Why set-comparison is vital in earlynumber learningKevin Muldoon1, Charlie Lewis2 and Norman Freeman3

1 School of Life Sciences, Heriot Watt University, Edinburgh, EH14 4AS, UK2 Centre for Research in Human Development, Psychology Department, Fylde College, Lancaster University, LA1 4YF, UK3 Department of Experimental Psychology, University of Bristol, BS8 1TU, UK

Opinion

Glossary

Cardinal number: the last numeral assigned to an item when counting.

Cardinal principle: the final numeral in the counting sequence represents the

items in a set as a whole, something the preceding numerals do not do.

Concept of Cardinality: conceptually, cardinal numbers serve two functions.

One is to tell us how many items are in any one set. The other is to tell us the

numerical relationship between multiple sets. A mature concept of cardinality

demands an understanding of both functions.

Successor function: for every item being counted (e.g., ‘n’) there is a unique

successor (‘n + 1’), and this is reflected in the values assigned to those

Cardinal numbers serve two logically complementaryfunctions. They tell us how many things are within aset, and they tell us whether two sets are equivalent ornot. Current modelling of counting focuses on therepresentation of number sufficient for the within-setfunction; however, such representations are necessarybut not sufficient for the equivalence function. We pro-pose that there needs to be some consideration of howthe link between counting and set-comparison isachieved during formative years of numeracy. We workthrough the implications to identify how this crucialchange in numerical understanding occurs.

Two ways of looking at numberThere are two particularly influential traditions in explain-ing the psychological origins of the number concept. Theearlier tradition explored the ability of children to recog-nize that two sets are numerically equivalent only if thesets can be placed in item-to-item correspondence [1]. Thework identified item-to-item correspondence between setsas the root of number, and downplayed the usefulness ofthe number word lexicon. By contrast, the currently domi-nant tradition identifies number with word-to-item corre-spondence and rules for adding number words to a lexicon[2–4]. Each tradition identifies a different but equallyimportant aspect of number. What is striking is that eachfalls short of providing a developmental model of numberthat is – by logical definition – a synthesis of item-to-itemwith word-to-item correspondence. We set out here whythis synthesis needs developmental and theoretical acti-vation, and how this might be achieved. Our account ofearly numeracy aims to correct the current oversight byfilling in this missing link in number research, in accordwith the general position that such linkages have to beexplicitly built into any account of number learning [5].

Preverbal and verbal systems for processing numberThere are at least two systems for non-verbal processing ofnumerical information; the ‘analog magnitudes’ and‘object-file’ systems. However, the absence of a symbolicmeans of representing the precise numerosity of sets ofmore than 4 items presents an impasse in the developmentof numerical reasoning, such that even simple numberproblems cannot be solved [6] (Box 1). The impasse hastwo aspects.

Corresponding author: Muldoon, K. (k.muldoon@hw.ac.uk).

1364-6613/$ – see front matter � 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.tics.2009.0

First, the barrier to identifying the precise numerosityof any single set can be breached by using a straightfor-ward rhythmic protocol. The unique token ‘1’ is applied toan item from a set, and then subsequent terms are appliedon a one-to-one basis to all other items in the same set. Thefinal number word assigned – the cardinal number (seeGlossary) – represents the number of items the set con-tains; this is the cardinal principle. If the concepts of ‘one’and ‘next number’ (sometimes referred to as the successorfunction) are tied together, then counting providesexogenous support for reasoning that the use of one moreor one less number word means one more or one less item,and that adding or subtracting one item changes thecardinal number by one. It also supports the notion of‘exact equality’, whereby the number of items in a set isrepresented by the same cardinal each time that set iscounted [4]. The past 30 years have witnessed multiple,often competing, accounts of how the cognitive systemmakes the link between written or spoken numerals anditems being counted, how it generates representationsthat conform to the set of natural integers and whatcontributes to the construction of a mental number line[3,4,6–8].

Second, even given a grasp of set-enumeration, theimpasse also forms a barrier to understanding the rela-tional property of number. Recognizing the full meaning ofcardinal numbers requires a grasp of how numerals definequantitative relationships between simultaneously pre-sented sets, and this is where the concept of cardinalityis at issue. After one set of items has been counted and thecardinal number arrived at, the process can be repeated foranother set (Figure 1a). Now, the two cardinals can becompared. If they are the same, then the sets are in item-to-item correspondence.

