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Towards a unified approach toteaching and learning mathematicsLeonard John Frobisher aa Tutor in Mathematics Education, Centre for Studies inScience and Mathematics Education , School of Education,The University of Leeds , Leeds LS2, UKPublished online: 07 Jul 2006.

To cite this article: Leonard John Frobisher (1992) Towards a unified approach toteaching and learning mathematics, Early Child Development and Care, 82:1, 5-26, DOI:10.1080/0300443920820102

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© 1992 Gordon and Breach Science Publishers S.A.Printed in the United Kingdom

Towards a unified approach to teaching andlearning mathematics

LEONARD JOHN FROBISHER

Tutor in Mathematics Education, Centre for Studies in Scienceand Mathematics Education, School of Education, TheUniversity of Leeds, Leeds LS2, UK

(Received 20 May 1992)

The National Curriculum presents mathematics as a fragmented collection ofunrelated topics. Children should experience mathematics as a coherent subject.This can be achieved by viewing Mathematics as a set of systems in which existrelations and operations on data in the systems. Every system has a structurewhich manifests itself through pattern. The study of mathematics involves aconsideration of such patterns.

Children should develop a consistent approach to learning mathematics. Suchan approach involves them constructing their own data for a system. The datathey produce is recorded in a "dynamic" form enabling them to move each itemof data from place to place assisting mathematical thinking and decision making.Relationships between data are explored using the "Operational Processes" ofSorting and Ordering. Children apply these processes to many of the topics in theProgramme of Study developing a unified approach to learning mathematics.

Key words: Communication, information, pattern, structures, operational pro-cesses

THE NATURE OF MATHEMATICS

The way in which the content of a mathematics curriculum is presented can tooreadily give the impression that mathematics comprises a collection of unrelatedtopics. Commercial textbooks often reinforce this perception as one chapter or sectionfollows another without any apparent relationships being developed between differenttopics. The Programmes of Study described in the initial National Curriculumdocument (DES, 1989) also presented mathematics in a format which encouraged theview that the subject is fragmented appearing to be no more than a collection ofseparate, discrete and unrelated topics.

Both the National Curriculum Council (NCC, 1989) and the Curriculum Councilfor Wales (CCW, 1989) attempted to redress this position by suggesting that a schoolpolicy document might include "a brief statement about the nature of mathematicsand how it contributes to the overall experience of pupils in schools". It is regrettable

Address correspondence to: L.J. Frobisher, 1 East Causeway Vale, Leeds LS16 8LG, UK.

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6 L. J. FROBISHER

that the vast majority of teachers in primary education have experienced only "schoolmathematics" and their view of mathematics has been determined by this limited andnarrow deformation of the nature of the subject. Unfortunately, the twoNon-Statutory documents do little to assist teachers in enlarging their experience andperception of the subject. Reference is made in paragraph 2.3 of the NCC publication(1989) to the creative nature of mathematics which can provide children with "ameans of creating new and imaginative worlds to explore", and to its affective natureby giving children the opportunity "for intellectual excitement and an appreciation ofthe essential creativity of mathematics". The nearest the NCC (1989) comes toproviding teachers with a possible working description of mathematics is thestatement in the same paragraph that exploring and investigating within mathemat-ics "encourages pupils to explore and explain the structure, patterns and relationshipswithin mathematics".

The CCW (1989) is more forthcoming in describing mathematics as "both a bodyof knowledge and a mode of enquiry, a 'product' and a 'process'". It lists three"complementary and interconnected ways" of categorising the nature of the processaspect of mathematics:

a. Mathematics is a search for patterns and relationships.b. Mathematics is a way of solving problems.c. Mathematics is a means of analysing and communicating information or ideas.

This paper is concerned with ways in which children can experience and developthese processes of enquiry within the framework of the National Curriculum.

The role of the teacher is crucial in providing young children with experienceswhich reflect the contribution of the three processes of enquiry to developing arecognition of what mathematics is and its power, elegance and beauty.Unfortunately, many teachers create mathematical learning experiences for theirpupils with little awareness, appreciation or understanding of the nature of math-ematics. This is a sad reflection on their previous experience of the subject, both as alearner in school and when training to teach in university or college. It is asking agreat deal of teachers of young children to provide pupils with mathematical learningactivities which are aimed at developing insights into "the structure, patterns andrelationships within mathematics" when they have had very few such experiences.

