young children's partitioning strategies

KATHY CHARLES and ROD NASON

YOUNG CHILDREN’S PARTITIONING STRATEGIES

ABSTRACT. Taxonomies for classifying children’s partitioning strategies have generallyfocused on the contexts in which the strategies occur and whether the strategies generatefair shares. Therefore, the overall aim of this study was to increase our knowledge aboutyoung children’s partitioning strategies by setting out not only to identify new partitioningstrategies but also to develop a taxonomy for classifying young children’s partitioningstrategies in terms of their ability to facilitate the abstraction of the partitive quotientfraction construct from the concrete activity of partitioning objects and/or sets of objects.Clinical interviews were conducted with twelve purposely-selected Year Three students.Each student was presented with a unique set of realistic partitioning tasks. The paperconcludes with a taxonomy for: (i) qualitatively evaluating a child’s progress towards theabstraction of the partitive quotient fraction construct, and (ii) planning and implementingteaching interventions commensurate with the child’s level of progress towards the ab-straction of the partitive quotient fraction construct. This taxonomy provides researchersand teachers with means of better utilising children’s informal partitioning strategies asthe foundation upon which to further develop their understandings of the partitive quotientfraction construct.

KEY WORDS: analog objects, children’s informal knowledge, conceptual mapping, fairshares, partitive quotient fraction construct, partitioning tasks, partitioning strategies, quan-tification, taxonomy, teaching interventions

INTRODUCTION

Many students experience difficulties with fractions. Indeed, a number ofeducators have stated that the learning of fractions is probably one of themost serious obstacles to the mathematical maturation of children (Behret al., 1992; Freudenthal, 1983; Kieren, 1988; Ohlsson, 1987, 1988; Post,Behr and Lesh, 1986; Streefland, 1991). Many students also have littleconceptual understanding of fractions and are dependent upon rote learn-ing procedures, which are often incorrect (Behr et al., 1983; Behr et al.,1985; Bennett, 1987; Kerslake, 1986; Kouba et al., 1988; Mack, 1990).

Many of these problems with the learning of fractions can be attributedto teaching efforts that have focussed almost exclusively on the part-wholeconstruct of a fraction (Kerslake, 1986; Streefland, 1991). In recent years,many mathematics educators such as Behr et al. (1992), Kieren (1993)and Streefland (1991) have questioned the wisdom of this practice. While

Educational Studies in Mathematics 43: 191–221, 2000.© 2001 Kluwer Academic Publishers. Printed in the Netherlands.

192 KATHY CHARLES AND ROD NASON

agreeing that the part-whole construct is central to mature functioning withfractions, they contended that understanding the part-whole construct isnot a sufficient condition for such functioning. They believe that the teach-ing of fractions also needs to focus on the other meanings of a fraction suchas the partitive quotient construct. Kieren (1993) went so far to say thatthe partitive quotient construct is central to the development of fractionalnumber knowledge.

The importance of the partitive quotient fraction construct has beenreflected in an increasing number of research studies that have investigatedyoung children’s strategies for partitioning. These studies have found that:(i) young children tend to use a variety of intuitive strategies when con-fronted with partitioning problems (Lamon, 1996; Pitkethly and Hunting,1996; Pothier and Sawada, 1983, Streefland, 1991), (ii) young children’sselection of partitioning strategies depends not only on their prior know-ledge and experiences but also on the context of the task, the type of analogobjects being shared, the number of analog objects being shared and num-ber of shares (Lamon; Pothier and Sawada; Streefland), and (iii) youngchildren’s use of partitioning strategies is situationally specific, demon-strating a strong adherence to social practice (Lamon, 1996).

Although the research literature contains much information about whatintuitive partitioning strategies are used by young children and how andwhen they use these strategies, few attempts seem to have been made toanalyse how young children’s intuitive partitioning strategies may facil-itate or possibly hinder the abstraction of the partitive quotient fractionconstruct from the concrete activity of partitioning objects and/or sets ofobjects. Therefore, the overall aim of this study was to increase our know-ledge about young children’s partitioning strategies by setting out not onlyto identify new partitioning strategies but also to develop a taxonomy forclassifying young children’s partitioning strategies in terms of their abilityto facilitate the abstraction of the partitive quotient fraction construct fromthe concrete activity of partitioning objects and/or sets of objects.

THE ABSTRACTION OF THE PARTITIVE QUOTIENT FRACTION

CONSTRUCT

In recent years, a number of educators such as Sfard (1994), Bereiter(1994a, 1994b), Prawatt (1993, 1995) and Dreyfus et al. (1997) have notedthat the abstraction of mathematical constructs from concrete situationsshould be perceived as a crucial outcome of mathematical learning activity.An abstraction is a lasting change that enables the identification of thesame concepts, structures and relationships in many different but structur-

YOUNG CHILDREN’S PARTITIONING STRATEGIES 193

ally similar tasks. Abstractions such as mathematical constructs are power-ful because they can be applied to many situations (Freudenthal, 1983)and thus facilitate the transfer of learning from specific contexts to generalcontexts outside of the classroom (Dreyfus et al., 1997). If abstraction doesnot occur, then the outcomes of mathematical activity can be the merecompletion of tasks and/or the performance of mathematical proceduresrather than the building of mathematical constructs (Bereiter, 1994a).

The partitive quotient fraction construct can be operationally definedas the process in which one starts with two quantities x and y, treats x asthe dividend and y as the divisor and by the operation of partitive divisionobtains a single quantity x /y (Behr et al., 1993). Although the partitivequotient fraction construct can be readily modelled with hands-on activitywith concrete materials and be closely related to many children’s ‘realworld’ experiences, the abstraction of this construct from concrete activ-ity presents children with many complex processes (English and Halford,1995). For example, the abstraction of the partitive quotient notion of thefraction 3/4 from the concrete activity of sharing three pizzas among fourpeople requires children to construct:

• a conceptual mapping between the number of people (4) to the frac-tion name of each share (fourths)

• a conceptual mapping between the number of pizza being shared (3)and the number of fourths in each share (3 fourths).

