paper 3: understanding rational numbers and intensive quantities

Key understandings in

mathematics learning

A review commissioned by the Nuffield Foundation

3

Paper 3: Understanding rational numbers and intensive quantitiesBy Terezinha Nunes and Peter Bryant, University of Oxford

2 SUMMARY – PAPER 2: Understanding whole numbers2 Paper 3: Understanding rational numbers and intensive quantities

In 2007, the Nuffield Foundation commissioned ateam from the University of Oxford to review theavailable research literature on how children learnmathematics. The resulting review is presented in aseries of eight papers:

Paper 1: OverviewPaper 2: Understanding extensive quantities and

whole numbersPaper 3: Understanding rational numbers and

intensive quantitiesPaper 4: Understanding relations and their graphical

representationPaper 5: Understanding space and its representation

in mathematicsPaper 6: Algebraic reasoningPaper 7: Modelling, problem-solving and integrating

conceptsPaper 8: Methodological appendix

Papers 2 to 5 focus mainly on mathematics relevantto primary schools (pupils to age 11 years), whilepapers 6 and 7 consider aspects of mathematics in secondary schools.

Paper 1 includes a summary of the review, which has been published separately as Introduction andsummary of findings.

Summaries of papers 1-7 have been publishedtogether as Summary papers.

All publications are available to download from our website, www.nuffieldfoundation.org

ContentsSummary of Paper 3 3

Understanding rational numbers and intensive quantities 7

References 28

About the authorsTerezinha Nunes is Professor of EducationalStudies at the University of Oxford. Peter Bryant is Senior Research Fellow in theDepartment of Education, University of Oxford.

About the Nuffield FoundationThe Nuffield Foundation is an endowedcharitable trust established in 1943 by WilliamMorris (Lord Nuffield), the founder of MorrisMotors, with the aim of advancing social wellbeing. We fund research and practicalexperiment and the development of capacity to undertake them; working across education,science, social science and social policy. Whilemost of the Foundation’s expenditure is onresponsive grant programmes we also undertake our own initiatives.

About this review

3 Key understandings in mathematics learning

Headlines• Fractions are used in primary school to represent

quantities that cannot be represented by a singlewhole number. As with whole numbers, childrenneed to make connections between quantities andtheir representations in fractions in order to beable to use fractions meaningfully.

• There are two types of situation in which fractionsare used in primary school. The first involvesmeasurement: if you want to represent a quantityby means of a number and the quantity is smallerthan the unit of measurement, you need a fraction– for example, a half cup or a quarter inch. Thesecond involves division: if the dividend is smallerthan the divisor, the result of the division isrepresented by a fraction. For example, when youshare 3 cakes among 4 children, each child receives¾ of a cake.

• Children use different schemes of action in thesetwo different situations. In division situations, theyuse correspondences between the units in thenumerator and the units in the denominator. Inmeasurement situations, they use partitioning.

• Children are more successful in understandingequivalence of fractions and in ordering fractions by magnitude in situations that involve division thanin measurement situations.

• It is crucial for children’s understanding of fractionsthat they learn about fractions in both types ofsituation: most do not spontaneously transfer whatthey learned in one situation to the other.

• When a fraction is used to represent a quantity,children need to learn to think about how the

numerator and the denominator relate to the value represented by the fraction. They must thinkabout direct and inverse relations: the larger thenumerator, the larger the quantity but the largerthe denominator, the smaller the quantity.

• Like whole numbers, fractions can be used torepresent quantities and relations betweenquantities, but in primary school they are rarelyused to represent relations. Older students often find it difficult to use fractions to represent relations.

There is little doubt that students find fractions achallenge in mathematics. Teachers often say that itis difficult to teach fractions and some think that itwould be better for everyone if children were nottaught about fractions in primary school. In orderto understand fractions as numbers, students mustbe able to know whether two fractions areequivalent or not, and if they are not, which one isthe bigger number. This is similar to understandingthat 8 sweets is the same number as 8 marbles andthat 8 is more than 7 and less than 9, for example.These are undoubtedly key understandings aboutwhole numbers and fractions. But even after theage of 11 many students have difficulty in knowingwhether two fractions are equivalent and do notknow how to order some fractions. For example, ina study carried out in London, students were askedto paint 2/3 of figures divided in 3, 6 and 9 equalparts. The majority solved the task correctly whenthe figure was divided into 3 parts but 40% of the11- to 12-year-old students could not solve itwhen the figure was divided into 6 or 9 parts,which meant painting an equivalent fraction (4/6 and 6/9, respectively).

Summary of paper 3: Understanding rational numbers andintensive quantities

4 SUMMARY – PAPER 2: Understanding whole numbers

Fractions are used in primary school to representquantities that cannot be represented by a singlewhole number. If the teaching of fractions were to beomitted from the primary school curriculum, childrenwould not have the support of school learning torepresent these quantities. We do not believe that itwould be best to just forget about teaching fractionsin primary school because research shows thatchildren have some informal knowledge that could beused as a basis for learning fractions. Thus the questionis not whether to teach fractions in primary schoolbut what do we know about their informal knowledgeand how can teachers draw on this knowledge.

There are two types of situation in which fractionsare used in primary school: measurement anddivision situations.

When we measure anything, we use a unit ofmeasurement. Often the object we are measuringcannot be described only with whole units, and weneed fractions to represent a part of the unit. In thekitchen we might need to use a ½ cup of milk andwhen setting the margins for a page in a documentwe often need to be precise and define the marginas, for example, as 3.17 cm. These two examplesshow that, when it comes to measurement, we use two types of notation, ordinary and decimalnotation. But regardless of the notation used, wecould not accurately describe the quantities in thesesituations without using fractions. When we speak of ¾ of a chocolate bar, we are using fractions in ameasurement situation: we have less than one unit,so we need to describe the quantity using a fraction.

In division situations, we need a fraction to represent aquantity when the dividend is smaller than the divisor.For example, if 3 cakes are shared among 4 children, itis not possible for each one to have a whole cake, butit is still possible to carry out the division and torepresent the amount that each child receives using a number, ¾. It would be possible to use decimalnotation in division situations too, but this is rarely thecase. The reason for preferring ordinary fractions inthese situations is that there are two quantities indivision situations: in the example, the number of cakes and the number of children. An ordinary fraction represents each of these quantities by a wholenumber: the dividend is represented by the numerator,the divisor by the denominator, and the operation ofdivision by the dash between the two numbers.

Although these situations are so similar for adults, wecould conclude that it is not necessary to distinguish

between them, however, research shows thatchildren think about the situations differently.Children use different schemes of action in each of these situations.

In measurement situations, they use partitioning. If a child is asked to show ¾ of a chocolate, the childwill try to cut the chocolate in 4 equal parts andmark 3 parts. If a child is asked to compare ¾ and6/8, for example, the child will partition one unit in 4 parts, the other in 8 parts, and try to compare thetwo. This is a difficult task because the partitioningscheme develops over a long period of time andchildren have to solve many problems to succeed in obtaining equal parts when partitioning. Althoughpartitioning and comparing the parts is not the onlyway to solve this problem, this is the most likelysolution path tried out by children, because theydraw on their relevant scheme of action.

In division situations, children use a different schemeof action, correspondences. A problem analogous tothe one above in a division situation is: there are 4children sharing 3 cakes and 8 children sharing 6identical cakes; if the two groups share the cakesfairly, will the children in one group get the sameamount to eat as the children in the other group?Primary school pupils often approach this problemby establishing correspondences between cakes andchildren. In this way they soon realise that in bothgroups 3 cakes will be shared by 4 children; thedifference is that in the second group there are twolots of 3 cakes and two lots of 4 children, but thisdifference does not affect how much each child gets.

From the beginning of primary school, many childrenhave some informal knowledge about division thatcould be used to understand fractional quantities.Between the ages of five and seven years, they arevery bad at partitioning wholes into equal parts but can be relatively good at thinking about theconsequences of sharing. For example, in one studyin London 31% of the five-year-olds, 50% of the six-year-olds and 81% of the seven-year-olds understoodthe inverse relation between the divisor and theshares resulting from the division: they knew that themore recipients are sharing a cake, the less each onewill receive. They were even able to articulate thisinverse relation when asked to justify their answers.It is unlikely that they had at this time made aconnection between their understanding ofquantities and fractional representation; actually, it isunlikely that they would know how to represent thequantities using fractions.

4 Paper 3: Understanding rational numbers and intensive quantities

The lack of connection between students’understanding of quantities in division situations andtheir knowledge about the magnitude of fractions isvery clearly documented in research. Students whohave no doubt that recipients of a cake sharedbetween 3 people will fare better than those of acake shared between 5 people may, nevertheless, saythat 1/5 is a bigger fraction than 1/3 because 5 is abigger number than 3. Although they understand theinverse relation in the magnitude of quantities in adivision situation, they do not seem to connect thiswith the magnitude of fractions. The link betweentheir understanding of fractional quantities andfractions as numbers has to be developed throughteaching in school.

There is only one well-controlled experiment whichcompared directly young children’s understanding ofquantities in measurement and division situations. Inthis study, carried out in Portugal, the children weresix- to seven-years-old. The context of the problemsin both situations was very similar : it was aboutchildren eating cakes, chocolates or pizzas. In themeasurement problems, there was no sharing, only partitioning. For example, in one of themeasurement problems, one girl had a chocolate barwhich was too large to eat in one go. So she cut herchocolate in 3 equal pieces and ate 1. A boy had anidentical bar of chocolate and decided to cut his into6 equal parts, and eat 2. The children were askedwhether the boy and the girl ate the same amountof chocolate. The analogous division problem wasabout 3 girls sharing one chocolate bar and 6 boyssharing 2 identical chocolate bars. The rate of correctresponses in the partitioning situation was 10% forboth six- and seven-year-olds and 35% and 49%,respectively, for six- and seven-year-olds in thedivision situation.

These results are relevant to the assessment ofvariations in mathematics curricula. Differentcountries use different approaches in the initialteaching of fraction, some starting from division and others from measurement situations. There is no direct evidence from classroom studies to show whether one starting point results in higherachievement in fractions than the other. The scarceevidence from controlled studies supports the ideathat division situations provide children with moreinsight into the equivalence and order of quantitiesrepresented by fractions and that they can learn howto connect these insights about quantities withfractional representation. The studies also indicatethat there is little transfer across situations: children

who succeed in comparing fractional quantities andfractions after instruction in division situations do no better in a post-test when the questions areabout measurement situations than other children in a control group who received no teaching. Theconverse is also true: children taught in measurementsituations do no better than a control group indivision situations.

A major debate in mathematics teaching is the relative weight to be given to conceptualunderstanding and procedural knowledge in teaching.The difference between conceptual understandingand procedural knowledge in the teaching offractions has been explored in many studies. Thesestudies show that students can learn procedureswithout understanding their conceptual significance.Studies with adults show that knowledge ofprocedures can remain isolated from understandingfor a long time: some adults who are able toimplement the procedure they learned for dividingone fraction by another admit that they have no ideawhy the numerator and the denominator exchangeplaces in this procedure. Learners who are able toco-ordinate their knowledge of procedures withtheir conceptual understanding are better at solvingproblems that involve fractions than other learnerswho seem to be good at procedures but show lessunderstanding than expected from their knowledgeof procedures. These results reinforce the idea that itis very important to try to make links betweenchildren’s knowledge of fractions and theirunderstanding of fractional quantities.

Finally, there is little, if any, use of fractions torepresent relations between quantities in primaryschool. Secondary school students do not easilyquantify relations that involve fractions. Perhaps this difficulty could be attenuated if some teachingabout fractions in primary school involvedquantifying relations that cannot be described by a single whole number.


6 SUMMARY – PAPER 2: Understanding whole numbers6 Paper 3: Understanding rational numbers and intensive quantities

Recommendations

Research about mathematicallearning

Children’s knowledge of fractionalquantities starts to develop before theyare taught about fractions in school.

There are two types of situation relevantto primary school teaching in whichquantities cannot be represented by asingle whole number: measurement anddivision.

Children do not easily transfer theirunderstanding of fractions from division tomeasurement situations and vice-versa.

Many students do no make links betweentheir conceptual understanding of fractionsand the procedures that they are taughtto compare and operate on fractions inschool.

Fractions are taught in primary school onlyas representations of quantities.

Recommendations for teaching and research

Teaching Teachers should be aware of children’s insightsregarding quantities that are represented by fractions andmake connections between their understanding of thesequantities and fractions.

Teaching The primary school curriculum should include thestudy of both types of situation in the teaching of fractions.Teachers should be aware of the different types of reasoningused by children in each of these situations.Research Evidence from experimental studies with largersamples and long-term interventions in the classroom areneeded to establish whether division situations are indeed a better starting point for teaching fractions.

Teaching Teachers should consider how to establish linksbetween children’s understanding of fractions in division andmeasurement situations.Research Investigations on how links between situations canbe built are needed to support curriculum development andclassroom teaching.

Teaching Greater attention may be required in the teachingof fractions to creating links between procedures andconceptual understanding.Research There is a need for longitudinal studies designed to clarify whether this separation between procedural andconceptual knowledge does have important consequences forfurther mathematics learning.

Teaching Consideration should be given to the inclusion ofsituations in which fractions are used to represent relations.Research Given the importance of understanding andrepresenting relations numerically, studies that investigateunder what circumstances primary school students can usefractions to represent relations between quantities areurgently needed.

