2002 Children Concept of Addition

Educational Psychology, Vol. 22, No. 5, 2002

Young Children’s Understanding of AdditionConcepts

KATHERINE H. CANOBI, ROBERT A. REEVE &PHILIPPA E. PATTISON, The University of Melbourne, Australia

ABSTRACT Children’s knowledge of concrete versions of additive composition, commutativityand associativity was investigated in two studies. In Study 1, 24 four- to � ve-year-olds and25 � ve- to six-year-olds judged the equivalence of conceptually related addition problemspresented using groups of objects. In Study 2, 45 � ve- to six-year-olds judged related problemsand solved addition problems. Both studies indicated that concrete versions of principles weresalient to most children although associativity was more dif� cult than commutativity and therewere considerable individual differences in children’s understanding. Study 1 results indicatedthat schoolchildren were more accurate at recognising additive composition than preschoolersand Study 2 results suggested that commutativity knowledge was related to using advancedcounting strategies for solving addition problems. Overall, the research supports the claim thatexamining early knowledge of addition principles provides important insights into children’semerging part–whole knowledge and mathematical development.

Introduction

The aim of the research was to explore children’s knowledge of concrete versions ofaddition principles in order better to understand the emergence and development ofpart–whole knowledge. Recognising the ways in which a whole is composed of differentparts is fundamental to number sense and underlies many relationships betweenaddition problems. For example, parts added in different orders still equal the whole,therefore a 1 b 5 b 1 a (commutativity). Principles such as additive composition, com-mutativity and associativity are fundamental properties of addition and exploring thesequence in which children learn about them is likely to shed light on the developmentof part–whole knowledge. However, despite the prominence of such principles in keytheories of mathematical development (Gelman & Gallistel, 1978; Piaget, 1952;Resnick, 1992), surprisingly little is known about how children learn about them. Forexample, some principles (such as associativity) are more complex than others (such as

ISSN 0144-3410 print; ISSN 1469-046X online/02/050513-20 Ó 2002 Taylor & Francis LtdDOI: 10.1080/0144341022000023608

514 K. H. Canobi et al.

commutativity) and may be acquired later but few studies have addressed the develop-mental sequence in children’s part–whole knowledge. The lack of research into additionprinciples is especially problematic given the evidence that individual differences inchildren’s knowledge of the principles are systematically related to their skill in solvingschool addition problems (Canobi, in press; Canobi, Reeve, & Pattison, 1998). Re-search into children’s knowledge of different principles is needed in order to understandthe emergence and development of conceptual understanding in addition.

Because the addition principles vary in complexity, they provide a useful frameworkfor investigating different forms of part–whole knowledge. Additive composition is theprinciple that larger sets are made up of smaller sets. Commutativity is the principlethat problems containing the same sets in a different order have the same answer,a 1 b 5 b 1 a. Associativity is the principle that problems in which sets are decomposed,and recombined in different orders, have the same answer, (a 1 b) 1 c 5 a 1 (b 1 c).

It seems likely that knowledge of addition principles emerges through noticingregularities in the ways in which physical objects can be combined. For example, theprocess of combining sets is commutative in the sense that the order in which groupsof objects are combined is irrelevant to the total number of objects in the combined set.An appreciation of principle-based regularities in interactions with sets of objects isviewed as important to conceptual development (Gelman & Gallistel, 1978; Piaget,1952; Resnick, 1992). For instance, Resnick (1986, 1992, 1994) argues that concep-tual development occurs as children map new forms of understanding onto an initially“protoquantitative” part–whole schema. Speci� cally, children may initially understandcommutativity and associativity in terms of how physical objects can be joined togetherand a crucial development occurs when counting knowledge is combined with thepart–whole schema so that children can reason using equations such as 2 apples 1 3apples 5 3 apples 1 2 apples. Resnick argues that, at a later stage, children begin toreason with numbers independently of their referential context (2 1 3 5 3 1 2) beforeunderstanding the principles as abstract rules (a 1 b 5 b 1 a). The claim that children� rst learn about addition principles in the context of physical objects (Gelman &Gallistel, 1978; Resnick, 1992) has important theoretical and educational implicationsand further research is needed to specify changes in children’s understanding ofconcrete versions of part–whole concepts.

Because the addition principles are likely to be important to children’s conceptualunderstanding, it is of interest to explore the sequence or sequences in which childrenlearn about them. However, not all researchers suggest a separation of additionprinciples in children’s representations. For example, Resnick (1992, 1994) claims thatassociativity and commutativity are not distinct in children’s understanding, citing alongitudinal study of Pitt, a seven-year-old who regarded commutativity and associativ-ity as self-evident permissions rooted in additive composition. However, although Pittmay have come to recognise the interdependency of the principles, it is possible that hecame to appreciate the principles at different stages. Moreover, there is some evidencethat commutativity may be acquired before associativity (Canobi et al., 1998; Close &Murtagh, 1986; Langford, 1981). For example, Close and Murtagh (1986) found thatchildren correctly solved more written problems designed to re� ect commutativity thanassociativity, but this difference may have been associated with the computationalrather than conceptual demands involved in solving three-addend problems. Canobi etal. (1998) measured conceptual knowledge separately from problem solving and foundthat children were more successful at recognising and explaining the relationshipbetween commuted problems than those depicting aspects of additive composition and

Children’s Understanding of Addition Concepts 515

associativity. However, the � ndings of both studies pertain to symbolically presentedproblems (2 1 3 5 3 1 2) and it is unclear how the results apply to children’s reasoningabout physical objects. Langford (1981) investigated concrete versions of the principlesin a longitudinal study in which children’s responses to an interviewer’s descriptions ofactions on covered boxes of beans suggested that knowledge of commutativity precedesassociativity. However, in this study, children needed to remember the interviewer’sdescriptions in order to respond correctly and associativity items involved more sets(and longer descriptions) than commutativity items. Therefore, further research intothe comparative dif� culty of recognising various addition principles in the context ofphysical objects is needed. Such research should control for possible confoundingfactors such as a reliance on verbal instructions and the potential use of computationprocedures on tasks designed to measure conceptual understanding.

A further interpretive dif� culty with previous research is that it is unclear whychildren � nd associativity [(a 1 b) 1 c 5 a 1 (b 1 c)] comparatively dif� cult. For exam-ple, Langford’s (1981) associativity task was the same as his commutativity task, exceptthat it involved three boxes of beans instead of two. Presented symbolically, Langford’sassociativity task may have been more closely analogous to the equationa 1 b 1 c 5 b 1 c 1 a than to the equation (a 1 b) 1 c 5 a 1 (b 1 c). However, in order toassess associativity understanding, it is necessary to assess knowledge of decomposingand recombining sets. For instance, Resnick and Omanson (1987) reported an exampleof associativity knowledge among school children who solved problems such as 23 1 8by decomposing 23 into 20 1 3 then recon� guring the problem into (20 1 8) 1 3(Resnick, 1992). In this example, children decomposed one addend and then recom-bined the resulting numbers in a new order. This re� ects aspects of the principle notassessed by Langford. Canobi et al. (1998) examined problem relationships of the form(a 1 b) 1 c 5 a 1 b 1 c and a 1 b 1 c 5 a 1 (b 1 c), thereby assessing aspects of associa-tivity related to additive composition but not the complete principle. Therefore, the roleof knowledge about three rather than two sets as well as the composition and orderingof sets in part–whole development is unclear.

