CHARACTERISTICS OF 5TH GRADERS’ LOGICAL DEVELOPMENT ...€¦ · children might learn division with decimals in the introductory lessons in a classroom, because all subjects in the

MASAKAZU OKAZAKI and MASATAKA KOYAMA

CHARACTERISTICS OF 5TH GRADERS’ LOGICALDEVELOPMENT THROUGH LEARNING DIVISION

WITH DECIMALS

ABSTRACT. When we consider the gap between mathematics at elementary and secondarylevels, and given the logical nature of mathematics at the latter level, it can be seen as im-portant that the aspects of children’s logical development in the upper grades in elementaryschool be clarified. In this study we focus on the teaching and learning of “division withdecimals” in a 5th grade classroom, because it is well known to be difficult for childrento understand the meaning of division with decimals, caused by certain conceptions whichchildren have implicitly or explicitly. In this paper we discuss how children develop theirlogical reasoning beyond such difficulties/misconceptions in the process of making senseof division with decimals in the classroom setting. We then suggest that children’s ex-planations based on two kinds of reversibility (inversion and reciprocity) are effective inovercoming the difficulties/misconceptions related to division with decimals, and that theyenable children to conceive multiplication and division as a system of operations.

KEY WORDS: cognitive conflict, division with decimals, equilibration, formal operationalthinking, implicit model, misconception

1. INTRODUCTION

In learning operations with decimals or fractions, children tend to acquireonly mechanistic procedures, like the procedure of “invert and multiply”in division with fractions. However, there are gaps between mathematicsat primary and secondary levels, and we have to be mindful of the logicalnature of mathematics at the latter level. Therefore it is necessary to encour-age children to develop their logical reasoning which is required in uppergrades while still at elementary school.1 This study focuses on divisionwith decimals and aims to clarify the process of extending the meaning ofdivision beyond the integer domain.

Over a period of twenty years, a considerable body of theoretical andempirical research has shown that children and even adults have diffi-culties in solving multiplicative word problems with decimals (e.g. Bellet al., 1981, 1984, 1989; Brown, 1981; Dickson et al., 1984; Fischbeinet al., 1985; Greer, 1987, 1988, 1992; Nesher, 1988; Ball, 1990; Tirosh andGraeber, 1990; Harel et al., 1994; De Corte and Verschaffel, 1996). Thesestudies have identified the factors of difficulty in students’ backgrounds

Educational Studies in Mathematics (2005) 60: 217–251DOI: 10.1007/s10649-005-8123-0 C© Springer 2005

218 MASAKAZU OKAZAKI AND MASATAKA KOYAMA

when they solved word problems, and stated some educational suggestionsfor overcoming the difficulties. However, it is still not so clear how suchdifficulties might be overcome. In particular, it has not been clarified howchildren might learn division with decimals in the introductory lessons ina classroom, because all subjects in the above mentioned studies were stu-dents or adults who had already learned multiplication and division withdecimals. Therefore it is important to examine the process of how divisionwith decimals is introduced to children in classroom lessons, as well ashow their misconceptions might be remedied afterwards.

Thus, the aim of this paper is to clarify how children’s difficulties couldemerge and might be overcome in introductory lessons of division withdecimals in a classroom, and how children might develop their logicalreasoning in the process of overcoming those difficulties. In this papersuch tasks are discussed by analyzing the characteristics of 5th graders’learning processes for understanding division with decimals.

We firstly examine the fundamental characteristics of division withdecimals, both mathematically and logically. Next we review the psy-chological difficulties that previous studies have highlighted, followed byconsidering the cognitive conflict approaches that have been developed forovercoming the difficulties and for firmly constructing concepts. After that,we present the classroom data of a teaching experiment designed by theauthors and implemented by a collaborating teacher. Finally, we discusschildren’s thinking processes and the logical characteristics of their rea-soning in extending the meaning of division beyond the integer domain.

2. THEORETICAL BACKGROUNDS

2.1. Fundamental characteristics of division with decimals andthe possibilities of growth in children’s logical reasoning

We begin by noting the well known distinction of two types of division; thepartitive division in which we divide the total by the number of groups tofind the number in each group, and the quotitive division in which we dividethe total by the number in each group to find the number of groups. Theseoperations may be redefined as “division by multiplier” and as “division bymultiplicand” respectively with the extension from integers to decimals,where the general structure of multiplication may be symbolized as “x[measure1] × y [measure2 per measure1] = xy [measure2]” (Greer, 1992).In this study, we will focus on the generalization of partitive division,because most Japanese textbooks introduce division with decimals in thepartitive situation.

CHARACTERISTICS OF 5TH GRADERS’ LOGICAL DEVELOPMENT 219

Here, we briefly examine the following two problems.

(A1) If 12 apples are fairly shared among 3 persons, how many applesdoes one person get?

(A2) The price of 2.8 meters of ribbon is 560 yen. How much does 1 metercost?

Both have the same structure because each quotient is a quantity-per-unit and both permit proportional reasoning, though they are very differentin the psychological sense. Mathematically saying, “If (a, b) is any orderedpair of rational numbers and (a, b) ∼ (c, d) is defined as ad = bc, therelation “∼” is an equivalence relation. Thus (partitive) division means totransform the element (a, b) into (quotient, 1) of the equivalent class (a:dividend, b: divisor)”. Vergnaud’s (1983, 1988) schematic representationis useful in clarifying the relationships among elements (Figure 1).

He terms “a structure that consists of a simple direct proportion betweenmeasure-spaces M1 and M2” an “isomorphism of measures” (Vergnaud,1983) where the relationship within each measure space is a scalar operatorand the relationship between measure spaces is a function operator f. Theschema enables us to conceive various multiplicative problems like equalsharing, constant price (goods and costs), and uniform speed (durations anddistances). Here partitive division is conceived as finding the unit value f (1)in the schema. This class of problems can be solved by applying the scalaroperator (÷a) within measure space M1 to the magnitude b. If we applythis to the above problem (A2), we can clarify the structure of divisionas dividing 560 by a scalar operator (÷2.8) within the measure space of‘length of ribbon’.

However, we also need to pay attention to at least two replacements asprevious steps or as useful strategies when children solve problems usingtheir existing knowledge on division with integers. These can be representedusing Vergnaud’s schema (Figure 2).

Namely, a pair of (560, 2.8) could once be transformed into other pairsof (5600, 28) by multiplying each by 10 or (20, 0.1) by dividing by 28,

Figure 1. Vergnaud’s schematic representation of isomorphism of measures.


Figure 2. Two replacements of the division problem using Vergnaud’s schema.

and successively into the pair of (200, 1) which is the quotient. Theseideas are consistent with the mathematical view of division, where a pairof dividend and divisor forms a representative element of an equivalentclass such that the quotient in any division is constant, rather than only twocomponents of an operation (Schwarz, 1988; Thompson and Saldanha,2003).

These views of division are important for the growth of children’s rea-soning towards secondary mathematics, which seems to be conceived interms of Piaget’s notion of formal operational thinking. Piaget’s theory hasbeen criticized over time because of the problems of age in his develop-mental stages, of the domain-specificity of tasks used in his experiments,and furthermore about giving little attention to educational, social and cul-tural aspects. However, we think these critics misunderstood Piaget, as DeCorte et al. (1996, p. 494) wrote, “Much of this criticism evaporates if it isrecognized that . . . these stages were intended as a classification of forms ofthinking, not of individual children” (see also Steffe and Thompson, 2000).Furthermore, we think he never lit up the roles of educational and socialfactors, but claimed the need for coordinating the factors of innateness,experience, and linguistics and education, and especially emphasized thefactor of equilibration or self regulation in cognitive development (Piaget,1977; see also Shayer’s (2003) discussion which compared Piaget withVygotsky).