IntroductionRational numbers, like natural numbers, can be usedto represent quantities. There are some quantitiesthat cannot be represented by a natural number, andto represent these quantities, we must use rationalnumbers. We cannot use natural numbers when thequantity that we want to represent numerically:• is smaller than the unit used for counting,

irrespective of whether this is a natural unit (e.g. we have less than one banana) or aconventional unit (e.g. a fish weighs less than a kilo)

• involves a ratio between two other quantities (e.g.the concentration of orange juice in a jar can bedescribed by the ratio of orange concentrate towater ; the probability of an event can be describedby the ratio between the number of favourablecases to the total number of cases).1

The term ‘fraction’ is often identified with situationswhere we want to represent a quantity smaller thanthe unit. The expression ‘rational number’ usuallycovers both sorts of examples. In this paper, we willuse the expressions ‘fraction’ and ‘rational number’interchangeably. Fractions are considered a basicconcept in mathematics learning and one of thefoundations required for learning algebra (Fennell,Faulkner, Ma, Schmid, Stotsky, Wu et al. (2008); sothey are important for representing quantities andalso for later success in mathematics in school.

In the domain of whole numbers, it has been knownfor some time (e.g. Carpenter and Moser, 1982;Ginsburg, 1977; Riley, Greeno and Heller, 1983) thatit is important for the development of children’smathematics knowledge that they establishconnections between the numbers and thequantities that they represent. There is littlecomparable research about rational numbers

(but see Mack, 1990), but it is reasonable to expectthat the same hypothesis holds: children should learnto connect quantities that must be represented byrational numbers with their mathematical notation.However, the difficulty of learning to use rationalnumbers is much greater than the difficulty oflearning to use natural numbers. This paper discusseswhy this is so and presents research that showswhen and how children have significant insights intothe complexities of rational numbers.

In the first section of this paper, we discuss whatchildren must learn about rational numbers and whythese might be difficult for children once they havelearned about natural numbers. In the second sectionwe describe research which shows that these areindeed difficult ideas for students even at the end ofprimary school. The third section compares children’sreasoning across two types of situations that havebeen used in different countries to teach childrenabout fractions. The fourth section presents a briefoverview of research about children’s understandingof intensive quantities. The fifth section considerswhether children develop sound understanding ofequivalence and order of magnitude of fractionswhen they learn procedures to compare fractions.The final section summarises our conclusions anddiscusses their educational implications.

What children must know inorder to understand rationalnumbers

Piaget’s (1952) studies of children’s understanding of number analysed the crucial question of whether


Understanding rationalnumbers and intensivequantities


young children can understand the ideas ofequivalence (cardinal number) and order (ordinalnumber) in the domain of natural numbers. He alsopointed out that learning to count may help thechildren to understand both equivalence and order.All sets that are represented by the same numberare equivalent; those that are represented by adifferent number are not equivalent. Their order ofmagnitude is the same as the order of the numberlabels we use in counting, because each number labelrepresents one more than the previous one in thecounting string.

The understanding of equivalence in the domain of fractions is also crucial, but it is not as simplebecause language does not help the children in the same way. Two fractional quantities that havedifferent labels can be equivalent, and in fact there is an infinite number of equivalent fractions: 1/3, 2/6,6/9, 8/12 etc. are different number labels but theyrepresent equivalent quantities. Because rationalnumbers refer, although often implicitly, to a whole, itis also possible for two fractions that have the samenumber label to represent different quantities: 1/3 of12 and 1/3 of 18 are not representations ofequivalent quantities.

In an analogous way, it is not possible simply totransfer knowledge of order from natural to rationalnumbers. If the common fraction notation is used,there are two numbers, the numerator and thedenominator, and both affect the order of magnitudeof fractions, but they do so in different ways. If thedenominator is constant, the larger the numerator,the larger is the magnitude of the fraction; if thenumerator is constant, the larger the denominator,the smaller is the fraction. If both vary, then moreknowledge is required to order the fractions, and it is not possible to tell which quantity is more bysimply looking at the fraction labels.

Rational numbers differ from whole numbers also inthe use of two numerical signs to represent a singlequantity: it is the relation between the numbers, nottheir independent values, that represents thequantity. Stafylidou and Vosniadou (2004) analysedGreek students’ understanding of this form ofnumerical representation and observed that moststudents in the age range 11 to 13 years did notseem to interpret the written representation offractions as involving a multiplicative relationbetween the numerator and the denominator: 20% of the 11-year-olds, 37% of the 12-year-olds and 48% of the 13-year-olds provided this type of

interpretation for fractions. Many younger students(about 38% of the 10-year-olds in grade 5) seemedto treat the numerator and denominator asindependent numbers whereas others (about 20%)were able to conceive fractions as indicating a part-whole relation but many (22%) are unable to offer a clear explanation for how to interpret thenumerator and the denominator.

Rational numbers are also different from naturalnumbers in their density (see, for example,Brousseau, Brousseau and Warfield, 2007;Vamvakoussi and Vosniadou, 2004): there are nonatural numbers between 1 and 2, for example, but there is an infinite number of fractions between1 and 2. This may seem unimportant but it is thisdifference that allows us to use rational numbers torepresent quantities that are smaller than the units.This may be another source of difficulty for students.

Rational numbers have another property which isnot shared by natural numbers: every non-zerorational number has a multiplicative inverse (e.g. theinverse of 2/3 is 3/2). This property may seemunimportant when children are taught aboutfractions in primary school, but it is important forthe understanding of the division algorithm (i.e. wemultiply the fraction which is the dividend by theinverse of the fraction that is the divisor) and will berequired later in school, when students learn aboutalgebra. Booth (1981) suggested that students oftenhave a limited understanding of inverse relations,particularly in the domain of fractions, and thisbecomes an obstacle to their understanding ofalgebra. For example, when students think offractions as representing the number of parts intowhich a whole was cut (denominator) and thenumber of parts taken (numerator), they find it verydifficult to think that fractions indicate a division andthat it has, therefore, an inverse.

Finally, rational numbers have two common writtennotations, which students should learn to connect: 1/2and 0.5 are conceptually the same number with twodifferent notations. There isn’t a similar variation innatural number notation (Roman numerals aresometimes used in specific contexts, such as clocksand indices, but they probably play little role in thedevelopment of children’s mathematical knowledge).Vergnaud (1997) hypothesized that differentnotations afford the understanding of differentaspects of the same concept; this would imply thatstudents should learn to use both notations forrational numbers. On the one hand, the common


fractional notation 1/2 can be used to help studentsunderstand that fractions are related to the operationof division, because this notation can be interpretedas ‘1 divided by 2’. The connection between fractionsand division is certainly less explicit when the decimalnotation 0.5 is used. It is reasonable to expect thatstudents will find it more difficult to understand whatthe multiplicative inverse of 0.5 is than the inverse of1/2, but unfortunately there seems to be no evidenceyet to clarify this.

On the other hand, adding 1/2 and 3/10 is acumbersome process, whereas adding the samenumbers in their decimal representation, 0.5 and 0.3,is a simpler matter. There are disagreements regardingthe order in which these notations should be taughtand the need for students to learn both notations inprimary school (see, for example, Brousseau,Brousseau and Warfield, 2004; 2007), but, to ourknowledge, no one has proposed that one notationshould be the only one used and that the other oneshould be banned from mathematics classes. There isno evidence on whether children find it easier tounderstand the concepts related to rational numberswhen one notation is used rather than the other.

Students’ difficulties with rational numbersMany studies have documented students’ difficultiesboth with understanding equivalence and order ofmagnitude in the domain of rational numbers (e.g.Behr, Harel, Post and Lesh, 1992; Behr, Wachsmuth,Post and Lesh, 1984; Hart, 1986; Hart, Brown,Kerslake, Küchermann and Ruddock, 1985; Kamiiand Clark, 1995; Kerslake, 1986). We illustrate herethese difficulties with research carried out in theUnited Kingdom.

The difficulty of equivalence questions varies acrosstypes of tasks. Kerslake (1986) noted that whenstudents are given diagrams in which the same shapesare divided into different numbers of sections andasked to compare two fractions, this task is relativelysimple because it is possible to use a perceptualcomparison. However, if students are given a diagramwith six or nine divisions and asked to mark 2/3 ofthe shape, a large proportion of them fail to markthe equivalent fractions, 4/6 and 6/9. Hart, Brown,Kerslake, Küchermann and Ruddock (1985), workingwith a sample of students (N =55) in the age range11 to 13 years, found that about 60% of the 11- to12-year-olds and about 65% of the 12- to 13-year-

olds were able to solve this task. We (Nunes, Bryant,Pretzlik and Hurry, 2006) gave the same item morerecently to a sample of 130 primary school studentsin Years 4 and 5 (mean ages, respectively, 8.6 and 9.6years). The rate of correct responses across theseitems was 28% for the children in Year 4 and 49% forthe children in Year 5. This low percentage of correctanswers could not be explained by a lack ofknowledge of the fraction 2/3: when the diagram wasdivided into three sections, 93% of the students inthe study by Hart el al. (1985) gave a correct answer;in our study, 78% of the Year 4 and 91% of the Year 5students’ correctly shaded 2/3 of the figure.

This quantitative information is presented here toillustrate the level of difficulty of these questions. A different approach to the analysis of how the levelof difficulty can vary is presented later, in the thirdsection of this paper.

Students often have difficulty in ordering fractionsaccording to their magnitude. Hart et al. (1985) askedstudents to compare two fractions with the samedenominator (3/7 and 5/7) and two with the samenumerator (3/5 and 3/4). When the fractions have thesame denominator, students can respond correctly byconsidering the numerators only and ordering them asif they were natural numbers. The rate of correctresponses in this case is relatively high but it does noteffectively test students’ understanding of rationalnumbers. Hart et al. (1985) observed approximately90% correct responses among their students in theage range 11 to 13 years and we (Nunes et al., 2006)found that 94% of the students in Year 4 and 87% ofthe students in Year 5 gave correct responses. Incontrast, when the numerator was the same and thedenominator varied (comparing 3/5 and 3/4), and thestudents had to consider the value of the fractions in away that is not in agreement with the order of naturalnumbers, the rate of correct responses wasconsiderably lower: in the study by Hart et al.,approximately 70% of the answers were correct,whereas in our study the percent of correct responseswere 25% in Year 4 and 70% among in Year 5.

These difficulties are not particular to U.K. students:they have been widely reported in the literature onequivalence and order of fractions (for examples inthe United States see Behr, Lesh, Post and Silver, 1983;Behr, Wachsmuth, Post and Lesh 1984; Kouba, Brown,Carpenter, Lindquist, Silver and Swafford, 1988).

Difficulties in comparing rational numbers are notconfined to fractions. Resnick, Nesher, Leonard,



Magone, Omanson and Peled (1989) have shown that students have difficulties in comparing decimalfractions when the number of places after the decimalpoint differs. The samples in their study were relativelysmall (varying from 17 to 38) but included studentsfrom three different countries, the United States, Israeland France, and in three grade levels (4th to 6th). Thechildren were asked to compare pairs of decimalssuch as 0.5 and 0.36, 2.35 and 2.350, and 4.8 and 4.63.The rate of correct responses varied between 36%and 52% correct, even though all students hadreceived instruction on decimals. A more recent study(Lachance and Confrey, 2002) of 5th grade students(estimated age approximately 10 years) who hadreceived an introduction to decimal fractions in theprevious year showed that only about 43% were ableto compare decimal fractions correctly.2 Rittle-Johnson,Siegler and Alibali (2001) confirmed students’difficulties when comparing the magnitude of decimals: the rate of correct responses by thestudents (N = 73; 5th grade; mean age 11 years 8months) in their study was 19%.

In conclusion, the very basic ideas about equivalenceand order of fractions by magnitude, without which wecould hardly say that the students have a good sensefor what fractions represent, seems to elude manystudents for considerable periods of time. In thesection that follows, we will contrast two situationsthat have been used to introduce the concept offractions in primary school in order to examine thequestion of whether children’s learning may differ as a function of these differences between situations.

Children’s schemas of action in division situationsMathematics educators and researchers may notagree on many things, but there is a clear consensusamong them on the idea that rational numbers arenumbers in the domain of quotients (Brousseau,Brousseau and Warfield, 2007; Kieren, 1988; 1993;1994; Ohlsson, 1988): that is, numbers defined by theoperation of division. So, it seems reasonable to seekthe origin of children’s understanding of rationalnumbers in their understanding of division.3

Our hypothesis is that in division situations childrencan develop some insight into the equivalence andorder of quantities in fractions; we will use the termfractional quantities to refer to these quantities.These insights can be developed even in theabsence of knowledge of representations for fractions,

either in written or in oral form. Two schemes of action that children use in division have beenanalysed in the literature: partitioning andcorrespondences (or dealing).

Behr, Harel, Post and Lesh (1992; 1993) pointed outthat fractions represent quantitites in a different wayacross two types of situation. The first type is the part-whole situation. Here one starts with a single quantity,the whole, which is divided into a certain number ofparts (y), out of which a specified number is taken (x);the symbol x/y represents this quantity in terms ofpart-whole relations. Partitioning is the scheme ofaction that children use in part-whole tasks. The mostcommon type of fraction problem that teachers giveto children is to ask them to partition a whole into afixed number of parts (the denominator) and show acertain fraction with this denominator. For example,the children have to show what 3/5 of a pizza is.4

The second way in which fractions representquantities is in quotient situations. Here one startswith two quantities, x and y, and treats x as thedividend and y as the divisor, and by the operation ofdivision obtains a single quantity x/y. For example, thequantities could be 3 chocolates (x) to be sharedamong 5 children (y).The fractional symbol x/yrepresents both the division (3 divided by 5) and thequantity that each one will receive (3/5). A quotientsituation calls for the use of correspondences as thescheme of action: the children establishcorrespondences between portions and recipients.The portions may be imagined by the children, notactually drawn, as they must be when the childrenare asked to partition a whole and show 3/5.5

When children use the scheme of partitioning inpart-whole situations, they can gain insights aboutquantities that could help them understand someprinciples relevant to the domain of rational numbers.They can, for example, reason that, the more partsthey cut the whole into, the smaller the parts will be.This could help them understand how fractions areordered. If they can achieve a higher level of precisionin reasoning about partitioning, they could develop some understanding of theequivalence of fractions: they could come tounderstand that, if they have twice as many parts,each part would be halved in size. For example, youwould eat the same amount of chocolate aftercutting one chocolate bar into two parts and eatingone part as after cutting it into four parts and eatingtwo, because the number of parts and the size of theparts compensate for each other precisely. It is an


empirical question whether children attain theseunderstandings in the domain of whole numbers andextend them to rational numbers.