Children’s responses to these mathematical principles may allow the identi� cation ofpro� les of part–whole knowledge. In support, Canobi et al. (1998) found that a keyaspect of individual differences in conceptual knowledge was a tendency for children to

1. understand both commutativity and associativity type relations2. understand only commutativity type relations or3. understand neither form of relation

Identifying the mathematical relationships that children understand is consistent withcalls for investigations of knowledge pro� les across mathematical tasks (Bisanz &Lefevre, 1992; Sophian, 1997) and claims that greater attention should be paid toindividual differences in children’s mathematics (Dowker 1998, Pellegrino & Goldman,1989; Siegler, 1987, 1996; Widaman & Little, 1992).

In addition to helping explain individual differences in children’s addition, examiningchildren’s emerging knowledge of addition principles has the potential to shed light onthe connections children make between informal knowledge and school mathematics.In particular, it may be useful to compare the addition concepts of children who havenot yet entered school with those who have begun to learn school mathematics. A studyby Canobi et al. (1998) suggests that the accuracy of 6- to 8-year-olds’ explanations ofproblem relationships based on additive composition, commutativity and associativity


is related to their problem solving skills; in particular, children with advancedpatterns of principle knowledge are faster, more accurate and more � exible at solvingschool addition problems than other children. Conceptually advanced children aremore likely to report retrieving problem answers from memory, as well as usingdecomposition (derived fact) and advanced counting strategies to solve problems(Canobi et al., 1998).

Although these � ndings suggest that older children’s problem solving is related toprinciple-type knowledge in the context of symbolic problems, less is known aboutyounger children’s knowledge of concrete versions of the principles and their earlycounting and problem solving skills. Indeed, a study by Sophian, Harley, and Martin(1995) suggests that children as young as three have some appreciation of principlesin physical contexts even when they cannot enumerate the sets to be compared.Sophian and colleagues argue that this research supports claims by Resnick (1992,1994) that for very young children, understanding of part–whole relations is indepen-dent of mental representations underlying quanti� cation and counting. Moreover,based on a study of � ve- to six-year-olds, Baroody and Gannon (1984) argue thatchildren’s use of the min strategy (counting on from the larger addend) does notnecessarily re� ect knowledge of commutativity. These two studies suggest that chil-dren’s early counting and problem solving skills may not be related to their knowledgeof addition principles.

Nonetheless, other studies suggest that conceptual knowledge underlies children’suse of advanced counting procedures to solve addition problems (Cowan & Renton,1996; Fuson, 1982, 1988; Martins-Mourao & Cowan, 1998; Siegler & Crowley, 1994).For example, some researchers suggest that children’s use of order-indifferent countingstrategies such as min re� ect a functional understanding of commutativity (Canobi etal., 1998; Cowan & Renton, 1996; Groen & Resnick, 1977). Similarly, based onchildren’s rearrangement of quantities in word problems and construction of amountswith different coins as compared with their counting on, Martins-Mourao & Cowan(1998) argue that counting on may be a consequence of understanding additivecomposition. Moreover, conceptual understanding of what constitutes a legitimateaddition strategy precedes children’s ability to count on from the larger addend insteadof counting all addends starting from one (Siegler & Crowley, 1994). Counting on isa more ef� cient strategy because children start their � nal count of the two addendsfrom one of the addends instead of starting their � nal count from zero (therefore tosolve 3 1 2, children count, “three, four, � ve”). Also, Fuson argues that counting onre� ects a signi� cant conceptual advance as it involves representing an addend using acardinal number in the � nal count (Fuson, 1982, 1988). However, although separatestudies suggest that forms of conceptual understanding may be related to usingparticular counting procedures when solving addition problems, this work has tendedto focus on isolated aspects of the relationship between conceptual knowledge andproblem solving skill in older children. Research into relations between children’spart–whole concepts and early problem solving is needed.

In the present research, children judged the equivalence of pairs of addition problemspresented using physical objects. Problem pairs varied in the order in which groups ofobjects were combined as well as the composition and number of groups. In order toexamine knowledge related to different principles, children judged the equivalence of:

1. two- and three-addend commuted problems such as a 1 b 5 b 1 a anda 1 b 1 c 5 a 1 c 1 b


2. two- and three-addend problems in which sets were decomposed or combined, forexample, (a 1 b) 5 a 1 b and (a 1 b) 1 c 5 a 1 b 1 c

3. problem pairs analogous to associativity: (a 1 b) 1 c 5 a 1 (b 1 c)

Parentheses in equations were represented by groups presented in combination (groupswere presented in a single container not separate containers). A large set of anunspeci� ed numerosity was used, in order to prevent children from using computa-tional procedures such as mental calculation, counting or subitising. An interviewermoved uncovered groups of objects so children did not need to remember verbaldescriptions.

The claim that commutativity knowledge precedes associativity knowledge (Canobiet al., 1998; Close & Murtagh, 1986; Langford, 1981) was explored by addressing threeissues. The � rst was whether this � nding would be supported for concrete versions ofthe principles. The second was whether there are differences between children’sresponses to problem pairs designed to re� ect additive composition and commutativityand between their responses to problem pairs designed to re� ect additive compositionand associativity. Exploring this issue would provide insight into whether children’sdif� culties in understanding associativity are due to a weakness in understandingadditive composition, and whether early forms of additive composition understandingprecede commutativity knowledge, as might be expected on the basis of Resnick’stheory (1986, 1992). The third issue was the involvement of three rather than two setsin associativity. If the relative dif� culty of associativity is only due to the need toconsider three sets, children should � nd it more dif� cult to judge additive compositionand commutativity problems involving three rather than two groups. Examining theseissues was expected to provide insight into the relationships between knowledge ofdifferent principles, and what aspects of the complex associativity principle are dif� cultfor children (the presence of three sets and/or the decomposition of sets and/orrecombination of sets). More generally, examining these issues was expected to provideinsight into the development of children’s part–whole knowledge in the context ofphysical objects.

In addition to exploring the relative dif� culty of part–whole concepts, the researchwas designed to compare the understanding of children who have just entered schoolto that of younger children. Examining age group differences was expected to provideinsight into how conceptual understanding develops after children enter school. How-ever, focussing on age group differences alone could lead to an inaccurate picture ofdevelopment because important individual differences among children of the same agecould be overlooked. Therefore, in addition to comparing the performance of agegroups, a cluster analysis exploring different pro� les of performance was conducted.

The relationship between children’s conceptual judgements and problem solvingwith the aid of counters was also examined. Given previous research indicating thatsophisticated counting strategies have conceptual underpinnings (Canobi et al., 1998;Cowan & Renton, 1996; Fuson, 1982, 1988; Martins-Mourao & Cowan, 1998; Siegler& Crowley, 1994), part–whole knowledge was expected to be related to using order-indifferent counting strategies and counting on strategies. Moreover, based on researchinto older children’s problem solving and judgements of symbolic problems (Canobi etal., 1998), it was hypothesised that children’s patterns of conceptual judgements wouldbe related to their problem solving accuracy.