A MODEL OF THE MATHEMATICS NATIONAL CURRICULUM

No attempt has been made in the more recent publication of the DES (1991) topresent the revised Programmes of Study in a way which emphasises the inter-relationships between the new Attainment Targets 2,3,4 and 5, and the role ATI canplay to highlight these relationships. Indeed, the reconstituted ATs have submergedthe importance of pattern and relationships, which in the "old" ATs had AT5devoted to its study with the aim that children were to "recognise and use patterns,relationships and sequences, and make generalisations" (DES, 1989). Figure 1

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TEACHING AND LEARNING MATHEMATICS

Figure 1 A model of the National Curriculum.

illustrates the claim that a major aspect of the nature of mathematics is a study ofpattern which exists as a thread weaving its way through ATs 2 to 5, and as such iscentral to and at the hub of mathematics learning.

The diagram also suggests that ATI, the "using and applying" AT, is allembracing in that it is an approach to learning mathematics that unifies the varietyand disparate content in the other four ATs. However, I will argue in this paper thatthe processes which bring about this unification are not made sufficiently explicit toenable the implementation of the ideas in ATI to succeed. These processes I will referto as Operational Processes which, together with the pattern searching andobserving, provide the means by which the aims of ATI can be achieved.

MATHEMATICS AS A STUDY OF PATTERN

Pattern which is to be found in all areas of mathematics occurs as a result of thestructures which exist in a mathematical system; indeed pattern is the manifestationof structure in either pictorial or symbolic form. The most primitive of mathematical

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8 L. J. FROBISHER

systems is a collection or set. Such a system offers little in the way of intellectualsatisfaction until relationships between the members of items of data which comprisethe set are explored. The simplest relations which exhibit pattern in a system arethose of sameness and difference. For children in the early years of schooling the word"sameness" is much preferred to that of "similarity" as it has a much more practicalmeaning for young children without suggestion same in every respect, i.e. identical. Italso has association with "same", just as "difference" is related to "different". Thus, iftwo logibloc shapes are blue, they are said to have the same colour; it is seldomclaimed that they are similar because they are both blue. Teachers of young childrenare familiar with and frequently use the relations sameness and difference whendiscussing with children colours, numbers and shapes. In this paper I wish to showthat these relations are fundamental to the study of mathematics at whatever level orage of the learner.

One of the earliest mathematical systems which children meet is the set of wholenumbers. These are initially seen by children as independent items of data in asystem, albeit that they are related by a counting order. Children do not immediatelyrecognise that the counting order is in fact the application of the order relation onemore than.

An excellent activity, which gives rise to the equivalence of the order of thecounting numbers and the order produced when the relation one more than is appliedthe set of whole numbers, involves children having a whole number on a card hungaround their neck with string. Each child has a different number and all the numbersup to the number in the class are included with no repeats. It is helpful if the activitytakes place in a large space such as a hall. The children walk around freely until theteacher claps her hands. They are then asked to make one line so that the numbersare in order. The children without hesitation interpret order to mean "countingorder". When the line is complete they are asked to look at and remember thenumbers on each side of them. The children are then told to walk around again untilthey hear the teacher again clap her hands. This time they are asked to hold in theirleft hand (they may need to hold up that hand first) the person who has the numberwhich is one more than the number they have. As before they are told to look at who isnext to them on each side. The activity needs to be repeated many times with thechildren changing numbers frequently. The observation of the numbers requireschildren to recognise that they have the same two numbers on either side of them onboth occasions. The activity can be repeated using the relation one less than. Thishowever, is a more difficult relation for children to understand than one more than.Young children often see the counting numbers when in order as making a pattern.This can be emphasised by referring to the counting number as making a patternbecause they are in order.