Inhelder and Piaget (1958) used the notion of “formal operation” tocharacterize the adolescent thinking starting from about 11 years of age.They noted that (a) it can proceed from some hypothesis or possibility, (b)it can be characterized as propositional logic by combining the statementsp and q, p or q, not p, p implies q (Jannson, 1986), (c) the object forthinking is the generality of the law, the proposition, etc., and (d) it includestwo kinds of reversibility. In exemplifying these characteristics in division


with decimals, it may be helpful to consider the following properties ofdivision.

a ÷ b = (a × m) ÷ (b × m) (1)a ÷ b = (a ÷ m) ÷ (b ÷ m) (2)(a × m) ÷ b = (a ÷ b) × m (3)(a ÷ m) ÷ b = (a ÷ b) ÷ m (4)a ÷ (b × m) = (a ÷ b) ÷ m (5)a ÷ (b ÷ m) = (a ÷ b) × m (6)

For example, why we may transform 560÷2.8 into 5600÷28 and whywe may transform it into (560 ÷ 28) × 10 can be respectively explainedby the property (1) where a = 560, b = 2.8 and m = 10, and by theproperty (6) where a = 560, b = 28 and m = 10. We think it is obviousthat such transformation into another imaginary situation includes somehypothesis or possibility and that the object for inquiry is then extendedtoward a general mechanism of division beyond just finding an answer.Furthermore, if why 560 ÷ 2.8 may be 200 is justified by mediating theabove properties, the reasoning may have a syllogistic character. Thus, wethink the learning of division with decimals is considerably related to thegrowth of formal operational thinking.

In the above (d), one reversibility is inversion, which enables one to “re-turn to the starting point by canceling an operation which has already beenperformed” (Inhelder and Piaget, 1958, p. 272), and the other is reciprocity,which is related to “compensating a difference” (p. 273) and is “required forequating operations which are oriented in opposite directions”(p. 154). Asto division, we think that the inversion corresponds to the thinking whichtransforms division into multiplication as the reverse operation, and thatthe reciprocity corresponds to the thinking which regards division itselfas multiplication as the equivalent operation. We will examine later howthese kinds of reversibility may emerge and become explicit in children’sthinking in the learning of division with decimals in a classroom.

In the follows, we review what conceptions children have in their mindsand how these conceptions make it difficult for children to learn divisionwith decimals.

2.2. Children’s difficulties regarding division with decimals

We can assume that in learning division with decimals, the mathematicalmeaning predominates and its difference from the child’s conception causes


his/her difficulty. Here, we shall return to the problems (A1) and (A2)described in the previous section.

Though we stated that both had same mathematical structure, it mustbe emphasized that in a child’s mind division with decimals is greatlydifferent from division with integers. That is, problem (A1) permits oneto imagine the situation where something is divided into equal parts andto have the conception that division always makes the answer smaller,however problem (A2) doesn’t permit one to think in the same way.Actually, a substantial amount of researches has shown that many chil-dren and even adults have difficulties in division with decimals (Bellet al., 1981, 1984, 1989; Brown, 1981; Dickson et al., 1984; Fischbeinet al., 1985; Greer, 1987, 1988, 1992; Ball, 1990; Tirosh and Graeber,1990; Mangan, 1989; Harel et al., 1994; De Corte and Verschaffel, 1996).These studies have clarified children’s difficulties through two kinds ofmethodology; choosing an operation for the given word problem out ofa set of choices; and making up word problems which fit to the givencalculation.

Since the early studies in this area, researchers have identified severalmisconceptions which resulted from children’s experiences in division withintegers and have examined these misconceptions extensively. The mosttypical misconception among children is that “multiplication makes theanswer bigger, and division makes it smaller” (Brown, 1981; Bell et al.,1981, 1984; Greer, 1987, 1988; Mangan, 1989; Tirosh and Graeber, 1990).

Fischbein et al. (1985) insisted that children’s misconceptions on mul-tiplicative word problems involving decimals resulted from their primitiveand implicit models. Then, with regard to division, two models were sug-gested; partitive and quotitive division models. In the former, there aresuch constraints as “the dividend must be larger than the divisor”, “thedivisor must be a whole number”, and “the quotient must be smaller thanthe dividend”, and in the latter there is only one constraint as “the dividendmust be larger than the divisor”. They demonstrated this theory throughdesigning tasks which violated these constraints, where many children haddifficulty in selecting an operation needed to solve the word problems orfor making up a word problem relevant to the given calculation. Graeberand Tirosh (1988) examined this theory with pre-service teachers, and clar-ified that the calculation procedures and informal methods concealed theirmisconceptions. Fischbein explained that students’ explicit models maybecome implicit “as an effect of routine or if their explicit use may con-tradict newly learned principles” and that “in principle, the student has togive up her primitive models and rely on formal definitions but this wouldrisk destroying the productivity of her reasoning processes”, and indicated


as an educational implication that “students should be helped to considerconsciously all the implications of the formal definitions taught and to tryto apply them to particular cases, especially those known by the teacher toproduce certain difficulties” (Fischbein, 1994, pp. 105–106). Furthermore,Fischbein (1989) indicated that the implicit models are robust and ableto survive long after they no longer correspond to the formal knowledgeacquired by the individual. This theory seems to provide us with a strongexplanation for many children’s difficulties; however, there are some casesthat can’t be explained by the theory (Bell et al., 1989; Harel et al., 1994;De Corte and Verschaffel, 1996).

As another factor of difficulty, several researchers have reported thatnumbers used or students’ numerical preferences had intervening effects(Bell et al., 1984, 1989; De Corte and Verschaffel, 1996; Harel et al.,1994), while logically speaking, the numbers used do not influence thechoice of operation (Schwarz, 1988). In particular, students’ performancessomewhat decrease when the multiplier or the divisor is a decimal, anddrop even more when they are less than 1.2 In Bell et al.’s (1989) in-vestigation, for example, students’ (15-year-olds) percentages of correctanswers were 50% for the problem “Henry puts 11 gms of powder into3.7 liter of water. How much powder should Julia use in one liter to havecoffee of the same strength?”, and 40% for “James uses 14 gms of coffeein 0.9 liters of water. How much coffee should Graham use in one liter,to make it the same strength?” in the choice-of-operation test. Further-more, it should be also noted that while the numbers involved in theseproblems influence the choice of operation, they do not influence the esti-mation of the answer. In fact, in the estimation-of-answer test the studentsachieved about 30% high scores more than that in the choice-of-operationtest.

Greer (1987, 1988, 1994) reported a phenomenon that remarkablyshows the effect of the types of number, and proposed the notion of“non-conservation of operations” that given two problems successivelywhich were equivalent except for the types of number involved, chil-dren did not recognize the invariance of the operations but changed theoperation. This result implies that the heuristic strategy of assuming aneasier problem has little effect because children may change their op-eration while substituting integers for decimals. As other variables, theinfluences of context, semantic structures of problems, types of quanti-ties etc. have been identified. The detailed discussion is given in Greer(1992).

In the following, we discuss the cognitive conflict approach that hasbeen developed for overcoming these difficulties for children.


2.3. What does the research tell us about interventionby cognitive conflict?

Fischbein indicated the robustness of children’s models, so we think thatit is insufficient to just acquire the division procedures for resolving thedifficulties, and that it is necessary for children to (re)construct the meaningof division. The intervention by cognitive conflict has been regarded as aneffective teaching approach (Bell and Purdy, 1986; Bell, 1986, 1987; Swan,1983, 2001; Steffe, 1990; Tirosh, 1990; Tirosh and Graeber, 1990). It seemsthat Piaget’s (1985) equilibration theory is the main theoretical basis forthe approach.

Equilibration refers to the regulating processes between assimilation andaccommodation that are not two separate but rather two functional polesof adaptation. Assimilation and accommodation are respectively aspects ofincorporating an object into a sensori-motor or conceptual scheme, and ofadjusting or modifying its assimilatory scheme according to the particulari-ties of the object. However, either of these aspects sometimes predominatesand they are not always in balance. Namely, there is often disequilibriumbetween assimilation and accommodation, where an equilibration processis needed. Piaget (1985) considers that this disequilibrium is essential forcognitive development as follows.