The relative dif� culty of concrete versions of addition principles and individualdifferences in conceptual knowledge were explored in two studies. Study 1 involved


preschool and school children, enabling an exploration of age group differences. InStudy 2 schoolchildren solved school addition problems as well as making conceptualjudgements, enabling an examination of relations between conceptual knowledge andproblem solving.

Study 1: Method

Participants

Participants attended a primary (elementary) school or kindergarten in multicultural,lower-to-middle socioeconomic status suburbs in a large Australian city. There were 49participants: 11 boys and 13 girls in kindergarten (preschool) whose mean ages were 5years 2 months (SD 5 3 months) and 5 years 2 months (SD 5 5 months) respectively,and 10 boys and 15 girls in preparatory (reception) grade whose mean ages were 5 years11 months (SD 5 4 months) and 6 years 2 months (SD 5 4 months) respectively. Theparents of the children gave written consent to their participation.

Materials and Procedure

An addition principles judgement task was administered in order to explore thechildren’s understanding of concrete versions of additive composition, commutativityand associativity. The task involved making judgements about the equivalence of pairsof addition problems presented using groups of objects. Similar to the procedureadopted by Sophian et al. (1995), children made judgements about the equivalence ofconceptually related and unrelated pairs of problems in the context of deciding whethertwo toys had been given the same number of objects.

A female experimenter interviewed children individually in two 15–25 minutevideotaped sessions because pilot work revealed that some of the younger childrenfound it dif� cult to concentrate throughout one long session. At the start of the � rstsession, the interviewer invited children to play a game in which two toy bears receivedsome smarties (sweets similar to M&Ms), asking them to judge whether the bears hadthe same number of smarties. The interviewer told children that they did not have tocount the smarties. She showed them two 5cm x 3.5 cm blue boxes, saying, “Look,these boxes are the same. They both have three blue smarties in them.” She thenrepeated the procedure with two green boxes, each containing four green smarties. Theinterviewer also showed children two red boxes, each containing 16 red smarties piledon top of each other so that they could not be counted. In order to prevent childrenfrom calculating mentally, the interviewer did not mention the numerosity of the redsmarties but said that the two red boxes contained the same number of red smarties.Children were not allowed to touch the displays.

Children sat facing two toy bears. Each toy had three empty containers in front of it.In order to familiarise children with the task and to check that they remembered thatmatching boxes contained equal numbers of smarties, the interviewer administeredpractice trials at the start of the sessions, in which she poured a single box of smartiesinto a container in front of each toy. As she distributed the smarties, the interviewerdescribed her actions (“Bill gets a box of red smarties and Kate gets a box of redsmarties.”). After distributing the smarties, she asked, “Do Bill and Kate have the samenumber of smarties?” She then informed children whether their judgement was correct,describing the display (“Yes! Bill’s got a box of red smarties and Kate’s got a box of red


smarties. So they do have the same number of smarties.”) After children respondedcorrectly to three consecutive randomly ordered practice trials, including one involvingthe same number of smarties and one involving a different number of smarties, theinterviewer administered the test trials.

The test trials were similar to the practice trials except that the interviewer gave twoor three groups of smarties to each toy, in effect presenting children with a pair ofaddition problems. Table I shows that in order to measure children’s knowledge ofaddition principles in a concrete context, some pairs of problems were conceptuallyrelated. In order to assess commutativity knowledge, trials involved judging the equiv-alence of two problems, each comprising two groups of smarties in separate containers,but with the groups presented in a different order. In order to assess knowledge aboutadditive composition, trials involved judging the equivalence of a problem in which twogroups were combined (in a single container) and one in which the equivalent groupswere presented in separate containers. Three-group order and composition trials wereexactly the same as these two-group trials, except that they involved three groups ofsmarties instead of two. The purpose of testing commutativity and additive compositionknowledge in the context of three groups of objects was to examine whether theinvolvement of three sets increases the dif� culty of principle judgements.

Order of composition trials involved changing the order in which two out of threegroups were combined in a single container. As well as examining commutativityknowledge, the trials were designed to test whether responses to composition trials wereassociated with comparing sets presented in combination with sets presented separately(additive composition knowledge) or with judging any trials involving sets presented incombination. Recomposition trials involved problems containing three groups in whichdifferent pairs of groups were combined. They were designed to address associativityknowledge. In one problem, the � rst two sets were combined (in a single container) whilethe third set was presented separately. In the other problem, the � rst set was presentedin a single container while the other two sets were combined. Children judged threeexamples of each of the six types of trials described in Table I.

All test trials included the group of 16 red smarties in order to measure conceptualknowledge independently of mental calculation or counting. The interviewer distributedsmarties from left to right in such a way that all groups remained visible to children, sothat they did not need to rely on their memory of her actions or words to judge theequivalence of two problems. For example, in order to test commutativity knowledge,the � rst toy received a box of reds in its � rst container and then four greens in its nextcontainer and the second toy received four greens in its � rst container and a box of redsin its next container. As she distributed the smarties, the interviewer described heractions (“Bill gets a box of reds, then he gets four greens. Kate gets four greens, thenshe gets a box of reds.”). Once the smarties were distributed, the interviewer asked, “DoBill and Kate have the same number of smarties?” After children judged whether the“addition problems” in each test display were equal, the interviewer asked them to justifytheir responses (“Why do you think that?”), but gave no feedback.

Children also made nine judgements about identity trials in which problems wereexactly the same (for example., “Bill gets three blues then he gets a box of reds. Kategets three blues then she gets a box of reds.”) and nine judgements about inequality trialsin which the problems were unrelated (for example “Bill gets a box of reds and threeblues. Kate gets three blues and four greens.”). These trials were employed becausejudging unequal problems as equal or judging identical problems as unequal wouldconstitute evidence for response bias. Each child was presented with the test


TABLE I. Examples of trials involving conceptually related problems

Trial type Equationa Interviewer’s description

2-group order r 1 3 Bill gets a box of reds (pours into container) then he getsthree blues (pours into second container). Kate gets

3 1 r three blues (pours into container) then she gets a box ofreds (pours into second container).

3-group order r 1 3 1 4 Bill gets a box of reds (pours into container) then he getsthree blues (pours into second container) then he gets

r 1 4 1 3 four greens (pours into second container). Kate gets a boxof reds (pours into container) then she gets four greens,(pours into second container) then she gets three blues(pours into third container)

Order-of- r 1 (3 1 4) Bill gets a box of reds (pours into container) then he getscomposition three blues and four greens (pours into second

r 1 (4 1 3) container). Kate gets a box of reds (pours intocontainer) then she gets four greens and three blues(pours into second container).

2-group (r 1 3) Bill gets a box of reds and three blues (pours intocomposition container). Kate gets a box of reds (pours into

r 1 3 container) then she gets three blues (pours into secondcontainer).