The developmental puzzle is that up to the age of six,even some two years after they have mastered procedural

numerosities, where ‘x’ has a unique successor (the next word in the string of

natural number terms). For example, if ‘x’ is ‘five’, then ‘x + 1’ must be ‘six’.

1.010 Available online 15 April 2009 203

Box 1. Pre-verbal and verbal systems for processing

number

The cognitive architecture comprises at least two systems for pre-

verbal processing of quantitative information [2,12,35] which

support iconic representations of numerosity [12]. Analogue

magnitude representations are iconic insofar as there is linear

relationship between a neural representation of magnitude and

some other physical indicator (e.g. area, length, brightness,

temporal duration and so on.), but the system is comparatively

crude because two analogs must be sufficiently different (e.g. a ratio

of 1:2 in infancy) to distinguish one numerosity from another [36].

Object file representations differentiate on difference of ‘1’, and are

iconic because there is one-to-one correspondence between a set of

‘n’ elements and its mental representation (e.g. between the eyes on

my face and an object file ‘ll’). However, this system is limited to

processing small sets of 3 or 4 items [2]. Accordingly, recognizing

number with any degree of accuracy becomes difficult when either

core system is exceeded [12].

Crucially, neither analog magnitudes nor object-files share an

obvious similarity with language-based representations of number.

There is nothing in the word ‘five’ or the written symbol ‘50, that

denotes ‘fiveness’ as a property of a set of elements, and so

language-based representations are largely non-iconic (‘largely non-

iconic’ because many written systems are iconic for numerosities up

to 3 and sometimes 4 [e.g. the Roman I, II, III] and therefore display

similar limits to object-file representations [37]). This is a major

challenge to a developing grasp of number: how to map between

numerosities and an arbitrary series of symbols that form part of

wider cultural language systems.

Opinion Trends in Cognitive Sciences Vol.13 No.5

counting, many children have yet to grasp that two setswith the same cardinal number must, by virtue of logicalnecessity, be equivalent, and that sets with different car-dinals must by the same logic be numerically different[9,10]. If a child counts two sets and agrees that eachcontains five items, but insists that one set is more numer-ous than the other (e.g. because the rows are differentlengths), there is a gap in that child’s grasp of numberwords. They have adhered to the cardinal principle for eachset separately, but there are doubts about just what theirconceptualization of cardinal number is [11].

The minimum criteria proposed to satisfy a concept ofnumber [4] and definitions of cardinality, as found incurrent theories of counting [2,3,7,8], are necessary butnot sufficient to recognize that two sets with the samecardinal must be equivalent. Although the reduction ofcardinality to a symbolic, verbal code solves the problem ofnot being able to store a representation of the itemsthemselves (unless the number of items is less than 4,in which case the object-file system will suffice), the evi-dence points towards the fact that the cognitive system orsystems that process numerical input still have work to do.Such work is likely to be challenging because synthesizingseparable forms of numerical representation and under-standing is what makes early number work hard for chil-dren [12].

Piaget [1] – who did focus on children’s grasp of item-to-item correspondence – was convinced that counting playedno part in numerical reasoning. Piaget emphasizedthat the concept for cardinality demands the abstractionof cardinal values to all sets that are in item-to-item correspondence. A cardinal number yields a repres-entation of set-size (jxj) that is, in each instance, unique todifferent numerosities, but the synthetic link requires the

204

understanding that jxj applies across equivalent sets suchthat if two sets both yield jxj, then n1 = jxj = n2, hencen1 = n2. Crucially, Piaget [1] never proposed an accountof the synthesis between his position on number and thedevelopmental problem, focused on in the later tradition, ofhow counting is understood.

How do children put cardinality and item-to-itemcorrespondence together?If counting provides input to a system attuned to theconcept of ‘one’ and the successor function, other beha-viours provide input for grasping the relevance of item-to-item correspondence between sets. Early – and often spon-taneous – experiences of using item-to-item correspon-dence to match sets occur when children group objects.Any two sets of items can be matched but sometimescorrespondence is actually forced upon a child duringactivities, like putting eggs in eggcups. The conceptualpairing of objects that clearly go together helps childrento recognize numerical equivalence [13]. Another commonform of correspondence occurs in children’s distribution ofobjects to people or toy animals. This skill emerges aroundthe same time as counting [14]. What distinguishes shar-ing from other activities involving item-to-item correspon-dence is that it is typically – from the outset – a domain fornegotiations over equivalence. Even if children first seesharing as a social action in which the arithmeticalimplications might be lost on them, the notion of socialfairness is usually not. Schemas for social equivalencebetween individuals are commonly used to organize beha-viour that conforms to culturally informed cognitivemodels for reciprocity, or when the goal is to achieve somesense of ‘balance’ [15].

