matalliotaki, e., (2012). resolution of division problems by young children: what are children...

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Resolution of Division Problems by Young Children: What

Are Children Capable of and under which Conditions?

Journal: European Early Childhood Education Research Journal

Manuscript ID: Draft

Manuscript Type: Research paper

Keywords: child, division, graphical representation, problem resolving,

reasoning

URL: http://mc.manuscriptcentral.com/recr

European Early Childhood Education Research Journal

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Resolution of Division Problems by Young Children: What Are Children

Capable of and under which Conditions?

1 Theoretical Framework: Concepts and Related Research

This section presents a theoretical approach to the operations of sharing and division,

followed by a review of related studies of young children.

1.1 Concepts on Division and Sharing

Division is an important arithmetical operation because even college students and adults

misunderstand this subject. Meanwhile, relatively less empirical research exists on the

beginnings of the learning of division than on other arithmetical operations. However, division is

interesting from the point of view of the schemas of action: the schema of sharing is at the base

of division. This idea is founded on Piaget’s claim that the sensory motor schemas are the bases

of subsequent formal constructions.

From the perspective of the mathematical definition of division, the idea that the activity

of sharing is a mathematical activity is contestable. Sharing concerns a form of socialisation.

However, from the developmental point of view, it is possible to consider that the schema of

sharing, which brings into play correspondences and the constitution of equivalent classes, can

constitute the first steps of division. At one time of their development, the children may be able

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to carry out division while being unaware of the mathematical components involved (Frydman

and Bryant, 1988); these are generally learned later.

1.2 Existing Studies on the Solution of Division Problems or Sharing by Young Children

This section will review empirical studies on partitive and quotitive division among

children between 5 and 6,5 years old.

1.2.1 Strategies of Resolution used by Children in Problems of Division

Kouba (1989) asked children between the ages of six and nine to solve division problems

that she presented to them verbally. She identifies three types of problems depending on the

required quantity.

1. Multiplication (unknown number of elements of the totality of the group)

2. Quotitive division (unknown number of subsets)

3. Partitive division (unknown number of elements constituting each subset)

Kouba suggested that the context of the relations between the quantities in a division

problem contributes to the difficulty of the problems more than it does in addition or subtraction

problems. Vergnaud (1983) and Fischbein et al. (1985) have stated that each arithmetical

operation is related to an intuitive, implicit, unconscious and primitive model. This model

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influences the decisions made when an operation is used. Fischbein et al. propose that—in

addition—the intuitive model is ‘put together’ while in subtraction it is ‘removed’; for

multiplication, it is the repeated addition, and for quotitive division, it is the model of repeated

subtraction. For partitive division, it is sharing.

The children in the study of Kouba were interviewed individually, and the strategies of

resolution were categorized as ‘inappropriate’, ‘appropriate’, and ‘not identifiable’, independent

of the accuracy of the calculation. The questioned children had concrete objects in case they

expressed the need to use them to help in answering the questions.

Kouba counted 333 suitable strategies of the 768 problems suggested and, among these,

56 different strategies, which is considerable compared to the standard procedures that were

taught. These 56 strategies were analysed according to two criteria: the degree of abstraction of

the step and the mode of use of the objects placed at participants’ disposal.

The results concerning the youngest children (six years old)—close to those of

Matalliotaki’s (2007) empirical study—indicate a 63 percent of the children solved the problems

correctly.

1.2.2 Solution to Partitive or Partitive and Quotitive Problems

Squire and Bryant’s (2002) study used material supports to highlight the importance of

the schemas of action in solving partitive problems.

In their study, children from five to eight years old were asked to solve problems

requiring them to distribute candies among dolls; the candies were divided among boxes in a

variety of ways. The number of candies in each box corresponded either to the divisor or the

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quotient. Before every experiment a “control” (“baseline trial”) condition was established in

which all of the candies were piled in front of dolls (Figure 1, right side).