3-group (r 1 3) 1 4 Bill gets a box of reds and three blues (pours intocomposition container) then he gets four greens (pours into second

r 1 3 1 4 container). Kate gets a box of reds (pours intocontainer) then she gets three blues (pours into secondcontainer) then she gets four greens (pours into thirdcontainer).

Recomposition (r 1 3) 1 4 Bill gets a box of reds and three blues (pours into intocontainer) then he gets four greens (pours into second

r 1 (3 1 4) container). Kate gets a box of reds (pours intocontainer) then she gets three blues and four greens(pours into second container).

a ‘r’ refers to 16 red smarties, ‘3’ refers to three blue smarties, ‘4’ refers to four green smarties,and parentheses refer to combined groups

trials, which included a mix of order, composition, identity and inequality trials, in oneof two random sequences. Each sequence of test trials was split into two sessions of 18test trials, in which children were asked to judge and to justify their judgements, andfor which the order was counterbalanced.

Children’s judgements were coded as correct if they stated that problems containingthe same groups of smarties were equal and problems containing different groups ofsmarties were not. Children’s justi� cations were coded according to whether they madereference to the experimental manipulation (such as noting that groups of smarties werecombined for one toy but placed in separate containers for the other toy) and thenumerosities (colours) of the groups in each problem (for example noting that problemscontained the same groups).

Study 1: Results and discussion

The purpose of Study 1 was to explore the relative dif� culty of judging differentaddition principles in the context of physical objects and to examine age-related


TABLE II. Means (and standard deviations) judgementscores in Study 1 as a function of Age Group

Trial Age 4–5 Age 5–6

Two-group order 70 (31) 85 (31)Three-group order 74 (30) 84 (33)Two-group composition 58 (38) 85 (29)Three-group composition 57 (37) 85 (31)Recomposition 54 (39) 74 (38)n 23 25

Note: Judgement scores are the percentage accuracy ofchildren’s judgements of conceptually related problemsadjusted for their incorrect judgements of nine in-equality problems. There were three judgements ofeach type of trial.

changes in children’s part–whole knowledge. Initially, children’s judgements of identityand inequality trials were examined in order to detect evidence of response bias. Onechild judged all problem pairs as unequal and was not included in further analyses.Other children’s judgements were fairly accurate for identity (mean 5 95%, SD 5 11)and inequality trials (mean 5 87%, SD 5 22), suggesting a good understanding of thetask. In order to adjust for a possible positive response bias and control for chanceresponding, each child’s percentage of false positive judgements (that inequality prob-lems were equal) was subtracted from his/her percentage of correct positive judgements(that conceptually related problems were equal). During the testing, children did notappear to notice the experimental manipulation in order of composition trials. This wasprobably because order of composition trials were the only trials in which the � naldisplay in the two problems was identical. Only two children failed to judge every orderof composition trial correctly so these trials were not analysed further.

The comparative dif� culty of concrete versions of additive composition, commutativ-ity and associativity principles was examined by comparing two-group order, two-groupcomposition and recomposition judgements. Repeated Wilcoxon tests were employedbecause this is a relatively powerful non-parametric approach to repeated measuresdesigns and is appropriate when planned pair-wise comparisons are of interest, pro-vided a conservative alpha level is adopted (Marascuilo & McSweeney, 1977), thereforean alpha level of .01 was used. Table II shows that the accuracy of compositionjudgements did not differ from recomposition or order judgements (z 5 2 1.99,P 5 0.05, and z 5 2 1.81, P 5 0.07, respectively), although the latter may be mainlydue to the older children’s similar performance on order and composition trials as theyounger children had considerably lower mean scores for composition than order trials.As expected, order judgements were more accurate than recomposition judgements(z 5 2 3.06, P 5 0.002). This supports previous research indicating that commutativityprecedes associativity understanding (Canobi et al., 1998; Close & Murtagh, 1986;Langford, 1981). However, group number had no effect on order or compositionjudgements (z 5 2 0.71, P 5 0.48 and z 5 2 0.49, P 5 0.96 respectively), suggestingthat the comparative dif� culty of associativity is not due to the need to consider threesets.

Mann-Whitney U tests were employed to examine differences in the part–wholeknowledge of the four- to � ve-year-old preschoolers and the � ve- to six-year-old


TABLE III. Means (and standard deviations) of thepercentages of different justi� cation types for correct

judgements in Study 1

Trial Age 4–5 Age 5–6

Two-group orderSame groups 83 (28) 89 (25)Different order 12 (31) 3 (9)

Three-group orderSame groups 86 (21) 79 (30)Different order 4 (21) 7 (16)

Two-group compositionSame groups 85 (25) 83 (33)Different combination 2 (7) 0

Three-group compositionSame groups 88 (27) 91 (24)Different combination 0 2 (7)

RecompositionSame groups 72 (38) 93 (24)Different combination 0 0

n 23 25

Note: There were three judgements and justi� cationsgiven for each type of trial.

school children. Because no differences were found in the accuracy of children’sjudgements of two- and three-group trials, these scores were combined, a Bonferroni-type adjustment was made to the alpha level of the three tests and a level of 0.017 wasadopted. Table II shows that children in both age groups tended to make extremelyaccurate commutativity (order) judgements and less accurate judgements of concreteversions of associativity (recomposition trials) and there were no age related improve-ments in order or recomposition judgements (z 5 2 1.96, P 5 0.05 and z 5 2 1.95,P 5 0.05, respectively). However, the accuracy of children’s judgements of compositiontrials suggests that the older children’s concrete understanding of additive compositionis superior to that of younger children (z 5 2 2.65, P 5 0.008).

Justi� cations for correct judgements are summarised in Table III. Justi� cations forincorrect judgements showed no discernable pattern. Table III shows that for bothorder and composition trials, children who made correct judgements tended to focus onwhether the same groups of objects were present in both problems, rather than ondifferences between problems. Speci� cally, most children justi� ed their correct judge-ments of order and composition trials by describing the sets that were present in bothadditions (for example, “Bill has greens and reds and Kate has the greens and redstoo.”) or simply stating that the same sets were present (for example, “Bill and Katehave the same smarties.”). Such justi� cations on recomposition trials appeared morecommon among older children. Few children made reference to the experimentalmanipulation that led to differences between the problems when explaining theircorrect judgements. That is, few children stated that groups had been distributed indifferent orders or combined in different ways. Thus, correct judgements appear tore� ect an ability to concentrate on the equivalence of groups of objects in pairs ofproblems, despite differences in the ways these groups were combined.

Overall, the results of Study 1 suggest that concrete versions of the addition


principles are quite salient even to preschoolers but as children enter school theydevelop in their understanding that groups of objects can be thought of as being madeup of combinations of smaller groups. The results also suggest that understandingassociativity in the context of physical objects is more dif� cult than understandingcommutativity and that this is due to differences in the conceptual reasoning involved(understanding how sets can be decomposed and recombined) not merely to differ-ences in the number of sets.

