This article was downloaded by: [Eirini Matalliotaki]On: 01 July 2012, At: 11:16Publisher: RoutledgeInforma Ltd Registered in England and Wales Registered Number: 1072954 Registeredoffice: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK

European Early Childhood EducationResearch JournalPublication details, including instructions for authors andsubscription information:http://www.tandfonline.com/loi/recr20

Resolution of division problems byyoung children: what are childrencapable of and under which conditions?Eirini Matalliotaki aa Sciences of Education Department, University of Caen/ BasseNormandie, France

Version of record first published: 19 Jun 2012

To cite this article: Eirini Matalliotaki (2012): Resolution of division problems by young children:what are children capable of and under which conditions?, European Early Childhood EducationResearch Journal, 20:2, 283-299

To link to this article: http://dx.doi.org/10.1080/1350293X.2012.681132

PLEASE SCROLL DOWN FOR ARTICLE

Full terms and conditions of use: http://www.tandfonline.com/page/terms-and-conditions

This article may be used for research, teaching, and private study purposes. Anysubstantial or systematic reproduction, redistribution, reselling, loan, sub-licensing,systematic supply, or distribution in any form to anyone is expressly forbidden.

The publisher does not give any warranty express or implied or make any representationthat the contents will be complete or accurate or up to date. The accuracy of anyinstructions, formulae, and drug doses should be independently verified with primarysources. The publisher shall not be liable for any loss, actions, claims, proceedings,demand, or costs or damages whatsoever or howsoever caused arising directly orindirectly in connection with or arising out of the use of this material.

Resolution of division problems by young children: what arechildren capable of and under which conditions?

Eirini Matalliotaki*

Sciences of Education Department, University of Caen/ Basse Normandie, France

ABSTRACT: In this article we explore the theoretical and experimental field ofsharing and of partitive and quotitive division problems, as illustrated by recentstudies in this field. The purpose was to explain and justify the utility ofpresenting quotitive division problems, accompanied by graphical representations,to young children. The current study presents six problems suggested to childrenof five- to six and a half-years-old within the framework of an empirical study.The studies prove that quotitive division is accessible to young children. Themanipulation of graphical representations in order to solve these problems provesto be more efficient than the oral presentation of the problems.

RESUME: Dans cet article nous explorons le champ théorique et expérimental desproblèmes de partage et de la division partitive et quotitive, comme illustrés par desétudes récentes dans ce domaine. Le but était d′expliquer et justifier l′utilité deprésenter des problèmes de la division quotitive, accompagnés desreprésentations graphiques, à des enfants de jeune âge. L′étude actuelle présentesix problèmes proposés à des enfants de 5 à 6,5 ans dans le cadre d′une étudeempirique. Les études montrent que la division quotitive est accessible auxenfants de jeune âge. La manipulation des représentations graphiques àfin derésoudre ces problèmes s’avère plus efficace que la présentation orale desproblèmes.

ZUSAMMENFASSUNG: In diesem Beitrag untersuchen wir das Feld destheoretischen und experimentellen Teilens und der partitiven und quantitativenDivision, wie diese durch die letzten Studien in diesem Bereich illustriertwurden. Das Hauptziel bestand darin zu erklären und zu rechtfertigen, warum esnützlich ist, für kleine Kinder quantitative Probleme in Verbindung mitgrafischen Darstellungen zu präsentieren. Die aktuelle Studie präsentiert sechsAufgaben, die Kindern von 5 bis 6,5 Jahren im Rahmen einer empirischenStudie vorgelegt wurden. Die Studien belegen, dass quantitative Divisionenkleinen Kinder zugänglich sind. Um diese Probleme zu lösen erweist sich dieManipulation von grafischen Darstellungen als effizienter als die mündlichePräsentation der Aufgaben.

RESUMEN: En este texto exploramos el campo teórico y experimental de ladistribución, partición y los problemas quotitive de la división, según lo ilustradopor estudios recientes en este campo. El propósito era explicar y justificar lautilidad de presentar problemas quotitive de la división, acompañada por lasrepresentaciones gráficas, a los niños pequeños. El estudio actual presenta seisproblemas sugeridos a los niños de 5 a 6.5 años en el marco de un estudioempírico. Los estudios prueban que la división quotitive es accesible para los

ISSN 1350-293X print/ISSN 1752-1807 online© 2012 EECERAhttp://dx.doi.org/10.1080/1350293X.2012.681132http://www.tandfonline.com

*Email: rmatalliotaki@yahoo.fr

European Early Childhood Education Research JournalVol. 20, No. 2, June 2012, 283–299

Dow

nloa

ded

by [

Eir

ini M

atal

liota

ki]

at 1

1:16

01

July

201

2

niños pequeños. La manipulación de representaciones gráficas para solucionarestos problemas demuestra ser más eficiente que la presentación oral de losproblemas.

Keywords: child; division; graphical representation; problem resolving andreasoning

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

Less empirical research exists on the beginnings of the learning of division than onother arithmetical operations. However, division is interesting from the point of viewof the schemas of action: the schema of sharing is at the base of division.

