramand then extending the baseline to reach the newvertical that is external to the parallelogram. TriangleA falls outside the original parallelogram, andtherefore is an addition to the f igure. Triangle S fallsinside the original parallelogram, and the nub of theteacher’s demonstration is to point out that theeffect of adding Triangle A and subtracting Triangle Swould be to transform the figure into a rectanglewith the same base and height as the or iginalparallelogram. Triangles A and S are congruent and so their areas are equal. Therefore, adding one andsubtracting the other tr iangle must produce a newfigure (the rectangle) of exactly the same size as theoriginal one (the parallelogram).

This is a neat demonstr ation, and it is an impor tantone from our point of view, because it is our f irstexample of the impor tance in geometr y ofunderstanding that there are systematic relationsbetween shapes. Rectangles and parallelograms aredifferent shapes but they are measured by the samebase-by-height rule because one can tr ansform anyrectangle into any parallelogram, or vice versa, withthe same base and height without changing thefigure’s area.

Some classic research by the well-known Austrianpsychologist Max Wertheimer (1945) suggests thatmany children learn the procedure for transformingparallelograms into rectangles quite easily, but apply itinflexibly. Wertheimer witnessed a group of 11-year-old children learning from their teacher why thesame base-by-height rule applied to parallelogramsas well. The teacher used the justif ication that wehave already described, which the pupils appeared tounderstand. However, Wertheimer was not cer tainwhether these children really had understood theunderlying idea. So, he gave them anotherparallelogram whose height was longer than its base(diagram 3 in Figure 5.4). When a parallelogram isoriented in this way, dropping two vertical lines fromits top two corners does not create two congruenttriangles. Wertheimer found that most of thechildren tried putting in the two vertical lines, butwere at a loss when they sa w the results of doing so .A few, however, did manage to solve the problem byrotating the new figure so that the base was longerthan its height, which made it possib le for them torepeat the teacher’s demonstration.

The fact that most of the pupils did not cope withWertheimer’s new figure was a clear demonstr ationthat they had learned more about the teacher’sprocedure than about the under lying idea abouttransformation that he had hoped to convey.Wertheimer argued that this was probably becausethe teaching itself concentrated too much on theprocedural sequence and too little on the idea of


Figure 5.4: Demonstrating by transforming a parallelogram into a rectangle that the base-by-height rule applies toparallelograms as well as to rectangles.

1

2

3


Figure 5.6: Piaget et al.’s irregular polygon whose area could be measured by decomposition.

Figure 5.5: An example of Wertheimer’s A and B figures. A figures could be easily transformed into a simple rectangle. B figures could not.

transformation. In later work, that is only repor tedrather informally (Luchins and Luchins, 1970),Wertheimer showed children two figures at a time,one of which could be easil y transformed into ameasurable rectangle while the other could not (for example, the A and B figures in Figure 5.5).Wertheimer repor ted that this is an eff ective way of preparing children for understanding the relationbetween the area of par allelograms and rectangles.

It is a regrettable irony that this extraordinarilyinteresting and ingenious research by a leadingpsychologist was done so long ago and is so widel yknown, and yet few researchers since then havestudied children’s knowledge of how to transformone geometric shape into another to f ind its area.

In fact, Piaget et al. (1960) did do a relevant study,also a long time ago. They asked children to measure

the area of an ir regular polygon (Figure 5.6). One good way to solve this difficult problem is to par tition the figure by imaging the divisionsrepresented by the dotted lines in the r ight handfigure. This creates the Triangle x and also a rectanglewhich includes another Triangle y. Since the twotriangles are congruent and Triangle x is par t of theoriginal polygon while Triangle y is not, the area ofthe polygon must add up to the area of therectangle (plus Triangle x minus Triangle y).

Piaget et al. report that the problem flummoxedmost of the children in their study, but repor t thatsome 10-year-olds did come up with the solutionthat we have just described. They also tell us thatmany children made no attempt to break up thefigure but that others, more advanced, were ready todecompose the figure into smaller shapes, but didnot have the idea of in eff ect adding to the f igure by


imagining the BD line which was exter nal to thefigure. Thus, the stumbling block for these relativelyadvanced children was in adding to the f igure. Thisvaluable line of research, long abandoned, needs tobe restarted.

Triangles: forming relations betweentriangles and parallelograms Other transformations from one shape to anotherare equally important. Triangles can be transformedinto parallelograms, or into rectangles if they areright-angle triangles, simply by being doubled (seeFigure 5.7). Thus the area of a tr iangle is half that of a parallelogram with the same base and height.

Hart et al.’s (1985) study shows us that 11- to 14-year-old children’s knowledge of the (base xheight)/2 rule for measuring triangles is distinctlysketchy. Asked to calculate the area of a r ight-angletriangle with a base of 3 cm and a height of 4 cm,only 48% of the children in their third y ear insecondary school (13- and 14-year-olds) gave theright answer. Only 31% of the f irst year secondaryschool children (11-year-olds) succeeded, whilealmost an equal number of them – 29% – ga ve theanswer 12, which means that they cor rectlymultiplied base by height but forgot to halve theproduct of that multiplication.

Children are usually taught about the relationshipbetween triangles, rectangles and parallelograms quiteearly on in their geometr y lessons. However, we knowof no direct research on how well children understandthe relationships between different shapes or on thebest way to teach them about these relations.

Summary

1 Learning about the measurement of the area ofdifferent shapes is a cumulative affair which isbased not just on formulas for measuring par ticularshapes but on grasping the relationships, throughtransformations, of different shapes to each other.Parallelograms can be transformed into rectanglesby adding and subtracting congruent triangles:triangles can be transformed into parallelogramsby being doubled.

2 There is little direct research on children’ sunderstanding of the impor tance of therelationships between shapes in theirmeasurement. Wertheimer’s observations suggestvery plausibly that their under standing dependsgreatly on the quality of the teaching.

3 We need more research on what are the mosteffective ways to teach children about theserelationships.


Figure 5.7: A demonstration that any two congruent triangles add up to a measurableparallelogram and any two congruent right angle triangles add up to a measurable rectangle.

Understanding newrelationships: the case of angle

Angle is an abstract relation. Sometimes it is thedifference between the orientations of two lines orrays, sometimes the change in your orientation fromthe beginning of a tur n that you are making to itsend, and sometimes the relation between a figure ora movement to permanent aspects of theenvironment: the angle at which an aeroplane r isesafter take-off is the relation between the slope of itspath and the spatial hor izontal. Understanding thatangles are a way of describing such a var iety ofcontexts is a basic par t of learning plane geometr y.Yet, research by psychologists on this impor tant andfascinating topic is remarkably thin on the ground.

Most of the relatively recent studies of children’s andadults’ learning about angles are about theeffectiveness of computer-based methods ofteaching. This is estimable and valuable work, but wealso need a great deal more inf ormation aboutchildren’s basic knowledge about angles and aboutthe obstacles, which undoubtedly do exist, toforming an abstract idea of what angles are .

In our everyday lives we experience angles in manydifferent contexts, and it may not at first be easy forchildren to connect information about anglesencountered in different ways. The obvious distinctionhere is between perceiving angles as conf igurations,such as the difference between perpendicular andnon-perpendicular lines in pictures and diagr ams, andas changes in movement, such as changing directionby making a turn. These forms of experiencing anglescan themselves be subdivided: it may not be obviousto school children that we make the same angularchange in our movements when we walk along apath with a r ight-angle bend as when we turn adoor-knob by 90o (Mitchelmore, 1998).

Another point that children might at first find hard tograsp is that angles are relational measures. When wesay that the angles in some of the f igures in Figure5.8a are 90o ones and in other s 45o, we are making astatement about the relation between theorientations of the two lines in each f igure and notabout the absolute or ientation of any of theindividual lines, which var y from each other. Also,angles affect the distance between lines, but only inrelation to the distance along the lines: in Figure 5.8bthe distance between the lines in the f igure with thelarger angle is greater than in the other f igure when


Figure 5.8: Angles as relations


the distance is measured at equivalent points alonglines in the two figures, but not necessarily otherwise.

A third possible obstacle is that children might f indsome representations of angle more understandablethan others. Angles are sometimes formed by themeeting of two clear lines, like the peak of a roof orthe corner of a table. With other angles, such as theinclination of a hill, the angle is clear ly represented byone line – the hill itself, but the other, a notionalhorizontal, is not so clear. In still other cases, such asthe context of turning, there are no clear ly definedlines at all: the angle is the amount of tur ning.Children may not see the connection between thesevery different perceptual situations, and they mayfind some much easier than other s.

There are few theories of how children learn about angles, despite the impor tance of the topic .The most comprehensive and in many ways themost convincing of the theor ies that do exist wasproduced by Mitchelmore and White (2000). Theproblem that these researcher s tried to solve is howchildren learn to abstract and classify angles despitethe large variety of situations in which theyexperience them. They suggest that children’sknowledge of angles develops in three steps:

1 Situated angle concepts Children first registerangles in completely specific ways, according toMitchelmore and White. They may realise that apair of scissors, for example, can be more open orless open, and that some playground slides havesteeper slopes than other s but they make no linkbetween the angles of scissor s and of slides, andwould not even recognise that a slide and a roofcould have the same slope.

2 Contextual angle concepts The next step thatchildren take is to realise that there are similar itiesin angles across different situations, but theconnections that they do make are alwaysrestricted to par ticular, fairly broad, contexts. Slope,which we have mentioned already, is one of thesecontexts; children begin to be ab le to compare theslopes of hills, roofs and slides, but they do notmanage to make any connection between theseand the angles of, for instance, turns in a road. Theybegin to see the connection between angles invery different kinds of turns – in roads and in abent nail, for example – but they do not link theseto objects turning round a fixed point, like a dooror a door-knob.

3 Abstract angle concepts Mitchelmore andWhite’s third step is itself a ser ies of steps. Theyclaim that children begin to compare angles acrosscontexts, for example, between slopes and turns,but that initially these connections across contextsare limited in scope; for example, theseresearchers repor t that even at the age of 11years many children cannot connect angles inbends with angles in tur ns. So, children form oneor more restr icted abstract angle ‘domains’ (e.g. adomain that links inter sections, bends, slopes andturns) before they finally develop this into acompletely abstract concept of standard angles.

To test this theoretical fr amework, Mitchelmore andWhite gave children of 7, 9 and 11 years pictures ofa wide range of situations (doors, scissors, bends inroads etc.) and asked them to represent the anglesin these, using a bent pipe cleaner to do so , and alsoto compare angles in pair s of different situations. Thestudy cer tainly showed different degrees ofabstraction among these children and provided someevidence that abstraction about angles increases withage. This is a valuable contribution, but we cer tainly need more evidence about thisdevelopmental change for at least two reasons.

One is methodological. The research that we have justdescribed was cross-sectional: the children in thedifferent age groups were different children. A muchbetter way of testing any hypothesis about a series ofdevelopmental cognitive changes is to do alongitudinal study of the ideas that the same childrenhold and then change over time as they get older. Ifthe hypothesis is also about what makes the changeshappen, one should combine this longitudinal researchwith an intervention study to see what provokes thedevelopment in question. We commented on theneed for combining longitudinal and inter ventionstudies in Paper 2. Once again, we commend this all-too-rarely adopted design to anyone planning to doresearch on children’s mathematics.

The second gap in this theory is its concentration onchildren learning what is ir relevant rather than whatis relevant to angle . The main claim is that childreneventually learn, for example, that the same anglesare defined by two clear lines in some cases b ut notin others, that some angular information is aboutstatic relations and some about movement, but thatit is still exactly the same kind of inf ormation. Thisclaim is almost cer tainly right, but it does not tell uswhat children learn instead.


One possibility, suggested by Piaget, Inhelder andSzeminska (1960), is that children need to be ab le torelate angles to the sur rounding Euclidean frameworkin order to reproduce them and compare them toother angles. In one study they asked children to copya triangle like the one in Figure 5.9.

They gave the children some rulers, strips of paperand sticks (but apparently no protractor) to helpthem do this. Piaget et al. wanted to find out howthe children set about reproducing the angles CAB,ABC and ACB. In the experimenters’ view the bestsolution was to measure all three lines, and alsointroduce an additional vertical line (KB in Figure 5.9)or to extend the hor izontal line and then introducea new vertical line (CK’ in Figure 5.9). It is not at allsurprising that children below the age of roughly tenyears did not think of this Euclidean solution. Sometried to copy the tr iangle perceptually. Others usedthe rulers to measure the length of the lines b ut didnot take any other measures. Both strategies tendedto lead to inaccurate copies.

The study is interesting, but it does not estab lish thatchildren have to think of angles in ter ms of theirrelationship to horizontal and vertical lines in orderto be able to compare and reproduce par ticularangles. The fact that the children were not given thechance to use the usual conventional measure forangles – the protractor – either in this study or inMitchelmore and White’s study needs to be noted.This measure, despite being quite hard to use , mayplay a significant and possibly even an essential par tin children’s understanding of angle.

Another way of approaching children’s understandingof angles is through what Mitchelmore and White

called their situated angle concepts. Magina andHoyles (1997) attempted to do this b y investigatingchildren’s understanding of angle in the context ofclocks and watches. They asked Brazilian children,whose ages ranged from 6 to 14 years, to showthem where the hands on a clock w ould be in halfan hour’s time and also half an hour bef ore the timeit registered at that moment. Their aim was to findout if the children could judge the cor rect degree of turn. Magina and Hoyles report that the youngerchildren’s responses tended to be either quiteunsystematic or to depend on the initial position of the minute hand: these latter children, mostly 8-to 11-year-olds, could move the minute hand torepresent half an hour’s difference well enough whenthe hand’s initial position was at 6 (half past) on theclock face, but not when it was at 3 (quar ter past).Thus, even in this highly familiar situation, manychildren seem to have an incomplete under standingof the angle as the degree of a tur n. This ratherdisappointing result suggests that the or igins ofchildren’s understanding may not lie in their informalspatial experiences.

One way in which children may learn about angles isthrough movement. The idea of children learningabout spatial relations by monitoring their own actionsin space fits well with Piaget’s framework, and it is thebasis for Logo, the name that Papert (1980) gave tohis well-known computer-system that has often beenused for teaching aspects of geometry. In Logo,children learn to write programmes to move a ‘turtle’around a spatial environment. These programmesconsist of a ser ies of instructions that determine asuccession of movements by the tur tle. Theinstructions are about the length and direction of each movement, and the instructions about direction


Figure 5.9: The triangle (represented by the continuous lines) that Piaget, Inhelder and Szeminska asked children to copy,with the vertical and horizontal (dotted) lines which some of the children created to help them to solve this problem.


take the form of angular changes e .g. L90 is aninstruction for the tur tle to make a 90o turn to theleft. Since the tur tle’s movements leave a trace,children effectively draw shapes by writing theseprogrammes.

There is evidence that exper ience with Logo doeshave an effect on children’s learning about angles,and this in turn supports the idea thatrepresentations of movement might be one effectiveway of teaching this aspect of geometr y. Noss(1988) gave a group of 8- to 11-y ear-old children,some of whom had attended Logo classes o ver awhole school year, a series of problems involvingangles. On the whole the children who had been tothe Logo classes solved these angular problemsmore successfully than those who had not. Therelative success of the Logo group was par ticularlymarked in a task in which the children had tocompare the size of the turn that people would haveto make at different points along a path, and this isnot surprising since this specific task resonated with the instructions that children make whendetermining the direction of the tur tle’s movements(see Figure 5.10).

However, the Logo group also did better than thecomparison in more static angular problems. This isan interesting result because it suggests that thechildren may have generalised what they learnedabout angles and movement to other angular tasks

which involve no movement at all. We need moreresearch to be sure of this conclusion and, as far asstudies of the effects of Logo and other computer-based programmes are concerned, we need studiesin which pre-tests are given before the children gothrough these programmes as well as post-tests thatfollow these classes.

Children do not just lear n about single angles inisolation from each other. In fact, to us, the mostinteresting question in this area is about theirlearning of the relations between different angles.These relations are a basic par t of geometr y lessons:pupils learn quite ear ly on in these lessons that, forexample, when two straight lines intersect oppositeangles are equal and that alter nate angles in a Z-shape figure are equal also, but how easily thisknowledge comes to them and how effectively theyuse it to solve geometric problems are interestingbut unanswered questions (at any rate, unansweredby psychologists). Some interesting educationalresearch by Gal and Vinner (1997) on 14-year-oldstudents’ reaction to perpendicular lines suggests thatthey have some difficulties in understanding therelation between the angles made by intersectinglines when the lines are perpendicular to each other.Many of the students did not realise at f irst that ifone of the four angles made by two intersectinglines is a r ight angle the other three must be so aswell. The underlying reason for this difficulty needsinvestigation.


Figure 5.10: The judgement about relative amount of turns in Noss’s study.

You are walking along this path. You start at point A and you finish at point G.

• At which point would you have to turn most?• At which point would you have to turn least?

finish

Summary

1 Although young children are aware of orientation,they seem to know little about angle (the relationbetween orientations) when they star t ongeometry.

2 The concept of angle is an abstr act one that cutsacross very different contexts. This is difficult forchildren to understand at first.

3 There is some evidence , mainly from work onLogo, that children can lear n about angle throughmovement.

4 Children’s understanding of the relation betweenangles within figures (e.g. when straight lines intersect,opposite angles are equivalent) is a basic par t ofgeometry lessons, but there seems to be no researchon their understanding of this kind of relation.

Spatial frameworks

Horizontal and vertical lines

Most children’s formal introduction to geometr y is a Euclidean one. Children are taught about str aightlines, perpendicular lines, and parallel lines, and theylearn how a quite complex system of geometr y canbe derived from a set of simple , comprehensible,axioms. It is an exercise in logic , and it must, for most children, be their first experience of a formaland explicit account of two-dimensional space. The principal feature of this account is the relationbetween lines such as par allel and perpendicular andintersecting lines.

These fundamental spatial relations are probab lyquite familiar, but in an implicit way, to the seven- andeight-year-old children when they star t classes ingeometry. In spatial environments, and par ticularly in‘carpentered’ environments, there are obvioushorizontal and vertical lines and surfaces, and theseare at r ight angles (perpendicular) to each other. Wereviewed the evidence on children’s awareness ofthese spatial relations in an ear lier par t of this paper,when we reached the following two conclusions.

1 Although quite young children can relate theorientation of lines to stable background features andoften rely on this relation to remember or ientations,they do not always do this when it would help them

to do so. Thus, many young children and some adultstoo (Howard, 1978) do not recognise that the levelof liquid is parallel to horizontal features of theenvironment like a table top.

2 Part of the difficulty that they have in Piaget’shorizontality and verticality tasks is that thesedepend on children being able to represent acuteand obtuse angles (i.e . non-perpendicular lines). Theytend to do this inaccurately, representing the linethat they draw as closer to perpendicular than itactually is. This bias towards the perpendicular mayget in the way of children’s representation of angle .

The role of horizontal and verticalaxes in the Cartesian system The Euclidean framework makes it possible to pinpointany position in a two-dimensional plane. We owe thisinsight to René Descar tes, the 17th century Frenchmathematician and philosopher, who was interested inlinking Euclid’s notions with algebra. Descartes devisedan elegant way of plotting positions by representingthem in terms of their position along two axes in atwo dimensional plane. In his system one axis wasvertical and the other hor izontal, and so the two axeswere perpendicular to each other. Descartes pointedout that all that you need to know in order to find aparticular point in two-dimensional space is its positionalong each of these two axes. With this informationyou can plot the point by extrapolating an imaginarystraight perpendicular line from each axis. The point atwhich these two lines intersect is the position inquestion. Figure 5.11 shows two axes, x and y, andpoints which are expressed as positions on these axes.

This simple idea has had a huge impact on scienceand technology and on all our dail y lives: forexample, we rely on Car tesian co-ordinates tointerpret maps, graphs and block diagrams. TheCartesian co-ordinate system is a good example of a cultural tool (Vygotsky, 1978) that has tr ansformedall our intellectual lives.

To understand and to use the Car tesian system toplot positions in two-dimensional space, one has tobe able to extrapolate two imaginary perpendicularstraight lines, and to co-ordinate the two in order towork out where they inter sect. Is this a difficult oreven an impossible barrier for young children?Teachers cer tainly need the answer to this questionbecause children are introduced to graphs and blockdiagrams in primary school, and as we have noted



these mathematical representations depend on theuse of Car tesian co-ordinates.

Earlier in this section we mentioned that, in socialcontexts, very young children do extrapolateimaginary straight lines. They follow their mother’sline of sight in order, apparently, to look at whateverit is that is attr acting her attention at the time(Butterworth, 1990). If children can extr apolatestraight lines in three-dimensional space , we canquite reasonably expect them to be able to do so in two-dimensional space as well. The Car tesianrequirement that these extrapolated lines areperpendicular to their baselines should not be aproblem either, since, as we have seen already,children usually find it easier to create per pendicularthan non-perpendicular lines. The only requirementthat this leaves is the ability to work out where thepaths of the two imaginary straight lines intersect.

A study by Somerville and Br yant (1985) establishedthat children as young as six years usually have thisability. In the most complex task in this study, youngchildren were shown a fair ly large square space on ascreen and 16 positions were clearly marked withinthis space, sometimes arranged in a regular grid andsometimes less regular ly than that. On the edge of

the square waited two characters, each just about toset off across the square . One of these char acterswas standing on a vertical edge (the right or left sideof the square) and the other on a horizontal edge of the square. The children were told that bothcharacters could only walk in the direction theywere facing (each character had a rather prominentnose to mark this direction, which was perpendicularto the depar ture line, clear), and that the two wouldeventually meet at one par ticular position in thesquare. It was the child’s task to say which positionthat would be.

The task was slightly easier when the choices werearranged in a gr id than when the ar rangement wasirregular, but in both tasks all the children chose theright position most of the time . The individualchildren’s choices were compared to chance (if achild followed just one extrapolated line instead ofco-ordinating both, he or she would be r ight bychance 25% of the time) and it turned out that the number of correct decisions made by everyindividual child was significantly above chance. Thus,all of these 6-year-old children were able to plot theintersection of two extrapolated, imaginary straightlines, which means that they were well equipped tounderstand Car tesian co-ordinates.


Figure 5.11: Descartes’ co-ordinates: three points (8, 8), (-3, 5) and (-9, -4) are plotted by their positions on the x- and y-axes

Piaget et al. (1960) were less optimistic aboutchildren’s grasp of co-ordinates, which they testedwith a copying task. They gave each child tworectangular sheets of paper, one with a small circleon it and the other a complete b lank. They alsoprovided the children with a pencil and a r uler andstrips of paper, and then they asked them to put acircle on the blank sheet in exactly the same positionas the circle on the other sheet. Piaget et al.’squestion was whether any of the children would usea co-ordinate system to plot the position of theexisting circle and would then use the co-ordinatesto position the circle that they had to dr aw on theother sheet. This was a difficult task, which children of six and seven years tended to fail. Most of themtried to put the new circle in the right place simplyby looking from one piece of paper to the other, andthis was a most unsuccessful str ategy.

There is no conflict between the success of all thechildren in the Somer ville and Br yant study, and thegrave difficulties of children of the same age in thePiaget et al. task. In the Piaget et al. task the childrenhad to decide that co-ordinates were needed andthen had to measure in order to estab lish theappropriate position on each axis. In the Somer villeand Bryant study, the co-ordinates were given and allthat the child had to do was to use them in order tofind the point where the two extrapolated lines met.So, six- and seven-year-old children can establish aposition given the co-ordinates but often cannot setup these co-ordinates in the f irst place.

Some older children in Piaget et al.’s study (all thesuccessful children given as examples in the bookwere eight- or-nine-years-old) did apparentlyspontaneously use co-ordinates. It seems unlikely to us that these children managed to in vent theCartesian system for themselves. How could eight-and nine-year-old children come up, in oneexperimental session, with an idea for which mankindhad had to wait till Descar tes had his br illiant insightin the middle of the 17th centur y? A more plausible reason for these children’s successis that, being among the older children in Piaget etal.’s sample, they had been taught about the use ofCartesian co-ordinates in maps or gr aphs already.

It appears that this success is not univ ersal. Manychildren who have been taught about Car tesian co-ordinates fail to take advantage of them or to usethem properly. Sarama, Clements, Swaminathan,McMillen and Gomez (2003) studied a group ofnine-year-old children while they were being given

intensive instruction, which the researcher sthemselves designed. In the tasks that they gave tothe children Sarama et al. represented the x and yco-ordinates as numbers, which was a good thing to include because it is a fundamental par t of theCartesian system, and they also imposed arectangular grid on many of the spaces that theygave the children to work with. Thus, one set ofmaterials was a rectangular grid, presented as a mapof a grid-plan city with ‘streets’ as the vertical and‘avenues’ as the hor izontal lines.

The children were given various tasks before, duringand after the instr uction. Some of these involvedrelating locations to x and y co-ordinates: thechildren had to locate positions given the co-ordinates and also to work out the co-ordinates ofparticular positions. Sarama et al. reported that mostof the children learned about this relation quicklyand well, as one might expect given their evidentability to co-ordinate extrapolated lines in arectangular context (Somerville and Bryant, 1985).However, when they had to co-ordinate inf ormationabout two or more locations, they were in greaterdifficulty. For example, some children found it hard towork out the distance between two locations in thegrid-like city, because they thought that the n umberof turns in a path affected its distance, and some didnot realise that the numerical differences in the xand y co-ordinate addresses between the twolocations represents the distance between them.

Thus, the children under stood how to find twolocations, given their coordinates, but struggled withthe idea that a compar ison between the two pairs of co-ordinates told them about the spatial relationbetween these locations. Fur ther observationsshowed that the problem that some of the childrenhad in working out the relations between two co-ordinate pairs was created by a cer tain tensionbetween absolute and relative information. The two co-ordinate pairs 10,30 and 5,0 represent twoabsolute positions: however, some children, whowere given first 10,30 and then 5,0 and ask ed towork out a path between the two, decided that 5,0represented the difference between the first and thesecond location and plotted a location f ive blocks tothe right of 10,30. They treated absolute informationabout the second position as relativ e informationabout the difference between the two positions.However, most of the children in this ingenious andimportant study seem to have overcome thisdifficulty during the period of instruction, and to have learned reasonably well that co-ordinate pair s



represent the relation between positions as well asthe absolute positions themselves.

Finally, we should consider children’s understanding ofthe use of Car tesian co-ordinates in graphs. Here,research seems to lead to much the same conclusionas we reached in our discussion of the use of co-ordinates to plot spatial positions. With graphs, too,children find it easy to locate single positions, butoften fail to take advantage of the information thatgraphs provide that is based on the relation betweendifferent positions. Bryant and Somerville (1986) gave six- and nine-year-old children a graph thatrepresented a simple linear function. Using much the same technique as in their previous research onspatial co-ordinates, these researchers measured thechildren’s ability to plot a position on x-axis given theposition on the y-axis and vice versa. This was quite aneasy task for the children in both age groups, and sothe main contribution of the study was to estab lishthat children can co-ordinate extr apolated straightlines in a graph-like task fair ly well even before theyhave had any systematic instruction about graphs.

The function line in a graph is formed from a ser iesof positions, each of which is deter mined byCartesian co-ordinates. As in the Sarama et al. study,it seems to be hard f or children to grasp what therelation between these different positions means. Aninteresting study by Knuth (2000) showed thatAmerican students ‘enrolled in 1st-year algebra’(Knuth does not say how old these students were)are much more likely to express linear functions asequations than graphically. We do not yet know thereasons for this preference

Summary

1 Cartesian co-ordinates seem to pose no basicintellectual difficulty for young children. They areable to extrapolate imaginary straight lines that areperpendicular to horizontal and vertical axes andto work out where these imaginar y lines wouldmeet in maps and in gr aphs.

2 However, it is harder for children to work out therelation between two or more positions that areplotted in this way, either in an map-like or in agraph-like task.

3 Students prefer expressing functions as equationsto representing them graphically.

4 Thus, although children have the basic abilities tounderstand and use co-ordinates well, there seemto be obstacles that prevent them using theseabilities in tasks which involve two or more plottedpositions. We need research on how to teachchildren to surmount these obstacles.

Categorising, composing anddecomposing shapesWe have chosen to end this chapter on lear ningabout space and geometr y with the question ofchildren’s ability to analyse and categorise shapes, butwe could just as easil y have star ted the section withthis topic, because children are in many ways exper tson shape from a very early age. They are born,apparently, with the ability to distinguish andremember abstract, geometric shapes, like squares,triangles, and circles, and with the capacity torecognise such shapes as constant even when theysee them from different angles on different occasionsso that the shape of the retinal image that they mak evaries quite radically over time. We left shape to theend because much of the learning that we havediscussed already, about measurement and angle andspatial co-ordinates, undoubtedly affects and changesschoolchildren’s understanding of shape.

There is near ly complete agreement among thosewho study mathematics education that children’sknowledge about shapes undergoes a series ofradical changes during their time at school. Differenttheories propose different changes but many ofthese apparent disagreements are really onlysemantic ones. Most claim, though in different terms,that school-children star t by being able to distinguishand classify shapes in a perceptual and implicit wa yand eventually acquire the ability to analyse theproperties of shapes conceptually and explicitly.

The model developed by the Dutch educationalistvan Hiele (1986) is cur rently the best knowntheoretical account of this kind of lear ning. This is agood example of what we called a ‘pragmatic theory’in our opening paper. Van Hiele claimed that childrenhave to take a sequence of steps in a f ixed order intheir geometric learning about shape. There are fivesuch steps in van Hiele’s scheme, but he agreed thatnot all children get to the end of this sequence:


Level 1: Visualization/recognition Studentsrecognise and learn to name cer tain geometricshapes but are usually only aware of shapes as awholes, and not of their proper ties or of theircomponents.

Level 2: Descriptive/analytic Students begin torecognise shapes by their proper ties.

Level 3: Abstract/relational Students begin to formdefinitions of shapes based on their commonproperties, and to under stand some proofs.

Level 4: Formal deduction Students understand thesignificance of deduction as a way of establishinggeometric theory within an axiomatic system, andcomprehend the interrelationships and roles ofaxioms, definitions, theorems, and formal proof.

Level 5: Rigour Students can themselves reasonformally about different geometric systems.

There have been several attempts to elaborate andrefine this system. For example, Clements and Sarama(2007 b) argue that it would better to rename theVisual/recognition stage as Syncretic, given itslimitations. Another development was Guttierez’s(1992) sustained attempt to extend the system to 3-D figures as well as 2-D ones. However, although vanHiele’s steps provide us with a useful and interestingway of assessing improvements in children’sunderstanding of geometry, they are descr iptive. Thetheory tells us about changes in what children do anddo not understand, but not about the under lyingcognitive basis for this understanding, nor about thereasons that cause children to move from one level to the next. We shall turn now to what is known andwhat needs to be known about these cognitive bases.

Composing and decomposing

If van Hiele is r ight, one of the most basic changes inchildren’s analysis of shape is the realisation thatshapes, and par ticularly complex shapes, can bedecomposed into smaller shapes. We have alreadydiscussed one of the reasons why children need tobe able to compose and decompose shapes, which isthat it is an essential par t of understanding themeasurement of the area of different shapes.Children, as we have seen, must learn, for example,that you can compose a par allelogram by puttingtogether two identical triangles (and thus that you

can decompose any parallelogram into two identicaltriangles) in order to under stand how to measurethe area of tr iangles. As far as we know, there hasbeen no direct research on the relationship betw eenchildren’s ability to compose and decompose shapesand their understanding of the rules for measuringsimple geometric figures, though such researchwould be easy to do.

The Hart et al. (1985) study included two itemswhich dealt with shapes that were decomposed intotwo parts and these par ts were then re-arranged. Inone item the re-arranged par ts were a rectangle andtwo triangles, which are simple and familiar geometr icshapes, and in the other they w ere unfamiliar andmore complex shapes. In both cases the childrenwere asked about the effect of the re-arrangementon the figure’s total area: was the new f igure’s areabigger or smaller or the same as the area of theoriginal one? A large proportion of the 11- to 14-year-old students in the study (over 80% in eachgroup) gave the correct answer to the first of thesetwo questions but the second was far harder : only60% of the 11-year-old group understood that thearea was the same after the rear rangement of par tsas before it. There are two reasons for beingsurprised at this last result. The first is that it is hardto see why there was such a lar ge difference in thedifficulty of the two problems when the logic forsolving both was exactly the same. The second is thatthe mistakes which the children made in suchabundance with the harder problem are in effectconservation failures, and yet these children are wellbeyond the age when conser vation of area shouldpose any difficulty for them. This needs fur ther study.

The insight that under standing composition anddecomposition may be a basic par t of children’slearning about shapes has been investigated inanother way. Clements, Wilson and Sarama (2004)looked at a group of three- to seven-year-oldchildren’s ability to assemble target patterns, like thefigure of a man, by assembling the r ight componentwooden shapes. This interesting study producedevidence of some sharp developmental changes: theyounger children tended to create the f igures bit bybit whereas the older children tended to create unitsmade out of several bits (an arm unit for example)and the oldest dealt in units made out of other units.The next step in this research should be to f ind outwhether there is a link between this developmentand the eventual progress that children make inlearning about geometr y. Once again we have to



make a plea for longitudinal studies (which are fartoo rare in research on children’s geometry) andintervention studies as well.

Summary

1 Although young school children are already veryfamiliar with shapes, they have some difficulty withthe idea of decomposing these into par ts, e.g. aparallelogram decomposed into two congruenttriangles or an isosceles tr iangle decomposed intotwo right-angle triangles, and also with the inverseprocess of composing new shapes by combiningtwo or more shapes to make a different shape.

2 The barrier here may be that these are unusualtasks for children who might lear n how to carrythem out easily given the r ight experience. This is a subject for future research.

Transforming shapes: enlargement,rotation and reflection We have already stressed the demands thatmeasurement of length and angle make on children’simagination, and the same holds f or their learningabout the basic transformations of shapes –translation, enlargement, rotation and reflection.Children have to learn to imagine how shapes wouldchange as a result of each of these tr ansformationsand we know that this is not always easy. The workby Har t (1981) and her colleagues on children’ ssolutions to reflection and rotation prob lemssuggests that these transformations are not alwayseasy for children to work out. They repor t that thereis a great deal of change in betw een the ages of 11and 16 years in students’ understanding of whatchanges and what stays constant as a result of thesetwo kinds of transformation.

One striking pattern repor ted by this research groupwas that the younger children in the group beingstudied were much more successful with rotationand reflection problems that involved horizontal andvertical figures than with sloping f igures: this resultmay be related to the evidence , mentioned ear lier,that much younger children discriminate andremember horizontal and vertical lines much betterthan sloping ones.

Psychologists, in contrast to educationalists, have notthrown a great deal light on children’ s learning about

transformations, even though some research onperceptual development has come close to doing so. They have shown that children remembersymmetrical figures better than asymmetr ical figures(Bornstein, Ferdinandsen and Gross, 1981), and there are obser vations of pre-school childrenspontaneously constructing symmetrical figures ininformal play (Seo and Ginsburg, 2004). However,the bulk of the psychological work on rotation andreflection has treated these tr ansformations in anegative sense. The researchers (Bomba, 1984;Quinn, Siqueland and Bomba, 1985; Bryant,1969,1974) were concerned with children’sconfusions between symmetrical, mirror-imagefigures (usually reflections around a vertical axis):they studied the development of children’s ability totell symmetrical figures apar t, not to under stand therelation between them. Here is another br idge stillto be crossed between psychology and education.

Enlargement raises some interesting issues aboutchildren’s geometric understanding. We know of nodirect research on teaching children or on childrenlearning about this transformation, at any rate in thegeometrical sense of shapes being enlar ged by adesignated scale factor. These scale factors of coursedirectly affect the perimeter of the shapes: thelengths of the sides of, for example, a r ight-angletriangle enlarged by a factor of 2 are twice those ofthe original triangle. But, of course, the relationbetween the areas of the two triangles is different:the area of the lar ger triangle is 4 times that of thesmaller one.

There is a danger that some children, and even someadults, might confuse these different kinds of relationbetween two shapes, one of which is an enlar gementof the other. Piaget et al. (1960) showed children a 3cm x 3 cm square which they said represented afield with just enough gr ass for one cow, and thenthey asked each child to dr aw a larger field of thesame shape which would produce enough for twocows. Since this area measured 9 cm 2 the newsquare would have to have an area of 18 cm.2 andtherefore sides of roughly 4.24 cm. since 4.24 is verynearly the square root of 18. In this study most ofthe children under the age of 10 y ears either actedquite unsystematically or made the mistake ofdoubling the sides of the original square in the newfigure that they drew, which meant that their newsquare (an enlargement of the original square by ascale factor of two) had 6 cm sides and an area ofaround 36 cm2 which is actually 4 times the area ofthe first square. Older children, however, recognised


the problem as a multiplicative one and calculatedeach of the two squares’ areas my multiplying itsheight by its width. The younger children’s difficultiesechoed those of the slave who, in Plato’s Meno, waslucky enough to be instr ucted about measuring areaby Socrates himself.

The widespread existence of this apparentl yprevailing belief that doubling the length of the sidesof a shape will double its area as well was recentlyconfirmed by a team of psychologists in Belgium((De Bock, Verschaffel and Janssens, 1998, 2002; DeBock, Van Dooren, Janssens and Verschaffel, 2002; DeBock, Verschaffel, Janssens, Van Dooren and Claes,2003). There could be an educational conflict herebetween teaching children about scale factors on theone hand and about propor tional changes in area onthe other. It is possible that the misconceptionsexpressed by students in the studies by Piaget et al.(1960) and also by the Belgian team may actuallyhave been the result of confusion betw een the useof scale factors in drawing and effect of doubling thesides of figures. In Paper 4 we discussed in detail thedifficulties of the studies car ried out by the Belgianteam but we still need to f ind out, by research,whether scale drawing does provoke this confusion.It would be easy to do such research.

Summary

1 Understanding, and being able to work out, thefamiliar transformations of reflection, rotation andenlargement are a basic par t of the geometry thatchildren learn at school. They are another instanceof the impor tance of grasping the relationsbetween shapes in learning geometry.

2 Psychology tells us little about children’sunderstanding of these relations, though it wouldbe easy enough for psychologists to do empir icalresearch on this basic topic. The reason forpsychologists’ neglect of transformations is thatthey have concentrated on children distinguishingbetween shapes rather than on their ability towork out the relations between them.

3 There is the possibility of a clash betw een learningabout scale factors in enlargement and about themeasurement of area. This should be investigated.

General conclusions onlearning geometry

1 Geometry is about spatial relations.

2 Children have become highly familiar with some ofthese relations, long before they learn about themformally in geometr y classes: others are new tothem.

3 In the case of the spatial relations that they kno wabout already, like length, orientation and positionrelations, the new thing that children ha ve to learnis to represent them numerically. The process ofmaking these numerical representations is notalways straightforward.

4 Representing length in standard units depends onchildren using one-to-one correspondencebetween the units on the ruler and imagined unitson the line being measured. This may seem to beeasy to do to adults, but some children find itdifficult to understand.

5 Representing the area of rectangles in standardunits depends on children understanding twothings: (a) why they have to multiply the base withthe height in centimetres (b) why thismultiplication produces a measure in differentunits, square centimetres. Both ideas are diff icultones for young children.

6 Understanding how to measure parallelograms andtriangles depends on children lear ning about therelation between these shapes and rectangles.Learning about the relations between shapes is asignificant par t of learning about geometr y anddeserves attention in research done bypsychologists.

7 The idea of angle seems to be new to mostchildren at the time that they begin to learn aboutgeometry. Research suggests that it takes childrensome time to a form an abstract concept of anglethat cuts across different contexts. More researchis needed on children’s understanding of therelations between angles in par ticular figures.

8 Children seem well-placed to learn about thesystem of Car tesian co-ordinates since they are , onthe whole, able to extrapolate imaginaryperpendicular lines from hor izontal and vertical co-ordinates and to work out where they intersect.They do, however, often find co-ordinate tasks



which involve plotting and working out therelationship between two or more positions quitedifficult. We need research on the reasons f or thisparticular difficulty.

9 There is a ser ious problem about the quality of theresearch that psychologists have done on childrenlearning geometry. Although psychologists havecarried out good work on children’s spatialunderstanding, they have done very little to extendthis work to deal with formal learning about themathematics of space. There is a special need f orlongitudinal studies, combined with inter ventionstudies, of the link between informal spatialknowledge and success in learning geometry.


Anderson, N. H., and Cuneo, D. (1978). The height +width rule in children’s judgments of quantity.Journal of Experimental Psychology- General, 107,335-378.

Bomba, P. C. (1984). The Development ofOrientation Categories between 2 and 4 Monthsof Age. Journal of Experimental Child Psychology,37, 609-636.

Bornstein, M.H., Ferdinandsen, K., and Gross, C.(1981) Perception of symmetry in infancy.Developmental Psychology, 17, 82-86.

Bremner, A., Bryant, P., and Mareschal, D. (2006).Object-centred spatial reference in 4-month-oldinfants. Infant Behavior and Development, 29(1), 1-10.

Bryant, P. E. (1969). Perception and memory of theorientation of visually presented lines by children.Nature, 224, 1331-1332.

Bryant, P. E. (1974). Perception and understanding inyoung children. London: Methuen.

Bryant, P. E., and Kopytynska, H. (1976). Spontaneousmeasurement by young children. Nature, 260, 773.

Bryant, P. E., and Somerville, S. C. (1986). The spatialdemands of graphs. British Journal of Psychology,77, 187-197.