Mathematical systems become more complex when the members (data) in thesystem are combined in some way, often by a predetermined rule. Thus the set ofwhole numbers are first combined by two members being added. The rule in this casefor addition is one which is readily accepted by children as it satisfies their previousexperiences with objects which are combined by putting two sets of objects togetherand counting the set so formed. There are many patterns to be observed which are the

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TEACHING AND LEARNING MATHEMATICS 9

outcomes of combining the whole numbers under the operation of "addition". Onesuch emerges when all the pairs of numbers which when added make a given number,say 6, are put in an order. (This is often known as "The story of 6"). The followingresults:

1 + 5 = 62 + 4 = 63 + 3 = 64 + 2 = 65 + 1 = 6

The knowledge of this and similar patterns is particularly useful for reducing achild's memory load. If 3 + 3 = 6 is known, then 2 + 4 can be quickly deduced.Such a child would be thinking mathematically.

There are obviously many ways in which data in a set can be combined. Each timea new rule of combination is applied to data in a set a new system is created. Thuswhen whole numbers are subtracted a different system to that which was formedwhen the numbers were added is established. The two systems themselves thenbecome data in a set of systems and higher-order relationships between the twosystems are explored and patterns such as

5 + 2 = 7 and 7 - 5 = 2

can be observed. Figure 2 shows two systems with the dots indicating data, thebroken lines a relation and the continuous lines a combining of two items of data. Theheavy line between the two systems is suggestive of a possible higher-order relationwhich may exist between the two systems. The diagram is only illustrative and doesnot purport to show all the relations which may exist within the two systems.

Whole numbers withthe operation of

addition.

Whole numbers withthe operation of

subtraction.

Figure 2 Mathematics as a collection of systems.

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10 L. J. FROBISHER

There is a danger that because of the way the National Curriculum Programmes ofStudy are presented, a very narrow view will be taken about the nature ofmathematical systems and their structures. This is likely to strengthen a teacher'sprevious experience of mathematics which was constrained by the content of theschool curriculum and the limited horizons of those who taught it. Thus themathematical systems most teachers will have experienced when learning mathemat-ics are likely to have been restricted to those involving numbers (whole, fractional,decimal and directed), together with the four conventional operations of addition,subtraction, multiplication and division. These are very particular instances of amathematical system and if children are to encounter the true nature of mathematicsthey must be provided with experiences which enable them to recognise that thesubject is made up of an infinity of fascinating and interrelated structures. They mustalso be given the opportunity to explore the relationships within a system andbetween different systems, and on occasions deciding that two apparently differentsystems have identical structures and are really one and the same system, butpresented in a different context. Recognition of relationships within and betweensystems is a result of observing patterns which are the outcomes of a system'sstructure.

INVESTIGATING CLOWN'S HATS

Exploring mathematical structures can and should begin from the moment childrenbegin to study mathematics in a reception class. At this early age it is important that astructure is set in a context which has appeal to young children. Such an activity,which has proved successful with children from 3 years upwards, is called Clown'sHats. The mathematical aspects of the activity involves placing 3 colours in 3positions with repeats of any colour permitted. This is embedded in the context ofmaking hats for a clown. However, many teachers have found it possible to place theproblem in different contexts appropriate to the cross-curricular topic they areteaching, e.g. Buttons on Jumpers, and Hat, Pipe and Button on Snowmen.

The activity is best presented orally, but is illustrated in Figure 3 as an activitysheet.

Very young children collect the data in the system by making hats to wear andsticking on each hat three poms-poms using coloured material or soft paper. Theychoose the colours from the three available, i.e. red, blue and green. A group of thechildren wear their hats at the front of the class and the remaining children comparethe hats to decide which ones have "samenesses" and which have "differences", i.e. inwhat way are the hats the same and in what way are they different. Interestingdiscussions arise about what is meant by "same" and "different" in this context.

Older children construct their hats on ready made representations of hats, asshown in Figure 4, colouring in the "pom-poms" using coloured pens or crayons.

If this is done as a group activity children pool their data and then compare thehats to find those which are the "same" and those which are "different". As the data isin a dynamic form it can be moved about making comparisons much easier. The group

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TEACHING AND LEARNING MATHEMATICS 11

TXOWN'S HATS

A circu'i clown needs lots of different hats.

The 3 pom-poms can be red. blue or green.They can be sewn on in aiy order.