Study 2: Method

Participants

Participants attended preparatory grade in two primary schools in multicultural, lower-to-middle socioeconomic status suburbs in a large Australian city. There were 45participants: 21 boys and 24 girls whose mean ages were 6 years (SD 5 4 months) and6 years 1 month (SD 5 3 months) respectively. The parents of the children gave writtenconsent to their participation. Preschool children were not included because they weregenerally unable to solve addition problems.

Materials and procedure

The materials and procedure for Study 2 were the same as those for Study 1, exceptthat in one of the judgement task sessions, children in Study 2 also completed aten-minute problem solving task. (Task order was counterbalanced.) The additionproblem solving task was designed to examine the ways in which children use countersto solve a set of addition problems in order to provide a basis for exploring therelationship between emerging conceptual knowledge and problem solving abilities.Fourteen single-digit addition problems were presented on separate 30 cm by 21 cmsheets in the format a 1 b 5 ?. The interviewer uncovered problems one after the other,and read them out aloud. Problems were constructed by randomly selecting twoaddends between one and ten. No number appeared twice in one problem. Theposition of the larger addend was counterbalanced. Half of the children solved theproblems in one random sequence and half in a different random sequence.

The interviewer drew children’s attention to counters placed on the table in front ofthem by saying “you can use these counters if you want.” The interviewer also toldthem that “it doesn’t matter how you work the problems out” so they would not regardany problem solving strategies as unacceptable (Siegler, 1987). As children solved eachproblem, the interviewer noted whether they used a covert or overt strategy. Childrenwere not asked to describe their solution procedures because piloting revealed that theyfound this quite dif� cult and the focus of the study was on their overt strategies withcounters. Overt procedures were coded according to whether children counted verbally,used their � ngers or used counters. Based on the range of strategies that children usedin pilot work, counting procedures were coded according to whether children:

· counted out each addend before conducting a � nal count (for 3 1 2 counting,“one, two, three,” (pause) “one, two,” then, “one, two, three, four, � ve.”)

· used counters/� ngers as tags without counting out each addend initially (for 3 1 2counting, “one, two, three,” then counting on, “four, � ve.”)

· started their � nal count from an addend rather than zero (for 3 1 2 counting on“three, four, � ve.”)


· represented addends using � ngers/ counters without overtly undertaking a � nalcount (for 3 1 2 counting “one, two, three,” (pause) “one, two,” then, “there are� ve altogether.”). This recognition strategy is similar to the “� ngers” strategyidenti� ed by Siegler and Robinson (1982) and Siegler and Shrager (1984).

The order in which children counted the addends was also recorded.

Study 2: Results and discussion

The analyses addressed three main issues. First, the relative dif� culty of judgingconcrete versions of addition principles was explored. Second, the relation betweenemerging conceptual understanding and early problem solving was addressed byexploring relations between children’s judgements of principles and aspects of theirproblem solving such as accuracy and the use of advanced counting strategies involvingcounting on and order-indifference. Third, the nature of individual differences inchildren’s knowledge about addition principles in both studies was explored using acluster analysis.

As in Study 1, children were accurate at judging identity trials (mean 5 98%,SD 5 5%) and inequality trials (mean 5 96%, SD 5 8%) and correct conceptual judge-ments were adjusted for false positives. As for Study 1 an alpha level of 0.01 wasadopted for the repeated Wilcoxon tests. Similarly to Study 1, Table IV shows that forStudy 2 although the accuracy of composition judgements did not differ from recompo-sition or order judgements (z 5 -1.06, P 5 0.29 and z 5 2 2.36, P 5 0.02, ns, respect-ively), children’s two-group order judgements were more accurate than theirrecomposition judgements (z 5 2 2.81, P 5 0.005). Accuracy level did not differ be-tween two-group and three-group judgements (z 5 2 1.67, P 5 0.10 for order trials andz 5 2 1.59, P 5 0.11 for composition trials). Therefore, like Study 1, the results ofStudy 2 suggest that concrete versions of associativity are more dif� cult than commuta-tivity because associativity involves decomposing and recombining sets—not because itinvolves three rather than two sets. Table IV shows that, similar to Study 1, children’sorder and composition justi� cations for correct judgements indicate that success in thejudgement task involved concentrating on the equivalence of groups of objects in pairsof problems, despite differences in the ways these groups were combined.

In order to explore relations between children’s emerging conceptual knowledge and

TABLE IV. Means (and standard deviations) of judgement scores and the percentage frequency ofdifferent types of justi� cations for correct judgements in Study 2

Justi� cations for correct judgements

Trial Judgement scores Same groups Different order/composition

Two-group order 79 (33) 87 (27) 16 (33)Three-group order 76 (35) 91 (23) 13 (28)Two-group composition 68 (41) 86 (30) 10 (29)Three-group composition 72 (40) 90 (24) 7 (21)Recomposition 66 (42) 88 (25) 2 (11)

Note: Judgement scores are percentage accuracy of judgements of three trials of conceptually relatedproblems for each trial type, adjusted for incorrect judgements of nine inequality problems made bythe 45 children in Study 2.


TABLE V. Means (and standard deviations) of the frequency andaccuracy of children’s problem solving strategies

Strategy % usea % strategy usersb % correct

Covert 32 (36) 69 67 (38)Advanced count 5 (10) 33 75 (41)Recognition 8 (10) 60 79 (37)Count-all 48 (35) 76 77 (24)Other 6 (18) 29 46 (48)

a Overall frequency of reported procedure use among the 45 children inStudy 2b Percentage of children who reported using a procedure to solve at leastone problem out of 14 and on whom accuracy scores are based

problem solving skills, their responses to the addition problem solving task wereanalysed. Children’s problem solving accuracy ranged from 0–100% (mean 5 64,SD 5 32). However, unexpectedly, there was no relationship between the accuracy ofchildren’s problem solving and their conceptual judgements (Spearman’s rho valueswere .12 for order, .05 for composition and 2 .04 for recomposition). This contrastswith the strong and systematic relationship between problem solving accuracy andknowledge of addition concepts found in older children (Canobi et al., 1998). Thedistribution of children’s problem solving procedures is presented in Table V, togetherwith accuracy for different procedures. (Because strategies involving � ngers wereextremely infrequent, they were grouped together with those involving counters.) Asexpected from earlier studies (Canobi et al., 1998; Siegler, 1987), children often usedmore than one problem solving strategy across the set of problems. Indeed, based onthe categories listed in Table V, only 16% of children used a single strategy to solve allproblems, while 27% used two, 40% used three, and 17% used more than threestrategies.