Partitioning is the scheme that is most often used to introduce children to fractions in the UnitedKingdom, but it is not the only scheme of actionrelevant to division. Children use correspondencesin quotient situations when the dividend is onequantity (or measure) and the divisor is anotherquantity. For example, when children share outchocolate bars to a number of recipients, thedividend is in one domain of measures – thenumber of chocolate bars – and the divisor is inanother domain – the number of children. Thedifference between partitioning and correspondencedivision is that in partitioning there is a single whole(i.e. quantity or measure) and in correspondencethere are two quantities (or measures).

Fischbein, Deri, Nello and Marino (1985) hypothesisedthat children develop implicit models of divisionsituations that are related to their experiences. Weuse their hypothesis here to explore what sorts ofimplicit models of fractions children may develop fromusing the partitioning or the correspondence schemein fractions situations. Fischbein and colleaguessuggested, for example, that children form an implicitmodel of division that has a specific constraint: thedividend must be larger than the divisor. We ourselveshypothesise that this implicit model is developed onlyin the context of partitioning. When children use thecorrespondence scheme, precisely because there aretwo domains of measures, young children readilyaccept that the dividend can be smaller than thedivisor: they are ready to agree that it is perfectlypossible to share one chocolate bar among three children.

At first glance, the difference between these twoschemes of action, partitioning and correspondence,may seem too subtle to be of interest when we arethinking of children’s understanding of fractions.Certainly, research on children’s understanding offractions has not focused on this distinction so far.However, our review shows that it is a crucialdistinction for children’s learning, both in terms of what insights each scheme of action affords and in terms of the empirical research results.

There are at least four differences between whatchildren might learn from using the partitioningscheme or the scheme of correspondences.

• The first is the one just pointed out: that, whenchildren set two measures in correspondence, thereis no necessary relation between the size of thedividend and of the divisor. In contrast, in partitioningchildren form the implicit model that the sum of theparts must not be larger than the whole. Therefore,it may be easier for children to develop anunderstanding of improper fractions when they formcorrespondences between two fields of measuresthan when they partition a single whole. They mighthave no difficulty in understanding that 3 chocolatesshared between 2 children means that each childcould get one chocolate plus a half. In contrast, inpartitioning situations children might be puzzled ifthey are told that someone ate 3 parts of achocolate divided in 2 parts.

• A second possible difference between the twoschemes of action may be that, when usingcorrespondences, children can reach the conclusionthat the way in which partitioning is carried out doesnot matter, as long as the correspondences betweenthe two measures are ‘fair’. They can reason, forexample, that if 3 chocolates are to be shared by 2children, it is not necessary to divide all 3 chocolatesin half, and then distribute the halves; giving a wholechocolate plus a half to each child would accomplishthe same fairness in sharing. It was argued in the firstsection of this paper that this an important insight inthe domain of rational numbers: different fractionscan represent the same quantity.

• A third possible insight about quantities that canbe obtained from correspondences more easilythan from partitioning is related to ordering ofquantities. When forming correspondences,children may realize that there is an inverserelation between the divisor and the quotient: themore people there are to share a cake, the lesseach person will get: Children might achieve thecorresponding insight about this inverse relationusing the scheme of partitioning: the more partsyou cut the whole into, the smaller the parts.However, there is a difference between theprinciples that children would need to abstractfrom each of the schemes. In partitioning, theyneed to establish a within-quantity relation (themore parts, the smaller the parts) whereas incorrespondence they need to establish abetween-quantity relation (the more children, the less cake). It is an empirical matter to find outwhether or not it is easier to achieve one ofthese insights than the other.



• Finally, both partitioning and correspondencescould help children to understand somethingabout the equivalence between quantities, but thereasoning required to achieve this understandingdiffers across the two schemes of action. Whensetting chocolate bars in correspondence withrecipients, the children might be able to reasonthat, if there were twice as many chocolates andtwice as many children, the shares would beequivalent, even though the dividend and thedivisor are different. This may be easier than thecomparable reasoning in partitioning. Inpartitioning, understanding equivalence is based on inverse proportional reasoning (twice as manypieces means that each piece is half the size)whereas in contexts where children use thecorrespondence scheme, the reasoning is based ona direct proportion (twice as many chocolates andtwice as many children means that everyone stillgets the same).

This exploratory and hypothetical analysis of howchildren can reach an understanding of equivalenceand order of fractions when using partitioning orcorrespondences in division situations suggests thatthe distinction between the two schemas is worthinvestigating empirically. It is possible that the schemeof correspondences affords a smoother transitionfrom natural to rational numbers, at least as far asunderstanding equivalence and order of fractionalquantities is concerned.

We turn now to an empirical analysis of this question.The literature about these schemes of action is vastbut this paper focuses on research that sheds light on whether it is possible to findcontinuities between children’s understanding ofquantities that are represented by natural numbersand fractional quantities. We review research oncorrespondences first and then research onpartitioning.

Children’s use of thecorrespondence scheme injudgements about quantitiesPiaget (1952) pioneered the study of how and whenchildren use the correspondence scheme to drawconclusions about quantities. In one of his studies,there were three steps in the method.

• First, Piaget asked the children to place one pinkflower into each one of a set of vases;

• next, he removed the pink flowers and asked thechildren to place a blue flower into each one of thesame vases;

• then, he set all the flowers aside, leaving on thetable only the vases, and asked the children to takefrom a box the exact number of straws required ifthey wanted to put one flower into each straw.

Without counting, and only using correspondences,five- and six-year old children were able to makeinferences about the equivalence between straws andflowers: by setting two straws in correspondence witheach vase, they constructed a set of straws equivalentto the set of vases. Piaget concluded that the children’sjudgements were based on ‘multiplicative equivalences’(p. 219) established by the use of the correspondencescheme: the children reasoned that, if there is a 2-to-1correspondence between flowers and vases and a 2-to-1 correspondence between straws and vases, thenumber of flowers and straws must be the same.

In Piaget’s study, the scheme of correspondence was used in a situation that involved ratio but notdivision. Frydman and Bryant (1988) carried out a series of studies where children establishedcorrespondences between sets in a division situationwhich we have described in more detail in Paper 2,Understanding whole numbers. The studies showedthat children aged four often shared pretend sweetsfairly, using a one-for-you one-for-me type ofprocedure. After the children had distributed thesweets, Frydman and Bryant asked them to countthe number of sweets that one doll had and thendeduce the number of sweets that the other dollhad. About 40% of the four-year-olds were able tomake the necessary inference and say the exactnumber of sweets that the second doll had; thisproportion increased with age. This result extendsPiaget’s observations that children can makeequivalence judgements not only in multiplication butalso in division problems by using correspondence.

Frydman and Bryant’s results were replicated in anumber of studies by Davis and his colleagues (Davisand Hunting, 1990; Davis and Pepper, 1992; Pitkethlyand Hunting, 1996), who refer to this scheme ofaction as ‘dealing’. They used a variety of situations,including redistribution when a new recipient comes,to study children’s ability to use correspondences indivision situations and to make inferences aboutequality and order of magnitude of quantities. Theyalso argue that this scheme is basic to children’sunderstanding of fractions (Davis and Pepper, 1992).


Correa, Nunes and Bryant (1998) extended thesestudies by showing that children can make inferencesabout quantities resulting from a division not onlywhen the divisors are the same but also when theyare different. In order to circumvent the possibilitythat children feel the need to count the sets afterdivision because they think that they could havemade a mistake in sharing, Bryant and his colleaguesdid not ask the children to do the sharing: thesweets were shared by the experimenter, outside thechildren’s view, after the children had seen that thenumber of sweets to be shared was the same.

There were two conditions in this study: samedividend and same divisor versus same dividend and different divisors. In the same dividend and samedivisor condition, the children should be able toconclude for the equivalence between the sets thatresult from the division; in the same dividend anddifferent divisor condition, the children shouldconclude that the more recipients there are, thefewer sweets they receive; i.e. in order to answercorrectly, they would need to use the inverserelation between the divisor and the result as aprinciple, even if implicitly.

About two-thirds of the five-year-olds, the vastmajority of the six-year-olds, and all the seven-year-olds concluded that the recipients had equivalentshares when the dividend and the divisor were thesame. Equivalence was easier than the inverserelation between divisor and quotient: 34%, 53% and 81% of the children in these three age levels,respectively, were able to conclude that the morerecipients there are, the smaller each one’s share willbe. Correa (1994) also found that children’s successin making these inferences improved if they solvedthese problems after practising sharing sweetsbetween dolls; this indicates that thinking about howto establish correspondences improves their abilityto make inferences about the relations between thequantities resulting from sharing.

In all the previous studies, the dividend wascomposed of discrete quantities and was larger than the divisor. The next question to consider iswhether children can make similar judgements aboutequivalence when the situations involve continuousquantities and the dividend is smaller than thedivisor : that is, when children have to think aboutfractional quantities.

Kornilaki and Nunes (2005) investigated thispossibility by comparing children’s inferences in

division situations in which the quantities werediscrete and the dividends were larger than thedivisors to their inferences in situations in which thequantities were continuous and dividends smallerthan the divisors. In the discrete quantities tasks, thechildren were shown one set of small toy fishes tobe distributed fairly among a group of white cats andanother set of fishes to be distributed to a group ofbrown cats; the number of fish was always greaterthan the number of cats. In the continuous quantitiestasks, the dividend was made up of fish-cakes, to bedistributed fairly among the cats: the number ofcakes was always smaller than the number of cats,and varied between 1 and 3 cakes, whereas thenumber of cats to receive a portion in each groupvaried between 2 and 9. Following the paradigmdevised by Correa, Nunes and Bryant (1998), thechildren were neither asked to distribute the fish norto partition the fish cakes. They were asked whether,after a fair distribution in each group, each cat in onegroup would receive the same amount to eat aseach cat in the other group.6

In some trials, the number of fish (dividend) and cats(divisor) was the same; in other trials, the dividendwas the same but the divisor was different. So in thefirst type of trials the children were asked aboutequivalence after sharing and in the second type thechildren were asked to order the quantities obtainedafter sharing.

The majority of the children succeeded in all theitems where the dividend and the divisor were thesame: 62% of the five-year-olds, 84% of the six-year-olds and all the seven-year-olds answered all thequestions correctly. When the dividend was the sameand the divisors differed, the rate of success was31%, 50% and 81%, respectively, for the three agelevels. There was no difference in the level of successattained by the children with discrete versuscontinuous quantities.

In almost all the items, the children explained theiranswers by referring to the type of relation betweenthe dividends and the divisors: same divisor, sameshare or, with different divisors, the more catsreceiving a share, the smaller their share. The use of numbers as an explanation for the relative size ofthe recipients’ shares was observed in 6% of answersby the seven-year-olds when the quantities werediscrete and less often than this by the youngerchildren. Attempts to use numbers to speak aboutthe shares in the continuous quantities trials werepractically non-existent (3% of the seven-year-olds’



explanations). Thus, the analysis of justificationssupports the idea that the children were reasoningabout relations between quantities rather than usingcounting when they made their judgments ofequivalence or ordered the quantities that would beobtained after division.

This study replicated the previous findings, which we have mentioned already, that young children canuse correspondences to make inferences aboutequivalences and also added new evidence relevantto children’s understanding of fractional quantities:many young children who have never been taughtabout fractions used correspondences to orderfractional quantities. They did so successfully whenthe division would have resulted in unitary fractionsand also when the dividend was greater than 1 andthe result would not be a unitary fraction (e.g. 2 fishcakes to be shared by 3, 4 or 5 cats).

A study by knowledge Mamede, Nunes and Bryant(2005) confirmed that children can make inferencesabout the order of magnitude of fractions in sharingsituations where the dividend is smaller than thedivisor (e.g. 1 cake shared by 3 children compared to 1 cake shared by 5 children). She worked withPortuguese children in their first year in school, whohad received no school instruction about fractions.Their performance was only slightly weaker than thatof British children: 55% of the six-year-olds and 71%of the seven-year-olds were able to make theinference that the larger the divisor, the smaller theshare that each recipient would receive.

These studies strongly suggest that children canlearn principles about the relationship betweendividend and divisor from experiences with sharingwhen they establish correspondences between thetwo domains of measures, the shared quantitiesand the recipients. They also suggest that childrencan make a relatively smooth transition fromnatural numbers to rational numbers when theyuse correspondences to understand the relationsbetween quantities. This argument is central toStreefland’s (1987; 1993; 1997) hypothesis aboutwhat is the best starting point for teachingfractions to children and has been advanced byothers also (Davis and Pepper, 1992; Kieren, 1993;Vergnaud, 1983).

This research tell an encouraging story aboutchildren’s understanding of the logic of division evenwhen the dividend is smaller than the divisor, butthere is one further point that should be considered

in the transition between natural and rationalnumbers. In the domain of rational numbers there isan infinite set of equivalences (e.g. 1/2 = 2/4 = 3/6etc) and in the studies that we have described so farthe children were only asked to make equivalencejudgements when the dividend and the divisor in theequivalent fractions were the same. Can they stillmake the inference of equivalence in sharingsituations when the dividend and the divisor aredifferent across situations, but the dividend-divisorratio is the same?