Disequilibria alone force the subject to go beyond his current state and strike outin new directions. (p. 10)

. . . Without them (disequilibrium and cognitive conflict: a note by authors), knowl-edge would remain static . . . . Progress is produced by reequilibration that leadsto new forms that are better than previous ones. We have called this process “op-timizing reequilibration”. Without disequilibrium, it would not occur. (p. 11)

This Piaget’s idea has been adopted as a principle in mathematics teach-ing, as Steffe (1990) stated, “the construction of reversible mathematicalschemes is critical to the mathematical progress of students, and cogni-tive conflict does and should play a role in these constructions and re-constructions” (p. 107). In particular, “the diagnostic teaching approach”that researchers in the Shell Center in England developed is probably aexemplary case which involves (1) assessing pupils’ existing conceptualframeworks, (2) making existing concepts and methods explicit in the class-room, (3) sharing methods and results and provoking conflict discussion,(4) resolving conflict through discussion and formulating new conceptsand methods, and (5) consolidating learning by using the new conceptsand methods through problem solving (Bell and Purdy, 1986; Bell, 1986,1987; Swan, 1983, 2001). They compared the diagnostic approach with apositive only approach3 and showed that only the diagnostic approachachieved significant longer term learning.


It may be appropriate to refer to studies which critically discussed thecognitive conflict approach in classroom lessons, because while we thinkit’s not a cure-all for remedying children’s misconceptions, such criticalviewpoints are really useful in designing and analyzing the lessons in aclassroom.

Even Steffe (1990), who recommended the strategy, stated “Granted,a teacher can attempt to activate cognitive conflict in his or her studentswith little or no success. . . . Experiencing conflict is the business of theactor and it is a bit naı̈ve for anyone to believe that a teacher has suchexperience under his or her control” (p. 107). Furthermore, Tirosh andGlaeber showed not only the effectiveness of the strategy, but also thatthere were some cases where “a learner may not interpret the apparentlyconflicting elements in a situation as conflicting” and that “learners whobecome aware of an inconsistency do not always respond to the conflictthey sense by progressing” (Tirosh, 1990; Tirosh and Graeber, 1990).4

Such critical points are reviewed by Limón (2001). She summarized thedifficulties to use the cognitive conflict strategy in the classroom as follows.

In our view, many of the difficulties found in the application of the cognitive conflictstrategy in the classroom are closely related to the complexity of factors in thecontext of school learning. Up to now, most of the theoretical models proposed toexplain conceptual change focused mainly on the individual’s cognitive processes,forgetting or, at least, not taking into account other individual’s characteristics,such as motivation, learning strategies, epistemological beliefs, attitudes, etc., notincluding variables as important as the teacher and his/her features (motivation,teaching strategies, training, beliefs about what learning and teaching is, etc.) andsocial factors, such as the role of peers. (Limón, 2001, pp. 364–365)

We agree with this statement that various factors are complicatedlyrelated to the emergence of cognitive conflict. Nevertheless we think Limóndidn’t deny the effectiveness of the strategy, and we see her commentis stipulating that the approach should be used in the classroom underconsiderations of the above factors.

We also pay attention to Limon’s statement, “Awareness of conflictwould be a first step of a process of integrating the new information . . .Conceptual change is a gradual process and not an “all-or-nothing” pro-cess.” (pp. 368–369). Thus, it must be clarified not only how the cognitiveconflict might emerge, but also how students might construct a higher levelof knowledge after they might be conscious of the conflict.

In the following, we describe the methods and the results, and theiranalysis, of fifth grade classroom lessons of division with decimals in ourteaching experiment.


3. METHODS

3.1. School and participants

Our teaching experiment was implemented in collaboration with a teacherin a university attached primary school. The school has a character of anexperimental school where the teaching and learning of mathematics isexpected to be practically developed; the classroom lessons are openedonce a year to a society of teachers for the study of primary education, sothat many teachers from other schools can attend and discuss the lessonswith the teachers at the school. The school also serves to provide manyuniversity students with pre-service teacher training. Teachers at the schoolare competent practitioners in general and our collaborative teacher wasan experienced male teacher. Thus, we left the orchestrating of classroompractices to him, except for the problems to be posed, the teaching materialsused, and the basic scenario for the lessons.

In our teaching experiment, six lessons were conducted for 38 fifthgraders (20 boys and 18 girls), who seemed to have somewhat higher abili-ties than those students in a general public school, because they had passeda specific entrance examination in order to enter the school. Further, wefelt that the children had more or less acquired abilities such as reasoning,communicating, comparing ideas, and getting their ideas in shape throughthe teachers’ instructions. That is, the teacher and many of the classroomchildren might more or less have the characteristics that Limón (2001)indicated. Thus, it was possible to concentrate the large part of our consid-eration on the cognitive processes without abandoning the idea that manyfactors are complicatedly related to the emergence of cognitive conflict.

Before the teaching experiment, during their fourth grade, the childrenhad already learned teaching units for “decimals” and “multiplying anddividing decimals by integers.” And in the fifth grade they learned theteaching unit for “multiplying by decimals” just before the teaching exper-iment, according to the Course of Study for primary school mathematicsin Japan.

3.2. Division problems designed for the teaching experiment

We designed three division problems to be given in the following sequenceto the classroom children during our teaching experiment.

Q1. The price of 2.5 meters of ribbon is 100 yen. How much does 1 metercost?

Q2. The price of 2.4 meters of ribbon is 108 yen. How much does 1 metercost?


Q3. The price of 0.8 liters of juice is 116 yen. How much does 1 liter cost?

We didn’t include any divisions with a decimal dividend, because pre-vious studies indicated that the influence of a divisor was more powerfulthan that of a dividend, and we considered that children could reconstructthe meaning of division in overcoming the influence of a divisor.

We considered that the problem in which the decimal part of the divisor is5 might be relatively easy and accessible by using the children’s knowledgeof division with integers. Moreover, because the children were expected totransform the problem into division with integers using just two pieces ofstrip (Figure 3), we situated the problem Q1 as the problem for introducingdivision with decimals in our teaching experiment.

Figure 3. A transformation from division with decimals to division with integers.

On the other hand, the problem Q3 in which the divisor is less than 1 wasdecided to be positioned as the final problem for the children in our teachingexperiment, because many research studies have shown that children feltparticular difficulties in solving such problems. We assumed that many ofour children might more or less feel cognitive state of disequilibrium in theproblem Q3. Therefore, we designed the problem Q2 as intermediate andput it between problem Q1 and Q3.

3.3. Didactical ideas for overcoming children’s difficulties

From the didactical point of view, in our teaching experiment, we devisedthe following three points for encouraging the children to construct themeanings of division with decimals through overcoming their difficulties.

First, the classroom teacher prompts the children to conceive the divisionproblem as having a lot of equivalent problem situations within it. Problem1 worked well to construct those situations, because children could makea new situation by using just two pieces of strip as is described in Figure3. For example, the teacher encouraged them to conceive a situation of“ribbon of 100 yen per 2.5 meters” as new situations of “200 yen per5 meters”, “300 yen per 7.5 meters”, “400 yen per 10 meters”, or “20yen per 0.5 meters”. From a mathematical viewpoint, it corresponds toidentifying the elements in an equivalent class. We think it can also be away to avoid the phenomenon of “non-conservation of operations” in whichchildren did not conceive division by decimals less than 1 as not division butmultiplication.


Second, the symbolizing or modeling processes (Gravemeijer andStephen, 2002) are assumed to develop the meaning of division with dec-imals (Figure 4). Therefore, in our teaching experiment we devised the“operational number line” ((b) in Figure 4) as an operative tool for the in-dividual children in order to develop their idea further by mediating betweenthe concrete materials ((a) in Figure 4) and the number line ((c) in Figure 4).