The clown can use more than one of each colour.

f M a k e a s n i a n y iI different ji hats as you canj

Figure 3 The Clown's Hats activity.

Figure 4 Representations of three hats.

attempt to discover any hats which they can make which they do not already have. Itis interesting to observe children finding "missing" hats: they intuitively collect hatstogether which are related in some way and use pattern to determine if it is possible tomake more hats.

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12 L J. FROBISHER

STRATEGIES FOR CONSTRUCTING DATA

It has been found that children, and indeed adults, fall into four categories accordingto the type of strategy they use to produce data in an investigative situation.

A Random Strategy

Children who adopt this approach do not relate any hat they make to any which theyhave previously made; each hat is made independently of any other hat. Possiblerelations or properties which hats may have in common are initially ignored.

An Incomplete Strategy Applied Consistently

This approach recognises relations between successive hats which are made, but thepattern used, although applied consistently, does not produce all the data. With theClowns' Hats children sometimes produce hats in a sequence as shown in Figure 5.

There is an obvious strategy adopted to produce this sequence of hats. However,after three applications of the "cyclic" strategy the fourth hat is the same as the first.(Will this always happen? Why? Can you see a pattern?). This can itself form thebasis of an investigation.

The strategy is incomplete unless it is repeated. Repeated use of the strategy with anew "hat" as the first of each set would eventually create all 27 hats. However, itbecomes increasingly difficult to find a hat which has not already been made as partof a previous sequence of three. Thus the strategy could complete all 27 hats, but it isdifficult for children to apply it consistently as they quickly become overwhelmed bythe amount of data they produce.

A Complete Strategy Applied Inconsistently

Many children begin with a strategy which if applied consistently would produce allthe data. However, it has been found that they become lost in the amount of data theyproduce. Young children find it difficult to sustain the application of a strategy for along period of time when the data they are producing begins to dominate their actionsand thought processes.

A Complete Strategy Applied Consistently

Although this category is one that all investigators would wish to use to produce datait frequently proves elusive at the early stages of an investigation. This is no doubtdue to it requiring some understanding of the structure of the system of which thedata is part. This understanding develops partly as a result of attempting to create thedata. Experience with young children has shown that it is not helpful to suggest thatthey should initially search for a complete strategy. Such an approach appears toinhibit children in their attempts to search for patterns and relationships whilstproducing data.

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TEACHING AND LEARNING MATHEMATICS 13

greenredblue

redbluegreen

Figure 5 A cyclic strategy.

INCOMPLETE DATA

Finding all the data is an activity in its own right and for some children is as far asthey take an investigation. For those children who are ready to explore the structureof a system it is essential that a complete set of data is available. A teacher shouldprepare data for children which should be in an attractive format retaining thedynamic nature in which the children originally created their own data. Investigatingan incomplete set of data leads to great frustration for some children. However, thereare occasions when activities with an incomplete set can lead children to recognisingwhich data are missing as they intuitively apply pattern to complete the set. Thedecision whether children should work with a complete or incomplete set of data mustbe left to each teacher to decide on the basis of their knowledge of the likely reaction ofthe children.

SORTING DATA

Sorting is a fundamental process in mathematical thinking which enables children toexplore patterns and relationships in a system. When first sorting data childrenrequire assistance in recognising possibilities. The Clown's Hats problem will be usedto illustrate the potential for sorting data which in turn highlights the structure of asystem.

Position Attributes

There are 3 positions in which the colours red, blue, and green can be placed. Thiscan be used to sort the 27 hats in many ways according to which position a particularcolour is in. Figures 6 and 7 show how the hats can be sorted using two differentsorting diagrams and the attribute or property red at the top.

The sorting takes place with the dynamic data on large grids. The most suitablesize for the grids is A3 which has sufficient room for the data and fits onto a child'sdesk. A matrix diagram is more useful than a Venn diagram as it places equalemphasis on those "hats" which do not have red at the top as those which do. It alsoallows space at the bottom of the regions to place number counters showing howmany "hats" are in each set.

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14 L. J. FROBISHER

redat the top

red notat the top

Figure 6 A matrix sorting diagram.

Figure 7 A Venn diagram.