Bryant, P. E., and Trabasso, T. (1971). Transitiveinferences and memory in young children. Nature,232, 456-458.

Butterworth, G. (2002). Joint visual attention ininfancy. In G. Bremner and A. Fogel (Eds.),Blackwell Handbook of Infant Development:Handbooks of Developmental Psychology (pp. 213-240). Oxford: Blackwell.

Butterworth, G., and Cochran, C. (1980). Towards amechanism of joint attention in human infancy.International Journal of Behavioural Development, 3, 253-272.

Butterworth, G., and Grover, L. (1988). The origins ofreferential communication in human infancy. In L.Weiskrantz (Ed.), Thought wihtout language (pp. 5-24). Oxford: Oxford University Press.

Butterworth, G., and Itakura, S. (2000). How the eyes,head and hand ser ve definite reference. BritishJournal of Developmental Psychology, 18, 25-50.

Butterworth, G., and Jarrett, N. (1991). What mindshave in common is space: spatial mechanismsserving joint visual attention in infancy. BritishJournal of Developmental Psychology, 9, 55-72.

Butterworth, G., and Morisette, P. (1996). Onset of pointing and the acquisition of language ininfancy. British Journal of Develomental Psychology,14, 219-231.

Buys, K., and Veltman, A. (2008). Measurement inKindergarten 1 and 2. In M. van den Heuvel andK. Buys (Eds.), Young children learn measurementand geometry: a learning-teaching trajectory withintermediate attainment targets for the lower gradesin Primary school (pp. 15-36). Rotterdam: SensePublishers.

Casey, B., Andrews, N., Schindler, H., Kersh, J. E.,Samper, A., and Copley, J. V. (2008). Thedevelopment of spatial skills throughinterventions involving block building activities.Cognition and Instruction, 26, 1-41.

Casey, B., Erkut, S., Ceder, I., and Young, J. M. (2008).Use of a stor ytelling context to improve girls’ andboys’ geometry skills in kindergarten. Journal ofApplied Developmental Psychology, 29, 29-48.

Clements, D. H., and Sarama, J. (2007 a.). Effects of apreschool mathematics curriculum: Summativeresearch on the Building Blocks project. .Journalfor Research in Mathematics Education, 38, 136-163

Clements, D. H., and Sarama, J. (2007 b.). Earlychildhood mathematics learning. In F. K. Lester(Ed.), Second handbook of research on mathematicsteaching and learning . (Vol. 1, pp. 461-555).Charlotte, NC:: Information Age Publishing.

Clements, D., Wilson, D., and Sarama, J. (2004). Youngchildren’s composition of geometr ic figures: alearning trajectory. Mathematical Thinking andLearning, 6(2), 163-184.


References


De Bock, D., Van Dooren, W., Janssens, D., andVerschaffel, L. (2002). Improper use of linearreasoning: An in-depth study of the nature andthe irresistibility of secondar y school students’errors. Educational Studies in Mathematics , 50,311–334.

De Bock, D., Verschaffel, L., and Janssens, D. (1998).The predominance of the linear model insecondary school students’ solutions of wordproblems involving length and area of similarplane figures. Educational Studies in Mathematics,35, 65–83.

De Bock, D., Verschaffel, L., and Janssens, D. (2002).The effects of different problem presentationsand formulations on the illusion of linearity insecondary school students. Mathematical Thinkingand Learning, 4, 65–89.

De Bock, D., Verschaffel, L., Janssens, D., Van Dooren,W., and Claes, K. (2003). Do realistic contexts andgraphical representations always have a beneficialimpact on students’ performance? Negativeevidence from a study on modeling non-lineargeometry problems. Learning and Instruction, 13,441-463.

Dembo, Y., Levin, I., and Siegler, R.S. (1997). Acomparison of the geometr ic reasoning of studentsattending Israeli ultra-orthodox and mainstreamschools. Developmental Psychology, 33, 92-103.

Guttierez, A. (1992). Exploring the link between vanHiele’s levels and 3-dimensional geometry.Structural Topology, 18, 31-48.

Hart, K. (1981). Children’s understanding ofmathematics: 11-16. London: John Murray.

Hart, K., Brown, M., Kerslake, D., Küchemann, D. andRuddock, G. (1985) Chelsea DiagnosticMathematics’ Test: Teacher’s Guide. Windsor: NFER-Nelson.

Howard, I. P. (1978). Recognition and knowledge ofthe water level principle. Perception, 7, 151-160.

Ibbotson, A., and Br yant, P. E. (1976). Theperpendicular error and the vertical effect inchildren’s drawing. Perception, 5, 319-326.

Kloosterman, P., Warfield, J., Wearne, D., Koc, Y.,Martin, G., and Strutchens, M. (2004). Knowledgeof learning mathematics and perceptions oflearning mathematics in 4th-grade students. In P.Kloosterman and F. Lester (Eds.), Resutlts andinterpetations of the 2003 mathematics assessmentof the NAEP . Reston, VA: national Council OfTeachers of Mathematics.

Knuth, E.J. (2000) Student under standing of theCartesian connection: an exploratory study.Journal for Research in Mathematics Education , 31, 500-508.

Lawrenson, W., and Bryant, P. E. (1972). Absolute andrelative codes in young children. Journal of ChildPsychology and Psychiatry, 12, 25-35.

Luchins, A. S., and Luchins, E. H. (1970). Wertheimer’sSeminars Re-visited: Problem solving and thinkingVols I to III . Albany, N.Y.: Faculty StudentAssociation, State University of New York.

Magina, S., and Hoyles, C. (1997). Children’sunderstanding of turn and angle. In T. Nunes andP. Bryant (Eds.), Learning and TeachingMathematics. An International Perspective (pp. 88-114). Hove (UK): Psychology Press.

Mitchelmore, M. (1998). Young students’ concepts ofturning. Cognition and Instruction, 16, 265-284.

Mitchelmore, M., and White, P. (2000). Developmentof angle concepts by progressive abstraction andgeneralisation. Educational Studies in Mathematics ,41, 209-238.

Noss, R. (1988). Children’s learning of geometricalconcepts through Logo. Journal for Research inMathematics Education, 18(5), 343-362.

Nunes, T., Light, P., and Mason, J. (1993). Tools forthought: the measurement of lenght and area.Learning and Instruction, 3, 39-54.

Outhred, L., and Mitchelmore, M. (2000). Youngchildren’s intuitive understanding of rectangulararea measurement. Journal for Research inMathematics Education, 31(2), 144-167.

Papert, S. (1980). Mindstorms: children, computers andpowerful ideas. Brighton: Harvester Press.

Piaget, J., and Inhelder, B. (1963). The Child’sConception of Space . London: Routledge. andKegan Paul.

Piaget, J., Inhelder, B., and Szeminska, A. (1960). Thechild’s conception of geometr y. London: Routledgeand Kegan Paul.

Quinn, P. C., Siqueland, E. R., and Bomba, P. C. (1985).Delayed recognition memory for orientation byhuman infants. Journal of Experimental ChildPsychology, 40, 293-303.

Rock, I. (1970). Towards a cognitive theory ofperceptual constancy. In A. Gilgen (Ed.),Contemporary Sceintific psychology (pp. 342-378).New York: Academic Press.

Sarama, J., Clements, D.H., Swaminathan, S., McMillen,S., and Gomez, G. (2003) Development ofMathematics Concepts of Two-DimensionalSpace in Grid Environments: an exploratorystudy. Cognition and Instruction, 21, 285-324.


Seo, K-H., and Ginsburg, H. (2004) What isdevelopmentally appropriate in ear ly childhoodmathematics education? In D.H. Clements, J.Sarama, and A-M. Dibias (Eds.) Engaging youngchildren in mathematics: Standards for earlychildhood mathematics education. (pp.91-104).Mahwah, NJ: Erlbaum.

Slater, A. (1999). Visual organisation and perceptualconstancies in ear ly infancy. In V. Walsh and J.Kulikowski (Eds.), Perceptual Constancy: why thingslook as they do (pp. 6-30). Cambridge: C.U.P.

Slater, A., Field, T., and Hernadez-Reif, M. (2002). Thedevelopment of the senses. In Slater and M. Lewis(Eds.), Introduction to Infant Development (pp. 83-98). Oxford: O.U.P.

Slater, A., and Lewis, M. (2002). Introduction to infantdevelopment. Oxford: O.U.P.

Slater, A., Mattock, A., and Brown, E. (1990). Sizeconstancy at bir th: newborn infants’ responses toretinal and real size. Journal of Experimental ChildPsychology, 49, 314-322.

Slater, A., and Morison, V. (1985). Shape constancyand slant perception at birth. Perception, 14, 337-344.

Somerville, S., and Br yant, P. E. (1985). Youngchildren’s use of spatial coordinates. ChildDevelopment, 56, 604-613.

Sowder, J., Wearne, D., Mar tin, G., and Strutchens, M.(2004). What grade 8 students know aboutmathematics: changes over a decade. In P.Kloosterman and F. Lester (Eds.), The 1990through 2000 mathematics assessment of theNAEP: Results and interpretations (pp. 105-143).Reston, VA: National Councilof Teachers ofMathematics.

van den Heuvel-Panhuizen, M. and Buys, K. (2008)Young children learn measurement and Geometr y: alearning-teaching trajectory with intermediateattainment targets for the lower grades in Pr imaryschool. Rotterdam: Sense Publishers.

van den Heuvel, M., Veltman, A., Janssen, C., andHochstenbach, J. (2008). Geometry inKindergarten 1 and 2. In M. van den Heuvel andK. Buys (Eds.), Young children learn measurementand geometry: a learning-teaching trajectory withintermediate attainment targets for the lower gradesin Primary school. Rotterdam: Sense Publishers.

van Hiele, P. M. (1986). Structure and insight. Orlando,Fl: Academic Press.

Vergnaud, G. (1983). Multiplicative structures. In R. Lesh and M. Landau (Eds.), Acquisition of mathematics concepts and processes(pp. 128-175). London: Academic Press.

Vygotsky, L. S. (1978). Mind in Society. Cambridge,Mass: Harvard University Press.

Wertheimer, M. (1945). Productive Thinking. New York:Harper and Row.

Wilkening, F. (1979). Combining of stimulusdimensions in children’s and adult’s judgements of area: an information integration analysis.Developmental Psychology, 15, 25-53.

Wilkening, F., and Anderson, N. H. (1982).Comparison of two-rule assessmentmethodologies for studying cognitivedevelopment and knowledge structure.Psychological Bulletin, 92, 215-237.




ISBN 978-0-904956-72-6





6

Paper 6: Algebraic reasoningBy Anne Watson, University of Oxford

2 SUMMARY – PAPER 2: Understanding whole numbers2 Paper 6: Algebraic reasoning













Algebraic reasoning 8

References 37

About the authorAnne Watson is Professor of MathematicsEducation at the University of Oxford.


About this review


Headlines

• Algebra is the way we express generalisationsabout numbers, quantities, relations and functions.For this reason, good understanding of connectionsbetween numbers, quantities and relations isrelated to success in using algebra. In particular,students need to understand that addition andsubtraction are inverses, and so are multiplicationand division.

• To understand algebraic symbolisation, studentshave to (a) understand the underlying operationsand (b) become fluent with the notational rules.These two kinds of learning, the meaning and thesymbol, seem to be most successful when studentsknow what is being expressed and have time tobecome fluent at using the notation.

• Students have to learn to recognise the differentnature and roles of letters as: unknowns, variables,constants and parameters, and also the meanings of equality and equivalence. These meanings are not always distinct in algebra and do not relateunambiguously to arithmetical understandings,Mapping symbols to meanings is not learnt in one-off experiences.

• Students often get confused, misapply, ormisremember rules for transforming expressionsand solving equations. They often try to applyarithmetical meanings to algebraic expressionsinappropriately. This is associated with over-emphasis on notational manipulation, or on‘generalised arithmetic’, in which they may try to get concise answers.

Understanding symbolisation

The conventional symbol system is not merel y anexpression of generalised arithmetic; to understand it students have to understand the meanings ofarithmetical operations, rather than just be able tocarry them out. Students have to understand ‘inverse’and know that addition and subtraction are inverses,and that division is the inverse of multiplication.Algebraic representations of relations betweenquantities, such as difference and ratio, encapsulate this idea of inverse. Using familiarity with symbolicexpressions of these connections, rather than thinkingin terms of generalising four arithmetical operations,gives students tools with which to under standcommutativity and distributivity, methods of solvingequations, and manipulations such as simplifying andexpanding expressions.

The precise use of notation has to be lear nt as well,of course, and many aspects of algebraic notationare inherently confusing (e.g. 2r and r 2). Over-reliance on substitution as a method of doing thiscan lead students to get stuck with ar ithmeticalmeanings and rules, rather than being able torecognise algebraic structures. For example, studentswho have been taught to see expressions such as:

97 – 49 + 49

as structures based on relationships betweennumbers, avoiding calculation, identifying variation, andhaving a sense of limits of var iability, are able to reasonwith relationships more securely and at a younger agethan those who have focused only on calculation. Anexpression such as 3x + 4 is both the answer to aquestion, an object in itself, and also an algor ithm orprocess for calculating a par ticular value. This hasparallels in arithmetic: the answer to 3 ÷ 5 is 3/5.

Summary of paper 6: Algebraic reasoning


Time spent relating algebraic expressions toarithmetical structures, as opposed to calculations,can make a difference to students’ understanding.This is especially important when understanding thatapparently different expressions can be equivalent,and that the processes of manipulation (often themain focus of algebra lessons) are actuallytransformations between equivalent forms.

Meanings of letters and signsLarge studies of students’ interpretation and use ofletters have shown a well-defined set of possibleactions. Learners may, according to the task andcontext:• try to evaluate them using irrelevant information• ignore them• used as shorthand for objects, e.g. a = apple• treat them as objects• use a letter as a specific unknown• use a letter as a generalised number• use a letter as a variable.

Teachers have to understand that students may useany one of these approaches and students need tolearn when these are appropr iate or inappropriate.There are conventions and uses of letter sthroughout mathematics that have to be under stoodin context, and the statement ‘letters stand fornumbers’ is too simplistic and can lead to confusion.For example:• it is not always true that different letters have

different values • a letter can have different values in the same

problem if it stands for a variable • the same letter does not have to have the same

value in different problems.

A critical shift is from seeing a letter as representingan unknown, or ‘hidden’, number defined within anumber sentence such as:

3 + x = 8

to seeing it as a var iable, as in y = 3 + x, or 3 = y – x.Understanding x as some kind of gener alizednumber which can take a range of values is seen bysome researchers to provide a bridge from the ideaof unknown to that of var iables. The use of boxes to indicate unknown numbers in simple ‘missingnumber’ statements is sometimes helpful, but canalso lead to confusion when used f or variables, or for more than one hidden number in a statement.

Expressions linked by the ‘equals’ sign might be not just numerically equal, but also equivalent, yetstudents need to retain the ‘unknown’ concept whensetting up and solving equations which ha ve finitesolutions. For example, 10x – 5 = 5(2x – 1) is astatement about equivalence, and x is a variable, but10x – 5 = 2x + 1 defines a value of the variable forwhich this equality is tr ue. Thus x in the second casecan be seen as an unknown to be found, but in thefirst case is a var iable. Use of graphical software canshow the difference visually and powerfully becausethe first situation is represented by one line, and thesecond by two intersecting lines, i.e. one point.

Misuse of rulesStudents who rely only on remembered r ules oftenmisapply them, or misremember them, or do notthink about the meaning of the situations in whichthey might be successfully applied. Many studentswill use guess-and-check as a first resor t whensolving equations, par ticularly when numbers aresmall enough to reason about ‘hidden numbers’instead of ‘undoing’ within the algebraic structure.Although this is sometimes a successful str ategy,particularly when used in conjunction with gr aphs,or reasoning about spatial str uctures, or practicalsituations, over-reliance can obstruct thedevelopment of algebraic understanding and more universally applicable techniques.

Large-scale studies of U.K. school children showthat, despite being taught the BIDMAS r ule and itsequivalents, most do not know how to decide onthe order of operations represented in an algebraicexpression. Some researchers believe this to be dueto not fully understanding the under lying operations,others that it may be due to misinter pretation ofexpressions. There is evidence from Australia andthe United Kingdom that students who are taughtto use flow diagrams, and inverse flow diagrams, toconstruct and reorganise expressions are betterable to decide on the order implied b y expressionsinvolving combinations of operations. However, it isnot known whether students taught this way cansuccessfully apply their knowledge of order insituations in which flow diagrams are inappropriate,such as with polynomial equations, those involvingthe unknown on ‘both sides’, and those with morethan one var iable. To use algebra effectively,decisions about order have to be fluent and accurate.

4 Paper 6: Algebraic reasoning

Misapplying arithmeticalmeanings to algebraicexpressionsAnalysis of children’s algebra in clinical studies with12- to 13-year-olds found that the main problems inmoving from arithmetic to algebra arose because: • the focus of algebra is on relations rather than

calculations; the relation a + b = c represents threeunknown quantities in an additive relationship

• students have to understand inverses as well asoperations, so that a hidden value can be foundeven if the answer is not obvious from knowingnumber bonds or multiplication facts; 7 + b = 4 canbe solved using knowledge of addition, but c + 63= 197 is more easily solved if subtraction is used asthe inverse of addition

• some situations have to be expressed algebraicallyfirst in order to solve them. ‘My brother is twoyears older than me, my sister is five years youngerthan me; she is 12, how old will my brother be inthree years’ time?’ requires an analysis andrepresentation of the relationships before solution.‘Algebra’ in this situation means constructing amethod for keeping track of the unknown asvarious operations act upon it.

• letters and numbers are used together, so thatnumbers may have to be treated as symbols in astructure, and not evaluated. For example, thestructure 2(3+b) is different from the structure of 6+ 2b although they are equivalent in computationalterms. Learners have to understand that sometimesit is best to leave number as an element in analgebraic structure rather than ‘work it out’.

• the equals sign has an expanded meaning; inarithmetic it is often taken to mean ‘calculate’ but inalgebra it usually means ‘is equal to’ or ‘is equivalentto’. It takes many experiences to recognise that analgebraic equation or equivalence is a statementabout relations between quantities, or betweencombinations of operations on quantities. Studentstend to want ‘closure’ by compressing algebraicexpressions into one term instead of understandingwhat is being expressed.

Expressing generalisationsIn several studies it has been found that studentsunderstand how to use algebra if they have focused on generalizing with numerical and spatialrepresentations in which counting is not an option.Attempts to introduce symbols to very youngstudents as tools to be used when they ha ve a need

to express known general relationships, have beensuccessful both for aiding their under standing ofsymbol use, and understanding the underlyingquantitative relations being expressed. For example,some year 1 children first compare and discussquantities of liquid in different vessels, and soonbecome able to use letter s to stand for unknownamounts in relationships, such as a > b; d = e; and soon. In another example , older primary children couldgeneralise the well-known questions of how manypeople can sit round a line of tab les, given that therecan be two on each side of each tab le and one ateach of the extreme ends. The ways in whichstudents count differ, so the forms of the generalstatement also differ and can be compared, such as:‘multiply the number of tables by 4 and add 2 or ‘it is two times one more than the n umber of tables’.

The use of algebra to express known arithmeticalgeneralities is successful with students who ha vedeveloped advanced mental strategies for dealingwith additive, multiplicative and proportionaloperations (e.g. compensation as in 82 – 17 = 87 –17 – 5). When students are allowed to use theirown methods of calculation they often f ind algebraicstructures for themselves. For example, expressing13 x 7 as 10 x 7 + 3 x 7, or as 2 x 72 –7, areenactments of distr ibutivity and learners canrepresent these symbolically once they know thatletters can stand for numbers, though this is nottrivial and needs several experiences. Explaining ageneral result, or structure, in words is often ahelpful precursor to algebraic representation.

Fortunately, generalising from experience is a naturalhuman propensity, but the everyday inductivereasoning we do in other contexts is not al waysappropriate for mathematics. Deconstruction ofdiagrams and physical situations, and identification ofrelationships between variables, have been found tobe more successful methods of developing a formulathan pattern-generalisation from number sequencesalone. The use of verbal descriptions has been shownto enable students to br idge between observingrelations and writing them algebraically.

Further aspects of algebra arise in the companionsummaries, and also in the main body of Paper 6:Algebraic reasoning.


6 SUMMARY – PAPER 2: Understanding whole numbers6 Paper 6: Algebraic reasoning

Recommendations


The bases for using algebraic symbolisationsuccessfully are (a) under standing theunderlying operations and relations and(b) being able to use symbolism cor rectly.

Children interpret ‘letter stands fornumber’ in a var iety of ways, according tothe task. Mathematically, letters haveseveral meanings according to context:unknown, variable, parameter, constant.

Children interpret ‘=’ to mean ‘calculate’;but mathematically ‘=’ means either ‘equalto’ or ‘equivalent to’.

Students often forget, misremember,misinterpret situations and misapply rules.

Everyone uses ‘guess-&-check’ if answersare immediately obvious, once algebraicnotation is understood.

Even very young students can use letter sto represent unknowns and variables insituations where they have reasoned ageneral relationship by relating proper ties.Research on inductive generalisation frompattern sequences to develop algebrashows that moving from expressing simpleadditive patterns to relating proper ties hasto be explicitly supported.

Recommendations for teaching

Emphasis should be given to reading numerical and algebraicexpressions relationally, rather than computationally. Foralgebraic thinking, it is more important to understand howoperations combine and relate to each other than how theyare performed. Teachers should avoid emphasising symbolismwithout understanding the relations it represents.

Developers of the cur riculum, advisory schemes of work andteaching methods need to be aware of children’s possibleinterpretations of letters, and also that when correctly used,letters can have a range of meanings. Teachers should avoidusing materials that oversimplify this variety. Hands-on ICT canprovide powerful new ways to understand these differences inseveral representations.

Developers of the cur riculum, advisory schemes of work andteaching methods need to be aware of the difficulties aboutthe ‘=’ sign and use multiple contexts and explicit language .Hands-on ICT can provide powerful new ways to understandthese differences in several representations.

Developers of the cur riculum, advisory schemes of work andteaching methods need to take into account that algebr aicunderstanding takes time, multiple experiences, and clar ity ofpurpose. Teachers should emphasise situations in whichgeneralisations can be identified and described to providemeaningful contexts for the use of algebr aic expressions. Useof software which car ries out algebraic manipulations shouldbe explored.

Algebra is meaningful in situations for which specific arithmeticcannot be easily used, as an expression of relationships. Focusingon algebra as ‘generalised arithmetic’, e.g. with substitutionexercises, does not give students reasons for using it.

Algebraic expressions of relations should be a commonplacein mathematics lessons, par ticularly to express relations andequivalences. Students need to have multiple experiences ofalgebraic expressions of general relations based in proper ties,such as arithmetical rules, logical relations, and so on as well asthe well-known inductive reasoning from sequences.

Recommendations for research

• The main body of Paper 6: Algebraic reasoningincludes a number of areas for which furtherresearch would be valuable, including the following.

• How does explicit work on understanding relationsbetween quantities enable students to movesuccessfully from arithmetical to algebraic thinking?

• What kinds of explicit work on expressinggenerality enable students to use algebra?

• What are the longer-term comparative effects ofdifferent teaching approaches to early algebra onstudents’ later use of algebraic notation andthinking?

• How do learners’ synthesise their knowledge ofelementary algebra to understand polynomialfunctions, their factorisation and roots, simultaneousequations, inequalities and other algebraic objectsbeyond elementary expressions and equations?

• What useful kinds of algebraic expertise could bedeveloped through the use of computer algebrasystems in school?



In this review of how students learn algebra we tryto balance an approach which f ocuses on whatlearners can do and how their generalising and useof symbols develop (a ‘bottom up’ developmentalapproach), and a view which states what is requiredin order to do higher mathematics (a ‘top down’hierarchical approach). The ‘top down’ view oftenframes school algebra as a list of techniques whichneed to be fluent. This is manifested in researchwhich focuses on errors made by learners in thecurriculum and small-scale studies designed toameliorate these. This research tells us aboutdevelopment of understanding by identifying theobstacles which have to be overcome, and alsoreveals how learners think. It therefore makes senseto star t by outlining the different aspects of algebra.However, this is not suggesting that all mathematicsteaching and learning should be directed towardspreparation for higher mathematics.

By contrast a ‘bottom up’ view usually focuses onalgebraic thinking, taken to mean the expression and use of general statements about relationshipsbetween variables. Lins (1990) sought a def inition ofalgebraic thinking which encompassed the differentkinds of engagement with algebr a that run throughmathematics. He concluded that algebr aic thinkingwas an intentional shift from context (which couldbe ‘real’, or a par ticular mathematical case) tostructure. Thus ‘algebraic thinking arises when peopleare detecting and expressing str ucture, whether inthe context of problem solving concerning numbersor some modelled situation, whether in the contextof resolving a class of prob lems, or whether in thecontext of studying str ucture more generally’ (Lins,1990). Thus a complementary ‘bottom up’ viewincludes consideration of the development of

students’ natural ability to discern patterns andgeneralise them, and their growing competence inunderstanding and using symbols; however thiswould not take us very far in consider ing all theaspects of school algebra. The content of schoolalgebra as the development of algebraic reasoning isexpressed by Thomas and Tall (2001) as the shiftsbetween procedure, process/concept, generalisedarithmetic, expressions as evaluation processes,manipulation, towards axiomatic algebra. In thisperspective it helps to see manipulation as thegeneration and transformation of equivalentexpressions, and the identification of specific values for variables within them.

At school level, algebra can be descr ibed as:• manipulation and transformation of symbolic

statements• generalisations of laws about numbers and patterns• the study of structures and systems abstracted

from computations and relations• rules for transforming and solving equations • learning about variables, functions and expressing

change and relationships• modelling the mathematical structures of situations

within and outside mathematics.

Bell (1996) and Kaput (1998; 1999) emphasise theprocess of symbolisation, and the need to oper atewith symbolic statements and the use them withinand outside algebra, but algebra is much more thanthe acquisition of a sign system with which toexpress known concepts. Vergnaud (1998) identifiesnew concepts that students will meet in algebr a as:equations, formulae, functions, variables andparameters. What makes them new is that symbolsare higher order objects than numbers and become


Algebraic reasoning

mathematical objects in their own right; arithmetichas to work in algebraic systems, but symbol systemsare not merely expressions of general arithmetic.Furthermore, ‘the words and symbols we use tocommunicate do not refer directly to reality but torepresented entities: objects, properties, relationships,processes, actions, and constructs, about which thereis no automatic agreement’ (p.167).

In this paper I dr aw on the research evidence aboutthe first five of the aspects above. In the next paper Ishall tackle modelling and associated issues, and theirrelation to mathematical reasoning and applicationmore generally at school level.

It would be naïve to write about algebraic reasoningwithout repor ting the considerable difficulties thatstudents have with adopting the conventions ofalgebra, so the first part of this review addresses therelationship between arithmetic and algebra, and theobstacles that have to be overcome to understandthe meaning of letter s and expressions and to usethem. The second par t looks at difficulties which areevident in three approaches used to developalgebraic reasoning: expressing generalities; solvingequations; and working with functions. The third part summarises the findings and makesrecommendations for practice and research.

Part 1: arithmetic, algebra,letters, operations, expressions

Relationships between arithmeticand algebraIn the United States, there is a strong commitmentto arithmetic, par ticularly fluency with fractions, to beseen as an essential precur sor for algebra: ‘Proficiencywith whole numbers, fractions, and par ticular aspectsof geometry and measurement are the Cr iticalFoundation of Algebra. …The teaching of fr actionsmust be acknowledged as critically important andimproved before an increase in student achievementin Algebra can be expected.’ (NMAP, 2008). Whilenumber sense precedes formal algebra in age-relateddevelopmental terms, this one-way relationship is farfrom obvious in mathematical terms. In the UnitedKingdom where secondar y algebra is not taughtseparately from other mathematics, integrationacross mathematics makes a two-way relationshippossible, seeing arithmetic as par ticular instances ofalgebraic structures which have the added featurethat they can be calculated. For example, rather thanknowing the procedures of fractions so that they canbe generalised with letters and hence make algebraicfractions, it is possible for fraction calculations to beseen as enactments of relationships between rationalstructures, those generalised enactments beingexpressed as algorithms.

For this review we see number sense as precedingformal algebra in students’ learning, but to imaginethat algebraic understanding is merely ageneralisation of ar ithmetic, or grows directly from it,is a misleading over-simplification.

Kieran’s extensive work (e.g. 1981, 1989, 1992)involving clinical studies with ten 12- to 13-y ear-oldsleads her to identify f ive inherent difficulties in makinga direct shifts between arithmetic and algebra.

• The focus of algebra is on relations rather thancalculations; the relation a + b = c represents twounknown numbers in an additive relation, and while3 + 5 = 8 is such a relation it is more usually seenas a representation of 8, so that 3 + 5 can becalculated whereas a + b cannot.

• Students have to understand inverses as well asoperations, so that finding a hidden number can be done even if the answer is not obvious fromknowing number bonds or multiplication facts; 7 +b = 4 can be done using knowledge of addition,



but c + 63 = 197 is more easil y done ifsubtraction is used as the in verse of addition1.Some writers claim that under standing thisstructure is algebraic, while others say that doingarithmetic to find an unknown is ar ithmeticalreasoning, not algebraic reasoning.

• Some situations have to be expressed algebraicallyin order to solve them, rather than starting asolution straight away. ‘I am 14 and my brother is 4years older than me’ can be solved by addition, but‘My brother is two years older than me, my sister isfive years younger than me; she is 12, how old willmy brother be in three years’ time?’ requires ananalysis and representation of the relationshipsbefore solution. This could be with letters, so thatthe answer is obtained by finding k where k – 5 =12 and substituting this value into (k + 2) + 3.Alternatively it could be done by mapping systemsof points onto a numberline, or using othersymbols for the unknowns. ‘Algebra’ in this situationmeans constructing a method for keeping track ofthe unknown as various operations act upon it.

• Letters and numbers are used together, so thatnumbers may have to be treated as symbols in a structure, and not evaluated. For example, thestructure 2(a + b) is different from the str uctureof 2a + 2b although they are equivalent incomputational terms.

• The equals sign has an expanded meaning; inarithmetic it often means ‘calculate’ but in algebrait more often means ‘is equal to’ or even ‘isequivalent to’.

If algebra is seen solely as generalised arithmetic (we take this to mean the expression of gener alarithmetical rules using letters), many problems arisefor learning and teaching. Some writers describethese difficulties as manifestations of a ‘cognitive gap’between arithmetic and algebra (Filloy and Rojano,1989; Herscovics and Linchevski, 1994). For example,Filloy and Rojano saw students dealing ar ithmeticallywith equations of the form ax + b = c , where a, band c are numbers, using inverse operations on thenumbers to complete the ar ithmetical statement.They saw this as ‘arithmetical’ because it dependedonly on using operations to find a ‘hidden’ number.The same students acted algebr aically with equationssuch as ax + b = cx + d , treating each side as anexpression of relationships and using directoperations not to ‘undo’ but to maintain the equationby manipulating the expressions and equality. If such a

gap exists, we need to know if it is developmental orepistemological, i.e. do we have to wait till lear nersare ready, or could teaching make a difference? Abottom-up view would be that algebraic thinking isoften counter-intuitive, requires good under standingof the symbol system, and abstract meanings whichdo not arise through normal engagement withphenomena. Nevertheless the shifts required tounderstand it are shifts the mind is ab le to makegiven sufficient experiences with new kinds of objectand their representations. A top-down view would bethat students’ prior knowledge, conceptualisations andtendencies create errors in algebra. Carraher andcolleagues (Carraher, Brizuela & Earnest, 2001;Carraher, Schliemann & Br izuela, 2001) show that theprocesses involved in shifting from an ar ithmeticalview to an algebraic view, that is from quantifyingexpressions to expressing relations between variables,are repeated for new mathematical structures athigher levels of mathematics, and hence arecharacteristics of what it means to lear n mathematicsat every level rather than developmental stages oflearners. This same point is made again and again b ymathematics educators and philosophers who pointout that such shifts are fundamental in mathematics,and that reification of new ideas, so that they can betreated as the elements for new levels of thought, is how mathematics develops both historically andcognitively. There is considerable agreement thatthese shifts require the action of teacher s andteaching, since they all involve new ways of thinkingthat are unlikely to arise naturally in situations (Filloyand Sutherland, 1996).

Some of the differences reported in research rest onwhat is, and what is not, described as algebraic. Forexample, the equivalence class of fr actions thatrepresent the rational number 3/5 is all fractions ofthe form 3k/5k (k Є N)2. It is a cur riculum decision,rather than a mathematical one, whether equivalentfractions are called ‘arithmetic’ or ‘algebra’ butwhatever is decided, learners have to shift from seeing3/5 as ‘three cakes shared between five people’ to aquantitative label for a general class of objectsstructured in a particular quantitative relationship. Thisis an example of the kind of shift lear ners have tomake from calculating number expressions to seeingsuch expressions as meaningful str uctures.

Attempts to introduce symbols to very youngstudents as tools to be used when they ha ve a needto express general relationships, can be successfulboth for them understanding symbol use, andunderstanding the under lying quantitative relations


being expressed (Dougher ty, 1996; 2001). InDougherty’s work, students star ting schoolmathematics first compare and discuss quantities ofliquid in different vessels, and soon become able touse letters to stand for unknown amounts. Arcavi(1994) found that, with a range of students frommiddle school upwards over several years, symbolscould be used as tools ear ly on to expressrelationships in a situation. The example he uses is thewell-known one of expressing how many people cansit round a line of tab les, given that there can be twoon each side and one at each of the extreme ends.The ways in which students count diff er, so the formsof the general statement also differ, such as: ‘multiplythe number of tables by 4 and add two’ or ‘it is twotimes one more than the number of tables’. In Brownand Coles’ work (e.g. 1999, 2001), several years ofanalysis of Coles’ whole-class teaching showed thatgeneralising by expressing structures was a powerfulbasis for students to need symbolic notation, whichthey could then use with meaning. For example, toexpress a number such that ‘twice the number plusthree’ is ‘three less’ than ‘add three and double thenumber’ a student who has been in a class of 12-year-olds where expression of gener al relationships isa normal and frequent activity introduced N forhimself without prompting when it is appropr iate.

When students are allowed to use their ownmethods of calculation they often f ind algebraicstructures for themselves. For example, expressing13 x 7 as 10 x 7 + 3 x 7, or as 2 x 7 2 - 7, areenactments of distr ibutivity (and, implicitly,commutativity and associativity) and can berepresented symbolically, though this shift is nottrivial (Anghileri, Beishuizen and van Putten,2002;Lampert, 1986). On the other hand, allowingstudents to develop a mindset in which any methodthat gives a r ight answer is as good as any other canlock learners into additive procedures wheremultiplicative ones would be more generalisable,multiplicative methods where exponential methodswould be more powerful, and so on. But somenumber-specific arithmetical methods do exemplifyalgebraic structures, such as the transformation of 13 x 7 descr ibed above. This can be seen either as‘deriving new number facts from known numberfacts’ or as an instance of algebr aic reasoning.

The importance of a link between the kinds oftransformations necessary for mental ar ithmetic andalgebraic thinking is demonstrated in a three-yearlongitudinal teaching and testing progr amme of 116students aged 12 to 14 (Br itt and Irwin, 2007).

Students who had developed advanced mentalstrategies, (e.g. compensation as in 82 – 17 = 87 –17 – 5) for dealing with additive, multiplicative andproportional operations, could use letter s inconventional algebra once they knew that they‘stood for’ numbers. Those who did best at algebrawere those in schools where teacher s had focusedon generalizing with numerical and spatialrepresentations in situations where counting was not a sensible option.

There are differences in the meaning of notation asone shifts between arithmetic and algebra. Wong(1997) tested and inter viewed four classes ofsecondary students to see whether they coulddistinguish between similar notations used forarithmetic and algebra. For example, in arithmetic theexpression 3(4 + 5) is both a str ucture of operationsand an invitation to calculate, but in algebra a(b + c)is only a structure of operations. Thus students getconfused when given mixtures such as 3(b + 5)because they can assume this is an in vitation tocalculate. This tendency to confuse what is possib lewith numbers and letters is subtle and depends onthe expression. For example, Wong found thatexpressions such as (2am)n are harder to simplify andsubstitute than (hk)n, possibly because the secondexpression seems very clearly in the realm of algebr aand rules about letters. Where Booth and Kieranclaim that it is not the symbolic conventions alonethat create difficulties but more often a lack ofunderstanding of the underlying operations, Wong’swork helpfully foregrounds some of the inevitableconfusions possible in symbolic conventions. Thestudent has to under stand when to calculate , whento leave an expression as a statement aboutoperations, what par ticular kind of number(unknown, general or variable) is being denoted, andwhat the structure looks like with numbers andletters in combination. As an example of the lastdifficulty, 2X is found to be harder to deal with than x2

although they are visually similar in form.

The question for this review is therefore not whetherlearners can make such shifts, or when they makethem, but what are the shifts they have to make, andin what circumstances do they make them.

Summary• Algebra is not just generalised arithmetic; there are

significant differences between arithmetical andalgebraic approaches.



• The shifts from arithmetic to algebra are the kinds of shifts of perception made throughoutmathematics, e.g. from quantifying to relationshipsbetween quantities; from operations to structuresof operations.

• Mental strategies can provide a basis forunderstanding algebraic structures.

• Students will accept letters and symbols standingfor numbers when they have quantitativerelationships to express; they seem to be able touse letters to stand for ‘hidden’ numbers and alsofor ‘any’ number.

• Students are confused by expressions that combinenumbers and letters, and by expressions in whichtheir previous experience of combinations arereversed. They have to learn to ‘read’ expressionsstructurally even when numbers are involved.

Meaning of letters

Students’ understanding of the meaning of letter s in algebra, and how they use letters to expressmathematical relationships, are at the root ofalgebraic development. Kuchemann (1981) identifiedseveral different ways adolescent students usedletters in the Chelsea diagnostic test instr ument(Hart, 1981). His research is based on test paper s of2900 students between 12 and 16 (see Appendix 1).

Letters were:• evaluated in some way, e.g. a = 1• ignored, e.g. 3a taken to be 3• used as shorthand for objects, e.g. a = apple• treated as objects• used as a specific unknown• used as a generalised number• used as a variable.

Within his categorisation there were correct andincorrect uses, such as students who ascr ibed a valueto a letter based on idiosyncr atic decisions or pastexperience, e.g. x = 4 because it was 4 in theprevious question. These interpretations appear tobe task dependent, so learners had developed asense of what sor ts of question were treated inwhat kinds of ways, i.e. generalising (sometimesidiosyncratically) about question-types throughfamiliarity and prior experience.

Booth (1984) inter viewed 50 students aged 13 to15 years, following up with 17 fur ther case studystudents. She took a subset of Kuchemann’ smeanings, ‘letters stand for numbers’, and fur therunpicked it to reveal problems based on students’test answers and follow-up interviews. She identifiedthe following issues which, for us, identify moreabout what students have to learn.

• It is not always true that different letters havedifferent values; for example one solution to 3x +5y = 8 is that x = y = 1.

• A letter can have different values in the sameproblem, but not at the same time, if it stands for avariable (such as an equation having multiple roots,or questions such as ‘find the value of y = x2 + x +2 when x = 1, 2, 3…’)

• The same letter does not have to have the samevalue in different problems.

• Values are not related to the alphabet (a = 1, b = 2…; or y > p because of relative alphabetic position).

• Letters do not stand for objects (a for apples)except where the objects are units (such as m formetres).

• Letters do not have to be presented in alphabeticalorder in algebraic expressions, although there aretimes when this is useful3.

• Different symbolic rules apply in algebra andarithmetic, e.g.: ‘2 lots of x’ is written ‘2x’ but twolots of 7 are not written ‘27’.

As well as in Booth’s study, paper and pencil teststhat were administered to 2000 students in aged 11to 15 in 24 Australian secondary schools in 1992demonstrated all the above confusions (MacGregorand Stacey, 1997).

These problems are not resolved easily, because lettersare used in mathematics in var ying ways. There is nosingle correct way to use them. They are used as labels for objects that have no numerical value, suchas vertices of shapes, or for objects that do havenumerical value but are treated as general, such aslengths of sides of shapes. They denote fixed constantssuch as g, e or π, also non-numerical constants such as i,and also they represent unknowns which have to befound, and variables. Distinguishing between thesemeanings is usually not taught explicitly, and this lack


of instruction might cause students some diff iculty. Onthe other hand it is very hard to explain how to knowthe difference between a parameter, a constant and avariable (e.g. when asked to ‘vary the constant’ toexplore a structure), and successful students may learn this only when it is necessar y to make suchdistinctions in par ticular usage. It is particularly hard toexplain that the O and E in O + E = O (to indicateodd and even numbers) are not algebraic, even thoughthey do refer to numbers. Interpretation is thereforerelated to whether students under stand the algebraiccontext, expression, equation, equivalence, function orother relation. It is not surprising that Furinghetti andPaola (1994) found that only 20 out of 199 studentsaged 12 to 17 could explain the diff erence betweenparameters and variables and unknowns (see alsoBloedy-Vinner, 1994). Bills’ (2007) longitudinal study ofalgebra learning in upper secondar y students noticedthat the letters x and y have a special status, so thatthese letters trigger certain kinds of behaviour (e.g.these are the var iables; or (x.y) denotes the generalpoint). Although any letter can stand for any kind ofnumber, in practice there are conventions, such as xbeing an unknown; x, y, z being variables; a, b, c beingparameters/coefficient or generalised lengths, and so on.