Figure 4. Symbolizing processes in developing the meaning of division with decimals: (a)pieces of strip, (b) operational number line, (c) number line, and (d) schema of proportion.The “operational number line” is made so that one can move the long strip back and forthonto the paper on which two number lines are written.


It was expected that the device could connect the operational and concreteview to the structural and proportional view of division in the lessons.

We think that the process of understanding mathematical conceptswould progress along with the symbolizing processes (Dörfler, 1991). Inparticular, the activities of symbolizing enable children to transit froman operational conception to a structural conception in the sense of Sfard(1991). The general characteristic of operational conception is that “a math-ematical entity is conceived as a product of a certain process or is identifiedwith the process itself”. On the other hand, the structural conception is that“a mathematical entity is conceived as a static structure – as if it was a realobject”. Sfard (1991) indicates that mathematical concepts develop fromoperational to structural conceptions historically and psychologically.

Furthermore, at a short meeting with the classroom teacher held after thethird lesson, we decided to use the schema of proportion ((d) in Figure 4).It was expected that the use of schema could help the children promotetheir structural conception of division.

Third, the teacher often consciously formed some situations ofcognitive disequilibrium that emerged from either intersubjective orintrasubjective conflicts (Cobb et al., 1993), for we expected the childrenmight develop their meaning of division with decimals in overcoming thedisequilibrium. We basically followed the diagnostic teaching approach.However, we decided not to make disequilibrium emerge intentionally bygiving conflicting data from the teacher, but to form the opposed ideaschildren presented as a situation that they should resolve, because weattach as much importance as possible to children’s constructions, as isseen in the constructivist perspective.

After considering the above three points comprehensively, we plannedour teaching experiment as shown in Table I. This table briefly summarizesthe problems and the objectives and activities in each lesson of theexperiment.

3.4. Data collection and method of analysis

In the teaching experiment, the classroom lessons were developed throughthe following cycles. First the authors designed the lesson, second-presented the design to the classroom teacher and discussed with him thecontents of each lesson mainly for anticipating the teacher’s utterancesand children’s activities, third-observed and recorded the lesson, especiallychildren’s activities, and lastly-adjusted the next lesson after consideringthe previous one. To do this, with the classroom teacher we had a relativelylong meeting before the teaching experiment started, and a short meetingwas held between each of the six lessons.


TABLE IProblems and objectives and activities in the teaching experiment

Lesson Problem Objectives and Activities

1 Q1: The price of 2.5 meters of ribbon is100 yen. How much does 1 metercost?

To find the answer by transformingthe given problem into divisionwith integers by using concretematerials

To know there are lots of situationsequivalent with the given problemin the sense that the answers aresame

To represent the activities asmathematical expressions

2 Q1 & Q2: The price of 2.4 meters ofribbon is 108 yen. How much does 1meter cost?

To translate the activities usingconcrete materials into theoperational number line

To find the answer by transformingthe problem into the equivalentdivisions with integers

To represent the activities usingmathematical expressions

3 Q3: The price of 0.8 liters of juice is116 yen. How much does 1 liter cost?

To find the answer using theoperational number line

To represent the activities usingmathematical expressions

To discuss why the answer is biggerthan the dividend

4 Q3: The price of 0.8 liters of juice is116 yen. How much does 1 liter cost?

To discuss what “divided by 0.8”means

To translate the activities usingoperational number line intoschemas of proportion

To understand the meaning of“divided by 0.8” as therelationships among four terms inthe schema of proportion

5–6 To discuss and summarize theactivities that have been done sofar

To reformulate the meaning ofdivision

The lessons were recorded by a video camera and field notes made,and the transcripts were also made of the video data. We could also usethe children’s writings in our analysis, as the teacher usually directed thechildren to write what they felt questionable and what they understood on


their notebooks during the last few minutes of every lesson. By using thesedata we conducted a qualitative analysis.

A characteristic of our analysis is to focus on the children’s disequi-librium that emerges in classroom lessons. We analyze it as being thedifference between the idea that the children are newly conscious of andtheir existing conceptions. For that purpose, we organize the meaningfulepisodes in the lessons chronologically in terms of both in what conditionsthe disequilibrium may emerge and how children may grapple with it.

We understand that the term ‘disequilibrium’ is not used for collective,but rather for individual cognition. Thus, when we speak of disequilibrium,we don’t mean that “all” children feel disequilibrium in the classroom. Thiswas the reason why the teacher checked the distributions of children whenvarious ideas were proposed or children felt disequilibrium. Then, we ex-amine the intersubjective conflicts that the children manage to resolve,because disequilibrium is usually made explicit as the differences amongideas in the collective. We agree with Cobb’s position that a classroompractice is reflexively related to children’s mathematical conceptions andactivity (Cobb, 2000). We think intersubjective conflicts among children inthe classroom can be an important index to ascertain individual children’sdisequilibrium and it’s difficult to separate them definitely. We also ascer-tain whether the phenomena occurred by checking the looks on children’sfaces in the video data. Our main analysis is then devoted to when andhow such cognitive disequilibria were formed, how they were discussedand overcome, and how they then develop the children’s logical reasoning.

4. RESULTS AND ANALYSES

The teaching experiment eventually needed six lessons, in which each les-son was 45 minutes, except for the fifth lesson which was just 30 minutes.However, basically the first four lessons were spent for the children’s solv-ing and discussing the three division problems designed by the authorsand for their constructing the meaning of division with decimals. And thelast two lessons were used to confirm and deepen children’s understandingof what had been constructed in the previous four lessons. Thus we willmainly analyze the teaching and learning activities during the first fourlessons.

4.1. Initial understanding of division with decimals

The first classroom lesson began with the problem “The price of 2 metersof ribbon is 100 yen. How much does 1 meter cost?”, and soon moved to


the problem Q1 “The price of 2.5 meters of ribbon is 100 yen. How muchdoes 1 meter cost?”. The teacher prompted the classroom children by pairsto solve it in various ways by using many pieces of strip each representing2.5 meters. After the activities by pairs, the children in turn presented theirways of solving in the whole class setting. Here the following three methodswere eventually presented by the children.

The method 1 was to combine two pieces of 2.5 meters of ribbon andto transform the problem Q1 into the division with integers as shown inFigure 5. For example, “Well, as this is 2.5 meters of ribbon and 2.5 isa decimal, I add one more piece and change it into 5 meters. I must alsodouble the price, because I double it (the length of ribbon). Then it becomes200 yen. So, I divide 200 by 5 and get the price of 1 meter”.

In the method 2, one piece was divided into the same lengths as shownin Figure 6. For example, “Well, first I divide it (2.5 meters) by 0.5 meterseach. (He drew the dotted lines on the material on the blackboard). As thetotal price is 100 yen, I divide 100 by 5 in order to get the price of 0.5meters. It is 20 yen and it is the price of 0.5 meters. So, we can solve whenwe multiply 20 by 2 in order to change it (0.5 meters) into 1 meter”.

The method 3 was to connect 10 pieces of 2.5 meters of ribbon as shownin Figure 7.

However, when a child presented this method, some other children ar-gued against it; e.g., “Though the method uses 10 pieces, we can solve itusing only 2 pieces”; “The method of two pieces is more rapid. It will beflawless if we combine some of two pieces into the longer unity”. Againstthese opinions, other children supporting method 3 said, “. . . If we use 10pieces, we can rapidly estimate 25 meters by multiplying by 10. Using 10

Figure 5. Method 1: To connect two pieces of 2.5 meters of ribbon.

Figure 6. Method 2: To divide one piece of 2.5 meters of ribbon into 5 equal parts.

