Successfully reported this slideshow.
We use your LinkedIn profile and activity data to personalize ads and to show you more relevant ads. You can change your ad preferences anytime.
ForPeerReviewOnlyResolution of Division Problems by Young Children: WhatAre Children Capable of and under which Conditions...
ForPeerReviewOnly1Resolution of Division Problems by Young Children: What Are ChildrenCapable of and under which Condition...
ForPeerReviewOnly2to carry out division while being unaware of the mathematical components involved (Frydmanand Bryant, 19...
ForPeerReviewOnly3influences the decisions made when an operation is used. Fischbein et al. propose that—inaddition—the in...
ForPeerReviewOnly4quotient. Before every experiment a “control” (“baseline trial”) condition was established inwhich all o...
ForPeerReviewOnly5improves accuracy. The older children perform better because that they have a greater capacity“to mental...
ForPeerReviewOnly6schemas of action1. The schemas of action would be primary and be the matrix for theconstruction of the ...
ForPeerReviewOnly71.3 Graphical Productions of the Children during Problem SolvingThere are virtually no studies on the gr...
ForPeerReviewOnly81.4 Justification of Empirical ProblemsThe studies presented in this article suggest that there is very ...
ForPeerReviewOnly9objects. Considering the fact that the youngest children have difficulty keeping several variablesin min...
ForPeerReviewOnly10To document the first interpretation, the current study conceived problems that seemed apriori adapted ...
ForPeerReviewOnly112.1.1 Subjects and DesignFifty-five nursery school children in good academic standing n the 13th arrond...
ForPeerReviewOnly122.1.2 Description of the ProblemsThe problems were first presented verbally and then with a graphic sup...
ForPeerReviewOnly13••••••The number of points of each group represents the divisor while the number of groupsrepresents th...
ForPeerReviewOnly14constitution of the regroupings (subsets) and their enumeration. The graphical representationwould prov...
ForPeerReviewOnly15Figure 3. Reduction of the Graphical Representation Accompanying the “Socks” Problem.Problem of the Foo...
ForPeerReviewOnly162 footballers 1 ball8 footballers X ballfrom which we get the equation b = f/g, where b is the sought q...
ForPeerReviewOnly17Problem of the Footballers: Pictorial Display. This problem was presented to thechildren along with a d...
ForPeerReviewOnly183 ResultsThree situations were presented to the children, first verbally, then in writing (through adra...
ForPeerReviewOnly19We observe the use of the strategy of one-to-one correspondence progressively in thelong-term performan...
ForPeerReviewOnly203.1 Improvement of Performance through the TasksWe observed (cf table III) a progressive improvement of...
ForPeerReviewOnly21This figurative drawing leads us to conclude which strategy that the pupil applies to solvethis type of...
ForPeerReviewOnly22Some children had difficulty in abandoning the semantic context. Fifteen of the 55 childrenstated that ...
ForPeerReviewOnly23As they continue working, more of the children build groups correctly, even if they donot reach the cor...
ForPeerReviewOnly24The children, even without experience with and formal capacities of this type of exercises,managed to s...
ForPeerReviewOnly25ReferencesCorrea, Jane, Terezinha Nunes, and Peter Bryant. 1998. Young children’s understanding ofdivis...
ForPeerReviewOnly26Lehrer, Richard and Leona Schauble. 2002. Symbolic Communication in Mathematics andScience: Co-Constitu...
ForPeerReviewOnly27---. 2002b. From sharing to dividing: The development of children’s understanding of division.Developme...
ForPeerReviewOnly1Tables and illustrations for Article: Resolution of Division Problems by YoungChildren: What Are Childre...
ForPeerReviewOnly2Figure 3. Reduction of the Graphical Representation Accompanying the “Socks” Problem.Figure 4. Reduction...
ForPeerReviewOnly3Problems Instructions1 Gloves oralquotitive-grouping divisionHow many children can one equip with 6glove...
ForPeerReviewOnly4Problems Strategies NumberofchildrenTotal ofchildrenOOC 17 55Gloves oralexamination G/CG 9 55OOC 20 55Gl...
ForPeerReviewOnly5ProblemsGroupingCorrectAnswers4,11-5,6yearsold5,7-5,11yearsold6-6,6yearsoldTotalG0 0 0 0Gloves oralexami...
ForPeerReviewOnlySummary of article: Resolution of Division Problems by Young Children: What Are ChildrenCapable of and un...