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TEACHING AND LEARNING MATHEMATICS 15

redat the top

blueat the fop

greenat the top

Figure 8 Sorting using the colour of the top pom-pom.

Children need to repeat sorting activities which specify one of the colours in a givenposition using the other colours in the same position. Many young children areunable to predict how many hats there will be in a set and find it surprising, forinstance, that there are 9 hats with red at the top, 9 hats with blue at the top and 9hats with green at the top. These three properties can be combined on an extendedmatrix diagram as shown in Figure 8.

Relative Position Attributes

These attributes for sorting involve the position relationship between two or three ofthe colours. Examples are

red next to green,blue between red and green,green above blue.

These sorting activities can be very demanding of children's observational skillsand their understanding of relative position words. Discussion is necessary to agreewhat each of the words means. Children are here experiencing the need to adoptdefinitions which are consistent within the system and acceptable to all. Sometimes itis necessary to explore the results of other definitions for the same word as this leadsto the need to express a relationship more precisely.

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16 L. J. FROBISHER

Quantity Attributes

The number of times an aspect occurs in an item of data can be used as an attributefor sorting. For example the number of reds in each Clown's Hat can be counted andthen sorted. Figure 9 shows a partly completed sort based upon the number of redpom-poms in a hat.

Any two attributes from the three types of attributes can be combined in the onesorting diagram. A two dimensional matrix diagram, sometimes called a Carrolldiagram, is shown in Figure 10, although a Venn diagram with two intersectingenclosures can also be used.

The number of hats in each column and row has been left with a question mark forthe reader to work out, at first mentally, and then with the assistance of the"dynamic" hats if they are needed. The helpfulness of having data which can bemoved into regions and then a decision made as to whether it is in the correct positionwill be readily appreciated. The National Curriculum refers to this approach as "trialand improvement".

WHY SORT?

Every area of human knowledge applies the principles of sorting to its informationand knowledge base. Mathematics is no exception. Indeed it could be claimed thatsorting is fundamental to the study of mathematics as every learner uses the processconsciously or unconsciously in some way or other.

When children learn mathematics sorting

develops observational skillsemphasises the relations "same" and "different"brings order out of "chaos"assists thinkingmakes it easier to place data in an orderencourages pattern spottinghighlights structures in systems

Whether teachers plan for it or not children sort and form classifications as theydevelop concepts, learn new knowledge and acquire a variety of mathematical skills.Recognition that this occurs whenever learning takes place suggests that teachersshould set out positively to develop children's ability to sort. The Handling Data ATsets this out in the Programme of Study at Levels 1 and 2.

INVESTIGATING ADDITION PAIRS

The activity Clown's Hats is not to be found in the National Curriculum, although itcould be claimed that the mathematical content involved in constructing the different

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TEACHING AND LEARNING MATHEMATICS 17

Ored 1 red 2 reds 3 reds

Figure 9 Sorting using the number of reds as criterion.

Oblue 1 blue 3 blueshow

many?

red

at the top

red not

at the top

how many? 27

Figure 10 Sorting using two attributes.

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18 L. J . FROBISHER

hats is similar to the Programme of Study (DES, 1991) at Level 6 of AT5 whichstates:

Pupils should engage in activities which involve identifying all the outcomes when dealing withtwo combined events which are independent, using diagrammatic, tabular or other forms.

An activity which is, perhaps, more recognisable as relating to the NationalCurriculum at Key Stage 1, involves investigating addition pairs. Reference to thisaspect of mathematical content can be found in AT3, Level 2, where the Programmeof Study specifies that:

Pupils should engage in activities which involve exploring and using patterns in addition andsubtraction facts to 10.

As with the Clown's Hats it is more appropriate to present the activity orally whenchildren are already involved in work on early ideas in addition. Figure 11 shows theinvestigation in an activity format.

It will be noted that a constraint is placed on the sum which an addition pair mustnot exceed. This is necessary, initially, in order that children are not overwhelmed bythe amount of data, i.e. the number of pairs is manageable. As an extension of theactivity the maximum value allowed for the sum can be increased, but this produces anew system as the data is now different.

[2 is an Addition Pair. Why do you think it isit is called that?