Figure 7. Method 3: To connect ten pieces of 2.5 meters of ribbon.


pieces makes our computation rapid”; “If we multiply by 10 or 100, we canalways get integers”. We can characterize this opposition as the empiricalversus theoretical. Namely those children supporting the use of two piecesconsidered that it was best to use the least numbers of strip practically, theothers for the method 3 clearly born in their mind the place value systemin computation.

Next, the teacher oriented all children to clearly represent these ideasusing mathematical expressions or their own language in their individualactivities. After that, three children who were designated by the teacherwrote their expressions on the blackboard (Figure 8). (Note: The figures inthis section are copies from the blackboard and the Japanese is translatedinto English by the authors.)

The teacher asked the other children to explain the expressions, for heintended that the ideas should be shared in the classroom. The classroomdiscussion led children to understand that methods 1 and 3 were basicallythe same and that it was more effective to use the method 3 if the decimalpart of the divisor was not 5.

Furthermore, the teacher directed all children to conceive the situationas lots of mathematical expressions, such as 300 ÷ 7.5, 400 ÷ 10, 600 ÷ 15and so on. The children seemed to begin understanding such situation,because they recited the expressions together and some of them said “Thereare many situations”, “Infinite situations”. However, not all children seemedto have a clear understanding that these mathematical expressions wereequivalent to the original problem represented as 100 ÷ 2.5 at this moment,because a child uttered, “When the number of the pieces is even, it is easierfor us to consider”.

Therefore, in the second half of the second lesson, similar activitieswere subsequently undertaken, mainly by using the operational numberline (Figure 9) for problem Q2 “The price of 2.4 meters of ribbon is 108yen. How much does 1 meter cost?”. Through these first two lessons inwhich the division problems with divisors bigger than 1 were solved anddiscussed, the following ideas seemed to be constructed and shared bychildren. We could confirm it not only by the fact that almost all children

Figure 8. Mathematical expressions children made based on the concrete activities.


Figure 9. Traces of activities using the operational number line.

raised their hands when the teacher asked children whether they agreedwith each of the ideas in the lesson, but also by the fact that the childrenwrote the following ideas as what they understood in their notebooks duringthe last few minutes of the second lesson.

(a) There are many situations that are the same as what 108 yen is to 2.4meters.

(e.g., 216 yen is to 4.8 meters; 324 yen is to 7.2 meters; 432 yen is to9.6 meters; 540 yen is to 12 meters; . . . 1080 yen is to 24 meters).

(b) If we multiply each number by 5 or 10, we can transform the probleminto a division with integers.

(e.g., 108 ÷ 2.4 = 1080 ÷ 24 = 45).(c) We can solve by first finding the price of 0.1 meter.

(e.g., 108 ÷ 24 × 10 = 45).In sum, through the first two lessons the classroom children became able

to consider the problem by transforming it into division with integers. Wecould say that it might be a first step for the children in order to conceivedivision as proportional.

4.2. The birth of disequilibrium

In the third lesson of our teaching experiment, the teacher presented theproblem Q3 in which the divisor was less than 1 (“The price of 0.8 litersof juice is 116 yen. How much does 1 liter cost?”). He first asked thechildren what the operation for the problem was. They unanimously said,“Multiplication and division”, “Multiplication”, “Division”, “We first domultiplication and next division”, “We can also first divide followed bymultiplying” etc. When the teacher confirmed the number of children infavor of each idea, multiplication was supported by five children, divi-sion was supported by seven children, and the others were for using bothmultiplication and division.

Here several children argued against the opinion of multiplication.(Note: In the transcripts in this paper, children’s names are different from


the real ones, and the words in the parentheses are notes by the authors.The numbers in front of the names are assigned in order for the sake ofconvenience.)

1 Katt: I think that has no meaning. Why can we find one liter whenmultiplying by 0.8? We should write like this. (He wrote “116 = 0.8”on the board.) 0.8 liters are 116 yen. Because 0.8 liters are 116 yen, it’sno sense to multiply.

2 Kawa: I calculated 116 times 0.8 actually, but it became 92.8. This isnot related with the price of 1 liter.

3 Naka: 0.8 liters are 116 yen. Why is it 92.8 yen when purchasing 1 liter?If so, 0.8 liters could be bigger than 1 liter.

Such discussion made children conceive that the operation was not amultiplication, though it was still unclear whether they might be consciousthat it was a division since they sought to use both multiplication anddivision. After this, the teacher encouraged all children to solve the problemin pairs by using the operational number line and to make mathematicalexpressions. Almost all children easily made expressions such as (116 ×10) ÷ (0.8 × 10), (116 ÷ 8) × 10 and found answers by using the thinkingin (b) or (c) above.

However, when the teacher then asked the children what the orig-inal expression was, a child answered “116 divided by 0.8”, and an-other child questioned that expression. With this opinion as a start, chil-dren began to feel uncertain and so brought about the cognitive state ofdisequilibrium.

4 Mits: If we do 116 divided by 0.8, we get 0.1 liters.5 Cs (Some children at the same time): What?6 T: Well? When we divide 116 by 0.8, do we get 0.1 liters?7 Mits: One liter? No, it’s incorrect.8 T: Is this expression wrong?9 Mits: It may be correct. Well, is it correct?

10 T: Everybody, is this expression right?11 Cs: Yes, it is.

. . . . . .

12 Arat: 116 divided by 0.8 . . . Why is the expression right? It might notbe 145.

13 Cs: It must be 145.14 Arat: It might be 145, but the answer for the problem is not 145 yen.15 Cs: Why? It must be 145 yen.16 T: If 116 yen is to 0.8 liters, then 145 yen is to 1 liter. Is that wrong?17 Naga: But, why do we get (the price of) 1 liter when dividing by 0.8.


18 C: I think the answer is the price of 0.1 liter.

. . . . . .

19 Yuki: I also think that if we do 116 ÷ 0.8, we get the price of 0.1 liter.20 Cs: I agree!21 Cs: No, it’s wrong!

It was observed from the video data that about seven children, in-cluding Arat, Naga and Yuki, then considered the answer 145 as theprice of 0.1 liter, while more children appeared to have the same opin-ion. We think that this is the influence of certain conceptions childrenhad previously gained of partitive division, because such phenomena didnot come up in previous lessons. In the following we will analyze thestages and characteristics of children’s reasoning for overcoming thisdifficulty.

4.3. Processes of overcoming the disequilibrium

4.3.1. Logical explanations and the robustness of the implicit modelThe opinion “the answer 145 yen to the problem is the price of 0.1 liter”was soon refuted by some children. For example, the following opinionswere presented.

22 Moto: If 145 yen were to 0.1 liter, then 0.1 liter might be more expensivethan 0.8 liters.

23 Mich: We must do 116 ÷ 8 in order to work out the price of 0.1 liter.24 Katt: (After pointing that both 116 ÷ 0.8 and 580 ÷ 4 have the same

answer, with reference to his activity on the operational number line)If we divide by 4 liters, of course the answer is the price of 1 liter. Theidea of 0.1 liter is strange.

25 Saku: 116 divided by 8 is 14.5, but 116 divided by 0.8 is . . . becauseof a tenth of eight, because the answer is 10 times 0.1 liters, namely 10times 14.5, I think 116 divided by 0.8 remains as it is.

In the above explanations we can find an initial form of deductive rea-soning. In fact, Katt’s utterance (24) may be interpreted as a syllogisticreasoning in the sense that the statement 3 is deduced from statements 1and 2 as follows.

Statement 1: If we divide 580 by 4, we get the price of 1 liter. (Agreed)Statement 2: 580÷4 can be equivalently changed into 116÷0.8. (Agreed)Statement 3: (Therefore) 116 ÷ 0.8 is the expression for finding the price

of 1 liter.


It should be noted that such syllogistic reasoning occurred through in-tersubjective conflict (Cobb et al., 1993) and was used in the classroomdiscussion. We found that such discussion could lead the children to theagreement among them, because almost all children seemed to see 116 di-vided by 0.8 should represent the price of 1 liter when the teacher checkedthe distribution of opinions at the end of the third lesson.