Nunes, Bryant, Pretzlik, Bell, Evans and Wade (2007)asked British children aged between 7.5 and 10 years,who were in Years 4 and 5 in school, to makecomparisons between the shares that would bereceived by children in sharing situations where thedividend and divisor were different but their ratio wasthe same. Previous research (see, for example, Behr,Harel, Post and Lesh, 1992; Kerslake, 1986) shows thatchildren in these age levels have difficulty with theequivalence of fractions. The children in this study hadreceived some instruction on fractions: they had beentaught about halves and quarters in problems aboutpartitioning. They had only been taught about one pairof equivalent fractions: they were taught that one halfis the same as two quarters. In the correspondenceitem in this study, the children were presented withtwo pictures: in the first, a group of 4 girls was going toshare fairly 1 pie; in the second, a group of 8 boys wasgoing to share fairly 2 pies that were exactly the sameas the pie that the girls had. The question was whethereach girl would receive the same share as each boy.The overall rate of correct responses was 73% (78%in Year 4 and 70% in Year 5; this difference was notsignificant). This is an encouraging result: the childrenhad only been taught about halves and quarters;nevertheless, they were able to attain a high rate ofcorrect responses for fractional quantities that couldbe represented as 1/4 and 2/8.

In the studies reviewed so far the children wereasked about quantities that resulted from divisionand always included two domains of measures; thusthe children’s correspondence reasoning wasengaged in these studies. However, they did notinvolve asking the children to represent thesequantities through fractions. The final study reviewedhere is a brief teaching study (Nunes, Bryant, Pretzlik,Evans, Wade and Bell, 2008), where the childrenwere taught to represent fractions in the context oftwo domains of measures, shared quantities andrecipients, and were asked about the equivalencebetween fractions. The types of arguments that the


children produced to justify the equivalence of thefractions were then analyzed and compared to theinsights that we hypothesized would emerge in thecontext of sharing from the use of thecorrespondence scheme.

Brief teaching studies are of great value in researchbecause they allow the researchers to know whatunderstandings children can construct if they aregiven a specific type of guidance in the interactionwith an adult (Cooney, Grouws and Jones, 1988;Steffe and Tzur, 1994; Tzur, 1999; Yackel, Cobb, Wood, Wheatley and Merkel, 1990). They also havecompelling ecological validity: children spend much of their time in school trying to use what they havebeen taught to solve new mathematics problems.Because this study has only been published in asummary form (Nunes, Bryant, Pretzlik and Hurry,2006), some detail is presented here.

The children (N = 62) were in the age range from7.5 to 10 years, in Years 4 and 5 in school. Children inYear 4 had only been taught about half and quartersand the equivalence between half and two quarters;children in Year 5 had been taught also about thirds.They worked with a researcher outside theclassroom in small groups (12 groups of between 4and 6 children, depending on the class size) and wereasked to solve each problem first individually, andthen to discuss their answers in the group. Thesessions were audio- and video-recorded. Thechildren’s arguments were transcribed verbatim; theinformation from the video-tapes was latercoordinated with the transcripts in order to help theresearchers understand the children’s arguments.

In this study the researchers used problemsdeveloped by Streefland (1990). The children solvedtwo of his sharing tasks on the first day and anequivalence task on the second day of the teachingstudy. The tasks were presented in booklets withpictures, where the children also wrote theiranswers. The tasks used in the first day were:

• Six girls are going to share a packet of biscuits. Thepacket is closed; we don’t know how many biscuitsare in the packet. (a) If each girl received onebiscuit and there were no biscuits left, how manybiscuits were in the packet? (b) If each girl receiveda half biscuit and there were no biscuits left, howmany biscuits were in the packet? (c) If some moregirls join the group, what will happen when thebiscuits are shared? Do the girls now receive more or less each than the six girls did?

• Four children will be sharing three chocolates. (a)Will each child be able to get one bar of chocolate?(b) Will each child be able to get at least a half barof chocolate? (c) How would you share thechocolate? (The booklets contained a picture withthree chocolate bars and four children and thechildren were asked to show how they would sharethe chocolates) Write what fraction each one gets.

After the children had completed these tasks, the researcher told them that they were going topractice writing fractions which they had not yetlearned in school. The children were asked to write‘half ’ with numerical symbols, which they knewalready. The researcher taught the children to writefractions that they had not yet learned in school inorder to help the children re-interpret the meaningof fractions. The numerator was to be used torepresent the number of items to be divided, thedenominator should represent the number ofrecipients, and the dash between them two numbersshould represent the sign for division (for a discussionof children’s interpretation of fraction symbols in thissituation, see Charles and Nason, 2000, and Empson,Junk, Dominguez and Turner, 2005).

The equivalence task, presented on the second day, was:

• Six children went to a pizzeria and ordered twopizzas to share between them. The waiter broughtone first and said they could start on it because itwould take time for the next one to come. (a)How much will each child get from the first pizzathat the waiter brought? Write the fraction thatshows this. (b) How much will each child get fromthe second pizza? Write your answer. (c) If you addthe two pieces together, what fraction of a pizzawill each child get? You can write a plus signbetween the first fraction and the second fraction,and write the answer for the share each child getsin the end. (d) If the two pizzas came at the sametime, how could they share it differently? (e) Arethese fractions (the ones that the children wrotefor answers c and d equivalent?

According to the hypotheses presented in theprevious section, we would expect children todevelop some insights into rational numbers bythinking about different ways of sharing the sameamount. It was expected that they would realize that:



1 it is possible to divide a smaller number by a largernumber

2 different fractions might represent the sameamount

3 twice as many things to be divided and twice asmany recipients would result in equivalent amounts

4 the larger the divisor, the smaller the quotient.

The children’s explanations for why they thoughtthat the fractions were equivalent provided evidencefor all these insights, and more, as described below.

It is possible to divide a smallernumber by a larger number There was no difficulty among the students inattempting to divide 1 pizza among 6 children. Inresponse to part a of the equivalence problem, allchildren wrote at least one fraction correctly (somechildren wrote more than one fraction for the sameanswer, always correctly).

In response to part c, when the children were askedhow they could share the 2 pizzas if both pizzascame at the same time and what fraction wouldeach one receive, some children answered 1/3 andothers answered 2/12 from each pizza, giving a totalshare of 4/12. The latter children, instead of sharing 1 pizza among 3 girls, decided to cut each pizza in12 parts: i.e. they cut the sixths in half.

Different fractions can represent thesame amountThe insight that different fractions can represent the same amount was expressed in all groups. Forexample, one child said that, ‘They’re the sameamount of people, the same amount of pizzas, andthat means the same amount of fractions. It doesn’tmatter how you cut it.’ Another child said, ‘Because itwouldn’t really matter when they shared it, they’d getthat [3 girls would get 1 pizza], and then they’d getthat [3 girls would get the other pizza], and then itwould be the same.’ Another child said, ‘It’s the sameamount of pizza. They might be different fractionsbut the same amount [this child had offered 4/12 asan alternative to 2/6].’ Another child said: ‘Erm, wellbasically just the time doesn’t make much difference,the main thing is the number of things.’

When the dividend is twice as largeand the divisor is also twice as large,the result is an equivalent amount

The principle that when the dividend is twice aslarge and the divisor is also twice as large, the resultis an equivalent amount was expressed in 11 of the12 groups. For example, one child said, ‘It’s half thegirls and half the pizzas; three is a half of six and oneis a half of two.’ Another child said, ‘If they have twopizzas, then they could give the first pizza to threegirls and then the next one to another three girls.(…) If they all get one piece of that each, and theyget the same amount, they all get the same amount’.

So all three ideas we thought that could appear inthis context were expressed by the children. But two other principles, which we did not expect toobserve in this correspondence problem, were alsomade explicit by the children.

The number of parts and size ofparts are inversely proportionalThe principle that the number of parts and size ofparts are inversely proportional was enunciated in 8 ofthe 12 groups. For example, one child who cut thepizzas the second time around in 12 parts each said,‘Because it’s double the one of that [total number ofpieces] and it’s double the one of that [number ofpieces for each], they cut it twice and each is half thesize; they will be the same’. Another child said, ‘Becauseone sixth and one sixth is actually a different way infractions [from 1 third] and it doubled [the number ofpieces] to make it [the size of the piece] littler, andhalving [the number of pieces] makes it [the size of thepiece] bigger, so I halved it and it became one third’.

The fractions show the same part-whole relationThe reasoning that the fractions show the samepart-whole relation, which we had not expected toemerge from the use of the correspondencescheme, was enunciated in only one group (out of12), initially by one child, and was then reiterated bya second child in her own terms. The first child said,‘You need three two sixths to make six [6/6 – heshows the 6 pieces marked on one pizza], and youneed three one thirds to make three (3/3 – showsthe 3 pieces marked on one pizza). [He wrote thecomputation and continued] There’s two sixths, add


two sixths three times to make six sixths. With onethird, you need to add one third three times tomake three thirds.’

To summarize: this brief teaching experiment wascarried out to elicit discussions between thechildren in situations where they could use thecorrespondence scheme in division. The first set ofproblems, in which they are asked about sharingdiscrete quantities, created a background for thechildren to use this scheme of action. Theresearchers then helped them to construct aninterpretation for written fractions where thenumerator is the dividend, the denominator is thedivisor, and the line indicates the operation ofdivision. This interpretation did not replace theiroriginal interpretation of number of parts takenfrom the whole; the two meanings co-existed andappeared in the children’s arguments as theyexplained their answers. In the subsequentproblems, where the quantity to be shared wascontinuous and the dividend was smaller than thedivisor, the children had the opportunity toexplore the different ways in which continuousquantities can be shared. They were not asked toactually partition the pizzas, and some made markson the pizzas whereas others did not. The mostsalient feature of the children’s drawings was thatthey were not concerned with partitioning per se,even when the parts were marked, but with thecorrespondences between pizzas and recipients.Sometimes the correspondences were carried outmentally and expressed verbally and sometimesthe children used drawings and gestures whichindicated the correspondences.

Other researchers have identified children’s use ofcorrespondences to solve problems that involvefractions, although they did not necessarily use thislabel in describing the children’s answers. Empson(1999), for example, presented the followingproblem to children aged about six to seven years(first graders in the USA): 4 children got 3 pancakesto share; how many pancakes are needed for 12children in order for the children to have the sameamount of pancake as the first group? She reportedthat 3 children solved this problem by partitioningand 3 solved it by placing 3 pancakes incorrespondence to each group of 4 children. Similarstrategies were reported when children solvedanother problem that involved 2 candy bars sharedamong 3 children.

Kieren (1993) also documented children’s use ofcorrespondences to compare fractions. In hisproblem, the fractions were not equivalent: therewere 7 recipients and 4 items in Group A and 4recipients and 2 items in Group B. The childrenwere asked how much each recipient would get in each group and whether the recipients in bothgroups would get the same amount. Kierenpresents a drawing by an eight-year-old, where the items are partitioned in half and thecorrespondences between the halves and therecipients are shown; in Group A, a line without arecipient shows that there is an extra half in thatgroup and the child argues that there should beone more person in Group A for the amounts tobe the same. Kieren termed this solution‘corresponding or ‘ratiolike’ thinking’ (p. 54).

Conclusion

The scheme of correspondences develops relativelyearly: about one-third of the five-year-olds, half of six-year-olds and most seven-year-olds can usecorrespondences to make inferences aboutequivalence and order in tasks that involve fractionalquantities. Children can use the scheme ofcorrespondences to:• establish equivalences between sets that have the

same ratio to a reference set (Piaget, 1952)• re-distribute things after having carried out one

distribution (Davis and Hunting, 1990; Davis andPepper, 1992; Davis and Pitkethly, 1990; Pitkethlyand Hunting, 1996);

• reason about equivalences resulting from divisionboth when the dividend is larger or smaller thanthe divisor (Bryant and colleagues: Correa, Nunesand Bryant, 1994; Frydman and Bryant, 1988; 1994;Empson, 1999; Nunes, Bryant, Pretzlik and Hurry,2006; Nunes, Bryant, Pretzlik, Bell, Evans and Wade,2007; Mamede, Nunes and Bryant, 2005);

• order fractional quantities (Kieren, 1993; Kornilakiand Nunes, 2005; Mamede, 2007).

These studies were carried out with children up tothe age of ten years and all of them producedpositive results. This stands in clear contrast with theliterature on children’s difficulties with fractions andprompts the question whether the difficulties mightstem from the use of partitioning as the startingpoint for the teaching of fractions (see also Lamon,1996; Streefland, 1987). The next section examinesthe development of children’s partitioning action andits connection with children’s concepts of fractions.



Children’s use of the scheme of partitioning in makingjudgements about quantitiesThe scheme of partitioning has been also namedsubdivision and dissection (Pothier and Sawada,1983), and is consistently defined as the process of dividing a whole into parts. This process isunderstood not as the activity of cutting somethinginto parts in any way, but as a process that must beguided from the outset by the aim of obtaining apre-determined number of equal parts.

Piaget, Inhelder and Szeminska (1960) pioneered thestudy of the connection between partitioning andfractions. They spelled out a number of ideas thatthey thought were necessary for children to developan understanding of fractions, and analysed them inpartitioning tasks. The motivation for partitioning wassharing a cake between a number of recipients, butthe task was one of partitioning. They suggested that‘the notion of fraction depends on two fundamentalrelations: the relation of part to whole (…) and therelation of part to part’ (p. 309). Piaget and colleaguesidentified a number of insights that children need toachieve in order to understand fractions:

1 the whole must be conceived as divisible, an ideathat children under the age of about two do notseem to attain

2 the number of parts to be achieved is determinedfrom the outset

3 the parts must exhaust the whole (i.e. thereshould be no second round of partitioning and noremainders)

4 the number of cuts and the number of parts arerelated (e.g. if you want to divide something in 2parts, you should use only 1 cut)

5 all the parts should be equal

6 each part can be seen as a whole in itself, nestedinto the whole but also susceptible of furtherdivision

7 the whole remains invariant and is equal to thesum of the parts.