Use the blank addition card a + • = aand the number'tablets to make as many addition sums as youcan BUT the answer must always be less than 6.

Write each Addition Pair you find on a card like this 11 +with the answer on the back like this.

r Find as manyADDITION PAIRS

I as you canIRS I

Figure 11 Addition Paris activity sheet.

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TEACHING AND LEARNING MATHEMATICS 19

The production of the addition pairs is assisted if the children are provided with ablank addition card and two sets of the numbers 1 to 9. Each number appears oncard, called a number tablet, which fits the frames on the blank card as illustrated inFigure 12.

The provision of the blank addition card and the number tablets enables childrento move the numbers around exploring possibilities. Working like this children arenot inhibited by the combination at one and the same time of the exploration and therecording in a static format before they have had the opportunity to consider the itemof data they have produced. In this way children naturally reflect upon what theyhave done, which is seldom the case when they record in a written form immediatelythey have constructed data. However, a record of the pairs they have produced is,obviously, essential. Children recognise the need to have a permanent record as theyeasily forget addition pairs which they have already made before they move thenumber tablets around to produce new pairs.

As with the Clown's Hats activity each individual addition pair is recorded on aseparate card so that the data is in a dynamic form for working with at a later stage.Thus each time a child makes an addition on the blank card it is recorded on apermanent "card" with the addition pair on one side and the answer to the additionon the reverse. Each child can work individually, or in a pair. Children can beobserved using strategies for producing pairs similar to those described in the earlierClown's Hats activity. When a random strategy is employed repeats of pairs are madeas children treat each item of data independently. Some children from the very startbegin to sort their data into sets as they construct enabling them to check whetherthey have made a pair before or not. Frequently this sorting also involves orderingmaking it easier to observe which pairs have been made and which pairs are still to bemade. Throughout the making of the data children are operating with the relationssameness and difference, and searching for patterns. Debate arises as to whether theaddition pairs, for example, 2 + 3 and 3 + 2, are the same or can be thought to bedifferent. This has to be reconciled by adopting an agreed definition. If, however, analternative definition is decided upon a different system is formed as the set of data isno longer the same.

At this stage children are seldom cognisant of their use of pattern as it is appliedintuitively while producing further data. To make them conscious of the patterns thatare part of the structure of the system of addition pairs children should be encouragedto explore ways of sorting their data. This can take place with an incomplete set orwith a full set provided for the children. The latter approach averts the frustrationmany children feel when not knowing whether they have all the data or not.

• + • =Figure 12 Number tablets and Addition card.

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20 L. J. FROBISHER

Children prove very creative in suggesting criteria or attributes for sorting thepairs, such as "7 is on the left", "the sum is 4", "the difference of the two numbers is 2". "a pairhas 0 even, or 1 even, or 2 evens". Figure 13 shows a two attribute sort which childrenoften develop for themselves.

This sort also produces an ordering of the number pairs and children observe manypatterns in the columns, in the rows and in the diagonals. The total number of pairsin the rows or the columns is the sum of the counting numbers from 1 to 4. The totalnumber of pairs is, therefore, a triangular number. Does this still pertain if themaximum sum of any pair is increased to 6, 7, 8 . . . ?

A natural extension of this activity is to ask " What would happen if 3 numbers were addedtogether with the answer being 5 or less?" Figure 14 shows all the triples sorted and orderedaccording to position critera.

Again, in the spirit of the nature of mathematics, children search for patterns in thesystem and attempt to communicate their observations to other children. Youngchildren when asked to explain a pattern respond by describing the pattern. It is as ifthey believe that to "describe" is to "explain". Teachers should however, not hesitateto ask children why they think the data has made a pattern. But they should beprepared to accept what is in effect a description of what the child observes.

1nn the left

2on the left on the left on the left

howmany?

fi'on the

right

2 + 1 3 + 1

2on theright

1+2 3 + 2 iJ

on theright

1 +3

on theright

1 +4

IIUW

many? HJ m

Figure 13 A two attribute sort using the positions of the numbers.