Many researchers in the fields of mathematics education (e.g., Mitchel-more and White, 1995; Sfard, 1991) and cognitive psychology (e.g., Derry,1996; Klahr and Wallace, 1976) have suggested that the process of ab-straction from concrete experience begins with children initially acquiringa body of disconnected knowledge situated in a large number of every-day experiences such as sharing objects like pizza and cakes with theirfamily or friends. They then note consistencies within different situations(Klahr and Wallace, 1976). Following the noting of these consistencies,children classify these situations into contexts in which the mathemat-ical construct is situated. During this phase, children begin to constructthe conceptual mappings which form the basis of the abstraction. Finallythey form abstract-general constructs by recognising consistencies and/orsimilarities across several different contexts.

In order for children to classify situations in which the partitive quotientfraction construct is situated and thus begin the construction of the con-ceptual mappings which underlie the abstraction of the partitive quotientfraction construct, children need to utilise partitioning strategies whichgenerate shares which are equal and able to be quantified. If the shares gen-erated by the partitioning strategies are not equal, then children will find


it hard to identify consistencies/similarities within the different sharingsituations. This will make the task of classifying situations into contextsin which the partitive quotient construct is situated problematical. If thestrategies generate shares which are not readily quantifiable, then childrenwill be unable to construct conceptual mappings between: (1) the numberof people (y) to the fraction name in each share (yths), and (2) the numberof objects being shared (x) and the number of yths in each share. If childrendo not construct these mappings, then the outcomes of their partitioningactivity may be the mere completion of the partitioning tasks rather thanthe building of the partitive quotient fraction construct.

METHOD

The overall aims of the study were achieved by identifying the strategiesa group of twelve purposely-selected young primary school students util-ised to solve realistic partitive quotient fraction problems; evaluating howwell each of these strategies facilitated the abstraction of the partitive quo-tient fraction construct; and then developing a taxonomy for evaluatinghow well partitioning strategies facilitated the abstraction of the partitivequotient fraction construct.

The data gathering technique used was the clinical or mixed methodtechnique of Ginsburg et al. (1983) which is a combination of Piaget’sclinical interview and the talk aloud procedures of Ericsson and Simon(1984). According to Ginsburg et al., this technique enables researchers tonot only elicit complex intellectual activity but also to identify the internalsymbolic mechanisms underlying the complex intellectual activity.

Participants

Twelve third grade students (aged between 7.9 and 8.3 years) from a primaryschool located in Eastern Australia participated in this study. The stu-dents were chosen through serial, contingent, purposeful and exhaustivesampling to produce maximum variation in cognitive functioning (Gubaand Lincoln, 1989). This sampling procedure relied heavily upon the teach-ers’ categorisation of the students into low, middle and high achievers.Students from all categories participated in the study to maximise the prob-ability that as many strategies as possible would emerge during the courseof the study. When it was found that no new partitioning strategies orother insights about young children’s partitioning emerged during the in-terviews with the eleventh and twelfth participants, data collection ceased.


The size of the sample thus was influenced by Guba and Lincoln’s criteriaof redundancy.

Instruments

A template of thirty realistic partitioning tasks based on prototypes cre-ated by Streefland (1991) was developed for this study (Charles, 1998). Ineach of the tasks, the children were asked to assume the roles of waiters/waitresses serving pizzas, pancakes, pikelets, icecream bars, apple pies orliquorice straps to a number of customers sitting at a restaurant table. Thesetasks brought the students into situations in which they could producefractions (cf. Streefland, 1991).

Three task variables were taken into consideration during the devel-opment of the set of partitioning tasks: (i) types of analog objects, (ii)number of analog objects, and (iii) number of people. The analog objectsused were circular region models, rectangular region models and lengthmodels (cf., Streefland, 1991). The number of analog objects ranged fromone to six and the number of people ranged from two to six. The circularregion models for the study included: pizza, pancake, pikelet and applepie. The rectangular region models included: cake and icecream barcake.The length model was represented by the liquorice strap (cf., Streefland,1991).

The template of thirty partitioning tasks was organised into five cat-egories. The first category of tasks was designed to produce shares ofone half and one quarter and involved the sharing of one pizza amongtwo or four people. This category was developed from the belief that:(i) the ‘pizza’ circular region model is an appropriate model for initialpartitioning activities (cf., Streefland, 1991); and (ii) children have soundinformal knowledge of a half (Behr et al., 1992; Ball, 1993) and powerfulstrategies for halving (Pothier and Sawada, 1990). A natural progressionfrom here was to investigate children’s knowledge of partitioning for threepeople. Accordingly, the second category of tasks was designed to involvethe children in partitioning of circular and rectangular region models intothirds. The rectangular region model was included in this category becauseBall (1993) found that partitioning of rectangular region models was easierfor children in her study than the partitioning of circular region models.The third category of tasks was designed to explore children’s partitioningstrategies for four people. These tasks involved four people sharing two,three, four, five, or six circular region models. The fourth category of tasksinvestigated children’s partitioning strategies for five people and involvedthe sharing of one, two or three circular region models. The final categoryof tasks explored children’s partitioning strategies when interacting with


the length model and the rectangular region model. A variety of tasks wereincluded here to explore children’s knowledge construction of quarters,thirds, fifths and sixths. Particular note was taken of children’s partitioningstrategies for the rectangular region model as Streefland (1991) noted thatthis model supported both one-directional (i.e., vertical or horizontal) andtwo-directional (i.e., vertical and horizontal) partitioning.

Procedure

Each child was interviewed individually in a mobile research vehicle thathad two sections: an interview room with two video cameras, and an ob-servation room with monitors and a videomixer. The interviewer sat withthe children and two observers watched and communicated with the inter-viewer via earphones. Each of the observers was an experienced teacher ora researcher with expertise in mathematics education.