Figure 1. Distribution of Dolls and Candies, according to Squire and Bryant (2002)

On the left side of Figure 1, the top figure represents the grouping by the divisor (the

number of boxes corresponding to the numbers of dolls); the bottom figure represents the

grouping by the quotient.

The principal result of this study is that, for all examined ages, the children were helped

more by the grouping by divisor than by the grouping by quotient. The ability of the children to

solve the problem by grouping by quotient improves with age. This can be explained, according

to Squire and Bryant by the improvement in the capacity to use the one-to-one correspondence

with age, or by the fact that older children eventually learn to understand the interchangeability

of the divisor and the quotient.

Including questions about the number of boxes and the number of candies actually

improves children’s scores when grouping by quotient. According to Squire and Bryant, this

made it possible for children to use additive instead of multiplicative reasoning. This was

confirmed by additional studies (Sophian et al., 1991) that found that, in children ages five and

six, passage by the addition plays a facilitating role.

The spatial arrangement of dolls and candies influences the children’s ability to solve the

problem. Squire and Bryant suggest that children’s performance might be better if they can

rearrange the objects themselves. They posit that handling of the objects or their images

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improves accuracy. The older children perform better because that they have a greater capacity

“to mentally rearrange” the whole under the condition of grouping per quotient.

This study also demonstrates that children’s early comprehension of division is

influenced by their comprehension of level sharing and the distribution of portions to recipients.

Consequently, we can assume that informal experiences affect the learning of mathematics. This

finding has important educational implications.

Another study carried out by Squire and Bryant (2002) compared the procedures used to

solve partitive and quotitive problems. In their study, two conditions were presented: a partitive

task in which the objects were grouped either by divisor or by quotient, and a quotitive task with

the same two groupings. The children found the grouping by divisor to be easier when solving

the partitive division problem and the grouping by quotient to be easier for quotitive division.

According to Squire and Bryant, such results must imply non-mathematical factors since in

mathematical terms there were no differences between the two conditions. Therefore, they

suggest that the most convincing reason for this difference between the two conditions is a

cognitive one.

The authors suggest that, even if certain models are strongly artificial and benefit only

from cultural drives (for example, in “pure” mathematics), others are probably acquired without

explicit instruction and are used by everyone. According to this approach, informal experience

can plausibly contribute to the formation of a “mental model” of the concepts and, consequently,

children can start to acquire a mental model of division by sharing. In other words, sharing can

be a schema of action by which a comprehension of division develops. This developmental

concept of the formation of the concepts supports Vergnaud, who granted a crucial role to the

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schemas of action1. The schemas of action would be primary and be the matrix for the

construction of the concepts.

Another interesting element arising from the work of Squire and Bryant is that the

difference between the children with better performances and those who encountered difficulties

in performing the tasks occurs only at the plan of the selected procedures of resolution (and not

the amount of time needed to arrive at a correct answer). This resulted in stressing the procedures

rather than speed or accuracy. This aspect was retained when designing tasks for the current

study.

1.2.3 The Inverse Divisor-quotient in Relation to Partitive and Quotitive Tasks

Correa, Nunes, and Bryant’s study (1998) pertained to partitive and quotitive tasks; it

drew a conceptual distinction between sharing and division. In sharing, children treat only the

equivalence of the shares. The concept of division, in contrast, implies the comprehension of the

relations of three values represented by the dividend, the divisor, and the quotient. The

equivalence of the shares is assumed, although in division children must understand that the

larger the number of shares, the smaller those shares will be.

In their studies the children understood well that, if the divisor (for example, the number

of rabbits) increases, the quotient (for example, the number of carrots per rabbit), decreases.

Comprehension improved with age, even though young children understood partitive tasks better

than they did quotitive ones, their abilities to perform both tasks was impressive.

1 The schemas of action (schèmes d'action) are familiar actions that can offer initial comprehension of the arithmetic operations (Vergnaud, 1985).

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1.3 Graphical Productions of the Children during Problem Solving

There are virtually no studies on the graphical productions of young children’s ability to

solve division problems, so we make use of the work that has been done in other fields of

problem solving.