From the perspective of the mathematical definition of division, the idea that theactivity of sharing in the everyday children’s environment is a mathematical activityis contestable. Sharing involves a form of socialisation (for instance, children learnthat we are all equal by sharing between them toys or equal parts of cakes).However, from the developmental point of view, it is possible to consider that theschema of sharing, which brings into play correspondences and the constitution ofequivalent classes, can constitute the first steps of division. At one point in time oftheir development, the children may be able to carry out division while beingunaware 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 youngchildren

This section will review empirical studies on partitive and quotitive division amongchildren between five- and six and a half-years-old.

1.2.1 Solution strategies used by children in problems of division

Kouba (1989) asked 128 children between the ages of six and nine (43 first graders, 35second graders and 50 third graders from one small Midwest (in USA) school whoreceived their mathematics instruction from their classroom teacher) to solve divisionproblems that she presented to them verbally. She identified three types of problemsdepending on the required quantity.

1. Multiplication (unknown number of elements of the totality of the group)2. Quotitive division – also referred to as measurement division in the literature –

(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 divisionproblem contributes to the difficulty of the problems more than it does in addition orsubtraction problems. Vergnaud (1983) and Fischbein et al. (1985) have stated thateach arithmetical operation is related to an intuitive, implicit, unconscious and primitive

284 E. Matalliotaki

Dow

nloa

ded

by [

Eir

ini M

atal

liota

ki]

at 1

1:16

01

July

201

2

model. This model influences the decisions made when an operation is used. Fischbeinet al. propose that – in addition – the intuitive model is ‘put together’; while in subtrac-tion it is ‘removed’; for multiplication, it is the repeated addition, and for quotitive div-ision, it is the model of repeated subtraction. For partitive division, it is sharing.

Kouba interviewed the children individually, and categorised the strategies of res-olution as ‘inappropriate,’ ‘appropriate,’ and ‘not identifiable,’ independent of the accu-racy of the calculation. The children questioned had concrete objects in case theyexpressed the need to use them to help in answering the questions.

Kouba counted 333 problems solved with appropriate strategies of the 768 multipli-cation and division problems given to children. Analysing these 333 problems, 56different strategies emerged, which is considerable compared to the standard pro-cedures taught. These 56 strategies were analysed according to two criteria: thedegree of abstraction of the steps (the smallest degree of abstraction involved represent-ing the data with objects, and the highest degree of abstraction involved giving thecorrect answer by remembering the appropriate multiplication or division combination)and the use or non-use of physical objects.

Kouba also found that the youngest children (six-years-old) solved 63% of the pro-blems. This success score is very close to the ones of Matalliotaki’s (2007) empiricalstudy for the same age group.

1.2.2 Solutions to partitive and quotitive problems

The study by Squire and Bryant (2002a) used material supports to highlight the impor-tance of the schemas of action in solving partitive problems. In this study, childrenbetween five- and eight-years-old were asked to solve problems requiring them to dis-tribute candies among dolls; the candies were divided among boxes in a variety ofways. The number of candies in each box corresponded either to the divisor or tothe quotient. Before every experiment a ‘control’ (‘baseline trial’) condition was estab-lished 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 (2002a).

European Early Childhood Education Research Journal 285

Dow

nloa

ded

by [

Eir

ini M

atal

liota

ki]

at 1

1:16

01

July

201

2

On the left side of Figure 1, the top figure represents the grouping by the divisor (thenumber of boxes corresponding to the number of dolls); the bottom figure represents thegrouping by the quotient (the number of candies in a box corresponding to the numberof dolls).

The principal result of this study is that, for all the ages studied, the children werehelped more by the grouping by divisor than by the grouping by quotient. The ability ofthe children to solve the problem by grouping by quotient improves with age. This canbe explained, according to Squire and Bryant by the improvement in the capacity to usethe one-to-one correspondence with age, or by the fact that older children eventuallylearn to understand the interchangeability of the divisor and the quotient.

Researchers asked the children: ‘when the candies are placed in the boxes, howmany boxes have been used and, then, how many candies are in each box?’ The actof asking questions on the number of boxes and the number of candies improved thescores of the children on the task of grouping by quotient. Since, according to otherauthors, this strategy enabled the children to use additive reasoning instead of multipli-cative reasoning. This finding is confirmed by other studies (Sophian 1991) that, forchildren aged five to six, the transition using addition (using addition strategies, andnot multiplication or division ones, to resolve multiplication or division problems)plays a facilitating role.

The spatial arrangement of dolls and candies influences the children’s ability tosolve the problem. Squire and Bryant suggest that children’s performance might bebetter if they can rearrange the objects themselves. They posit that handling of theobjects or their images improves accuracy. The older children perform better becausethat they have a greater capacity ‘to mentally rearrange’ the whole under the conditionof grouping per quotient.

This study also demonstrates that children’s early comprehension of division isinfluenced by their comprehension of level sharing (an activity that involves a formof socialisation, due to the fact that in every day environment children learn to share,for instance, parts of cakes or candies, and has mathematical aspects) and the distri-bution of portions to recipients. Consequently, we can assume that informal experi-ences affect the learning of mathematics. This finding has important educationalimplications.