The interview and analysis of children’s responses were hermeneutic.Four children were interviewed in each of three sessions. Each of thetwelve children was presented with a series of partitioning tasks. For eachchild in turn, the interviewer began by administering a selection of tasksfrom the first category of tasks. During the course of an interview, theinterviewer and the observers used the children’s actions and verbalisationsto identify strategies and inform the selection of further tasks to be admin-istered in the interview. The children were encouraged to think aloud asthey attempted to solve the tasks. A similar procedure was adopted for eachsuccessive interview. Each child was thus administered a unique sequenceof partitive quotient fraction tasks.

In order to produce a more dynamic assessment of the children’s know-ledge structures, the clinical interviews were often but not always extendedto include limited teaching episodes. According to Hunting (1980), theinclusion of limited teaching episodes within clinical interviews takes theclinical interview a step beyond merely assessing the status of a child’scognitive functioning.

Limited teaching episodes

During these limited teaching episodes, there was no direct teaching ofpartitioning strategies. The teaching episodes instead focused on: (i) ex-ploring and extending children’s knowledge construction, or (ii) helpingthe children to overcome cognitive impasses (VanLehn, 1990) in order toinvestigate the extent of the child’s knowledge.

Many of the limited teaching episodes which aimed at exploring andextending the children’s knowledge of the partitive quotient fraction con-struct tried to encourage the children to make generalisations about the


mathematical ideas they had been investigating. This type of limited teach-ing episode was attempted when it was noted that during and/or at thecompletion of the process of solving a particular task, a child seemed tobe on the verge of making a conceptual leap about partitioning. Duringlimited teaching episodes such as these, a child was administered a care-fully selected sequence of partitioning tasks which maximised the chancesof the child extending an insight about partitioning (s)he had generatedduring the solution of the set task. Sometimes, tables were used to recordthe results from the investigation of each of the teaching episode tasks inorder to facilitate the process of looking for patterns and relationships. Oneexample of this type of teaching intervention occurred just after Caitlinhad successfully shared three liquorice straps between four people. Theresearchers noted that she seemed to be on the verge of abstracting thepartitive quotient fraction construct. She then was asked in turn to predictand then test how much liquorice each of the four people would get iffive, six, eight straps of liquorice were shared. Each time, she was askedto justify her predictions. Caitlin was soon able to articulate (and justify) ageneral rule for the solution of partitive quotient fraction tasks.

Most of the other limited teaching episodes which aimed at exploringand extending the children’s knowledge of the partitive quotient fractionconstruct tried to encourage the children to add to and/or modify theirrepertoires of partitioning strategies. This type of limited teaching episodeoften occurred when a child did not seem to be making progress towardsthe abstraction of the partitive quotient construct. In cases such as this, thechild was presented with tasks where the application of his or her presentrepertoire of strategies led to impasses. For example, when it was notedthat Thomas always began partitioning by cutting the objects into halves,he was then presented with a sequence of tasks where the use of thisstrategy made the process of partitioning very difficult and cumbersome.He soon began to explore non-halving partitioning strategies.

The limited teaching episodes which focused on helping the childrento overcome cognitive impasses utilised teaching strategies such as: (a)providing learning activities which enabled the children to construct theknowledge necessary for overcoming the impasse(s), and (b) breaking atask into a sequence of simpler sub-tasks. One example of the former oc-curred when Sophie was unable to correctly state how much each personwould receive after she had correctly shared two ice cream bars betweenthree people. She incorrectly stated that each person’s share was two quar-ters. Accordingly, the teaching intervention focussed on helping Sophiedevelop an understanding of the connection between the number of piecesin the whole and the fraction name. An example of the latter occurred when


TABLE I

Partitioning strategies

Hitherto Reported Strategies Hitherto Unreported Strategies

Partitive quotient foundational studyStreefland (1991)English and Halford (1995)

Partition and quantify by part-whole no-tion strategy

Proceduralised partitive quotient strategyStreefland (1991)

Regrouping strategy

Horizontal partitioning strategyStreefland (1991)Pothier and Sawada (1983)

People by objects strategy

Halving the object then halving again andagain strategyStreefland (1991)Pothier and Sawada (1983)

Half the objects between half the peoplestrategy

Half to each person and then quarter toeach person strategyStreefland (1991)

Repeated sizing strategy

Whole to each person then half the re-maining objects between half the peopleLamon (1996)

Repeated halving/repeated sizingstrategy

Thomas was unable to successfully share three pancakes between fourpeople. The teaching intervention employed was to tell Thomas that thekitchen was very slow today and that the three pancakes would come outfrom the kitchen one at a time (and not all three at once). This encouragedThomas to partition and share each pancake one at a time. The initial taskthus was changed into three simpler sub-tasks.

CHILDREN’S STRATEGIES FOR SOLVING PARTITIONING FRACTION

TASKS

During the course of the study, twelve partitioning strategies emerged. Asindicated in Table I above, six of the strategies used by the children in thisstudy had been hitherto reported in Streefland (1991), English and Halford


(1995), Pothier and Sawada (1983), and Lamon (1996). However, six ofthe strategies used by the children in this study were hitherto unreported inthe research literature.

The twelve strategies were classified into three categories:

1. Partitive Quotient Construct Strategies;2. Multiplicative Strategies; and3. Iterative Sharing Strategies

1. Partitive quotient construct strategies

All the strategies in this category had one common characteristic; they util-ised the relationship between number of people sharing and the fractionalname to generate the denomination for each share. For example, if two,three or more pizzas were being shared between five people, the valueof each share was expressed in fifths. Five partitive quotient constructstrategies were identified: (i) the partitive quotient foundational strategy,(ii) the proceduralised partitive quotient strategy, (iii) the partition andquantify by part-whole notion strategy, (iv) the regrouping strategy, and(v) the horizontal partitioning strategy.