Table V indicates that the most common problem solving procedure was countingall, in which children counted counters or � ngers for each addend, and then countedthe total number of counters in the combined set. This procedure, though laborious,was quite accurate. Children used recognition and other counting strategies much lessoften. More advanced counting procedures involved either counting both addendsseparately then initiating the � nal count starting from the total number in one of theseaddends (counting on), or using the counters as tags, counting one addend and thencounting on in a single count. Because these two advanced counting strategies wererelatively infrequent but similar, they were grouped together. The two strategies werejudged as similar because they both involve using counters to simultaneously representthe separate addends and the combined set in an addition problem and this reduces thenumber of counts required to solve the problem. Mann-Whitney tests were used toexplore whether children who used advanced counting strategies differed in theirconceptual knowledge from those who did not, a Bonferroni-type adjustment to thealpha level was made and a level of 0.017 was used. In support of research indicatingthat advanced counting strategies have conceptual underpinnings (Fuson, 1982, 1988;Martins-Mourao & Cowan, 1998; Siegler & Crowley, 1994), Table VI shows thatchildren who used advanced counting procedures (involving using counters to simul-taneously represent the parts and the whole) made more accurate order judgements


TABLE VI. Means (and standard deviations) of children’s judgment scores as a functionof their use of advanced counting problem-solving strategies

Advanced counting strategies Order Composition Recomposition n

Absent 55 (41) 52 (43) 59 (43) 30Present 85 (24) 68 (40) 79 (38) 15

Note: Judgement scores are percentage accuracy of children’s judgements of three trialsof conceptually related problems for each trial type, adjusted for their incorrect judgementsof nine inequality problems.

than other children (z 5 2 2.39, P 5 0.017). However, this difference did not extend tocomposition (z 5 2 1.03, P 5 0.30) or recomposition judgements (z 5 2 1.80,P 5 0.07). This � nding suggests that commutativity knowledge is related to under-standing that counters can be used to signify both the addends and the total simul-taneously. The result supports previous � ndings that children who use advancedcounting strategies such as min (counting on from the larger addend) tend to have agood understanding of commutativity (Canobi et al., 1998; Cowan & Renton, 1996).

Table V shows that some children used a recognition strategy in which they countedout each set, and then named the total number of items in the combined set withoutovertly undertaking a � nal count. Children using this strategy may have arrived at the� nal number of counters or � ngers through covert counting, subitising or a kinaestheticstrategy. Table V also shows that covert problem solving procedures were quitecommon. However, the accuracy of covert procedures was quite low, suggesting thatthey may have involved guessing or inaccurate covert counting and should not beaccepted as evidence of retrieval of answers from memory (as is sometimes assumed inolder children). The use of counters appears to have assisted children who used therelatively unsophisticated counting all strategy to achieve accuracy rates as high as thoseachieved by children who used more sophisticated counting strategies. In contrast,Canobi et al. (1998) found counting all to be the least accurate strategy among olderchildren. This difference may imply that mean accuracy levels are not as useful an indexof problem solving skill in research involving young children with access to counters asin studies of older children who do not have access to counters.

A measure of order-indifference was calculated as the percentage of problems forwhich an overt counting strategy was used with addends counted in a different orderfrom that in which they appeared. In keeping with previous research (Canobi et al.,1998), the measure was computed using the seven problems for which the largeraddend was presented second as there is no apparent ef� ciency associated withorder-indifferent counting strategies when the larger addend is presented � rst. Unfortu-nately, such strategies were used on a very small percentage of problems (mean 5 4,SD 5 10), making it dif� cult to test whether they were related to conceptual measures.Most children were very rigid, always operating on problem addends in the order inwhich they were presented. Moreover, of the nine children who used order-indifferentstrategies, six only used these strategies once, which may only suggest occasionalinattention rather than genuine � exibility in treatment of addend order. However, giventhe debate over whether commutativity knowledge precedes the use of order-indifferentcounting strategies (Baroody & Gannon, 1984; Cowan & Renton, 1996), it is of interestto note that, with a single exception, all children who used an order-indifferent strategyjudged the majority of commuted addition problems correctly.


Combined Analysis of Studies 1 and 2

In order to explore individual differences in conceptual understanding, a clusteranalysis was conducted on the judgement scores of children in both studies. (The samejudgement task was used in Studies 1 and 2.) Children in both studies were includedin a single analysis in order to increase the likelihood of identifying small subgroupswith distinctive patterns of conceptual knowledge. Children’s patterns of judgementswere identi� ed using Ward’s clustering algorithm (SPSS, 1999). The clustering algor-ithm was applied to children’s judgement scores for two- and three-group order andcomposition trials and recomposition trials. A � ve-cluster solution, which accounted for88% of the total variation in children’s judgement scores, was selected. The six-clustersolution comprised small groups and accounted for 90% of the variance and thefour-cluster solution accounted for 85% of the variance. Table VII shows that thepatterns of judgements associated with the clusters were distinct and partially ordered.Children in Cluster 1 displayed the most sophisticated pattern of performance, per-forming close to ceiling level. It is noteworthy that the majority of school children (� ve-to six-year-olds) and about a third of the preschool children (four- to � ve-year-olds)were in the Cluster 1. This indicates that for many children in the sample, butparticularly for school children, the addition principles presented in the context of thephysical objects were very salient. Children in Clusters 2 and 3 had similar pro� lesexcept that Cluster 3 children were much less accurate overall. They had similarjudgement scores on composition and order trials but were less accurate at judgingrecomposition trials, supporting earlier � ndings that associativity judgements are lessaccurate than commutativity judgements. Cluster 4 children judged order trials accu-rately but judged composition and recomposition trials inaccurately. Children in thissubgroup appear to have some concrete version of commutativity, but not additivecomposition—a relatively uncommon pattern of results that was not re� ected in theWilcoxon tests for differences in the order and composition scores in each study.Children in Cluster 5 performed close to � oor level, making mainly incorrect judge-ments.

The cluster analysis supports the results of the Wilcoxon tests by indicating thatmany children judged order trials more accurately than recomposition trials. Thepresence of such children supports previous � ndings that concrete versions of theassociativity principle are relatively dif� cult for young children to understand (Lang-ford, 1981). In addition, the cluster analysis indicates that a small group of childrenmay understand relations between groups of objects based on commutativity but notadditive composition. Another interesting � nding is the tendency for older children to

TABLE VII. Means (and standard deviations) of judgement scores as a function of cluster membership

Number of children

Trial type AgeTwo-group Three-group Two-group Three-group

Cluster order order composition composition Recomposition 4–5 5–6

1 98 (4) 95 (10) 96 (9) 98 (4) 97 (6) 7 432 82 (19) 87 (13) 82 (19) 80 (20) 62 (17) 6 93 40 (25) 34 (16) 37 (25) 48 (18) 12 (16) 6 64 79 (29) 92 (12) 3 (8) 3 (8) 14 (24) 4 45 8 (15) 0 (0) 4 (12) 0 (0) 0 (0) 0 8


respond to the principles in an all or nothing manner. Table VII shows that the vastmajority of � ve- to six-year-olds were in Clusters 1, 4 and 5, indicating a consistentacceptance or rejection of any given principle. In contrast, the majority of four- to� ve-year-olds were in Clusters 2 and 3, indicating less consistent judgements for at leastsome principles.