A critical shift is from seeing a letter as representingan unknown, or ‘hidden’, number defined within anumber sentence such as:

3 + x = 8

to seeing it as a var iable, as in y = 3 + x, or 3 = y – x .While there is research to show how quasi-variablessuch as boxes can help students understand the useof letters in relational statements (see Car penter andLevi, 2000) the shift from unknown to variable whensimilar letters are used to have different functions isnot well-researched. Understanding x as some kind ofgeneralised number which can take a range of valuesis seen by some researchers to provide a bridge fromthe idea of unknown to that of var iables (Bednarz,Kieran and Lee, 1996).

The algebra of unknowns is about using solutionmethods to find mystery numbers; the algebra ofvariables is about expressing and tr ansforming relationsbetween numbers. These different lines of thoughtdevelop throughout school algebra. The ‘variable’ viewdepends on the idea that the expressions linked by the‘equals’ sign might be not just n umerically equal, butalso equivalent, yet students need to retain the‘unknown’ concept when setting up and solvingequations which have finite solutions. For example, 10x

– 5 = 5(2x – 1) is a statement about equivalence , andx is a variable, but 10x – 5 = 2x + 1 defines a value ofthe variable for which this equality is true. Thus x in thesecond case can be seen as an unknown to be found.

It is possible to address some of the prob lems by givingparticular tasks which force students to sor t out thedifference between parameters and variables (Drijvers,2001). A parameter is a value that def ines the structureof a relation. For example, in y = mx + c the variablesare x and y, while m and c define the relationship andhave to be fixed before we can consider thecovariation of x and y. In the United Kingdom this isdealt with implicitly, and finding the gradient andintercept in the case just descr ibed is seen as a specialkind of task. At A-level, however, students have to findcoefficients for partial fractions, or the coefficients ofpolynomials which have given roots, and after manyyears of ‘finding x’ they can find it hard to use par ticularvalues for x to identify parameters instead. By that timeonly those who have chosen to do mathematics needto deal with it, and those who ear lier could only findthe m and c in y=mx + c by using formulae withoutcomprehension may have given up maths. Fortunately,the dynamic possibilities of ICT offer tools to fullyexplore the variability of x and y within the constantbehaviour of m and c and it is possible that moreextensive use of ICT and modelling approaches mightdevelop the notion of var iable further.

Summary• Letters standing for numbers can have many

meanings.

• The ways in which operations and relationships arewritten in arithmetic and algebra differ.

• Learners tend to fall into well-known habits andassumptions about the use of letters.

• A particular difficulty is the difference betweenunknowns, variables, parameters and constants,unless these have meaning.

• Difficulties in algebra are not merely about usingletters, but about understanding the underlyingoperations and structures.

• Students need to learn that there are different usesfor different letters in mathematical conventions; forexample, a, b and c are often used as parameters,or generalised lengths in geometry, and x, y and zare often used as variables.



Recognising operations

In several intervention studies and textbooksstudents are expected to use algebr aic methods forproblems for which an answer is required, and forwhich ad hoc methods work perfectly well. Thisarises when solving equations with one unkno wn onone side where the answer is a positive integer (suchas 3x + 2 = 14); in word problems which can beenacted or represented diagrammatically (such as ‘Ihave 15 fence posts and 42 metres of wire; how farapart must the fence posts be to use all the wireand all the posts to make a straight fence?’); and inthese and other situations in which tr ial-and-adjustment work easily. Students’ choice to use non-algebraic methods in these contexts cannot be taken as evidence of problems with algebra.

In a teaching exper iment with 135 students age 12 to 13, Bednarz and Janvier (1996) found that amathematical analysis of the operations required forsolution accurately predicted what students would finddifficult, and they concluded that prob lems where onecould start from what is known and work towardswhat is not known, as one does in arithmeticalcalculations, were significantly easier than problems inwhich there was no obvious bridge between knownsand the unknown, and the relationship had to beworked out and expressed before any calculationscould be made. Many students tried to workarithmetically with these latter kinds of problem,starting with a fictional number and working forwards,generating a structure by trial and error rather thanidentifying what would be appropriate. This study isone of many which indicate that under standing themeaning of arithmetical operations, rather than merelybeing able to carry them out, is an essential precursornot only to deciding what operation is the r ight one todo, but also to expressing and under standing structuresof relations among operations (e.g. Booth, 1984). Theimpact of weak arithmetical understanding is alsoobserved at a higher level, when students can confusethe kinds of propor tionality expressed in y = k/x and y= kx, thinking the former must be linear because itinvolves a ratio (Baker, Hemenway and Trigueros,2001).The ratio of k to x in the first case is specific for eachvalue of x, but the ratio of y to x in the second case isinvariant and this indicates a propor tional relationship.

Booth (1984) selected 50 students from f our schoolsto identify their most common er rors and tointerview those who made cer tain kinds of error. Thisled her to identify more closel y how their weaknesswith arithmetic limited their progress with algebr a. The

methods they used to solve word problems werebound by context, and depended on counting, adding,and reasoning with whole and half n umbers. Theywere unable to express how to solve problems interms of arithmetical operations, so that algebraicexpressions of such operations were of little use ,being unrelated to their own methods. Similarly, theirmethods of recording were not conducive toalgebraic expression, because the roles of differentnumbers and signs were not clear in the layout. Forexample, if students calculate as they go along, ratherthan maintaining the ar ithmetical structure of aquestion, much information is lost. For example, 42 –22

becomes 16 – 4 and the ‘difference between twosquares’ is lost; similarly, turning rational or irrationalnumbers into decimal fractions can lose both accuracyand structure.

In Booth’s work it was not the use of letter s that is difficult, but the under lying arithmeticalunderstanding. This again suppor ts the view that it is not until ad hoc, number fact and guess-and-testmethods fail that students are lik ely to see a needfor algebraic methods, and in a cur riculum based onexpressions and equations this is lik ely occur whensolving equations with non-integer answers, where afull understanding of division expressed as fr actionswould be needed, and when working with theunknown on both sides of an equation. Alternatively,if students are tr ying to express general relationships,use of letters is essential once they realise thatparticular examples, while illustrating relationships, donot fully represent them. Nevertheless students’invented methods give insight into what they mightknow already that is formalisable, as in the 13 x 7example given above.

Others have also obser ved the persistence ofarithmetic (Kieran, 1992; Vergnaud, 1998). Vergnaudcompares two student protocols in solving adistance/time problem and comments that theadditive approach chosen by one is not conceptuall ysimilar to the multiplicative chosen by the other, eventhough the answers are the same , and that this linearapproach is more natural for students than themultiplicative. Kieran (1983) conducted clinicalinterviews with six 13-year-old students to find outwhy they had difficulty with equations. The studentstended to see tasks as about ‘getting answers’ andcould not accept an expression as meaningful initself. This was also obser ved by Collis (1971) andmore recently by Ryan and Williams in their largescale study of students’ mathematical understanding,drawing on a sample of about 15 000 U .K. students


(2007). Stacey and Macgregor (2000, p. 159) talk ofthe ‘compulsion to calculate’ and comment that atevery stage students’ thinking in algebraic problemswas dominated by arithmetical methods, whichdeflected them from using algebra. Fur thermore,Bednarz and Janvier (1996) showed that even thosewho identified structure during interviews were likelyto revert to arithmetical methods minutes later. Itseemed as if testing par ticular numbers was anapproach that not only overwhelmed any attemptsto be more analytical, but also preventeddevelopment of a str uctural method.

This suggests that too much focus on substitution inearly algebra, rather than developing understandingof how structure is expressed, might allow a‘calculation’ approach to per sist when working withalgebraic expressions. If calculation does per sist, thenit is only where calculation breaks down thatalgebraic understanding becomes crucial, or, as inBednarz and Janvier (1996), where word problemsdo not yield to str aightforward application ofoperations. For a long time in Soviet education wordproblems formed the core of algebr a instruction.Davydov (1990) was concerned that arithmetic doesnot necessarily lead to awareness of generality,because the approach degenerates into ‘letterarithmetic’ rather than the expression of gener ality.He developed the approach used by Dougher ty(2001) in which young students have to expressrelationships before using algebra to generalizearithmetic. For example, students in the f irst year of school compare quantities of liquid (‘do y ou havemore milk than me?’) and express the relationship as, say, G < R. They understand that adding the sameamount to each does not make them equal, but thatthey have to add some to G to mak e them equal.They do not use numbers until relationshipsbetween quantities are established.

Substituting values can, however, help students tounderstand and verify relationships: it matters if thisis for an unknown: 5 = 2x – 7 where only one valuewill do; or for an equation where var iables will berelated: y = 2x – 7; or to demonstrate equivalence:e.g. does 5(x + y) – 3 = 5x + 5y – 3 or 5x + y – 3?But using substitution to under stand whatexpressions mean is not helpful. Fur thermore thechoice of values offered in many textbooks canexacerbate misunderstandings about the valuesletters can have. They can reinforce the view that aletter can only take one value in one situation, andthat different letters have to have different values,and even that a = 1, b = 2 etc.

Summary• Learners use number facts and guess-and-check

rather than algebraic methods if possible.

• Doing calculations, such as in substitution andguess-and-check methods, distracts from thedevelopment of algebraic understanding.

• Substitution can be useful in exploring equivalenceof expressions.

• Word problems do not, on their own, scaffold ashift to algebraic reasoning.

• Learners have to understand operations and theirinverses.

• Methods of recording arithmetic can scaffold a shiftto understanding operations.

What shifts have to be madebetween arithmetic and algebra?

Changing focus slightly, we now turn to what thelearner has to see differently in order to overcomethe inherent problems discussed above. A key shiftwhich has to be made is from f ocusing on answersobtained in any possible way, to focusing onstructure. Kieran (1989, 1992), reflecting on her long-term work with middle school students, classifies‘structure’ in algebra as (1) surface str ucture ofexpression: arrangement of symbols and signs; (2)systemic: operations within an expression and theiractions, order, use of brackets etc.; (3) structure of anequation: equality of expressions and equivalence .

Boero (2001) identifies transformation andanticipation as key processes in algebraic problemsolving, drawing on long-term research in authenticclassrooms, reconstructing learners’ meanings fromwhat they do and say. He obser ved two kinds oftransformation, firstly the contextual ar ithmetical,physical and geometric transformations students doto make the problem meaningful within their currentknowledge (see also Filloy, Rojano and Robio,2001);secondly, the new kinds of tr ansformation madeavailable by the use of algebr a. If students’anticipation is locked into arithmetical activity: findinganswers, calculating, proceeding step-by-step fromknown to unknown (see also Dettor i, Garutti, andLemut, 2001), and if their main exper ience of algebrais to simplify expressions, then the shift to using thenew kinds of transformation afforded by algebra is



hindered. Thus typical secondar y school algebraicbehaviour includes reaching for a formula andsubstituting numbers into it (Arzarello, Bazzini and Chiappini,1994), as is often demonstrated instudents’ meaningless approaches to f inding areasand perimeters (Dickson, 1989 a). Typically studentswill multiply every available edge length to get area, and add everything to get perimeter. Theseapproaches might also be manifestations of learners’difficulties in understanding area (see Paper 5,Understanding space and its representation inmathematics) which cause them to rely on methodsrather than meaning.

The above evidence confirms that the relationshipbetween arithmetic and algebra is not a directconceptual hierarchy or necessarily helpful. Claims that arithmetical understanding has to precede theteaching of algebra only make sense if the focus is on the meaning of oper ations and on ar ithmeticalstructures, such as inverses and fractional equivalence,rather than in correct calculation. A focus on answersand ad hoc methods can be a distr action unless theunderlying structures of the ad hoc methods aregeneralisable and expressed structurally. Booth (1984)found that inappropriate methods were sometimestransferred from arithmetic; students often did notunderstanding the purpose of conventions andnotations, for example not seeing a need f or bracketswhen there are multiple operations. The possibilities ofnew forms of expression and tr ansformation have tobe appreciated, and the visual format of algebraicsymbolism is not always obviously connected to itsmeanings (Wertheimer, 1960; Kirschner, 1989). Forexample, the meaning of index notation has to belearnt, and while y3 can be related to its meaning insome way, y1/2 is rather harder to interpret withoutunderstanding abstract structure.

In the U.K. context of an integrated curriculum, a non-linear view of the shift between arithmetic and algebracan be considered. Many researchers have shown that middle-school students can develop algebraicreasoning through a focus on relationships, rather than calculations4. Coles, Dougherty and Arcavi havealready been mentioned in this respect, and Blantonand Kaput (2005) showed in an inter vention-and-observation study of cohort of 20 pr imary teachers,in particular one self-defined as ‘not a maths per son’ inher second year of teaching, could integrate algebraicreasoning into their teaching successfully, particularlyusing ICT as a medium for providing bridges betweennumbers and structures. Fujii and Stephens (2001,2008) examined the role of quasi-var iables (signs

indicating missing values in number sentences) as aprecursor to understanding generalization. Brown andColes (1999, 2001) develop a classroom environmentin a U.K. secondary school in which relationships aredeveloped which need to be expressed str ucturally,and algebraic reasoning becomes a tool to make newquestions and transformations possible. These studiesspan ages 6 to lower secondary and provide school-based evidence that the development of algebraicreasoning can happen in deliber ately-designededucational contexts. In all these contexts, calculationis deliberately avoided by focusing on, quantifiable butnot quantified, relationships, and using Kieran’s firstlevel of structure, surface structure, to expressphenomena at her third level, equality of expressions.A study with 105 11- and 12-y ear-olds suggests thatexplaining verbally what to do in gener al terms is aprecursor to understanding algebraic structure(Kieran’s third level) (Reggiani 1994). In this section Ihave shown that it is possible for students to makethe necessary shifts given certain circumstances, andcan identify necessary experiences which can suppor tthe move.

Summary of what has to be learnt to shift from arithmetic and algebra• Students need to focus on relations and

expressions, not calculations.

• Students need to understand the meaning ofoperations and inverses.

• Students need to represent general relations whichare manifested in situations

• In algebra letters and numbers are used together;algebra is not just letters.

• The equals sign means ‘has same value as’ and ‘isequivalent to’ – not ‘calculate’.

• Arithmetic can be seen as instances of generalrelationships between quantities.

• Division is a tool for constructing a rationalexpression.

• The value of a number is less important than itsrelation to other numbers in an expression.

• Guessing and checking, or using known numberfacts, has to be put aside for more generalmethods.


• A letter does not always stand for a particularunknown.

Without explicit attention to these issues, learnerswill use their natural and quasi-intuitive reasoning to:• try to match their use of letters to the way they

use numbers• try to calculate expressions• try to use ‘=’ to mean ‘calculate’• focus on value rather than relationship• try to give letters values, often based on

alphabetical assumptions.

Understanding expressionsAn expression such as 3x + 4 is both the answer toa question, an object in itself, and also an algor ithm or process for calculating a par ticular number. This isnot a new way of thinking in mathematics that onl yappears with algebra: it is also tr ue that the answer to3 ÷ 5 is 3/5, something that students are expected tounderstand when they learn about intensivequantities and fractions. Awareness of this kind ofdual meaning has been called proceptual thinking(Gray and Tall, 1994), combining the process with itsoutcome in the same way as a multiple is a numberin itself and also the outcome of m ultiplication. Thenotions of ‘procept’ and ‘proceptual understanding’signify that there is a need f or flexibility in how weact towards mathematical expressions.

Operational understandingMany young students understand, at least undersome circumstances, the inverse relation betweenaddition and subtraction but it takes studentslonger to understand the inverse relation between multiplication and division5. This may be par ticularly difficult when the division is notsymbolized by the division sign ÷ b ut by means ofa fraction, as in 1/3. Understanding division when itis symbolically indicated as a fr action would requirestudents to realise that a symbol such as 1/3represents not only a quantity (e .g. the amount ofpizza someone ate when the pizza was cut intothree parts) but also as an operation. Kerslake(1986) has shown that older primary and youngersecondary students in the United Kingdom rarelyunderstand fractions as indicating a division. Afurther difficulty is that multiplication, seen asrepeated addition, does not provide a ready imageon which to build an understanding of the inverseoperation. An array can be split up vertically or

horizontally; a line of repeated quantities can only be split up into commensurate lengths. Thelanguage of division in schools is usually ‘sharing’ or ‘shared by’ rather than divide, thus triggering anassignment metaphor. This is a long way from thenotion of number required in order to, forexample, find y when 6y = 7. There is evidencethat students understand some proper ties ofoperations better in some contexts than in other s(e.g. Nunes and Br yant, 1995).

As well as knowing about operations and theirinverses, students need to know that only additionand multiplication are commutative in arithmetic, sothat with subtraction and division it matter s whichway round the numbers go. Also in subtraction andmultiplication it makes a difference if an unknownnumber or variable is not the number being actedon in the operation. For example, if 7 – p = 4, thento find p the appropriate inverse operation is 7 – 4.In other words ‘subtract from n’ is self-inverse. Asimilar issue arises with ‘divide into n’.

We are unconvinced by the U.S. NationalMathematics Advisory Panel’s suggestion that fractionsmust be understood before algebra is taught (NMAP,2008). Their argument is based on a ‘top-down’curriculum view and not on research about ho w suchideas are learnt. The problems just described arealgebraic, yet contribute to a full under standing offractions as rational structures. There is a strongargument for seeing the mathematical str ucture offractions as the unifying concept which draws togetherparts, wholes, divisions, ratio, scalings and multiplicativerelationships, but it may only be in such situations assolving equations, algebraic fractions, and so on thatstudents need to extend their view of division andfractions, and see these as related.

To understand algebraic notation requires anunderstanding that terms made up of additive,multiplicative and exponential operations, e.g.(4a3b – 8a), are var iables rather than instructionto calculate, and have a structure and equivalentforms. It has been suggested that spending timerelating algebraic terms to arithmetical structurescan provide a br idge between arithmetic andalgebra (Banerjee and Subramaniam, 2004). Moreresearch is needed, but working this way round,rather than introducing terms by reverting tosubstitution and calculation, seems to havepotential.



Summary• Learners tend to persist in additive methods rather

than using multiplicative and exponential whereappropriate.

• It is hard for students to learn the nature ofmultiplication and division – both as inverse ofmultiplication and as the structure of fractions andrational numbers.

• Students have to learn that subtraction and divisionare non-commutative, and that their inverses arenot necessarily addition and multiplication.

• Students have to learn that algebraic terms canhave equivalent forms, and are not instructions tocalculate. Matching terms to structures, rather thanusing them to practice substitution, might be useful.

Relational reasoning

Students may make shifts between arithmetic andalgebra, and between operations and relations,naturally with enough exper ience, but researchsuggests that teaching can make a difference to thetiming and robustness of the shift. Carpenter andLevi (2000) have worked substantially over decadesto develop an approach to ear ly algebra based onunderstanding equality, making generalisationsexplicit, representing generalisations in various waysincluding symbolically, and talking about justif icationand proof to validate gener alities. Following thiswork, Stephens and other s have demonstrated thatstudents can be taught to see expressions such as:

97 – 49 + 49

as structures, in Kieran’s second sense ofrelationships among operations (see also the paper on natural numbers). In international studies,students in upper pr imary in Japan generally tackledthese relationally, that is they did not calculate all theoperations but instead combined operations andinverses, at a younger age than Australian studentsmade this shift. Chinese students generally appearedto be able to choose between rapid computationand relational thinking as appropr iate, while 14-year-old English students var ied between teachers in theirtreatment of these tasks (Fujii and Stephens, 2001,2008; Jacobs, Franke, Carpenter, Levi and Battey,2007). This ‘seeing’ relationally seems to depend onthe ability to discern details (Piaget, 1969 p. xxv) and application of an intelligent sense of str ucture

(Wertheimer, 1960) and also to know when andhow to handle specifics and when to stay withstructure. The power of such approaches isillustrated in the well-known story of the youngGauss’ seeing a str uctural way to sum an ar ithmeticprogression. In Fujii and Stephens’ work, seeingpatterns based on relationships between numbers,avoiding calculation, identifying variation, having asense of limits of var iability, were all found to bepredictors of an ability to reason with relationshipsrather than numbers.

These are fundamental algebraic shifts. Seeing algebraas ‘generalised arithmetic’ is not achieved by inductivereasoning from special cases, but by developing astructural perspective on number sentences.

Summary• Learners naturally generalise, they look for patterns

and habits, and familiar objects.

• Inductive reasoning from several cases is a naturalway to generalise, but it is often more important tolook at expressions as a whole.

• Learners can shift from ‘seeing’ number expressionsas instructions to calculate to seeing them asrelationships.

• This shift can be scaffolded by teaching whichencourages students not to calculate but to identifyand use relations between numbers.

• Learners who are fluent in both ways of seeingexpressions, as structures or as instructions tocalculate, can choose which to use.

Combining operations

Problems arise when an expression contains morethan one operation, as can be seen in our paper onfunctional relations where young children cannotunderstand the notion of relations betweenrelations, such as differences of differences. Inarithmetical and algebraic expressions, some relationsbetween relations appear as combinations ofoperations, and learners have to decide what has to be ‘done’ first and how this is indicated in thenotation. Carpenter and Levi (2000), Fujii andStephens (2001, 2008), Jacobs et al. (2007), drawattention to this in their work on how students readnumber sentences. Linchevski and Her scovics (1996)


studied how 12- and13-year-olds decided on theorder of operations. They found that students tendedto overgeneralise the order, usually giving additionpriority over subtraction; or using operations in leftto right order ; they can show lack of awareness ofpossible internal cancellations; they can see br acketsas merely another way to write expressions ratherthan an instruction to act first, for example: 926 –167 – 167 and 926 – (167 + 167) yielded diff erentanswers (Nickson, 2000 p. 120); they also did notunderstand that signs were somehow attached tothe following number.

Apart from flow diagrams, a common way to teachabout order in the United Kingdom is to off er‘BODMAS6’ and its var iants as a r ule. However, it isunclear whether such an approach adequatel yaddresses typical errors made by students in theiruse of expressions.

The following expression errors were manifested in the APU tests (Foxman et al., 1985). These testsinvolved a cohor t of 12 500 students age 11 to 15years. There is also evidence in more recent studies(see Ryan and Williams, 2007) that these arepersistent, especially the first.

• Conjoining: e.g. a + b = ab

• Powers are interpreted as multiplication, an errormade by 20% of 15-year-olds

• Not understanding that having no coefficient meansthe coefficient is 1

• Adding all three values when substituting in, say, u + gt

• Expressing the cost of a packet of sweets where xpackets cost 90p as x/90

The most obvious explanation of the conjoining erroris that conjoining is an attempt to express and ‘answer’by constructing closure, or students may just not knowthat letters together in this notation mean ‘multiply’.

Ryan and Williams (2007) found a significant numberof 14-year-olds did not know what to do with anexpression; they tr ied to ‘solve’ it as if it is anequation, again possibly a desire for an ‘answer’. Theyalso treated subtraction as if it is commutative, andignored signs associated with numbers and letters.Both APU (Foxman et al., 1985) and Har t, (1981)concluded that understanding operations was agreater problem than the use of symbols to indicate

them, but it is clear from Ryan and Williams’ studythat interpretation is also significantly problematic.The prevalence of similar er rors in studies 20 yearsapart is evidence that these are due to students’normal sense-making of algebra, given their previousexperiences with arithmetic and the inherent non-obviousness of algebraic notation.

Summary• Understanding operations and their inverses is a

greater problem than understanding the use ofsymbols.

• Learners tend to use their rules for reading andother false priorities when combining operations,i.e. interpreting left to right, doing addition first,using language to construct expressions, etc. Theyneed to develop new priorities.

• New rules, such as BODMAS (which can bemisused), do not effectively and quickly replace oldrules which are based on familiarity, habit, andarithmetic.

Equals sign

A significant body of research repor ts on difficultiesabout the meaning of the equals sign Sfard andLinchevski (1994) find that students who can do 7 x+ 157 = 248 cannot do 112 = 12 x + 247, butthese questions include two issues: the position andmeaning of the equals sign and that algor ithmicapproaches lead to the temptation to subtr actsmaller from larger, erroneously, in the secondexample. They argue that the root prob lem is thefailure to understand the inverse relation betweenaddition and subtraction, but this research showshow conceptual difficulties, incompleteunderstandings and notations can combine to mak emultiple difficulties. If students are taught to mak echanges to both sides of an equation in order tosolve it (i.e . transform the equation y – 5 = 8 into y– 5 + 5 = 8 + 5) and they do not see the need tomaintain equivalence between the values in the twosides of the equation, then the method that theyare being taught is mysterious to them, par ticularlyas many of the cases they are off ered at first can beeasily solved by arithmetical methods. Booth (1984)shows that these er rors combine problems withunderstanding operations and inverses andproblems understanding equivalence.



There are two possible ways to tackle these problems:to identify all the separate problems, treat themseparately, and expect learners to apply the relevantnew understandings when combinations occur ; or totreat algebraic statements holistically and semantically,so that the key feature of the above examples isequality. There is no research which shows conclusivelythat one approach is better than the other (astatement endorsed in NMAP’s review (2008)).

There is semantic and syntactic confusion about themeaning of ‘=’ that goes beyond learning a notation(Kieran, 1981; 1992). Sometimes, in algebra, it is usedto mean that the two expressions are equal in aparticular instance where their values are equal; othertimes it is used to mean that tw o expressions areequivalent and one can be substituted for another inevery occurrence. Strictly speaking, the latter isequivalence and might be wr itten as ‘ ’ but we are notarguing for this to become a new ‘must do’ for the curriculum as this would cut across so muchcontextual and historical practice. Rather, theunderstanding of algebraic statements must besituational, and this includes learning when to use ‘==’to mean ‘calculate’; when to use it to mean ‘equal inspecial cases’ and when to mean ‘equivalent’; and whento indicate that ‘these two functions are related in thisway’ (Saenz-Ludlow and Walgamuth, 1998). Thesedifferent meanings have implications for how the letteris seen: a quantitative placeholder in a structure; amystery number to be found to make the equalitywork; or a var iable which co-varies with others withinrelationships. Saenz-Ludlow and Walgamuth showed,over a year-long study with children, that the shifttowards seeing ‘=’ to mean ‘is the same as’ rather than‘find the answer’ could be made within ar ithmetic withconsistent, intentional, teaching. This was a teachingexperiment with eight-year-olds in which children wereasked to find missing sums and addends in additiongrids. The verb ‘to be’ was used instead of the equalssign in this and several other tasks. Another taskinvolved finding several binary calculations whoseanswer was 12, this time using ‘=’. Word problems,including some set by the children, were also used.Children also devised their own ways to represent andsymbolise equality. We do not have space here todescribe more of the exper iment, but at the end thechildren had altered their initial view that ‘=’ was aninstruction to calculate. They understood ‘=’ as givingstructural information. Fujii and Stephens’ (2001)research can be interpreted to show that students doget better at using ‘new’ meanings of the equals signand this may be a product of repeated exper ience ofwhat Boero called the ‘new transformations’ made

possible by algebra, combined with ‘new anticipations’also made possible by algebra.

Alibali and colleagues (2007) studied 81 middle schoolstudents over three years to map their under standingof equations. They found that those who had, ordeveloped, a sophisticated understanding of the equalssign were able to deal with equivalent equations, usingequivalence to transform equations and solve forunknowns. Kieran and Saldanha (2005) used aComputer Algebra System to enable five classes ofupper secondary students to explore differentmeanings of ‘=’ and found that given suitable tasks theywere able to understand equivalence, generating forthemselves two different understandings: equivalenceas meaning that expressions would give them equalvalues for a range of input values of the var iables, andequivalence as meaning that the expressions werebasically transformations of the same form. Both ofthese understandings contribute to meaningfulmanipulation from one form to another. Also focusingon equivalence, Kieran and Sfard (1999) used agraphical function approach and thus enabled studentsto recognise that equivalent algebraic representations offunctions would generate the same graphs, and hencerepresent the same relationships between variables.

The potential for confusion between equality andequivalence relates to confusion between findingunknowns (such as values of variables when twonon-equivalent expressions are temporarily madeequal) and expressing relationships betweenvariables. Equivalence is seen when gr aphs coincide;equality is seen when graphs intercept.

Summary• Learners persist in using ‘=’ to mean ‘calculate’

because this is familiar and meaningful for them.

• The equals sign has different uses withinmathematics; sometimes it indicates equivalenceand sometimes equality; learners have to learnthese differences.

• Different uses of the equals sign carry differentimplications for the meaning of letters: they canstand for hidden numbers, or variables, orparameters.

• Equivalence is seen when graphs coincide, and canbe understood either structurally or as generationof equal outputs for every input; equality is seenwhen graphs intercept.


Equations and inequality

In the CMF study (Johnson, 1989), 25 classes in 21schools in United Kingdom were tested to find outwhy and how students between 8 and 13 cling toguess-and-check and number-fact methods ratherthan new formal methods offered by teachers. Thestudy focused on several topics, including linearequations. The findings, dependent on large scaletests and additional inter views in four schools, aresummarised here and can be seen to include severaltendencies already described in other, related,algebraic contexts. That the same tendencies emer gein several algebraic contexts suggest that these arenatural responses to symbolic stimuli, and hence taketime to overcome.

Students tended to:• calculate each side rather than operate on them • not use inverse operations with understanding• use ad hoc number-specific methods• interpret a box or triangle to mean ‘missing

number’ but could not interpret a letter for thispurpose

• not relate a method to the symbolic form of amethod

• be unable to explain steps of their procedures• confuse a ‘changing sides’ method with a ‘balance’

metaphor, particularly not connecting what is saidto what is done, or to what is written

• test actual numbers rather than use an algebraicmethod

• assume different letters had different values• think that a letter could not have the value zero.

They also found that those who used the language‘getting rid of ’ were more likely to engage insuperficial manipulation of symbols. They singled out‘get rid of a minus’ for par ticular comment as it hasno mathematical meaning. These findings have beenreplicated in United Kingdom and elsewhere , andhave not been refuted as evidence of commondifficulties with equations.

In the same study, students were then taught using a‘function machine’ approach and this led to betterunderstanding of what an equation is and the variablenature of x. However, this approach only makes sensewhen an input-output model is appropr iate, i.e. notfor equating two functions or for higher orderfunctions (Vergnaud 1997). Ryan and Williams (2007)found that function machines can be used b y moststudents age 12 to 14 to solv e linear equations, butonly when provided. Few students chose to introduce

them as a method. Most 12-year-olds could reverseoperations but not their order when ‘undoing’ to findunknowns in this approach. Booth (1984) and Piagetand Moreau (2001) show that students whounderstand inversion might not under stand that,when inverting a sequence of oper ations, the inverseoperations cannot just be car ried out in any order :the order in which they are car ried out influences theresult. Robinson, Ninowski and Gray (2006) alsoshowed that coordinating inversion with associativityis a greater challenge than using either inversion orassociativity by themselves in problem solving.Associativity is the property that x + (y + z) is equalto (x + y) + z, so that we can add either the f irsttwo terms, and then the last, or the last two and thenthe first. This property applies to multiplication also.(Incidentally, note that the automatic application ofBODMAS here would be unnecessar y.) Students get confused about how to ‘undo’ such relatedoperations, and how to undo other paired operationswhich are not associative. As in all such matter s,teaching which is based on meaning has differentoutcomes (see Brown and Coles, 1999, 2001).

Once learners understand the meaning of ‘=’ there isa range of ‘intuitive’ methods they use to f indunknown numbers: using known facts, counting,inverse operations, and trial substitution (Kieran,1992). These do not generalize for situations in whichthe unknown appears on both sides, so formalmethods are taught. Formal methods each carrypotential difficulties: function machines do not extendbeyond ‘one-sided’ equations; balance methods do notwork for negative signs or for non-linear equations;change-side/change-sign tends to be misapplied ratherthan seen as a special kind of tr ansformation.

Many errors when solving equations appear tocome from misapplication of r ules and processesrather than a flawed understanding of the equalssign. Filloy describes several ‘cognitive tendencies’observed over several studies of studentsprogressing from concrete to abstr actunderstandings (e.g. Filloy and Suther land, 1996).These tendencies are: to cling to concrete models;to use sign systems inappropr iately; to makeinappropriate generalizations; to get stuck whennegatives appear ; to misinterpret concrete actions.Problems with the balance metaphor could be amanifestation of the general tendency to cling toconcrete models (Filloy and Rojano, 1989), and thenegative sign cannot be related to concreteunderstandings or even to some syntactic r uleswhich may have been learnt (Vlassis, 2002). Another



problem is that when the ‘unknown’ is on both sidesit can no longer be treated with simple in versiontechniques as finding ‘the hidden number’; 3x = 12entails answering the question ‘what number must Imultiply 3 by to get 12?’. But when balancing ‘4m +3’ with ‘3m + 8’ the balance metaphor can suggesttesting and calculating each side until they match,rather than solving by filling-in arithmetical facts.Vlassis devised a teaching experiment with 40 lowersecondary students in two classes. The first task wasa word problem which would have generated twoequal expressions in one variable, and students onlyapplied trial-and-error to this. The second task was a sequence of balance prob lems with diagramsprovided, and all students could solv e these. Thefinal task was a sequence of similar prob lemsexpressed algebraically, two of which used negativesigns. These generated a range of erroneousmethods, including failure to identify when to use aninverse operation, misapplication of r ules, syntacticalmistakes and manipulations whose meaning washard to identify. In subsequent exercises errors ofsyntax and meaning diminished, but errors withnegative integers persisted. Eight months later, in adelayed interview, Vlassis’ students were still usingcorrectly the pr inciples represented in the balancemodel, though not using it explicitly, but still hadproblems when negatives were included. In Filloyand Rojano (1989) a related tendency is descr ibed,that of students creating a per sonal sense ofconcrete action (e .g. ‘I shall move this from here tohere’) and using them as if they are algebr aic rules(also observed by Lima and Tall, 2008). More insightinto how learners understand equations is given byEnglish and Sharry (1996) who asked students toclassify equations into similar types. Some classifiedthem according to superficial syntactic aspects, andothers to under lying algebraic structure. English andSharry draw attention to the need f or students tohave experience of suitable structures in order toreason analogically and identify deeper similar ities.

There is little research in students’ understanding ofinequality in algebra. In number, children may knowabout ranges of smaller, or larger, or ‘between’numbers from their position on a n umberline, andchildren often know that adding the same quantityto two unequal quantities maintains the inequality.There are well-known confusions about relative sizeof decimal numbers due to misunder standings aboutthe notation, but beyond the scope of this review(Hart, 1981). Research by Tsamir and othersdescribe common problems which appear to relateto a tendency to act procedurally with unequal

algebraic expressions without maintaining anunderstanding of the inequality (Linchevski and Sfard,1991; Tsamir and Bazzini, 2001; Tsamir and Almog,2001). One of these studies compares theperformance of 170 Italian students to that of 148Israeli students in higher secondar y school (Tsamirand Bazzini, 2001). In both countr ies students hadbeen formally taught about a r ange of inequalities.They were asked whether statement about the set S ={ x Є R: x = 3} could be tr ue or not: ‘S can bethe solution set of an equality and an inequality’.Only half the students under stood that it could bethe solution set of an inequality, and those fewItalians who gave examples chose a quadr aticinequality that they already knew about. Somestudents offered a linear inequality that could besolved to include 3 in the answ er. The researchersconcluded that unless an inequality question wasanswerable using procedural algebra it was too hardfor them. Another task asked if par ticular solutionsets satisfied 5x4 < 0. Only half were able to say thatx = 0, the next most popular answer being x < 0.The researchers compared students’ responses toboth tasks. It seems that the image of ‘imbalance’often used with algebraic inequalities is abandonedwhen manipulation is done . The ‘imbalance’ imagedoes not extend to quadratic inequalities, for whicha graphical image works better, but again aprocedural approach is preferred by many studentswho then misapply it.

Summary• Once students understand the equals sign, they are

likely to use intuitive number-rules as a first resort.

• The appearance of the negative sign creates needfor a major shift to abstract meanings of operationsand relations, as concrete models no longeroperate.

• The appearance of the unknown on both sides ofan equation creates the need for a major shifttowards understanding equality and variables.

• Students appear to use procedural manipulationswhen solving equations and inequalities without amental image or understanding strong enough toprevent errors.

• Students appear to develop action-based ruleswhen faced with situations which do not haveobvious concrete manifestations.


• Students find it very hard to detach themselvesfrom concrete models, images and instructions andfocus on structure in equations.

Manipulatives

It is not only arithmetical habits that can causeobstacles to algebra. There are other algebraicactivities in which too strong a memor y for processmight create obstacles for future learning. Forexample, a popular approach to teaching algebr a is the provision of mater ials and diagrams whichascribe unknown numerical (dimensional) meaningto letters while facilitating their manipulation tomodel relationships such as commutativity anddistributivity. These appear to have some success inthe shor t term, but shifts from physical appearanceto mental abstraction, and then to symbolism, arenot made automatically by learners (Boulton-Lewis,Cooper, Atweh, Pillay, Wilss and Mutch,1997). Thesemanipulatives provide persistent images andmetaphors that may be obstructions in futurework. On the other hand, the or iginal approach todealing with var iables was to represent them asspatial dimensions, so there are strong histor icalprecedents for such methods. There are repor tedinstances of success in teaching this, relating toBruner’s three perspectives, enactive-iconic-symbolic (1966), where detachment from themodel has been under stood and scaffolded byteaching (Filloy and Suther land, 1996; Simmt andKieren, 1999). Detachment from the model has tobe made when values are negative and can nolonger be represented concretely, and also withfractional values and division oper ations. Spatialrepresentations have been used with success where the image is used per sistently in a r ange of algebraic contexts, such as expressions andequations and equivalence , and where teacher s use language to scaffold shifts between concrete,numerical and relational per spectives.

Use of rod or bar diagr ams as in Singapore (NMAP,2008; Greenes and Rubenstein, 2007) to representpart/whole comparisons, reasoning, and equations,appears to scaffold thinking from actual numbers tostructural relationships, so long as they onl y involveaddition and/or repeated addition. Statements in theproblem are translated into equalities betweenlengths. These equal lengths are constructed fromrods which represent both the actual and theunknown numbers. The rod arrangements or valuescan then be manipulated to f ind the value of the

unknown pieces. Equations with the var iable on bothsides are taught to 11 and 12 y ear-olds in Singaporeusing such an approach. The introduction of suchmethods into classrooms where teacher s are notexperienced in its use has not been researched. Ithas some similar ities to the approach based onCuisenaire rods championed by Gattegno in Europe .Whereas use for numbers was widespread in U.K.primary schools, use for algebra was not, possiblybecause the curriculum focus on substitution andsimplification, rather than meaning and equivalence ,provided an obstacle to sustained use .

Summary• Manipulatives can be useful for modelling algebraic

relationships and structures.

• Learners might see manipulatives as ‘just somethingelse to learn’.

• Teachers can help learners connect the use ofobjects, the development of imagery and the use ofsymbols through language.

• Students have to appreciate the limitations ofconcrete materials and shift to mental imagery and abstract understandings.

Application of formulae withinmathematicsDickson’s study with three classes of ten-y ear-olds(1989 a) into students’ use of formulae and formalmethods is based on using the f ormula for area ofrectangle in various contexts. In order to besuccessful in such tasks, students have to understandwhat multiplication is and how it relates to area, e.g.through an array model, how to use the formula bysubstitution and how the measuring units for areaare applied. Some students can then work out aformula for themselves without formal teaching.From this study, Dickson (1989a, 1989b) and hercolleagues found several problems in how studentsapproach formal methods in ear ly secondary school.A third of her subjects did not use a f ormal methodat all; a third used it in a test b ut could not explain itin interviews; a third used it and explained it. Shefound that they:

• may not have underlying knowledge on which tobase formalisation (note that formalisation canhappen spontaneously when they do have such



knowledge)• base their reasoning on incorrect method• have a sound strategy that may not match formal

method• may be taught methods leading up to formal, but

not matching the formal method• may retain other methods, which may have limited

application• may retain formalisation but lose meaning, then

misapply a formal method in future• pre-formal enactive or iconic experiences may have

been forgotten• might be able to use materials to explain formal

method• may interpret formal notation inadequately.

The research described above, taken as a whole ,suggests that the problems students have with usingformulae in subjects other than mathematics are due to: not being fluent with the notation; notunderstanding the under lying operations; experienceof using such formulae in mathematics lessons beinglimited to abstract or confusing situations, or even tosituations in which an algebr aic formula is notnecessary. In addition, of course, they may notunderstand the intended context.

Summary• Learners are able to construct formulae for

themselves, at least in words if not symbols, if theyhave sufficient understanding of the relationshipsand operations.

• Learners’ problems using formulae have severalpossible root causes.1 Underlying knowledge of the situation or

associated concepts may be weak.2 Existing working strategies may not match the

formal method.3 Notational problems with understanding how

to interpret and use the formula.

Part 2: problems arising indifferent approaches todeveloping algebraic reasoningSince the CSMS study (Har t, 1981) there has beenan expansion of teaching approaches to dev elopmeaningful algebra as: • expressing generalities which the child already

knows, therefore is expressing something that hasmeaning, and comparing equivalent expressions

• describing relationships between expressions asequations, which can then be solved to findunknown values (as in word problems)

• a collection of techniques for transformingequations to either find unknown values orrepresent relationships between variables indifferent ways

• expressing functions and their inverses, in whichinputs become outputs according to a sequence of operations; using multiple representations

• modelling situations by identifying variables andhow they co-vary.

Each of these offers more success in some aspectsthan an approach based on r ules for manipulatingexpressions, but also highlights fur ther obstacles toreasoning. Research is patchy, and does not examinehow students learn across contexts and materials(Rothwell-Hughes, 1979). Indeed, much of theresearch is specifically about learning in par ticularcontexts and materials.