Partitive quotient foundational strategyThe partitive quotient foundational strategy involved the application of thefollowing six steps:

Step 1. recognition of number of people (y)Step 2. generation of fraction name from number of people (yths)Step 3. recognition of relationship between fraction name and number of

equal pieces in each whole object (y equal pieces)Step 4. partitioning of each whole object into equal pieces (y equal pieces)Step 5. sharing the pieces (1/y of each object to each person)Step 6. quantifying each share (addition of 1/y pieces)

This strategy was utilised by eleven of the twelve children for the parti-tioning of one object between four people. It was not so widely adoptedfor tasks involving the sharing of more than one object between a numberof people. However, Sally employed this strategy when she was asked toshare two icecream barcakes among three people. After cutting the firsticecream barcake into three equal pieces, Sally shared 1/3 of the icecreambar to each person. While she was cutting the icecream bar into three equalpieces, the investigator asked her what she was doing. She replied thatshe was “Cutting it into thirds”. When she had completed sharing the firsticecream bar, she then proceeded to cut the second icecream bar into thirdsand share the thirds to each person. When asked how much icecream each


person received, she replied without hesitation, that each person receivedtwo thirds.

Proceduralised partitive quotient strategyThis strategy is a truncated version of the Foundational Strategy in whichthe number of steps required to generate a solution to a problem has beenreduced from 6 to 3. The steps for this truncated, proceduralised strategyare as follows:

Step 1. recognition of number of people (y)Step 2. recognition of number of analog objects (x)Step 3. quantification of each person’s share as x/y

Only four of the students, Joshua, Caitlin, Mark and Sally applied thisproceduralised strategy. Caitlin and Sally demonstrated knowledge of thisstrategy without actually partitioning and sharing the objects (i.e., withoutexecuting Steps 4 and 5 of the ‘foundational strategy’). For example, whenasked what each person would receive if there were 10 liquorice strapsto share between 4 people, Caitlin replied, ‘Ten quarters.’ When askedwhat each person would receive if there were 100 pieces of liquorice toshare between 4 people, she replied, ‘A hundred quarters?’ Sally used thisstrategy to determine each person’s share when two, three and six icecreambarcakes were shared between six people.

Mark and Joshua executed all three steps of the strategy and then util-ised the concrete analogs to confirm the correctness of their solutions. Forexample, when sharing three pizzas between four people, Joshua and Markdid not cut each pizza into quarters but instead they cut 1/4 from each ofthree pizzas leaving 3/4 of each of the three pizzas intact. They then placedthe three 1/4 pieces together to form the fourth share. Their applicationof this strategy thus has much in common with Lamon’s (1996) ‘preservedpieces strategy’. It should be noted that Joshua employed this strategy onlyafter having successfully completed the same problem previously and waswell aware of the share each person would receive.

Partition and quantify by part-whole notion strategyThis hitherto unreported strategy omits Steps 2 and 3 from the ‘partitivequotient foundational strategy’. The steps for this strategy are:

Step 1. recognition of number of people (y)Step 2. partitioning of each whole object into equal pieces (y equal pieces)Step 3. sharing one piece from each object to each person,Step 4. quantification of each person’s share by application of part-whole

system mapping to determine fraction name.


Candice, Emma and Da used this strategy to share one pizza among threepeople and Sophie used it to share two pizzas among three people. Afternoting that there were three people to share the pizza, the children parti-tioned each pizza into three equal portions. After all the portions had beenshared, the children attempted to quantify how much pizza was in eachshare.

Whenever this strategy was used, the children were unable to recognisethe relationship between the number of people and the fraction name; thenumber of people gave the number of pieces each whole was to be parti-tioned into rather than the fraction name. Thus, the quantification of eachshare could not be achieved until after the sharing had been completed.This was done by applying the part-whole notion of a fraction to determinethe quantity in each share. Unfortunately, whenever the children employedthis strategy, they were unable to accurately quantify each share duringStep 4 due to their inability to successfully apply the part-whole systemmapping process in this context.

Regrouping strategyLike the ‘partition and quantify by part-whole notion strategy’, this strategyalso does not seem to parallel any of the partitioning strategies hithertoreported in the literature. The steps for this strategy are shown below.

Step 1. recognition of the number of people (y)Step 2. recognition that the number of people gives the fraction name (yths)Step 3. a realisation that the fraction name gives the number of pieces in

the whole (y)Step 4. a realisation that the total number of yths to be shared can be gen-

erated by multiplying the number of objects (x) by the number of ythsin each whole

Step 5. quantification of each share through a recognition that the numberof yths for each person can be calculated by dividing the total numberof yths by the number of people (x)

This strategy was only used by one child, Sophie, but only after she hadsuccessfully completed six partitioning tasks. She employed this strategyto share five pizzas between four people. When asked to predict what eachperson would receive, she reasoned that there were20/4 in total which whenshared between the four people would give each person5/4.

Horizontal partitioning strategyThis strategy involved the following steps.

Step 1. recognition of number of people (y)


Figure 1. Horizontal partitioning strategy.

Step 2. generation of fraction name from number of people (yths)

Step 3. recognition of relationship between fraction name and number ofpieces in each whole object (y)

Step 4. horizontal partitioning of each whole circular object into pieces (ypieces)

Step 5. quantification of each share (1/y piece)

Step 6. recognition that shares are unequal

This strategy has much in common with some of the partitioning strategiesused by many participants in Streefland’s (1991) study and with the ‘al-gorithmic halving strategy’ reported in Pothier and Sawada (1983). Claudiaused this strategy to share one pancake between three people. Prior to thepresentation of this task, Claudia had successfully partitioned one pancakebetween four people. She was then presented with the task of partitioningone pancake between three people and employed the horizontal partition-ing strategy illustrated in Figure 1. Later on, tasks involving the sharingof two and three pancakes between four people were presented to Claudia.She partitioned these tasks successfully but when presented with the taskof partitioning two pancakes between three people, she was again unableto successfully partition the objects. Claudia indicated that she knew thatthe shares were unequal and therefore, that none of the shares were equalto 1/3.