However, at the beginning of the fourth lesson when the teacher askedthe children whether they felt uncertain about the expression, about 70%children showed a feeling of uneasiness. Here again, those children whowere confident began to speak.

26 Kawa: (Referring to the expression 116 ÷ 8 × 10) If we put this 10 infront, it is the same as 1160 ÷ 8. So, I think this (116 ÷ 0.8) summarizesthese expressions which were made to get the price of 1 liter of juice.

27 Mori: . . . This is what no one has learned, but division means . . . If thedivisor is a decimal, we must transform it into an integer by moving adecimal point by some numbers, and add zero here by multiplying this(dividend) by same numbers too. So, we can deduce such expression.

However after these explanations, when the teacher again asked thechildren whether they could explain why the expression 116 ÷ 0.8 is theprice of 1 liter, about 80% of them responded that it was difficult to explain.

28 Sasa: Though I don’t know the reason, even with the division the quo-tient is bigger . . . than the dividend.

29 Naka: I can understand that if we divide something by 2, we get half.But I don’t know how we get 145 when we divide by 0.8.

30 T: Do you think that “divided by 0.8” is a problem?31 Cs: Yes. It’s unclear and strange.

We found that children in the classroom implicitly experienced a cogni-tive state of disequilibrium. We can see that the opinions of Sasa (28) andNaka (29) were stated in terms of the violations of the implicit model ofpartitive division. Here it may be interesting to note that as in the transcriptNaka (3) he was the boy who had objected to that the operation in theproblem Q3 was a multiplication. Thus, at that time when Naka objected,he was not convinced about why the operation was a division. This episodesuggests that even if logical explanations are given, they are not sufficientto overcome the disequilibrium resulting from the children’s conceptionsbased on their experiences of division with integers. Therefore even if theexpression 116 ÷ 0.8 was transformed into other mathematical expressions(e.g. 1160 ÷ 8), it seems that the children’s disequilibrium does not resolvewithout discussing what “divided by 0.8” itself means.


4.3.2. The process of equilibration based on reversibility “inversion”The equilibration was initiated from a child’s utterance based on the inverseoperation.

32 Saka: It is not good to consider 116 ÷ 0.8. By reversing it, if we thinkof the problem as “The price of 1 liter of juice is 145 yen. How muchdo 0.8 liters cost?”, it will be 116 (He calculated it) . . . I got it. So,division means . . . even if the divisor may be a decimal or an integer,the answer is . . . 1 liter . . . to get 1 liter.

This opinion was very powerful and many children began to regard116 ÷ 0.8 as being a valid expression for finding the price of 1 liter. Here itshould also be noted that this explanation included the meaning of division(the quantity-per-unit). But Saka’s opinion was soon rejected because ithad the character of checking after solving the division.

33 Arat: Saka said that it was all right if we changed it to a multiplication.However we used the division (the quotient) and so it’s questionablewhether it (the product) becomes 116. The explanation is not so mean-ingful.

34 T: Arat said that the multiplication (145 × 0.8) was the story afterfinding the answer of the division (116 ÷ 0.8).

35 Saka: But, I proved it in such a way.

Here, Naka made the point more explicit.

36 Naka: Well, if it is 116 times 0.8, I can regard it to take 0.8 pieces of116. But, please tell me how we do 116 ÷ 0.8.5

It seemed that Naka, and probably also other children in the classroom,wanted to conceive the division as a concrete operation. Saka’s explanationbased on the inverse operation was strong. We think here the teacher couldhave asked children whether the answer was bigger or less than dividendand activated the discussions on the explanation, but he might lose thechance here. Anyway, the children still somewhat remained in a concreteworld, and they seemed to need further explorations and explanations inorder to reach a state of equilibrium.

4.3.3. The process of equilibration based on reversibility “reciprocity”In the second half of the fourth lesson, the teacher reflected on the previousactivities on the number line, and proposed to rewrite them by using theschema of proportion (Figure 10). After that, he furthermore asked childrento consider the meaning of “divided by 0.8” in the abbreviated schema(Figure 11).


Figure 10. Activity on the number line and its translation to the schema of proportion.

Figure 11. Abbreviated version of the schema of proportion.

37 T: Let us discuss by using these two parts (the left and right blanks inFigure 11).

38 Tach: (He wrote on the blackboard “×1.25” beside the left blank)39 T: Really? And what is it here?40 Tach: (He wrote “×1.25” beside the right blank.) 0.8 liters are 116 yen

and 1 liter is 145 yen. I think some multiple of 0.8 liters is 1 liter. Icalculated “1 divided by 0.8”. I found 1.25.

41 T: Wait. 1 divided by 0.8? Oh, it’s 1.25.42 Tach: If we multiply 0.8 by 1.25, then of course we must also multiply

the price by 1.25. So 116 times 1.25 is 145.43 Cs: Yes. We agree to it.

Though the teacher had expected the children to put “÷0.8” into theleft blank, actually “×1.25” was natural for them. He then asked them toconsider it without using the number 1.25 after he told that the number1.25 was not written in the text of the problem Q3. However, they persistedwith the idea of using the number 1.25.

44 Yama: We find how many 0.8 liters are into 1 liter. The price of 1 literis 145 yen and 0.8 liters are 116 yen, so 145 divided by 116.

45 T: What is 145 divided by 116? Is it 1.25?46 Yama: Yes, it is.47 T: Then, what is 116 divided by 145?48 Yama: Let me calculate it. (He actually calculated it.) It’s 0.8.49 Arat: In the case of 5 divided 4, we get 1.25, and 10 divided by 8 is

1.25 too.

The children reinterpreted the expressions familiar to them, e.g. 116 ×10 ÷ 8 as 116 × 1.25 and gained more confidence in the idea “×1.25”.


Here, the teacher again tried to direct their focus to the operation “÷0.8”and then to the relation between “×1.25” and “÷0.8”.50 T: This is “×1.25”. Can you represent it by using division? By what

do you divide 0.8 liters in order to get 1 liter?51 Naga: We divide it by 0.8.52 T: If you divide 0.8 by 0.8, you get 1. Then by what do you multiply 1

liter in order to get 0.8?53 Naka: We multiply it by 0.8.54 T: If we multiply 145 by some number, we get 116. What is the number?55 Sako: Oh, it’s 0.8.56 T: Times 0.8. . . . If we divide 116 by some number, it becomes 145.

This is 0.8, isn’t it? If we multiply 145 by some number, it becomes116. By what do you multiply it?

57 Hira: Well, 0.8.58 T: Is there anything you notice?59 Arat: “×1.25” and “÷0.8” are same.60 T: Are they same? Everyone, check whether “÷0.8” is the same to

“×1.25”.61 Cs: (Children individually checked mainly whether the answer of mul-

tiplication 116 × 1.25 was 145.) Oh, they are same.. . . . . .

62 T: Really? . . .. Can you say that this (116 ÷ 0.8) is the same to “116 ×1.25”?

63 Arat: They are a little bit different. No, oh, yes, same.64 T: (He wrote “A × 1.25 = , A ÷ 0.8 = ”.) I’m asking everyone

whether multiplying some number by 1.25 is the same to dividing itby 0.8 or not.

65 Cs: Yes, same.66 T: Do you think it’s sure that the answer is bigger than the dividend?67 Cs: Absolutely yes! It’s sure.

The children made sense of “÷0.8” in terms of “×1.25”, in which theyhad confidence. It then seemed that the children were more or less consciousof the reciprocal relations and understood why they should divide by 0.8and why the answer would then be bigger than the dividend. To our regret,we could not check the number of children that clearly understood theserelationships at the end of the fourth lesson, because the lesson time wasalready over. However, in the fifth and six lessons the teacher and childrenagain discussed those relations fully, and summarized them as in Figure 12.