Another study carried out by Squire and Bryant (2002b) compared the proceduresused to solve partitive and quotitive problems. In their study, two conditions were pre-sented: a partitive task in which the objects were grouped either by divisor or by quo-tient, and a quotitive task with the same two groupings. The children found thegrouping by divisor to be easier when solving the partitive division problem and thegrouping by quotient to be easier for quotitive division. According to Squire andBryant, such results must imply non-mathematical factors since in mathematicalterms there were no differences between the two conditions. Therefore, they suggestthat the most convincing reason for this difference between the two conditions is a cog-nitive one, the following:

The authors suggest that, even if certain models are strongly artificial and benefitonly from cultural drives (for example, in ‘pure’ mathematics), others are probablyacquired without explicit instruction and are used by everyone. According to thisapproach, informal experience can plausibly contribute to the formation of a ‘mentalmodel’ of the concepts and, consequently, children can start to acquire a mentalmodel of division by sharing. In other words, sharing can be a schema of action bywhich a comprehension of division develops. This developmental concept of the

286 E. Matalliotaki

Dow

nloa

ded

by [

Eir

ini M

atal

liota

ki]

at 1

1:16

01

July

201

2

formation of the concepts supports Vergnaud, who granted a crucial role to the schemasof action.1 The schemas of action would be primary and be the matrix for the construc-tion of the concepts.

Another interesting element arising from the work of Squire and Bryant is that thedifference between the children with better performances and those who encountereddifficulties in performing the tasks occurs only at the plan of the selected proceduresof resolution (and not the amount of time needed to arrive at a correct answer). Thisresulted in stressing the procedures rather than speed or accuracy. This aspect wasretained 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) relate to partitive and quotitive tasks; it drewa conceptual distinction between sharing and division. In sharing, children treat onlythe equivalence of the shares. The concept of division, in contrast, implies the compre-hension of the relations of three values represented by the dividend, the divisor, and thequotient. The equivalence of the shares is assumed, although in division children mustunderstand 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, thenumber of rabbits) increases, the quotient (for example, the number of carrots perrabbit) decreases. Comprehension improved with age, even though young childrenunderstood partitive tasks better than they did quotitive ones, their abilities toperform both tasks was impressive.

1.3 Graphical productions of the children during problem solving

There are virtually no studies on the graphical productions of young children’s ability tosolve division problems, so we make use of the work that has been done in other fieldsof problem solving.

Weil-Barais and Resta-Schweitzer (2006, 2008) showed that the graphical pro-ductions of children between five and six years of age years reveal the degree of con-ceptualisation in physical phenomena. The drawings are one means by which childrenexpress a complex phenomenon, since they facilitate the expression of the spatialrelations of the objects. Tantaros et al. (2005) studied the production of graphical rep-resentations by young children in order to communicate the solution of a problem andconcluded that children’s productions improve with age and that drawings can consti-tute a cognitive tool. Lehrer and Schauble (2002) consider that a representation is notonly a copy of reality. That implies inventing and adapting conventions of a system ofrepresentation in order to choose, compose and transport information. Childrenlearned that a system of representation that could represent information was notimmediately perceptible. The authors conclude that developing adequate graphical rep-resentations for information represented conventionally, promotes the learning of themathematical concepts that are needed for comprehension of the properties of thisinformation.

Based on the research that has been cited, we expect the use of graphical represen-tations by children will make it easier for them to solve division problems. In thegraphical representations, children can visualise the strategies to follow, since the pos-ition of the mathematical elements and the relations between them are visible. Weexpect that the children will use the graphical representations as a tool to envisagethe connections among the elements.

European Early Childhood Education Research Journal 287

Dow

nloa

ded

by [

Eir

ini M

atal

liota

ki]

at 1

1:16

01

July

201

2

1.4 Justification of empirical problems

The studies presented in this article suggest that there is very little empirical research onyoung children’s ability to solve division problems. However, it is an interesting oper-ation as the schema of action that it mobilises is level sharing.

Even if the level sharing belongs to the informal experience of the child, division isregarded as a more complex operation than addition or subtraction due to the relationsbetween the quantities. However even if it is more complex arithmetical operation, chil-dren participating in the studies, even those as young as five, showed remarkablesuccess with certain division problems. This supports the idea that the direct instructionof division is not essential in the formation of the concept, at least in its least concep-tually elaborate form (level sharing).

The characteristics of division raised in the studies reported in this article justify thechoice to use graphical representations. In Squire and Bryant’s (2002a) study, the chil-dren could not rearrange the objects or modify their distribution. However, the use ofgraphical representations makes it possible to consider and preserve several types ofregroupings. Thus, as suggested by Squire and Bryant the physical handling ofobjects can help children to solve complex problems; thus, it would be interesting toexamine in what contexts the use of graphical representations plays a facilitatingrole. Furthermore, Kouba (1989) wonders if the categories of resolution that she distin-guished in her study would be the same if the problems had employed graphic supportsinstead of physical objects. Considering the fact that the youngest children have diffi-culty keeping several variables in mind and performing complex operations, the graphi-cal representations of a problem can indeed facilitate the handling of such situations byyoung children.