2. Multiplicative strategies

In multiplicative strategies, a multiplication algorithm is used to gener-ate the number of parts in each whole object. One of the best-knownmultiplicative strategies is Pothier and Sawada’s (1983) ‘composition shar-ing strategy’ which children used to generate an odd number of parti-tions. In this strategy, fifteenths were generated by either trisecting fifthsor by partitioning each third into five equal parts. According to Pothier andSawada, multiplicative strategies are very efficient strategies for partition-ing an object into an odd number of equal parts. None of the children inthis study used Sawada and Pothier’s ‘composition sharing strategy’ whenpartitioning an object into an odd number of parts.

However, one child, Thomas, used a very similar type of multiplicat-ive strategy to generate shares when he was asked to partition two pizzasbetween three people. We have labelled this strategy the ‘people by objectsstrategy’.

People by objects strategyThe steps for this strategy are:

Step 1. recognition of the number of people (y)Step 2. recognition of number of objects (x)Step 3. the number of pieces in each whole is generated by multiplying (y)

by (x)Step 4. Partitioning each whole into (yx) piecesStep 5. sharing the pieces between peopleStep 6. quantification of each share

Thomas had difficulty partitioning a circular model into three equal sizedpieces. However, he appeared very confident when cutting a semicircularmodel into three equal pieces. He thus used this strategy to share two pizzasamong three people. He cut the first pizza in half, then each half into threepieces so that the whole was divided into six equally sized pieces. He thenshared two sixths to each person. He cut and shared the second pizza thesame as the first. When asked why he chose sixths, he replied, “Becausethere are three plates and two pizzas.” He quantified each person’s share as4/6.

3. Iterative sharing strategies

Four iterative sharing strategies were noted during this investigation: (i)halving the object then halving again and again, (ii) half the objects betweenhalf the people, (iii) repeated sizing strategy, and (iv) repeated halving andrepeated sizing strategy.


Halving object then halving again and again strategyWith this strategy, the halving process was only repeated twice (i.e., eachwhole was partitioned into eighths). It involved the following steps.

Step 1. iterative halving of each whole object into eight equal piecesStep 2. recognition that each piece is 1/8Step 3. sharing out the piecesStep 4. quantification of each share

Ben and Thomas used this strategy. When sharing one pizza between twopeople, Ben stated that he would, “Cut it in 1/2, then again and again.”However, he failed to successfully apply Step 2. He incorrectly quantifiedeach person’s share as 1/4 instead of 4/8. He seemed to confuse the numberof pieces in each person’s share (4) with the number of pieces in the whole(8) when he was generating the fraction denominator. Thomas employedthis strategy to share five pancakes between four people. He proceededto cut each pizza into eighths and shared 2/8 to each person. He correctlyquantified each person’s share as 10/8. Thus, in contrast to Ben, Thomaswas able to successfully apply all four steps in this strategy.

Half the objects between half the people strategyAlthough this is a halving strategy, it is qualitatively quite different fromthe halving strategies reported hitherto in the research literature. It shouldbe noted, however, that this strategy was only ever utilised for the sharingof two objects between four people. It involved the following six steps:

Step 1. recognition of the number of people (4)Step 2. recognition of the number of objects (2)Step 3. realisation that halving the objects will generate enough pieces to

share between all the peopleStep 4. partition the objects into halvesStep 5. sharing the halves between all the peopleStep 6. quantification of each share

Joshua, Claudia, Sophie, Thomas, Emma, Sally, Da and Belinda utilisedthis strategy. When asked how he solved the problem, Joshua highlightedthe iterative nature of the process by stating, “Two people so I cut in1/2 togive one piece for two people and then I do it again.” Belinda recognisedthat half the people could share half the objects. This is illuminated by herstatement, “Those two people share a whole and these two people sharea whole.” When asked how he solved the problem, Thomas replied, “Twohalves in a whole and there are four people.”

After Claudia had partitioned the two wholes and distributed the fourhalf objects, she incorrectly assumed that each person had received one


part out of four equal parts. It appeared that she had converted the prob-lem from one being represented by region analog models to one beingrepresented by a set analog model consisting of four separate elements.Therefore, she incorrectly quantified each person’s share as 1/4 (i.e., onepart out of four equal parts) instead of either1/2 or 2/4 of a pizza. The othereight children who utilised this strategy were able to accurately quantifyeach share as 1/2.

Extensions of this strategyExtension 1: Whole to each person then half remaining objects betweenhalf people strategyThis strategy was developed in response to the need to partition sharesgreater than one whole. It involved the following seven steps:

Step 1. recognition of the number of people (y)Step 2. recognition of the number of objects (x)Step 3. realisation that the number of objects (x) was greater than the num-

ber of people (y)Step 4. sharing of one whole object to each person (cf., Lamon’s (1996)

Preserved-Pieces Strategy)

[Step 3 and 4 repeated until x < y]

Step 5. application of ‘half objects between half the people strategy’Step 6. quantification of each share

Emma employed this strategy to share six pizzas between four people. Sheshared one whole to each person. Then she tiered the two ‘left-over’ pizzas,cut them into four halves and shared1/2 to each person. She quantified eachperson’s share as 1 1/2.

Extension 2:Half to each person then a quarter to each person strategyThis strategy is a combination of the ‘half the objects between half thepeople strategy’ and the ‘partitive quotient foundational strategy’. It in-volved the following steps:

Step 1. application of the ‘half objects between half the people strategy’Step 2. realisation that there are not enough remaining objects to reapply

the ‘half the objects between half the people strategy’Step 3. application of the ‘partitive quotient foundational strategy’Step 4. quantification of each share

Both Joshua and Sally used this strategy. When asked to share three piz-zas between four people, Joshua applied the ‘half the objects betweenhalf the people strategy’ to partition the first two pizzas into halves and


share them fairly between the four people. He realised that there were notenough pizzas leftover to reapply the same strategy. Therefore, he utilisedthe ‘partitive quotient foundational strategy’ to share the remaining pizzabetween the four people. Finally, he quantified the total amount each per-son received by adding the 1/2 (generated by the application of the ‘halfthe objects between half the people strategy’) to the 1/4 (generated by theapplication of the ‘partitive quotient foundational strategy’) to get3/4.