Expressing generalisations from patternsOne approach to address inherent diff iculties inalgebra is to dr aw on our natur al propensity toobserve patterns, and to impose patter ns ondisparate experiences (Reed 1972). In thisapproach, sequences of patterns are presented and students asked to deduce formulae todescribe quantitative aspects of a gener al term in the sequence . The expectation is that thisgenerates a need for algebraic symbolisation, whichis then used to state what the student can alreadyexpress in other ways, numerical, recursive,diagrammatically or enactively.

This approach is prevalent in the United Kingdom,Australia and par ts of Nor th America. The NMAP(2008) review finds no evidence that expressinggenerality contributes to algebraic understanding, yetothers would say that this depends on the definition


of algebraic understanding. Those we offered at thestart of this chapter include expression of gener ality asan indication of understanding. In Australia, there arecontradictory findings about the value of such tasks.

The following is an example of one of the itemswhich was used in the lar ge scale test administeredto students by MacGregor and Stacey (repor ted inMason and Suther land, 2002).

Look at the numbers in this table and answer thequestions:

x y

1 5

2 6

3 7

4 8

5 9

6 ..

7 11

8 ...

.... ...

...

(i) When x is 2, what is y?(ii) When x is 8, what is y?(iii) When x is 800, what is y?(iv) Describe in words how you would find y if you

were told that x is ………(v) Use algebra to write a rule connecting x and y

………..

MacGregor and Stacey found performance on theseitems varied from school to school. The success of14-year-old students in wr iting an algebraic ruleranged from 18% in one school to 73% in another. Ingeneral students searched for a term-to-term rule(e.g. Stacey, 1989). They also tested the same studentswith more traditional items involving substitution toshow the meaning of notation and tr ansformation, toshow equivalence and finding unknowns. From thisstudy they concluded that students taught with a

pattern-based approach to algebra did no better andno worse on traditional algebra items than studentstaught with a more traditional approach (MacGregorand Stacey, 1993, 1995).

Redden (1994) studied the work of 1400 10- to 13-year-olds to identify the stages through whichstudents must pass in such tasks. First they mustrecognise the number pattern (which might bemultiplicative), then there must be a stimulus toexpression, such as being asked for the next termand then the value of uncountab le term; they mustthen express the general rule and use symbols toexpress it. Some students could only process onepiece of data, some could process more pieces ofdata, some gave only a specific example, some gavethe term-to-term formula and a few gave a fullfunctional formula. A major shift of perception has totake place to express a functional f ormula and this ismore to do with ‘seeing’ the functional relationship, ashift of perception, than symbolising it. Rowland andBills (1996) describe two kinds of generalisation:empirical and structural, the first being moreprevalent than the second. Amit and Neria (2007)use a similar distinction and f ound that students whohad followed a pattern-generalisation curriculumwere able to switch representations meaningfull y,distinguish between variables, constants and theirrelationships, and shift voluntarily from additive tomultiplicative reasoning when appropr iate.

Moss, Beatty and Macnab (2006) worked with nine-year-old students in a longitudinal study and foundthat developing expressions for pattern sequenceswas an effective introduction to understanding thenature of rules in ‘guess the rule’ problems. Nearly all of the 34 students were then able to ar ticulategeneral descriptions of functions in the classichandshake problem7 which is known to be hard forstudents in ear ly secondary years. By contrast, Ryanand Williams (2007) found in large-scale testing thatthe most prevalent error in such tasks for 12- and14-year-olds was giving the term-to-term formularather than the functional formula, and giving anactual value for the nth term. Cooper and Warren(2007, and Warren and Cooper, 2008), worked forthree years in five elementary classrooms, usingpatterning and expressing patterns, to teach studentsto express generalisations to use var iousrepresentations, and to compare expressions andstructures. Their students learnt to use algebraicconventions and notations, and also under stood thatexpressions had under lying operational meanings.Clearly, students are capable of learning these aspects



of algebra in cer tain pedagogic conditions. Amongother aspects common to most such studies, Cooperand Warren’s showed the value of compar ingdifferent but equivalent expressions that ar ise fromdifferent ways to generalise the patterns, and alsointroduced inverse operations in the context offunction machines, and a range of mental ar ithmeticmethods. If other research about gener alising patternsapplies in this study, then it must be the combinationof pattern-growth with these other aspects of algebr athat made the difference in the learning of theirstudents.They point to ‘the importance ofunderstanding and communicating aspects ofrepresentational forms which allowed commonalitiesto be seen across or between representations’.

As Carraher, Martinez & Schliemann (2007) show, itis important to nurture the transition from empirical(term-to-term) generalizations (called naïveinduction by Radford, 2007), to generalisations thatfollow from explicit statements about mathematicalrelations between independent and dependentvariables, and which might not be ‘seen’ in the data.Steele (2007) indicates some of the wa ys in which afew successful 12 to 13 year old students go aboutthis transition when using var ious forms of data,pictorial, diagrammatic and numerical, but biggerstudies show that this shift is not automatic andbenefits from deliberate tuition. Radford furtherpoints out that once a functional relation isobserved, expressing it is a fur ther process involvingintegration of signs and meaning. Stephens’ work(see Mason, Stephens and Watson, in press) showsthat the oppor tunity and ability to exemplifyrelationships between variables as number pairs, and to express the relationship within the pair s, arenecessary predictors of the ability to f ocus on andexpress a functional relationship. This research alsoillustrates that such abilities are developmental, andhence indicates the kind of lear ning experiencesrequired to make this difficult shift.

Rivera and Becker (2007), looking longitudinally atmiddle school students’ understanding of sequencesof growing diagrammatic patterns in a teachingexperiment, specify three forms of generalizationthat students engage with: constructive standard,constructive nonstandard, and deconstructive. It isthe deconstruction of diagrams and situations thatleads most easily to the functional formula, theyfound, rather than reasoning inductively fromnumbers. However, their students generally revertedto arithmetical strategies, as reported in many otherstudies of this and other shifts to wards algebra.

Reed (1972) hypothesised that classifying is a naturalact that enables us to make distinctions, clump ideas,and hence deal with lar ge amounts of newinformation. It is therefore useful to think of what sor tof information learners are tr ying to classify in thesekinds of task. Reed found that people extractprototypes from the available data and then see howfar other cases are from this prototype . Applying thisto pattern-growth and sequence tasks makes itobvious that term-to-term descriptions are far easierand likely to be dominant when the data is expressedsequentially, such as in a table. We could legitimatelyask the question: is it worth doing these kinds ofactivity if the shifts to seeing and then expressingfunctional relationships are so hard to make? Doesthis just add more diff iculties to an already diff icultsubject? To answer this, we looked at some studies inwhich claims are made of improvements in seeing andexpressing algebraic relationships, and identifyingfeatures of pedagogy or innovation which may haveinfluenced these improvements. Yeap and Kaur (2007)in Singapore found a wider range of factors influencingsuccess in unfamiliar generalisation tasks than has beenreported in studies which focus on rehearsedprocedures. In a class of 38 ten-y ear-old students theyset tasks, then observed and interviewed studentsabout the way they had worked on them. Their aimwas to learn more about the str ategies students hadused and how these contributed towards success. Thetask was to find the sums of consecutive oddnumbers: 1 + 3 + 5 + …+ (2 n – 1). Students werefamiliar with adding integers from 1 to 100, and alsowith summing multiples. They were given a sequenceof subtasks: a table of values to complete , to find thesum or 1 + 3 + … + 99 and to f ind the sum of 51 +53 + … + 99. The researchers helped students byoffering simpler versions of the same kinds ofsummation if necessary. Nearly all students were ableto recognize and continue the pattern of sums (theyturn out to be the square n umbers); two-thirds wereable to transfer their sense of str ucture to the ‘sum to99’ task, but only one-third completed the ‘sum from51’ task – the one most dissimilar to the table-fillingtasks, requiring adaptation of methods and use ofprevious knowledge to make an argument. Theresearcher had a series of designed prompts to helpthem, such as to f ind the sum from 1 to 49, and thensee what else they needed to get the sum to 99.Having found an answer, students then had to f ind itagain using a different method. They found thatsuccess depended on: • the ability to see structures and relationships• prior knowledge• metacognitive strategies


• critical-thinking strategies• the use of organizing heuristics such as a table• the use of simplifying heuristics such as trying out

simpler cases• task familiarity• use of technology to do the arithmetic so that large

numbers can be handled efficiently.

As with all mathematics teaching, limited experienceis unhelpful. Some students only know one way toconstruct cases, one way to accomplish generalisation(table of values and patter n spotting), and have onlyever seen simple cases used to star t sequencegeneration, rather than deliberate choices to aidobservations. Students in this situation may beunaware of the necessity for critical, reflective thinkingand the value of simplifying and or ganising data.Furthermore, this collection of studies on expressinggenerality shows that construction, design, choice andcomparison of various representational means doesnot happen spontaneously for students who arecapable of using them. Choosing when and why toswitch representations has long been known to be amark of successful mathematics students (Krutetskii,1976) and therefore this is a str ategy which needs tobe deliberately taught. Evidence from Blanton andKaput’s intervention study with 20 teacher s (2005) isthat many primary children were able to invent andsolve ‘missing number’ sentences using letter s asplaceholders, symbolize quantities in patterns, deviseand use graphical representations for single variables,and some could wr ite simple relations using letter s,codes, ‘secret messages’ or symbols. The interventionwas supportive professional development whichhelped teachers understand what algebraic reasoningentails, and gave them resources, feedback, and othersupport over five years. Ainley (1996) showed thatsupportive technology can display the purpose offormal representations, and also remove the technicaldifficulties of producing new representations. Ten-year-old students in her study had w orked for a few yearsin a computer-rich environment and usedspreadsheets to collect data from pur posefulexperiments. They then generated graphs from thedata and studied these , in relation to the data, tomake conjectures and test them. One task wasdesigned to lead to a prob lematic situation so thatstudents would have to look for a shor tcut, and sheobserved that the need to ‘teach the computer’ howto perform a calculation led to spontaneous f ormalrepresentation of a var iable.

So, if it is possib le for students to learn to makethese generalizations only with a great deal of

pedagogic skill and technical know-how, why shouldit be pursued? The reason is that skill in the meaningand use of algebra enables fur ther generalizations tobe made, and transformations of mathematicalrelationships to be used and studied. The workrequired to understand the functional relationship isnecessary to operate at a higher level than merelyusing algebra to symbolize what you do, as withterm-to-term formulae. It is algebra that provides themeans to building concepts upon concepts, a keyaspect of secondar y mathematics, by providingexpression of abstract relationships in ways that canbe manipulated. In algebra, the products are notanswers, but structures, relationships, and informationabout relationships and special instances of them.These tasks provide contexts for that kind of shift,but do not guarantee that it will take place.

Assumptions, such as that which appear s to be madein Redden’s study, that understanding term-to-termrelationships is a route to under standing functionalrelations contradict the experience ofmathematicians that algebra expresses the str uctureof relations, and this can be adduced from singlecases which are gener ic enough to illustrate therelationship through diagrams or other spatialrepresentations. Numerical data has to be backed upwith fur ther information about relationships. Forexample, consider this data set:

x y

1 1

2 4

3 9

... ...

While it is possible for these values to be examplesof the function y = x2 it is also possible that theyexemplify y = x2 + (x – 1)(x – 2)(x – 3). Withoutfurther information, such as x being the side of asquare and y being its area, we cannot deduce afunctional formula, and inductive reasoning ismisapplied. There is much that is mathematicallyinteresting in the connection between term-to-termand functional formulae, such as application of themethod of differences, and students have to learnhow to conjecture about algebr aic relationships, butto only approach generalisation from a sequenceperspective is misleading and, as we have seen from



these studies, very hard without the support ofspecially-designed tasks comparing and transformingequivalent structural generalisations.

Summary• Learners naturally make generalisations based on

what is most obviously related; this depends on thevisual impact of symbols and diagrams.

• Seeing functional, abstract, relationships is hard andhas to be supported by teaching.

• Deconstruction of diagrams, relationships, situationsis more helpful in identifying functional relationshipsthan pattern-generation.

• Development of heuristics to support seeingstructural relationships is helpful.

• There is a further shift from seeing to expressingfunctional relationships.

• Learners who can express relationships correctlyand algebraically can also exemplify relationshipswith number pairs, and express the relationshipswithin the pairs; but not all those who can expressrelationships within number pairs can express therelationship algebraically.

• Learners who have combined pattern-generalisation with function machines and otherways to see relationships can become more fluentin expressing generalities in unfamiliar situations.

• Conflicting research results suggest that the natureof tasks and pedagogy make a difference to success.

• Functional relationships cannot be deduced fromsequences without further information aboutstructure.

Using an equation-centred approachto teaching algebra

There are new kinds of prob lem that arise in anequation-centred approach to teaching algebra inaddition to those descr ibed earlier: the solution ofequations to find unknown values, and the constructionof equations from situations. The second of these newproblems is considered in Paper 7. Here we look atdifficulties that students had in teaching studies designedto focus on typical problems in finding unknowns.

Students in one class of Booth’ s (1984) inter ventionstudy (which took place with f our classes in lowersecondary school) had a teacher who emphasisedthroughout that letters had numerical value. Thesestudents were less likely than others to treat a letteras merely an object. In her study, discussion aboutthe meaning of statements before formal activityseemed to be beneficial, and those students whowere taught a formal method seemed to under standit better some time after the lesson, maybe afterrepeated experiences. However, some students didnot understand it at all. As with all inter ventionstudies, the teaching makes a difference. Linchevskiand Herscovics (1996) taught six students to collectlike terms and then decompose additive terms inorder to focus on ‘sides’ or equations as expressionswhich needed to be equated. While this led to thembeing better able to deal with equations, there werelingering problems with retaining the sign precedingthe letter rather than attaching the succeeding sign.

Several other inter vention studies (e.g. van Ameron,2003; Falle, 2005) confirm that the type of equationand the nature of its coeff icients often make non-formal methods available to learners, even if theyhave had significant recent teaching in formalmethods. These studies fur ther demonstrate thatstudents will use ad hoc methods if they seem moreappropriate, given that they under stand the meaningof an equation; where they did not understand theyoften misapplied formal methods. Falle’s studyincluded more evidence that the structure a/x = bcaused par ticular problems as learners interpreted‘division’ as if it were commutative. As with otherapproaches to teaching algebra, using equations asthe central focus is not trouble-free.

Summary• As with all algebraic expressions, learners may react

to the visual appearance without thinking about themeaning.

• Learners need to know what the equation is tellingthem.

• Learners need to know why an algebraic method isnecessary; this is usually demonstrated when theunknown is negative, or fractional, and/or when theunknown is on both sides. They are likely to choosead hoc arithmetic methods such as guess- and-check, use of known number facts, compensation ortrial-and-adjustment if these are more convenient.


• Learners’ informal methods of making the sidesequal in value may not match formal methods.

• ‘Undoing’ methods depend on using inverseoperations with understanding.

• Fluent technique may be unconnected to explainingthe steps of their procedures.

• Learners can confuse the metaphors offered to‘model’ solving equations, e.g. ‘changing sides’ with‘balance’.

• Metaphors in common use do not extend tonegative coefficients or ‘unknowns’ or non-linearequations.

• Non-commutative and associative structures arenot easily used with inverse reasoning.

• As in many other contexts, division and rationalstructures are problematic.

Spreadsheets

Learners have to know how to recognise str uctures(based on understanding arithmetical operationsand what they do), express structures in symbols,and calculate par ticular cases (to stimulate inductiveunderstanding of concepts) in order to use algebr aeffectively in other subjects and in highermathematics. Several researchers have usedspreadsheets as a medium in which to explorewhat students might be ab le to learn (e.g. Schwar tz and Yerushalmy, 1992; Sutherland andRojano, 1993; Friedlander and Tabach, 2001). Theadvantages of using spreadsheets are as f ollows.

• In order to use spreadsheets you have to know thedifference between parameters (letters and numbersthat structure the relationship) and variables, and thespreadsheet environment is low-risk since mistakesare private and can easily be corrected.

• The physical act of pointing the cursor provides anenactive aspect to building abstract structures.

• Graphical, tabular and symbolic representations arejust a click away from each other and are updatedtogether.

• Correspondences that are not easy to see in othermedia can be aligned and compared on a

spreadsheet, e.g. sequences can be laid side by side,input and output values for different functions canbe compared, and graphs can be related directly tonumerical data.

• Large data sets can be used so that questionsabout patterns and generalities become moremeaningful.

In Sutherland and Rojano’s work, two small groupsof students 10- to 11-years-old with no formalalgebraic background were given some algebraicspreadsheet tasks based on area. It was found thatthey were less likely to use ar ithmetical approacheswhen stuck than students repor ted in non-spreadsheet research, possibly because thesearithmetical approaches are not easil y available in aspreadsheet environment. Sutherland and Rojanoused three foci known to be difficult for students: therelation between functions and inverse functions, thedevelopment of equivalent expressions and wordproblems. The arithmetic methods used includedwhole/part approaches and tr ying to work fromknown to unknown. Most of the problems, however,required working from the unknown to the knownto build up relationships. In a similar follow-up study15-year-old students progressively modified thevalues of the unknowns until the given totals werereached (Sutherland and Rojano, 1993). There wassome improvement in post-tests over pre-tests forthe younger students, but most still found the tasksdifficult. One of the four intervention sessionsinvolved students constructing equivalentspreadsheet expressions. Some students started byconstructing expressions that generated equality inspecific cases, rather than overall equivalence.Students who had star ted out by using par ticulararithmetical approaches spontaneously derivedalgebraic expressions in the pencil-and-paper tasks of the post-test. This appears to confound evidencefrom other studies that an ar ithmetical approachleads to obstacles to algebr aic generalization. Thegeneration of numbers, which can be compared tothe desired outputs, and adjusted through adaptingthe spreadsheet formula, may have made the needfor a formula more obvious. The researchersconcluded that comparing expressions whichreferred only to numbers, to those which referred to variables, appeared to have enabled students tomake this cr itical shift.

A recent area of research is in the use of computeralgebra systems (CAS) to develop algebraicreasoning. Kieran and Saldanha (2005) have had



some success with getting students to deal withequations as whole meaningful objects within CAS.

SummaryUse of spreadsheets to build formulae:• allows large data sets to be used• provides physical enactment of formula

construction• allows learners to distinguish between variables and

parameters• gives instant feedback• does not always lock learners into arithmetical and

empirical viewpoints.

Functional approach

Authors vary in their use of the w ord ‘function’.Technically, a function is a relationship ofdependency between variables, the independentvariables (input) which var y by some externalmeans, and the dependent var iables (output) whichvary in accordance with the relationship . It is therelationship that is the function, not a par ticularrepresentation of it, however in practice authorsand teachers refer to ‘the function’ when indicating a graph or equation. An equivalence such astemperature conversion is not a function, becausethese are just different ways to express the samething, e.g. t = 9/5 C + 32 where t is temperature indegrees Fahrenheit and C temperature in degreesCelsius (Janvier 1996). Thus a teaching approachwhich focuses on comparing different expressions ofthe same generality is concerned with structure andwould afford manipulation, while an approach whichfocuses on functions, such as using functionmachines or multiple representations, is concernedwith relationships and change and would affordthinking about pairs of values, critical inputs andoutputs, and rates of change.

Function machinesSome researchers repor t that students find it hardto use inverses in the r ight order when solvingequations. However, in Booth’s work (1984) withfunction machines she found that lower secondarystudents were capable of instructing the ‘machine’ bywriting operations in order, using proper algebraicsyntax where necessar y, and could make the shift tounderstanding the whole expression. They couldthen reverse the flow diagram, maintaining order, to ‘undo’ the function.

We have discussed the use of function machines tosolve equations above.

Multiple representationsA widespread attempt to overcome the obstacles oflearning algebra has been to offer learners multiplerepresentations of functions because:• different representations express different aspects

more clearly• different representations constrain interpretations –

these have to be checked out against each other • relating representations involves identifying and

understanding isomorphic structures (Goldin 2002).

By and large these methods offer graphs, equations,and tabular data and maybe a physical situation ordiagram from which the data has been generated. The fundamental idea is that when the main f ocus ison meaningful functions, rather than mechanicalmanipulations, learners make sensible use ofrepresentations (Booth, 1984; Yerushalmy, 1997; Ainley,Nardi and Pratt, 1999; Hollar and Norwood, 1999).

A central issue is that in most contexts f or a letterto represent anything, the student must understandwhat is being represented, yet it is often only by theuse of a letter that what is being represented can beunderstood. This is an essential shift of abstr action. Itmay be that seeing the use of letter s alongside otherrepresentations can help develop meaning, especiallythrough isomorphisms.

This line of thought leads to a substantial body ofwork using multiple representations to developunderstandings of functions, equations, graphs andtabular data. All these studies are teachingexperiments with a range of students from upperprimary to first year undergraduates. What we learnfrom them is a r ange of possibilities for learning andnew problems to be overcome. Powell and Maher(2003) have suggested that students can themselvesdiscover isomorphisms. Others have found thatlearners can recognise similar structures (English andSharry, 1996) but need experience or prompts inorder to go beyond surface features. This is becausesurface features contribute to the first impact of anysituation, whether they are visual, aural, the way thesituation is first ‘read’, or the first recognition ofsimilarity.

Hitt (1998) claims that ‘A central goal ofmathematics teaching is taken to be that thestudents be able to pass from one representation


type to another without falling into contradictions.’(p. 134). In experiments with teachers on a cour sehe asked them to match pictures of v essels withgraphs to represent the relationship between thevolume and height of liquid being poured into them.The most common er rors in the choice of functionswere due to misinterpretation of the graphicalrepresentation, and misidentification of theindependent variable in the situation. Understandingthe representation, in addition to under standing thesituation, was essential. The choice of representation,in addition to under standing, is also influential insuccess. Arzarello, Bazzini and Chiappini, (1994) gave137 advanced mathematics students this prob lem:‘Show that if you add a 4-digit number to the 4-digitnumber you get if you reverse the digits, the answeris a multiple of eleven’. There were three strategiesused by successful students, and the most-used wasto devise a way to express a 4-digit number as thesum of multiples of powers of ten. This strategy leadsimmediately to seeing that the ter ms in the sumcombine to show multiples of eleven. Therelationship between the representation and itsmeaning in terms of ‘eleven’ was very close. ‘Talk’ can structure a choice of representations that mostclosely resemble the mathematical meaning (see also Siegler and Ster n, 1998).

Even (1998) points to the ability to select, use, movebetween and compare representations as a cr ucialmathematical skill. She studied 162 early students in 8 universities (the findings are informative forsecondary teaching) and found a difference betweenthose who could only use individual data points andthose who could adopt a global, functional approach.Nemirovsky (1996) demonstrates that the Car tesianrelationship between graphs and values is mucheasier to understand pointwise, from points to lineperhaps via a table of values, than holistically, everypoint on a line representing a par ticular relationship.

Some studies such as Computer-Intensive Algebra(e.g. Heid, 1996) and CARAPACE (Kieran, Boileauand Garancon,1996) go some way towardsunderstanding how learners might see the duality ofgraphs and values. In a study of 14 students agedabout 13, the CARAPACE environment (of graphs,data, situations and functions) seemed to suppor tthe understanding of equality and equivalence of two functions. This led to findings of a significantimprovement in dealing with ‘unknown on bothsides’ equations over groups taught moreconventionally. The multiple-representation ICTenvironment led to better performance in word

problems and applications of functions, but studentsneeded additional teaching to become as fluent inalgebra as ‘conventional’ students. But teaching tofluency took only six weeks compared to one yearfor others. This result seems to conf irm that ifalgebra is seen to have purpose and meaning thenthe technical aspects are easier to lear n, eitherbecause there is motivation, or because the lear nerhas already developed meanings for algebraicexpressions, or because they have begun to developappropriate schema for symbol use. When studentsfirst had to express functions, and only then had toanswer questions about par ticular values, they hadfewer problems using symbols.

There were fur ther benefits in the CARAPACEstudy: they found that their students could switchfrom variable to unknown correctly more easily thanhas been found in other studies; the students saw asingle-value as special case of a function, but theirjustifications tended to relate to tabular data andwere often numerical, not relating to the overallfunction or the context. The students had toconsciously reach for algebraic methods, even to usetheir own algorithms, when the situations becameharder. Even in a multi-representational environment,using functions algebraically has to be taught; this isnot spontaneous as long as numerical or graphicaldata is available. Students preferred to movebetween numerical and graphical data, not symbolicrepresentations (Brenner, Mayer, Moseley, Brar, Duran,Reed and Webb,1997). This finding must depend ontask and pedagogy, because by contrast Lehrer,Strom and Confrey,(2002) give examples wherecoordinating quantitative and spatial representationsappears to develop algebraic reasoning throughrepresentational competence. Even (1998) arguesthat the flexibility and ease with which we hope students will move fromrepresentation to representation depends on whatgeneral strategy students bring to mathematicalsituations, contextual factors and previousexperience and knowledge. We will look fur ther at this in the next paper.

Further doubts about a multiple representationapproach are raised by Amit and Fried (2005) inlessons on linear equations with 13 – 14 y ear-olds:‘students in this class did not seem to get the ideathat representations are to be selected, applied, andtranslated’. The detail of this is elabor ated throughthe failed attempts of one student, who did makethis link, to persuade her peers about it. Hirschhorn(1993) repor ts on a longitudinal compar ative study



at three sites in which those taught using m ultiplerepresentations and meaningful contexts didsignificantly better in tests than other s taught moreconventionally, but that there was no diff erence inattitude to mathematics. All we really learn from thisis that the confluence of oppor tunity, task andexplanation are not sufficient for learning. Overall the research suggests that there are some gains inunderstanding functions as meaningful expressions ofvariation, but that symbolic representation is still hardand the least preferred choice.

The effects of multiple representationalenvironments on students’ problem-solving andmodeling capabilities are descr ibed in the next paper.

Summary• Learners can compare representations of a

relationship in graphical, numerical, symbolic anddata form.

• Conflicting research results suggest that the natureof tasks and pedagogy make a difference to success.

• The hardest of these representations for learners isthe symbolic form.

• Previous experience of comparing multiplerepresentations, and the situation being modelled,helps students understand symbolic forms.

• Learners who see ‘unknowns’ as special cases ofequality of two expressions are able to distinguishbetween unknowns and variables.

• Teachers can scaffold the shifts betweenrepresentations, and perceptions beyond surfacefeatures, through language.

• Some researchers claim that learners have tounderstand the nature of the representations inorder to use them to understand functions, whileothers claim that if learners understand thesituations, then they will understand therepresentations and how to use them.

What students could do if taught, butare not usually taught

Most research on algebra in secondar y school is of an innovative kind, in which par ticular tasks orteaching approaches reveal that learners of a

particular age are, in these circumstances, able todisplay algebraic behaviour of par ticular kinds. Usuallythese experiments contradict curriculumexpectations of age, or order, or nature, of learning.For example, in a teaching experiment over severalweeks with 8-year-old students, Carraher, Brizuelaand Schliemann (2000) repor t that young learnersare able to engage with problems of an algebraicnature, such as expressing and f inding the unknownheights in problems such as: Tom is 4 inches tallerthan Maria, Maria is 6 inches shor ter than Leslie;draw their heights. They found that young learnerscould learn to express unknown heights with letter sin expressions, but were sometimes puzzled by theneed to use a letter f or ‘any number’ when they hadbeen given a par ticular instance. This is a real sourcefor confusion, since Maria can only have one height.Students can naturally generalise about operationsand methods using words, diagrams and actionswhen given suitable support (Bastable and Schifter,2008). They can also see operators as objects(Resnick, Lesgold and Bill, 1990). These and otherstudies appear to indicate that algebr aic thinking candevelop in primary school.

In secondary school, students can work with a widerrange of examples and a greater degree of complexityusing ICT and graphical approaches than whenconfined to paper and pencil. For example, Kieran andSfard (1999) used graphs successfully to help 12- and13-yearold students to appreciate the equivalence ofexpressions. In another example, Noss, Healy andHoyles (1997) constructed a matchsticks microworldwhich requires students to build up LOGOprocedures for drawing matchstick sequences. Theyreport on how the software suppor ted some 12- to13-year-old students in finding alternative ways toexpress patterns and structures of Kieran’s secondand third kinds. Microworlds provide support forstudents’ shifts from par ticular cases to what has to betrue, and hence suppor t moves towards using algebraas a reasoning tool.

In a teaching study with 11 y ear-old students, Noble,Nemirovsky, Wright and Tierney, (2001) suggest thatstudents can recognise core mathematical str ucturesby connecting all representations to per sonally-constructed environments of their own, relevant forthe task at hand. They asked pairs of students toproceed along a linear measure , using steps ofdifferent sizes, but the same number of steps each,and record where they got to after each step . Theyused this data to predict where one w ould be afterthe other had taken so many steps. The aim was to


compare rates of change. Two further tasks, one anumber table and the other a software-suppor tedrace, were given and it was noticed from the ways inwhich the students talked that they were bringing toeach new task the language, metaphors andcompetitive sense which had been gener ated in theprevious tasks. This enabled them to progress fromthe measuring task to compar ing rates in multiplecontexts and representations. This still suppor ts thefact that students recognize similarities and look foranalogical prototypes within a task, but questionswhether this is related to what the teacher expects inany obvious way. In a three-year study with 16 lowersecondary students, Lamon (1998) found that a year’steaching which focused on modelling sequentialsituations was so effective in helping studentsunderstand how to express relationships that theycould distinguish between unknowns, variables andparameters and could also choose to use algebr awhen appropriate – normally these aspects were notexpected at this stage , but two years further on.

Lee (1996) descr ibes a long ser ies of teachingexperiments: 50 out of 200 first year universitystudents committed themselves to an extra studygroup to develop their algebraic awareness. Thisstudy has implications for secondary students, astheir algebraic knowledge was until then r ule-basedand procedural. She forced them, from the star t, totreat letters as variables, rather than as hiddennumbers. By many measures this group succeededin comparative tests, and there was also evidence ofsuccess beyond testing, improvements in attitudeand enjoyment. However, the impact ofcommitment to extra study and ‘belonging’ to a special group might also have played a par t.Whatever the causal factors, this study shows thatthe notion of var iable can be taught to those whohave previously failed to under stand, and can form a basis for meaningful algebra.

SummaryWith teaching:

• Young children can engage with missing numberproblems, use of letters to represent unknownnumbers, and use of letters to representgeneralities that they have already understood.

• Young children can appreciate operations asobjects, and their inverses.

• Students can shift towards looking at relationships ifencouraged and scaffolded to make the shift,through language or microworlds, for example.

• Students can shift from seeing letters as unknownsto using them as variables.

• Students will develop similarities and prototypes tomake sense of their experience and support futureaction.

• Students can shift from seeing cases as particular toseeing algebraic representations as statement aboutwhat has to be true.

• Comparison of cases and representations cansupport learning about functions and learning howto use algebra to support reasoning.



Part 3: Conclusions and recommendations

Conclusions

Error research about elementar y algebra and pre-algebra is uncontentious and the f indings aresummarised above. However, it is possible for younglearners to do more than is nor mally expected inthe curriculum, e.g. they will accept the use of letter sto express generalities and relationships which theyalready understand. Research about secondar yalgebra is less coherent and more patch y, but broadlycan be summarised as follows.

Teaching algebra by offering situations in whichsymbolic expressions make mathematical sense, and what learners have to find is mathematicallymeaningful (e.g. through multiple representations,expressing generality, and equating functions) is moreeffective in leading to algebr aic thinking and skill usethan the teaching of technical manipulation andsolution methods as isolated skills. However, thesemethods need to be combined through complexpedagogy and do not in themselves bring about allthe necessary learning. Technology can play a big par tin this. There is a difference between using ICT in the learner’s control and using ICT in the teacher’scontrol. In the learner’s control the physical actionsof moving around the screen and choosing betweenrepresentations can be easily connected to theeffects of such moves, and feedback is personalisedand instant.

There is a tenuous relationship between what itmeans to understand and use the affordances ofalgebra, as described in the previous paragraph, andunderstanding and using the symbolic f orms ofalgebra. Fluency in under standing symbolicexpressions seems to develop through use, and also contributes to effective use – this is a tw o-wayprocess. However, this statement ignores themessages from research which is purel y aboutprocedural fluency, and which suppor ts repetitivepractice of procedures in carefull y constructedvarying forms. Procedural research focuses onobstacles such as dealing with negativ e signs andfractions, multiple operations, task complexity andcognitive load but not on meaning, use, relationships,and dealing with unfamiliar situations.

Recommendations

For teachingThese recommendations require a change from afragmented, test-driven, system that encourages anemphasis on fluent procedure followed byapplication.

• Algebra is the mathematical tool for workingwith generalities, and hence should permeatelessons so that it is used wherever mathematicalmeaning is expressed. Its use should becommonplace in lessons.

• Teachers and writers must know about theresearch about learning algebra and take it intoaccount, particularly research about commonerrors in understanding algebraic symbolisation andhow they arise.

• Teachers should avoid using published and web-based materials which exacerbate the difficulties byover-simplifying the transition from arithmetic toalgebraic expression, mechanising algebraictransformation, and focusing on algebra as‘arithmetic with letters’.

• The curriculum, advisory schemes of work, andteaching need to take into account how shifts fromarithmetical to algebraic understanding take time,multiple experiences, and clarity of purpose.

• Students at key stage 3 need support in shiftingto representations of generality, understandingrelationships, and expressing these inconventional forms.

• Students have to change focus from calculation,quantities, and answers to structures of operationsand relations between quantities as variables. Thisshift takes time and multiple experiences.

• Students should have multiple experiences ofconstructing algebraic expressions for structuralrelations, so that algebra has the purpose ofexpressing generality.

• The role of ‘guess-the-sequence-rule’ tasks in thealgebra curriculum should be reviewed: it ismathematically incorrect to state that a finitenumber of numerical terms indicates a uniqueunderlying generator.


• Students need multiple experiences over time tounderstand: the role of negative numbers and thenegative sign; the role of division as inverse ofmultiplication and as the fundamental operationassociated with rational numbers; and the meaningof equating algebraic expressions.

• Teachers of key stage 3 need to understand howhard it is for students to give up their arithmeticalapproaches and adopt algebraic conventions.

• Substitution should be used purposefully forexemplifying the meaning of expressions andequations, not as an exercise in itself. Matchingterms to structures, rather than using them topractice substitution, might be more useful.

• The affordances of ICT should be exploited fully, inthe learner’s control, in the teacher’s control, and inshared control, to support the shifts ofunderstanding that have to be made includingconstructing objects in order to understandstructure.

• Teachers should encourage the use of symbolicmanipulation, using ICT, as a set of tools to supporttransforming expressions for mathematicalunderstanding.

For policy• The requirements listed above signal a training

need on a national scale, focusing solely on algebraas a key component in the drive to increasemathematical competence and power.

• There are resource implications about the use ofICT. The focus on providing interactive whiteboardsmay have drawn attention away from the need forstudents to be in control of switching betweenrepresentations and comparisons of symbolicexpression in order to understand the syntax andthe concept of functions. The United Kingdom maybe lagging behind the developed world in exploringthe use of CAS, spreadsheets and other softwareto support new kinds of algebraic thinking.

• In several other countries, researchers have beenable to develop differently-sequenced curricula inwhich students have been able to use algebra as away of expressing general and abstract notions asthese arise. Manipulation, solution of equations,and other technical matters to do with symbolsdevelop as well as with formal teaching, but are

better understood and applied. Similardevelopment in the United Kingdom has not beenpossible due to an over-prescriptive curriculumand frequent testing which forces a focus ontechnical manipulation.

• Textbooks which promulgate an ‘arithmetic withletters’ approach should be avoided; this approachleads inevitably to the standard, obvious errors andhence turns students off algebra and mathematicsin favour of short-term gains.

• Symbolic manipulators, graph plotters and otheralgebraic software are widely available and used toallow people to focus on meaning, application andimplications. Students should know how to usethese and how to incorporate them intomathematical explorations and extended tasks.

• We need to be free to draw on research andexplore its implications in the United Kingdom, andthis may include radical re-thinking of the algebracurriculum and how it is tested. This may happen aspart of the ‘functional maths’ agenda but itsfoundations need to be established when studentsare introduced to algebra.

For research• Little is known about school learning of algebra in

the following areas.

• The experiences that an average learner needs, ineducational environments conducive to change, toshift from arithmetical to algebraic thinking.

• The relationship between understanding the natureof the representations in order to use them tounderstand functions, and understanding thesituations as an aid to understanding therepresentations and how to use them.

• Whether teaching experiments using functional,multi-representational, equation or generalisationapproaches have an impact on students’ typicalnotation-related difficulties. In other words, we donot know if and how semantic-focused approachesto algebra have any impact on persistent and well-known syntactic problems.

• How learners’ synthesise their knowledge tounderstand quadratic and other polynomials, theirfactorisation and roots, simultaneous equations,



inequalities and other algebraic objects beyondelementary expressions and equations.

• Whether and how the use of symbolicmanipulators to transform syntax supportsalgebraic understanding in school algebra.

• Using algebra to justify and prove generalities,rather than generate and express them.

• How students make sense of different metaphorsfor solving equations (balance, doing-undoing,graphical, formal manipulation).

Endnotes1 The importance of inverses was discussed in the paper on

natural numbers

2 In the paper on r ational numbers we talk more about therelationship between fractions and rational numbers, and weoften use these words interchangeably.

3 The advantage of this is that spotting lik e terms might be easier,but this can also mask some other characteristics such asphysical meaning (e.g. E = mc2) and symmetry e.g. x2y + y2x.

4 This should be contrasted with the problems young learnershave with expressing relations using number, described in ourpaper on functional relations. Knowing that relations arethemselves number-like objects does not necessar ily mean wehave to calculate them.

5 This is discussed in detail in our paper s on whole numbers andrational numbers and outlined here .

6 A very common mnemonic to remind people to do: brackets,‘of ’, division, multiplication, addition and subtraction in thatorder. It does not always work.

7 If n people all shake hands with each other, how manyhandshakes will there be?

AcknowledgementsThis chapter was produced with the help of NicholaClarke who did much of the technical work.


Ainley, J. (1996) Purposeful Contexts for FormalNotation in a Spreadsheet Environment. Journal ofMathematical Behavior, 15(4) 405-422

Ainley, J., Nardi, E., and Pratt, D. (1999). ConstructingMeaning for Formal Notation in Active Graphing,in I. Scwank (Ed.) Proceedings of the Fir stConference of the European Society f or Research inMathematics Education, vol. 1 Forschungsinstitutfuer Mathematikdidaktik, Osnabrueck,

Alibali, M., Knuth, E., Hattikudur, S., Mcneil, N. andStephens, A. (2007) A Longitudinal Examinationof Middle School Students’ Understanding of theEqual Sign and Equivalent Equations MathematicalThinking and Learning, Volume 9, Issue 3, 2007,221 – 247

Amit, M. and Fr ied, M. (2005) Multiplerepresentations in 8th grade algebra lessons:Are learners really getting it? In Chick, H. L., andVincent, J. L. (Eds.). (2005). Proceedings of theInternational Group for the Psychology ofMathematics Education 2-57 to 2-64Melbourne: PME

Amit, M. and Neria, D. (2007) ‘‘Rising to thechallenge’’: using generalization in patternproblems to unearth the algebraic skills oftalented pre-algebra students ZDM MathematicsEducation 40:111–129

Anghileri J., Beishuizen, M. and van Putten, C. (2002)From informal strategies to structuredprocedures: Mind the gap! Educational Studies inMathematics 49(2) 149-170.

Arcavi A. (1994) Symbol sense: The informal sense-making in formal mathematics, For the Learning ofMathematics, 14 (3) p24-35.

Arzarello, F., Bazzini, L., Chiappini, G. (1994) Theprocess of naming in algebraic problem solving, inJ. Ponte and J. Matos (Eds.) Proc. PME XVIII , Lisbon,II, p40-44.

Baker, B., Hemenway, C., Trigueros, M. (2001) Ontransformations of basic functions. In Chick, H.,Stacey, K., Vincent, J. and Vincent, J. (eds.)Proceedings of the 12th ICMI study conference: Thefuture of the teaching and lear ning of algebra.pp.41-47 University of Melbourne, Australia, Dec9-14, 2001.

Banerjee, R. and Subramaniam, K. (2004) ‘Term’ as abridge concept between algebra and ar ithmetic.Paper presented at Episteme inter nationalconference December 13-17, InternationalCentre, Dona Paula, Goa

Bastable, V. and Schifter D. (2008) Classroom stor ies:examples of elementar y students engaged inearly algebra. In J. Kaput, D. Carraher and M.Blanton (eds.) Algebra in the ear ly grades. 165-184. New York: Erlbaum

Bednarz, N. and Janvier, B. (1996). Emergence anddevelopment of algebra as a problem solvingtool: Continuities and discontinuities witharithmetic. In N. Bednarz, C. Kieran and L. Lee, L.(Eds.), Approaches to Algebra (pp. 115 136).Dordrecht/Boston/London: Kluwer AcademicPublishers.

Bednarz, N., Kieran C., Lee, L. (1996) Approaches toalgebra: perspectives for research on teaching. InBednarz, N., Kieran C., Lee, L. (eds.) Approaches toalgebra: perspectives for research on teaching . pp.3-14 Dordrecht: Kluwer

Bell, A. (1996) Algebraic thought and the role ofmanipulable symbolic language. In Bednarz, N.,Kieran C., Lee, L. (eds.) Approaches to algebra:perspectives for research on teaching . 151-154Kluwer, Dordrecht

Bills, L. (2007) Stereotypes of liter al symbol use insenior school algebra. In E.Pehkonen (ed.)Proceedings of the annual conference of theInternatioanl Group for the Psychology ofMathematics Education. (2) 73-80. Lahti:University of Helsinki.