The allocation of attentional resources to two sets ofobjects in space, or even across modalities, is, from thestandpoint of our definition of the concept of cardinality,complementary to counting. Sharing (Figure 1b) affordsthe monitoring of item-to-item correspondence, and set-equivalence is assured every time two items are assignedto respective sets. Moreover, numerical equivalence isknown without the need to count both sets. But, just ascounting is no guarantee that children will rely on numberwords to judge set-correspondence, nor does the ability toestablish set-correspondence guarantee that children willrecognize that the resulting sets have the same cardinalvalue (or just as important, different cardinals if error orcheating occurs when sharing out). In Frydman and Bry-ant’s [16] classic study, 4-year-olds were very good atsharing items out and counting one of the resulting shares.However, rather than directly inferring the number ofitems, each child started unnecessarily to count theremaining share. Did the children have yet to understandthat quantitative equivalence entails identical cardinalvalues, or did they lack trust in that insight? As Frydmanand Bryant [16] found, under conditions that force childrento draw on their understanding (by precluding theunnecessary counting), many 4-year-olds can infer thenumber of items in an uncounted set by way of cardinalextension (i.e. extending the cardinal number to any setknown to be in item-to-item correspondence) [13]. See alsoRef. [17] for a congruent finding for 3-year-olds in which

Figure 1. In (a) the line depicting ‘Time’ shows the temporal independence when determining cardinality for two sets of items; the challenge is to recognize that the

resulting cardinals denote item-to-item correspondence (even when sets are not in spatial alignment). In (b) the temporal distribution ensures item-to-item correspondence;

the challenge is to recognize that such correspondence entails identical cardinal values for the respective sets.

Opinion Trends in Cognitive Sciences Vol.13 No.5

children refrained from extending a cardinal if the equiv-alence of the two sets had not been established via item-to-item correspondence, thereby confirming that such chil-dren were not merely using a heuristic such as last-word-responding.

Another paradox that characterises early numberknowledge can be observed when children who can counttwo sets to get identical cardinal numbers insist that thesets are numerically non-equivalent if they look different.Yet the same children might share items out, or pair-upconceptually related items, and know that in such cases,the same cardinal applies to both sets. These twin states ofunderstanding and failing to understand about the sameconcept can co-exist, and must ultimately be reconciled.

What gets in the way of children recognizing the pri-macy of cardinal numbers when comparing sets, eventhough they seem to know that sharing produces set-equivalence? One plausible explanation is that children

believe that relative numerical magnitude must be con-gruent with relative spatial magnitude. Alternatively, itmight be that counting and sharing are, from the outset,treated as discrete routines with no conceptual overlap.Why should a child spontaneously make the connectionbetween word-to-item correspondence and item-to-itemcorrespondence, even if they are keenly attuned to eachwhen following the relevant procedure?

It is the question of how this conceptual overlap comes tobe recognized and reflected in arithmetical problem-sol-ving that we turn to in the next section. For the moment,note that we take it as given that the mind must have ameans of representing sets precisely [18]. For small sets,the object-file system can support this [12], whereas forlarger sets the count words suffice. For precise judgmentsabout set-equivalence, the cognitive systemmust also haveinputs that afford representations of item-to-item corre-spondence, and this is what procedures like sharing, or

205

Figure 2. A young child showing a puppet how to share things out. Once all the

items have been distributed, the child can be asked about the number of items in

each set, bringing both types of correspondence together.

Box 2. Error-detection and functional organization of

principles

A popular means of testing children’s conceptual understanding of a

procedural routine is to use error-detection tasks [19,38,39]. For

example, counting can be checked by asking a child to watch

carefully as a puppet skips or double-counts an item and asking

‘was the count ok or not ok?’ Such error-detection tasks on their own

do not necessarily reveal conceptual understanding of number; it is

possible that the child can simply recognize a violation to a well-

rehearsed routine (in much the same way that ‘A, B, C, E’ would be

recognizably incorrect). To examine their grasp of number, the child

must be tested on their recognition that the miscount impacts on the

cardinal value for the set (i.e. has consequences for the application

of the cardinal principle). A more stringent examination is achieved

by also asking ‘how many does the puppet think there are?’ and

‘how many are there really?’ A child who correctly answers these

additional questions has gone beyond simply noticing a miscount

(the numerical implication of a violation which may or may not be

recognized) to appreciating the impact of the error (i.e. the error is

recognized as something that jeopardizes the validity of the cardinal

number). The counting principles can now be said to be ‘function-

ally organized’. The true cardinal has as its juxtaposition the false

cardinal derived from the miscount, and the means by which the

respective cardinals were derived can thus be compared. Moreover,

such a comparison rests on the fact they recognize that there are

two possible mental representations of number for the same

quantity; one true and one ‘counterfactual’ [39], and the contem-

poraneous developments leading to ‘theory of mind’ are not merely

coincidental [40,41].