When attempting the same problem, Sally cut the first and second pizzain halves and the third pizza in quarters. She shared1/2 to each person andthen 1/4 to each person. She quantified each share in a similar manner tothat used by Joshua.

Repeated sizing strategyThis strategy was hitherto unreported. It involved the realistic sizing ofa pizza into unequal sized pieces such as it would be served in manyAustralian restaurants. In these restaurants, pizzas are rarely partitionedinto equal sized pieces. The use of this strategy thus tends to confirm La-mon’s (1996) theory that young children’s partitioning strategies demon-strate a strong adherence to social practice. The steps for this strategy areas follows:

Step 1. partitioning of each whole object into an even number of unequalpieces

Step 2. sharing of pieces using attribute of area rather than attribute ofnumber in an attempt to achieve equal shares

Ben used this strategy to share one pizza among four people. He cut thepizza into many unequal pieces and shared them as evenly by area ashe could. Thus, the quantification of each share was not possible. Thediagrams in Figures 2 demonstrates Ben’s application of this strategy.

Repeated halving/repeated sizing strategyThis composite strategy has not been reported previously in the researchliterature. It involved the following two steps;

Step 1. application of the ‘halving the object again and again strategy’Step 2. application of the ‘repeated sizing strategy’

Because of the application of the repeated sizing strategy in Step 2, thequantification of each share was not possible. Joshua used this strategyto share a rectangular cake between three people. He cut the cake in halfhorizontally. Then he cut each half in half and shared 1/4 to each person.He resized the final 1/4 as demonstrated in Figure 3 and shared the piecesas evenly by the attribute of area as possible.


Figure 2. Repeated sizing strategy.

Claudia employed this strategy to share two pancakes between threepeople. She cut both pancakes into halves then quarters and dealt 2/4 toeach person. The remaining 2/4 were halved and one piece (1/8) was sharedto each person. The left-over piece (1/8) was halved and halved again intofour pieces. One piece (1/32) was shared to each person and the final piece(1/32) was resized into three unequal pieces. At the end of this laborious


Figure 3. Joshua’s application of Repeated halving/repeated sizing strategy.


Figure 4. Claudia’s application of Repeated halving/repeated sizing strategy.


process, Claudia was unable to tell the interviewer how much pancake eachperson received. Her application of the strategy is illustrated in Figure 4.

A TAXONOMY FOR EVALUATING THE ABSTRACTION-ABILITY OF

CHILDREN’S PARTITIONING STRATEGIES

Each of the twelve strategies was evaluated in terms of how well it facil-itated the abstraction of the partitive quotient fraction construct from theconcrete activity of partitioning objects or sets of objects.

In our initial conceptualisation of the process of abstraction, it wasstated that in order for children to be able to abstract the partitive quotientfraction construct from concrete partitioning activity, they need to utilisepartitioning strategies that generate shares which are equal and able to beaccurately quantified. However, during our evaluation of the partitioningstrategies utilised by children in this research study, it was found thatgeneration of equal and quantifiable shares were necessary but not suf-ficient conditions for the full abstraction of the partitive quotient fractionconstruct. In addition to these two conditions, the partitioning strategiesalso needed to facilitate direct mappings:

• between the number of people (y) to the fraction name of each share(yths)

• between the number of objects being shared (x) and the number ofyths in each share (x yths).

It was found that if the partitioning strategies did not meet all three condi-tions, then the outcomes of the concrete activity often were the completionof the tasks rather than the building of the partitive quotient construct.

Only two strategies were found to meet all three conditions. Thesewere categorised as Class 1 strategies. Three other classes of partitioningstrategies were generated during the evaluation of the partitioning strategies.Class 2 strategies met the first two conditions, namely that of generatingequal and quantifiable shares. Class 3 strategies only met the conditionof generating equal shares. Class 4 strategies met none of the conditions.These classes can be viewed as being on a continuum of abstract-abilitywith Class 1 demonstrating full abstraction, Classes 2 and 3 demonstratinglesser degrees of abstraction, and Class 4 demonstrating nil abstraction.The taxonomy consisting of four classes of strategies generated by theevaluation of the partitioning strategies is presented in Table II below.


TABLE II

Classes of strategies

Class Strategies in each class Characteristics of each class

Class 1 • Partitive quotientfoundational strategy

• Generates fair shares

• Proceduralised partitive quo-tient strategy

• Accurate quantification ofshares

• Conceptual mapping

Class 2 • Regrouping strategy • Generates fair shares

• People by objects strategy • Accurate quantification ofshares

• Half to each person thenquarter to each personstrategy

• No conceptual mapping

Class 3 • Partition and quantify by part-whole notion strategy

• Generates fair shares

• Halving the objects betweenhalf the people strategy

• Little or no accurate quantific-ation of shares

• Whole to each person thenhalf the remaining objectsbetween half the peoplestrategy


Class 4 • Horizontal partitioningstrategy

• Does not generate fair shares

• Repeated sizing strategy • Little or no accurate quantific-ation of shares

• Repeated halving/repeatedsizing strategy


Class 1 strategies

When children applied Class 1 strategies, they were able to immediatelyabstract the partitive quotient fraction construct from the concrete activityinvolved in the solution of a partitioning sharing task. That is, they wereable to abstract that if a quantity x was partitioned into y equal shares,then the amount in each share was x/y. A close analysis of each of thesestrategies revealed that they met all of the three conditions necessary and


sufficient for the full abstraction of the partitive quotient fraction construct.That is, they:

• generated fair (equal) shares;• facilitated the accurate quantification of each of the shares; and• facilitated direct mappings:

– between the number of people (y) to the fraction name of each share(yths)

– between the number of objects being shared (x) and the number ofyths in each share (x yths)

Class 2 strategies

Although this class of strategies did not facilitate an immediate abstrac-tion of the partitive quotient fraction construct from the concrete activ-ity involved in the solution of a partitioning sharing task, they never-the-less provided very sound springboards for the construction of the partitivequotient fraction construct. When children utilised these strategies, it wasfound that they were very close to abstracting the partitive quotient fractionconstruct. An analysis of each of these strategies revealed that they:

• generated fair (equal) shares; and• facilitated the accurate quantification of each of the shares.