References


Blanton, M and Kaput, J (2005) Characterizing aclassroom practice that promotes algebraicreasoning. Journal for Research in MathematicsEducation, 36 (5) 412-446, pp. 35-25

Bloedy-Vinner, H. (1994) The analgebraic mode ofthinking: the case of the parameter. In J.Ponte andJ. Matos (eds.) Proceedings of the annualconference of the International Group for thePsychology of Mathematics Education. (2) 88-95.Lisbon, University of Lisboa.

Boero, P. (2001). Transformation and anticipation askey processes in algebraic problem solving. InSutherland, R., Rojano, T., Bell, A. and Lins, R.(eds.)Perspectives on School Algebra. pp. 99-119Dordrecht, Kluwer.

Booth, L. (1984) Algebra: Children’s strategies anderrors, a report of the strategies and errors in thesecondary school project, NFER-Nelson, London.

Boulton-Lewis, G. M., Cooper, T. J., Atweh, B., Pillay, H.,Wilss, L., and Mutch, S. (1997). The transition fromarithmetic to algebra: A cognitive perspective. InE. Pehkonen (ed.) Proceedings of the InternationalGroup for the Psychology of Mathematics Education ,21(2), 185-192.

Brenner, M. E., Mayer, R. E., Moseley, B., Brar, T., Duran,R., Reed, B. S. and Webb, D. (1997) Learning byunderstanding: The role of multiplerepresentations in learning algebra. AmericanEducational Research Journal, 34(4), 663-689.

Britt, M. and Irwin, K. (2007) Algebraic thinking withand without algebraic representation: a three-yearlongitudinal study. ZDM Mathematics Education40, 39-53

Brown, L. and Coles, A. (1999) Needing to usealgebra – a case study. In O Zaslavsky (ed.)Proceedings of the 23rdt annual conference of theInternational Group for the Psychology ofMathematics Education, Technion; Haifa, 2, 153-160.

Brown, L. and Coles, A. (2001) Natural algebraicactivity. In Chick, H., Stacey, K., Vincent, J. andVincent, J. (eds.) Proceedings of the 12th ICMI studyconference: The future of the teaching and lear ningof algebra. pp.120-127 University of Melbourne,Australia, Dec 9-14, 2001.

Bruner, J. (1966). Toward a Theory of Instruction.Cambridge, MA: Harvard University Press

Carpenter, T. and Levi, L. (2000) Developingconceptions of algebraic reasoning in the pr imarygrades. Research Report, Madison WI: NationalCenter for Improving Student Learning andAchievement in Mathematics and Science .www.wcer.wisc.edu/ncisla/publications

Carraher, D. Brizuela, B. and Schliemann A. (2000)Bringing out the algebraic character of ar ithmetic:Instantiating variables in addition and subtr action,in T. Nakahara and M. Koyama (Eds.), Proceedingsof PME–XXIV, Vol. 2. Hiroshima, Japan, p 145-152.

Carraher, D., Brizuela, B. M., and Earnest, D. (2001).The reification of additive differences in earlyalgebra. In H. Chick, K. Stacey, J. Vincent, and J.Vincent (Eds.), The future of the teaching andlearning of algebra: Proceedings of the 12th ICMIStudy Conference (vol. 1). The University ofMelbourne, Australia.

Carraher, D., Mar tinez, M. and Schliemann, A. (2007)Early algebra and mathematical generalizationZDM Mathematics Education 40, 3-22

Carraher, D., Schliemann, A., and Brizuela, B. (2001).Can young students operate on unknowns?Proceedings of the XXV Conference of theInternational Group for the Psychology ofMathematics Education, Utrecht, The Netherlands.Vol. 1, 130-140

Collis, K. (1971) A study of concrete and f ormalreasoning in school mathematics. Australian Journalof Psychology. 23, 289-296.

Cooper, T. and Warren, E. (2007) The effect ofdifferent representations on Years 3 to 5 students’ability to generalize. ZDM Mathematics Education40 29-37

Davydov V. (1990) Types of generalisation ininstruction: Logical and psychological prob lems in thestructuring of school curricula. Reston, VA: NCTM.

Dettori, G. Garutti, R. Lemut, E. (2001) Fromarithmetic to algebraic Thinking by using aspreadsheet, in R. Sutherland, T. Rojano, A. Bell andR. Lins (Eds) Perspectives on School Algebra, p 191-208, Kluwer, Dordrecht.

Dickson, L. (1989a) Area of rectangle. In D. Johnson(ed.) Children’s Mathematical Frameworks 8 – 13:a study of classroom teaching . 88-125, Windsor:NFER-Nelson.

Dickson, L. (1989b) Equations. In D. Johnson (ed.)Children’s Mathematical Frameworks 8 to 13: astudy of classroom teaching. pp. 151-190 Windsor:NFER- Nelson

Dougherty, B. (1996) The write way: a look at jour nalwriting in first-year algebra. Mathematics Teacher89, 556-560

Dougherty, B. (2001) Access to algebra: a processapproach. In Chick, H., Stacey, K., Vincent, J. andVincent, J. (eds.) Proceedings of the 12th ICMI studyconference: The future of the teaching and lear ningof algebra. pp. 207-212 University of Melbourne,Australia, Dec 9-14, 2001.


Drijvers, P. (2001) The concept of parameter in acomputer algebra environment. In Chick, H.,Stacey, K., Vincent, J. and Vincent, J. (eds.)Proceedings of the 12th ICMI study conference: Thefuture of the teaching and lear ning of algebra. pp.221-227 University of Melbourne, Australia, Dec9-14, 2001.

English, L. and Sharry, P. (1996) Analogical reasoningand the development of algebraic abstraction.Educational Studies in Mathematics, 30(2), 135-157.

Even, R. (1998). Factors involved in linkingrepresentations of functions. Journal ofMathematical Behavior, 17(1), 105-121.

Falle, J. (2005) From ar ithmetic to algebra: novicestudents’ strategies for solving equations.Proceedings of annual conference of MathematicsEducation Research Group of Australasia.Downloaded fromhttp://www.merga.net.au/search.php April 2008

Filloy E. and Rojano, T. (1989) Solving equations: thetransition from arithmetic to algebra. For theLearning of Mathematics , 9(2) 19-25.

Filloy, E. and Suther land, R. (1996), Designingcurricula for teaching and learning algebra, in A.Bishop, K. Clements, C. Keitel, J. Kilpatrick and C .Laborde (Eds.) International Handbook ofMathematics Education Part 1, Chapter 4, Kluwer,Dordrecht, p 139-160.

Filloy, E. Rojano, T. and Robio, G. (2001) Propositionsconcerning the resolution of ar ithmetical-algebraicproblems, in R. Sutherland (Ed.) Perspectives onAlgebra, Kluwer, Dordrecht, p155-176.

Foxman, D., Ruddock, G., Joffe, L., Mason, K., Mitchell, P.,and Sexton, B . (1985) A Re view of Monitor ing inmathematics 1978 to 1982, Assessment ofPerformance Unit, Depar tment of Education andScience, London.

Friedlander, A. and Tabach, M. (2001) Developing acurriculum of beginning algebra in a spreadsheetenvironment. In Chick, H., Stacey, K., Vincent, J. andVincent, J. (eds.) Proceedings of the 12th ICMI studyconference: The future of the teaching and lear ningof algebra. Pp.252-257. University of Melbourne,Australia.

Fujii, T. and Stephens, M. (2001) Fostering anunderstanding of generalisation through numericalexpressions: the role of quasi-var iables. In Chick,H., Stacey, K., Vincent, J. and Vincent, J. (eds.)Proceedings of the 12th ICMI study conf erence: Thefuture of the teaching and lear ning of algebra.Pp.258-264.University of Melbourne, Australia.

Fujii, T. and Stephens, M. (2008) Using numbersentences to introduce the idea of var iable. In.C.Greenes and R. Rubenstein (eds.) Algebra andAlgebraic Thinking in School Mathematics . 70th

Yearbook. pp. 127-140. Reston,VA: NCTM.Furinghetti, F. and Paola, D. (1994). Parameters,

unknowns and variables: a little difference?, inProceedings of the XVIII Inter national Conference forthe Psychology of Mathematics Education, pp. ? ?University of Lisbon, Portugal.

Goldin, G. (2002) Representation in mathematicallearning in English, L (ed.) Handbook ofinternational research in mathematics educationpp.197-218, Mahwah NJ, Erlbaum.

Gray, E. and Tall, D. (1994). Duality, ambiguity andflexibility: A proceptual view of simple ar ithmetic,Journal for Research in Mathematics Education, 26(2), 115-141

Greenes, C. and Rubenstein, R. (2007) (eds.) Algebraand algebraic thinking in school mathematics . 70th

Yearbook. Reston,VA: NCTM.Hart, K. (Ed.) (1981) Children’s understanding of

mathematics 11-16, Murray, London..Heid, M. (1996) Computer Intensive Algebra:

Reflections on a functional approach to beginningalgebra. In Bednarz, N., Kieran C., Lee, L. (eds.)Approaches to algebra: perspectives for research onteaching. Kluwer, Dordrecht

Herscovics, N. and Linchevski, L. (1994) A cognitivegap between arithmetic and algebra, Educationalstudies in mathematics, Vol. 27, pp. 59-78.

Hirschhorn, D. (1993) A longitudinal study ofstudents completing four years of UCSMPmathematics Journal for Research in MathematicsEducation 24 (2), 136-158

Hitt, F. (1998) Difficulties in the articulation of differentrepresentations linked to the concept of function.Journal of Mathematical Behavior , 17(1), 123-134.

Hollar, J. and Norwood, K. (1999) The effects of agraphing-approach intermediate algebracurriculum on students’ understanding of functionJournal for Research in Mathematics Education 30(2) 220-226.

Jacobs, V., Franke, M., Carpenter, T., Levi, L., Battey, D.(2007) Professional development focused onchildren’s algebraic reasoning in elementar yschool. Journal of Research in MathematicsEducation 38(3) 258-288

Janvier, C. (1996) Modeling and the initiation intoalgebra. In Bednarz, N., Kieran C., Lee, L. (eds.)Approaches to algebra: perspectives for research onteaching. pp. 225-238. Kluwer, Dordrecht



Johnson, D. (1989) (ed.) Children’s MathematicalFrameworks 8 – 13: a study of classroom teaching .Windsor: NFER-Nelson.

Kaput, J. (1998) Transforming algebra from an engineof inequity to an engine of mathematical po werby ‘algebrafying’ the K–12 cur riculum. In NationalCouncil of Teachers of Mathematics andMathematical Sciences Education Board (Eds.).Thenature and role of algebra in the K–14 cur riculum:Proceedings of a National Symposium (pp. 25–26).Washington, DC: National Research Council,National Academy Press.

Kaput, J. (1999). Teaching and learning a new algebra.In E. Fennema and T. A. Romberg (Eds.),Mathematics classrooms that promoteunderstanding (pp. 133-155). Mahwah, NJ:Erlbaum.

Kerslake, D. (1986) Fractions: Children’s strategies anderrors. Windsor: NFER-Nelson.

Kieran, C. (1981) Concepts associated with theequality symbol. Educational Studies inMathematics 12, 317-326

Kieran, C. (1983) Relationships between novices’views of algebraic letters and their use ofsymmetric and asymmetric equation-solvingprocedures. In J. Bergeron and B. Herscovics(eds.) Proceedings of the International Group for the Psychology of Mathematics Educationpp.1.161-1-168, Montreal: Universite deMontreal.

Kieran, C. (1989) The early learning of algebra: astructural perspective. In S. Wagner and C . Keiran(eds. ) Research issues in the learning and teachingof algebra. pp. 33-56, Reston, VA, NCTM.

Kieran, C. (1992). The learning and teaching ofalgebra, in D.A. Grouws (Ed.), Handbook ofResearch on Mathematics Teaching and Learning,Macmillan, New York, p390-419.

Kieran, C. and Saldanha, L. (2005) Computer algebrasystems (CAS) as a tool for coaxing theemergence of reasoning about equations ofalgebraic expressions. In Chick, H. L., and Vincent,J. L. (Eds.). (2005). Proceedings of the InternationalGroup for the Psychology of Mathematics Educationpp.3-193-3-200 Melbourne: PME

Kieran, C. and Sfard, A. (1999), Seeing throughsymbols: The case of equivalent expressions, Focuson Learning Problems in Mathematics, 21 (1) p1-17

Kieran, C., Boileau, A.and Garancon, M. (1996)Introducing algebra by means of a technology-supported, functional approach. In Bednarz, N.,Kieran C., Lee, L. (eds.) Approaches to algebra:perspectives for research on teaching . Pp. 257-294Kluwer, Dordrecht

Kirshner David (1989)The Visual Syntax of AlgebraJournal for Research in Mathematics Education20(3) 274-287

Krutetskii, V. (1976). The psychology of mathematicalabilities in school children. Chicago: University ofChicago Press.

Küchemann, D. (1981) Algebra, in K. Har t (Ed.)Children’s Understanding of Mathematics 11-16 , p102-119 Murray, London..

Lamon, S. (1998) Algebra: meaning throughmodelling, in A. Olivier and K. Newstead (Eds)22nd Conference of the Inter national Group forthe Psychology of Mathematics Education, vol. 3,pp.167-174. Stellenbosch: International Groupfor the Psychology of Mathematics Education.

Lampert, M. (1986) Knowing, doing and teachingmultiplication. Cognition and Instruction 3 ,305-342.

Lee, L. (1996) An incitation into algebraic culturethrough generalization activities. In Bednarz, N.,Kieran C., Lee, L. (eds.) Approaches to algebra:perspectives for research on teaching. pp. 87-106.Kluwer, Dordrecht

Lehrer, R., Strom, D. and Confrey, J. (2002) Groundingmetaphors and inscriptional resonance: children’semerging understanding of mathematicalsimilarity. Cognition and Instruction 20(3) 359-398.

Lima, R. and Tall, D. (2008) Procedural embodimentand magic in linear equations. Educational Studiesin Mathematics 67(1) 3-18

Linchevski, L., and Herscovics, N. (1996). Crossing thecognitive gap between arithmetic and algebra:Operating on the unknown in the context ofequations. Educational Studies in Mathematics ,30,1, 39-65.

Linchevski, L. and Sfard, A. (1991) Rules withoutreasons as processes without objects: The case ofequations and inequalities. In F. Furinghetti (ed.)Proceedings of the 15th International Group for thePsychology of Mathematics Education. 2, 317-324.

Lins, R.L. (1990). A framework of understanding whatalgebraic thinking is. In G. Booker, P.Cobb and T.Mendicuti (eds.) Proceedings of the 14thConference of the International Group for thePsychology of Mathematics Education, 14, 2, 93-101. Oaxtepec, Mexico.

MacGregor, M. and Stacey, K. (1993) CognitiveModels Underlying Students’ Formulation ofSimple Linear Equations Journal for Research inMathematics Education 24(3) 274-232

MacGregor, M. and Stacey, K. (1995), The Effect ofDifferent Approaches to Algebra on Students’Perceptions of Functional Relationships,Mathematics Education Research Journal, 7 (1)p69-85.


MacGregor M.and Stacey K. (1997) Students’understanding of algebraic notation 11-15,Educational Studies in Mathematics. 33(1), 1-19

Mason, J. and Suther land, R. (2002), Key Aspects ofTeaching Algebra in Schools, QCA, London.

Mason, J., Stephens, M. and Watson, A. (in press)Appreciating mathematical structure for all.Mathematics Education Research Journal.

Moss, J., R. Beatty, L. Macnab.(2006) Design for thedevelopment and teaching of an inte gratedpatterning curriculum. Paper present to AERA(April 2006.)

Nemirovsky, R. (1996) A functional approach toalgebra: two issues that emerge. In Bednarz, N.,Kieran C., Lee, L. (eds.) Approaches to algebra:perspectives for research on teaching . pp. 295-316.Kluwer, Dordrecht

Nickson, M. (2000) Teaching and learningmathematics: a teacher’s guide to recent researchand its applications. Cassell London

NMAP, National Mathematics Advisory Panel (2008)downloaded April 2008 from:http://www.ed.gov/about/bdscomm/list/mathpanel/index.html

Noble, T., Nemirovsky, R., Wright, T., and Tierney, C.(2001) Experiencing change: The mathematics ofchange in multiple environments. Journal forResearch in Mathematics Education, 32(1) 85-108.

Noss, R., Healy, L., Hoyles, C. (1997) The constructionof mathematical meanings: Connecting the visualwith the symbolic . Educational Studies inMathematics 33(2), 203-233

Nunes, T. and Bryant, P. (1995) Do problem situationsinfluence children’s understanding of thecommutativity of multiplication? MathematicalCognition, 1, 245-260.

O’Callaghan, B. (1998) Computer-Intensive Algebraand Students’ Conceptual Knowledge ofFunctions, Journal For Research in MathematicsEducation, 29 (1) p21-40.

Piaget, J. (1969) The mechanisms of perception.London: RKP

Piaget, J., & Moreau, A. (2001). The inversion ofarithmetic operations (R. L. Campbell, Trans.). In J.Piaget (Ed.), Studies in reflecting abstraction (pp.69–86). Hove, UK: Psychology Press.Powell, A. B.and Maher, C. A. (2003). Heuristics of TwelfthGraders’ Building Isomorphism. In Pateman, N.,Dougherty, B. and Zilliox, J. (eds.) Proceedings ofthe 27th conference of the international group forthe psychology of mathematics education 4, pp. 23-30. Honolulu, Hawaii: University of Hawaii.

Radford, L. (2007) Iconicity and contraction: asemiotic investigation of forms of algebraicgeneralizations of patterns in different contextsZDM Mathematics Education 40 83-96

Redden, T. (1994) Alternative pathways in thetransition from arithmetical thinking to algebraicthinking. In J. da Ponte and J. Matos (eds.)Proceedings of the 18th annual conference of theInternational Group for the Psychology ofMathematics Education vol. 4, 89-96, University ofLisbon, Portugal.

Reed, S. (1972) Pattern recognition andcategorization. Cognitive Psychology 3, 382-407

Reggiani, M. (1994) Generalization as a basis foralgebraic thinking: observations with 11-12 yearold pupils Proceedings of PME XVIII, IV, p97 - 104.

Resnick, L. B., Lesgold, S., and Bill, V. (1990). Fromprotoquantities to number sense. In G. Booker, J.Cobb, and T. N. de Mendicuti (Eds.), Proceedings ofthe Fourteenth Psychology of MathematicsEducation Conference (Vol. 3, pp. 305-311). MexicoCity, Mexico: International Group for thePsychology of Mathematics Education.

Rivera, F. and Becker, J. (2007) Middle school children’scognitive perceptions of constructive anddeconstructive generalizations involving linear figuralpatterns ZDM Mathematics Education 40 65-82

Robinson, K., Ninowski, J. and Gray, M. (2006)Children’s understanding of the ar ithmeticconcepts of inversion and associativity, Journal ofExperimental Child Psychology 94, pp. 349–362.

Rothwell-Hughes, E.(1979) Conceptual Powers ofChildren: an approach through mathematics andscience. Schools Council Publications.

Rowland, T. and Bills, L. (1996) ‘Examples,Generalisation and Proof ’ Proceedings of BritishSociety for Research in Learning MathematicsLondon Institute of Education, pp. 1-7

Ryan, J. and Williams, J. (2007) Children’s Mathematics4 -15: learning from errors and misconceptionsMaidenhead Open University Press.

Sáenz-Ludlow, A.and Walgamuth, C. (1998) ThirdGraders’ Interpretations of Equality and the EqualSymbol Educational Studies in Mathematics , 35(2)153-187.

Schwartz, J. and Yerushalmy, M. (1992) Gettingstudents to function in and with algebr a. In G.Harel and E. Dubinsky (eds.) The concept offunction: Aspects of epistemology and pedagogy.MAA Notes, 25, 261-289. Washington DC: MAA.

Sfard, A. and Linchevski, L. (1994) . The gains and thepitfalls of reification: The case of algebra.Educational Studies in Mathematics , 26, 191-228.



Siegler, R. S., and Stern, E. (1998). Conscious andunconscious strategy discoveries: a microgeneticanalysis. Journal of Experimental Psychology 127,377-397.

Simmt, E. and Kieren, T. (1999). Expanding thecognitive domain: the role(s) and consequence ofinteraction in mathematics knowing. In F. Hitt andM. Santos (Eds.) Proceedings of the Twenty FirstAnnual Meeting Psychology of MathematicsEducation North American Chapter, pp.299-305.

Stacey, K. (1989) Finding and Using Patterns in LinearGeneralising Problems, Educational Studies inMathematics, 20, p147-164.

Stacey, K., and MacGregor, M. (2000). Learning thealgebraic method of solving problems. Journal ofMathematical Behavior, 18(2), 149-167.

Steele, D. (2007) Seventh-grade students’representations for pictorial growth and changeproblems 97-110 ZDM Mathematics Education40 97-110

Sutherland R. and Rojano T. (1993) A SpreadsheetAlgebra Approach to Solving Algebra Problems,Journal of Mathematical Behavior, 12, 353-383.

Thomas, M. and Tall, D. (2001) The long-termcognitive development of symbolic algebra. InChick, H., Stacey, K., Vincent, J. and Vincent, J. (eds.)Proceedings of the 12th ICMI study conference: Thefuture of the teaching and lear ning of algebrapp.590-597. University of Melbourne, Australia,Dec 9-14, 2001.

Tsamir, P., and Almog, N. (2001). Students’ strategiesand difficulties: The case of algebraic inequalities.International Journal of Mathematics Education inScience and Technology, 32, 513-524.

Tsamir, P., and Bazzini, L. (2001). Can x=3 be thesolution of an inequality? A study of Italian anIsraeli students. In M. van den Heuvel-Panhuizen(Ed.), Proceedings of the 25th Annual Meeting forthe Psychology of Mathematics Education,Utrecht: Holland. (Vol IV, pp. 303-310).

van Ameron. B. (2003) Focusing on informalstrategies when linking arithmetic to ear ly algebra.Educational Studies in Mathematics, 2003, vol. 54,no. 1, p. 63-75..

Vergnaud, G. (1997) The nature of mathematicalconcepts. In T. Nunes and P. Bryant (eds.) Learningand teaching mathematics: an internationalperspective. pp. 5-28 Hove: Psychology Press

Vergnaud, G. (1998). A comprehensive theory ofrepresentation for mathematics Journal ofMathematical Behavior, 17(2), 167-181

Vlassis, J., (2002) The balance model: Hindrance orsupport for the solving of linear equations withone unknown. Educational Studies in Mathematics49(3), 341-359.

Warren, E. and Cooper, T. (2008) Patterns thatsupport early algebraic thinking in the elementaryschool. In.C. Greenes and R. Rubenstein (eds.)Algebra and Algebraic Thinking in SchoolMathematics. 70th Yearbook. pp.113-126,Reston,VA: NCTM.

Wertheimer, M. (1960) Productive thinking. New York:Harper

Wong, M. (1997) Numbers versus letters in algebraicmanipulation: which is more difficult? InE.Pehkonen (ed.) Proceedings of the annualconference of the Internatioanl Group for thePsychology of Mathematics Education. (4) 285-290.Lahti: University of Helsinki.

Yeap, B-H. and Kaur, B. (2007) Elementar y schoolstudents engaging in making gener alisation:aglimpse from a Singapore classroom ZDMMathematics Education 40, 55-64

Yerushalmy, M. (1997) Designing representations:Reasoning about functions of two variables.Journal for Research in Mathematics Education,28 (4) 431-466.




ISBN 978-0-904956-73-3





7

Paper 7: Modelling, problem-solving and integrating conceptsBy Anne Watson, University of Oxford

2 SUMMARY – PAPER 2: Understanding whole numbers2 Paper 7: Modelling, problem-solving and integrating concepts













Modelling, problem-solving andintegrating concepts 8

References 35

About the authorAnne Watson is Professor of MathematicsEducation at the University of Oxford.


About this review


Headlines

We have assumed a general educational contextwhich encourages thinking and problem-solvingacross subjects. A key difference about mathematicsis that empirical approaches may solve individualproblems, and offer directions for reasoning, but donot themselves lead to new mathematical knowledgeor mathematical reasoning, or to the power thatcomes from applying an abstract idea to a situation.

In secondary mathematics, the major issue is nothow children learn elementary concepts, but whatexperiences they have had and how these enable orlimit what else can be lear nt.That is why we havecombined several aspects of secondar y mathematicswhich could be exemplified by par ticular topics.

• Students have to be fluent in understanding methodsand confident about using them to know why andwhen to apply them, but such application does notautomatically follow learning procedures. Studentshave to understand the situation as well as being ableto call on a familiar repertoire of ideas and methods.

• Students have to know some elementary conceptswell enough to apply them and combine them toform new concepts in secondary mathematics, butlittle is known from research about what conceptsare essential in this way. Knowledge of a range offunctions is necessary for modelling situations.

• Students have to learn when and how to useinformal, experiential reasoning and when to useformal, conventional, mathematical reasoning.Without special attention to meanings manystudents tend to apply visual reasoning, or betriggered by verbal cues, rather than to analysesituations mathematically.

• In many mathematical situations in secondarymathematics, students have to look for relationsbetween numbers and variables and relationsbetween relations and properties of objects, andknow how to represent them.

How secondary learners tacklenew situations

In new situations students f irst respond to familiar ityin appearance, or language, or context. They bringearlier understandings to bear on new situations,sometimes erroneously. They naturally generalise fromwhat they are offered, and they often over-generaliseand apply inappropriate ideas to new situations. Theycan learn new mathematical concepts either asextensions or integrations of ear lier concepts, and/oras inductive generalisations from examples, and/or asabstractions from solutions to problems.

Routine or context?

One question is whether mathematics is lear ntbetter from routines, or from complex contextualsituations. Analysis of research which compares howchildren learn mathematics through being taughtroutines efficiently (such as with computer ised andother learning packages designed to minimisecognitive load) to learning through problem-solvingin complex situations (such as through RealisticMathematics Education) shows that the significantdifference is not about the speed and retention oflearning but what is being learnt. In each approachthe main question for progression is whether thestudent learns new concepts well enough to use

Summary of paper 7:Modelling, problem-solving and integratingconcepts


and adapt them in future lear ning and outsidemathematics. Both approaches have inherentweaknesses in this respect. These weaknesses willbecome clear in what follows. However, there areseveral studies which show that those who developmathematical methods of enquir y over time canthen learn procedures easily and do as well, orbetter, in general tests.

Problem-solving and modelling

To learn mathematics one has to lear n to solvemathematical problems or model situationsmathematically. Studies of students’ problem solvingmainly focus in the successful solution ofcontextually-worded problems using mathematicalmethods, rather than using problem solving as acontext for learning new concepts and developingmathematical thinking. To solve unfamiliar problems in mathematics, a meta-analysis of 487 studiesconcluded that for students to be maximallysuccessful:• problems need to be fully stated with supportive

diagrams• students need to have previous extensive

experience in using the representations used• they have to have relevant basic mathematical skills

to use• teachers have to understand problem-solving

methods.

This implies that fluency with representations andskills is impor tant, but also depends on how clearlythe problem is stated. In some studies the difficulty isalso to do with the under lying concept, for example,in APU tests area problems were difficult with orwithout diagrams.

To be able to solve problems whose wording doesnot indicate what to do, students have to be ableto read the problem in two ways: firstly, theirtechnical reading skills and under standing ofnotation have to be good enough; secondly, theyhave to be able to interpret it to under stand thecontextual and mathematical meanings. They haveto decide whether and how to br ing informalknowledge to bear on the situation, or, if theyapproach it formally, what are the variables andhow do they relate . If they are approaching itformally, they then have to represent therelationships in some way and decide how tooperate on them.

International research into the use of ICT to pro videnew ways to represent situations and to seerelationships, such as by comparing spreadsheets,graph plotters and dynamic images appear to speedup the process of relating representations throughisomorphic reasoning about covariation, and hencethe development of understanding aboutmathematical structures and relations.

Application of earlier learning

Knowing methodsStudents who have only routine knowledge may notrecognize that it is relevant to the situation. Or theycan react to verbal or visual cues without referenceto context, such as ‘how much?’ triggeringmultiplication rather than division, and ‘how many?’always triggering addition. A fur ther problem is thatthey may not understand the under lying relationshipsthey are using and how these relate to each other.For example, a routine approach to 2 x 1/3 x 3/2may neither exploit the meaning of fractions nor themultiplicative relation.

Students who have only experience of applyinggeneric problem-solving skills in a r ange ofsituations sometimes do not recognize underlyingmathematical structures to which they can appl ymethods used in the past. Indeed, given the well-documented tendency for people to use ad hocarithmetical trial-and-adjustment methods whereverthese will lead to reasonab le results, it is possiblethat problem-solving experience may not result inlearning new mathematical concepts or workingwith mathematical structures, or in becoming fluentwith efficient methods.

Knowing conceptsStudents who have been helped to lear n concepts,and can define, recognise and exemplify elementaryideas are better able to use and combine theseideas in new situations and while learning newconcepts. However, many difficulties appear to bedue to having too limited a r ange of understanding.Their understanding may be based on exampleswhich have irrelevant features in common, such asthe parallel sides of parallelograms always beingparallel to the edges of a page . Understanding isalso limited by examples being similar to aprototype, rather than extreme cases. Anotherproblem is that students may recognize examples ofa concept by focusing too much on visual or verbal

4 Paper 7: Modelling, problem-solving and integrating concepts

aspects, rather than their proper ties, such asbelieving that it is possib le to construct anequilateral triangle on a nine-pin geoboard because it ‘looks like one’.

Robust connections between and within ear lier ideas can make it easier to engage with new ideas,but can also hinder if the ear lier ideas are limited andinflexible. For example, learning trigonometry involvesunderstanding: the definition of triangle; right-angles;recognizing them in different orientations; what anglemeans and how it is measured; typical units formeasuring lines; what ratio means; similarity oftriangles; how ratio is written as a fraction; how tomanipulate a multiplicative relationship; what ‘sin’(etc.) means as a symbolic representation of afunction and so on. Thus knowing about ratios can support learning trigonometry, but if theunderstanding of ‘ratio’ is limited to mixing cakerecipes it won’t help much. To be successful studentshave to have had enough exper ience to be fluent,and enough knowledge to use methods wisely.

They become better at problem-solving andmodelling when they can:• draw on knowledge of the contextual situation to

identify variables and relationships and/or, throughimagery, construct mathematical representationswhich can be manipulated further

• draw on a repertoire of representations, functions,and methods of operation on these

• have a purpose for the modelling process, so thatthe relationship between manipulations in themodel and changes in the situation can bemeaningfully understood and checked forreasonableness.

Knowing how to approachmathematical tasks

To be able to decide when and how to use informalor formal approaches, and how to use pr iorknowledge, students need to be able to thinkmathematically about all situations in mathematicslessons. This develops best as an all-encompassingperspective in mathematics lessons, rather thanthrough isolated experiences.

Students have to:• learn to avoid instant reactions based in superficial

visual or verbal similarity• practice using typical methods of mathematical

enquiry explicitly over time

• have experience of mulling problems over time inorder to gain insight.

With suitable environments, tools, images andencouragement, learners can and do use theirgeneral perceptual, comparative and reasoningpowers in mathematics lessons to:• generalise from what is offered and experienced• look for analogies• identify variables• choose the most efficient variables, those with

most connections• see simultaneous variations• understand change• reason verbally before symbolising• develop mental models and other imagery• use past experience of successful and unsuccessful

attempts• accumulate knowledge of operations and situations

to do all the above successfully

Of course, all the tendencies just descr ibed can alsogo in unhelpful directions and in par ticular peopletend to: • persist in using past methods and applying

procedures without meaning, if that has been theirprevious mathematical experience

• get locked into the specific situation and do not, bythemselves, know what new mathematical ideascan be abstracted from these experiences

• be unable to interpret symbols, text, and otherrepresentations in ways the teacher expects

• use additive methods; assume that if one variableincreases so will another; assume that all change islinear ; confuse quantities.


6 SUMMARY – PAPER 2: Understanding whole numbers6 Paper 7: Modelling, problem-solving and integrating concepts

RECOMMENDATIONS


Learning routine methods and lear ningthrough complex exploration lead todifferent kinds of knowledge and cannotbe directly compared; neither methodnecessarily enables learning new conceptsor application of powerful mathematicsideas. However, those who have the habitof complex exploration are often able tolearn procedures quickly.

Students naturally respond to familiaraspects of mathematics; try to apply priorknowledge and methods, and generalizefrom their experience.

Students are more successful if they have a fluent reper toire of conceptualknowledge and methods, includingrepresentations, on which to dr aw.

Multiple experiences over time enablestudents to develop new ways to work onmathematical tasks, and to develop theability to choose what and how to applyearlier learning.

Students who work in computer-supported multiple representationalcontexts over time can understand anduse graphs, variables, functions and themodelling process. Students who canchoose to use available technology arebetter at problem solving, and havecomplex understanding of relations, andhave more positive views of mathematics.

Recommendations for teaching

Developers of the cur riculum, advisory schemes of work andteaching methods need to be aware of the impor tance ofunderstanding new concepts, and avoid teaching solely to passtest questions, or using solely problem-solving mathematicalactivities which do not lead to new abstr act understandings.

Students should be helped to balance the need f or fluencywith the need to work with meaning.

Teaching should take into account students’ natural ways ofdealing with new perceptual and verbal information, and thelikely misapplications. Schemes of work and assessment shouldallow enough time for students to adapt to new meanings andmove on from ear lier methods and conceptualizations.

Developers of the cur riculum, advisory schemes of work andteaching methods should give time for new experiences andmathematical ways of working to become familiar in severalrepresentations and contexts before moving on.

Students need time and multiple experiences to develop arepertoire of appropriate functions, operations,representations and mathematical methods in order to solv eproblems and model situations.

Teaching should ensure conceptual under standing as well as‘knowing about’, ‘knowing how to’, and ‘knowing how to use’.

Schemes of work should allow for students to have multipleexperiences, with multiple representations over time todevelop mathematically appropriate ‘habits of mind’.

There are resource implications about the use of ICT. Studentsneed to be in control of switching betw een representations andcomparisons of symbolic expression in order to under stand thesyntax and the concept of functions. The United Kingdom maybe lagging behind the developed world in exploring the use ofspreadsheets, graphing tools, and other software to suppor tapplication and authentic use of mathematics.

Recommendations for research

Application of research findings about problem-solving, modelling and conceptual lear ning to currentcurriculum developments in the United Kingdomsuggests that there may be different outcomes interms of students’ ability to solve quantitative andspatial problems in realistic contexts. However, thereis no evidence to convince us that the new NationalCurriculum in England will lead to better conceptualunderstanding of mathematics, either at theelementary levels, which are necessar y to learnhigher mathematics, or at higher levels which providethe confidence and foundation for fur thermathematical study. Where contextual andexploratory mathematics, integrated through thecurriculum, do lead to fur ther conceptual learning itis related to conceptual lear ning being a r igorousfocus for curriculum and textbook design, and inteacher preparation, such as in China, Japan,Singapore, and the Nether lands, or in specificallydesigned projects based around such aims.

In the main body of Paper 7: Modelling, solvingproblems and learning new concepts in secondar ymathematics there are several questions for futureresearch, including the following.• What are the key conceptual understandings for

success in secondary mathematics, from the pointof view of learning?

• How do students learn new ideas in mathematicsat secondary level that depend on combinations ofearlier concepts?

• What evidence is there of the characteristics ofmathematics teaching at higher secondary levelwhich contribute both to successful conceptuallearning and application of mathematics?



Introduction

By the time students enter secondar y school, theypossess not only intuitive knowledge from outsidemathematics and outside school, but also a range ofquasi-intuitive understandings within mathematics,derived from ear lier teaching and from theirmemory of generalisations, metaphors, images,metonymic associations and strategies that haveserved them well in the past. Many of these typicalunderstandings are described in the previouschapters. Tall and Vinner (1981) called theseunderstandings ‘concept images’, which are a r agbagof personal conceptual, quasi-conceptual, perceptualand other associations that relate to the language ofthe concept and are loosely connected by thelanguage and observable artefacts associated withthe concept. Faced with new situations, students willapply whatever familiar methods and associationscome to mind relatively quickly – perhaps notrealising that this can be a r isky strategy. If ‘doingwhat I think I know how to do’ leads so easily toincorrect mathematics it is hardly surprising thatmany students end up seeing school mathematics as the acquisition and application of methods, and asite of failure, rather than as the development of arepertoire of adaptable intellectual tools.

At secondary level, new mathematical situations areusually ideas which ar ise through mathematics andcan then be applied to other areas of activity; it isless likely that mathematics involves the formalisationof ideas which have arisen from outside exper ienceas is common in the primary phase. Because of thisdifference, learning mathematics at secondar y levelcannot be understood only in terms of overallcognitive development. For this review, wedeveloped a perspective which encompasses both

the ‘pure’ and ‘applied’ aspects of learning atsecondary level, and use research from bothtraditions to devise some common implications and overall recommendations for practice.

Characteristics of learningsecondary mathematics

We justify the broad scope of this chapter byindicating similarities between the learning of the newconcepts of secondary mathematics and learning howto apply mathematics to analyse, express and solveproblems in mathematical and non-mathematicalcontexts. Both of these aspects of lear ningmathematics depend on interpreting new situationsand bringing to mind a reper toire of mathematicalconcepts that are understood and fluent to someextent. In this review we will show that learningsecondary mathematics presents core commondifficulties, whatever the curriculum approach beingtaken, which need to be addressed through pedagogy.

In all teaching methods, when presented with a newstimulus such as a symbolic expression on the board,a physical situation, or a statement of a complex ill-defined ‘real life’ problem, the response of anengaged learner is to wonder:

What is this? This entails ‘reading’ situations, usually reading mathematical representations orwords, and interpreting these in conventionalmathematical ways. It involves perception, attention,understanding representations and being able todecipher symbol systems.


Modelling, problem-solving and integratingconcepts

What is going on here? This entails identifyingsalient features including non-visual aspects,identifying variables, relating par ts to each other,exploring what changes can be made and the eff ectsof change, representing situations in mathematicalways, anticipating what might be the pur pose of amathematical object. It involves attention, visualisation,modelling, static and dynamic representations,understanding functional, statistical and geometr icalrelationships, focusing on what is mathematicall ysalient and imagining the situation or arepresentation of it.

What do I know about this? This entails recognisingsimilarities, seeking for recognisable structuresbeyond visual impact, identifying variables, proposingsuitable functions, drawing on reper toire of pastexperiences and choosing what is lik ely to be useful.Research about memory, problem-solving, conceptimages, modelling, functions, analysis and analogicalreasoning is likely to be helpful.

What can I do? This entails using past exper ience totry different approaches, heuristics, logic, controllingvariables, switching between representations,transforming objects, applying manipulations andother techniques. It involves analogical reasoning,problem-solving, tool-use, reasoning, generalisation andabstraction, and so on.

Thus, students being presented with the task ofunderstanding new ideas draw on past exper ience, ifthey engage with the task at all, just as they would ifoffered an unfamiliar situation and asked to expressit mathematically. They may only get as far as the f irststep of ‘reading’ the stimulus. The alternative is towait to be told what to do and treat ev erything asdeclarative, verbatim, knowledge. A full review ofrelevant research in all these areas is bey ond thescope of this paper, and much of it is gener ic ratherthan concerned with mathematics.

We organise this Paper into three par ts: Part 1 looks at what learners have to be able to do to be successful in these aspects of secondar ymathematics; Part 2 considers what learners actuallydo when faced with new complex mathematicalsituations; and Part 3 reviews what happens withpedagogic intervention designed to address typicaldifficulties. We end with recommendations for futureresearch, curriculum development and practice.

Part 1: What learners have to be able to do in secondarymathematicsIn this chapter we describe what learners have to be able to do in order to lear n new concepts, solveproblems, model mathematical situations, and engagein mathematical thinking.

Learning ‘new’ concepts

Extension of meaningThroughout school, students meet familiar ideas usedin new contexts which include but extend their olduse, often through integrating simpler concepts intomore complex ideas. Sfard (1991) descr ibes thisprocess of development of meaning as consisting of‘interiorization’ through acting on a new idea withsome processes so that it becomes familiar andmeaningful; understanding and expressing theseprocesses and their effects as manageable units(condensation), and then this new str ucturebecomes a thing in itself (reified) that can be acted on as a unit in future .

In this way, in algebra, letters standing for numbersbecome incorporated into terms and expressionswhich are number-like in some uses and yet cannotbe calculated. Operations are combined to descr ibestructures, and expressions of str uctures becomeobjects which can be equated to each other. Variablescan be related to each other in ways that representrelationships as functions, rules for mapping onevariable domain to another (see the ear lier chapterson functional relations and algebr aic thinking).

Further, number develops from counting, wholenumbers, and measures to include negativ es,rationals, numbers of the form a + b √n where aand b are rationals, irrationals and transcendentals,expressions, polynomials and functions which arenumber-like when used in expressions, and the usesof estimation which contradict earlier shifts towardsaccuracy. Eventually, two-dimensional complexnumbers may also have to be under stood. Possiblediscontinuities of meaning can ar ise between discreteand continuous quantities, monomials andpolynomials, measuring and two-dimensionality, anddifferent representations (digits, letters, expressionsand functions).