For the problems used in the present research, quotitive division was chosenbecause studies have demonstrated that this type of division appears more complex(and is thus more interesting to study) for children than partitive division. The intui-tive model of quotitive division is the repeated subtraction. The children havealready experienced the schema of the action of sharing, which applies to partitivedivision, while the schemas of the action of subtraction, which apply to quotitivedivision, 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 ‘divisionquotitive-set’ category.

The current research conceived problems that seemed a priori adapted to the chil-dren in a nursery school, aged five- to six-years-old. In these problems, the childrenmust make inferences related to quantities. More precisely, these situations focus onquotitive division: a number of objects in total and a number of objects by recipientsbeing indicated to determine the number of recipients. In mathematical terms, thisinvolves giving the quantity of a group and the number of elements constituting thesubsets (or parts) so as to find the number of subsets (or of parts). When not havingeasy-to-handle objects, young children involved in this study could solve these pro-blems by the use of graphical representations.

2 Empirical study

In this section, we present the results of the empirical study that was conducted in aFrench nursery school.

288 E. Matalliotaki

Dow

nloa

ded

by [

Eir

ini M

atal

liota

ki]

at 1

1:16

01

July

201

2

2.1 Methodological framework

The methodological framework of the empirical study is set out below. This consists ofthe population chosen, the research design, and the description of the problems pre-sented to the children.

2.1.1 Subjects and design

Fifty-five nursery school children, all from the same school and in good academicstanding (not having to repeat a class, nor having major cognitive problems) in the13th arrondissement (district) of Paris took part in this research. The context of thestudy carried out in France did not require distinguishing the results we wouldobtain depending on socio-economical status or ethnic origin. For that reason we donot have any concrete percentages concerning these two parameters. We may notethat the 13th district of Paris is considered to be of middle class socio-economicstatus, with a big presence of families of Asian ethnic origin (even though childrenof all ethnic origin are included in the sample). Children ranged from five to 6 and ahalf years in age. The 31 boys and 24 girls were divided among three classes. Theirparents signed a consent form authorizing their children’s participation in the study.Children were asked if they wished or not to participate in this study. All childrenasked agreed to participate.

The children were interviewed individually. The child and the researcher sat face toface at a table with sheets of paper and coloured pencils.

In the Gloves-Socks-Footballers test, the boards (Figures 2–4) are presented to thechild progressively. In addition, for any answer given by the child, the researcher askshim or her to explain how he or she arrived at the answer, and noted the explanationwithout 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.

2.1.2 Description of the problems

The problems were first presented verbally and then with a graphic support, whichreflected the facts of the case. The introduction of the graphic support is regarded asa new problem (not as a means of assisting with the verbal one), even if the quantitativeinformation and the structure of the problem were the same. The purpose of the exercisewas to examine the types of strategies that the children use both with and without thegraphic support, not only to ascertain if they were able to solve a division problem, asprevious studies have demonstrated.

Problem of the Gloves: Oral Presentation. The first question was ‘how manychildren can wear six gloves?’ This first problem implies minor amounts, whichchildren of this age can process either mentally or with the use of fingers. Thesolution of such a problem can be based on the child’s cultural experience and onawareness of his/her body (demonstrated in Matalliotaki 2001), which indicates thata person with two hands needs two gloves. The children must thus deduce that, aseach child needs two gloves (cardinal of the subsets), three children can wear sixgloves. The problem is a quotitive-grouping division one and can be formalised asfollows:

2 gloves→ 1 child

European Early Childhood Education Research Journal 289

Dow

nloa

ded

by [

Eir

ini M

atal

liota

ki]

at 1

1:16

01

July

201

2

6 gloves→ X children

For children, at an age where they are not yet able to produce formal mathematical nota-tions, it is possible to await an analogue representation (mental or written) close to thefollowing:

••

••

••

The number of points of each group represents the divisor while the number of groupsrepresents the quotient. We note that these problems (and the ones that follow) involveadditive reasoning (prepare three pairs and then add them to find the correct answer),which, according to Squire and Bryant (2002a and b) and Sophian (1991), plays a facil-itating role for children of this age.

Problem of the gloves: pictorial display. The second problem proposes a schemati-sation of the data likely to facilitate the choice of a strategy of resolution. The sameproblem is verbally presented, but this time accompanied by a board (with theformat 21 x 29.7) that graphically represents the objects. It is also a quotitive-groupingdivision problem. Figure 2 presents a reduced graphical representation that was pre-sented to the children.

In a study of tasks in children’s books intended for children of five and a half-years-old to six and a half-years-old, Matalliotaki (2007) found that very few graphical rep-resentations accompanying problems help children to make inferences. That is why weconceived this schematisation so that we facilitate the grouping of the elements (con-struction of the groups of gloves of same pattern), and help them graphically toproduce inferences. In this manner, introducing the strategy to be followed for the res-olution of the problem involves the constitution of the regroupings (subsets) and theirenumeration. The graphical representation would provide an inferential function as itmakes it possible to calculate a quotient.

The elements drawn are not perfectly ordered because the presence of the identicalfeatures between the gloves of each pair is expected to be sufficient for completion ofthe groupings.