This class of strategies thus enabled the children to effectively and ef-ficiently (if only a small number of people and objects were involved)quantify how much each share would be in a partitioning task.

However, these strategies did not promote conceptual mapping betweenthe concrete activity of partitioning and the partitive quotient fraction con-struct. For example, the ‘regrouping’ and the ‘people by objects’ strategiesdid not prescribe the sharing of 1/y of each object (where y = number ofpeople) to each person. The ‘half to each person then a quarter to eachperson’ strategy when applied to sharing two objects between four peopleprescribed the sharing of 1/2 of one object instead of 1/4 of each object.Therefore, the ‘regrouping’ and the ‘people by objects’ strategies did notaddress the relationship between the number of people, the fraction nameand the number of equal pieces in each whole and the ‘half to each personthen a quarter to each person’ strategy only did this to a limited degree.

Class 3 strategies

This class of strategies enabled the children to generate fair (equal) sharesbut seemed to make the process of abstracting the partitive quotient frac-tion construct from the concrete activity of solving a partitioning task very


difficult. Whenever a child used one of these strategies without the aid oflimited teaching episodes which focused on rebuilding the whole and onhaving the child conceptualise the size of the share before, during and afterthe sharing process (cf., Lamon, 1996), (s)he was unable to go beyond gen-erating equal shares. A close analysis of each of these strategies revealedthey:

• made the process of generating correct quantification of shares verydifficult, and

• did not facilitate direct mappings:


– a conceptual mapping between the number of objects being shared(x) and the number of yths in each share (x yths)

For example, the ‘partitioning and quantifying by part-whole notion’ stra-tegy did not enable children to make the link between number of peopleand fraction name. When children applied this strategy, they partitioned theobject according to the number of people in the sharing situation. However,they did not know the name of each piece and thus were unable to quantifyeach person’s share.

Neither the ‘half the objects between half the people’ strategy nor the‘whole to each person then half remaining objects between half people’strategy fostered the notion that the number of people in the sharing situ-ation can be used to generate the fraction name. Nonetheless, childrenwho employed these two strategies did recognise some kind of relation-ship between the number of people and the number of pieces requiredfor fair sharing. These two strategies represented truncated versions of the‘partitive quotient foundational strategy’ (but not truncated to the degreedemonstrated by the ‘proceduralised partitive quotient strategy’). In ap-plying these truncated strategies, children minimised the number of cutsand therefore, the number of pieces in each whole. When applying thesetwo strategies, part-whole notions were sometimes misconstrued and thechildren therefore experienced difficulties quantifying each piece and eachperson’s share.

For children who utilised the ‘halving the object then halving againand again strategy’, the number of people in the sharing situation was notsalient to the partitioning process. Children who employed this strategyalways adopted a ‘repeated halving’ strategy regardless of the number ofpeople in the sharing situation. This strategy was only ever efficient whenthe sharing situation involved an even number of people.


Figure 5. Fraction name taken from total number of pieces in total number of analogobjects.

Figure 6. Fraction name taken from total number of pieces in total number of analogobjects.

Children applying Class 3 strategies regularly suffered ‘loss of whole’.That is, when attempting to quantify each person’s share, they were notthinking in terms of the number of pieces in each one whole unit, but ratherin terms of: (i) the number of pieces in all the units; or (ii) the numberof pieces in each half of a unit. For instance, if two objects were eachpartitioned into four pieces, the total number of pieces would be eight;therefore the fraction name would be eighths (Figure 5). Similarly, if two


Figure 7. Half the object is assumed to be the whole therefore fraction name is taken fromthe number of pieces in each half object.

objects were each partitioned in half, there would be four pieces altogether,therefore each half would be seen as 1/4 (Figure 6). Alternatively, if anobject was partitioned into eighths and shared among four people, eachperson would receive one piece from each of the four pieces in each half.Each half was seen as a whole, therefore each piece was considered to be1/4 instead of 1/8 (Figure 7).

CLASS 4 STRATEGIES

This class of strategies made the process of abstracting the partitive quo-tient fraction construct from the concrete activity of solving a partitioningtask impossible. Whenever a child used one of these strategies, (s)he wasunable to even generate fair (equal) shares. Furthermore, unlike the casewith Class 3 strategies, the cognitive impasses associated with this classof strategies were found not to be amenable to limited teaching episodeswhich focused on rebuilding the whole and on having the child concep-tualise the size of the share before, during and after the sharing process.A close analysis of each of these strategies revealed why. Each of thesestrategies:

• produced unequal shares,• produced unquantifiable shares, and• counteracted the process of direct mapping.



– between the number of objects being shared (x) and the number ofyths in each share (x yth)

DISCUSSION

The overall aim of this study was to increase our knowledge about youngchildren’s partitioning strategies by setting out not only to identify newpartitioning strategies but also to develop a taxonomy for classifying youngchildren’s partitioning strategies in terms of their ability to facilitate theabstraction of the partitive quotient fraction construct from the concreteactivity of partitioning objects and/or sets of objects.