Piaget and colleagues observed that children rarelyachieved correct partitioning (sharing a cake) beforethe age of about six years. There is variation in the

level of success depending on the shape of the whole(circular areas are more difficult to partition thanrectangles) and on the number of parts. A majorstrategy in carrying out successful partitioning was theuse of successive divisions in two: so children are ableto succeed in dividing a whole into fourths beforethey can succeed with thirds. Successive halvinghelped the children with some fractions: dividingsomething into eighths is easier this way. However, itinterfered with success with other fractions: somechildren, attempting to divide a whole into fifths,ended up with sixths by dividing the whole first inhalves and then subdividing each half in three parts.

Piaget and colleagues also investigated children’sunderstanding of their seventh criterion for a trueconcept of fraction, i.e. the conservation of thewhole. This conservation, they argued, would requirethe children to understand that each piece could not be counted simply as one piece, but had to beunderstood in its relation to the whole. Somechildren aged six and even seven years failed tounderstand this, and argued that if someone ate acake cut into 1/2 + 2/4 and a second person ate acake cut into 4/4, the second one would eat morebecause he had four parts and the first one only hadthree. Although these children would recognise thatif the pieces were put together in each case theywould form one whole cake, they still maintainedthat 4/4 was more than 1/2 + 2/4. Finally, Piaget andcolleagues also observed that children did not haveto achieve the highest level of development in thescheme of partitioning in order to understand theconservation of the whole.

Children’s difficulties with partitioning continuouswholes into equal parts have been confirmed manytimes in studies with pre-schoolers and children intheir first years in school (e.g. Hiebert andTonnessen, 1978; Hunting and Sharpley, 1988 b,)observed that children often do not anticipate thenumber of cuts and fail to cut the whole extensively,leaving a part of the whole un-cut. These studies alsoextended our knowledge of how children’s expertisein partitioning develops. For example, Pothier andSawada (1983) and Lamon (1996) proposed moredetailed schemes for the analysis of the developmentof partitioning schemes and other researchers(Hiebert and Tonnessen, 1978; Hunting and Sharpley,1988 a and b; Miller, 1984; Novillis, 1976) found thatthe difficulty of partitioning discrete and continuousquantities is not the same, as hypothesized by Piaget.Children can use a procedure for partitioningdiscrete quantities that is not applicable to


continuous quantities: they can ‘deal out’ the discretequantities but not the continuous ones. Thus theyperform significantly better with the former than thelatter. This means that the transition from discrete tocontinuous quantities in the use of partitioning is notdifficult, in contrast to the smooth transition noted inthe case of the correspondence scheme.

These studies showed that the scheme ofpartitioning continuous quantities develops slowly,over a longer period of time. The next question toconsider is whether partitioning can promote theunderstanding of equivalence and ordering offractions once the scheme has developed.

Many studies investigated children’s understanding ofequivalence of fractions in partitioning contexts (e.g.Behr, Lesh, Post and Silver, 1983; Behr, Wachsmuth,Post and Lesh, 1984; Larson, 1980; Kerslake, 1986),but differences in the methods used in these studiesrender the comparisons between partitioning andcorrespondence studies ambiguous. For example, ifthe studies start with a representation of thefractions, rather than with a problem aboutquantities, they cannot be compared to the studiesreviewed in the previous section, in which childrenwere asked to think about quantities withoutnecessarily using fractional representation. We shallnot review all studies but only those that usecomparable methods.

Kamii and Clark (1995) presented children withidentical rectangles and cut them into fractions usingdifferent cuts. For example, one rectangle was cuthorizontally in half and the second was cut across adiagonal. The children had the opportunity to verifythat the rectangles were the same size and that thetwo parts from each rectangle were the same in size.They asked the children: if these were chocolatecakes, and the researcher ate a part cut from the first rectangle and the child ate a part cut from thesecond, would they eat the same amount? Thismethod is highly comparable to the studies byKornilaki and Nunes (2005) and by Mamede (2007),where the children do not have to carry out theactions, so their difficulty with partitioning does notinfluence their judgements. They also use similarlymotivated contexts, ending in the question of whetherrecipients would eat the same amount. However, thequestion posed by Kamii and Clark draws on thechild’s understanding of partitioning and the relationsbetween the parts of the two wholes because eachwhole corresponds to a single recipient.

The children in Kamii’s study were considerably olderthan those in the correspondence studies: they werein the fifth or sixth year in school (approximately 11and 12 years). Both groups of children had beentaught about equivalent fractions. In spite of havingreceived instruction, the children’s rate of successwas rather low: only 44% of the fifth graders and51% of the sixth graders reasoned that they wouldeat the same amount of chocolate cake becausethese were halves of identical wholes.

Kamii and Clark then showed the children twoidentical wholes, cut one in fourths using a horizontaland a vertical cut, and the other in eighths, using onlyhorizontal cuts. They discarded one fourth from thefirst ‘chocolate cake’, leaving three fourths be eaten,and asked the children to take the same amountfrom the other cake, which had been cut intoeighths, for themselves. The percentage of correctanswers was this time even lower: 13% of the fifthgraders and 32% of the sixth graders correctlyidentified the number of eighths required to take thesame amount as three fourths.

Recently, we (Nunes and Bryant, 2004) included asimilar question about halves in a survey of Englishchildren’s knowledge of fractions. The children in ourstudy were in their fourth and fifth year (mean ageseight and a half and nine and a half, respectively) inschool. The children were shown pictures of a boyand a girl and two identical rectangular areas, the‘chocolate cakes’. The boy cut his cake along thediagonal and the girl cut hers horizontally. Thechildren were asked to indicate whether they ate thesame amount of cake and, if not, to mark the childwho ate more. Our results were more positive thanKamii and Clark’s: 55% of the children in year four(eight and a half year olds) in our study answeredcorrectly. However, these results are weak bycomparison to children’s rate of correct responseswhen the problem draws on their understanding ofcorrespondences. In the Kornilaki and Nunes study,100% of the seven-year-olds (third graders) realizedthat two divisions that have the same dividend andthe same divisor result in equivalent shares. Ourresults with fourth graders, when both the dividendand the divisor were different, still shows a higherrate of correct responses when correspondences areused: 78% of the fourth graders gave correct answerswhen comparing one fourth and two eighths.

In the preceding studies, the students had to thinkabout the quantities ignoring their perceptualappearance. Hart et al. (1985) and Nunes et al.



(2004) presented students with verbal questions,which did not contain drawings that could lead toincorrect conclusions based on perception. In bothstudies, the children were told that two boys hadidentical chocolate bars; one cut his into 8 parts andate 4 and the other cut his into 4 parts and ate 2.Combining the results of these two studies, it ispossible to see how the rate of correct responseschanged across age: 40% at ages 8 to 9 years, 74% at10 to11 years, 60% at 11 to 12 years, and 64% at 12to 13 years. The students aged 8 to 10 wereassessed by Nunes et al. and the older ones by Hartet al. These results show modest progress on theunderstanding of equivalence questions presented inthe context of partitioning even though thequantities eaten were all equivalent to half.

Mamede (2007) carried out a direct comparisonbetween children’s use of the correspondence andthe partitioning scheme in solving equivalence andorder problems with fractional quantities. In this well-controlled study, she used story problems involvingchocolates and children, similar pictures andmathematically identical questions; the divisionscheme relevant to the situation was the onlyvariable distinguishing the problems. Incorrespondence problems, for example, she askedthe children: in one party, three girls are going toshare fairly one chocolate cake; in another party, sixboys are going to share fairly two chocolate cakes.The children were asked to decide whether eachboy would eat more than each girl, each girl wouldeat more than each boy, or whether they wouldhave the same amount to eat. In the partitioningproblems, she set the following scenario: This girl andthis boy have identical chocolate cakes; the cakes aretoo big to eat at once so the girl cuts her cake in 3 identical parts and eats one and the boy cuts hiscake in 6 identical parts and eats 2. The childrenwere asked whether the girl and the boy ate thesame amount or whether one ate more than theother. The children (age range six to seven) werePortuguese and in their first year in school; they hadreceived no instruction about fractions.

In the correspondence questions, the responses of35% of the six-year-olds and 49% of the seven-year-olds were correct; in the partitioning questions, 10%of the answers of children in both age levels werecorrect. These highly significant differences suggestthat the use of correspondence reasoning supportschildren’s understanding of equivalence betweenfractions whereas partitioning did not seem to affordthe same insights.

Finally, it is important to compare students’ argumentsfor the equivalence and order of quantitiesrepresented by fractions in teaching studies wherepartitioning is used as the basis for teaching. Manyteaching studies that aim at promoting students’understanding of fractions through partitioning havebeen reported in the literature (e.g., Behr, Wachsmuth,Post and Lesh, 1984; Brousseau, Brousseau andWarfield, 2004; 2007; Empson, 1999; Kerslake, 1986;Olive and Steffe, 2002; Olive and Vomvoridi, 2006;Saenz-Ludlow, 1994; Steffe, 2002). In most of thesestudies, students’ difficulties with partitioning arecircumvented either by using pre-divided materials(e.g. Behr, Wachsmuth, Post and Lesh, 1984) or byusing computer tools where the computer carries out the division as instructed by the student (e.g. Olive and Steffe, 2002; Olive and Vomvoridi, 2006).

Many studies combine partitioning withcorrespondence during instruction, either becausethe researchers do not use this distinction (e.g.Saenz-Ludlow, 1994) or because they wish toconstruct instruction that combines both schemes in order to achieve a better instructional program(e.g. Brousseau, Brousseau and Warfield, 2004; 2007).These studies will not be discussed here. Two studiesthat analysed student’s arguments focus theinstruction on partitioning and are presented here.

The first study was carried out by Berhr, Wachsmuth,Post and Lesh (1984). The researchers used objectsof different types that could be manipulated duringinstruction (e.g. counters, rectangles of the same sizeand in different colours, pre-divided into fractionssuch as halves, quarters, thirds, eighths) but alsotaught the students how to use algorithms (divisionof the denominator by the numerator to find aratio) to check on the equivalence of fractions. Thestudents were in fourth grade (age about 9) andreceived instruction over 18 weeks. Behr et al.provided a detailed analysis of children’s argumentsregarding the ordering of fractions. In summary, theyreport the following insights after instruction.

• When ordering fractions with the same numeratorand different denominators, students seem to be ableto argue that there is an inverse relation between thenumber of parts into which the whole was cut andthe size of the parts. This argument appears eitherwith explicit reference to the numerator (‘there aretwo pieces in each, but the pieces in two fifths aresmaller.’ p. 328) or without it (‘the bigger the numberis, the smaller the pieces get.’ p. 328).


• A third fraction can be used as a reference pointwhen two fractions are compared: three ninths isless than three sixths because ‘three ninths is …less than half and three sixths is one half ’ (p. 328). It is not clear how the students had learned that3/6 and 1/2 are equivalent but they can use thisknowledge to solve another comparison.

• Students used the ratio algorithm to verify whetherthe fractions were equivalent: 3/5 is not equivalentto 6/8 because ‘if they were equal, three goes intosix, but five doesn’t go into eight.’ (p. 331).

• Students learned to use the manipulative materialsin order to carry out perceptual comparisons: 6/8equals 3/4 because ‘I started with four parts. Then Ididn’t have to change the size of the paper at all. Ijust folded it, and then I got eight.’ (p. 331).

Behr et al. report that, after 18 weeks of instruction,a large proportion of the students (27%) continuedto use the manipulatives in order to carry ourperceptual comparisons; the same proportion (27%)used a third fraction as a reference point and asimilar proportion (23%) used the ratio algorithmthat they had been taught to compare fractions.

Finally, there is no evidence that the students wereable to understand that the number of parts and sizeof parts could compensate for each other precisely in a proportional manner. For example, in thecomparison between 6/8 and 3/4 the students couldhave argued that there were twice as many partswhen the whole was cut into 8 parts in comparisonwith cutting in 4 parts, so you need to take twice asmany (6) in order to have the same amount.

In conclusion, students seemed to develop someinsight into the inverse relation between the divisorand the quantity but this only helped them when thedividend was kept constant: they could not extendthis understanding to other situations where thenumerator and the denominator differed.

The second set of studies that focused onpartitioning was carried out by Steffe and hiscolleagues (Olive and Steffe, 2002; Olive andVomvoridi, 2006; Steffe, 2002). Because the aim ofmuch of the instruction was to help the childrenlearn to label fractions or compose fractions thatwould be appropriate for the label, it is not possibleto extract from their reports the children’sarguments for equivalence of fractions.

However, one of the protocols (Olive and Steffe,2002) provides evidence for the student’s difficultywith improper fractions, which, we hypothesise, couldbe a consequence of using partitioning as the basisfor the concept of fractions. The researcher asked Joeto make a stick 6/5 long. Joe said that he could notbecause ‘there are only five of them’. After prompting,Joe physically adds one more fifth to the five alreadyused, but it is not clear whether this physical actionconvinces him that 6/5 is mathematically appropriate.In a subsequent example, Joe labels a stick made with9 sticks, which had been defined as ‘one seventh’ ofan original stick, 9/7, but according to the researchers‘an important perturbation’ remains. Joe later counts8 of a stick that had been designated as ‘one seventh’but doesn’t use the label ‘eight sevenths’. When theresearcher proposes this label, he questions it: ‘Howcan it be EIGHT sevenths?’ (Olive and Steffe, 2002, p. 426). Joe later refused to make a stick that is 10/7,even though the procedure is physically possible.Subsequently, on another day, Joe’s reaction toanother improper fraction is: ‘I still don’t understandhow you could do it. How can a fraction be biggerthan itself? (Olive and Steffe, 2002, p. 428; emphasis in the original).