Upcoming SlideShare
Loading in …5
×

MATALLIOTAKI, E., (2012). Resolution of Division Problems by Young Children: What are children capable of and in which conditions? European Early Childhood Education Research Journal, 20(2), 283-299 (DRAFT)

355 views

Published on

Dans cet article nous explorons le champ théorique et expérimental des problèmes de division partitive, de quotition et de partage, comme illustrés par des études récentes dans ce domaine. Le but était d'expliquer et justifier l'utilité de présenter des problèmes de quotition, accompagnés des représentations graphiques, à des enfants de jeune âge. L'étude actuelle présente six problèmes proposés à des enfants de 5 à 6,5 ans dans le cadre d'une étude empirique. Les études montrent que la division quotition est accessible aux enfants de jeune âge. La manipulation des représentations graphiques afin de résoudre ces problèmes s'avère plus efficace que la présentation orale des problèmes.

Published in: Education, Technology
  • Be the first to comment

  • Be the first to like this

MATALLIOTAKI, E., (2012). Resolution of Division Problems by Young Children: What are children capable of and in which conditions? European Early Childhood Education Research Journal, 20(2), 283-299 (DRAFT)

  1. 1. ForPeerReviewOnlyResolution of Division Problems by Young Children: WhatAre Children Capable of and under which Conditions?Journal: European Early Childhood Education Research JournalManuscript ID: DraftManuscript Type: Research paperKeywords:child, division, graphical representation, problem resolving,reasoningURL: http://mc.manuscriptcentral.com/recrEuropean Early Childhood Education Research Journal
  2. 2. ForPeerReviewOnly1Resolution of Division Problems by Young Children: What Are ChildrenCapable of and under which Conditions?1 Theoretical Framework: Concepts and Related ResearchThis 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 SharingDivision is an important arithmetical operation because even college students and adultsmisunderstand this subject. Meanwhile, relatively less empirical research exists on thebeginnings of the learning of division than on other arithmetical operations. However, division isinteresting from the point of view of the schemas of action: the schema of sharing is at the baseof division. This idea is founded on Piaget’s claim that the sensory motor schemas are the basesof subsequent formal constructions.From the perspective of the mathematical definition of division, the idea that the activityof 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 ofsharing, which brings into play correspondences and the constitution of equivalent classes, canconstitute the first steps of division. At one time of their development, the children may be ablePage 1 of 33URL: http://mc.manuscriptcentral.com/recrEuropean Early Childhood Education Research Journal123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960
  3. 3. ForPeerReviewOnly2to carry out division while being unaware of the mathematical components involved (Frydmanand Bryant, 1988); these are generally learned later.1.2 Existing Studies on the Solution of Division Problems or Sharing by Young ChildrenThis section will review empirical studies on partitive and quotitive division amongchildren between 5 and 6,5 years old.1.2.1 Strategies of Resolution used by Children in Problems of DivisionKouba (1989) asked children between the ages of six and nine to solve division problemsthat she presented to them verbally. She identifies three types of problems depending on therequired 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 divisionproblem contributes to the difficulty of the problems more than it does in addition or subtractionproblems. Vergnaud (1983) and Fischbein et al. (1985) have stated that each arithmeticaloperation is related to an intuitive, implicit, unconscious and primitive model. This modelPage 2 of 33URL: http://mc.manuscriptcentral.com/recrEuropean Early Childhood Education Research Journal123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960
  4. 4. ForPeerReviewOnly3influences the decisions made when an operation is used. Fischbein et al. propose that—inaddition—the intuitive model is ‘put together’ while in subtraction it is ‘removed’; formultiplication, it is the repeated addition, and for quotitive division, it is the model of repeatedsubtraction. For partitive division, it is sharing.The children in the study of Kouba were interviewed individually, and the strategies ofresolution were categorized as ‘inappropriate’, ‘appropriate’, and ‘not identifiable’, independentof the accuracy of the calculation. The questioned children had concrete objects in case theyexpressed 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 weretaught. These 56 strategies were analysed according to two criteria: the degree of abstraction ofthe 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 ofMatalliotaki’s (2007) empirical study—indicate a 63 percent of the children solved the problemscorrectly.1.2.2 Solution to Partitive or Partitive and Quotitive ProblemsSquire and Bryant’s (2002) study used material supports to highlight the importance ofthe schemas of action in solving partitive problems.In their study, children from five to eight years old were asked to solve problemsrequiring them to distribute candies among dolls; the candies were divided among boxes in avariety of ways. The number of candies in each box corresponded either to the divisor or thePage 3 of 33URL: http://mc.manuscriptcentral.com/recrEuropean Early Childhood Education Research Journal123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960