Weil-Barais and Resta-Schweitzer (2008, 2006) showed that the graphical productions of

children between five and six years of age years reveal the degree of conceptualisation in

physical phenomena. The drawings are one means by which children express a complex

phenomenon, since they facilitate the expression of the spatial relations of the objects. Tantaros

et al. (2005) studied the production of graphical representations by young children in order to

communicate the solution of a problem and concluded that children’s productions improve with

age and that drawings can constitute a cognitive tool. Lehrer and Schauble (2002) consider that a

representation is not only a copy of reality. That implies inventing and adapting conventions of a

system of representation in order to choose, compose and transport information. Children learned

that a system of representation could represent information that was not immediately perceptible.

The authors conclude that developing adequate graphical representations for information

represented conventionally, promotes the learning of the mathematical concepts that are needed

for comprehension of the properties of this information.

Based on the research that has been cited, we expect the use of graphical representations

by children will make it easier for them to solve division problems. The drawings can include the

graphic strategies to follow since the icons let children visualize the position of the mathematical

elements and the relations between them. We expect that the children will use the drawings as a

tool to envision the connections among the elements.

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1.4 Justification of Empirical Problems

The studies presented in this article suggest that there is very little empirical research on

young children’s ability to solve division problems. However, it is an interesting operation as the

schema of action that it mobilises is level sharing—an activity that concerns a form of

socialisation and has mathematical aspects.

Even if the level sharing belongs to the informal experience of the child, division is

regarded as a more complex operation than addition or subtraction due to the relations between

the quantities. However, even if it is more complex arithmetical operation, children—even those

as young as five—participating in the studies showed remarkable success with certain division

problems. This supports the idea that the direct instruction of division is not essential in the

formation of the concept—at least in its least conceptually elaborate form (level sharing).

The characteristics of division raised in the studies reported in this article justify the

choice of graphical representations. Indeed, if the regroupings that need to be realised in order to

carry out a quotitive division can be materialised by a distribution in boxes (cf experiments by

Squire and Bryant, 2002), they can also be the subject of graphical representations.

In Squire and Bryant’s (2002) study, the children could not rearrange the objects or to

modify their distribution. However, the use of graphical representations makes it possible to

consider and preserve several types of regroupings. Thus, as suggested by Squire and Bryant the

physical handling of objects can help children to solve complex problems; thus, it would be

interesting to examine in what contexts the use of graphical representations plays a facilitating

role. Furthermore, Kouba wonders if the categories of resolution that she distinguished in her

study would be the same if the problems had employed graphic supports instead of physical

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objects. Considering the fact that the youngest children have difficulty keeping several variables

in mind and performing complex operations, the graphical representations of a problem can

indeed facilitate the handling of such situations by young children.

For the problems designed herein, quotitive division was chosen because studies have

demonstrated that this type of division appears more complex (and is thus more interesting to

study) for children than partitive division. The intuitive model of quotitive division is the

repeated subtraction. The children have already experienced the schema of the action of sharing,

which applies to partitive division, while the schemas of the action of subtraction, which apply to

quotitive division, are more difficult to follow. Employing the categorisation of Kouba (1989),

the first four problems proposed to the children belong to the category “division quotitive-

grouping” and the last two problems belong to the “division quotitive-set” category.

Furthermore, a study of tasks intended for the children of 5,5 to 6,5 years old,

Matalliotaki (2007) found that very few problems presented to children ask them to make

inferences. Several reasons may explain this fact. Children of this age may not have developed

the capacity to draw inferences from graphic information. Moreover, these inferences may rely

on formal knowledge that the children have not yet acquired, since the literature does not

envisage this type of exercise.

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To document the first interpretation, the current study conceived problems that seemed a

priori adapted to the children of five to six years old of a nursery school, although quasi non-

existent in the consulted school exercises (Matalliotaki, 2007). In these problems, the children

must make inferences related to quantities. More precisely, these situations focus on quotitive

division: a number of objects in total and a number of objects by recipients being indicated to

determine the number of recipients. In mathematical terms, this involves giving the quantity of a

group and the number of elements constituting the subsets (or parts) so as to find the number of

subsets (or of parts). When not having easy-to-handle objects, young children involved in this

study could solve these problems by the use of graphical representations.