According to the researchers, Joe only sees thatimproper fractions are acceptable when theypresented a problem where pizzas were to be sharedby people. When 12 friends ordered 2 slices each ofpizzas cut into 8 slices, Joe realized immediately thatmore than one pizza would be required; the traditionalpartitioning situation, where one whole is divided intoequal parts, was transformed into a less usual one,where two wholes are required but the size of thepart remains fixed.

This example illustrates the difficulty that studentshave with improper fractions in the context ofpartitioning but which they can overcome bythinking of more than one whole.

Conclusion

Partitioning, defined as the action of cutting a whole into a pre-determined number of equal parts,shows a slower developmental process thancorrespondence. In order for children to succeed,they need to anticipate the solution so that the rightnumber of cuts produces the right number of equalparts and exhausts the whole. Its accomplishment,however, does not seem to produce immediateinsights into equivalence and order of fractional



quantities. Apparently, many children do not see it asnecessary that halves from two identical wholes areequivalent, even if they have been taught about theequivalence of fractions in school.

In order to use this scheme of action as the basis for learning about fractions, teaching schemes andresearchers rely on pre-cut wholes or computertools to avoid the difficulties of accurate partitioning.Students can develop insight into the inverse relationbetween the number of parts and the size of theparts through the partitioning scheme but there isno evidence that they realize that if you cut a wholein twice as many parts each one will be half in size.Finally, improper fractions seem to cause uneasinessto students who have developed their conception offractions in the context of partitioning; it is importantto be aware of this uneasiness if this is the schemechosen in order to teach fractions.

Rational numbers and children’sunderstanding of intensivequantitiesIn the introduction, we suggested that rationalnumbers are necessary to represent quantities thatare measured by a relation between two otherquantities. These are called intensive quantities andthere are many examples of such quantities both in everyday life and in science. In everyday life, weoften mix liquids to obtain a certain taste. If you mix fruit concentrate with water to make juice, theconcentration of this mixture is described by arational number: for example, 1/3 concentrate and2/3 water. Probability is an intensive quantity that isimportant both in mathematics and science and ismeasured as the number of favourable cases dividedby the number of total cases.7

The conceptual difficulties involved in understandingintensive quantities are largely similar to thoseinvolved in understanding the representation ofquantities that are smaller than the whole. In order to understand intensive quantities, students must form a concept that takes two variablessimultaneously into account and realise that there isan inverse relation between the denominator andthe quantity represented.

Piaget and Inhelder described children’s thinkingabout intensive quantities as one of the manyexamples of the development of the scheme ofproportionality, which they saw as one of the

hallmarks of adolescent thinking and formaloperations. They devoted a book to the analysis ofchildren’s understanding of probabilities (Piaget andInhelder, 1975) and described in great detail thesteps that children take in order to understand thequantification of probabilities. In the mostcomprehensive of their studies, the children wereshown pairs of decks of cards with different numbersof cards, some marked with a cross and othersunmarked. The children were asked to judge whichdeck they would choose to draw from if theywanted to have a better chance of drawing a card marked with a cross.

Piaget and Inhelder observed that many of theyoung children treated the number of marked andunmarked cards as if they were independent:sometimes they chose one deck because it hadmore marked cards than the other and sometimesthey chose a deck because it had fewer blank cardsthan the other. This approach can lead to correctresponses when either the number of marked cardsor the number of unmarked cards is the same inboth decks, and children aged about seven yearswere able to make correct choices in such problems.This is rather similar to the observations of children’ssuccesses and difficulties in comparing fractionsreported earlier on: they can reach the correctanswer when the denominator is constant or whenthe numerator is constant, as this allow them tofocus on the other value. When they must think ofdifferent denominators and numerators, thequestions become more difficult.

Around the age of nine, children started makingcorrespondences between marked and unmarkedcards within each deck and were able to identifyequivalences using this type of procedure. Forexample, if asked to compare a deck with onemarked and two unmarked cards (1/3 probability)with another deck with two marked and fourunmarked cards (2/6 probability), the children wouldre-organise the second deck in two lots, setting onemarked card in correspondence with two unmarked,and conclude that it did not make any differencewhich deck they picked a card from. Piaget andInhelder saw these as empirical proportionalsolutions, which were a step towards the abstractionthat characterises proportional reasoning.

Noelting (1980 a and b) replicated these results withanother intensive quantity, the taste of orange juicemade from a mixture of concentrate and water. Inbroad terms, he described children’s thinking and its


development in the same way as Piaget and Inhelderhad done. This is an important replication of Piaget’sresults results considering that the content of theproblems differed marked across the studies,probability and concentration of juice.

Nunes, Desli and Bell (2003) compared students’ability to solve problems about extensive andintensive quantities that involved the same type ofreasoning. Extensive quantities can be representedby a single whole number (e.g. 5 kilos, 7 cows, 4days) whereas intensive quantities are representedby a ratio between two numbers. In spite of thesedifferences, it is possible to create problems whichare comparable in other aspects but differ withrespect to whether the quantities are extensive orintensive. Intensive quantities problems alwaysinvolve three variables. For example, three variablesmight be amount of orange concentrate, amount ofwater, and the taste of the orange juice, which isthe intensive quantity. The amount of orangeconcentrate is directly related to how orangey thejuice tastes whereas the amount of water isinversely related to how orangey the juice tastes. A comparable extensive quantities problem wouldinvolve three extensive quantities, with the oneunder scrutiny being inversely proportional to oneof the variables and directly proportional to theother. For example, the number of days that thefood bought by a farmer lasts is directlyproportional to the amount of food purchased andinversely proportional to the number of animalsshe has to feed. In our study, we analysed students’performance in comparison problems where theyhad to consider either intensive quantities (e.g. howorangey a juice would taste) or extensive quantities(e.g. the number of days the farmer’s food supplywould last). Students performed significantly betterin the extensive quantities problems even thoughboth types of problem involved proportionalreasoning and the same number of variables. So,although the difficulties shown by children acrossthe two types of problem are similar, their level ofsuccess was higher with extensive than intensivequantities. This indicates that students find it difficultto form a concept where two variables must becoordinated into a single construct,8 and thereforeit may be important for schools and teachers toconsider how they might promote thisdevelopment in the classroom.

We shall not review the large literature on intensivequantities here (see, for example, Erickson, 1979;Kaput, 1985; Schwartz, 1988; Stavy, Strauss, Orpaz

and Carmi, 1982; Stavy and Tirosh, 2000), but there islittle doubt that students’ difficulties in understandingintensive quantities are very similar to those thatthey have when thinking about fractions whichrepresent quantities smaller than the unit. They treatthe values independently, they find it difficult to thinkabout inverse relations, and they might think of therelations between the numbers as additive instead ofmultiplicative.

There is presently little information to indicatewhether students can transfer what they havelearned about fractions in the context ofrepresenting quantities smaller than the unit to therepresentation and understanding of intensivequantities. Brousseau, Brousseau, and Warfield (2004)suggest both that teachers believe that students willeasily go from one use of fractions to another, andthat nonetheless the differences between these twotypes of situation could actually result in interferencerather than in easy transfer of insights acrosssituations. In contrast, Lachance and Confrey (2002)developed a curriculum for teaching third gradestudents (estimated age about 8 years) about ratiosin a variety of problems, including intensive quantitiesproblems, and then taught the same students infourth grade (estimated age about 9 years) aboutdecimals. Their hypothesis is that students wouldshow positive transfer from learning about ratios tolearning about decimals. They claimed that theirstudents learned significantly more about decimalsthan students who had not participated in a similarcurriculum and whose performance in the samequestions had been described in other studies.

We believe that it is not possible at the moment toform clear conclusions on whether knowledge offractions developed in one type of situation transferseasily to the other, shows no transfer, or actuallyinterferes with learning about the other type ofsituation. In order to settle this issue, we must carry out the appropriate teaching studies andcomparisons.

However, there is good reason to conclude that theuse of rational numbers to represent intensivequantities should be explicitly included in thecurriculum. This is an important concept in everydaylife and science, and causes difficulties for students.



Learning to use mathematicalprocedures to determine theequivalence and order of rational numbersPiaget’s (1952) research on children’s understandingof natural numbers shows that young children, agedabout four, might be able to count two sets ofobjects, establish that they have the same number,and still not conclude that they are equivalent if thesets are displayed in very different perceptualarrangements. Conversely, they might establish theequivalence between two sets by placing theirelements in correspondence and, after counting theelements in one set, be unable to infer what thenumber in the other set is (Piaget, 1952; Frydmanand Bryant, 1988). As we noted in Paper 2,Understanding whole numbers, counting is aprocedure for creating equivalent sets and placingsets in order but many young children who knowhow to count do not use counting when asked tocompare or create equivalent sets (see, for example,Michie, 1984, Cowan and Daniels, 1989; Cowan,1987; Cowan, Foster and Al-Zubaidi, 1993; Saxe,Guberman and Gearhart, 1987).

Procedures to establish the equivalence and order of fractional quantities are much more complex thancounting, particularly when both the denominatorand the numerator differ. Students are taughtdifferent procedures in different countries. Theprocedure that seems most commonly taught inEngland is to check the equivalence by analysing the multiplicative relation between or within thefractions. For example, when comparing 1/3 with4/12, students are taught to find the factor thatconnects the numerators (1 and 4) and then applythe same factor to the denominators. If thenumerator and the denominator of the secondfraction are the product of the numerator and thedenominator of the first fraction by the samenumber, 4 in this case, they are equivalent. Analternative approach is to find whether themultiplicative relation between the numerator and the denominator of each fraction is the same (3 in this case): if it is, the fractions are equivalent.

If students learned this procedure and applied itconsistently, it should not matter whether the factoris, for example, 2, 3 or 5, because these are well-known multiplication associations. It should also notmatter whether the fraction with larger numeratorand denominator is the first or the second. However,research shows that these variations affect students’

performance. Hart et al. (1985) presented studentswith the task of identifying the missing values inequivalent fractions. The children were presentedwith the item below and asked which numbersshould replace the square and the triangle:

2/7 = �/14 = 10/�

The rate of correct responses by 11 to 12 and 12- to13-year-olds for the second question was about halfthat for the first one: about 56% for the first questionand 24% for the second. The within-fraction methodcannot be easily applied in these cases but the factorsare 2 and 5, and these multiplication tables should bequite easy for students at this age level.

We recently replicated these different levels ofdifficulty in a study with 8- to 10-year-olds. Theeasiest questions were those where the commonfactor was 2; the rates of correct responses for 1/3 = 2/� and 6/8 = 3/� were 52% and 45%,respectively. The most difficult question was 4/12 = 1/� this was only answered correctly by 16% of the students. It is unlikely that the difficulty ofcomputation could explain the differences inperformance: even weak students in this age rangeshould be able to identify 3 as the factor connecting4 and 12, if they had been taught the within-fractionmethod, or 4 as the factor connecting 1 and 4, ifthey were taught the between-fractions method.

A noteworthy aspect of our results was the lowcorrelations between the different items: althoughmost were significant (due to the large sample size;N = 188), only two of the nine correlations wereabove .4. This suggests that the students were notable to use the procedure that they learnedconsistently to solve five items that had the sameformat and could be solved by the same procedure.

Our assessment, like the one by Hart et al. (1985),also included an equivalence question set in thecontext of a story: two boys have identical chocolatebars, one cuts his into 8 equal parts and eats 4 andthe other cuts his into 4 equal parts and eats 2; thechildren are asked to indicate whether the boys eatthe same amount of chocolate and, if not, who eatsmore. This item is usually seen as assessing children’sunderstanding of quantities as it is not expressed infraction terms. In our sample, no student wrote thefractions 4/8 and 2/4 and compared them by meansof a procedure. We analysed the correlationsbetween this item and the five items described inthe previous paragraph. If the students used the


same reasoning or the same procedure to solve theitems, there should be a high correlation betweenthem. This was not so: the highest of the correlationsbetween this item and each of the five previous oneswas 0.32, which is low. This result exemplifies theseparation between understanding fractionalquantities and knowledge of procedures in thedomain of rational numbers. This is much the sameas observed in the domain of natural numbers.

It is possible that understanding the relations betweenquantities gives students an advantage in learning theprocedures to establish the equivalence of fractions,but it may not guarantee that they will actually learn itif teachers do connect their understanding with theprocedure. When we separated the students into twogroups, one that answered the question about theboys and the chocolates correctly and the other thatdid not, there was a highly significant differencebetween the two groups in the rate of correctresponses in the procedural items: the group whosucceeded in the chocolate question showed 38%correct responses to the procedural items whereasthe group who failed only answered 18% of theprocedural questions correctly.

A combination of longitudinal and interventionstudies is required to clarify whether students whounderstand fractional quantities benefit more whentaught how to represent and compare fractions.There are presently no studies to clarify this matter.