  5. 5. ForPeerReviewOnly4quotient. Before every experiment a “control” (“baseline trial”) condition was established inwhich 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 (thenumber of boxes corresponding to the numbers of dolls); the bottom figure represents thegrouping by the quotient.The principal result of this study is that, for all examined ages, the children were helpedmore by the grouping by divisor than by the grouping by quotient. The ability of the children tosolve the problem by grouping by quotient improves with age. This can be explained, accordingto Squire and Bryant by the improvement in the capacity to use the one-to-one correspondencewith age, or by the fact that older children eventually learn to understand the interchangeabilityof the divisor and the quotient.Including questions about the number of boxes and the number of candies actuallyimproves children’s scores when grouping by quotient. According to Squire and Bryant, thismade it possible for children to use additive instead of multiplicative reasoning. This wasconfirmed by additional studies (Sophian et al., 1991) that found that, in children ages five andsix, passage by the addition plays a facilitating role.The spatial arrangement of dolls and candies influences the children’s ability to solve theproblem. Squire and Bryant suggest that children’s performance might be better if they canrearrange the objects themselves. They posit that handling of the objects or their imagesPage 4 of 33URL: http://mc.manuscriptcentral.com/recrEuropean Early Childhood Education Research Journal123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960
  6. 6. ForPeerReviewOnly5improves 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 isinfluenced 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. Thisfinding has important educational implications.Another study carried out by Squire and Bryant (2002) compared the procedures used tosolve partitive and quotitive problems. In their study, two conditions were presented: a partitivetask in which the objects were grouped either by divisor or by quotient, and a quotitive task withthe same two groupings. The children found the grouping by divisor to be easier when solvingthe 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 inmathematical terms there were no differences between the two conditions. Therefore, theysuggest that the most convincing reason for this difference between the two conditions is acognitive one.The authors suggest that, even if certain models are strongly artificial and benefit onlyfrom cultural drives (for example, in “pure” mathematics), others are probably acquired withoutexplicit instruction and are used by everyone. According to this approach, informal experiencecan 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 canbe a schema of action by which a comprehension of division develops. This developmentalconcept of the formation of the concepts supports Vergnaud, who granted a crucial role to thePage 5 of 33URL: http://mc.manuscriptcentral.com/recrEuropean Early Childhood Education Research Journal123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960
  7. 7. ForPeerReviewOnly6schemas of action1. The schemas of action would be primary and be the matrix for theconstruction 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 encountered difficultiesin performing the tasks occurs only at the plan of the selected procedures of resolution (and notthe amount of time needed to arrive at a correct answer). This resulted in stressing the proceduresrather than speed or accuracy. This aspect was retained when designing tasks for the currentstudy.1.2.3 The Inverse Divisor-quotient in Relation to Partitive and Quotitive TasksCorrea, Nunes, and Bryant’s study (1998) pertained to partitive and quotitive tasks; itdrew a conceptual distinction between sharing and division. In sharing, children treat only theequivalence of the shares. The concept of division, in contrast, implies the comprehension of therelations of three values represented by the dividend, the divisor, and the quotient. Theequivalence of the shares is assumed, although in division children must understand that thelarger the number of shares, the smaller those shares will be.In their studies the children understood well that, if the divisor (for example, the numberof rabbits) increases, the quotient (for example, the number of carrots per rabbit), decreases.Comprehension improved with age, even though young children understood partitive tasks betterthan they did quotitive ones, their abilities to perform both tasks was impressive.1The schemas of action (schèmes daction) are familiar actions that can offer initial comprehension of the arithmeticoperations (Vergnaud, 1985).Page 6 of 33URL: http://mc.manuscriptcentral.com/recrEuropean Early Childhood Education Research Journal123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960
  8. 8. ForPeerReviewOnly71.3 Graphical Productions of the Children during Problem SolvingThere 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 fields ofproblem solving.Weil-Barais and Resta-Schweitzer (2008, 2006) showed that the graphical productions ofchildren between five and six years of age years reveal the degree of conceptualisation inphysical phenomena. The drawings are one means by which children express a complexphenomenon, since they facilitate the expression of the spatial relations of the objects. Tantaroset al. (2005) studied the production of graphical representations by young children in order tocommunicate the solution of a problem and concluded that children’s productions improve withage and that drawings can constitute a cognitive tool. Lehrer and Schauble (2002) consider that arepresentation is not only a copy of reality. That implies inventing and adapting conventions of asystem of representation in order to choose, compose and transport information. Children learnedthat a system of representation could represent information that was not immediately perceptible.The authors conclude that developing adequate graphical representations for informationrepresented conventionally, promotes the learning of the mathematical concepts that are neededfor comprehension of the properties of this information.Based on the research that has been cited, we expect the use of graphical representationsby children will make it easier for them to solve division problems. The drawings can include thegraphic strategies to follow since the icons let children visualize the position of the mathematicalelements and the relations between them. We expect that the children will use the drawings as atool to envision the connections among the elements.Page 7 of 33URL: http://mc.manuscriptcentral.com/recrEuropean Early Childhood Education Research Journal123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960