2 Empirical Study

In this section we present the results of an empirical study that we conducted in a nursery

school in France.

2.1 Methodological Framework

We present the methodological framework of the empirical study, consisting of the

population chosen, the study design, and the description of the problems presented to the

children.

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2.1.1 Subjects and Design

Fifty-five nursery school children in good academic standing n the 13th arrondissement

(district) of Paris took part in this research. The children ranged from 5 years to 6 years 6 months

in age. The 31 boys and 24 girls were divided among three classes. Their parents signed a

consent form authorising their children’s participation in the study.

The children were interviewed individually. The child and the researcher sat face to face

at table with sheets of paper and coloured pencils.

In the Gloves-Socks-Footballers test, some boards (figures 2 to 4) are presented to the

child progressively. In addition, for any answer given by the child, the researcher asks him or her

to explain how he or she arrived at the answer, and noted the explanation without expressing

approval or disapproval.

Each child was given as much time as he or she needed to answer each questions. The

meetings were video recorded.

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2.1.2 Description of the Problems

The problems were first presented verbally and then with a graphic support, which

reflected the facts of the case. The introduction of the graphic support is regarded as a new

problem (not as a means of assisting with the verbal one), even if the quantitative information

and the structure of the problem were the same. The purpose of the exercise was to examine the

types of strategies that the children use both with and without the graphic support, not only to

ascertain if they were able to solve a division problem, as previous studies have demonstrated.

Problem of the Gloves: Oral Presentation. The first question was “how many children

can one equip with six gloves?” This first problem implies minor amounts, which children of this

age can process either mentally or with the use of fingers. The solution of such a problem can be

based on the child’s cultural experience and on body consciousness (demonstrated in

Matalliotaki 2001), which indicates that a person with two hands needs two gloves. The children

must thus deduce that, as each child needs two gloves (cardinal of the subsets), six gloves can

equip three children. The problem can be formalised as follows.

2 gloves 1 child

6 gloves X children

For children, at an age where they are not yet able to produce formal mathematical

notations, it is possible to await an analogue representation (mental or written) close to that the

following:

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••

••

••

The number of points of each group represents the divisor while the number of groups

represents the quotient. We note that these problems (and the ones that follow) involve additive

reasoning (prepare three pairs and then add them to find the correct answer), which, according to

Squire and Bryant (2002) and Sophian et al. (1991), plays a facilitating role for children of this

age.

Problem of the Gloves: Pictorial Display. The second problem proposes a schematisation

of the data likely to facilitate the choice of a strategy of resolution. The same problem is verbally

presented, but this time accompanied by a board (with the format 21x 29,7) that graphically

represents the objects. Figure 1 presents a reduced graphical representation that was presented to

the children.

Figure 2. Reduction of the Graphic Support Accompanying the Pictorial Display of the “Gloves” Problem

This schematisation was conceived in order to facilitate the grouping of the elements

(construction of the groups, which indicate the number of elements of each group). In this

manner, introducing the strategy to be followed for the resolution of the problem involves the

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constitution of the regroupings (subsets) and their enumeration. The graphical representation

would provide an inferential function as it makes it possible to calculate a quotient. Few

drawings accompanying the exercises by mathematics perform this function (Matalliotaki, 2007),

which is why this study examines whether or not this type of drawing could help children

complete a calculation.

The drawn elements are not perfectly ordered because the presence of the identical

features between the gloves of each pair is expected to be sufficient for completion of the

groupings.

Problem of the Socks: Oral Presentation. The third problem utilised the same structure as

the first, but with larger quantities: how many children can one equip with twelve socks? Here,

the quantities are not readily processed with the use of fingers, thus legitimating the use of a

schematisation of the data.