It seemed that most of the children were clearly conscious of the rela-tions, as far as we could tell from our inspection of the video. Also, whenin the sixth lesson we asked children to choose the ways of thinking which


Figure 12. Reciprocal relations in the proportional schema of division.

clearly represented that 116 ÷ 0.8 was the expression for finding the priceof 1 liter from the given choices that were presented from children duringthe lessons (it was permitted for children to choose more than one choice),16 children chose the thinking of the reciprocal relations, and 31 childrenchose the proportional thinking using the number line like on the left ofFigure 10. Here, it should not be thought that this fact shows how manychildren understood the relations, but rather how many of them felt somemerits in the ways of thinking beyond just making sense of them. We thinkthat the children’s conceptions of division with decimals enhanced theircertainty towards secondary school mathematics.

5. DISCUSSION

When we look back on the lessons of division with decimals in the class-room, we see that children could make sense of division with decimalsthrough experiencing the disequilibrium and repeatedly overcoming it, andthey could develop their own logical reasoning in the process of lessons. Wefind out that the process can be conceived as being in three developmentalstages. In this section we first show the stages, followed by a discussionof the logical characteristics of children’s reasoning at each stage. Finallywe discuss in what conditions the cognitive conflict could function for thedevelopment of logical reasoning.

5.1. Three stages and the role of representations in the developmentof making sense of division with decimals

It may be natural to consider that in the teaching experiment the children’sunderstandings of division with decimals were influenced by the design andthe orchestration of the classroom lessons. Thus, in this section, we willdistinguish children’s conceptions of division found in our experimentalclassroom.

First, the classroom children conceived division with decimals by ma-nipulating concrete objects. For example, children replaced the situation


of “100 yen per 2.5 meters” with the situation of “200 yen per 5 meters”by connecting 2 pieces of strip (one piece represented the 2.5 meters ofribbon) and solved it as division with integers (“200 ÷ 5”). Moreover, theyseemed to conceive a problem situation not as a different situation but asone of many situations equivalent with the original one (100÷2.5) throughconnecting some pieces of strip in various ways (e.g., 200÷5, 1000÷25).This conception might be promoted through the individual children’s ac-tivities using the “operational number line”, as some children referred tothe possibility of infinite situations that could be assumed as equivalentto the original one. Moreover, this activity seemed to enable children toacquire their own ways of finding an answer to the given problem situation,through judging which operation should be relevant to the given situationby transforming it into other (equivalent) situations that would be easier forthe children to consider. We think that such activities had the character ofconcrete operational thinking, and that they were important for the childrenin order to connect division with decimals to their existing knowledge ofdivision and to prepare their further learning.

Second, children in the classroom began to conceive division as the rela-tionship between the equivalent expressions at the hypothetical-deductivelevel detached from the concrete one. It seemed to be accelerated by thetransition from the operational number line to the number line and then tothe schema of proportion in the classroom. In the process the children’s ob-ject for thinking was gradually changed from the answer to the mechanismof the mathematical expression. The change occurred through children’strials of refuting the idea that 116 ÷ 0.8 = 145 was representing the priceper 0.1 liter, which was influenced by the constraints of their implicit model.Since the children had realized that the expression 116÷0.8 was equivalentto the expression e.g. 580 ÷ 4 that they felt realistic, they seemed to feelthat the expression 116 ÷ 0.8 = 145 was right, but instead they changedtheir interpretation of the answer 145. We think that this is one remarkableexample of the influence of numbers (Bell et al., 1984, 1989; De Corteand Verschaffel, 1996; Harel et al., 1994), in particular the phenomenon ofnon-conservation of operations (Greer, 1987, 1988), though what was notconserved at this point is not an operation but a meaning of an expression.Thus, we may be able to see a new way of looking at the phenomenon asincluding non-conservation of the interpretation of expressions.

The conceptions of division that the children proposed here were not somuch concerning what answer could be found by the expressions 1160÷8 or116÷8×10, as the relationships among the expression 116÷0.8, the trans-formed expressions, and the answer. Thus, the expressions for children thenwere tools for justifying a statement, which was compared with the toolsfor finding the answer at the first stage. The shift of children’s conception


was, we think, more or less enabled based on their conception acquiredat the first stage that a given problem could be solved by transforming itinto other equivalent situations. However, the children’s conception had thecharacter of a concrete operation at the same time, like Naka’s utterance(29). That is, the children didn’t attain the cognitive state of equilibriumbecause of the obstinacy of the constraints of their implicit model.

Third, the children constructed two explanations; each correspondedto two kinds of reversibility. One explanation was based on the inverseoperation. It was when Saka (32) inversed the division into the form ofmultiplication that many children firstly seemed to realize the correctnessof the expression. However, more explanations were needed by childrenbecause the multiplication had the character of checking after solving thedivision problem Q3. When the expression was inverted into a multipli-cation form, the teacher could have activated their discussion whether theanswer was bigger or less than the dividend, but actually he did not doso. In that sense, the first chance to overcome the constraints of children’simplicit model might be lost.

Next the children made sense of the expression by using multiplicationin another way. It was to consider “÷0.8” as equivalent to “×1.25”, whichwere so-called two sides of the same coin. It was easier for the childrento consider the operation of changing 0.8 to 1 as “×1.25” than as “÷0.8”,because they had already learned multiplication with decimals and appreci-ated that “×1.25” makes the answer bigger. Though the way of consideringthe transformation from 0.8 to 1 as the operation ÷0.8 was mainly led bythe teacher, the children seemed to understand the transformation throughregarding it as not dividing by a decimal less than 1 but as dividing somenumber by the same number. And, since the children had already appreci-ated that the answer of 116 ÷ 0.8 was 145 by the end of the second stage,they could check that the result of 116 × 1.25 was same to that of thedivision 116 ÷ 0.8.

The use of the schema of proportion, especially its abbreviated version,in the classroom could promote children’s thinking, though it had not beensupposed by the authors and the teacher when we had designed the lesson.Through reflecting on the descriptions summarized in the schema on ablackboard (Figure 12), the children seemed to be able to make sensemore deeply of the relationships among the numbers and operations in theschema. We can deduce that this eventually led the children to conceivemultiplication and division as a system of operations, in other words toacquire formal operational thinking.

As Naka (36) stated “I can regard it to take 0.8 pieces of 116”, somechildren might conceive multiplication with decimals by extending theirconcrete operational thinking. But we think that division with decimals


loses the meaning of a concrete operation in nature, since the divisor isno longer to divide in its operational sense. Thus, it seems natural withhindsight that the children first contrasted “division with decimals” withmultiplication, and next conceived it in terms of multiplication in two waysof inverse and reciprocity.

Next, we discuss the logical characteristics at the second and the thirdstages.

5.2. Logical characteristics in children’s making sense of division

We find that children’s reasoning was more or less logical at the secondstage which was the transitional stage from the concrete to the formaloperational thinking.

If we mathematically observe children’s explanations in terms of sixproperties of division (1) to (6) described in Section 2.1, we can see thatproperties (1) and (6) were frequently used in their reasoning. The property(1) was found in transforming the expression 116 ÷ 0.8 into 1160 ÷ 8, andthe property (6) was used when 116 ÷ 0.8 was changed into 116 ÷ 8 × 10.The other properties are also found, though maybe implicitly. For example,we can see properties (2) and (3) in Kawa’s utterance (26).

When we logically examine the children’s reasoning in the process ofjustifying why 116 ÷ 0.8 = 145 represented the price of 1 liter, the reason-ing seemed to have some characteristics of formal operational thinking. Forexample, Katt (24) showed the syllogistic reasoning by combining somegiven facts as described in Section 4.3.1. And Moto’s reasoning (22) couldbe an origin of the formal reasoning in the reductive absurdity at the hy-pothetical level. Furthermore, Kawa (26) implicitly used the commutativelaw and operated on the expression itself, as she stated “If we put this 10 infront, it (116 ÷ 8 × 10) becomes the same as 1160 ÷ 8”. In other words, shejustified it by using just symbolic operations in the expression. Thereforeit could be said that at this point the children’s reasoning had some of thecharacteristics of formal operational thinking.