General Discussion

The purpose of the research was to use the mathematical properties of whole numberaddition to explore children’s conceptual knowledge of how groups of objects can becombined. The results are consistent with arguments that young children develop anunderstanding of additive composition, commutativity and associativity in the contextof physical objects (Gelman & Gallistel, 1978; Resnick, 1992). In addition, the resultssupport the usefulness of exploring different patterns in children’s conceptual knowl-edge. Indeed, analyses of children’s patterns of judgements and of the comparativedif� culty of different principles suggest a developmental progression in part–wholeunderstanding. One group of children appears to have a concrete knowledge that setscan be combined in different orders without understanding that sets presented incombination are equal to the same sets presented separately. This suggests that somechildren might acquire a primitive form of commutativity before understanding thatgroups of objects are additively composed of smaller groups. The results also supportclaims that commutativity knowledge precedes associativity knowledge, and indicatethat it is the conceptual relations involved in concrete versions of associativity ratherthan the mere presence of three sets that makes the principle more dif� cult. Consistentwith previous � ndings, the results also suggest that although children’s understandingof concrete versions of additive composition improves after they enter school, manypreschoolers understand that problems are equal when the same groups of objects arecombined in different orders or are decomposed and recombined (Cowan & Renton,1996; Sophian et al., 1995). Finally, while the systematic relationship between problemsolving and conceptual knowledge found among older children by Canobi et al. (1998)was not present, the emergence of advanced counting strategies was related to commu-tativity knowledge.

Although the majority of children who participated in the study had a surprisinglygood understanding of the addition principles presented in a concrete context, differentpatterns of partial knowledge were identi� ed. The results support previous researchinvolving symbolic problems, which indicates that some children who understandcommutativity have dif� culties with associativity (Canobi et al., 1998). The results alsoindicate that some children in the present research had a concrete understanding ofcommutativity but not additive composition although the reverse pattern was notfound.

The cluster solution suggests that some children acquire some understanding thatparts combined in different orders are equivalent before they fully appreciate the waysin which the parts can be combined to form a whole (although this � nding was notapparent in the separate analyses of judgement scores for each study). Theidenti� cation of such a conceptual pro� le appears inconsistent with Resnick’s (1992,1994) claim that additive composition knowledge is fundamental to children’s under-standing of other principles. Moreover, failing to understand that a combination of twogroups of objects is equal to those groups of objects presented separately indicates afailure to understand addition in a fundamental sense. Therefore, it may seem surpris-


ing that some children seem to have acquired an early version of commutativity withoutsuch a basic addition understanding. Nonetheless, Baroody and Ginsburg (1986) haveargued that even children who have a very primitive notion of addition may have someunderstanding of the role of order in adding physical objects. They claim that on thebasis of their computational experience, children learn that it does not matter whetheryou start with one set of blocks and add another set, or start with the second set andadd the � rst set: the result is still the same. Baroody and Ginsburg argued that thiscomputation-based knowledge does not necessarily imply that children understand thataddition is a binary operation involving the union of two sets. Indeed, it is possible thatan understanding the order-irrelevance counting principle (Gelman & Gallistel, 1978)helps at least some children appreciate of the role of order in combining groups ofobjects. Knowledge that the objects can be counted in any order to arrive at the totalmay lead to recognition that two groups of objects combined in different orders havethe same total.

The present investigation into children’s knowledge of the ways in which physicalobjects can be combined supports the suggestion that commutativity knowledge pre-cedes associativity knowledge (Canobi et al., 1998; Close & Murtagh, 1986; Langford,1981). The current research suggests that differences in children’s performance oncommutativity and associativity items are not restricted to symbolic problems andcannot be explained by appealing to non-conceptual factors such the involvement ofmore dif� cult computational, memorial or verbal demands. Interestingly, the presenceof three rather than two groups in order and composition trials did not affect theaccuracy of children’s judgements. This suggests that the mere presence of three sets isnot responsible for children’s dif� culties in understanding concrete versions of associa-tivity; instead, these dif� culties arise due to the conceptual demands involved inunderstanding that sets can be decomposed and recombined in different ways.

Although some age group differences in conceptual understanding were found, thepresent results also support previous research indicating that many preschool childrenunderstand that problems are equal when the same groups of objects are combined indifferent orders or are decomposed and recombined (Cowan & Renton, 1996; Sophianet al., 1995). The accuracy with which even preschoolers judged the principles suggeststhat young children have a rich knowledge of regularities in the ways in which groupsof objects can be joined together and pulled apart. These � ndings are in keeping withtheoretical accounts suggesting that children’s understanding of part–whole conceptscrucial to school arithmetic emerges in the context of the physical world (Resnick,1992, 1994). Moreover, the � ndings suggest that using the formal principles as aframework allows the identi� cation of important developmental changes in children’sknowledge. For example, the results suggest that one important conceptual gain thatmany children make after entering school is a greater appreciation that larger groups ofobjects are additively composed of smaller groups.

The current research into children in their � rst year of school did not reveal thestrong systematic relationship between conceptual knowledge and problem solvingfound in children who had been attending school for more than a year (Canobi, inpress; Canobi et al., 1998). The � nding that conceptual judgements were not relatedto problem solving accuracy is in keeping with research suggesting that people often� nd it dif� cult to coordinate their knowledge of formal and informal mathematics(Nunes, Schliemann, & Carraher, 1993) and supports Resnick’s (1992, 1994) claimthat a protoquantitative part–whole schema emerges separately from knowledge relatedto counting and quanti� cation. It may be the case that many children in the sample had


not reached the stage in their mathematical development in which protoquantitativepart–whole knowledge and quanti� cation knowledge become integrated.

Nonetheless, there was some indication that the forms of part–whole knowledgeexplored in the current study interact with problem solving skills. In support of previousresearch indicating that conceptual understanding precedes children’s use of countingon procedures (Fuson, 1982, 1988; Martins-Mourao & Cowan, 1998; Siegler &Crowley, 1994), the results suggest that children who used advanced counting strate-gies to solve addition problems had a greater understanding of concrete versions ofcommutativity than those who did not. Although it is dif� cult to explain why thechildren who used advanced counting strategies did not also have a better under-standing of additive composition and associativity than their less skilled counterparts,the results do suggest that some forms of principle understanding are related to anability to use counters to simultaneously represent the parts as well as the whole inproblem solving. Considered alongside the other � ndings, this result implies that, forthe most developmentally advanced children in the sample, there may have been aninteraction between knowledge of part–whole relations and counting and problemsolving skills. However, the majority of children in the sample did not appear to makeconnections between their relatively strong understanding of part–whole concepts inthe context of the physical world and school addition problems.

The results have important theoretical and educational implications. They suggestthat addition principles presented in the context of physical objects are quite salient toyoung children and that many children have a surprisingly advanced concrete knowl-edge of such principles even before they enter school. It seems likely that such childrenwould bene� t from instruction in which explicit links are made between this emergingunderstanding and the problem solving skills that they are taught in school. Indeed, thepresent study indicates that the few children who employ strategies involving a moreef� cient use of counters to solve addition problems have a relatively good under-standing of commutativity-type relations in the physical world. Further research exam-ining the kinds of experiences that facilitate the emergence of knowledge aboutmathematical properties such as commutativity is likely to prove very useful. Inparticular, it may be helpful to explore interventions designed to help children torecognise patterns in the ways in which objects can be combined, progress fromconcrete to more abstract versions of the principles, and apply part–whole knowledgeto school-based mathematical problems. Differences in the accuracy of children’sjudgements of trials related to commutativity, additive composition, and associativitysuggest that examining children’s knowledge of different mathematical principles willprovide insight into changes in their domain knowledge over time. Therefore, longitudi-nal research tracking the emergence of these forms of conceptual understanding is alsolikely to prove informative.