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TEACHING AND LEARNING MATHEMATICS 21

1on the left

I 2Ian the left

3on the left

howmany?

li+r+il1 +2 + 1

n[2 + 2 + 11

+ 1 + 2|

+ 2 + 2|

2 + 1 + 2

1 + 1 + 3

nowmany?

Figure 14 Sorting Addition Triples.

INVESTIGATING TRIANGLES

A 3 by 3 pin-board, sometimes called a geoboard, provides excellent opportunities foryoung children to construct triangles which form the data of a simple mathematicalsystem. The limited size of the board places sufficient constraints on the number oftriangles which can be made to make the exploration of the system manageable, yetfull of valuable mathematical concepts, relationships and patterns.

Children make as many "different" triangles as they can using an elastic band anda 3 by 3 pin-board. As with previous activities children quickly realise the need torecord their findings. They should be provided with separate representations of the 3by 3 pins on which each triangle can be drawn. These drawings of triangles can thenbe used to explore the structure of the system.

Children soon recognise that it is necessary to decide when two triangles are thesame or can be considered to be different. For example, are the two triangles shown inFigure 15 the same or different?

Young children usually decide that triangles which we would say are congruent aredifferent because they are in a different position. Thus the two triangles in Figure 15are initially seen by children as not being the same. When this definition is agreed andapplied consistently the investigation can take two directions. Children can eitherexplore how many triangles there are which have the same shape and size, but are in

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22 L. J. FROBISHER

• • •

Figure 15 Are the triangles the same?

a different position, thus making them different, or by attempting to construct asmany triangles as they can which are different in shape and size. The first activitysearches for the number of triangles which can be made that belong to each congruentset; the second activity concentrates on finding the number of different congruent setswhich can be made. Figure 16 shows the eight non-congruent triangles which it ispossible to make on a 3 by 3 pin-board.

Each one of these triangles is a representative of the set of congruent triangles towhich the triangle belongs. These eight triangles become the data for exploring thesystem which the children have created. As with every new system the first process tobe applied to data is that of sorting. Children consider different ways of sorting theeight triangles. The concepts involved which provide criteria for sorting are those oflength, size (area), angle and similarity ("same" shape), but children frequentlysuggest sorting based upon the number of pins which the elastic bands touches, i.e.the number of dots on the edge or perimeter. Figure 17 shows the result of sorting thetriangles in this way.

Area, in terms of accurate measurement is inappropriate for young children. Butthey are able to sort the triangles according to the size of a triangle and whether it issmall, middling (or medium) or large. This involves them in considerable debate andjudgement comparing the relative sizes of triangles. Very often children try to put thetriangles into order from smallest to largest. Thus the idea of area is experiencedwithout difficulties associated with measurement. Children's ability to give reasonsabout the relative sizes of triangles often shows considerable mathematical thoughtand insight into the concept of area.

INVESTIGATING SYSTEMS INTERNALLY

The three activities which have been described do have solutions to particularproblems and they could in essence be considered to be problems. The emphasis,

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TEACHING AND LEARNING MATHEMATICS 23

m • • • o •

Figure 16 The 8 different triangles.

1 l1 on thoins

edglo ins

on the edge the edgon the edge6 pinsthe edg

pon the edge

Figure 17 Sorting using the number of pins on the edge as criteria.

however, in the activities is not on solving problems, i.e. that there are 27 hats or 8different triangles, but on the construction of the data that makes up the system ofwhich the solution to the problem is only one very small part. When the total data in asystem has been created it is explored by sorting it in a variety of ways. On occasionsthe data, either in the whole system, or in a subset, is ordered according to somecriterion. At all times children are encouraged to look for patterns and to describetheir observations.

The processes which have been used to explore the systems embedded in the threeactivities are:

• constructing the data# sorting the data in a variety of ways

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24 L. J. FROBISHER

0 ordering the data by some criterion0 searching for patterns

These four processes provide the beginnings of investigating the structure of asystem. As they are applied directly to data in the system and do not consider anyextensions beyond the system, they are referred to as internal processes and whenchildren apply them to data they are said to be investigating a system internally.