  9. 9. ForPeerReviewOnly81.4 Justification of Empirical ProblemsThe 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 operation as theschema of action that it mobilises is level sharing—an activity that concerns a form ofsocialisation and has mathematical aspects.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 relations betweenthe quantities. However, even if it is more complex arithmetical operation, children—even thoseas young as five—participating in the studies showed remarkable success with certain divisionproblems. This supports the idea that the direct instruction of division is not essential in theformation 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 thechoice of graphical representations. Indeed, if the regroupings that need to be realised in order tocarry out a quotitive division can be materialised by a distribution in boxes (cf experiments bySquire 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 tomodify their distribution. However, the use of graphical representations makes it possible toconsider and preserve several types of regroupings. Thus, as suggested by Squire and Bryant thephysical handling of objects can help children to solve complex problems; thus, it would beinteresting to examine in what contexts the use of graphical representations plays a facilitatingrole. Furthermore, Kouba wonders if the categories of resolution that she distinguished in herstudy would be the same if the problems had employed graphic supports instead of physicalPage 8 of 33URL: http://mc.manuscriptcentral.com/recrEuropean Early Childhood Education Research Journal123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960
  10. 10. ForPeerReviewOnly9objects. Considering the fact that the youngest children have difficulty keeping several variablesin mind and performing complex operations, the graphical representations of a problem canindeed facilitate the handling of such situations by young children.For the problems designed herein, quotitive division was chosen because studies havedemonstrated that this type of division appears more complex (and is thus more interesting tostudy) for children than partitive division. The intuitive model of quotitive division is therepeated 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 toquotitive 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 makeinferences. Several reasons may explain this fact. Children of this age may not have developedthe capacity to draw inferences from graphic information. Moreover, these inferences may relyon formal knowledge that the children have not yet acquired, since the literature does notenvisage this type of exercise.Page 9 of 33URL: http://mc.manuscriptcentral.com/recrEuropean Early Childhood Education Research Journal123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960
  11. 11. ForPeerReviewOnly10To document the first interpretation, the current study conceived problems that seemed apriori 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 childrenmust make inferences related to quantities. More precisely, these situations focus on quotitivedivision: a number of objects in total and a number of objects by recipients being indicated todetermine the number of recipients. In mathematical terms, this involves giving the quantity of agroup and the number of elements constituting the subsets (or parts) so as to find the number ofsubsets (or of parts). When not having easy-to-handle objects, young children involved in thisstudy could solve these problems by the use of graphical representations.2 Empirical StudyIn this section we present the results of an empirical study that we conducted in a nurseryschool in France.2.1 Methodological FrameworkWe present the methodological framework of the empirical study, consisting of thepopulation chosen, the study design, and the description of the problems presented to thechildren.Page 10 of 33URL: http://mc.manuscriptcentral.com/recrEuropean Early Childhood Education Research Journal123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960
  12. 12. ForPeerReviewOnly112.1.1 Subjects and DesignFifty-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 monthsin age. The 31 boys and 24 girls were divided among three classes. Their parents signed aconsent form authorising their children’s participation in the study.The children were interviewed individually. The child and the researcher sat face to faceat table with sheets of paper and coloured pencils.In the Gloves-Socks-Footballers test, some boards (figures 2 to 4) are presented to thechild progressively. In addition, for any answer given by the child, the researcher asks him or herto explain how he or she arrived at the answer, and noted the explanation without expressingapproval or disapproval.Each child was given as much time as he or she needed to answer each questions. Themeetings were video recorded.Page 11 of 33URL: http://mc.manuscriptcentral.com/recrEuropean Early Childhood Education Research Journal123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960