In conclusion, the present study supports the claim that the mathematical propertiesof whole number addition provide a useful framework for exploring children’s concep-tual development (Gelman & Gallistel, 1978; Langford, 1981; Resnick, 1992). Theresults suggest that many preschoolers know about relationships based on additionprinciples in the context of combining sets of objects, and that they are particularlyadept in recognising the consequences of combining groups of objects in differentorders. Moreover, at least for the current sample, a key change in children’s knowledgeafter they enter school is a greater recognition that large groups of objects can bethought of as being made up of combinations of smaller groups. In addition, an analysisof individual differences suggests that some children recognise the equivalence of


problems in which groups of objects are combined in different orders but not problemsin which groups of objects are decomposed and recombined in different ways. Theseresults support claims that the addition principles represent fundamental properties ofthe addition operation and that young children learn about these properties throughtheir experiences with physical objects. The � ndings also indicate that exploringconcrete versions of part–whole concepts based on formal addition principles providesimportant information about children’s conceptual development in early mathematics.

Correspondence: Katherine H. Canobi, Department of Psychology, University ofMelbourne, Victoria, Australia, 3010 (e-mail: [email protected]).

REFERENCES

Baroody, A. J., & Gannon, K. E. (1984). The development of the commutativity principle andeconomical addition strategies. Cognition and Instruction, 1, 321–339.

Baroody, A. J., & Ginsburg, H. P. (1986). The relationship between initial meaningful and mechanicalknowledge of arithmetic. In J. Hiebert (Ed.), Conceptual and procedural knowledge of mathematics(pp. 75–112). Hillsdale: Erlbaum.

Bisanz, J., & Lefevre, J. (1992). Understanding elementary mathematics. In J. I. D. Campbell (Ed.),The nature and origins of mathematical skills (pp. 113–136). Amsterdam: Elsevier.

Canobi, K. H. (in press). Individual differences in children’s addition and subtraction knowledge.Cognitive Development.

Canobi, K. H., Reeve, R. A., & Pattison, P. E. (1998). The role of conceptual understanding inchildren’s addition problem solving. Developmental Psychology, 34, 882–891.

Close, J. S., & Murtagh, F. (1986). An analysis of the relationships among computation-related skillsusing a hierarchical-clustering technique. Journal for Research in Mathematics Education, 17, 112–129.

Cowan, R., & Renton, M. (1996). Do they know what they are doing? Children’s use of economicaddition strategies and knowledge of commutativity. Educational Psychology, 16, 407–420.

Dowker, A. (1998). Individual differences in normal arithmetical development. In C. Donlan (Ed.),The development of mathematical skills (pp. 275–301). Hove: Psychology Press.

Fuson, K. C. (1982). An analysis of the counting-on procedure in addition. In T. P. Carpenter & T.A. Romberg (Eds.), Addition and subtraction: A cognitive perspective (pp. 67–82). Hillsdale: Erlbaum.

Fuson, K. C. (1988). Children’s counting and concepts of number. New York: Springer-Verlag.Gelman, R., & Gallistel, C. R. (1978). The child’s understanding of number. Cambridge: Harvard

University Press.Groen, G. J., & Resnick, L. B. (1977). Can preschool children invent addition algorithms? Journal of

Educational Psychology, 69, 645–652.Langford, P. E. (1981). A longitudinal study of children’s understanding of logical laws in arithmetic

and Boolean algebra. Educational Psychology, 1, 119–139.Marascuilo, M., & McSweeney, M. (1977). Nonparametric and distribution-free methods for the social

sciences. Monteroy: Brooks/Cole.Martins-Mourao, A., & Cowan, R. (1998). The emergence of the additive composition of number.

Educational Psychology, 18, 377–389.Nunes, T., Schliemann, A. D., & Carraher, D. W. (1993). Street mathematics and school mathematics.

Cambridge: Cambridge University Press.Pellegrino, J. W., & Goldman, S. R. (1989). Mental chronometry and individual differences in

cognitive processes: common pitfalls and their solutions. Learning and Individual Differences, 1,203–225.

Piaget, J. (1952). The child’s conception of number. London: Routledge & Kegan Paul Limited.Resnick, L. B. (1986). The development of mathematical intuition. In M. Perlmutter (Ed.), Perspectives

on intellectual development: The Minnesota Symposia on Child Psychology (Vol. 3, pp. 159–194).Hillsdale: Erlbaum.

Resnick, L. B. (1992). From protoquantities to operators: Building mathematical competence on afoundation of everyday knowledge. In G. Leinhardt, R. Putnam, & R. A. Hattrup (Eds.), Analysis ofarithmetic for mathematics teaching (Vol. 19, pp. 275–323). Hillsdale: Erlbaum.


Resnick, L. B. (1994). Situated rationalism: Biological and social preparation for learning. In L. A.Hirschfeld & S. A. Gelman (Eds.), Mapping the mind: Domain-speci� city in cognition and culture (pp.474–493). Cambridge: Cambridge University Press.

Resnick, L. B., & Omanson, S. F. (1987). Learning to understand arithmetic. In R. Glaser (Ed.),Advances in instructional psychology (Vol. 3, pp. 41–95). Hillsdale: Erlbaum.

Siegler, R. S. (1987). The perils of averaging data over strategies: an example from children’s addition.Journal for Experimental Psychology, General, 116, 250–264.

Siegler, R. S. (1996). Emerging minds: The process of change in children’s thinking. New York: OxfordUniversity Press.

Siegler, R. S., & Crowley, K. (1994). Constraints on learning in nonprivileged domains. CognitivePsychology, 27, 194–226.

Siegler, R. S., & Robinson, M. (1982). The development of numerical understandings. In H. W. Reese& L. P. Lipsitt (Eds.), Advances in child development and behaviour (pp. 242–32). New York:Academic Press.

Siegler, R. S., & Shrager, J. (1984). Strategy choices in addition and subtraction: How do childrenknow what to do? In C. Sophian (Ed.), Origins of cognitive skills (pp. 229–293). Hillsdale: Erlbaum.

Sophian, C. (1997). Beyond competence: the signi� cance of performance for conceptual development.Cognitive Development, 12, 281–303.

Sophian, C., Harley, H., & Martin, C. S. M. (1995). Relational and representational aspects of earlynumber development. Cognition and Instruction, 13, 253–268.

SPSS Inc. (1999). SPSS for Windows, Version 9. Chicago: SPSS Inc.Widaman, K. F., & Little, T. D. (1992). The development of skill in mental arithmetic: an individual

differences perspective. In J. I. D. Campbell (Ed.), The nature and origins of mathetical skills(pp. 189–253). New York: Elsevier.

2002 Children Concept of Addition

Documents

Transcript of 2002 Children Concept of Addition