Graphs are used first to compare values of variousdiscrete categories, then are used to express two-



dimensional discrete and continuous data as inscatter-graphs, or algebraic relationships betweencontinuous variables, and later such relationships,especially linear ones, might be fitted to statisticalrepresentations.

Shapes which were familiar in pr imary school have to be defined and classified in new ways, and newproperties explored, new geometric configurationsbecome important and descriptive reasoning basedon characteristics has to give way to logicaldeductive reasoning based on relational proper ties.Finally, all this has to be applied in the three-dimensional contexts of everyday life.

The processes of learning are sometimes said tofollow historical development, but a better analogywould be to compare lear ning trajectories with theconceptual connections, inclusions and distinctions ofmathematics itself.

Integration of conceptsAs well as this kind of extension, there are new ‘topics’that draw together a range of ear lier mathematics.Typical examples of secondar y topics are quadraticfunctions and trigonometry. Understanding each ofthese depends to some extent on under standing arange of concepts met ear lier.

Quadratic functions: Learning about quadraticfunctions includes understanding:• the meaning of letters and algebraic syntax; • when letters are variables and when they can be

treated as unknown numbers; • algebraic terms and expressions; • squaring and square rooting; • the conventions of coordinates and graphing

functions; • the meaning of graphs as representing sets of

points that follow an algebraic rule; • the meaning of ‘=’; • translation of curves and the ways in which they

can change shape; • that for a product to equal zero at least one of its

terms must equal zero and so on.

Trigonometry: Learning this includes knowing:• the definition of triangle; • about right-angles including recognizing them in

different orientations; • what angle means and how it is measured; • typical units for measuring lines; • what ratio means; • similarity of triangles;

• how ratio is written as a fraction; • how to manipulate a multiplicative relationship; • what ‘sin’ (etc.) means as a symbolic representation

of a function and so on.

New concepts therefore develop both throughextension of meaning and combination of concepts.In each of these the knowledge learners bring to the new topic has to be adaptab le and usable, not so strongly attached to previous contexts in which it has been used that it cannot be adapted. Ahierarchical ‘top down’ view of learning mathematicswould lead to thinking that all contr ibutory conceptsneed to be fully understood before tackling newtopics (this is the view taken in the NMAP review(2008) but is unsuppor ted by research as far as wecan tell from their document). By contrast, if we takelearners’ developing cognition into account we seethat ‘full understanding’ is too vague an aim; it is theprocesses of applying and extending prior knowledgein the context of working on new ideas thatcontribute to understanding.

Whichever view is taken, learners have to bringexisting understanding to bear on new mathematicalcontexts. There are conflicting research conclusionsabout the process of bringing existing ideas to bearon new stimuli: Halford (e.g. 1999) talks of conceptualchunking to describe how earlier ideas can be dr awnon as packages, reducing to simpler objects ideaswhich are initially formed from more complex ideas,to develop further concepts and argues for suchchunking to be robust before moving on. He focusesparticularly on class inclusion (see Paper 5,Understanding space and its representation inmathematics) and transitivity, structures of relationsbetween more than two objects, as ideas which arehard to deal with because they in volve several levelsof complexity. Examples of this diff iculty werementioned in Paper 4, Understanding relations andtheir graphical representation, showing how relationsbetween relations cause problems. Chunking includesloss of access to lower level meanings, which may beuseful in avoiding unnecessary detail of specificexamples, but can obstruct meaningful use.Freudenthal (1991, p. 469) points out that automaticconnections and actions can mask sources of insight,flexibility and creativity which arise from meanings. Heobserved that when students are in the flo w ofcalculation they are not necessarily aware of whatthey are doing, and do not monitor their work. It is also the case that m uchof the chunking that has taken place in ear liermathematics is limited and hinders and obstructs


future learning, leading to confusion withcontradictory experiences. For example, theexpectation that multiplication will make ‘things’ biggercan hinder learning that it only means this sometimes– multiplication scales quantities in a var iety of ways.The difficulties faced by students whose understandingof the simpler concepts learnt in primary school islater needed for secondary mathematical ideas, aretheorised by, among others, Trzcieniecka-Schneider(1993) who points out that entrenched and limitedconceptual ideas (including what Fischbein calls‘intuitions’ (1987) and what Tall calls ‘metbefores’(2004)) can hinder a student’s approach to unfamiliarexamples and questions and create resistance , ratherthan a willingness to engage with new ideas whichdepend on adapting or giving up strongl y-held notions.This leads not only to problems understanding newconcepts which depend on ear lier concepts, but alsomakes it hard for learners to see how to applymathematics in unfamiliar situations. On the otherhand, it is impor tant that some knowledge is fluentand easily accessible, such as number bonds,recognition of multiples, equivalent algebraic forms, the shape of graphs of common functions and so on. Learners have to know when to apply ‘old’understandings to be extended, and when to givethem up for new and different understandings.

Inductive generalisationLearners can also approach new ideas b y inductivegeneralisation from several examples. English andHalford (1995 p. 50) see this inductive process1 asthe development of a mental model which fits theavailable data (the range of examples and instanceslearners have experienced) and from whichprocedures and conjectures can be generated. Forexample, learners’ understanding about what a lineargraph can look like is at first a generalisation of thelinear graphs they have seen that have been namedas such. Similarly, learners’ conjectures about therelationship between the height and volume of waterin a bottle, given as a data set, depends on reasoningboth from the data and from general knowledge ofsuch changes. Leading mathematicians often remarkthat mathematical generalisation also commonlyarises from abductive reasoning on one genericexample, such as conjectures about relationshipsbased on static geometrical diagrams. For both theseprocesses, the examples available as data, instances,and illustrations from teachers, textbooks and othersources play a crucial role in the process. Learnershave to know what features are salient and generalisefrom them. Often such reasoning depends onmetonymic association (Holyoak and Thagard, 1995),

so that choices are based on visual, linguistic and cueswhich might be misleading (see also earlier chapteron number) rather than mathematical meaning. Asexamples: the prototypical parallelogram has itsparallel edges horizontal to the page; and x2 and 2xare confused because it is so common to use ‘x = 2’as an example to demonstr ate algebraic meaning.

Abstraction of relationshipsA fur ther way to meet new concepts is through aprocess of ‘vertical mathematisation’ (Treffers, 1987)in which experience of solving complex problemscan be followed by extracting general mathematicalrelationships. It is unlikely that this happens natur allyfor any but a few students, yet school mathematicsoften entails this kind of abstr action. The FreudenthalInstitute has developed this approach throughteaching experiments and national roll-out over aconsiderable time, and its Realistic MathematicsEducation (e.g. Gravemeijer and Doorman, 1999)sees mathematical development as • seeing what has to be done to solve the kinds

of problems that involve mathematics• from the solutions extracting new mathematical

ideas and methods to add to the repertoire• these methods now become available for future

use in similar and new situations (as with Piaget’snotion of reflective abstraction and Polya’s ‘looking back’).

Gravemeijer and Doorman show that this approach,which was developed for primary mathematics, isalso applicable to higher mathematics, in this casecalculus. They refer to ‘the role models can play in ashift from a model of situated activity to a model f ormathematical reasoning. In light of this model-of/model-for shift, it is argued that discrete functionsand their graphs play a key role as an intermediarybetween the context problems that have to besolved and the formal calculus that is developed.’

Gravemeijer and Doorman’s observation explainswhy, in this paper, we are treating the lear ning ofnew abstract concepts as related to the use ofproblem-solving and modelling as forms ofmathematical activity. In all of these examples of newlearning, the fundamental shift lear ners are expectedto make, through instruction, is from informal,experiential, engagement using their existingknowledge to formal, conventional, mathematicalunderstanding. This shift appears to have threecomponents: construction of meaning; recognition innew contexts; playing with new ideas to build fur therideas (Hershkowitz, Schwar tz and Dreyfus, 2001).



It would be wrong to claim, however, that learningcan only take place through this route, becausethere is considerable evidence that learners canacquire routine skills through progr ammes ofcarefully constructed, graded, tasks designed to dealeducatively with both right answers and commonerrors of reasoning, giving immediate feedback(Anderson , Corbett, Koedinger and Pelletier, 1995).The acquisition of routine skills without explicitwork on their meaning is not the f ocus of thispaper, but the automatisation of routines so thatlearners can focus on structure and meaning byreflection later on has been a successful route forsome in mathematics.

There is recent evidence from controlled tr ialsthat learning routines from abstract presentationsis a more eff icient way to learn about under lyingmathematical structure than from contextual,concrete and stor y-based learning tasks (Kaminski,Sloutsky and Heckler, 2008). There are severalproblems with their f indings, for example in onestudy the sample consisted of undergraduates forwhom the under lying arithmetical concept beingtaught would not have been new, even if it hadnever been explicitly formalised for them before.In a similar study with 11-year-olds, additionmodulo 3 was being taught. For one group amodel of filling jugs with three equal doses wasused; for the other group abstr act symbols wereused. The test task consisted of spuriouscombinatorics involving three unrelated objects2.Those who had been taught using abstractunrelated symbols did better, those who had beentaught using jug-filling did not so well. While thesestudies suggest that abstr act knowledge aboutstructures is not less applicab le than exper ienceand ad hoc knowledge, they also illuminate theinterpretation difficulties that students have inlearning how to model phenomenamathematically, and how familiar meanings (e.g. about jug-filling) dominate over abstractengagement. What Kaminski’s results say toeducationists is not ‘abstract rules are better’ but‘be clear about the lear ning outcomes you arehoping to achieve and do not expect easy transfer between abstract procedures andmeaningful contexts’.

Summary• Learners have to understand new concepts as

extensions or integrations of earlier concepts, asinductive generalisations from examples, and asabstractions from solutions to problems.

• Robust chunking of earlier ideas can make it easierto engage with new ideas, but can also hinder if theearlier ideas are limited and inflexible.

• Routine skills can be adopted through practise to fluency, but this does not lead to conceptualunderstanding, or ability to adapt to unfamiliarsituations, for many students.

• Learners have to know when and how to bringearlier understandings to bear on new situations.

• Learners have to know how and when to shiftbetween informal, experiential activity to formal,conventional, mathematical activity.

• There is no ‘best way’ to teach mathematicalstructure: it depends whether the aim is to becomefluent and apply methods in new contexts, or tolearn how to express structures of given situations.

Problem-solving

The phrase ‘problem-solving’ has many meanings and the research literature often fails to makedistinctions. In much research solving word problemsis seen as an end in itself and it is not clear whetherthe problem introduces a mathematical idea,formalises an informal idea, or is about translation ofwords into mathematical instructions. There areseveral interpretations and the ways students learn,and can learn, differ accordingly. The following are themain uses of the phrase in the literature.

1 Word problems with arithmetical steps used tointroduce elementary concepts by harnessinginformal knowledge, or as situations in whichlearners have to apply their knowledge ofoperations and order (see Paper 4, Understandingrelations and their graphical representation).

These situations may be modelled with concretematerials, diagrams or mental images, or might drawon experiences outside school. The purpose maybe either to learn concepts through familiarsituations, or to learn to apply formal or informalmathematical methods. For example, upper primary


students studied by Squire, Davies and Bryant(2004) were found to handle commutativity muchbetter than distributivity, which they could only do if there were contextual cues to help them. Forteaching purposes this indicates that distr ibutivesituations are harder to recognise and handle, and amathematical analysis of distributivity supports thisbecause it entails encapsulation of one operationbefore applying the second and recognition of theimportance of order of oper ations.

2 Worded contexts which require the learner to decide touse standard techniques, such as calculating area, time,and so on. Diagrams, standard equations and gr aphsmight offer a bridge towards deciding what to do.For instance, consider this word problem: ‘The areaof a triangular lawn is 20 square metres, and oneside is 5 metres long. If I walk in a str aight line fromthe vertex opposite this side , towards this side, tomeet it at r ight angles, how far have I walked?’ Thestudent has to think of how area is calculated,recognise that she has been given a ‘base’ length andasked about ‘height’, and a diagram or mental imagewould help her to ‘see’ this. If a diagram is givensome of these decisions do not ha ve to be made,but recognition of the ‘base’ and ‘height’ (notnecessarily named as such) and knowledge of areaare still crucial.

In these first two types of problem, Vergnaud’sclassification of three types of multiplicative problems(see Paper 4, Understanding relations and theirgraphical representation) can be of some help if theyare straightforwardly multiplicative. But the secondtype often calls for application of a standard formulawhich requires factual knowledge about the situation,and understanding the derivation of formulae so thattheir components can be recognised.

3 Worded contexts in which there is no standardrelationship to apply, or algor ithm to use, but ananswer is expected. Typically these require settingup an equation or formula which can then beapplied and calculated. This depends onunderstanding the variables and relationships; thesemight be found using knowledge of the situation,knowledge of the meaning of oper ations, mentalor graphical imagery. For example, consider thequestion, ‘One side of a rectangle is reduced inlength by 20%, the other side in increased b y 20%;what change takes place in the area?’ The studentis not told exactly what to do, and has to developa spatial, algebraic or numerical model of thesituation in order to proceed. She might decide

that this is about representing the changed lengthsin terms of the old lengths, and that these lengthshave to be multiplied to understand what happensto the area. She might ascr ibe some arbitrarynumbers to help her do this, or some letters, orshe might realise that these are not really relevant– but this realisation is quite sophisticated.Alternatively she might decide that this is anempirical problem and generate several numericalexamples, then using inductive reasoning to give ageneral answer.

4 Exploratory situations in which there is an ill-definedproblem, and the learner has to mathematise byidentifying variables and conjecturing relationships,choosing likely representations and techniques.Knowledge of a range of possible functions may be helpful, as is mental or gr aphical imagery.

In these situations the problem might have beenposed as either quasi-abstract or situated. There maybe no solution, for example: ‘Describe the advantagesand disadvantages of raising the price of cheese rollsat the school tuck shop by 5p, given that cheeseprices have gone down by 5% but rolls have goneup by 6p each’. Students may even have posed theentire situation themselves. They have to treat this asa real situation, a real problem for them, and mightuse statistical, algebraic, logical or ad hoc methods.

5 Mathematical problems in which a situation ispresented and a question posed f or which there isno obvious method. This is what a mathematicianmeans by ‘problem’ and the expected line of attackis to use the forms of enquir y and mathematicalthinking specific to mathematics. For example:‘What happens to the relationship between thesum of squares of the two shor ter sides oftriangles and the square on the longer side if w eallow the angle between them to var y?’ We leavethese kinds of question for the later section onmathematical thinking.

Learning about students’ solution methods forelementary word problems has been a major focus in research on learning mathematics. This researchfocuses on two stages: translation into mathematicalrelations, and solution methods3. A synthesis can befound in Paper 4, Understanding relations and theirgraphical representation. It is not always clear in theresearch whether the aim is to solv e the originalproblem, to become better at mathematisingsituations, or to demonstrate that the student can usealgebra fluently or knows how to apply arithmetic.



Students have to understand that there will be severallayers to working with worded problems and cannotexpect to merely read and know immediately what to do. Problems in which linguistic str ucture matchesmathematical structure are easier because they onl yrequire fluent replacement of words and numbers byalgebra. For example, analysis according to cognitiveload theory informs us that problems with fewerwords, requiring fewer operations, and where thelinguistic structure matches the mathematical structureclosely, are easier for learners to solve algebraically(Kintsch, 1986), but this is tautologous as suchproblems are necessarily easier since they avoid theneed for interpretation and translation. Suchinterpretation may or may not be related tomathematical understanding. This research does,however, alert us to the need for students to learnhow to tackle problems which do not tr anslate easily– simply knowing what to do with the algebr aicrepresentation is not enough. In Paper 4,Understanding relations and their gr aphicalrepresentation, evidence is given that rephrasing thewords to make meaning more clear might hinderlearning to transform the mathematicalrelationships in problems.

Students might star t by looking at the numbersinvolved, thinking about what the var iables are andhow they relate, or by thinking of the situation andwhat they expect to happen in it. Whether thechoice of approach is appropr iate depends oncurriculum aims, and this obser vation will crop upagain and again in this chapter. It is illustrated in theassumptions behind the work of Bassler, Beers andRichardson (1975). They compared two approachesto teaching 15-year-olds how to solve verbalproblems, one more conducive to constructingequations and the other more conducive to graspingthe nature of the problem. Of course, differentemphases in teaching led to diff erent outcomes in theways students approached word problems. If the aimis for students to construct symbolic equations, thenstrategies which involve identification of variables andrelationships and understanding how to express themare the most appropr iate. If the aim is f or learners tosolve the problem by whatever method then a moresuitable approach might be for them to imagine thesituation and choose from a r ange of representations(graphical, numerical, algebraic, diagrammatic) possiblyshifting between them, which can be manipulated toachieve a solution.

Clements (1980) and other s have found that withelementary students reading and comprehension

account for about a quar ter of the errors of lowerachieving students. The initial access to such prob lemsis therefore a separate issue before students have toanticipate and represent (as Boero (2001) descr ibesthe setting-up stage) the mathematics they are goingto use. Ballew and Cunningham (1982) with a sampleof 217 11-year-olds found that reading andcomputational weaknesses were to blame fordifficulties alongside interpretation – but they mayhave underestimated the range of problems lurkingwithin ‘interpretation’ because they did not probe any fur ther than these two variables and the linksbetween reading, understanding the relations, anddeciding what to compute were not analysed.Verschaffel, De Cor te and Vierstraete (1999)researched the problem-solving methods anddifficulties experienced by 199 upper pr imarystudents with nine word problems which combinedordinal and cardinal numbers. Questions werecarefully varied to require different kinds ofinterpretation. They found, among othercharacteristics, that students tended to chooseoperations according to the relative size of thenumbers in the question and that choice of f ormalstrategies tended to be er roneous while informalstrategies were more likely to be correct.Interpretation therefore depends on under standingoperations sufficiently to realize where to apply them,recognizing how variables are related, as well asreading and computational accuracy. Success alsoinvolves visualising, imagining, identifying relationshipsbetween variables. All these have to be employedbefore decisions about calculations can be made .(This process is descr ibed in detail in Paper 2 for thecase of distinguishing between additive andmultiplicative relations.) Then learners have to knowwhich variable to choose as the independent var iable,recognise how to express other var iables in relationto it, have a reper toire of knowledge of operationsand functions to draw on, and think to dr aw onthem. Obviously elementary arithmetical skills arecrucial, but automatisation of procedures only aidssolution if the structural class is properly identified inthe first place. Automatisation of techniques canhinder solution of problems that are slightly differentto prior experience because it can lead to o ver-generalisation and misapplication, and attention tolanguage and layout cues rather than the structuralmeaning of the stated problem. For example, iflearners have decided that ‘how many…?’ questionsalways indicate a need to use m ultiplication (as in ‘Iffive children have seven sweets each, how many dothey have in total?’) they may find it hard to answerthe question ‘If 13 players drink 10 litres of cola, how


many should I buy for 22 players?’ because theanswer is not a str aightforward application ofmultiplication. The ‘automatic’ association of ‘howmany’ with a multiplication algorithm, whether it istaught or whether learners have somehow devised itfor themselves, would lead to misapplication.

Learners may not know how and when to br ingother knowledge into play; they may not have hadenough experience of producing representations tothink to use them; the problem may offer arepresentation (e.g. diagram) that does not for themhave meaning which can match to the situation. Ifthey cannot see what to do, they may decide to tr ypossible numbers and see what happens. A difficultywith successive approximation is that young learnersoften limit themselves to natural numbers, and donot develop facility with fractions which appear as aresult of division, nor with decimals which arenecessary to deal with ‘a little bit more than’ and ‘alittle bit less than’. An area which is well-known toteachers but is under-researched is how learnersshift from thinking about only about natural numbersin trial-and-adjustment situations.

Caldwell and Goldin (1987) extended what wasalready well-known for primary students into thesecondary phase, and found that abstract problemswere, as for primary, significantly harder thanconcrete ones for secondary students in general, butthat the differences in difficulty became smaller forolder students. ‘Concrete problems’ were thosecouched in terms of material objects and realisticsituations, ‘abstract’ problems were those whichcontained only abstract objects and/or symbols. Theyanalysed the scripts of over 1000 students who tooka test consisting of 20 prob lems designed along theconcrete-abstract dimension in addition to someother variables. Lower secondary studentssucceeded on 55% of the concrete prob lems and43% abstract, whereas higher secondar y studentssucceeded on 69% of concrete and 66% of abstr act.Whether the narrowing of the gap is due toteaching (as Vygotsky might suggest) or natur almaturation (as some interpretations of Piaget mightsuggest) we do not know. They also found thatproblems which required factual knowledge areeasier than those requir ing hypotheses for secondarystudents, whereas for primary students the reverseappeared to be tr ue. This shift might be due toadolescents being less inclined to enter imaginar ysituations, or to adolescents knowing more facts, or itmay be educative due to the emphasis teachers puton factual rather than imaginative mathematical

activity. However, it is too simplistic to sa y ‘applyingfacts is easy’. In this study, fur ther analysis suggeststhat the questions posed may not have beencomparable on a str uctural measure of difficulty,number of variables and operations for example,although comparing ‘level of difficulty’ in differentquestion-types is not robust.

In a well-replicated result, the APU sample of 15-year-olds found area and per imeter problemsequally hard both in abstr act and diagrammaticpresentations (Foxman et al. 1985). A contextualquestion scored 10% lower than abstract versions.The only presentation that was easier for area was‘find the number of squares in…’ which vir tually tellsstudents to count squares and par ts of squares. In ateaching context, this indication of method is notnecessarily an over-simplification. Dickson’s study ofstudents’ interpretation of area (in four schools)showed that, given the square as a measuring unit,students worked out how to evaluate area and inthen went on to formalise their methods and evendevise the rectangle area formula themselves (1989).

The research findings are therefore inconclusive aboutshifts between concrete and abstract approacheswhich can develop in the normal conditions of schoolmathematics, but the role of pedagogy indicates thatmore might be done to suppor t abstract reasoningand hypothesising as important mathematical practicesin secondary school.

Hembree’s meta-analysis (1992) of 487 studies ofproblem-solving gives no surprises – the factor s thatcontribute to success are: • that problems are fully stated with supportive

diagrams• that students have previous extensive experience in

using the representations used• that they have relevant basic mathematical skills to

use• that teachers who understand problem-solving

methods are better at teaching them• that heuristics might help in lower secondary.

Hembree’s analysis seems to say that learners get tothe answer easiest if there is an ob vious route tosolution. While Hembree did a great ser vice inproducing this meta-analysis, it fails to help with thequestions: How can students learn to create theirown representations and choose between them?How can students learn to devise new methods tosolve new problems? How can students learn to actmathematically in situations that are not full y defined?



How do students get the exper ience that makesthem better at problem-solving? An alternativeapproach is to view problem-solving as far fromclear-cut and instead to see each prob lem as asituation requiring modelling (see next section).

SummaryTo solve problems posed for pedagogic purposes,secondary mathematics learners have to:• be able to read and understand the problem• know when they are expected to use formal

methods• know which methods to apply and in what order

and how to carry them out• identify variables and relationships, choosing which

variable to treat as independent• apply appropriate knowledge of situations and

operations• use mental, graphical and diagrammatic imagery• choose representations and techniques and know

how to operate with them• know a range of useful facts, operations and

functions• decide whether to use statistical, algebraic, logical

or ad hoc methods.

Modelling

In contrast to ‘problem-solving’ situations in which theaim and purpose is often ambiguous, modelling refersto the process of expressing situations in con ventionalmathematical representations which affordmanipulation and exploration. Typically, learners areexpected to construct an equation, function or

diagram which represents the var iables in the situationand then, perhaps, solve an equation or answer someother related question based on their model4. Thusmodelling presents many of the oppor tunities andobstacles described under ‘problem-solving’ above butthe emphasis of this section is to f ocus on theidentification of variables and relationships and thetranslation of these into representations. Carpenter,Ansell, Franke, Fennema and Weisbeck (1993) showthat even very young children can do far moresophisticated quantitative reasoning when modellingsituations for themselves than is expected if we thinkof it solely as application of known operations,because they bring their knowledge of acting in similarsituations to bear on their reasoning.

A typical modelling cycle involves representing arealistic situation in mathematical symbols and thenusing isomorphism between the model and thesituation, manipulate variables either in the model orthe situation and obser ve how such transformationsre-translate between the model and the situation. Thisduality is encapsulated in the ideas of model-of andmodel-for. The situation is an instantiation of an abstr actmodel. The abstract model becomes a model-for beingused to provide new insights and possibilities for theoriginal situation. This isomorphic duality is a moregeneral version of Vergnaud’s model described inPaper 4, Understanding relations and their gr aphicalrepresentation. For learners, the situation can provideinsight into possibilities in the mathematics, or themathematics can provide insights into the situation. Forexample, a graphical model of temperature changescan afford prediction of future temperatures, whileactual temperature changes can afford understandingof continuous change as expressed by graphs.


Figure 7.1: Typical modelling cycle with two-way relationship between situation and representations.

Research literature in this area gives primacy todifferent features. We are limited to looking atteaching experiments which are necessar ily influencedby par ticular curriculum aims. Either the researchlooks at the learning of functions (that is extendingthe learners’ repertoire of standard functions andtheir understanding of their features and proper ties)and sees modelling, interpreting and reifying functionsas components of that lear ning (e.g. O’Callaghan,1998), or the research sees skill in the modellingprocess as the goal of lear ning and sees knowledge of functions (their types and behaviour) as anessential component of that. In either approach thereare similar difficulties. O’Callaghan, (1998) using acomputer-intensive approach, found that whilestudents did achieve a better understanding offunctions through modelling than compar ablestudents pursuing a traditional ‘pure’ course, and weremore motivated and engaged in mathematics, theywere no better at reifying what they had lear nt thanthe traditional students. In pre- and post-testsstudents were asked to: model a situation using afunction; interpret a function in a realistic situation;translate between representations; and use andtransform algebraic functions which represent afinancial situation. Students’ answers improved in allbut the last task which required them to understandthe role of var iables in the functions and the relationbetween the functions. In other words, they weregood at modelling but not at knowing more aboutfunctions as objects in their own right.

MacGregor and Stacey (1993) (281 lower-secondarystudents in free response format and 1048 similarstudents who completed a multiple-choice item)show that the relationship between words, situationsand making equations is not solely one of translatinginto symbols and correct algebra, rather it involvestranslating what is read into some kind of modeldeveloped from an existing schema and thenrepresenting the model – so there are tw o stages atwhich inappropriate relationships can be introduced,the mental model and the expressions of thatmodel. The construction of mental models isdependent on:• what learners know of the situation and how they

imagine it• how this influences their identification of variables,

and • their knowledge of possible ways in which variables

can vary together.

What is it that students can see? Car lson andcolleagues (2002) investigated students’ perceptions

and images of covariation, working mainly withundergraduates. The task is to work out how onevariable varies in relation to another variable. Theirfindings have implications for younger students,because they find that their students can constr uctand manipulate images of how a dependent var iablerelates to the independent var iable in dynamicevents, such as when variation is positional, or visuallyidentifiable, or can be seen to increase or decreaserelative to the dependent variable, but the rate atwhich it changes change is harder to imagine . Forour purposes, it is impor tant to know thatadolescents can construct images of relationships, butO’Callaghan’s work shows that more is required forthis facility to be used to dev elop knowledge offunctions. When distinguishing between linear andquadratic functions, for example, rate of change is auseful indicator instead of some par ticular values, theturning point or symmetr ical points, which may notbe available in the data.

Looking at situations with a mathematicalperspective is not something that can be directl ytaught as a topic , nor does it ar ise naturally out ofschool mathematical learning. Tanner and Jones(1994) worked with eight schools introducingmodelling to their students. Their aim was not toprovide a vehicle to learning about functions but todevelop modelling skills as a f orm of mathematicalenquiry. They found that modelling had to bedeveloped over time so that learners developed arepertoire of experience of what kinds of things tofocus on. Trelinski (1983) showed that of 223graduate maths students only 9 could constructsuitable mathematical models of non-mathematicalsituations – it was not that they did not know therelevant mathematics, but that they had never beenexpected to use it in modelling tasks before. It doesnot naturally follow that someone who is good atmathematics and knows a lot about functionsautomatically knows how to develop models.

So far we have only talked about what happens whenlearners are asked to produce models of situations.Having a use for the models, such as a problem tosolve, might influence the modelling process. Campbell, Collis and Watson (1995) extended the findings ofKouba’s research (1989) (repor ted in Paper 4) andanalysed the visual images produced and used by fourgroups of 16-year-olds as aids to solving prob lems. Thegroups were selected to include students who hadhigh and low scores on a test of vividness of visualimagery, and high and low scores on a test ofreasoning about mathematical operations. They were



then given three problems to solve: one involvingdrink-driving, one about cutting a painted cube intosmaller cubes and one about three people consuminga large bag of apples by successively eating 1/3 of whatwas left in it. The images they developed differed intheir levels of generality and abstraction, and successrelated more to students’ ability to operate logicallyrather than to produce images, but even so there wasa connection between the level of abstraction affordedby the images, logical operational facility and the use ofvisually based strategies. For example, graphicalvisualisation was a successful method in the dr ink-driving problem, whereas images of three men withbeards sleeping in a hut and eating the apples w erevivid but unhelpful. The creation of useful mathematicalimages needs to be learnt. In Campbell’s study,questions were asked for which a model was needed,so this purpose, other than producing the model itself,may have influenced the modelling process. Modelswere both ‘models of ’ and ‘models for’, the formerbeing a representation to express str uctures and thelatter being related to a fur ther purpose (e.g. van denHeuvel-Panhuizen, 2003). Other writers have alsopointed to the positive effects of purpose: Ainley, Prattand Nardi (2001) and Fr iel, Curcio and Br ight(2001) allfound that having a purpose contributes to students’sense-making of graphs.

Summary• Modelling can be seen as a subclass of problem-

solving methods in which situations arerepresented in formal mathematical ways.

• Learners have to draw on knowledge of thesituation to identify variables and relationships and,through imagery, construct mathematicalrepresentations which can be manipulated further.

• There is some evidence that learners are better atproducing models for which they have a fur therpurpose.

• To do this, they have to have a repertoire ofmathematical representations, functions, andmethods of operation on these.

• A modelling perspective develops over time andthrough multiple situational experiences, and canthen be applied to given problems – the processesare similar to those learners do when faced withnew mathematical concepts to understand.

• Modelling tasks do not necessarily lead toimproved understanding of functions without the

development of repertoire and deliberatepedagogy.

Functions

For learners to engage with secondar y mathematicssuccessfully they have to be able to decipher andinterpret the stimuli they are offered, and thisincludes being orientated towards looking forrelations between quantities, noticing structures,identifying change and generalising patterns ofbehaviour. Kieran (1992) lists these as goodapproaches to ear ly algebra. They also have to know the difference between statistical and algebraicrepresentations, such as the difference between abar char t and an algebraic graph.

Understanding what a function is, a mapping thatrelates values from one space into values in anotherspace, is not a straightforward matter for learners. InPaper 4, Understanding relations and their gr aphicalrepresentation, evidence that the exper ience oftransforming between values in the same space isdifferent from transforming between spaces isdescribed, and for this paper we shall move on andassume that the purpose of simple additive, scalarand multiplicative functions is under stood, and thetask is now to understand their nature, a range ofkinds of function, their uses, and the ways in whichthey arise and are expressed.

Whereas in ear ly algebra learners need to shift from seeing expressions as things to be calculated to seeing them as expressing str uctures, they thenhave to shift fur ther to seeing functions as relationsbetween expressions, so that functions becomemathematical objects in themselves and numerical‘answers’ are likely to be pair s of related values(Yerushalmy and Schwar tz, 1993). Similarly equationsare no longer situations which hide an unkno wnnumber, but expressions of relationships betweentwo (or more) var iables. They have also tounderstand the difference between a point-wiseview of functional relationships (as expressed b ytables of values) and a holistic view (reinf orcedespecially by graphs).

Yerushalmy and Gilead, in a teaching exper imentwith lower-secondary students over a few years(1999) found that knowledge of a range of functionsand the nature of functions was a good basis f orsolving algebraic problems, par ticularly those thatinvolved rate because a graph of a function allows


rates to be obser ved and compared. Thus functionsand their graphs support the focus on rate thatCarlson’s students found difficult in situations anddiagrams. Functions appeared to provide a bridgethat turned intractable word problems intomodelling tasks by conjecturing which functionsmight ‘fit’ the situation. However, their students couldmisapply a functional approach. This seems to be anexample of the well-known phenomenon of over-generalizing an approach beyond its appropriatedomain of application, and arises from studentspaying too much attention to what has recentl y beentaught and too little to the situation.

Students not only have to learn to think aboutfunctional relationships (and consider non-linearrelationships as possibilities), which have an input towhich a function is applied gener ating some specificoutput, but they also need to think about relationsbetween relations in which there is no immediateoutput, rather a structure which may involve severalvariables. Halford’s analysis (e.g.1999) closely followsInhelder and Piaget’s (1959) theories about thedevelopment of scientific reasoning in adolescence .He calls these ‘quaternary’ relationships because theyoften relate four components appearing as two pairs.Thus distributivity is quaternary, as it involves twobinary operations; proportion is quaternary as itinvolves two ratios. So are rates of change, in whichtwo variables are compared as they both var y inrelation to something else (their functional relation,or time, for example). This complexity mightcontribute to explaining why Carlson and colleaguesfound that students could talk about co variationrelationships from graphs of situations but not ratesof change. Another reason could be the opacity ofthe way rate of change has to be read from gr aphs:distances in two directions have to be selected andcompared to each other, a judgement or calculationmade of their ratio, and then the same process hasto be repeated around other points on the gr aphsand the ratios compared. White and Mitchelmore(1996) found that even after explicit instr uctionstudents could only identify rates of change in simplecases, and in complex cases tr ied to use algebraicalgorithms (such as a given formula for gradient)rather than relate quantities directly.

One area for research might be to f ind out whetherand how students connect the ‘method ofdifferences’, in which rates of change are calculatedfrom tables of values, to graphical gradients. One ofthe problems with understanding functions is thateach representation brings cer tain features to the

fore (Goldin, 2002). Graphical representationsemphasise linearity, roots, symmetry, continuity,gradient; domain; ordered dataset representationsemphasise discrete covariation and may distractstudents from star ting conditions; algebraicrepresentations emphasise the structure of relationsbetween variables, and the family of functions towhich a par ticular one might be related. Tounderstand a function fully these have to beconnected and, fur ther, students have to think aboutfeatures which are not so easy to visualiz e but haveto be inferred from, or read into, the representationby knowing its proper ties, such as growth rate(Confrey and Smith, 1994; Slavit, 1997). Confrey and Smith used data sets to in vite unit-by-unitcomparison to focus on rate-of-change, and deducedthat rate is different from ratio in the ways that it islearnt and understood (1994). Rate depends onunderstanding the covariation of variables, and beingable to conceptualise the action of change , whereasratio is the compar ison of quantities.

Summary To understand the use of functions to describesituations secondary mathematics learners have to:• distinguish between statistical and algebraic

representations• extend knowledge of relations to understanding

relations between relations• extend knowledge of expressions as structures to

expressions as objects• extend knowledge of equations as defining

unknown numbers to equations as expressingrelationships between variables

• relate pointwise and holistic understandings andrepresentations of functions

• see functions as a new kind of mathematical object• emphasize mathematical meaning to avoid over-

generalising• have ways of understanding rate as covariation.

Mathematical thinking

In this section we mention mathematical problems –those that arise in the exploration of mathematicsrather than problems presented to learners for themto exercise methods or develop ‘problem-solving’skills. In mathematical problems, learners have to usemathematical methods of enquir y, some of which arealso used in word problems and modelling situations,or in learning about new concepts. To learnmathematics in this context means two things:



to learn to use methods of mathematical enquir y and to learn mathematical ideas which ar ise in such enquiry.

Descriptions of what is entailed in mathematicalthinking are based mainly on Polya’s work (1957), in which mathematical thinking is descr ibed as aholistic habit of enquir y in which one might dr aw on any of about 70 tactics to make progress with a mathematical question. For example, the tacticsinclude make an analogy, check a result, look forcontradictions, change the problem, simplify,specialise, use symmetr y, work backwards, and soon. Although some items in Polya’s list appear indescriptions of problem-solving and modellingtactics, others are more likely to be helpful in purel ymathematical contexts in which facts, logic, andknown proper ties are more impor tant than merelydealing with current data. Cuoco, Goldenberg andMark (1997) have devised a typography of aspectsof mathematical habits of mind. For example,mathematicians look at change, look at stability,enjoy symbolisation, invent, tinker, conjecture,experiment, relate small things to big things, and soon. The typography encompasses the per spectiveswhich exper ts bring to bear on mathematics – thatis they br ing ideas and relationships to bear onsituations rather than merely use current data andspecific cases. Both of these lists contain dozens ofdifferent ‘things to do’ when faced with mathematics.Mason, Bur ton and Stacey (1982) condensed theseinto ‘specialise-generalise; conjecture-convince’ whichfocuses on the shifts between specific cases andgeneral relationships and proper ties, and thereasoning shift between demonstrating and proving.All of these reflect the processes of mathematicalenquiry under taken by experienced mathematicians.Whereas in modelling there are clear stages ofwork to be done , ‘mathematical thinking’ is not anordered list of procedures, rather it is a way ofdescribing a cast of mind that views any stimulus asan object of mathematical interest, encapsulatingrelationships between relationships, relationshipsbetween proper ties, and the potential for moresuch relationships by varying variables, parametersand conditions.

Krutetskii (1976) conducted clinical inter views with130 Soviet school children who had been identif iedas strong mathematicians. He tested themqualitatively and quantitatively on a wide r ange ofmathematical tasks, looked for common factors inthe way they tackled them, and found that thosewho are better at mathematics in gener al were

faster at grasping the essence of a mathematicalsituation and seeing the str ucture through theparticular surface features. They generalised moreeasily, omitted intermediate steps of reasoning,switched between solution methods quickly, tried toget elegant solutions, and were able to reverse trainsof thought. They remembered relationships andprinciples of a problem and its solution r ather thanthe details and tended to explain their actions r atherthan describe them. Krutetskii’s methods were clinicaland grounded and dependent on case studies withinhis sample, nevertheless his work over many yearsled him to form the view that such ‘abilities’ wereeducable as well as innate and drew strongl y onnatural propensities to reason spatially, perceptually,computationally, to make verbal analogies, mentalassociations with remembered exper iences, andreasoning. Krutetskii, along with mathematiciansreporting their own experiences, observed the needto mull, that is to leave unsolved questions alone fora while after effortful attempts, to sleep, or do otherthings, as this often leads to fur ther insights whenreturning to them. This commonly observedphenomenon is studied in neuroscience which isbeyond the scope of this paper, but does haveimplications for pedagogy.

Summary• Successful mathematics learners engage in

mathematical thinking in all aspects of classroomwork. This means, for example, that they see what isvarying and what is invariant, look for relationships,curtail or reverse chains of reasoning, switchbetween representations and solution methods,switch between examples and generalities, andstrive for elegance.

• Mathematical ‘habits of mind’ draw on abilities orperception, reasoning, analogy, and mentalassociation when the objects of study aremathematical, i.e. spatial, computational, relational,variable, invariant, structural, symbolic.

• Learners can get better at using typical methods ofmathematical enquiry when these are explicitlydeveloped over time in classrooms.

• It is a commonplace among mathematicians thatmulling over time aids problem-solving andconceptualisation.


Part 2: What learners do whenfaced with complex situations in mathematicsIn this section we collect research findings thatindicate what school students typicall y do whenfaced with situations to model, solve, or makemathematical sense of.

Bringing outside knowledge to bear on mathematical problems

Real-life problems appear to invite solutions which arewithin a ‘human sense’ framework rather than amathematical frame (Booth 1981). ‘Wrong’approaches can therefore be seen not as er rors, butas expressing a need for enculturation into what doesand does not count in mathematical problem-solving.Cooper and Dunne (2000) show that in tests theappropriate use of outside knowledge and ways ofreasoning, and when and when not to br ing it intoplay, is easier for socially more advantaged students tounderstand than less advantaged students who ma yuse their outside knowledge inappropriately. This isalso true for students working in languages otherthan their first, who may only have access to formalapproaches presented in standard ways. Cooper andHarries (2002) worked on this problem fur ther andshowed how typical test questions for 11- to 12-yearolds could be rewr itten in ways which encouragemore of them to reason about the mathematics,rather than dive into using handy but inappropriateprocedures. Vicente, Orrantia and Verschaffel (2007)studied over 200 primary school students’ responsesto word problems and found that elaboratedinformation about the situation was much lesseffective in improving success than elaborating theconceptual information. Wording of questions, as wellas the test environment, is therefore significant indetermining whether students can or can not solv eunfamiliar word problems in appropriate ways.Contrary to a common assumption that givingmathematical problems in some context helpslearners understand the mathematics, analysis oflearners’ responses in these research studies sho wsthat ‘real-life’ contexts can:• lead to linguistic confusion • create artificial problems that do not fit with their

experience• be hard to visualise because of unfamiliarity, social

or emotional obstacles• structure mathematical reasoning in ways which are

different from abstract mathematics

• obscure the intended mathematical generalisation• invite ad hoc rather than formal solution methods• confuse students who are not skilled in deciding

what ‘outside’ knowledge they can bring to thesituation.