Figure 2. Reduction of the graphic support (of format 21cm x 29.7cm) accompanying the pic-torial display of the ‘gloves’ problem.

290 E. Matalliotaki

Dow

nloa

ded

by [

Eir

ini M

atal

liota

ki]

at 1

1:16

01

July

201

2

Problem of the socks: oral presentation. The third problem utilised the same structureas the first, but with larger quantities: how ‘how many children can wear twelve socks?’Here, the quantities are not readily processed with the use of fingers, thus legitimatingthe use of a schematisation of the data. The problem is a quotitive-grouping divisionone.

Problem of the socks: pictorial display. The same problem was given to children, with agraphical representation of six pairs of socks (with the format 21 x 29.7). It is also aquotitive-grouping division problem. Two versions were proposed: one in which thepairs of socks are locatable by graphic characteristics (see Figure 3, left side) andone in which all the socks were drawn in the same way (see Figure 3, right side).These two versions were presented to two groups of children in the same age cohort.It is assumed that the first graphical representation performs two functions – referentialand inferential – as it provides indices for regrouping by two: the child can find theanswer by counting only the different socks. The second graphical representation per-forms only one referential function. It can perform an inferential function if the childrenproduce graphical indices of regrouping. These two versions were proposed in order toexamine the impact of the inferential function on children’s performances in a concretemanner. In these terms, this study also examines whether or not the inferential type ofgraphical representation could facilitate children complete a calculation comparing toreferential one.

Problem of the footballers: oral presentation. It is a quotitive-set division problem.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 incorporatedwhen some children started to solve the problem under the assumption that there would betwo groups. The sentence ‘they will practice in groups of two’ did not appear clear enoughto describe the formation of the groups.) Each group will have a ball. How many balls willbe needed?

The children are expected to solve this problem using the former experience (graphicalresolution of the problems). A possible schematisation of the procedure used to solvethis problem might be as follows:

Figure 3. Reduction of the graphical representation (of format 21 cm x 29.7cm) accompanyingthe ‘socks’ problem.

European Early Childhood Education Research Journal 291

Dow

nloa

ded

by [

Eir

ini M

atal

liota

ki]

at 1

1:16

01

July

201

2

Or more formally:

2 footballers→ 1 ball8 footballers→ X ball

From which we get the equation b = f/g, where b is the sought quotient, f the full numberof footballers, and g the number of footballers in each group.

For children of this age, who have not yet received formal instruction in mathemat-ics, an analogue representation seems more suitable:

In this representation, the squares represent the footballers and the circles the balls. Thetwo footballers in each group represent the divisor and the number of built footballercouples represents the quotient. This problem represents for the children a morecomplex situation than does preceding problems because the bond with the humanbody no longer exists, thereby eliminating a familiar context to which the child canrefer. The context is more formal even if football is an interest of the children. Thenumber of balls per player is arbitrary, which is not the case in regards to the glovesor 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 associatewith balls.

Problem of the footballers: pictorial display. It is a quotitive-set division problem. Thisproblem was presented to the children along with a graphical representation (with theformat 21 x 29.7) (see Figure 4). Unlike the second and fourth problems, both balls andplayers are represented here. After the grouping of the players in pairs, the children canconnect each group to a balloon and count the number of connections.

The suggested schematisation does not provide an inferential function, as thegraphical representation does not propose elements that would support the groupingor the connection of each group with one ball. This last problem was conceived totest if a representation very close to an environment familiar to the child (e.g., storybooks with illustrations, images in media) can be seen and handled in an abstractcontext of mathematics when the children have not received formal education in math-ematics. According to Squire and Bryant (2002a), informal experiences can affect theacquisition of mathematical skills. In our study we determine whether or not an infor-mal context that is familiar to children facilitates their problem solving.

Table 1 provides, in a synthetic form, the problems suggested to the children withinthe framework of this empirical study.

3 Results

Three situations were presented to the children (gloves, socks, footballers), first verb-ally, then graphically (through a graphical representation of the objects). The graphical

292 E. Matalliotaki

Dow

nloa

ded

by [

Eir

ini M

atal

liota

ki]

at 1

1:16

01

July

201

2

Table 1. Problems given to children and their instructions

Problems Instructions

1➪Gloves oralquotitive-grouping division

How many children can one equip with 6gloves?

2➪Gloves pictorial displayquotitive-grouping division

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

3➪Socks oralquotitive-grouping division

How many children can one equip with 12socks?

4➪Socks pictorial displayquotitive-grouping division

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

5➪Footballers oralquotitive-set division

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

6➪Footballers pictorial displayquotitive-set division

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

Figure 4. Reduction of the graphical representation (of format 21cm x 29.7cm) accompanyingthe ‘footballers’ problem.

European Early Childhood Education Research Journal 293

Dow

nloa

ded

by [

Eir

ini M

atal

liota

ki]

at 1

1:16

01

July

201

2

form is associated with a greater number of correct answers (11% of correct answers tothe oral examination compared to 40% in the written form). In the graphical form, theproblem of the footballers proves to be more difficult but it is nevertheless solved cor-rectly by approximately one-third of the children. These percentages were based strictlyon correct answers. Children that followed a correct strategy but miscounted or madeerrors in calculating were considered to have provided the wrong answer.