Research that analyses students’ knowledge ofprocedures used to find equivalent fractions and itsconnection with conceptual knowledge of fractionshas shown that there can be discrepancies betweenthese two forms of knowledge. Rittle-Johnson, Sieglerand Alibali (2001) argued that procedural andconceptual knowledge develop in tandem butKerslake (1986) and Byrnes and colleagues (Byrnes,1992; Byrnes and Wasik, 1991), among others,identified clear discrepancies between students’conceptual and procedural knowledge of fractions.Recently we (Hallett, Nunes and Bryant, 2007)analysed a large data set (N = 318 children in Years4 and 5) and observed different profiles of relativeperformance in items that assess knowledge ofprocedures to compare fractions and understandingof fractional quantities. Some children show greatersuccess in procedural questions than would beexpected from their performance in conceptualitems, others show better performance in conceptualitems than expected from their performance in theprocedural items, and still others do not show any

discrepancy between the two. Thus, some studentsseem to learn procedures for finding equivalentfractions without an understanding of why theprocedures work, others base their approach tofractions on their understanding of quantities withoutmastering the relevant procedures, and yet othersseem able to co-ordinate the two forms ofknowledge. Our results show that the third group ismore successful not only in a test about fractions butalso in a test about intensive quantities, which didnot require the use of fractions in the representationof the quantities.

Finally, we ask whether students are better at usingprocedures to compare decimals than to compareordinary fractions. The students in some of the gradelevels studied by Resnick and colleagues (1989)would have been taught how to add and subtractdecimals: they were in grades 5 and 6 (the estimatedage for U.S. students is about 10 and 11 years) andone of the early uses of decimals in the curriculumin the three participating countries is addition andsubtraction of decimals. When students are taught toalign the decimal numbers by placing the decimalpoints one under the other before adding – forexample, when adding 0.8, 0.26 and 0.361 you needto align the decimal points before carrying out theaddition – they may not realise that they are using aprocedure that automatically converts the values tothe same denominator: in this case, x/1000. It ispossible that students may use this procedure ofaligning the decimal point without fully understandingthat this is a conversion to the same denominatorand thus that it should help them to compare thevalue of the fractions: after learning how to add andsubtract with decimals, they may still think that 0.8 isless than 0.36 but probably would not have said that0.80 is less than 0.36.

To conclude, we find in the domain of rationalnumbers a similar separation betweenunderstanding quantities and learning to operatewith representations when judging the equivalenceand order of magnitude of quantities. Students aretaught procedures to test whether fractions areequivalent but their knowledge of these proceduresis limited, and they do not apply it across itemsconsistently. Similarly, students who solveequivalence problems in context are not necessarilyexperts in solving problems when the fractions arepresented without context.

The significance of children’s difficulties inunderstanding equivalence of fractions cannot be



overstressed: in the domain of rational numbers,students cannot learn to add and subtract withunderstanding if they do not realise that fractionsmust be equivalent in order to be added. Adding 1/3and 2/5 without transforming one of these into anequivalent fraction with the same denominator asthe other is like adding bananas and tins of soup: itmakes no sense. Above and beyond the fact thatone cannot be said to understand numbers withoutunderstanding their equivalence and order, in thedomain of rational numbers equivalence is a coreconcept for computing addition and subtraction.Kerslake (1986) has shown that students learn toimplement the procedures for adding andsubtracting fractions without having a glimpse at why they convert the fractions into commondenominators first. This separation between themeaning of fractions and the procedures cannotbode well for the future of these learners.

Conclusions and educationalimplications• Rational numbers are essential for the

representation of quantities that cannot berepresented by a single natural number. For thisreason, they are needed in everyday life as well asscience, and should be part of the curriculum in theage range 5 to 16.

• Children learn mathematical concepts by applyingschemes of action to problem solving and reflectingabout them. Two types of action schemes areavailable in division situations: partitioning, whichinvolves dividing a whole into equal parts, andcorrespondence situations, where two quantities(or measures) are involved, a quantity to be sharedand a number of recipients of the shares.

• Children as young as five or six years in age arequite good at establishing correspondences toproduce equal shares, whereas they experiencemuch difficulty in partitioning continuous quantities.Reflecting about these schemes and drawing insightsfrom them places children in different paths forunderstanding rational number. When they use thecorrespondence scheme, they can achieve someinsight into the equivalence of fractions by thinkingthat, if there are twice as many things to be sharedand twice as many recipients, then each one’s shareis the same. This involves thinking about a directrelation between the quantities. The partitioningscheme leads to understanding equivalence in a

different way: if a whole is cut into twice as manyparts, the size of each part will be halved. Thisinvolves thinking about an inverse relation betweenthe quantities in the problem. Research consistentlyshows that children understand direct relationsbetter than inverse relations.

• There are no systematic and controlledcomparisons to allow for unambiguous conclusionsabout the outcomes of instruction based oncorrespondences or partitioning. The availableevidence suggests that testing this hypothesisappropriately could result in more successfulteaching and learning of rational numbers.

• Children’s understanding of quantities is oftenahead of their knowledge of fractionalrepresentations when they solve problems usingthe correspondence scheme. Schools could makeuse of children’s informal knowledge of fractionalquantities and work with problems aboutsituations, without requiring them to use formalrepresentations, to help them consolidate thisreasoning and prepare them for formalization.

• Research has identified the arguments that childrenuse when comparing fractions and trying to seewhether they are equivalent or to order them bymagnitude. It would be important to investigatenext whether increasing teachers’ awareness ofchildren’s own arguments would help teachersguide children’s learning more effectively.

• In some countries, greater attention is given todecimal representation than to ordinary fractionsin primary school whereas in others ordinaryfractions continue to play an important role. Theargument that decimals are easier to understandthan ordinary fractions does not find support insurveys of students’ performance: students find itdifficult to make judgements of equivalence andorder both with decimals and with ordinaryfractions.

• Some researchers (e.g. Nunes, 1997; Tall, 1992;Vergnaud, 1997) argue that differentrepresentations shed light onto the same conceptsfrom different perspectives. This would suggest that a way to strengthen students’ learning ofrational numbers is to help them connect bothrepresentations. Moss and Case (1999) analysedthis possibility in the context of a curriculum basedon measurements, where ordinary fractions andpercentages were used to represent the same


information. Their results are encouraging, but thestudy does not include the appropriate controlsthat would allow for establishing firmer conclusions.

• Students can learn procedures for comparing,adding and subtracting fractions without connectingthese procedures with their understanding ofequivalence and order of fractional quantities,independently of whether they are taught withordinary or decimal representation. This is not adesired outcome of instruction, but seems to be aquite common one. Research that focuses on theuse of children’s informal knowledge suggests that itis possible to help students make connections (e.g.Mack, 1990), but the evidence is limited. There isnow considerably more information regardingchildren’s informal strategies to allow for newteaching programmes to be designed and assessed.

• Finally, this review opens the way for a freshresearch agenda in the teaching and learning offractions. The source for the new researchquestions is the finding that children achieve insightsinto relations between fractional quantities beforeknowing how to represent them. It is possible toenvisage a research agenda that would not focuson children’s misconceptions about fractions, buton children’s possibilities of success when teachingstarts from thinking about quantities rather thanfrom learning fractional representations.

Endnotes

1 Rational numbers can also be used to represent relations thatcannot be described by a single whole number but therepresentation of relations will not be discussed here.

2 The authors report a successful programme of instructionwhere they taught the students to establish connectionsbetween their understanding of ratios and decimals. Thestudents had received two years of instruction on ratios. A fulldiscussion of this very interesting work is not possible here asthe information provided in the paper is insufficient.

3 There are different hypotheses regarding what types ofsubconstructs or meanings for rational numbers should bedistinguished (see, for example, Behr, Harel, Post and Lesh,1992; Kieren, 1988) and how many distinctions are justifiable.Mathematicians and psychologists may well use differentcriteria and consequently reach different conclusions.Mathematicians might be looking for conceptual issues inmathematics and psychologists for distinctions that have animpact on children’s learning (i.e. show different levels ofdifficulty or no transfer of learning across situations). We havedecided not to pursue this in detail but will consider thisquestion in the final section of the paper.

4 Steffe and his colleagues have used a different type of problem,where the size of the part is fixed and the children have toidentify how many times it fits into the whole.

5 This classification should not be confused with the classificationof division problems in the mathematics education literature.Fischbein, Deri, Nello and Marino (1985) define partitivedivision (which they also term sharing division) as a model forsituations in which ‘an object or collection of objects is dividedinto a number of equal fragments or sub-collections. Thedividend must be larger than the divisor ; the divisor (operator)must be a whole number; the quotient must be smaller thanthe dividend (operand)… In quotative division or measurementdivision, one seeks to determine how many times a givenquantity is contained in a larger quantity. In this case, the onlyconstraint is that the dividend must be larger than the divisor. Ifthe quotient is a whole number, the model can be seen asrepeated subtraction.’ (Fischbein, Deri, Nello and Marino, 1985,p.7). In both types of problems discussed by Fishbein et al., thescheme used in division is the same, partitioning, and thesituations are of the same type, part-whole.

6 Empson, Junk, Dominguez and Turner (2005) have stressed that‘the depiction of equal shares of, for example, sevenths in apart–whole representation is not a necessary step tounderstanding the fraction 1/7 (for contrasting views, seeCharles and Nason, 2000; Lamon, 1996; Pothier and Sawada,1983). What is necessary, however, is understanding that 1/7 isthe amount one gets when 1 is divided into 7 same-sized parts.’

7 Not all intensive quantities are represented by fractions;speed, for example, is represented by a ratio, such as in 70miles per hour.

8 Vergnaud (1983) proposed this hypothesis in his comparisonbetween isomorphism of measures and product of measuresproblems. This issue is discussed in greater detail in anotherpaper 4 of this review.



Behr, M., Harel, G., Post, T., and Lesh, R. (1992).Rational number, ratio, proportion. In D. A.Grouws (Ed.), Handbook of Research onMathematics Teaching and Learning (pp. 296-333).New York: Macmillan.

Behr, M. J., Harrel, G., Post, R., & Lesh, R. (1993).Toward a unified discipline of scientific enquiry. InT. Carpenter, P. E. Fennema & T. A. Romberg (Eds.),Rational Numbers: an integration of research (pp.13-48). Hillsdale, NJ: Lawrence Erlbaum Ass.

Behr, M., Lesh, R., Post, T., and Silver, E. A. (1983).Rational number concepts. In R. Lesh and M.Landau (Eds.), Acquisition of MathematicalConcepts and Processes (pp. 91-126). New York:Academic Press.

Behr, M. J., Wachsmuth, I., Post, T. R., and Lesh, R. (1984).Order and equivalence of rational numbers: Aclinical teaching experiment. Journal for Research inMathematics Education, 15(5), 323-341.

Brousseau, G., Brousseau, N., and Warfield, V. (2004).Rationals and decimals as required in the schoolcurriculum Part 1: Rationals as measurement.Journal of Mathematical Behavior, 23, 1–20.

Brousseau, G., Brousseau, N., and Warfield, V.(2007). Rationals and decimals as required inthe school curriculum Part 2: From rationals to decimals. Journal of Mathematical Behavior 26 281–300.

Booth, L. R. (1981). Child-methods in secondarymathematics. Educational Studies in Mathematics,12 29-41.

Byrnes, J. P. (1992). The conceptual basis ofprocedural learning. Cognitive Development, 7,235-237.

Byrnes, J. P. and Wasik, B. A. (1991). Role ofconceptual knowledge in mathematicalprocedural learning. Developmental Psychology,27(5), 777-786.

Carpenter, T. P., and Moser, J. M. (1982). Thedevelopment of addition and subtractionproblem solving. In T. P. Carpenter, J. M. Moser andT. A. Romberg (Eds.), Addition and subtraction: Acognitive perspective (pp. 10-24). Hillsdale (NJ):Lawrence Erlbaum.

Charles, K. and Nason, R. (2000). Young children’spartitioning strategies. Educational Studies inMathematics, 43, 191–221.

Cooney, T. J., Grouws, D. A., and Jones, D. (1988). Anagenda for research on teaching mathematics. InD. A. Grouws and T. J. Cooney (Eds.), Effectivemathematics teaching (pp. 253-261). Reston, VA:National Council of Teachers of Mathematics andHillsdale, NJ: Erlbaum.

Correa, J. (1994). Young children’s understanding of thedivision concept. University of Oxford, Oxford.

Correa, J., Nunes, T., and Bryant, P. (1998). Youngchildren’s understanding of division: Therelationship between division terms in anoncomputational task. Journal of EducationalPsychology, 90, 321-329.

Cowan, R., and Daniels, H. (1989). Children’s use ofcounting and guidelines in judging relative number.British Journal of Educational Psychology, 59, 200-210.

Cowan, R. (1987). When do children trust countingas a basis for relative number judgements? Journalof Experimental Child Psychology, 43, 328-345.

Cowan, R., Foster, C. M., and Al-Zubaidi, A. S. (1993).Encouraging children to count. British Journal ofDevelopmental Psychology, 11, 411-420.

Davis, G. E., and Hunting, R. P. (1990). Spontaneouspartitioning: Preschoolers and discrete items.Educational Studies in Mathematics, 21, 367-374.

Davis, G. E., and Pepper, K. L. (1992). Mathematicalproblem solving by pre-school children.Educational Studies in Mathematics, 23, 397-415.

Davis, G. E., and Pitkethly, A. (1990). Cognitive aspectsof sharing. Journal for Research in MathematicsEducation, 21, 145-153.


References

Empson, S. B. (1999). Equal sharing and sharedmeaning: The development of fraction concepts ina First-Grade classroom. Cognition and Instruction,17(3), 283-342.

Empson, S. B., Junk, D., Dominguez, H., and Turner, E.(2005). Fractions as the coordination ofmultiplicatively related quantities: a cross-sectionalstudy of children’s thinking. Educational Studies inMathematics, 63, 1-28.

Erickson, G. (1979). Children’s conceptions of heatand temperature. Science Education, 63, 221-230.

Fennell, F. S., Faulkner, L. R., Ma, L., Schmid, W., Stotsky,S., Wu, H.-H., et al. (2008). Report of the TaskGroup on Conceptual Knowledge and SkillsWashington DC: U.S. Department of Education,The Mathematics Advisory Panel.