Opinion Trends in Cognitive Sciences Vol.13 No.5

‘provoked correspondences’ (i.e. conceptually paired items)support. Once separate behavioural mastery over countingand sharing (or careful 1–1 matching between paireditems) has been achieved, the two forms of correspondenceintegral to the concept of cardinality that we argue for canbe welded together.

What leads to insights about complementary sets?Children must engage with the notion that numericalequivalence for sets of discrete items is predicated onitem-to-item correspondence and that number words arerelevant. This is necessary because the procedure thatproduces a cardinal representation of numerosity is boundtightly to the relationship between multiple numerositiesdefined by between-item correspondence. This is logicallytrue because sets in item-to-item correspondence musthave the same cardinal number, and sets out of correspon-dence cannot. Although word-to-item correspondence isnecessary for the application of the cardinal principle toa single set, item-to-item correspondence is necessary forgeneralizing between sets. The twin forms of correspon-dence are linked because correspondence between items isconserved in the correspondence between words assignedto those items. Thus, sets of cups and saucers each with thecardinal ‘3’ must be numerically equivalent because eachset has a ‘1’, a ‘2’ and a ‘3’ assigned to it.

Children engaging as previously mentioned also need todisengage from the notion that counting and cardinalnumbers are only relevant to questions about a singleset (e.g. ‘How many are there? If I add one, how manydo I have now?’). One useful way of promoting these twinadvances is to ask children to judge whether incorrectsharing produces fair shares, and whether two sets createdby sharing have the same cardinal number (Figure 2). Oncechildren begin to reason about the answers, the link be-tween count words and item-to-item correspondence can beexplicitly represented.

Evidence for this picture of development comes from ourrecent longitudinal study [19]. Using Frydman and Bry-ant’s [16] set-inference task as our outcome measure, wefound that gains in set-inference were predicted not byprocedural mastery of counting or sharing (these skillsremained fairly stable over a 6-month period) but byincreased sensitivity to violations of sharing. Importantly,an emerging sensitivity to miscounts did not predict suc-cess, but counting tasks only tap knowledge of the word-to-item correspondence (i.e. within a set), not the numericalrelationship between sets. We take this as evidence that agrasp of item-to-item correspondence is involved in chil-dren’s developing understanding of set-equivalence, andwhereas a causal account demands complementary evi-dence obtained from intervention studies [20], the presentdata establish a naturalistic basis for a focus on sharing, orat least item-to-item matching, which it has largely failedto secure in the extant literature on number.

This raises the importance of error-detection in earlynumeracy. Detecting someone else’s mistakes when theycarry out familiar routines like counting can, under theright conditions, act as a catalyst for ‘re-conceptualisation’of that routine (Box 2). The same applies when children areattuned to instances of sharing incorrectly. If items are

206

Opinion Trends in Cognitive Sciences Vol.13 No.5

shared out fairly, a childmight recognize that the resultingsets should be called ‘equivalent’ but not understand whythis must be arithmetically true. If the sets are counted toproduce identical cardinals, this might be taken as hap-penstance. However, if the item-to-item correspondence isviolated and the resulting sets counted, the child is in aposition – if they have detected the error – to gain insightabout the role of item-to-item correspondence in determin-ing the respective cardinal values. This is because thedistinction between same or different cardinals rests onwhether item-for-item correspondence has been main-tained. We would be less confident of inferring this insightif a child is only able to say that fair sharing producesfair shares, and that they thus have the same number(which might be produced simply by using a strategy oflast-word-responding; simply repeating the cardinal forone share).