In order to see more deeply how children thought for resolving these problems andif they actually were capable to find the correct strategy but just failed in counting, wedistinguished the strategies they followed. Two strategies common to all problems wereidentified: pairing with enumeration of the groups (correct strategy) and the one-to-onecorrespondence (incorrect strategy). The first strategy includes pairing (e.g. A child hastwo hands so we put together two gloves for each child) and enumeration of the groups(e.g. How many pairs of gloves did we create corresponding to one child?). In thesecond strategy, children just gave one glove to each child without having the priorreasoning that one child needs two gloves. We chose to present these two strategiesin this article in the effort to understand young children’s mathematical reasoningbut of course, there have been children that did not find any solution strategy(correct or erroneous). This category of children is not presented here. No other classi-fiable strategies were identified.

Table 2 presents the number of children who followed the correct mathematicalstrategy to solve the problems (pairing and counting of the built groups) and thenumber of children that followed the incorrect strategy (the one-to-one correspon-dence). 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 six, it means that the child distributed a glove to each child instead of buildingpairs of gloves. In the following table, OOC means one-to-one correspondence andG/CG means Grouping and Counting of the built groups.

We observe the progressive use of the strategy of one-to-one correspondence in thelong-term performance of the tasks. According to Table 2, the children seem to give itup gradually. In oral version of socks, 18 children follow this strategy, in the nextproblem (graphical socks) 15, in the following (oral footballers) 12 and in the finalproblem (graphical footballers) only four. This means that as problems went by,

Table 2. Summary of the mathematical strategies used by the children.

Problems Strategies Number of children Total of children

Gloves oral examination OOC 17 55G/CG 9 55

Gloves graphical examination OOC 20 55G/CG 24 55

Socks oral examination OOC 18 55G/CG 5 55

Socks graphical examination OOC 15 55G/CG 24 55

Footballers oral examination OOC 12 55G/CG 5 55

Footballers graphical examination OOC 4 55G/CG 19 55

294 E. Matalliotaki

Dow

nloa

ded

by [

Eir

ini M

atal

liota

ki]

at 1

1:16

01

July

201

2

understanding that this strategy was not the convenient one occurred and children knewit by themselves since we never showed approval or disapproval in the performances ofchildren. (We also note here that the different graphical presentations of the ‘socks’problem – see Figure 3 – did not have any impact in children’s success or mathematicalchoice).

Some children seem to have understood the strategy or building groups in order tosolve the problems but for some reason, they could not make this strategy succeed. Forinstance, in the socks oral problem, they started making pairs of socks representingthem with their fingers (they made the reasoning that one child needs two socks) butsince the number of socks was 12, there were not enough fingers to count. The childrendid not think of writing down their data, so in the end they failed in counting. Even inthe graphical version of the problems, children sometimes miscounted even if they hadcorrectly created their groups. We thus present in Table 3 the number of children whostarted correctly the grouping (pairing) but failed to give the right answer (by age)through the problems, in comparison to the number of children who answered it cor-rectly (after having proved to having used the correct answer). In the table, G meansGrouping and represents the number of children who carried out the grouping andCA represents the number of children who answered correctly.

Authors such as Squire and Bryant (2002a) discovered that even a few months ofdifference in the age of the subjects could play a significant part in their ability to under-stand and produce symbolic notations (notations that include conventional or non con-ventional symbols like alphabet, marks, numerals, drawings, icons, etc). We thereforecarried out an analysis of the performances of the children depending their age.

In problems such as the graphical problem of gloves, nine children carried out thegrouping correctly but could not to finish the procedure and solve the problem. Thesechildren knew which strategy to follow but were not ready to complete this application.We can suppose that even more children than the percentages we presented previouslywould be able to resolve these problems even though ‘difficult’ for their cognitive level,if other difficulties as miscounting, or coordinating different parameters of a problemand keeping them in memory, were not an obstacle.

We can also observe the absence of ‘grouping’ without arriving at the correctanswer in the first problem, which can make us suppose that children did not haveany idea of which strategy to apply to resolve the first oral problem. The childrenthat responded correctly in the first oral problem calculated mentally without expressingthe use of any strategy. Only when graphical data is given to them (graphical version ofgloves) can we observe children being able to find a strategy to resolve the problem,(even if in some cases they could not finish it). That is why we assume that graphicaldata helps children learn how to conceive solution strategies that they could eventuallyapply in other similar problems. Since in all following problems (even oral ones) somechildren succeed to find and express the correct strategy (even if not able to apply it tillthe end) we assume that there was learning from previous problems, (since all six pro-blems needed the same strategy of resolution), of how to treat data. We then considerthat giving children the chance to treat graphical data helped them create a mathemat-ical model of a strategy on which they could rely even in following problems.

We did not observe any major differences in the performances of children accordingto age categories, except possibly in the last two problems where older children performbetter. A test with a larger sample would possibly reveal more clearly differences incapacities in each age group (see Table 3).