  13. 13. ForPeerReviewOnly122.1.2 Description of the ProblemsThe 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 as a newproblem (not as a means of assisting with the verbal one), even if the quantitative informationand the structure of the problem were the same. The purpose of the exercise was to examine thetypes of strategies that the children use both with and without the graphic support, not only toascertain 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 childrencan one equip with six gloves?” This first problem implies minor amounts, which children of thisage can process either mentally or with the use of fingers. The solution of such a problem can bebased on the child’s cultural experience and on body consciousness (demonstrated inMatalliotaki 2001), which indicates that a person with two hands needs two gloves. The childrenmust thus deduce that, as each child needs two gloves (cardinal of the subsets), six gloves canequip three children. The problem can be formalised as follows.2 gloves 1 child6 gloves X childrenFor children, at an age where they are not yet able to produce formal mathematicalnotations, it is possible to await an analogue representation (mental or written) close to that thefollowing:Page 12 of 33URL: http://mc.manuscriptcentral.com/recrEuropean Early Childhood Education Research Journal123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960
  14. 14. ForPeerReviewOnly13••••••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) involve additivereasoning (prepare three pairs and then add them to find the correct answer), which, according toSquire and Bryant (2002) and Sophian et al. (1991), plays a facilitating role for children of thisage.Problem of the Gloves: Pictorial Display. The second problem proposes a schematisationof the data likely to facilitate the choice of a strategy of resolution. The same problem is verballypresented, but this time accompanied by a board (with the format 21x 29,7) that graphicallyrepresents the objects. Figure 1 presents a reduced graphical representation that was presented tothe children.Figure 2. Reduction of the Graphic Support Accompanying the Pictorial Display of the “Gloves” ProblemThis 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 thismanner, introducing the strategy to be followed for the resolution of the problem involves thePage 13 of 33URL: http://mc.manuscriptcentral.com/recrEuropean Early Childhood Education Research Journal123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960
  15. 15. ForPeerReviewOnly14constitution of the regroupings (subsets) and their enumeration. The graphical representationwould provide an inferential function as it makes it possible to calculate a quotient. Fewdrawings 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 childrencomplete a calculation.The drawn elements are not perfectly ordered because the presence of the identicalfeatures between the gloves of each pair is expected to be sufficient for completion of thegroupings.Problem of the Socks: Oral Presentation. The third problem utilised the same structure asthe 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 aschematisation of the data.Problem of the Socks: Pictorial Display. The same problem was put to the children, witha drawing of six pairs of socks (with the format 21x 29,7). Two versions were proposed: one inwhich the pairs of socks are locatable by graphic characteristics (see Figure 2, left side) and onein which all the socks were drawn in the same way (see Figure 2, right side). These two versionswere presented to two groups of children in the same age cohort. It is assumed that the firstdrawing performs two functions—referential and inferential—as it provides indices forregrouping by two: the child can find the answer by counting only the different socks. Thesecond drawing performs only one referential function. It can perform an inferential function ifthe children produce graphical indices of regrouping. These two versions were proposed in orderto examine the impact of the inferential function on children’s performances in a concretemanner.Page 14 of 33URL: http://mc.manuscriptcentral.com/recrEuropean Early Childhood Education Research Journal123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960
  16. 16. ForPeerReviewOnly15Figure 3. Reduction of the Graphical Representation Accompanying the “Socks” Problem.Problem of the Footballers: Oral Presentation. Unlike previous problems, this problemproposes a more formal mathematical problem:Eight footballers will practice in pairs (i.e., two by two: This explanation wasincorporated when some children started to solve the problem under theassumption that there would be two groups. The sentence “they will be involvedin groups of two” did not appear clear enough to describe the formation of thegroups. ). Each group will have a ball. How many balls will be used?The children are expected to solve this problem using the former experience (graphicalresolution of the problems). A possible schematisation of the procedure used to solve thisproblem might be as follows:8 footballers 2+2+2+22 + 2 + 2 + 2 footballers1+ 1+ 1+ 1 ballsor more formally:Page 15 of 33URL: http://mc.manuscriptcentral.com/recrEuropean Early Childhood Education Research Journal123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960
  17. 17. ForPeerReviewOnly162 footballers 1 ball8 footballers X ballfrom which we get the equation b = f/g, where b is the sought quotient, f the full number offootballers, and g the number of footballers in each group.For children of this age, who have not yet received formal instruction in mathematics, ananalogue representation seems more suitable:• • • •In this representation, the squares represent the footballers and the circles the balls. The twofootballers in each group represent the divisor and the number of built footballer couplesrepresents the quotient. This problem represents for the children a more complex situation thandoes preceding problems because the bond with the human body no longer exists, therebyeliminating a familiar context to which the child can refer. The context is more formal even iffootball is an interest of the children. The number of balls per player is arbitrary, which is not thecase in regards to the gloves or socks, which are always in pairs. The children must thusmemorise information (i.e., one ball for every two players), which may contradict what childrenusually associate with balls.Page 16 of 33URL: http://mc.manuscriptcentral.com/recrEuropean Early Childhood Education Research Journal123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960