What are the right conditions for changing conceptualunderstanding? There is evidence that discourse withothers about the importance of procedural accuracy isinvolved [10]. Although feedback can sometimes lead chil-dren to change their responses in the search for the correctanswer, it typically fails to promote re-conceptualization ofthe task [21,22], whereas asking children for explanationsabout causal relationships is effective across a range ofdomains [23–27]. In the context of number, the causalrelationship concerning the association between word-to-item correspondence and item-to-item correspondenceseems to be particularly informative. The two forms ofcorrespondence can be brought together by asking childrento judge whether two sets created by inaccurate item-to-item matching or sharing have the same cardinal number.Knowledge acquisition in this case reflects the rational-constructivist view of learning [28]; skeletal outlines forthe twin forms of correspondence essential to the concept ofcardinality are reflected – however implicitly – in theroutines of counting and sharing. The degree of structuraloverlap between the two procedures – and hence the twoforms of correspondence – will support cognitive synthesisbetween the different forms of correspondence when and ifchildren are prompted or encouraged to reflect on suchconceptual overlapping.

An interesting question for modelling is where suchenrichment of the early system of representation fits intocurrent controversy on induction over physical items innatural number concepts [29–33]. Our approach wasnoted as one that is perhaps right in that multi-viewcontroversy [33]. We claim no more than it is a view worthconsidering for its focus on an issue elided by othermodels. What is the developmental relationship betweenconcepts of equivalence and non-equivalence, predicatedon item-to-item correspondence and the natural numbersystem, in particular the concept of cardinality? We haveargued that although a mature conceptualization ofnatural number is ultimately abstract, mental operationsover physical items and physical operations with thoseitems in childhood can provide the basis for arithmeticalinductions under the right conditions [33]. Bringingtogether the twin forms of correspondence is, we believe,an essential step on the road to more mature mathemat-ical problem-solving.

ConclusionSensitivity to item-to-item correspondence provides sup-port for a key aspect of number; the relationship that oneset has with another. Provoked-correspondences (e.g. be-tween conceptually paired items) and the routine of shar-ing are set-equivalence tools par excellence, and it isprobably no accident that they are mastered so earlyand universally; indeed, precise item-for-item distributionmight be a product of evolutionary forces that favoured thesharing of resources [34]. We are not proposing that theconcept of item-to-item correspondence is more importantthan the cardinal principle in a developing sense of num-ber. Rather, attunement to item-to-item correspondencemight advance the development of the child’s ability toabstract cardinal values on the basis of numerical proper-ties of sets. The conjunction between these two conceptsties the two traditions in number research together andallows for the cardinal principle to be understood in termsof the abstraction of the concept ‘cardinal number’. This is adriving force of development that current theories of num-ber need to incorporate.

AcknowledgementsWe thank Michael Siegal and seven reviewers (including Rochel Gelman)for insightful readings and comments on earlier versions of this paper.C.L.’s contribution to the paper was supported by the Economic andSocial Research Council (RES-576–25–5020).

References1 Piaget, J. (1941/1952) The Child’s Conception of Number, Routledge

and Kegan Paul2 Carey, S. (2004) Bootstrapping and the origin of concepts. Daedalus

133, 59–683 Gelman, R. and Gallistel, C.R. (1978) The Child’s Understanding of

Number. Harvard University Press4 Leslie, A.M. et al. (2008) The generative basis of natural number

concepts. Trends Cogn. Sci. 12, 213–2185 Dowker, A. (2008) Individual differences in numerical abilities in

preschoolers. Dev. Sci. 11, 650–6546 Simon, T.J. (1999) The foundations of numerical thinking in a brain

without numbers. Trends Cogn. Sci. 3, 363–3657 Le Corre, M. and Carey, S. (2007) One, two, three, four, nothing more:

an investigation of the conceptual sources of the verbal countingprinciples. Cognition 105, 395–438

8 LeFevre, J.A. et al. (2006) What counts as knowing? The developmentof conceptual and procedural knowledge of counting from kindergartenthrough grade 2. J. Exp. Child Psychol. 93, 285–303

9 Cowan, R. (1987)When do children trust counting as a basis for relativenumber judgments? J. Exp. Child Psychol. 43, 328–345

10 Muldoon, K. et al. (2007) Using cardinality to compare quantities: therole of social-cognitive conflict in the development of basic arithmeticalskills. Dev. Sci. 10, 694–711

11 Frydman, O. (1995) The concept of number and the acquisition ofcounting concepts: the ‘when’, the ‘how’, and the ‘what’ of it. Cah.Psychol. Cogn. 14, 653–684

12 Feigenson, L. et al. (2004) Core systems of number.Trends Cogn. Sci. 8,307–314

13 Muldoon, K.P. et al. (2005) Because it’s there! Why children count,rather than infer, corresponding sets. Cogn. Dev. 20, 472–491