Problem of the Socks: Pictorial Display. The same problem was put to the children, with

a drawing of six pairs of socks (with the format 21x 29,7). Two versions were proposed: one in

which the pairs of socks are locatable by graphic characteristics (see Figure 2, left side) and one

in which all the socks were drawn in the same way (see Figure 2, right side). These two versions

were presented to two groups of children in the same age cohort. It is assumed that the first

drawing performs two functions—referential and inferential—as it provides indices for

regrouping by two: the child can find the answer by counting only the different socks. The

second drawing performs only one referential function. It can perform an inferential function if

the children produce graphical indices of regrouping. These two versions were proposed in order

to examine the impact of the inferential function on children’s performances in a concrete

manner.

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Figure 3. Reduction of the Graphical Representation Accompanying the “Socks” Problem.

Problem of the Footballers: Oral Presentation. Unlike previous problems, this problem

proposes a more formal mathematical problem:

Eight footballers will practice in pairs (i.e., two by two: This explanation was

incorporated when some children started to solve the problem under the

assumption that there would be two groups. The sentence “they will be involved

in groups of two” did not appear clear enough to describe the formation of the

groups. ). Each group will have a ball. How many balls will be used?

The children are expected to solve this problem using the former experience (graphical

resolution of the problems). A possible schematisation of the procedure used to solve this

problem might be as follows:

8 footballers 2+2+2+2

2 + 2 + 2 + 2 footballers

1+ 1+ 1+ 1 balls

or more formally:

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2 footballers 1 ball

8 footballers X ball

from which we get the equation b = f/g, where b is the sought quotient, f the full number of

footballers, and g the number of footballers in each group.

For children of this age, who have not yet received formal instruction in mathematics, an

analogue representation seems more suitable:

��

• • • •

In this representation, the squares represent the footballers and the circles the balls. The two

footballers in each group represent the divisor and the number of built footballer couples

represents the quotient. This problem represents for the children a more complex situation than

does preceding problems because the bond with the human body no longer exists, thereby

eliminating a familiar context to which the child can refer. The context is more formal even if

football is an interest of the children. The number of balls per player is arbitrary, which is not the

case in regards to the gloves or socks, which are always in pairs. The children must thus

memorise information (i.e., one ball for every two players), which may contradict what children

usually associate with balls.

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Problem of the Footballers: Pictorial Display. This problem was presented to the

children along with a drawing (with the format 21 x 29,7) (see Figure 4). Unlike the second and

fourth problems, both balls and players are represented here. After the grouping of the players in

pairs, the children can connect each group to a balloon and count the number of connections.

Figure 4. Reduction of the Graphical Representation Accompanying the “Footballers” Problem.

The suggested schematisation does not provide an inferential function as the drawing

does not propose elements that would support the grouping or the connection of each group with

one ball. This last problem was conceived to test if a representation very close to an environment

familiar to the child (e.g., story books with illustrations, images in media) can be seen and

handled in an abstract context of mathematics when the children have not received formal

education in mathematics. According to Squire and Bryant (2002), informal experiences can

affect the acquisition of mathematical skills. In our study we determine whether or not an

informal context that is familiar to children facilitates their problem solving.

Table I provides, in a synthetic form, the problems suggested to the children within the

framework of this empirical study.

Table I: Problems Addressed to Children and Their Instructions

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3 Results

Three situations were presented to the children, first verbally, then in writing (through a

drawing of the objects). The written form is associated with a greater number of correct answers

(11 percent of correct answers to the oral examination compared to 40 percent in the written

form). In the written form, the problem of the footballers proves to be more difficult but it is

nevertheless solved correctly by approximately one-third of the children.

In order to arrive at the correct answer the children inevitably followed the suitable strategy.

Two strategies were identified: pairing with enumeration of the groups (correct strategy) and the

one-to-one correspondence (incorrect strategy).

Table II presents the number of children who followed the correct mathematical strategy to

solve the problems (pairing and counting of the built groups) and the number of children that

followed the incorrect strategy (the one-to-one correspondence). To determine if the children

followed the one-to-one correspondence strategy, it was enough to see their answer. If the

answer to the problem of gloves, for example, were 6, it means that the child distributed a glove

to each child instead of building pairs of gloves. In following table, OOC means one-to-one

correspondence and G/CG means Grouping and Counting of the built groups.