  18. 18. ForPeerReviewOnly17Problem of the Footballers: Pictorial Display. This problem was presented to thechildren along with a drawing (with the format 21 x 29,7) (see Figure 4). Unlike the second andfourth problems, both balls and players are represented here. After the grouping of the players inpairs, 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 drawingdoes not propose elements that would support the grouping or the connection of each group withone ball. This last problem was conceived to test if a representation very close to an environmentfamiliar to the child (e.g., story books with illustrations, images in media) can be seen andhandled in an abstract context of mathematics when the children have not received formaleducation in mathematics. According to Squire and Bryant (2002), informal experiences canaffect the acquisition of mathematical skills. In our study we determine whether or not aninformal context that is familiar to children facilitates their problem solving.Table I provides, in a synthetic form, the problems suggested to the children within theframework of this empirical study.Table I: Problems Addressed to Children and Their InstructionsPage 17 of 33URL: http://mc.manuscriptcentral.com/recrEuropean Early Childhood Education Research Journal123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960
  19. 19. ForPeerReviewOnly183 ResultsThree situations were presented to the children, first verbally, then in writing (through adrawing 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 writtenform). In the written form, the problem of the footballers proves to be more difficult but it isnevertheless 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 theone-to-one correspondence (incorrect strategy).Table II presents the number of children who followed the correct mathematical strategy tosolve the problems (pairing and counting of the built groups) and the number of children thatfollowed the incorrect strategy (the one-to-one correspondence). To determine if the childrenfollowed the one-to-one correspondence strategy, it was enough to see their answer. If theanswer to the problem of gloves, for example, were 6, it means that the child distributed a gloveto each child instead of building pairs of gloves. In following table, OOC means one-to-onecorrespondence and G/CG means Grouping and Counting of the built groups.Table II: Summary of the Mathematical Strategies Used by the Children according to AgePage 18 of 33URL: http://mc.manuscriptcentral.com/recrEuropean Early Childhood Education Research Journal123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960
  20. 20. ForPeerReviewOnly19We observe the use of the strategy of one-to-one correspondence progressively in thelong-term performance of the tasks. According to table II, the children seem to give it upgradually.Some children seem to have understood the strategy or building groups in order to solvethe problems but for some reason, they could not make this strategy succeed. We thus present intable III the number of these children (by age) through the problems, in comparison to thenumber of children who answered it correctly. In the table, G means Grouping and represents thenumber of children who carried out the grouping and CA represents the number of children whoanswered correctly.Since authors like Squire and Bryant discovered that even a few months of difference inthe age of the subjects could play a significant part in their ability to understand symbolicnotations, we carried out an analysis of the performances of the children depending their age.Table III: Mathematical Strategies Used by the Children by AgeIn problems such as the graphical problem of gloves, nine children carried out the groupingcorrectly but could not to finish the procedure and solve the problem. This leads us to supposethat the children knew which strategy to follow (thanks to the preceding problem) but were notready to apply until the end. This can explain the absence of “grouping” without arriving at thecorrect answer in the first problem, where the children did not have yet a mathematical model ofa strategy on which they could rely.Page 19 of 33URL: http://mc.manuscriptcentral.com/recrEuropean Early Childhood Education Research Journal123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960
  21. 21. ForPeerReviewOnly203.1 Improvement of Performance through the TasksWe observed (cf table III) a progressive improvement of the children’s performance of thetasks. The children abandoned the erroneous strategy, by gradually applying the strategy ofgrouping the elements.The children spontaneously used an analogue representation (their fingers) when theproblems were verbally explained to them, but not for the first problem. Moreover, when theyused their fingers, the use that they made of their fingers (which represented the elements) wasinfluenced by the drawings that we presented to them in the preceding problems thanks to whichthey managed to find the right strategy to solve the problem. This shows that the children wereinfluenced by our suggestion to use a graphical representation, which ensures the inferentialfunction to solve problems; they seem to trust this tool.Indeed, the inferential function ensured by the first drawing (gloves) and generally, thefact of suggesting to children the use of an analogical system, influenced them to find ananalogical strategy to solve the following problem. David had a typical example of a graphicalproduction (cf figure 5): he used graphical representations that were very similar to the drawingthat had been presented to him to solve the problem of the footballers. David did not use suchdrawings for the first problems that had been presented to him, so we can conclude that thegraphical representations that we introduced influenced him to choose such a technique to solvethe problem.Figure 5: Graphical Production of David for the Resolution of the Problem Footballers Exposed OrallyPage 20 of 33URL: http://mc.manuscriptcentral.com/recrEuropean Early Childhood Education Research Journal123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960