Clearly students (and their teacher s) need to beclear about how to distinguish between situations inwhich everyday knowledge is, or is not, preferable toformal knowledge and how these relate. In Boaler’scomparative study of two schools (1997) somestudents at the school, in which mathematics wastaught in exploratory ways, were able to recognisethese differences and decisions. However, it is alsotrue that students’ outside knowledge usedappropriately might:• enable them to visualise a situation and thus

identify variables and relationships• enable them to exemplify abstract relationships

as they are manifested in reality• enable them to see similar structures in different

situations, and different structures in similarsituations

• be engaged to generate practical, rather thanformal, solutions

• be consciously put aside in order to perform asmathematically expected.

Information processing

In this section we will look at issues about cognitiv eload, attention, and mental representations. At thestart of the paper we posed questions about what alearner has to do at f irst when faced with a newsituation of any kind. Information processing theoriesand research are helpful but there is little research inthis area within mathematics teaching except interms of cognitive load, and as we have said before it is not helpful for cognitive load to be minimized ifthe aim is to lear n how to work with complexsituations. For example, Sweller and Leung-Martin(1997) used four experiments to find out whatcombinations of equations and words were moreeffective for students to deal successfull y withequivalent information. Of course, in mathematicslearning students have to be able to do both and allkinds of combinations, but the researchers did findthat students who had achieved fluency withalgebraic manipulations were slowed down by having to read text. If the aim is merel y to doalgebraic manipulations, then text is an extr a load.Automaticity, such as fluency in algebr aicmanipulation, is achievable efficiently if differences



between practice examples are minimized.Automaticity also frees up working memory forother tasks, but as Freudenthal and others havepointed out, automaticity is not always a suitable goalbecause it can lead to thoughtless application ofmethods. We would expect a learner to read textcarefully if they are to choose methods meaningfull yin the context. The information-processing tutorsdeveloped in the work of Anderson and hiscolleagues (e.g. 1995) focused mainly onmathematical techniques and processes, but includedunderstanding the effects of such processes. We arenot arguing for adopting his methods, but we dosuggest that information processing has something to offer in the achievement of fluency, and thegeneration of multiple examples on which thelearner can then reflect to under stand the patternsgenerated by mathematical phenomena.

Most of the research on attention in mathematicseducation takes an affective and motivational view,which is beyond the scope of this paper (see NMAP,2008). However, there is much that can be doneabout attention from a mathematical per spective.The deliberate use of var iation in examples offeredto students can guide their focus towards par ticularvariables and differences. Learners have to knowwhen to discern par ts or wholes of what is off eredand which par ts are most cr itical; manipulation ofvariables and layouts can help direct attention. Whatis available to be learnt differs if different relationsare emphasised by different variations. For example,students learning about gradients of straight linefunctions might be offered exercises as follows:

Gradient exercise 1: find the gradients between eachof the following pairs of points.

(4, 3) and (8, 12) (-2, -1) and (-10, 1)(7, 4) and (-4, 8) (8, -7) and (11, -1)(6, -4) and (6, 7) (-5, 2) and (10, 6)(-5, 2) and (-3, -9) (-6, -9) and (-6, -8)

Gradient exercise 2:

(4, 3) and (8, 12) (4, 3) and (4, 12)(4, 3) and (7, 12) (4, 3) and (3, 12)(4, 3) and (6, 12) (4, 3) and (2, 12)(4, 3) and (5, 12) (4, 3) and (1, 12)

In the first type, learners will typically focus on themethods of calculation and dealing with negativ enumbers; in the second type , learners typicallygesture to indicate the changes in gr adient. Research

in this area shows how learners can be directedtowards different aspects by manipulating variables(Runesson and Mok, 2004; Chik and Lo, 2003).

Theories of mental representations claim thatdeclarative knowledge, procedural knowledge andconceptual knowledge are stored in different ways inthe brain and also draw distinctions betweenverbatim memory and gist memor y (e.g. Brainerdand Reyna, 1993). Such theories are not much helpwith mathematics teaching and lear ning, becausemost mathematical knowledge is a combination of allthree kinds, and in a typical mathematical situationboth verbatim and gist memor y would be employed.At best, this knowledge reminds us that providing‘knowledge’ only in verbatim and declarative form is unlikely to help learners become adaptablemathematical problem-solvers. Learners have tohandle different kinds of representation and knowwhich different representations represent differentideas, different aspects of the same ideas, and afforddifferent interpretations.

Summary• Learners’ attention to what is offered depends

on variation in examples and experiences.

• Learners’ attention can be focused on criticalaspects by deliberate variation.

• Automaticity can be helpful, but can also hinderthought.

• If information is only presented as declarativeknowledge then learners are unlikely to developconceptual understanding, or adaptive reasoning.

• The form of representation is a critical influence on interpretation.

What learners do naturally thatobstructs mathematicalunderstanding

Most of the research in secondar y mathematics isabout student errors. These are persistent over time,those being found by Ryan and Williams (2007)being similar to those found by APU in the late1970s. Errors do not autocorrect because ofmaturation, experience or assessment. Rather theyare inherent in the ways learners engage withmathematics through its formal representations.


Persistence of ‘child-methods’ pervades mathematics at secondary level (Booth, 1981). Whether ‘child-methods’ are seen as intuitive, quasi-intuitive,educated, or as over-generalisations beyond thedomain of applicability, the implication for teaching isthat students have to experience, repeatedly, thatnew-to-them formal methods are more widelyapplicable and offer more possibilities, and that ear lierideas have to be extended and, perhaps, abandoned.If students have to adopt new methods withoutunderstanding why they need to abandon ear lierones, they are likely to become confused and evendisaffected, but it is possible to demonstrate this needby offering par ticular examples that do not yield tochild-methods. To change naïve conceptualisations isharder as the next four ‘persistences’ show.

Persistence of additive methods This ‘child method’ is worthy of separate treatmentbecause it is so per vasive. The negative effects ofthe persistence of additive methods show up againand again in research. Bednarz and Janvier (1996)conducted a teaching exper iment with 135 12- to13-year-olds before they had any algebra teaching tosee what they would make of word problems whichrequired several operations: those with multiplicativecomposition of relationships turned out to be muchharder than those which involved composing mainlyadditive operations. The tendency to use additivereasoning is also found in reasoning about r atios andproportion (Har t, 1981), and in students’expectations about relationships between variablesand sequential predictions. That it occurs naturallyeven when students know about a var iety of otherrelationships is an example of how intuitiveunderstandings persist even when more formalalternatives are available (Fischbein, 1987)

Persistence of more-more, same-same intuitions Research on the interference from intuitive rules givesvaried results. Tirosh and Stavy (1999) found that theiridentification of the intuitive rules ‘more-more’ and‘same-same’ had a strong predictive power forstudents’ errors and their deduction accords with thegeneral finding that rules which generally work atprimary level persist. For example, students assumethat shapes with larger perimeters must have largerareas; decimals with more digits must be larger thandecimals with fewer digits, and so on. Van Dooren, DeBok, Weyers and Verschaffel (2004), with a sample of172 students from upper secondar y found that,contrary to the findings of Tirosh and Stavy, students’errors were not in general due to consistentapplication of an intuitive rule of ‘more-more’ ‘same-

same’. Indeed the more er rors a student made the lesssystematic their errors were. This was in a multiple-choice context, and we may question the assumptionthat students who make a large number of errors insuch contexts are engaging in any mathematicalreasoning. However, they also sampled wr ittencalculations and justifications and found that errorswhich looked as if they might be due to ‘more-more’and ‘same-same’ intuitions were often due to othererrors and misconceptions. Zazkis, however (1999),showed that this intuition per sisted when thinkingabout how many factors a number might have, largenumbers being assumed to have more factors.

Persistence of confusions between different kindsof quantity, counting and measuringAs well as the per sistence of additive approaches to multiplication, being taught ideas and beingsubsequently able to use them are not immediatel yconnected. Vergnaud (1983) explains that theconceptual field of intensive quantities, thoseexpressed as ratio or in terms of other units (see Paper 3, Understanding rational numbers andintensive quantities), and multiplicative relationshipdevelopment continues into adulthood. Nesher andSukenik (1991) found that only 10% of studentsused a model based on under standing ratio afterbeing taught to do so f ormally, and then only forharder examples.

Persistence of the linearity assumption Throughout upper primary and secondary students act as if relationships are always linear, such as believingthat if length is multiplied by m then so is area, or if the10th term in a sequence is 32, then the 100th must be 320 (De Bock,Verschaffel and Janssens1998; VanDooren, De Bock, Janssens and Verschaffel, 2004,2007). Results of a teaching exper iment with 93 upper-primary students in the Nether lands showed that,while linearity is persistent, a non-linear realistic contextdid not yield this er ror. Their conclusion was that thelinguistic structure of word problems might invitelinearity as a first, flawed response. They also found thata single experience is not enough to change this habit.A related assumption is that functions increase as theindependent variable increases (Kieran, Boileau and Garancon 1996).Students’ habitual ways ofattacking mathematical questions and problems also cause problems.

Persistence with informal and language-basedapproachesMacgregor and Stacey (1993) tested over 1300upper-secondary students in total (in a r ange of



studies) to see how they mathematised situations.They found that students tend f irst of all to tr y toexpress directly from the natural language of asituation, focusing on in equalities between quantities.Engaging with the under lying mathematical meaningis not a natural response. Students wanted thealgebraic expression to be some kind of linguisticcode, rather than a relational expression. Manyresearchers claim this is to do with translating wordorder inappropriately into symbols (such as ‘thereare six students to each professor, so 6s = p’) butMacgregor and Stacey suggest that the cause is moreto with inadequate models of multiplicativerelationships and ratio. However, it is easy to see thatthis is compounded by an unfortunate choice ofletters as shor thand for objects rather than asvariables. For example, Wollman (1983) and Clement(1982) demonstrate that students make this classic‘professor-student’ error because of haste , failure tocheck that the meaning of the equation matches themeaning of the sentence , over-reliance on linguisticstructure, use of non-algebraic symbols (such as pfor professor instead of p for number of professors)and other reasons. At least one of these is aprocessing error which could be resolved with a‘read out loud’ strategy for algebra.

Persistence of qualitative judgements in modellingLesh and Doerr (2003) repor ted on the modellingmethods employed by students who had not hadspecific direction in what to do. In the absence ofspecific instructions, students repeat patterns oflearning that have enabled them to succeed in othersituations over time. They tend to start on eachproblem with qualitative judgements based on theparticular context, then shift to additive reasoning, thenform relationships by pattern recognition or repeatedaddition, and then shift to propor tional and relationalthinking. At each stage their students resor ted tochecking their arithmetic if answers conflicted ratherthan adapting their reasoning by seeing if answersmade sense or not. This repetition of naïve strategiesuntil they break down is inefficient and not what themost successful mathematics students do.

In modelling and problem-solving students confuseformal methods with contextual methods; they clingstrongly to limited prototypes; they over-generalise;they read left-to-r ight instead of interpreting themeaning of symbolic expressions. In word problemsthey misread; miscomprehend; make errors intransformation into operations; errors in processes;and misinterpret the solution in the problem context(see also Ryan and Williams 2007).

Persistent application of procedures.Students can progress from a manipulativ e approachto algebra to understanding it as a tool f or problem-solving over time, but still tend over-rely onautomatic procedures (Knuth, 2000). Knuth’s sampleof 178 first-year undergraduates’ knowledge of therelationship between algebraic and graphicalrepresentations was superficial, and that they reachedfor algebra to do automatised manipulations r atherthan use graphical representations, even when thelatter were more appropriate.

SummaryLearners can create obstacles for themselves byresponding to stimuli in particular ways:• persistence of past methods, child methods, and

application of procedures without meaning• not being able to interpret symbols and other

representations• having limited views of mathematics from their past

experience• confusion between formal and contextual aspects• inadequate past experience of a range of examples

and meanings• over-reliance on visual or linguistic cues, and on

application of procedures• persistent assumptions about addition, more-

more/same-same, linearity, confusions aboutquantities

• preferring arithmetical approaches to those basedon meaning.

What learners do naturally that is useful

Students can be guided to explore situations in asystematic way, learning how to use a typicallymathematical mode of enquir y, although it is hard tounderstand phenomena and change in dynamicsituations. Carlson, Jacobs, Coe, Larson and Hsu.(2002) and Yerushalmy (e.g. 1997) have presentedconsistent bodies of work about modelling andcovariation activities and their work, with that of Kaput(e.g. 1991), has found that this is not an inherentlymaturation problem, but that with suitable tools andrepresentations such as those available in SimCalcchildren can learn not only to understand change byworking with dynamic images and models, but also tocreate tools to analyse change. Carlson and hercolleagues in teaching exper iments have developed aframework for describing how students learn aboutthis kind of co-var iation. First they learn how to


identify variables; then they form an image of how thevariables simultaneously vary. Next, one variable has tobe held still while the change in another is observed.This last move is at the core of mathematics andphysics, and is essential in constr ucting mathematicalmodels of multivariate situations, as Inhelder andPiaget also argued more generally.

In these supported situations, students appear toreason verbally before they can operate symbolically(Nathan and Koedinger, 2000). The usual ‘order’ ofteaching suggested in most cur ricula (arithmetic,algebra, problem-solving) does not match students’development of competence in which verbal modestake precedence5. This fits well with Swafford andLangrall’s study of ten 11-year-olds (2000) in which it was clear that even without formal teaching aboutalgebra, students could identify var iables and ar ticulatethe features of situations as equations where theywere familiar with the under lying operations. Studentswere asked to work on six tasks in inter views. Thetasks were realistic problems that could berepresented by direct propor tion, linear relations in a numerical context, linear relations in a geometr iccontext, arithmetic sequences, exponential relationsand inverse proportion. Each task consisted ofsubtasks which progressed from str ucturedexploration of the situation, verbal description of how to find some unknown value, write an equationto express this given certain letters to representvariables, and use the equation to f ind out somethingelse. The ability to express their verbal descriptions as equations was demonstrated across the tasks;everyone was able to do at least one of thesesuccessfully and most did more than one . The onlysituation for which no one produced an equation wasthe exponential one. The study also showed that, givensuitably-structured tasks, students can avoid the usualassumptions of linearity. This shows some intuitivealgebraic thinking, and that formal symbolisation cantherefore be introduced as a tool to expressrelationships which are already understood fromsituations. Of course, as with all teaching exper iments,this finding is specific to the teaching and task andwould not automatically translate to other contexts,but as well as supporting the teaching of algebr a asthe way to express generality (see Paper 6 Algebraicreasoning) it contributes to the substantial practicalknowledge of the value of star ting with what studentssee and getting them to ar ticulate this as a foundationfor learning.

It is by looking at the capabilities of successfulstudents that we learn more about what it takes to

learn mathematics. In Krutetskii’s study of suchstudents, to which we referred earlier, (1976) hefound that they exhibited what he called a‘mathematical cast of mind’ which had analytical,geometric, and harmonic (a combination of the two)aspects. Successful students focused on structure and relationships rather than par ticular numbers of asituation. A key result is that memor y about pastsuccessful mathematical work, and its associatedstructures, is a stronger indicator of mathematicalsuccess than memory about facts and techniques. He did not find any common aspects in theircomputational ability.

Silver (1981) reconstructed Krutetskii’s claim that 67lower-secondary high-achieving mathematics studentsremembered structural information aboutmathematics rather than contextual information. Heasked students to sor t 16 problems into groups thatwere mathematically-related. They were then giventwo problems to work on and asked to write downafterwards what they recalled about the prob lems.The ‘writing down’ task was repeated the next day,and again about four weeks later. There was acorrelation between success in solving problems anda tendency to focus on under lying mathematicalstructure in the sor ting task. In addition, studentswho recalled the str ucture of the problems were themore successful ones, but others who hadperformed near average on the problems could talkabout them structurally immediately after discussion.The latter effect did not last in the f our-week recalltask however. Silver showed, by these and othersimilar tasks, that structural memory aided transfer ofmethods and solutions to new, mathematically similar,situations. A question arises, whether this is teachableor not, given the results of the f our-week recall.Given that we know that mathematical strategies canbe taught in general (Vos, 1976; Schoenfeld, 1979;1982, and others) it seems likely that structuralawareness might be teachable, however this mayhave to be sustained over time and students alsoneed knowledge of a reper toire of structures tolook for.

We also know something about how studentsidentify relationships between variables. While manywill choose a var iable which has the mostconnections within the problem as the independentone, and tended also to start by dealing with thelargest values, thus showing that they can anticipateefficiency, there are some who pref er the least valueas the star ting point. Nesher, Hershkovitz andNovotna, (2003) found these tendencies in the



modelling strategies of 167 teachers and 132 15-year-old students in twelve situations which all hadthree variables and a comparative multiplicationrelationship with an additive constraint. This is arelatively large sample with a high number of slightly-varied situations for such studies and could provide amodel for fur ther research, rather than small studieswith a few highly varied tasks.

Whatever the disposition towards identifyingstructures, variables and relationships, it is widelyagreed that the more you know, the better equippedyou are to tackle such tasks. Alexander and others(1997) worked with very young children (26 three-to five-year-olds) and found that they could reasonanalogically so long as they had the necessar yconceptual knowledge of objects and situations torecognise possible patterns. Analogical reasoningappears to be a natural everyday power even forvery young children (Holyoak and Thagard, 1995)and it is a valuable source of hypotheses, techniques,and possible translations and transformations.Construction of analogies appear s to help withtransfer, since seeking or constr ucting an analogyrequires engagement with str ucture, and it isstructure which is then sought in new situations thus enabling methods to be ‘carried’ into new uses. English and Shar ry (1996) provide a gooddescription of the processes of analogical reasoning:first seeing or working out what relations areentailed in the examples or instances being offered(abductively or inductively), this relational str ucture is extracted and represented as a model, mental,algebraic, graphical i.e. constructing an analogy insome familiar, relationally similar form. They observed,in a small sample , that some students act ‘pseudostructurally’ i.e. emphasising syntax hindered themseeking and recognising relational mappings. A criticalshift is from focusing on visual or contextualsimilarity to structural similarity, and this has to besupported. Without this, the use of analogies canbecome two things to learn instead of one .

Past experience is also valuable in the interpretationof symbols and symbolic expressions, as well as whatattracts their attention and the inter-relationbetween the two (Sfard and Linchevski, 1994). Inaddition to past exper ience and the effects of layoutand familiarity, there is also a diff erence in readingsmade possible by whether the student perceives a statement to be oper ational (what has to becalculated), relational (what can be expressedalgebraically) or structural (what can be gener alized).Generalisation will depend on what students see and

how they see it, what they look for and what theynotice. Scheme-theory suggests that what they lookfor and notice is related to the ways they havealready constructed connections between pastmathematical experiences and the concept imagesand example spaces they have also constructed andwhich come to mind in the current situation. Thusgeneralisations intended by the teacher are notnecessarily what will be noticed and constr ucted bystudents (Steele and Johanning, 2004).

SummaryThere is evidence to show that, with suitableenvironments, tools, images and encouragement,learners can and do:• generalise from what is offered and experienced• look for analogies• identify variables• choose the most efficient variables, those with most

connections• see simultaneous variations• observe and analyse change• reason verbally before symbolising• develop mental models and other imagery• use past experience• need knowledge of operations and situations to do

all the above successfully• particularly gifted mathematics students also:

• quickly grasp the essence of a problem• see structure through surface features• switch between solution methods• reverse trains of thought• remembered the relationships and

principles of a problem • do not necessarily display computational

expertise.


Part 3: What happens withpedagogic interventiondesigned to address typicaldifficulties?

We have described what successful and unsuccessfullearners do when faced with new and complexsituations in mathematics. For this section we showhow particular kinds of teaching aim to tackle thetypical problems of teaching at this level. This dependson reports of teaching exper iments and, as with thedifferent approaches taken to algebra in the ear lierpaper, they show what it is possib le for secondarystudents to learn in par ticular pedagogic contexts.

It is worth looking at the successes and newdifficulties introduced by researchers and developerswho have explored ways to influence learningwithout exacerbating the diff iculties describedabove. We found broadly five approaches, thoughthere are overlaps between them: focusing ondevelopment of mathematical thinking; task design;metacognitive strategies; the teaching of heur istics;and the use of ICT.

Focusing on mathematical thinking

Experts and novices see problems differently; andsee different similarities and differences betweenproblems, because experts have a wider repertoireof things to look for, and more exper ience aboutwhat is, and is not, worthwhile mathematically.Pedagogic intervention is needed to enable alllearners to look for underlying structure orrelationships, or to devise subgoals and reflect on theoutcomes of pursuing these as successful studentsdo. In a three-year course for 12- to 15-year-olds,Lamon educated learners to understand quantitativerelationships and to mathematise exper ience bydeveloping the habits of identifying quantities, makingassumptions, describing relationships, representingrelationships and classifying situations (1998). It isworth emphasising that this development of habitstook place over three years, not over a few lessonsor a few tasks.

• Students can develop habits of identifying quantitiesand relationships in situations, given extendedexperience.

Research which addresses development ofmathematical thinking in school mathematics

includes: descriptive longitudinal studies of cohor ts of students who have been taught in ways whichencourage mathematical enquir y and proof andcomparative studies between classes taught inthrough enquiry methods and traditional methods.Most of these studies focus on the development ofclassroom practices and discourse, and how socialaspects of the classroom influence the nature ofmathematical knowledge. Other studies are ofstudents being encouraged to use specificmathematical thinking skills, such as exemplification,conjecturing, and proof of the eff ects of a focus onmathematical thinking over time. These focusedstudies all suffer to some extent from the typical‘teaching experiment’ problem of being designed toencourage X and students then are obser ved to doX. Research over time would be needed todemonstrate the effects of a focus on mathematicalthinking on the nature of long-ter m learning.Longitudinal studies emphasise development ofmathematical practices, but the value of these isassumed so they are outside the scope of this paper .However, it is worth mentioning that the CAMEinitiative appeared to influence the development ofanalytical and complex thinking both withinmathematics and also in other subjects, evidenced innational test scores rather than only in study-specifictests (Johnson, Adhami and Shayer, 1988; Shayer,Johnson and Adhami, 1999). In this initiative teacherswere trained to use mater ials which had beendesigned to encourage cycles of investigation:problem familiarity, investigating the problem,synthesising outcomes of investigation, abstractingthe outcomes, applying this new abstraction to afurther problem, and so on.

• Students can get better at thinking about andanalysing mathematical situations, given suitableteaching.

Task design

Many studies of the complexity of tasks and theeffect of this on solving appear to us to be thewrong way round when they state that prob lemsare easier to solve if the tasks are stated moresimply. For a mathematics cur riculum the purposeof problem-solving is usually to learn how tomathematise, how to choose methods andrepresentations, and how to contact bigmathematical ideas – this cannot be achiev ed bysimplifying problems so that it is obvious what todo to solve them.



Students who have spent time on complexmathematical activity, such as modelling and prob lem-solving, are not disadvantaged when they are testedon procedural questions against students who havehad more preparation for these. This well-knownresult arises from several studies, such as that ofThompson and Senk (2001) in connection with theUniversity of Chicago School Mathematics Project:those given a curriculum based on problems and avariety of exploratory activities did better on open-ended and complex, multistage tasks, thancomparable groups taught in more conventionalways, and also did just as well on traditionalquestions. Senk and Thompson (2003) went on tocollect similar results from eight mathematics teachingprojects in the United States in which they lookedspecifically for students’ development of ‘basic skills’alongside problem-solving capabilities. The skills theylooked for at secondary level included traditionalareas of difficulty such as fractions computations andalgebraic competence. Each project evaluated itsfindings differently, but overall the result was thatstudents did as well or better than compar ativestudents in basic mathematical skills at theappropriate level, and were better at applying theirknowledge in complex situations. Additionally, severalprojects reported improved attainment for studentsof previously low attainment or who were ‘at risk’ insome sense. In one case , algebraic manipulation wasnot as advanced as a compar ison group taught froma traditional textbook but teachers were able tomake adjustments and restore this in subsequentcohorts without returning to a more limitedapproach. New research applying one of thesecurriculum projects in the United Kingdom is showingsimilar findings (Eade and Dickenson, 2006 a; Eadeand Dickenson, Hough and Gough, 2006 b). A U.K.research project comparing two similar schools, inwhich the GCSE results of matched samples w erecompared, also showed that those who were taughtthrough complex mathematical activity, solvingproblems and enquiring into mathematics, did betterthan students who were taught more procedurallyand from a textbook. The GCSE scripts showed thatthe former group was more willing to tackleunfamiliar mathematics questions as problems to besolved, where the latter group tended to not attemptanything they had not been taught explicitl y (Boaler,1997). Other research also supports these results(Hembree, 1992; Watson and De Geest, 2005).

• Students who spend most of their time oncomplex problems can also work out how to do ‘ordinary’ maths questions.

Recent work by Swan (2006) shows how taskdesign, based on introducing information whichmight conflict with students’ current schema andwhich also includes pedagogic design to enab le theseconflicts to be explored collabor atively, can make asignificant difference to learning. Students who hadpreviously been failing in mathematics were able toresolve conflict through discussion with other s inmatching, sor ting, relating and generating tasks. Thisled directly to improvements in conceptualunderstanding in a var iety of traditionally problematicdomains.

• Students can sort out conceptual confusions withothers if the tasks encourage them to confronttheir confusion through contradiction.

Metacognitive strategies

Success in complex mathematical tasks is associatedwith a range of metacognitive orientation andexecution decisions, but mostly with deliberateevaluating the effects of cer tain actions (Stillman andGalbraith, 1998). Reflecting on the effects of activity(to use Piaget’s ar ticulation) makes sense in themathematics context, because often the ultimategoal is to under stand relationships betweenindependent and dependent variables. It makessense, therefore, to wonder if teaching thesestrategies explicitly makes a difference to learning.Kramarski, Mevarech and Arami (2002) showed thatexplicitness about metacognitive strategies isimportant in success not only in complex authentictasks but also in quite ordinar y mathematical tasks.Kramarski (2004) went on to show that explicitmetacognitive instruction to small groups providedthem with ways to question their approach tographing tasks. They were taught to discussinterpretations of the problem, predict the outcomesof using various strategies, and decide if theiranswers were reasonable. The groups who had beentaught metacognitive methods engaged in discussionsthat were more mathematically focused, and didbetter on post-tests of gr aph interpretation andconstruction, than control groups. Discussionappeared to be a factor in their success. The value of metacognitive prompts also appear s to bestronger if students are asked to write about theirresponses; students in a randomized trial tried morestrategies if they were asked to write about themthan those who were asked to engage in think-aloudstrategies (Pugalee, 2004). In both these studies, therequirement and oppor tunity to express


metacognitive observations turned out to beimportant. Kapa (2001) studied 441 students in f ourcomputer-instruction environments which offereddifferent kinds of metacognitive prompting whilethey were working on mathematical questions:during the solution process, during and after theprocess, after the process, none at all. Those withprompts during the process were more successful,and the prompts made more diff erence to thosewith lower previous knowledge than to other s.While this was an ar tificial environment with specialproblems to solve, the finding appears to suppor tthe view that teaching (in the f orm of metacognitivereminders and suppor t) is impor tant and thatstudents with low prior knowledge can do better ifencouraged to reflect on and monitor the eff ects oftheir activity. An alternative to explicit teaching andrequests to apply metacognitive strategies is toincorporate them implicitly into the waysmathematics is done in classrooms. While there isresearch about this, it tends to be in studiesenquiring into whether such habits are adopted b ylearners or not, rather than whether they lead tobetter learning of mathematics.

• Students can sometimes do better if they arehelped to use metacognitive strategies.

• Use of metacognitive strategies may be enhanced:in small group discussion; if students are asked towrite about them; and/or if they are promptedthroughout the work.

Teaching problem-solving heuristics

The main way in which educator s and researchershave explored the question of how students can get better at problem solving is by constructingdescriptions of problem-solving heuristics, teachingthese explicitly, and comparing the test performanceof students who have and have not received thisexplicit teaching. In general, they have found thatstudents do learn to apply such heuristics, andbecome better at problem-solving than those whohave not had such teaching (e .g. Lucas, 1974). Thisshould not surprise us.

A collection of clinical projects in the 1980s (e.g.Kantowski, 1977; Lee, 1982) which appear to showthat students who are taught problem-solvingheuristics get better at using them, and those whouse problem-solving heuristics get better at problemsolving. These results are not entirely tautologous if

we question whether heur istics are useful for solvingproblems. The evidence suggests that they are (e.g.Webb, 1979 found that 13% of var iance among 40students was due to heur istic use), yet we do notknow enough about how these help or hinderapproaches to unfamiliar problems. For example, aheuristic which involves planning is no use if thesituation is so unfamiliar that the students cannotplan. For this situation, a heuristic which involvescollecting possible useful knowledge together (e.g.‘What do I know? What do I want?’ Mason, Bur tonand Stacey , 1982) may be more useful but requiressome initiative and effort and imagination to apply.The ultimate heuristic approach was probablySchoenfeld’s (1982) study of seven students in whichhe elaborated heuristics in a multi-layered way, thusshowing the things one can do while doingmathematical problem-solving to be fractal in nature,impossible to learn as a list, so that truemathematical problem-solving is a creative taskinvolving a mathematical cast of mind (Kr utetskii,1976) and range of mathematical habits of mind(Cuoco, Goldenberg and Mark, 1997) rather than alist of processes.

Schoenfeld (1979; 1982) and Vos (1976) found thatlearners taught explicit problem-solving strategies arelikely to use them in new situations compared tosimilar students who are expected to abstr actprocesses for themselves in practice examples. Thereis a clear tension here between explicit teaching andthe development of general mathematical awareness.Heuristics are little use without knowledge of when,why and how to use them. What is cer tainly true isthat if learners perform learnt procedures, then wedo not know if they are acting meaningfull y or not.Vinner (1997) calls this ‘the cognitive approach fallacy’– assuming that one can anal yse learnt behaviouralprocedures as if they are meaningful, when perhapsthey are only imitative or gap-filling processes.

Application of learnt heuristics can be seen asmerely procedural if the heur istics do not requireany interpretation that draws on mathematicalrepertoire, example spaces, concept images and soon. This means that too close a procedur al approachto conceptualization and analysis of mathematicalcontexts is merely what Vinner calls ‘pseudo’. There isno ‘problem’ if what is presented can be processedby heuristics which are so specif ic they can beapplied like algorithmically. For example, findingformulae for typical spatial-numeric sequences (acommon feature of the U.K. curriculum) is oftentaught using the heur istic ‘generate a sequence of



specific examples and look for patterns’. No initialanalysis of the situation, its variables, and relevantchoice of strategy is involved.

On the other hand, how are students to lear n howto tackle problems if not given ideas about tacticsand strategies? And if they are taught, then it is likelythat some will misapply them as they do any learntalgorithm. This issue is unresolved, but working withunfamiliar situations and being helped to reflect onthe effects of par ticular choices seem to be usefulways forwards.

There is little research evidence that students taughta new topic using problems with the explicit use oftaught heuristics learn better, but Lucas (1974) didthis with 30 students lear ning early calculus and theydid do significantly better that a ‘normal’ group whentested. Learning core curriculum concepts throughproblems is under-researched. A recent findingreported by Kaminski, Sloutsky and Heckler (2006;2008) is that learning procedurally can give fasteraccess to under lying structure than working throughproblems. Our reading of their study suggests thatthis is not a robust result, since the way theycategorise contextual problems and formalapproaches differs from those used by the researchthey seek to refute .

• Students can apply taught problem-solvingheuristics, but this is not always helpful in unfamiliarsituations if their learning has been procedural.

One puzzle which arose in the U .S. Task Panel’sreview of comparative studies of students taught indifferent ways (NMAP, 2008) is that those who havepursued what is often called a ‘problem-solvingcurriculum’ turn out to be better at tacklingunfamiliar situations using problem-solving strategies,but not better at dealing with ‘simple’ given wordproblems. How students can be better atmathematising real world problems and resolvingthem, but not better at solving given word problems?This comment conceals three impor tant issues: firstly,‘word problems’, as we have shown, can be of avariety of kinds, and the ‘simple’ kinds call on differentskills than complex realistic situations; secondly, thataccording to the studies repor ted in Senk andThompson (2003) performance on ‘other aspects’ ofmathematics such as solving word problems may nothave improved, but neither did it decline; thirdly, thatinterpretation of these findings as good or baddepends on curriculum aims6. Fur thermore, the panelconfined its enquir ies to the U.S. context and did not

take into account the Nether lands research in whichthe outcomes of ‘realistic’ activity are scaffoldedtowards formality. The familiar phrase ‘use of realworld problems’ is vague and can include a range of practices.

The importance of the difference in curriculum aimsis illustrated by Huntley, Rasmussen, Villarubi, Sangtogand Fey (2000) who show, along with other studies,that students following a curriculum focusing onalgebraic problem solving are better at prob lemsolving, especially with suppor t of graphicalcalculators, but comparable students who havefollowed a traditional course did better in a test forwhich there were no graphical calculators availableand were also more fluent at manipulatingexpressions and working algebraically without acontext. In the Boaler (1997) study, one schooleducated students to take a problem-solving view of all mathematical tasks so that what students‘transferred’ from one task to another was notknowledge of facts and methods but a generalapproach to mathematics. We described earlier howthis helped them in examinations.

• There is no unique answer to the questions of whyand when students can or cannot solve problems –it depends on the type of problem, the curriculumaim, the tools and resources, the experience, andwhat the teacher emphasises.

How can students become more systematic atidentifying variables and applying operations andinverses to solve problems? One aspect is to beclear about whether the aim is f or a formal methodof solution or not. Another is experience so thatheuristics can be used flexibly because of exposureto a range of situations in which this has to be done– not just being given equations to be solved; notjust constructing general expressions fromsequences; etc. The value of repeated exper iencemight be what is behind a f inding from Blume andSchoen (1988) in which 27 14-year-old studentswho had learnt to programme in Basic were testedagainst 27 others in their ability to solve typicalmathematical word problems in a pen and paperenvironment. Their ability to wr ite equations was no different but their ability to solve problemssystematically and with frequent review wassignificantly stronger for the Basic group. Presumablythe frequent review was an attempt to replicate thequick feedback they would get from the computeractivity. However, another Basic study which hadbroader aims (Hatfield and Kieren, 1972) implied


that strengths in problem-solving while using Basic as a tool were not universal across all kinds ofmathematics or suitable for all kinds of lear ning goal.

A subset of common problem-solving heuristics arethose that relate specifically to modelling, andmodelling can be used as a prob lem-solving strategy.Verschaffel and De Cor te (1997) working with 11-year-olds show that rather than seeing modelling-ofand modelling-for as two separate kinds of activity, acombination of the two, getting learners to framereal problems as word problems through modelling,enables learners to do as well as other groups inboth ‘realistic’ mathematical problem-solving and withword problems when compared to other groups.Their students developed a disposition towardsmodelling in all situational problems.

• Students may understand the modelling processbetter if they have to construct models ofsituations which then are used as models for new situations.

• Students may solve word problems more easily if they have experience of expressing realisticproblems as word problems themselves.

Using ICT

Students who are educated to use a vailablehandheld technology appear to be better prob lemsolvers. The availability of such technology removesthe need to do calculations, gives immediatefeedback, makes reverse checking less tedious, allowsdifferent possibilities to be explored, and gives moresupport for risk taking. If the purpose of complextasks is to show assessors that students can ‘do’calculations than this result is negativ e; if the purposeis to educate students to deal with non-routinemathematical situations, then this result is positiv e.

Evidence of the positive effects of access to and useof calculators is provided by Hembree and Dessar t(1986) whose meta-analysis of 79 research studiesshowed conclusively that students who had sustainedaccess to calculators had better pencil-and-paperand problem-solving skills and more positiveattitudes to mathematics than those without. Theonly years in which this result was not f ound wasgrade 4 in the United States, and we assume thatthis is because calculator use may make studentsreluctant to learn some algorithmic approacheswhen this is the main f ocus of the cur riculum. In the

United Kingdom, these positive results were alsofound in the 1980s in the C AN project, with theadded finding that students who could choose whichmethod to use, paper, calculator or mental, hadbetter mental skills than other s.

We need to look more closely at why this is, whatnormal obstacles to learning are overcome by usingtechnology and what other forms of learning areafforded? Doerr and Zangor (2000) recognized thathandheld calculators offered speed and facility incomputation, transformation of tasks, data collectionand analysis, visualisation, switching representations,checking at an individual level but hinderedcommunication between students. Graham andThomas (2000) achieved significant success usinggraphical calculators in helping students under standthe idea of var iable. The number of situations,observation of variation, facility for experimentation,visual display, instant feedback, dynamicrepresentation and so on contr ibuted to this.

• Students who can use available handheldtechnology are better at problem solving and havemore positive views of mathematics.

We do not know if it is only in interactive computerenvironments that school students can develop adeep, flexible and applicable knowledge of functions,but we do know that the affordances of such ICTenvironments allow all students access to a widevariety of examples of functions, and gives them the exploratory power to see what these mean inrelation to other representations and to see theeffects on one of changing the other. Thesepossibilities are simply not available within thenormal school time and place constr aints withouthands-on ICT. For example, Godwin andBeswetherick (2002) used graphical software toenhance learners’ understanding of quadraticfunctions and point out that the ICT enab les thelearning environment to be str uctured in ways thatdraw learners’ attention to key characteristics andvariation. Schwarz and Her shkowitz (1999) find thatstudents who have consistent access to such toolsand tasks develop a strong reper toire of prototypicalfunctions, but rather than being limited by these canuse these as levers to develop other functions, applytheir knowledge in other contexts and lear n aboutthe attributes of functions as objects in themselv es.

Software that allows learners to model dynamicexperiences was developed by Kaput (1999) and theintegration of a range of physical situations,



represented through ICT, with mental modellingencouraged very young students to use algebr a topose questions, model and solve questions. Enteringalgebraic formulae gave them immediate feedbackboth from graphs and from the representations ofsituations. In extended teaching exper iments withupper primary students, Yerushalmy encouraged themto think in terms of the events and processes inherentin situations. The software she used emphasisedchange over small intervals as well as overall shape.This approach helped them to understandrepresentations of quantities, relationships amongquantities, and relationships among therepresentations of quantities in single var iablefunctions (Yerushalmy, 1997). Yerushalmy claims thatthe shifts between pointwise and holistic views offunctions are more easily made in technologicalenvironments because, perhaps, of the easy availabilityof several examples and feedback showing translationbetween graphs, equations and data sets. She thengave them situations which had more than one inputvariable, for example the cost of car rental which ismade up of a daily rate and a mileage r ate. This kindof situation is much harder to analyse and representthan those which have one independent and onedependent variable. To describe the effects of the firstvariable the second var iable has to be invariant, andvice versa. In discussion, a small sample of studentstried out relations between various pairs of variablesand decided, for themselves, that two of the var iableswere independent and the f inal cost depended onboth of them. They then tried to draw separategraphs in which one of the var iables was controlled.We are not claiming that all students can do this bythemselves, but that these students could do it, isremarkable. This study suggests that students forwhom the ideas of var iables, functions, graphs andsituations are seen as connected have the skills toanalyse unfamiliar and more complex situationsmathematically. Nemirovsky (1996) suggests anotherreason is that students can relate diff erentrepresentations to understand the story the graph isrepresenting. He undertook a multiple representationteaching experiment with 15- and 16-year-olds inwhich graphs were generated using a toy car and amotion detector. Having seen the connection betweenone kind of movement and the graph, students werethen asked to predict graphs for other movements,showing how their telling of the stor y of themovement related to the gr aphs they were drawing.Students could analyse continuous movement thatvaried in speed and direction by seeing it to be asequence of segments, then relate segments ofmovement to time, and then integrate the segments

to construct a continuous graph. Additionally,comparing the real movement, their descriptions of it,and graphs also enabled them to correct and adjusttheir descriptions. Nemirovsky found that switchingbetween these representations helped them to seethat graphs told a continuous story about situations.Rather than expressing instances of distance atparticular times, the students were talking aboutspeed, an interpretation of rate from the graph ratherthan a pointwise use of it. Using a similar approachwith nine- and ten-year-olds Nemirovsky found thatthese students were more likely to ‘read’ symbolicexpressions as relationships between variables ratherthan merely reading them from left to right as childrentaught traditionally often do.

Students at all levels can achieve deep understandingof concepts and also lear n relevant graphing andfunction skills themselves, given the power to see theeffects of changes in multiple representations, takingmuch less time than students taught onl y skills andprocedures through pencil-and-paper methods(Heid, 1988; Ainley and her colleagues, e.g. 1994).

• Computer-supported multiple representationalcontexts can help students understand and usegraphs, variables, functions and the modellingprocess.


Recommendations

For curriculum and practice

The following recommendations for secondarymathematics teaching draw on the conclusionssummarized above.

Learning new concepts• Teaching should take into account students’ natural

ways of dealing with new perceptual and verbalinformation (see summaries above), including thoseways that are helpful for new mathematical ideasand those that obstruct their learning.

• Schemes of work and assessment should allowenough time for students to adapt to newmeanings and move on from earlier methods andconceptualisations; they should give time for newexperiences and mathematical ways of working tobecome familiar in several representations andcontexts before moving on.

• Choice of tasks and examples should bepurposeful, and they should be constructed to help students shift towards understanding newvariations, relations and properties. Such guidanceincludes thinking about learners’ initial perceptionsof the mathematics and the examples offered.Students can be guided to focus on critical aspectsby the use of controlled variation, sorting andmatching tasks, and multiple representations.

• Students should be helped to balance the need forfluency with the need to work with meaning.

Applications, problem-solving, modelling,mathematical thinking• As above, teaching should take into account

students’ natural ways of dealing with newperceptual and verbal information (see summariesabove), including those ways that are helpful for new mathematical ideas and those that obstructtheir learning.