Fischbein, E., Deri, M., Nello, M., and Marino, M. (1985).The role of implicit models in solving verbalproblems in multiplication and division. Journal forResearch in Mathematics Education, 16, 3-17.

Frydman, O., and Bryant, P. E. (1988). Sharing and theunderstanding of number equivalence by youngchildren. Cognitive Development, 3, 323-339.

Frydman, O., and Bryant, P. (1994). Children’sunderstanding of multiplicative relationships in theconstruction of quantitative equivalence. Journal ofExperimental Child Psychology, 58, 489-509.

Ginsburg, H. (1977). Children’s Arithmetic: The LearningProcess. New York: Van Nostrand.

Gréco, P. (1962). Quantité et quotité: nouvellesrecherches sur la correspondance terme-a-termeet la conservation des ensembles. In P. Gréco andA. Morf (Eds.), Structures numeriques elementaires:Etudes d’Epistemologie Genetique Vol 13 (pp. 35-52). Paris: Presses Universitaires de France.

Hart, K. (1986). The step to formalisation. Insititute ofEducation, University of London: Proceedings ofthe 10th International Conference of thePsychology of Mathematics Education, pp 159-164.

Hart, K., Brown, M., Kerslake, D., Kuchermann, D., &Ruddock, G. (1985). Chelsea DiagnosticMathematics Tests. Fractions 1. Windsor (UK):NFER-Nelson.

Hallett, D., Nunes, T., and Bryant, P. (2007). Individualdifferences in conceptual and proceduralknowledge when learning fractions. under review.

Hiebert, J., and Tonnessen, L. H. (1978). Developmentof the fraction concept in two physical contexts:An exploratory investigation. Journal for Researchin Mathematics Education, 9(5), 374-378.

Hunting, R. P. and Sharpley, C. F. (1988 a).Preschoolers’ cognition of fractional units. BritishJournal of Educational Psychology, 58, 172-183.

Hunting, R. P. and Sharpley, C. F. (1988 b). Fractionalknowledge in preschool children. Journal forResearch in Mathematics Education, 19(2), 175-180.

Kamii, C., and Clark, F. B. (1995). Equivalent Fractions:Their Difficulty and Educational Implications.Journal of Mathematical Behavior, 14, 365-378.

Kaput, J. (1985). Multiplicative word problems andintensive quantities: An integrated software response(Technical Report 85-19). Cambridge (MA):Harvard University, Educational Technology Center.

Kerslake, D. (1986). Fractions: Children’s Strategies andErrors: A Report of the Strategies and Errors inSecondary Mathematics Project. Windsor: NFER-Nelson.

Kieren, T. (1988). Personal knowledge of rationalnumbers: Its intuitive and formal development. InJ. Hiebert and M. Behr (Eds.), Number conceptsand operations in the middle-grades (pp. 53-92).Reston (VA): National Council of Teachers ofMathematics.

Kieren, T. E. (1993). Rational and Fractional Numbers:From Quotient Fields to RecursiveUnderstanding. In T. Carpenter, E. Fennema and T.A. Romberg (Eds.), Rational Numbers: AnIntegration of Research (pp. 49-84). Hillsdale, N.J.:Lawrence Erlbaum Associates.

Kieren, T. (1994). Multiple Views of MultiplicativeStructures. In G. Harel and J. Confrey (Eds.), TheDevelopment of Multiplicative Reasoning in theLearning of Mathematics (pp. 389-400). Albany,New York: State University of New York Press.

Kornilaki, K., and Nunes, T. (2005). GeneralisingPrinciples in spite of Procedural Differences:Children’s Understanding of Division. CognitiveDevelopment, 20, 388-406.

Kouba, V. L., Brown, C. A., Carpenter, T. P., Lindquist, M.M., Silver, E. A., and Swafford, J. (1988). Results ofthe Fourth NAEP Assessment of Mathematics:Number, operations, and word problems.Arithmetic Teacher, 35 (4), 14-19.

Lachance, A., and Confrey, J. (2002). Helping studentsbuild a path of understanding from ratio andproportion to decimal notation. Journal ofMathematical Behavior, 20 503–526.

Lamon, S. J. (1996). The Development of Unitizing: ItsRole in Children’s Partitioning Strategies. Journal forResearch in Mathematics Education, 27, 170-193.

Larson, C. N. (1980). Locating proper fractions onnumber lines: Effect of length and equivalence.School Science and Mathematics, 53, 423-428.

Mack, N. K. (1990). Learning fractions withunderstanding: Building on informal knowledge.Journal for Research in Mathematics Education, 21,16-32.



Mamede, E., Nunes, T., & Bryant, P. (2005). Theequivalence and ordering of fractions in part-wholeand quotient situations. Paper presented at the29th PME Conference, Melbourne; Volume 3,281-288.

Mamede, E. P. B. d. C. (2007). The effects ofsituations on children’s understanding of fractions.Unpublished PhD Thesis, Oxford BrookesUniversity, Oxford.

Michie, S. (1984). Why preschoolers are reluctant to count spontaneously. British Journal ofDevelopmental Psychology, 2, 347-358.

Miller, K. (1984). Measurement procedures and thedevelopment of quantitative concepts. In C.Sophian (Ed.), Origins of cognitive skills. TheEighteen Annual Carnegie Symposium onCognition (pp. 193-228). Hillsdale, New Jersey:Lawrence Erlbaum Associates.

Moss, J., and Case, R. (1999). Developing children’sunderstanding of rational numbers: A new modeland an experimental curriculum. Journal forResearch in Mathematics Education, 30, 122-147.

Noelting, G. (1980 a). The development ofproportional reasoning and the ratio conceptPart I - Differentiation of stages. EducationalStudies in Mathematics, 11, 217-253.

Noelting, G. (1980 b). The development ofproportional reasoning and the ratio conceptPart II - Problem-structure at sucessive stages:Problem-solving strategies and the mechanism ofadaptative restructuring. Educational Studies inMathematics, 11, 331-363.

Novillis, C. F. (1976). An Analysis of the FractionConcept into a Hierarchy of SelectedSubconcepts and the Testing of the HierarchicalDependencies. Journal for Research in MathematicsEducation, 7, 131-144.

Nunes, T. (1997). Systems of signs and mathematicalreasoning. In T. Nunes and P. Bryant (Eds.),Learning and Teaching Mathematics. AnInternational Perspective (pp. 29-44). Hove (UK):Psychology Press.

Nunes, T., and Bryant, P. (2004). Children’s insights andstrategies in solving fractions problems. Paperpresented at the First International Conference ofthe Greek Association of Cognitive Psychology,Alexandropolis, Greece.

Nunes, T., Bryant, P., Pretzlik, U., and Hurry, J. (2006).Fractions: difficult but crucial in mathematicslearning. London Institute of Education, London:ESRC-Teaching and Learning ResearchProgramme.

Nunes, T., Bryant, P., Pretzlik, U., Bell, D., Evans, D., &Wade, J. (2007). La compréhension des fractionschez les enfants. In M. Merri (Ed.), Activitéhumaine et conceptualisation. Questions à GérardVergnaud (pp. 255-262). Toulouse, France:Presses Universitaires du Mirail.

Nunes, T., Bryant, P., Pretzlik, U., Evans, D., Wade, J., andBell, D. (2008). Children’s understanding of theequivalence of fractions. An analysis of argumentsin quotient situations. submitted.

Nunes, T., Desli, D., and Bell, D. (2003). Thedevelopment of children’s understanding ofintensive quantities. International Journal ofEducational Research, 39, 652-675.

Piaget, J. (1952). The Child’s Conception of Number.London: Routledge and Kegan Paul.

Piaget, J., and Inhelder, B. (1975). The Origin of the Ideaof Chance in Children. London: Routledge andKegan Paul.

Piaget, J., Inhelder, B., and Szeminska, A. (1960). TheChild’s Conception of Geometry. New York: Harperand Row.

Ohlsson, S. (1988). Mathematical meaning andapplicational meaning in the semantics offractions and related concepts. In J. Hiebert andM. Behr (Eds.), Number Concepts and Operationsin the Middle Grades (pp. 53-92). Reston (VA):National Council of Mathematics Teachers.

Olive, J., and Steffe, L. P. (2002). The construction ofan iterative fractional scheme: the case of Joe.Journal of Mathematical Behavior, 20 413-437.

Olive, J., and Vomvoridi, E. (2006). Making sense ofinstruction on fractions when a student lacksnecessary fractional schemes: The case of Tim.Journal of Mathematical Behavior, 25 18-45.

Piaget, J. (1952). The Child’s Conception of Number.London: Routledge and Kegan Paul.

Piaget, J., Inhelder, B., and Szeminska, A. (1960). TheChild’s Conception of Geometry. New York: Harperand Row.

Pitkethly, A., and Hunting, R. (1996). A Review ofRecent Research in the Area of Initial FractionConcepts. Educational Studies in Mathematics, 30,5-38.

Pothier, Y., and Sawada, D. (1983). Partitioning: TheEmergence of Rational Number Ideas in YoungChildren. Journal for Research in MathematicsEducation, 14, 307-317.

Resnick, L. B., Nesher, P., Leonard, F., Magone, M.,Omanson, S., and Peled, I. (1989). Conceptualbases of arithmetic errors: The case of decimalfractions. Journal for Research in MathematicsEducation, 20, 8-27.


Riley, M., Greeno, J. G., and Heller, J. I. (1983).Development of children’s problem solving abilityin arithmetic. In H. Ginsburg (Ed.), Thedevelopment of mathematical thinking (pp. 153-196). New York: Academic Press.

Rittle-Johnson, B., Siegler, R. S., and Alibali, M. W.(2001). Developing Conceptual Understandingand Procedural Skill in Mathematics: An IterativeProcess. Journal of Educational Psychology, 93(2),46-36.

Saenz-Ludlow, A. (1994). Michael’s FractionSchemes. Journal for Research in MathematicsEducation, 25, 50-85.

Saxe, G., Guberman, S. R., and Gearhart, M. (1987).Social and developmental processes in children’sunderstanding of number. Monographs of the Societyfor Research in Child Development, 52, 100-200.

Schwartz, J. (1988). Intensive quantity and referenttransforming arithmetic operations. In J. Hiebertand M. Behr (Eds.), Number concepts andoperations in the middle grades (pp. 41-52).Hillsdale,NJ: Erlbaum.

Stafylidou, S., and Vosniadou, S. (2004). Thedevelopment of students’ understanding of thenumerical value of fractions. Learning andInstruction 14, 503-518.

Stavy, R., Strauss, S., Orpaz, N., and Carmi, G. (1982).U-shaped behavioral growth in rationcomparisons, or that’s funny I would not havethought you were U-ish. In S. Strauss and R. Stavy(Eds.), U-shaped Behavioral Growth (pp. 11-36).New York: Academic Press.

Stavy, R., and Tirosh, D. (2000). How Students (Mis-)Understand Science and Mathematics. IntuitiveRules. New York: Teachers College Press.

Steffe, L. P. (2002). A new hypothesis concerningchildren’s fractional knowledge. Journal ofMathematical Behavior, 102, 1-41.

Steffe, L. P., and Tzur, R. (1994). Interaction andchildren’s mathematics. Journal of Research inChildhood Education, 8, 99- 91 16.

Streefland, L. (1987). How lo Teach Fractions So As toBe Useful. Utrecht, The Netherlands: The StateUniversity of Utrecht.

Streefland, L. (1990). Free productions in Teachingand Learning Mathematics. In K. Gravemeijer, M.van den Heuvel & L. Streefland (Eds.), ContextsFree Production Tests and Geometry in RealisticMathematics Education (pp. 33-52). UtrechtNetherlands: Research group for MathematicalEducation and Educational Computer Centre -State University of Utrecht.

Streefland, L. (1993). Fractions: A Realistic Approach.In T. P. Carpenter, E. Fennema and T. A. Romberg(Eds.), Rational Numbers: An Integration of Research(pp. 289-326). Hillsdale, NJ: Lawrence ErlbaumAssociates.

Streefland, L. (1997). Charming fractions or fractionsbeing charmed? In T. Nunes and P. Bryant (Eds.),Learning and Teaching Mathematics. AnInternational Perspective (pp. 347-372). Hove (UK):Psychology Press.

Tall, D. (1992). The transition to advancedmathematical thinking: Functions, limits, infinity, andproof. In D. A. Grouws (Ed.), Handbook ofResearch on Mathematics Teaching and Learning(pp. 495-511). New York: Macmillan.

Tzur, R. (1999). An Integrated Study of Children’sConstruction of Improper Fractions and theTeacher’s Role in Promoting That Learning. Journalfor Research in Mathematics Education, 30, 390-416.

Vamvakoussi, X., and Vosniadou, S. (2004).Understanding the structure of the set of rationalnumbers: a conceptual change approach. Learningand Instruction, 14 453-467.

Vergnaud, G. (1997). The nature of mathematicalconcepts. In T. Nunes and P. Bryant (Eds.), Learningand Teaching Mathematics. An InternationalPerspective (pp. 1-28). Hove (UK): PsychologyPress.

Vergnaud, G. (1983). Multiplicative structures. In R. Lesh & M. Landau (Eds.), Acquisition ofmathematics concepts and processes(pp. 128-175). London: Academic Press.

Yackel, E., Cobb, P., Wood, T., Wheatley, G., andMerkel, G. (1990). The importance of socialinteraction in children’s construction ofmathematical knowledge. In T. J. Cooney and C. R.Hirsch (Eds.), Teaching and learning mathematics inthe 1990s. 1990 Yearbook of the National Councilof Teachers of Mathematics (pp. 12-21). Reston,VA: National Council of Teachers of Mathematics.


paper 3: understanding rational numbers and intensive quantities

Documents