Table II: Summary of the Mathematical Strategies Used by the Children according to Age

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We observe the use of the strategy of one-to-one correspondence progressively in the

long-term performance of the tasks. According to table II, the children seem to give it up

gradually.

Some children seem to have understood the strategy or building groups in order to solve

the problems but for some reason, they could not make this strategy succeed. We thus present in

table III the number of these children (by age) through the problems, in comparison to the

number of children who answered it correctly. In the table, G means Grouping and represents the

number of children who carried out the grouping and CA represents the number of children who

answered correctly.

Since authors like Squire and Bryant discovered that even a few months of difference in

the age of the subjects could play a significant part in their ability to understand symbolic

notations, we carried out an analysis of the performances of the children depending their age.

Table III: Mathematical Strategies Used by the Children by Age

In problems such as the graphical problem of gloves, nine children carried out the grouping

correctly but could not to finish the procedure and solve the problem. This leads us to suppose

that the children knew which strategy to follow (thanks to the preceding problem) but were not

ready to apply until the end. This can explain the absence of “grouping” without arriving at the

correct answer in the first problem, where the children did not have yet a mathematical model of

a strategy on which they could rely.

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3.1 Improvement of Performance through the Tasks

We observed (cf table III) a progressive improvement of the children’s performance of the

tasks. The children abandoned the erroneous strategy, by gradually applying the strategy of

grouping the elements.

The children spontaneously used an analogue representation (their fingers) when the

problems were verbally explained to them, but not for the first problem. Moreover, when they

used their fingers, the use that they made of their fingers (which represented the elements) was

influenced by the drawings that we presented to them in the preceding problems thanks to which

they managed to find the right strategy to solve the problem. This shows that the children were

influenced by our suggestion to use a graphical representation, which ensures the inferential

function to solve problems; they seem to trust this tool.

Indeed, the inferential function ensured by the first drawing (gloves) and generally, the

fact of suggesting to children the use of an analogical system, influenced them to find an

analogical strategy to solve the following problem. David had a typical example of a graphical

production (cf figure 5): he used graphical representations that were very similar to the drawing

that had been presented to him to solve the problem of the footballers. David did not use such

drawings for the first problems that had been presented to him, so we can conclude that the

graphical representations that we introduced influenced him to choose such a technique to solve

the problem.

Figure 5: Graphical Production of David for the Resolution of the Problem Footballers Exposed Orally

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This figurative drawing leads us to conclude which strategy that the pupil applies to solve

this type of problem:

• Distinguishes each footballer as a mathematical unit

• Gathers two footballers

• Allots to each pair of footballers a balloon.

3.2 Errors of Children: Not to Escape the Aesthetic or Pragmatic Context

We tried to locate the error children most often made during problem solving in order to

understand their reasoning and to identify the possible reasons for their inability to reach correct

solutions.

The children answer graphically or verbally by drawing or colouring the sketch given to

them, or while producing conclusions relating to their cultural knowledge of the problem. For

example, the footballers will need only one ball because football is played with one ball.

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Some children had difficulty in abandoning the semantic context. Fifteen of the 55 children

stated that the footballers would need one ball in the oral problem of footballers (by justifying

their answer “the football game is played with one ball”). In the graphic version of this same

problem only seven of the children answered that one ball would be used.

We saw that children were influenced by the graphic data in giving their answer and that they

had difficulty in remembering the instructions. We suppose that children of this age, as affirmed

by Squire and Bryant (2002), often reinterpret problems in a way that makes sense for them. The

pragmatic context of children influences their interpretation of the instruction. We can assume,

then, that with these kinds of problems, the informal context of children did not facilitate the

solution.

4 General Discussion

For the three situations, the drawing allowed the children to solve the problem by gathering

pairs of objects. In the verbal form of the problems we were not able to identify the strategies

used by the children; without paper and pencil, the children are not able to keep track of their

train of thought in order to solve problems that had been presented to them orally.