14 Mix, K. (2002) The construction of number concepts. Cogn. Dev. 17,1345–1363

15 Fiske, A.P. and Haslam, N. (1996) Social cognition is thinking aboutrelationships. Curr. Dir. Psychol. Sci. 5, 143–148

16 Frydman, O. and Bryant, P. (1988) Sharing and understanding ofnumber equivalence by young children. Cogn. Dev. 3, 323–339

17 Sophian, C. et al. (1995) Making numbers count: the early developmentof numerical inferences. Dev. Psychol. 31, 263–273

18 Gallistel, C.R. (2007) Commentary on LeCorre&Carey.Cognition 105,439–445

207

Opinion Trends in Cognitive Sciences Vol.13 No.5

19 Muldoon, K. et al. (2007) Predictors of early numeracy: is there a placeformistakes when learning about number?Br. J. Dev. Psychol. 25, 543–

55820 Bryant, P. and Alegria, J. (1989) The transition from spoken to written

language. In Transition Mechanisms in Child Development: TheLongitudinal Perspective (De Ribaupierre, A., ed.), pp. 126–144,Cambridge University Press

21 Michie, S. (1984) Why preschoolers are reluctant to countspontaneously. Br. J. Dev. Psychol. 2, 347–358

22 Cowan, R. et al. (1993) Encouraging children to count. Br. J. Dev.Psychol. 11, 411–420

23 Chi, M.T.H. et al. (1989) Self-explanations: how students study and useexamples in learning to solve problems. Cogn. Sci. 13, 145–182

24 Chi, M.T.H. et al. (1994) Eliciting self-explanations improvesunderstanding. Cogn. Sci. 18, 439–477

25 Ferguson-Hessler, M.G.M. and de Jong, T. (1990) Studying physicstexts: differences in study processes between good and poor performers.Cogn. Instr. 7, 41–54

26 Pine, K.J. and Messer, D.J. (2000) The effect of explaining another’sactions on children’s implicit theories of balance.Cogn. Instr. 18, 35–51

27 Siegler, R.S. (1995) How does change occur: a microgenetic study ofnumber conservation. Cognit. Psychol. 28, 225–273

28 Gelman, R. and Lucariello, J. (2002) Role of learning in cognitivedevelopment. In Stevens’ Handbook of Experimental PsychologyVolume 3: Learning, Motivation and Emotion (3rd edn) (Pashler, H.and Gallistel, R., eds), pp. 395–443, Wiley & Sons

29 Rips, L.J. et al. (2006) Giving the boot to the bootstrap: how not to learnthe natural numbers. Cognition 101, B51–B60

208

30 Margolis, E. and Laurence, S. (2008) How not to learn the naturalnumbers: inductive inference and the acquisition of number concepts.Cognition 106, 924–939

31 Gelman, R. and Butterworth, B. (2005) Number and language: how arethey related? Trends Cogn. Sci. 9, 6–10

32 Rips, L.J. et al. (2008) From numerical concepts to concepts of number.Behav. Brain Sci. 31, 623–642

33 Muldoon, K. et al. (2008) Don’t throw the baby out with themath water:why discounting the developmental foundations of early numeracy ispremature and unnecessary. Behav. Brain Sci. 31, 663–664

34 Fehr, E. et al. (2008) Egalitarianism in young children. Nature 454,1079–1084

35 Wiese, H. (2003) Iconic and non-iconic stages in number development:the role of language. Trends Cogn. Sci. 7, 385–390

36 Xu, F. and Spelke, E.S. (2000) Large number discrimination in 6-montholds infants. Cognition 74, B1–B11

37 Dehaene, S. (1997) The Number Sense: How the Mind CreatesMathematics. Oxford University Press

38 Gelman, R. et al. (1986) Young children’s numerical competence. Cogn.Dev. 1, 1–29

39 Freeman, N.H. et al. (2000) Representation of the cardinality principle:early conception of error in a counterfactual test. Cognition 74, 71–89

40 Carpendale, J.I.M. and Lewis, C. (2004) Constructing anunderstanding of mind: the development of children’s socialunderstanding within social interaction. Behav. Brain Sci. 27, 79–151

41 Kuhn, D. (2002) What is scientific thinking and how does it develop? InBlackwell’s Handbook of Childhood Cognitive Development (Goswami,U., ed.), pp. 371–393, Blackwell