  22. 22. ForPeerReviewOnly21This figurative drawing leads us to conclude which strategy that the pupil applies to solvethis 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 ContextWe tried to locate the error children most often made during problem solving in order tounderstand their reasoning and to identify the possible reasons for their inability to reach correctsolutions.The children answer graphically or verbally by drawing or colouring the sketch given tothem, or while producing conclusions relating to their cultural knowledge of the problem. Forexample, the footballers will need only one ball because football is played with one ball.Page 21 of 33URL: http://mc.manuscriptcentral.com/recrEuropean Early Childhood Education Research Journal123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960
  23. 23. ForPeerReviewOnly22Some children had difficulty in abandoning the semantic context. Fifteen of the 55 childrenstated that the footballers would need one ball in the oral problem of footballers (by justifyingtheir answer “the football game is played with one ball”). In the graphic version of this sameproblem 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 theyhad difficulty in remembering the instructions. We suppose that children of this age, as affirmedby Squire and Bryant (2002), often reinterpret problems in a way that makes sense for them. Thepragmatic 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 thesolution.4 General DiscussionFor the three situations, the drawing allowed the children to solve the problem by gatheringpairs of objects. In the verbal form of the problems we were not able to identify the strategiesused by the children; without paper and pencil, the children are not able to keep track of theirtrain 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-onecorrespondence. This means that children received some training in the resolution of theproblems that required the same strategy. Children of this age do not have experience in solvingproblems of quotitive division. The choice of the erroneous mathematical strategy thus seems tomake sense. What is astonishing is the fact that children have been able to find the correctstrategy despite their lack of experience in division. This shows that the children of this age havethe ability to solve such abstract problems.Page 22 of 33URL: http://mc.manuscriptcentral.com/recrEuropean Early Childhood Education Research Journal123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960
  24. 24. ForPeerReviewOnly23As they continue working, more of the children build groups correctly, even if they donot reach the correct answer. This enables us to say that there is an improvement in theirexecution of a mathematical strategy, while passing from one problem to the other. The firstimprovement 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 graphicalproblem of gloves for example, among the oldest children, ten came up with the solution to theproblem; 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, whichis acquired with age: the capacity to coordinate the parameters of a problem and to retain them inmemory (Squire and Bryant 2002). The youngest children can conceive the correct mathematicalstrategy to solve the problem but their ability to perform complex operations and to store partialproducts remains immature. Therefore in general, the idea to accompany a problem with agraphical representation would affect the performances of the youngest children. In order toconfirm this assumption, a test with a larger number of children in each age cohort would benecessary.The graphic version given to the children helped them not only to find the correctstrategy 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 problems which frequentlyconstrains children of this age.Page 23 of 33URL: http://mc.manuscriptcentral.com/recrEuropean Early Childhood Education Research Journal123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960
  25. 25. ForPeerReviewOnly24The 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. Thechildren’s observed successes at division at an early age should encourage educators to make themost 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 cultivatedsooner instead of later. This justifies that research among pre-school age children on this topicshould continue.Page 24 of 33URL: http://mc.manuscriptcentral.com/recrEuropean Early Childhood Education Research Journal123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960
  26. 26. ForPeerReviewOnly25ReferencesCorrea, Jane, Terezinha Nunes, and Peter Bryant. 1998. Young children’s understanding ofdivision: The relationship between division terms and a non-computational task. Journalof Educational Psychology 2: 321-329.Fischbein, Efraim, Maria Deri, Maria Sainati Nello, and Maria Sciolis Marino. 1985. The role ofimplicit models in solving verbal problems in multiplication and division. Journal forResearch in Mathematics Education 16: 3-17.Frydman, Oliver and Peter Bryant. 1988. Sharing and the understanding of number equivalenceby young children. Cognitive Development,3: 323-339.Gaux, Christine, Lydie Iralde, Annick Weil-Barais, and Aline Ferte. 2005. Evolution del’utilisation des systèmes de notation pour communiquer à autrui la construction d’unobjet, entre le cours élémentaire 1èreannée et le cours moyen 2èmeannée (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 divisionword problems. Journal for Research in Mathematics Education 20: 147-158.Page 25 of 33URL: http://mc.manuscriptcentral.com/recrEuropean Early Childhood Education Research Journal123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960