Six hitherto reported and six hitherto unreported partitioning strategieswere identified. During the course of the study, it was found that eachchild tended to use a variety of partitioning strategies when confrontedby partitioning problems. Thus, in addition to identifying six hitherto un-reported partitioning strategies, this study also confirmed one of the moreimportant findings from previous research studies, namely that young chil-dren’s selection of partitioning strategies depends not only on their priorknowledge and experiences but also on the context of the task, the type ofanalog objects being shared, the number of analog objects being shared andnumber of shares (cf., Lamon, 1996; Pitkethly and Hunting, 1996; Pothierand Sawada, 1983; Streefland, 1991).

However, the major outcome of this study was the development ofthe taxonomy. Unlike previous taxonomies, this taxonomy provides re-searchers and teachers with the means for utilising, in more systematicand effective ways, young children’s informal partitioning strategies as thefoundation upon which to develop the young children’s understanding ofthe partitive quotient fraction construct. It does this by providing research-ers and teachers with a framework for: qualitatively evaluating a child’sprogress towards the abstraction of the partitive quotient fraction construct;and planning and implementing teaching interventions commensurate withthe child’s level of progress towards the abstraction of the partitive quotientfraction construct.

For example, children who utilise Class 1 strategies can be evaluated ashaving constructed advanced conceptions of the partitive quotient fractionconstruct. Very little teaching would be required to facilitate the processof consolidating and generalising the partitive quotient fraction construct.This would entail: (i) the sequencing of tasks and activities to consolidate


the conceptual mapping of the partitive quotient fraction construct fromthe concrete activities, and (ii) the presentation of more complex inter-related (number of people by number of objects) tasks to generalise thepartitive quotient fraction construct from the concrete activities throughthe application of the proceduralised partitive quotient strategy (cf. La-mon, 1996). It would be anticipated that such teaching interventions wouldextend children’s informal knowledge structures of fractions (cf., Mack,1990).

In a similar manner, children who utilise Class 2 strategies can be eval-uated as having constructed sound foundations for the acquisition of thepartitive quotient fraction construct. However, they have not as yet madethe crucial process of constructing mappings between:

• the number of people (y) to the fraction name of each share (yths)• the number of objects being shared (x) and the number of yths in each

share (x yths).

Therefore, the findings from this study seem to indicate that teaching in-terventions such as the use of questions during the course of partitioningtasks which focus the children’s attention on establishing conceptual linksbetween the quantity being shared, the number of shares and the quantityin each share would be most appropriate.

Similarly, children utilising Class 3 strategies can be evaluated as beingless advanced in the process of constructing an understanding of the par-titive quotient fraction construct. Because of the problems which childrenapplying Class 3 strategies have with accurately quantifying the shares, itseems that initial learning activities need to focus on:

• rebuilding the whole (cf., Lamon, 1996) and other part/whole notions;and

• constructing an understanding of the relationship between partition-ing and unitising (where unitising is a cognitive process for concep-tualising the size of a share before, during and after sharing).

Children who adopt Class 4 strategies can be evaluated as being the leastadvanced in the process of constructing an understanding of the partitivequotient fraction construct. These children would require extensive teach-ing interventions in which they would be extensively exposed to presym-bolic experiences to develop what Lamon (1996) refers to as “a flexibleconcept of the unit and a firm foundation for quantification” (p. 192).Based on the findings from Steffe and Olive (1993) and Charles, Nasonand Cooper (1999), it seems that these initial experiences should be con-ducted with continuous analog objects which are easy to partition such aslength models and long, thin rectangular region models. The findings from


this study and from Ball (1993) also indicate that the initial focus of thelearning activities for these children should involve sharing among two orfour people rather than three or five people.

In conclusion, it is important to emphasise that the application of thetaxonomy in the planning and implementation of learning activities com-mensurate with a child’s level of progress towards the construction of anunderstanding of the partitive quotient fraction construct was not a majorfocus of this research study. Therefore, in addition to investigating theefficacy of the taxonomy for evaluating a child’s progress towards theabstraction of the partitive quotient fraction construct, our claims abouthow the taxonomy can be applied in the planning and implementationof learning activities also need to be investigated in future research andteaching studies.

During the course of this study, we often found that the level of strategiesutilised by the children were not consistent across all of the analog objects.For example, many of the children reverted to Class 3 strategies whenconfronted with the task of partitioning a rectangular cake or a circularregion analog object immediately after they successfully had used Class1 or 2 strategies on tasks involving easier to partition analog objects suchas long narrow rectangles and length models (see Charles et al. (1999) fora more detailed discussion of this phenomenon). This is consistent withfindings from Lamon (1996), Pothier and Sawada (1983) and Streefland(1991) that young children’s selection of partitioning strategies dependson the context of the task and the type of analog objects being shared andwith findings from Clough and Driver (1986), Noss and Hoyles (1996) andTirosh (1990) that students tend to respond inconsistently to tasks relatedto the very same mathematical or scientific concept.

Therefore, another issue which also needs to be investigated in futureresearch and teaching studies is the relationship between concrete analogobjects and the emergence of classes of partitioning strategies. The find-ings from this line of investigation could have important implications forthe future application of the taxonomy in the planning and implementa-tion of learning activities which build on children’s intuitive partitioningstrategies and also for the sequencing of analog objects during programsof instruction. This would prepare the way for future research studies inwhich teachers would be able to utilise, in more systematic and effectiveways than is possible with types of taxonomies developed by previousresearchers in this field, children’s intuitive partitioning strategies duringthe teaching of the partitive quotient fraction construct.


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Centre for Mathematics and Science Education,Queensland University of Technology,Victoria Park Road,Kelvin Grove Q 4059,Brisbane Queensland Australia

KATHY CHARLES

Editorial Assistant: Mathematical Thinking and Learning,Telephone: 617 3864 3441; Fax: 617 3864 3643,E-mail: [email protected],http://www.fed.qut.edu.au/charles,http://www.fed.qut.edu.au/mtl

ROD NASON

Senior Lecturer,Telephone: 617 3864 5556; Fax: 617 3864 3643,E-mail: [email protected],Http://www.fed.qut.edu.au/nason

young children's partitioning strategies

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