Working with these processes children actively reflect upon the data and thestructure of a system and in doing so they

0 observe samenesses and differences in the data0 develop insights into the structure of the system0 improve their data construction strategies0 begin to recognise similarities in structures0 transfer knowledge from one system to another0 work towards a total mastery of the system

OPERATIONAL PROCESSES

Sorting and ordering are only two of seven Operational Processes which can be appliedto data in a system to explore its structure. Together they provide a unifyingapproach to the exploration of the variety of systems which comprise mathematics.Figure 18 shows a model which, when the processes are combined with a continuoussearch for patterns, illustrates how children can apply the same approach toinvestigating mathematical problems and the systems which arise when the problemdata is constructed.

It has only been possible in this paper to describe the use of sorting and, to a lesserextent, ordering with three examples which have proved successful with early years'children. The operational processes of matching, comparing and sequencing are alsovery relevant and appropriate for children working within Key Stage 1 of theNational Curriculum. There are many ways in which data in the Clown's Hats, theAddition Pairs and Pin-board Triangles systems can be operated on using matching,comparing and sequencing to develop further insight and understanding of thesystems. Combining and Changing are processes more suited to Key Stage 2activities, although they must not be dismissed for the younger children. Experiencewith the activities described has shown that very young children raise ideas and issueswhich give rise to the application of combining and changing data according to themost unusual rules. When this occurs it should be explored to the depth to which thechildren are capable and for as long as their interest holds.

INVESTIGATING SYSTEMS EXTERNALLY

T h e N a t i o n a l C u r r i c u l u m in m a t h e m a t i c s r equ i r e s t h a t pup i l s a t Level 3 of theP r o g r a m m e of S t u d y in A T I s h o u l d ask the q u e s t i o n " W h a t w o u l d h a p p e n if . . . ?"

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TEACHING AND LEARNING MATHEMATICS 25

Figure 18 The seven operational processes.

In effect this is demanding that a system is modified or extended in some way, thusbecoming a different and external system, but related to the original system in that itretains many commonalities.

For example with the Clown's Hats' activity a question which could be asked is"What happens if each hat has only two pom-poms?" The data would now be different andconsequently a new system would be created. There is nothing special about 2 or 3poms-poms, so the number of pom-poms could be changed to any number. Theinvestigation would then consider relations between the different systems formed witha view to forming a generalisation which applies to any of the systems whatever thenumber of pom-poms on the hats. In a similar way the number of colours availablecan be changed. The ultimate aim is to investigate Clown's Hats which have ppositions and c colours.

Externally investigating a system considers related problems and explores relation-ships between solution to such problems. Although much of this work is beyond thatof the vast majority of young children teachers should not refrain from asking andencouraging children to ask themselves "What would happen if . . . ?"

CONCLUSION

School mathematics is a distorted image of the true nature of mathematics.Mathematics can be viewed as a study of systems which are themselves collections of

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26 L. J . FROBISHER

data, relations between data and ways of combining data. Systems have structureswhich can be investigated by children constructing the data which comprises asystem and exploring the relations between data by sorting and ordering data in avariety of ways. The data which children construct should be represented as separateand dynamic items which can be moved and compared with each other. As each topicin the mathematics curriculum is itself made up of an infinity of systems children candevelop a unifying approach to their learning of mathematics by the application ofoperational processes to each system, particularly the processes of sorting andordering.

The teacher's role becomes one of a creator of activities which give rise to the needto construct and explore data. The activities which should lead to an exploration of amathematical system can be embedded in a context which has appeal for youngchildren. It is necessary to place constraints on the quantity of data in a system inorder that the data is manageable and does not overwhelm children. Each activity ortask should be related to ongoing work and not be seen as bolted on to the normalmathematics which takes place in the classroom. In this way children develop anapproach to learning not school mathematics, but mathematics, enabling them tomeet future unknown situations with a confidence which at present many childrensadly lack.

References

Curriculum Council for Wales (1989) Mathematics in the National Curriculum: Non-Statutory Guidance forTeachers. Cardiff: Curriculum Council for Wales

Department of Education and Science (1989) Mathematics in the National Curriculum. London: HMSODepartment of Education and Science (1991) Mathematics in the National Curriculum. London: HMSONational Curriculum Council (1989) Mathematics: Non-Statutory Guidance. York: National Curriculum

Council

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