  27. 27. ForPeerReviewOnly26Lehrer, Richard and Leona Schauble. 2002. Symbolic Communication in Mathematics andScience: Co-Constituting Inscription and Thought. In: E. Amsel, J. Byrnes (Ed),Language, Literacy, and Cognitive Development. London: Lawrence ErlbaumAssociates.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 deproblèmes. Thèse de Doctorat en Sciences de l’Education, Université René Descartes-Paris 5.Resta-Schweitzer, Marcela and Annick Weil-Barais. 2006. Education scientifique etdéveloppement intellectuel du jeune enfant. Review of Science, Mathematics and ICTEducation 1: 63-82Sophian, Catherine. 1991. Le nombre et sa genèse avant l’école primaire. Comment s’en inspirerpour enseigner les mathématiques. In : Jacqueline Bideaud, Claire Meljac and Jean- PaulFischer (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’sunderstanding of division. Journal of Experimental Child Psychology 81: 1-43.Page 26 of 33URL: http://mc.manuscriptcentral.com/recrEuropean Early Childhood Education Research Journal123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960
  28. 28. ForPeerReviewOnly27---. 2002b. From sharing to dividing: The development of children’s understanding of division.Developmental Science 5: 452-466.---. 2003. Children’s models of division. Cognitive Development 18 : 355-376.Tantaros, Spyridon, Kalypso Sarigianni, Evmorfia Sotiropoulou, Dimitris Koliopoulos, andKonstantinos Ravanis. 2005 Etude des notations à visée communicationnelle par desenfants d’une école primaire en Grèce dans le cadre d’une activité scientifique. ColloqueNoter pour Penser, Angers, 26-27 January 2005.Vergnaud, Gérard. 1983. L’enfant, la mathématique et la réalité. Bern : Peter Lang.---. 1985. Concepts et schèmes dans une théorie opératoire de la représentation. Psychologiefrançaise 30 : 245-252.Weil-Barais, Annick and Marcela Resta-Schweitzer. 2008. Approche cognitive etdéveloppementale de la médiation en contexte d’enseignement – apprentissage. Nouvelle RevueAIS 42 : 83-98.Page 27 of 33URL: http://mc.manuscriptcentral.com/recrEuropean Early Childhood Education Research Journal123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960
  29. 29. ForPeerReviewOnly1Tables and illustrations for Article: Resolution of Division Problems by YoungChildren: 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” ProblemPage 28 of 33URL: http://mc.manuscriptcentral.com/recrEuropean Early Childhood Education Research Journal123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960
  30. 30. ForPeerReviewOnly2Figure 3. Reduction of the Graphical Representation Accompanying the “Socks” Problem.Figure 4. Reduction of the Graphical Representation Accompanying the “Footballers” Problem.Page 29 of 33URL: http://mc.manuscriptcentral.com/recrEuropean Early Childhood Education Research Journal123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960
  31. 31. ForPeerReviewOnly3Problems Instructions1 Gloves oralquotitive-grouping divisionHow many children can one equip with 6gloves?2 Gloves pictorial displayquotitive-grouping divisionHow many children can one equip withthese gloves? (showing the “gloves” board)3 Socks oralquotitive-grouping divisionHow many children can one equip with 12socks?4 Socks pictorial displayquotitive-grouping divisionHow many children can one equip withthese socks? (showing the “socks” board)5 Footballers oralquotitive-set divisionEight footballers will practice in groups of 2.Each group will have 1 ball. How manyballs will be used?6 Footballers pictorial displayquotitive-set divisionThe 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)Table I: Problems Addressed to Children and Their InstructionsPage 30 of 33URL: http://mc.manuscriptcentral.com/recrEuropean Early Childhood Education Research Journal123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960
  32. 32. ForPeerReviewOnly4Problems Strategies NumberofchildrenTotal ofchildrenOOC 17 55Gloves oralexamination G/CG 9 55OOC 20 55GlovesgraphicalexaminationG/CG2455OOC 18 55Socks oralexaminationG/CG 5 55OOC 15 55SocksgraphicalexaminationG/CG2455OOC 12 55FootballersoralexaminationG/CG555OOC 4 55FootballersgraphicalexaminationG/CG1955Table II: Summary of the Mathematical Strategies Used by the Children according to AgePage 31 of 33URL: http://mc.manuscriptcentral.com/recrEuropean Early Childhood Education Research Journal123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960
  33. 33. ForPeerReviewOnly5ProblemsGroupingCorrectAnswers4,11-5,6yearsold5,7-5,11yearsold6-6,6yearsoldTotalG0 0 0 0Gloves oralexaminationCA 3 3 3 9G4 2 3 9GlovesgraphicalexaminationCA5 9 10 24G1 1 0 2Socks oralexaminationCA 1 1 3 5G1 2 1 4SocksgraphicalexaminationCA8 7 9 24G1 1 2 4FootballersoralexaminationCA0 2 3 5G2 0 0 2FootballersgraphicalexaminationCA4 8 7 19Table III: Mathematical Strategies Used by the Children by AgeFigure 5: Graphical Production of David for the Resolution of the Problem Footballers Exposed OrallyPage 32 of 33URL: http://mc.manuscriptcentral.com/recrEuropean Early Childhood Education Research Journal123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960
  34. 34. ForPeerReviewOnlySummary of article: Resolution of Division Problems by Young Children: What Are ChildrenCapable of and under which Conditions?In this paper we explore the theoretical and experimental field of sharing and partitive andquotitive division problems, as illustrated by recent studies in this field. The purpose was toexplain and justify the utility of presenting quotitive division problems, accompanied bygraphical representations, to young children. The current study presents six problemssuggested to children of 5 to 6,5 years old within the framework of an empirical study. Thestudies prove that quotitive division is accessible to young children. The manipulation ofgraphical representations in order to solve these problems proves to be more efficient than theoral presentation of the problems.Key words: child, division, graphical representation, problem resolving and reasoningPage 33 of 33URL: http://mc.manuscriptcentral.com/recrEuropean Early Childhood Education Research Journal123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

×