European Early Childhood Education Research Journal 295

Dow

nloa

ded

by [

Eir

ini M

atal

liota

ki]

at 1

1:16

01

July

201

2

3.1 Improvement of performance through the tasks

We observed (see Table 2) a progressive improvement of the children’s performance inthe tasks. The children abandoned the erroneous strategy, by gradually applying thestrategy of grouping the elements.

The children that succeededGrouping spontaneously used an analogue representation(their fingers) when the problems were verbally explained to them, but not for the firstproblem. Moreover, when they used their fingers (which represented the elements ofthe data), they placed them in a way similar to how elements were placed in graphical rep-resentation of the previous problem. This means they were influenced by the graphicalrepresentations we presented to them in the preceding problems thanks to which theymanaged to find the right strategy to solve the problem. This shows that the childrenwere influenced by our suggestion to use a graphical representation, which ensures theinferential function to solve problems; they seem to trust this tool.

Furthermore, we had one example of a child that actually produced a graphical rep-resentation to resolve the oral footballer’s problem that shows the efficiency of theinferential function ensured by the first graphical representation (gloves). Generally,the fact of suggesting to children the use of an analogical system influenced them(most children by using their fingers, one child by producing a graphical representation)to find an analogical strategy to solve the following problem. David used agraphical representation (see Figure 5) as a support for resolving the footballers oralproblem. He did not use such graphical representations for the first problems thatwere been presented to him, so we can conclude that he learnt how to choose a tech-nique to solve such a problem by using the graphical representations that we introducedearlier.

This figurative drawing reveals which strategy David follows to solve this type ofproblem:

. Distinguishes each footballer as a mathematical unit

. Gathers two footballers

. Allots a balloon to each pair of footballers.

Table 3. Mathematical strategies used by the children by age

ProblemsGrouping Correct

Answers4,11-5,6years old

5,7-5,11years old

6-6,6years old Total

Gloves oralexamination

G 0 0 0 0CA 3 3 3 9

Gloves graphicalexamination

G 4 2 3 9CA 5 9 10 24

Socks oral examination G 1 1 0 2CA 1 1 3 5

Socks graphicalexamination

G 1 2 1 4CA 8 7 9 24

Footballers oralexamination

G 1 1 2 4CA 0 2 3 5

Footballers graphicalexamination

G 2 0 0 2CA 4 8 7 19

296 E. Matalliotaki

Dow

nloa

ded

by [

Eir

ini M

atal

liota

ki]

at 1

1:16

01

July

201

2

Wewill now try to locate the errors children most often made during problem solving inorder to understand their reasoning and to identify the possible reasons for theirinability to reach correct solutions.

Errors of children: not to escape the aesthetic or pragmatic context

Children answer graphically or verbally by drawing or colouring the sketch given tothem, or while producing conclusions relating to their cultural knowledge of theproblem. For example, the footballers will need only one ball because football isplayed with one ball. Some children had difficulty in abandoning the semanticcontext. Fifteen of the 55 children stated that the footballers would need one ball inthe oral problem of footballers (by justifying their answer ‘the football is a gameplayed with one ball’). In the graphic version of this same problem only seven of thechildren answered that one ball would be used.

We saw that children were influenced by the graphic data in giving their answer andthat they had difficulty in remembering the instructions. We suppose that children ofthis age, as suggested by Squire and Bryant (2002a), often reinterpret problems in away that makes sense for them. The pragmatic context of children influences theirinterpretation of the instruction. We can assume, then, that with these kinds of pro-blems, the informal context of children did not facilitate the solution.

4. General discussion

For the three contextual situations (gloves, socks, footballers), the graphical represen-tation allowed the children to solve the problem by gathering pairs of objects. In theverbal form of the problems, children find it difficult to conceive and express thecorrect mathematical strategy. Without a graphical support, children are not able tokeep track of their train of thought in order to solve problems that had been presentedto them orally. This means that maybe they start creating groups (correct strategy) but it

Figure 5. Graphical production of David for the resolution of the problem footballers exposedorally.

European Early Childhood Education Research Journal 297

Dow

nloa

ded

by [

Eir

ini M

atal

liota

ki]

at 1

1:16

01

July

201

2

is not easy for them to memorise all the data and so resolve the problem. With graphicalrepresentation, they can see and manipulate the data; their cognitive energy is not spentin the effort of memorising but in the effort of coordinating the different parameters ofthe problem and in reasoning which is already not easy for children of this age. Accord-ing to Squire and Bryant, the capacity to coordinate the parameters of a problem and toretain them in memory evolves with age (Squire and Bryant 2002a).

According to Table 2, after the graphical version of socks, children gradually reducethe use of one-to-one correspondence strategy. This means that children received sometraining in the resolution of the problems that required the same strategy. Children ofthis age do not have experience in solving problems of quotitive division. Thechoice of the erroneous mathematical strategy thus seems to make sense. What is aston-ishing is the fact that children have been able to find the correct strategy despite theirlack of experience in division. This shows that children of this age have the ability tosolve such abstract problems.

After the first problem (oral gloves), more children build groups correctly, even ifthey do not reach the correct answer. This enables us to say that there is an improvementin their execution of a mathematical strategy, while passing from one problem to theother. The first improvement seen was the gradual abandonment of the one-to-one cor-respondence strategy.