• Schemes of work should allow for students to havemultiple experiences, with multiple representations,over time to develop mathematically appropriate‘habits of mind’.

• The learning aims and purpose of tasks should beclear : whether they are to develop a broadermathematical repertoire; to learn modelling and

problem-solving skills; to understand the issueswithin the context better etc.

• Students need help and experience to know whento apply formal, informal or situated methods.

• Students need a repertoire of appropriatefunctions, operations, representations andmathematical methods in order to become goodapplied mathematicians. This can be gained throughmultiple experiences over time.

• Student-controlled ICT supports the developmentof knowledge about mathematics and itsapplications; student-controlled ICT also providesauthentic working methods.

For policy

• These recommendations indicate a trainingrequirement based on international research aboutlearning, rather than merely on implementation of anew curricula.

• There are resource implications about the use ofICT. Students need to be in control of switchingbetween representations and comparisons ofsymbolic expression in order to understand thesyntax and the concept of functions. The UnitedKingdom may be lagging behind the developedworld in exploring the use of spreadsheets,graphing tools, and other software to supportapplication and authentic use of mathematics.

• The United Kingdom is in the forefront of newschool mathematics curricula which aim to preparelearners better for using mathematics in theireconomic, intellectual and social lives. Uninformedteaching which focuses only on methods and test-training is unlikely to achieve these goals.

• Symbolic manipulators, graph plotters and otheralgebraic software are widely available and used toallow people to focus on meaning, application andimplications. Students should know how to usethese and how to incorporate them intomathematical explorations and extended tasks.

• A strong message emerging about learningmathematics at this level is that students needmultiple experiences over time for new-to-themways of thinking and working to become habitual.



For research

• There are few studies focusing on the introduction of specific new ideas, based on students’ existingknowledge and experience, at the higher secondarylevel. This would be a valuable research area. Thisrelates particularly to topics which combine conceptsmet earlier in new ways, such as: trigonometry,quadratics and polynomials, and solving simultaneousequations. (There is substantial research aboutcalculus beyond the scope of this paper.)

• There are many studies on the development ofmodelling and problem-solving skills, but a valuablearea for research, particularly in the new U.K.context at 14–19, would be the relationshipbetween these and mathematical conceptualdevelopment which, as we have shown above,involves similar – not separate – learning processesif it is to be more than trial-and-error.

• There is little research which focuses on thetechnicalities of good mathematics teaching, and it would be valuable to know more about: use ofimagery, the role of visual and verbal presentations,development of mathematical thinking,development of geometrical reasoning, howrepresentations commonly used in secondarymathematics influence learning, and how and whysome students manage to avoid over-generalisingabout facts, methods, and approaches.

• There is very little research on statistical reasoning,non-algebraic modelling, and learning mathematicswith and without symbolic manipulators.

Endnotes1 ‘Induction’ here is the process of devising plausib le

generalisations from several examples, not mathematicalinductive reasoning.

2 They claimed that the post-test was contextual because objectswere used, but the relations between the objects werespurious so the objects functioned as symbols r ather than ascontextual tools.

3 There a little research on inter preting problems in statisticalterms, but this is beyond the scope of this paper.

4 Modelling has other meanings as well in mathematicseducation, such as the provision or creation of visual and tactilemodels of mathematical ideas, but here we are sticking to whatmathematicians mean by modelling.

5 Also, as is recognised in the Realistic Mathematics Educationand some other projects, students are able to engage in ad hocproblem solving from a young age.

6 Meta-analysis of the studies they used is bey ond the scope ofthis review.

Acknowledgements

This paper was produced with the help of Nusr atRizvi who did much of the technical work.


Ainley, J. (1994) Building on children’s intuitions aboutline graphs. In J. da Ponte and J. Matos (eds.)Proceedings of the 18th annual conference of the International Group for the Psychology ofMathematics Education vol. 1, pp. 1-8, Universityof Lisbon, Portugal.

Ainley, J., Pratt, D., and Nardi, E. (2001). Normalising:children’s activity to construct meanings fortrend.’ Educational Studies in Mathematics , 45 (1-3), 131-146.

Alexander, P., White, C. and Daugherty, M. (1997)Analogical reasoning and early mathematicslearning. Pp. 117-148. In English (ed.)Mathematical reasoning: analogies, metaphors andimages. Mahwah, NJ: Erlbaum.

Anderson, J. R., Corbett, A. T., Koedinger, K., and Pelletier,R. (1995). Cognitive tutors: Lessons learned. TheJournal of Learning Sciences, 4, 167-207.

Ballew, Hunter ; Cunningham, James W. (1982)Diagnosing strengths and weaknesses of sixth-grade students in solving word problems Journal forResearch in Mathematics Education 13(3) 202-210

Bassler, O., Beers, M. and Richardson, L. (1975)Comparison of two instructional strategies forteaching the solution of verbal problems Journal forResearch in Mathematics Education 6(3)170-177.

Bednarz, N. and Janvier, B. (1996). Emergence anddevelopment of algebra as a problem solvingtool: Continuities and discontinuities witharithmetic. In N. Bednarz, C. Kieran and L. Lee, L.(Eds.), Approaches to Algebra (pp. 115-136).Dordrecht/Boston/London: Kluwer AcademicPublishers.

Blume, G. and Schoen, H. (1988).MathematicalProblem-Solving Performance of Eighth-GradeProgrammers and Nonprogrammers Journal forResearch in Mathematics Education 19(2) 142-156

Boaler, J. (1997) Experiencing School Mathematics:Teaching Styles, Sex and Setting. Open UniversityPress: Buckingham, England

Bock, D. De; Verschaffel, L.; Janssens, D. (1998) ThePredominance of the Linear Model in Secondar ySchool Students’ Solutions of Word ProblemsInvolving Length and Area of Similar Plane FiguresEducational Studies in Mathematics 35(1) 65-83

Boero, P. (2001). Transformation and Anticipation asKey Processes in Algebraic Problem Solving. InSutherland, R., Rojano, T., Bell, A. and Lins, R.(eds.)Perspectives on School Algebra. pp. 99-119Dordrecht, Kluwer.

Booth, Lesley R.(1981) Child-Methods in Secondar yMathematics Educational Studies in Mathematics12(1) 29-41

Brainerd, C. J., and Reyna, V. F. (1993). Memoryindependence and memory interference incognitive development. Psychological Review, 100,42-67.

Caldwell, J., Goldin, G. (1987) Variables AffectingWord Problem Difficulty in Secondar y SchoolMathematics Journal for Research in MathematicsEducation 18(3) 187-196.

Campbell, K.; Collis, K.; Watson, J. (1995) VisualProcessing during Mathematical Problem SolvingEducational Studies in Mathematics 28(2) 177-194

Carlson, M., Jacobs, S., Coe, E., Larsen, S. and Hsu, E.(2002) Applying Covariational Reasoning WhileModeling Dynamic Events: A Framework and aStudy Journal for Research in Mathematics 33(5)352-378

Carpenter, T.; Ansell, E.; Franke, M.; Fennema, E.;Weisbeck, L. (1993) Models of Problem Solving: AStudy of Kindergarten Children’s Problem-Solving Processes Journal for Research inMathematics Education 24(5) 428-441

Chik, P. and Lo. M. (2003) Simultaneity and theEnacted Object of Learning. In F. Mar ton and A.B. M. Tsui (Eds.), Classroom discourse and the spaceof learning (pp.89-110) Mahwah, New Jersey:Lawrence


References


Clement, J. (1982) Algebra Word Problem Solutions:Thought Processes Under lying a CommonMisconception Journal for Research in MathematicsEducation 13(1) 16-30

Clements, M. (1980) Analyzing children’s errors onwritten mathematical tasks. Educational Studies inMathematics, 11 (1), 1-21

Confrey, J. and Smith, E. (1994) ExponentialFunctions, Rates of Change , and the MultiplicativeUnit Educational Studies in Mathematics 26(3)135-164

Cooper, B. and Dunne, M) (2000) Assessing Children’sMathematical Knowledge: Social Class , Sex andProblem-Solving, Maidenhead: Open UniversityPress

Cooper, B. & Harries, T. (2002) Children’s Responsesto Contrasting ‘Realistic’ Mathematics Problems:Just How Realistic Are Children Ready to Be?Educational Studies in Mathematics 49(1) 1-23

Cuoco, A., Goldenberg, E. P., and J. Mark. (1997).‘Habits of Mind: an organizing principle formathematics curriculum.’ Journal of MathematicalBehavior, 15(4), 375-402.

Dickson, L. (1989) Equations. In D. Johnson (ed.)Children’s Mathematical Frameworks 8 to 13: astudy of classroom teaching. (p.151-190) Windsor:NFER-nelson

Doerr, H. and Zangor, R. (2000) Creating meaning forand with the graphing calculator EducationalStudies in Mathematics 41(2) 143-163

Eade, F. and Dickinson, P. (2006a) Exploring RealisticMathematics Education in English Secondar ySchools. Proceedings PME July 2006

Eade, F., Dickinson, P., Hough S. and Gough S (2006b)Developing Mathematics through Contexts: year 2.Annual Report September 2006www.partnership.mmu.ac.uk/cme

English, L. and Halford, G. (1995) MathematicsEducation: Models and Processes . Mahwah, NJ:Lawrence Erlbaum Associates

English, L. and Sharry, P. (1996) Analogical Reasoningand the development of algebraic abstraction.Educational Studies in Mathematics, Vol. 30, No.2.135-157.

Fischbein, E. (1987) Intuition in science andmathematics : an educational approach . Dordrecht;D. Reidel

Foxman, D., Ruddock, G., Joffe, L., Mason, K., Mitchell,P., and Sexton, B. (1985) A Review of Monitor ing inmathematics 1978 to 1982, Assessment ofPerformance Unit, Depar tment of Education andScience, London.

Freudenthal, H. (1991) Didactical Phenomenology ofMathematical Structures, Dordrecht: D. Reidel.

Friel, S., Curcio, F. and Bright, G. (2001) Making Senseof Graphs: Critical Factors InfluencingComprehension and Instructional ImplicationsJournal for Research in Mathematics Education32(2), 124-158.

Godwin,S. and Beswetherick, R. (2002) Reflectionson the role of task str ucture in learning theproperties of quadratic functions with ICT - adesign initiative. In S. Pope (ed.) Proceedings of theBritish Society for Research into LearningMathematics day conference 16 November,Nottingham University, British Society forResearch into Learning Mathematics. pp. 43-48.

Goldin, G. (2002) Representation in mathematicallearning in English, L (ed.) Handbook ofinternational research in mathematics education(pp. 197-218) Mahwah, NJ: Erlbaum.

Graham, A. and ; Thomas, M. (2000) Building aversatile understanding of algebraic variables witha graphic calculator Educational Studies inMathematics 41(3) 265-282

Gravemeijer, K. and Doorman, M. (1999) ContextProblems in Realistic Mathematics Education: ACalculus Course as an Example EducationalStudies in Mathematics 39(1-3) 111-129

Halford, G. (1999) The Development of IntelligenceIncludes the Capacity to Process Relations ofGreater Complexity.in M. Anderson (ed.) TheDevelopment of Intelligence pp. 193-213 Hove:Psychology Press

Hart, K. (Ed.) (1981) Children’s Understanding ofMathematics 11-16, Murray, London..

Hatfield, L. and Kieren, T. (1972) Computer-AssistedProblem Solving in School Mathematics Journal forResearch in Mathematics Education 3(2) 99-112

Heid, M. (1988) Resequencing Skills and Concepts inApplied Calculus Using the Computer as a ToolJournal for Research in Mathematics Education19(1) 3-25

Hembree, Ray (1992) Experiments and RelationalStudies in Problem Solving: A Meta-AnalysisJournal for Research in Mathematics Education23(3) 242-273

Hembree, Ray; Dessar t, Donald J. (1986) Effects ofHand-Held Calculators in Precollege MathematicsEducation: A Meta-Analysis Journal for Research inMathematics Education 17(2) 83-99

Hershkowitz, R., Schwarz, B.and Dreyfus, T. (2001)Abstraction in Context: Epistemic Actions. Journalfor Research in Mathematics Education , 32 (2)195-222


Heuvel-Panhuizen, M. van den (2003) The DidacticalUse of Models in Realistic MathematicsEducation: An Example from a LongitudinalTrajectory on Percentage Educational Studies inMathematics 54(1) 9-35

Holyoak, K. and Thagard, P. (1995) Mental Leaps:Analogy in Creative Thought. Cambridge, MA:Bradford Book, MIT Press

Huntley, M., Rasmussen, C., Villarubi, R., Sangtong, J.and Fey, J. (2000) Effects of Standards-BasedMathematics Education: A Study of the Core-plusMathematics Project Algebra and FunctionsStrand Journal for Research in MathematicsEducation 31(3) 328-361

Inhelder, B. & Piaget, J. (1959). The Growth of LogicalThinking From Childhood to Adolescence. NY: BasicBooks

Johnson, D.C., Adhami, M. and Shayer, M. (1998).Does CAME work? Summary repor t on phase 2of the Cognitive Acceleration in MathematicsEducation, CAME, project. In Proceedings of theBritish Society for Research into LearningMathematics Day Conference, Bristol, 15 Nov. 1997.London: BSRLM, 26-31.

Kaminski, J., Sloutsky, V. and Heckler, A. (2006) DoChildren Need Concrete Instantiations to Lear nan Abstract Concept? Proceedings of the XXVIIIAnnual Conference of the Cognitive Science Society,1167-1172. Mahwah, NJ: Erlbaum

Kaminski, J., Sloutsky, V. and Heckler, A. (2008) TheAdvantage of Abstract Examples in LearningMath Science 320 (454-455).

Kantowski, M. (1977) Processes Involved inMathematical Problem Solving Journal for Researchin Mathematics Education 8(3) 163-180

Kapa, E. (2001) A Metacognitive Support during theProcess of Problem Solving in a ComputerizedEnvironment Educational Studies in Mathematics47(3) 317-336

Kaput, J. (1999) Teaching and Learning a New AlgebraWith Understanding Dartmouth: University ofMassachusetts.

Kaput, J. J. (1991). Notations and representations asmediators of constructive processes. In E. vonGlasersfeld (Ed.), Radical Constructivism inMathematics Education (pp. 53-74). Netherlands:Kluwer Academic Publishers.

Kieran, C. (1992). The learning and teaching ofalgebra, in D.A. Grouws (Ed.), Handbook ofResearch on Mathematics Teaching and Learning,Macmillan, New York, p390-419.

Kieran, C., Boileau, A.and Garancon, M. (1996)Introducing algebra by means of a technology-supported, functional approach. In Bednarz, N.,Kieran C., Lee, L. (eds.) Approaches to algebra:perspectives for research on teaching . Pp. 257-294Kluwer, Dordrecht

Kintsch, W. (1986) Learning from text. Cognition andInstruction 3(2).

Knuth, E. J. (2000). Student understanding of thecartesian connection: An exploratory study.Journal for Research in Mathematics Education,31(4), 500-508.

Kouba, V. L. (1989). Children’s solution strategies forequivalent set multiplication and divisionproblems. Journal for Research in MathematicsEducation, 20, 147-158.

Kramarski, B. (2004) Making sense of graphs: Doesmetacognitive instruction make a difference onstudents’ mathematical conceptions andalternative conceptions? Learning and Instruction14, 593-619.

Kramarski, B., Mevarech, Z. and Arami, M. (2002) TheEffects of Metacognitive Instruction on SolvingMathematical Authentic Tasks Educational Studiesin Mathematics 49(2) 225-250

Krutetskii, j, V. (1976). The psychology of mathematicalabilities in school children. Chicago: University ofChicago Press.

Lamon, S. (1998) Algebra: Meaning throughmodeling. In A. Oliver and K. Newstead (eds.)Proceedings of the 22nd Conference of theInternational Group for the Psychology ofMathematics Education, vol. 3, pp.167-174,University of Stellenbosch: Stellenbosch.

Lee, K. (1982) Four th Graders’ Heuristic Problem-Solving Behavior Journal for Research inMathematics Education 13(2) 110-123

Lesh, R. and Doerr, H. (2003). Foundations of amodels and modelling per spective onmathematics teaching, learning and problemsolving. In R. Lesh and H. Doerr (Eds.) Beyondconstructivisim: A models and modeling per spectiveson mathematics problem solving, learning andteaching.. Mahwah, NJ: Lawrence Earlbaum.

Lucas, J. (1974) The Teaching of Heuristic Problem-Solving Strategies in Elementar y Calculus Journalfor Research in Mathematics Education 5(1) 36-46.

MacGregor, M. and Stacey, K. (1993) CognitiveModels Underlying Students’ Formulation ofSimple Linear Equations Journal for Research inMathematics Education, Vol. 24, No. 3.,217-232.

Mason, J. Bur ton L. and Stacey K. (1982) ThinkingMathematically, Addison Wesley, London.



Nathan, M. and Koedinger, K. (2000) Teachers’ andResearchers’ Beliefs about the Development ofAlgebraic Reasoning Journal for Research inMathematics Education 168-190

Nemirovsky, R. (1996) Mathematical nar ratives. InN. Bednarz, C. Kieran and L. Lee, L. (Eds.),Approaches to Algebra (pp.197-223)Dordrecht/Boston/London: Kluwer AcademicPublishers.

Nesher, P. and M. Sukenik (1991). The Effect ofFormal Representation on the Lear ning of RatioConcepts. Learning and Instruction 1: 161 - 175.

Nesher, P., Hershkovitz, S. and Novotna, Jarmila(2003) Situation Model, Text Base and What Else?Factors Affecting Problem Solving EducationalStudies in Mathematics 52(2) 151-176


O’Callaghan, B. (1998) Computer-Intensive Algebraand Students’ Conceptual Knowledge ofFunctions Journal for Research in MathematicsEducation 29(1) 21-40

Polya, G. (1957) How to Solve It , 2nd ed., PrincetonUniversity Press.

Pugalee, D. (2004) A Comparison of Verbal andWritten Descriptions of Students’ ProblemSolving Processes Educational Studies inMathematics 55(1) 27-47

Runesson, U., and Mok, I. A. C. (2004). Discernmentand the Question, ‘What can be learned?’ In F.Marton and A. B. M. Tsui (Eds.), Classroomdiscourse and the space of lear ning (pp. 63-87).Mahwah, New Jersey: Lawrence ErlbaumAssociates, Inc., Publishers.

Ryan, J. and Williams, J. (2007) Children’s Mathematics4 -15: learning from errors and misconceptionsMaidenhead Open University Press.

Schoenfeld, A. (1979) Explicit Heur istic Training as aVariable in Problem-Solving Performance Journal forResearch in Mathematics Education 10(3) 173-187.

Schoenfeld, A. (1982) Measures of Problem-SolvingPerformance and of Problem-Solving Instruction.Journal for Research in Mathematics Education 13(1) 31-49

Schwarz, B., Hershkowitz, R> (1999) Prototypes:Brakes or Levers in Learning the FunctionConcept? The Role of Computer Tools Journal forResearch in Mathematics Education 30(4) 362-389

Senk, S. L. & Thompson, D. R. (2003). Standards-basedschool mathematics curricula: What are they? Whatdo students learn? Mahwah, NJ: Lawrence Erlbaum

Sfard, A. (1991). On the dual nature of mathematicalconceptions: reflections on processes and objectsas different sides of the same coin. EducationalStudies in Mathematics , 22, 1-36.

Sfard, A. and Linchevski, L. (1994) The Gains and thePitfalls of Reification: The Case of AlgebraEducational Studies in Mathematics 26(2-3) 191-228

Shayer, M., Johnson, D.C. and Adhami, M. (1999).Does CAME work (2)? Report on Key Stage 3results following the use of the CognitiveAcceleration in Mathematics Education, CAME,project in Year 7 and 8. In Proceedings of theBritish Society for Research into LearningMathematics Day Conference, St. Martin’s College,Lancaster, 5 June London: BSRLM, 79-84.

Silver, E. (1981) Recall of Mathematical ProblemInformation: Solving Related Problems Journal forResearch in Mathematics Education 12(1) 54-64

Slavit, D. (1997) An Alternate Route to theReification of Function Educational Studies inMathematics 33(3) 259-281

Squire, S., Davies, C. and Bryant, P. (2004) Does thecue help? Children’s understanding of multiplicativeconcepts in different problem contexts BritishJournal of Educational Psychology, 515-532

Steele, D. and Johanning, D. (2004) A Schematic-Theoretic View of Problem Solving andDevelopment of Algebraic Thinking EducationalStudies in Mathematics 57(1) 65-90

Stillman, G. and Galbraith, P. (1998) ApplyingMathematics with Real World Connections:Metacognitive Characteristics of SecondaryStudents Educational Studies in Mathematics36(2) 157-195

Swafford, J. and Langrall, C. (2000).PreinstructionalUse of Equations to Describe and RepresentProblem Situations Journal for Research inMathematics Education 31(1) 89-112

Swan, M. (2006) Collaborative Learning inMathematics: A Challenge to Our Beliefs andPractices London: National Institute of AdultContinuing Education.

Sweller, J. and Leung-Mar tin, L. (1997) Learning fromequations or words. Instructional Science, 25(1) 37-70.

Tall, D. O., (2004), Thinking through three worlds ofmathematics, Proceedings of the 28th Conferenceof the International Group for the Psychology ofMathematics Education, Bergen, Norway.

Tall, D.O. & Vinner S. (1981), Concept image andconcept definition in mathematics, with specialreference to limits and continuity, EducationalStudies in Mathematics , 12, 151-169.


Tanner, H. and Jones, S. (1994) Using Peer and Self-Assessment to Develop Modelling Skills withStudents Aged 11 to 16: A Socio-ConstructiveView Educational Studies in Mathematics 27(4)413-431

Thompson, D. and Senk, S. (2001) The Effects ofCurriculum on Achievement in Second-YearAlgebra: The Example of the University ofChicago School Mathematics Project Journal forResearch in Mathematics Education 32(1) 58-84

Tirosh, D. and Stavy, R. (1999) Intuitive rules: A wayto explain and predict students’ reasoningEducational Studies in Mathematics 38(1-3) 51-66

Treffers, A. (1987) Three dimensions. A model of goaland theory description in mathematics instr uction –The Wiskobas project. Dordrecht: Reidel

Trelinski, G (1983) Spontaneous mathematization ofrsituations outside mathematics EducationalStudies in Mathematics 14(3) 275-284

Trzcieniecka-Schneider, I. (1993) Some remarks oncreating mathematical concepts EducationalStudies in Mathematics 24(3) 257-264.

Van Dooren, W., De Bock, D., Janssens, D. andVerschaffel, L. (2007) Pupils’ over-reliance onlinearity: A scholastic effect? British Journal ofEducational Psychology, 77 (2) 307-321.

Van Dooren, W.; De Bock, D.,Weyers, D. andVerschaffel, L. (2004) The predictive power ofintuitive rules: A critical analysis of the impact of‘More A-More B’ and ‘Same A-Same B’ EducationalStudies in Mathematics 56(2/3) 179-207

Vergnaud, G. (1983). Multiplicative structures. In R.Lesh and M. Landau (Eds.), Acquisition ofmathematics concepts and processes. New York:Academic Press.

Verschaffel, L. and De Cor te, E. (1997) TeachingRealistic Mathematical Modeling in theElementary School: A Teaching Experiment withFifth Graders Journal for Research in MathematicsEducation 28(5) 577-601

Verschaffel, L., De Corte, E. and Vierstraete, H. (1999)Upper Elementary School Pupils’ Difficulties inModeling and Solving Nonstandard AdditiveWord Problems Involving Ordinal NumbersJournal for Research in Mathematics Education30(3) 265-285

Vicente, S., Orrantia, J. and Verschaffel, L. (2007)Influence of situational and conceptual rewordingon word problem solving British Journal ofEducational Psychology, 77(4) 829-848

Vinner, S. (1997) The Pseudo-Conceptual and thePseudo-Analytical Thought Processes inMathematics Learning Educational Studies inMathematics 34(2) 97-129

Vos, K. (1976) The Effects of Three InstructionalStrategies on Problem-Solving Behaviors inSecondary School Mathematics Journal forResearch in Mathematics Education 7(5) 264-275

Watson, A. & De Geest, E. (2005) Principled teachingfor deep progress: improving mathematical learningbeyond methods and materials. Educational Studiesin Mathematics, 58 (2), 209-234.

Webb, N. (1979) Processes, Conceptual Knowledge,and Mathematical Problem-Solving Ability Journalfor Research in Mathematics Education 10(2) 83-93.

White, P. and Mitchelmore, M. (1996) ConceptualKnowledge in Introductory Calculus Journal forResearch in Mathematics Education 27(1) 79-95.

Wollman, W. (1983) Determining the Sources ofError in a Translation from Sentence to EquationJournal for Research in Mathematics Education14(3) 169-181

Yerushalmy, M. (1997) Designing representations:Reasoning about functions of two variables.Journal for Research in Mathematics Education 28(4), 431-466

Yerushalmy, M. and Gilead, S. (1999) Structures ofConstant Rate Word Problems: A FunctionalApproach Analysis Educational Studies inMathematics 39(1-3) 185-203

Yerushalmy, M.. and Schwartz, J. (1993), Seizing theOpportunity to Make Algebra Mathematically andPedagogically Interesting, in T. Romberg, E.Fennema and T. Carpenter, (Eds.), IntegratingResearch on the Graphical Representation ofFunctions, Lawrence Erlbaum Associates, Hillsdale,New Jersey.

Zazkis, R. (1999).Intuitive rules in number theory:example of the more of A, the more of B r uleimplementation. Educational Studies inMathematics, 40(2)197-209




ISBN 978-0-904956-74-0


Paper 8: Methodological appendixBy Terezinha Nunes, Peter Bryant, and Anne Watson, University of Oxford




8

2 SUMMARY – PAPER 2: Understanding whole numbers2 Paper 8: Methodological appendix












ContentsAppendix to Papers 1 to 7 2

List of journals consulted for Papers 2 to 5 6

List of journals consulted for Papers 6 and 7 6

Reviews and collections used for algebra 6

Large-scale studies used for Papers 6 and 7 6

References for Appendix 7

About the authorsTerezinha Nunes is Professor of EducationalStudies at the University of Oxford. Peter Bryant is Senior Research Fellow in theDepartment of Education, University of Oxford.Anne Watson is Professor of MathematicsEducation at the University of Oxford.


About this review


This review was conceived as standing between aresearch synthesis and a theoretical review. ‘Researchsyntheses focus on empirical studies and seek tosummarize past research by drawing overallconclusions from many separate investigations thataddress related or identical hypotheses. The researchsynthesis hopes to present the state of kno wledgeconcerning the relation(s) of interest and to highlightimportant issues that research has left unresolved’(Cooper, 1998, p. 3). In a theoretical review, the aimis to present theor ies offered to explain a par ticularphenomenon and to compare them in breadth,internal consistency, and the empir ical support thatthey find in empir ical studies. ‘Theoretical reviews willtypically contain descriptions of cr itical experimentsalready conducted or suggested, assessments ofwhich theory is most powerful and consistent withknown relations, and sometimes reformulations orinteractions or both of abstr act notions fromdifferent theories.’ (Cooper, 1998, p. 4).

It was quite clear to us that a review that aims to answer the question ‘how children learnmathematics, ages 5 to 16’ could not be treated as a straightforward research synthesis. The aim of aresearch synthesis is usually more restr icted than this.For example, a research synthesis in education mighttry to examine the effect of one var iable on another(e.g. the effect of reading aloud on children’s literacylearning; Blok,1999; Bus, van IJzendoorn, andPellegrini, 1995) or the conditions under which aparticular educational practice can be said to work(e.g. the effect of phonological or morphologicalinstruction on literacy learning; Bus, and vanIJzendoorn, (1999); Ehri, Nunes, Stahl, and Willows,2001; Reed, 2008). Such searches star t frompreviously defined variables, the incorporation ofwhich in a study can easil y be identified in a searchthrough the literature. A review of the literature that

starts with a much broader question cannot use thesame conception of how the literature search will be carried out. The variables to be analysed are notconceived from the start and one of the aims ofaddressing such a broad question is in fact to clar ifyhow mathematics learning could be conceptualised.

Theoretical syntheses have broader aims, which are in some ways similar to the aims adopted in thissynthesis, but the current conception of theoreticalsyntheses can only be par tially adopted in this review.Although there are occasionally alternative views ofhow a par ticular aspect of children’s mathematicslearning can be explained, the notion of cr iticalexperiments to assess which theor y is more powerfulcannot easily be met when we try to understandhow children learn mathematics. The very conceptionof what it is that one is tr ying to explain var ies evenwhen the same words are used to descr ibe the focusof the research. In the second paper in this review ,we try to show exactly this. There are two alternativetheories about children’s understanding of number indevelopmental psychology but the phenomenon thatthey are tr ying to explain is not the same: Piaget’stheory focuses on children’s understanding ofrelations between quantities and Gelman’s theory on children’s counting skills. For older children, theproblem becomes even more complex becausethere are alternative views of the nature and contentof mathematical learning, and the role of pedagogymakes the notion of cr itical experiment eitherimpossible or inapplicable. This is true of all researchinto secondary mathematics and reflects a changefrom seeing mathematics as the formalisation andextension of children’s quantitative and spatialdevelopment to seeing learning mathematics ascoming to understand abstract tools which canprovide new formal and analytical perspectives onthe world.

Appendix to papers 1 to 7


We did not approach this synthesis as a systematicreview but as an attempt to summar ise and developsome of the main ideas that are par t of research and theory about how children learn mathematics.Within this perspective, we defined some inclusionand exclusion criteria from the outset.

Inclusion criteria

1 Theoretical explanations regarding how childrenlearn mathematics which have been suppor ted by research. There are theoretical explanations in the domain of mathematics lear ning which were proposed without their author s providingsystematic empirical evidence. We did not considerthese latter theories in the review except asframeworks to structure the approach in theabsence of other explanations.

2 Research about children’s mathematics learning inthe age range 5 to 11 was considered when itfocused on the four domains defined as the focusof this research: children’s understanding of naturaland rational numbers, relations between quantitiesand functions, and space and its representation.These were considered the cornerstones forfurther mathematics learning in the domains of algebra, modelling and applications to highermathematical concepts; the focus of these twopapers was on students aged 12 to 16. For algebrathe available research on learning focuses onidentifying typical errors, hence showing criticalaspects of successful learning but not how thatlearning might take place. Fur ther than this welooked at teaching exper iments showing howstudents respond to different pedagogicalapproaches designed to overcome these typicaldifficulties. For modelling we intended to follow a similar approach but little was available exceptsmall-scale teaching experiments.

3 Research published in books and book chapters,journals and refereed conference proceedingswhich aim at under standing how children learnmathematics. Considering the constraints of time, the search in jour nals was limited to thoseavailable electronically and otherwise in theUniversity of Oxford. A list of journals and theiraims and scope is appended. The refereedconference proceedings of the Inter national Groupfor the Study of the Psychology of MathematicsEducation will be the only proceedings included inthe review.

Exclusion criteria

1 There are domains of research, such as histor y ofmathematics, mathematics teacher development,neuropsychological studies of adults with br aindamage who have developed mathematicsdifficulties, and studies of mathematical abilities inanimals and infants, which have not been so farconnected to a theor y of how children learnmathematics between 5 and 16 years. Thesedomains of research are excluded.

2 Research that focused on learning how to usespecific technologies rather than on howtechnologies are used by students to learnmathematics. There is a relatively large number of publications on how students learn to useparticular tools that are relevant to mathematics(e.g. calculators, number line, spreadsheets, LOGOand Cabri). Considering our aim of under standinghow children learn mathematics, we will only referto research that uses these tools when the focus ison mathematics learning (e.g. using spreadsheets tohelp students understand the concept of variable).

We did not use methodological cr iteria in the choiceof papers. Descriptive as well as experimentalresearch, qualitative or quantitative studies wereconsidered when we went through the search. Inview of the brevity of the period dedicated to thissynthesis, we did exclude mater ials that could neitherbe obtained by electronic means or in the libr ariesof the University of Oxford. There is, therefore, a biastowards papers published in English languagejournals, even though we could have readpublications in three other languages.

The search process was systematic . We used theBritish Educational Index as a star ting point for thesearch of papers in the four chapters about childrenin the age range 5 to 11. Three searches werecarried out, one for natural and rational numbers,one for geometry and one for understandingrelations and functions. We included in thesesearches three sets of key-words, the first definingthe domain of research (mathematics education andother key words from the thesaurus), the seconddefining the topic area (e .g. natural number, rationalnumber and other options from the thesaur us), andthe third defining the age parameters (throughschooling levels). Theses and one-page abstractswere excluded from the output list of ref erences at this point. The references were then checked for availability and to see whether they repor ted

4 Paper 8: Methodological appendix

research results and excluded if they were notavailable or did not repor t any research results. Werepeated this search process using Psych-inf o, a database which includes psychological research, whichhad been poor ly represented in the previous database. Finally, this initial search was complemented b ya journal by journal search of the titles listed at theend of this note . This search seemed to yield mostl yrepeated references so we considered this the endof the process of search. We also consulted booksand book chapters of works that are recognised inthe literature and previous syntheses presented inthe Handbook of Research on Mathematics Teachingand Learning. Two the Task Group Repor ts of theNational Mathematics Advisory Panel, USA, werealso consulted: the repor ts on learning processes and on conceptual knowledge. These were used as sources of references rather than for theirconclusions. In the end, approximately 200 paperswere downloaded and read by the authors.However, not all of these paper s are cited in thechapters. The references used are those which didcontribute to the development of the concepts andempirical results used in the synthesis.

For algebra, we conducted a systematic search inelectronic journals in English for refereed researcharticles using algebra as the keyword. Journals arelisted below. We did not define an age range sincewe were interested in how algebraic understandingdevelops throughout school, although this happensmainly in secondar y education. We also usedrefereed innovation studies, which show what it ispossible for learners to do, given par ticular kinds ofteaching or technology; this tells us aboutpossibilities. We restricted our use of these to studiesfor which the learning aims clearly relate to a broadview of algebra given above. For example, we didnot include self-referential studies in which, forexample, it is assumed that patter n-spotting is animportant aspect of algebra, so teaching and lear ningpattern-spotting is researched, but we would forexample include a study of teaching patter n-spottingwhere students’ ability to use pattern-spotting for ahigher level algebraic purpose was discussed as anoutcome. We also used refereed studies of students’typical errors and methods (see below). These tell uswhat needs to be lear nt and hence descr ibe thedevelopment of algebraic understanding, but nothow successful students learn it. We also drew onsignificant overviews and compilations of research onalgebra. These reviews were used as gateways toother research literature. We excluded studies whichfocus only on shor t-term fluent performance of

algebraic procedures in familiar situations unless this was linked specifically to the development ofalgebraic reasoning. Most of the studies we usedbase their claims to success on the complementar yneeds both to act fluently with symbolic expressionsand to understand them. We accessed 174 paper splus 78 references in books in addition to thereviews and studies mentioned above. Of these,about 95 were read but not all are included asreference. Some of these overlapped in theirconclusions, or added nothing or only a little to themain references.

For Paper 7, Modelling, problem-solving andintegrating concepts, an initial search using U.S. andU.K. spellings gave very few relevant results. Wetherefore broadened the search to include:modelling, problem-solving, realistic, real-life, variableand word problems. This process was iter ative asthe search for explanations for what could beinferred about students’ learning led us into otherrelated areas. Later we did fur ther searches onsome other terms which emerged as impor tant:linearity, linear assumption, equation. Finally, wesearched for papers which addressed how studentslearned combinations of concepts which b uild onelementary concepts, such as tr igonometry. In allwe located over 3200 references using Br itishEducation Index, ERIC and other sources.Fortunately many of these were not research-based, or used the ter ms in irrelevant ways, oraddressed the focus in limited ways related toyoung children. The final relevant list consisted of125 papers and a journal special issue. We usedthese papers to point to other sources. Most ofthese papers were repor ts of teaching experiments.Teaching experiments usually have a par ticularcommitment to the nature of an aspect ofmathematics and how it is best lear nt. Theexperiment is constructed to see if students will be able to do X in cer tain circumstances, and X is measured as an outcome b ut in this processknowledge of how X is learnt, and what can gowrong, can be found. In reading this literature wefound an overall coherence about students’ learningof higher mathematics and the f inal version of thepaper was constructed to show these similar ities. A list of journals accessed is included in thisappendix. There were only four reviews of researchused, two meta-analyses by Hembree, (1986; 1992)used as summaries of literature and the U.S. TaskPanel (NMAP 2008) was used as a gatewa y toother sources.



List of journals consulted forPapers 2 to 5

British Journal of Developmental Psychology British Journal of Educational Psychology Child DevelopmentCognition and InstructionEducational Studies in MathematicsEurasia Journal of Mathematics, Science andTechnology EducationInternational Electronic Journal of Mathematics

EducationInternational Journal for Mathematics and LearningInternational Journal of Science and Mathematics

EducationJournal for Research in Mathematics Education Journal for Research in Mathematics EducationMonographJournal of Educational PsychologyLearning and Instruction

List of journals consulted forPapers 6 and 7

British Journal of Developmental PsychologyBritish Journal of Educational PsychologyChild DevelopmentCognition and InstructionEducational Studies in MathematicsInternational Journal of Mathematical Education in

Science and TechnologyJournal for Research in Mathematics Education Learning and Instruction Mathematical Thinking and LearningProceedings of International Group for the

Psychology of Mathematics Education

Reviews and collections used for algebra

Bednarz, N., Kieran C., Lee, L. (eds.) Approaches toalgebra: perspectives for research on teaching .Kluwer, Dordrecht

Chick, H., Stacey, K., Vincent, J. and Vincent, J. (eds.)Proceedings of the 12th ICMI study conference: Thefuture of the teaching and learning of algebra.University of Melbourne, Australia, Dec 9-14, 2001.

Greenes, C. and Rubenstein, R. (eds.) Algebra andAlgebraic Thinking in School Mathematics. 70th

Yearbook. Reston,VA: NCTM.

Kaput, J., Carraher, D. and Blanton, M. (eds.) Algebra in the ear ly grades. New York: Erlbaum

Mason, J. and Suther land, R. (2002), Key Aspects ofTeaching Algebra in Schools, QCA, London

Nickson, M. (2000) Teaching and learningmathematics: a teacher’s guide to recent researchand its applications. London: Cassell


ZDM The International Journal of MathematicsEducation volume 40 (2008)

Large-scale studies used for Papers 6 and 7

Concepts in Secondary Mathematics and ScienceProject (CSMS) (see Har t et al., 1981)Diagnostic tests derived from clinical inter views with 30 children age 11 to 16. In these inter viewsthe test items were trialled and revised, and students’own methods and typical er rors were observed.Common errors and methods were found acrossschools which were not teacher-taught but hadarisen through students’ own reasoning. The samplefor testing was from urban, rural and city areasacross England. It was selected from volunteerschools according to IQ distr ibutions in order torepresent the countr y as a whole. About 3000students took the Algebra test.

Strategies and Errors in Secondary MathematicsProject (SESM) focused on a small number of errorsarising in the CSMS study. There used a large numberof individual inter views and some teachingexperiments involving several classes of students.

Ryan and WilliamsRyan and Williams randomly-sampled 13 000 Englishschool children from ages 4 to 15 using diagnostictests designed to reveal typical errors and child-methods, as CSMS, but with the express pur pose of identifying progress made by students inmathematics. They found little progress madebetween ages 11 to 14, and that many errors weresimilar to those found by Har t et al. 20 years earlier.See Ryan, J. and Williams, J. (2007) Children’sMathematics 4-15: learning from errors andmisconceptions, Maidenhead: Open University Press. More details of the tests can be found inMathematics Assessment for Learning and Teaching,(2005) London: Hodder and Stoughton.

6 Paper 8: Methodological appendix

Mollie MacGregor and Kaye StaceyA series of pencil and paper tests were administeredto 2000 students from a representativ e sample ofvolunteer schools in Years 7–10 (ages 11 –15) in 24Australian secondary schools.

Assessment of Performance Unit (APU) test resultsof 1979 (Foxman et al., 1981). These tests involved acohort of 12 500 students age 11 to 15 and w eredesigned to track development of mathematicalunderstanding by sampling across schools and regions.

Children’s Mathematical Frameworks study (CMF)(Johnson, 1989), 25 classes in 21 schools in theUnited Kingdom were tested to find out why andhow students between 8 and 13 cling to guess-and-check and number-fact methods rather than newformal methods offered by teachers.

References for appendix

Bus, A. G., and van IJzendoorn, M. H. (1999).Phonological awareness and ear ly reading: Ameta-analysis of experimental training studies.Journal of Educational Psychology, 91, 403-414.

Bus, A. G., van IJzendoorn, M. H., and Pellegrini, A. D.(1995). Joint book reading makes for success inlearning to read: A meta-analysis onintergenerational transmission of literacy. Reviewof Educational Research, 65, 1-21.

Blok, H. (1999). Reading to young children ineducational settings: A meta-analysis of recentresearch. Language Learning, 49, 343-371.

Cooper, H. M. (1998). Synthesizing research : a guidefor literature reviews. Thousand Oaks, CA: SagePublications.

Ehri, L. C., Nunes, S. R., Stahl, S. A., and Willows, D. M.(2001). Systematic Phonics Instr uction HelpsStudents Learn to Read: Evidence from theNational Reading Panel’s Meta-Analysis. Review ofEducational Research, 71, 393-447.

Reed, D. K. (2008). A Synthesis of MorphologyInterventions and Effects on Reading Outcomesfor Students in Grades K–12. Learning DisabilitiesResearch and Practice, 23, 36-49.




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