According to table II, the children gradually abandoned the strategy of one-to-one

correspondence. This means that children received some training in the resolution of the

problems that required the same strategy. Children of this age do not have experience in solving

problems of quotitive division. The choice of the erroneous mathematical strategy thus seems to

make sense. What is astonishing is the fact that children have been able to find the correct

strategy despite their lack of experience in division. This shows that the children of this age have

the ability to solve such abstract problems.

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As they continue working, more of the children build groups correctly, even if they do

not reach the correct answer. This enables us to say that there is an improvement in their

execution of a mathematical strategy, while passing from one problem to the other. The first

improvement seen was the gradual abandonment of the one-to-one correspondence strategy.

In table 3 we also observe that the number of the correct answers increases with age,

which can show an improvement of certain capacities of children with age. In the graphical

problem of gloves for example, among the oldest children, ten came up with the solution to the

problem; only five of the youngest children.

We also observed that the youngest children found it more difficult to transform a

“grouping” into a ‘correct answer’. Thus the correct strategy might depend on a capacity, which

is acquired with age: the capacity to coordinate the parameters of a problem and to retain them in

memory (Squire and Bryant 2002). The youngest children can conceive the correct mathematical

strategy to solve the problem but their ability to perform complex operations and to store partial

products remains immature. Therefore in general, the idea to accompany a problem with a

graphical representation would affect the performances of the youngest children. In order to

confirm this assumption, a test with a larger number of children in each age cohort would be

necessary.

The graphic version given to the children helped them not only to find the correct

strategy for the resolution, but it also gave them an analogical strategy with which to solve the

verbal problems and to escape from the semantic context of the problems which frequently

constrains children of this age.

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The children, even without experience with and formal capacities of this type of exercises,

managed to solve the problems and to acquire the training to improve their performance. The

children’s observed successes at division at an early age should encourage educators to make the

most of their competences, even though official school programmes do not do so. Indeed,

younger children are fully capable of learning division, and this capacity should be cultivated

sooner instead of later. This justifies that research among pre-school age children on this topic

should continue.

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References

Correa, Jane, Terezinha Nunes, and Peter Bryant. 1998. Young children’s understanding of

division: The relationship between division terms and a non-computational task. Journal

of Educational Psychology 2: 321-329.

Fischbein, Efraim, Maria Deri, Maria Sainati Nello, and Maria Sciolis Marino. 1985. The role of

implicit models in solving verbal problems in multiplication and division. Journal for

Research in Mathematics Education 16: 3-17.

Frydman, Oliver and Peter Bryant. 1988. Sharing and the understanding of number equivalence

by young children. Cognitive Development,3: 323-339.

Gaux, Christine, Lydie Iralde, Annick Weil-Barais, and Aline Ferte. 2005. Evolution de

l’utilisation des systèmes de notation pour communiquer à autrui la construction d’un

objet, entre le cours élémentaire 1ère année et le cours moyen 2ème année (de 7 à 11 ans).

Colloque Noter pour penser, Angers, 26-27 January 2005.

Kouba, Vicky. 1989. Children’s solution strategies for equivalent set multiplication and division

word problems. Journal for Research in Mathematics Education 20: 147-158.

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Lehrer, Richard and Leona Schauble. 2002. Symbolic Communication in Mathematics and

Science: Co-Constituting Inscription and Thought. In: E. Amsel, J. Byrnes (Ed),

Language, Literacy, and Cognitive Development. London: Lawrence Erlbaum

Associates.

Matalliotaki, Eirini. 2001. L’utilisation du dessin comme outil cognitif à l’école maternelle.

Mémoire de D.E.A en sciences de l’éducation, Université René Descartes-Paris 5.

---. 2007. Les pratiques graphiques à l’école maternelle dans un contexte de résolution de

problèmes. Thèse de Doctorat en Sciences de l’Education, Université René Descartes-

Paris 5.