In Table 3, we observe that the number of the correct answers in graphical glovesincreases with age, which can show an improvement of certain capacities of childrenwith age. In the graphical problem of gloves, 10 of the oldest children came up withthe solution to the problem, whilst only five of the youngest children reached the sol-ution. In order to confirm this assumption, a test with a larger number of children ineach age cohort would be necessary.

We also observed that children found it sometimes difficult to transform a ‘group-ing’ into a ‘correct answer.’ Thus, the use of the correct strategy might depend on acapacity, which is acquired with age: the capacity to coordinate the parameters of aproblem and to retain them in memory (Squire and Bryant 2002a). Some childrencan conceive the correct mathematical strategy to solve the problem but theirability to perform complex operations and to store partial products remains immature.Therefore, we observed improvement in this capacity from the second problem andon, which means that training children in class with such tasks may improve thesecapacities.

The graphic version given to the children helped themnot only tofind the correct strat-egy for the resolution, but it also gave them an analogical strategy with which to solve theverbal problems and to escape from the semantic context of the problemswhich frequentlyconstrains children of this age. The children, even without experience with and formalcapacities of this type of exercises,managed to solve the problems and to acquire the train-ing to improve their performance. The children’s observed successes at division at an earlyage should encourage educators tomake themost of their competences, even though offi-cial school programmes do not do so. Indeed, younger children are fully capable of learn-ing division, and this capacity should be cultivated sooner instead of later. Giving suchtasks to children of this age helps them create mathematical models of problem resolvingthat can be of great use in future formal learning of mathematics. It also helps themimprove their ability to recognise the important variables in a problem and encouragethem to reason flexibly in diverse contexts. This provides a justification for continuingresearch on this topic amongst pre-school age children should continue.

298 E. Matalliotaki

Dow

nloa

ded

by [

Eir

ini M

atal

liota

ki]

at 1

1:16

01

July

201

2

Note1. The schemas of action (schèmes d′action) are familiar actions that can offer initial compre-

hension of the arithmetic operations (Vergnaud 1985).

ReferencesCorrea, J., T. Nunes, and P. Bryant. 1998. Young children’s understanding of division: The

relationship between division terms and a non-computational task. Journal ofEducational Psychology 2: 321–29.

Fischbein, E., M. Deri, M.Sainati Nello, and Sciolis M. Marino. 1985. The role of implicitmodels in solving verbal problems in multiplication and division. Journal for Research inMathematics Education 16: 3–17.

Frydman, O., and P. Bryant. 1988. Sharing and the understanding of number equivalence byyoung children. Cognitive Development 3: 323–39.

Kouba, V. 1989. Children’s solution strategies for equivalent set multiplication and divisionword problems. Journal for Research in Mathematics Education 20: 147–58.

Lehrer, R., and L. Schauble. 2002. Symbolic communication in mathematics and science:Co-constituting inscription and thought. In Language, literacy, and cognitive development,ed. E. Amsel and J. Byrnes, 173–199. London: Lawrence Erlbaum Associates.

Matalliotaki, E. 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.

Matalliotaki, E. 2007. Les pratiques graphiques à l’école maternelle dans un contexte derésolution de problèmes, Thèse de Doctorat en Sciences de l’Education, Université RenéDescartes-Paris 5.

Resta-Schweitzer, M., and A. Weil-Barais. 2006. Education scientifique et développement intel-lectuel du jeune enfant. Review of Science, Mathematics and ICT Education 1: 63–82.

Sophian, C. 1991. Le nombre et sa genèse avant l’école primaire. Comment s’en inspirer pourenseigner les mathématiques. In Les chemins du nombre, ed. J. Bideaud, C. Meljac and J.-P.Fischer, 35–58. Lille: Presses Universitaires de Lille.

Squire, S., and P. Bryant. 2002a. The influence of sharing in young children’s understanding ofdivision. Journal of Experimental Child Psychology 81: 1–43.

Squire, S., and P. Bryant. 2002b. From sharing to dividing: The development of children’sunderstanding of division. Developmental Science 5: 452–66.

Squire, S. 2003. Children’s models of division. Cognitive Development 18: 355–76.Tantaros, S., K. Sarigianni, E. Sotiropoulou, D. Koliopoulos, and K. Ravanis. 2005. Etude des

notations à visée communicationnelle par des enfants d’une école primaire en Grèce dans lecadre d’une activité scientifique. Colloque Noter pour Penser, Angers, 26–27 January 2005.

Vergnaud, G. 1983. L’enfant, la mathématique et la réalité. Bern: Peter Lang.Vergnaud, G. 1985. Concepts et schèmes dans une théorie opératoire de la représentation.

Psychologie franca̧ise 30: 245–52.Weil-Barais, A., and M. Resta-Schweitzer. 2008. Approche cognitive et développementale de la

médiation en contexte d’enseignement – apprentissage. Nouvelle Revue AIS 42: 83–98.

European Early Childhood Education Research Journal 299

Dow

nloa

ded

by [

Eir

ini M

atal

liota

ki]

at 1

1:16

01

July

201

2