However, again from a mathematical point of view, we think that thescalar operator for transforming the elements (a, b) with each other wasstill integers in the children’s reasoning; so they did not attain equilibriumat the second stage. At the third stage, however, children recognized therole of decimal operator “÷0.8” that transformed 0.8 meters into 1 meter(quantity per unit) and at the same time 116 into the answer. Moreover,the operator “÷0.8” seemed to be reconceived by children both as thereverse operation of ×0.8 and as the equivalent operation of “×1.25”. Atthis point, we think, the children attained a higher level of understandingof the equilibrated system of multiplication and division.


5.3. The aspects of the emergence of cognitive conflicts

We can say that the intervention by cognitive conflicts was successful inour teaching experiment. In this section, as the last point we will brieflydiscuss in what conditions such conflicts or disequilibria occurred in thelessons.

We think that our classroom lessons basically followed the processof a diagnostic teaching approach (Swan, 2001), and that the process waseffective to some extent. However, we must consider other studies in whichlearners didn’t interpret the conflicting elements as conflicting, and didn’talways respond to the conflict progressively (Tirosh, 1990; Tirosh andGraeber, 1990). Thus, we need to better clarify the characteristics of ourteaching experiments.

A most remarkable characteristic is that the conflicting data were not thepreviously decided ones, and the teacher instead formed the ideas childrenpresented in opposition or comparison. This characteristic is somewhatdifferent from the diagnostic teaching approach. In order to make the con-flict explicit, it may be quicker to give children the conflicting data thatare opposed to their conceptions, by diagnosing it in advance, like Tiroshand Glaeber (1990) who gave their subjects a problem 4 ÷ 0.5 and askedthem to calculate it, and furthermore to work out 4 ÷ 1/2 if moving thedecimal point in the algorithm obscured the conflict when they indicatedthe misconception “the quotient must be less than the divisor”. However,we need to consider that the conflicting data may really be realistic tosubjects.

It’s difficult to judge didactically whether a teacher should give con-flicting data intentionally, but we think two situations in opposition need tobe fairly realistic to children. From this point of view, the disequilibriummight not have emerged if the teacher had given children some calculationproblems in which the divisors were less than 1, for they just began tounderstand calculations at that time because of the introductory lessons.

Another characteristic of the teaching experiment is related to our as-sumption that there were various semiotic devices for overcoming the con-flict and also that the classroom children already had some abilities. FromLimón’s (2001) viewpoint, the children might have already acquired somegood tendencies such as individual motivation, a belief in learning, an at-titude of engaging in arguments, and some reasoning ability because theyactively took part in the discussions, and could express their own opinionswell. However, we can also think that a series of activities in the lessons,especially the symbolizing processes to connect the concrete activity tothe abstract thinking, reciprocally enhanced their motivation, attitude, andreasoning.


However, we doubt whether children had the epistemological belief ofthe consistent and changing nature of knowledge, since fifth graders couldhardly have experienced such an extension of knowledge. Nevertheless,they didn’t separate division with decimals from their existing knowledge,but tried to understand by using their knowledge. Hence, we may considerthat a series of activities, especially the initial activity making equiva-lent situations, functioned well for avoiding the separation, and that thebelief might be enhanced by classroom discussions because the disequilib-ria were activated by thoughtfully engaging in discussions, which in turnmight be activated by other factors children had. However, it goes withoutsaying that further teaching experiments must be implemented to under-stand better these aspects in the intervention by cognitive conflict in theclassroom.

6. FINAL REMARKS

It has been several decades since researchers began to analyze factors pre-venting children from learning multiplication and division with decimals.However, the process by which children might overcome the difficultiesand how we might be able to help them have still not been fully clarifiedin the field of mathematics education. In this study, we tried to better clar-ify the processes through which children might overcome difficulties anddevelop logical reasoning by analyzing classroom lessons in our teachingexperiment.

The main outcome of this study was the clarification that the rea-soning based on two kinds of reversibility contributed to overcomingthe difficulties. In particular, it was the awareness of the reciprocal re-lation of multiplication and division for children that made their adher-ence to the constraints from the implicit model vanish. We could describethese processes as three stages of making sense of division with deci-mals, in which children developed their reasoning logically and math-ematically. Here it should be noted that in our experimental classroomthese stages emerged not linearly, but as a process of attaining equilib-rium in which temporal regressions (disequilibria) were often involved,and more coherent ideas were constructed by coordinating some ideaswith each other every time a temporary state of equilibrium was achieved.We believe that what the children developed in the lessons would func-tion as strong foundations for further mathematical learning at secondaryschool.

It must be needed to investigate whether we can get the same resultswith the children in other classrooms at a local public school, because


the children involved in our teaching experiment had higher mathematicalabilities. However, we believe that the design and findings of our teach-ing experiment will have some important suggestions for the investiga-tion. Moreover, the followings still remain as the tasks to be addressed.First, regarding the conditions in which disequilibrium may emerge, sofurther investigations will be needed under some classroom settings basedon Limón’s (2001) viewpoint. We definitely ascertained that the factorsLimón indicated were working in our classroom lessons. However, it isnot clear from our data if all factors were essential for the emergenceof cognitive conflict. Second, we couldn’t explore and discuss how thesymbolizing processes (Gravemeijer and Stephen, 2002) assisted chil-dren’s making sense of division with decimals. Third, though we dealtwith the generalization of partitive division in this paper, the logical de-velopment in the generalization of quotitive division also needs to beclarified. As these generalizations are related to each other, and withother proportional concepts (Thompson and Saldanha, 2003), we needto examine the issue as being relationships in a web of related con-cepts. We believe that addressing these tasks will be shed some lighton ways to bridge the gap between elementary and secondary schoolmathematics.

ACKNOWLEDGMENTS

This paper is the elaboration of Okazaki (2003) presented at the PME27 conference as a Research Report. We would like to thank the refereesfor their insightful and helpful comments on this paper in the process ofreviewing. We also thank the school staff and children who participatedin the teaching experiment. This research was funded by the Grant-in-Aidfor Scientific Research in Japan Society for the Promotion of Science (No.14780097 and No. 16700539).

NOTES

1. In Japanese educational system, the elementary school continues from 1st to 6th grades,lower secondary school the 7th through 9th, and upper secondary school the 10th through12th.

2. It was also reported that the multiplicand or dividend has some effects by Harel et al.(1994) and De Corte and Verschaffel (1996).

3. The positive only approach makes no attempt to examine errors, and in fact avoids themwherever possible by teaching the pupils to use simple and efficient methods from thestart (Swan, 1983, p. 211).


4. It can be helpful to refer to Piaget’s (1985) distinction among three types of compensa-tion; alpha, beta, and gamma. Type alpha means that when a new fact surfaces, it canproduce no modification of the system at all. Type beta reactions transform perturbationsinto internal variations by integrating or internalizing them into the cognitive system inplay. However, these may be capable only of partial compensations. Type gamma is aperfect compensation, where the possible variations are predictable or deducible andlose their character as perturbations and become instead, potential transformations ofthe system.

5. In the Japanese notational system, we write 300 × 5 as the expression for the problem“The price of one apple is 300 yen. How much do 5 apples cost?” which is differentfrom the other countries’ notational system of 5 × 300.

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MASAKAZU OKAZAKI

Department of MathematicsJoetsu University of Education1, Yamayashiki, Joetsu City943-8512, JapanE-mail: [email protected]


MASATAKA KOYAMA

Department of Mathematics EducationGraduate School of EducationHiroshima University1-1-1, Kagamiyama, Higashi-Hiroshima City739-8524, JapanE-mail: [email protected]

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