Resta-Schweitzer, Marcela and Annick Weil-Barais. 2006. Education scientifique et

développement intellectuel du jeune enfant. Review of Science, Mathematics and ICT

Education 1: 63-82

Sophian, Catherine. 1991. Le nombre et sa genèse avant l’école primaire. Comment s’en inspirer

pour enseigner les mathématiques. In : Jacqueline Bideaud, Claire Meljac and Jean- Paul

Fischer (Eds.), Les chemins du nombre (pp. 35-58). Lille: Presses Universitaires de Lille.

Squire, Sarah and Peter Bryant. 2002a. The influence of sharing in young children’s

understanding of division. Journal of Experimental Child Psychology 81: 1-43.

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---. 2002b. From sharing to dividing: The development of children’s understanding of division.

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Konstantinos Ravanis. 2005 Etude des notations à visée communicationnelle par des

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Tables and illustrations for Article: Resolution of Division Problems by Young

Children: What Are Children Capable of and under which Conditions?

Figure 1. Distribution of Dolls and Candies, according to Squire and Bryant (2002)

Figure 2. Reduction of the Graphic Support Accompanying the Pictorial Display of the “Gloves” Problem

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Figure 3. Reduction of the Graphical Representation Accompanying the “Socks” Problem.

Figure 4. Reduction of the Graphical Representation Accompanying the “Footballers” Problem.

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Problems Instructions

1 �Gloves oral quotitive-grouping division

How many children can one equip with 6 gloves?

2�Gloves pictorial display quotitive-grouping division

How many children can one equip with these gloves? (showing the “gloves” board)

3�Socks oral quotitive-grouping division

How many children can one equip with 12 socks?

4�Socks pictorial display quotitive-grouping division

How many children can one equip with these socks? (showing the “socks” board)

5�Footballers oral quotitive-set division

Eight footballers will practice in groups of 2. Each group will have 1 ball. How many balls will be used?

6�Footballers pictorial display quotitive-set division

The footballers you see here will practice in groups of 2. Each group will have 1 ball. How many balls will be used? (showing the “footballers” board)

Table I: Problems Addressed to Children and Their Instructions

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Problems Strategies Number

of

children

Total of

children

OOC 17 55 Gloves oral

examination G/CG 9 55

OOC 20 55 Gloves

graphical

examination G/CG

24 55

OOC 18 55 Socks oral

examination G/CG 5 55

OOC 15 55 Socks

graphical

examination G/CG

24 55

OOC 12 55 Footballers

oral

examination G/CG

5 55

OOC 4 55 Footballers

graphical

examination G/CG

19 55

Table II: Summary of the Mathematical Strategies Used by the Children according to Age

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Problems

Grouping

Correct

Answers

4,11-5,6

years

old

5,7-5,11

years

old

6-6,6

years

old

Total

G 0 0 0 0 Gloves oral

examination CA 3 3 3 9

G 4 2 3 9 Gloves

graphical

examination CA

5 9 10 24

G 1 1 0 2

Socks oral

examination CA 1 1 3 5

G 1 2 1 4 Socks

graphical

examination CA

8 7 9 24

G 1 1 2 4

Footballers

oral

examination CA

0 2 3 5

G 2 0 0 2 Footballers

graphical

examination CA

4 8 7 19

Table III: Mathematical Strategies Used by the Children by Age

Figure 5: Graphical Production of David for the Resolution of the Problem Footballers Exposed Orally

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Summary of article: Resolution of Division Problems by Young Children: What Are Children

Capable of and under which Conditions?

In this paper we explore the theoretical and experimental field of sharing and partitive and

quotitive division problems, as illustrated by recent studies in this field. The purpose was to

explain and justify the utility of presenting quotitive division problems, accompanied by

graphical representations, to young children. The current study presents six problems

suggested to children of 5 to 6,5 years old within the framework of an empirical study. The

studies prove that quotitive division is accessible to young children. The manipulation of

graphical representations in order to solve these problems proves to be more efficient than the

oral presentation of the problems.

Key words: child, division, graphical representation, problem resolving and reasoning

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