do young children acquire number words through subitizing or

Cognitive Development 19 (2004) 291–307

Do young children acquire number words throughsubitizing or counting?

Laurent Benoita, Henri Lehallea,b,∗, François Jouenb

a Université Paul Valéry, Montpellier III (EA 3021: Mémoire et Cognition), Route de Mende,F 34199 Montpellier, Cedex 5, France

b Laboratoire Développement et Complexité, EPHE, 41, rue Gay-Lussac, F75005, Paris, France

Received 1 March 2003; received in revised form 1 March 2004; accepted 1 March 2004

Abstract

Two alternative hypotheses can be used to explain how young children acquire the cardinal meaningof small-number words. The first stresses the role of counting and predicts better performance whenthe items are presented in succession. The second considers the role of subitizing and predicts betterperformance when the items are presented simultaneously. In this experiment, collections of red dots(from 1 to 6 dots) were presented to three groups of young children (ages 3–5 years) under strictlycontrolled conditions: simultaneous short presentations versus consecutive short presentations (onedot at a time), and familiar versus unfamiliar configurations. The children had to indicate how manydots they had seen. The results showed that, for small numbers, the children were better at giving theright number–word in the simultaneous task. Performance generally improved when familiar config-urations were used, except for small numbers in simultaneous presentation. As a whole, subitizingappears to be the developmental pathway for acquiring the meaning of the first few number words,since it allows the child to grasp the whole and the elements at the same time.© 2004 Elsevier Inc. All rights reserved.

Keywords: Number words; Subitizing; Counting; Number development

1. Introduction

As soon as children reach a semiotic level of functioning, they begin to use numberwords (Gelman & Gallistel, 1978). But we know little about how young children are able tograsp the referents of small-number words, especially two and three. Two kinds of model

∗ Corresponding author.E-mail address: [email protected] (H. Lehalle).

0885-2014/$ – see front matter © 2004 Elsevier Inc. All rights reserved.doi:10.1016/j.cogdev.2004.03.005

292 L. Benoit et al. / Cognitive Development 19 (2004) 291–307

have been used to explain this early development of ‘number’ awareness. Although bothexplanations take preverbal knowledge of numerosity into account, they differ as to howthey do so, and as to how counting and subitizing are implicated in numerical development.

The first explanation follows Gelman and Gallistel’s interpretation of early number de-velopment (Gallistel & Gelman, 1992; Gelman & Gallistel, 1978). The main idea in thisinterpretation is the precociousness of counting principles, which are assumed to constitutetruly implicit knowledge in infant numerical competence. Like animals (at least in somespecies), infants can react to magnitude differences, in accordance with the so-called accu-mulator model originally proposed byMeck and Church (1983). In this model, each focuson an item is paired with a burst of impulses that dumps a magnitude into a hypothetical“accumulator.” After successive focuses on a series of items, the magnitude of the wholecan be evaluated in terms of how “full” the accumulator is. This mechanism is sufficient fordiscriminating small numbers. Moreover, according toGallistel and Gelman (1992, p. 51),the accumulator model “conforms to the principles that define counting processes.” Thus,if children can represent magnitude in a preverbal form, the acquisition of the first fewnumber words simply results, through counting, from a mapping between the verbal formsand the magnitudes already represented at the preverbal level.

This explanation is supported by empirical findings showing that by the age of 3, childrenalready behave in accordance with the counting principles (Gelman, Meck, & Merkin, 1986).But other authors (Baroody, 1991; Rittle-Johnson & Siegler, 1998) have stressed that, whilecounting principles are indeed embedded in effective early procedures for counting, it doesnot follow that these principles constitute underlying concepts that guide the acquisition ofverbal counting (and number words). Moreover,Wynn (1992a)andCarey (2002)arguedthat if Gallistel and Gelman were right, the acquisition of the number–word sequence viathe mapping of preverbal magnitudes to verbal names, would be easier than it appears inFuson’s and Wynn’s data (Fuson, 1988; Wynn, 1992a).

Furthermore, recent findings suggest that infants’ “numerical” competence is not trulynumerical. When infants seem to discriminate small “numerosities,” they probably useperceptual or spatiotemporal cues that are not really numerical (Bideaud, 2002; Feigenson,Carey, & Spelke, 2002; Mix, Huttenlocher, & Levine, 2002; Tan & Bryant, 2000; Uller,Carey, Huntley-Fenner, & Klatt, 1999; Wakeley, Rivera, & Langer, 2000). As a consequence,if young children learn the meaning of the first few number words through counting, they stillhave to learn the cardinal referents without relying on a preverbal awareness of numerosity.

The second explanation of early number–word acquisition is grounded both on the“object-file model” (Kahneman, Treisman, & Gibbs, 1992) and the subitizing mode ofquantitative evaluation (Klahr & Wallace, 1976). Subitizing, is generally defined as anaccurate quantitative evaluation of small sets without explicit (neither internal nor exter-nal) counting (see for instanceCarey, 2002, p. 315). But, beyond this common definition,subitizing processes and the underlying competence may vary greatly from one author tothe next.

Thus, it is useful to differentiate between three more specific subitizing skills.

(a) The first is strictly perceptual, where the only requirement is that individuals (in partic-ular, infants) can differentiate perceptual arrays differing by only one item. In this case,the focus may not be on numerosity, since many alternative cues can be used to pro-

L. Benoit et al. / Cognitive Development 19 (2004) 291–307 293

duce such a behavioral differentiation (Feigenson et al., 2002; Mix et al., 2002; Tan &Bryant, 2000).

(b) The second is perceptual–preverbal subitizing, given, asTan and Bryant (2000, p. 1162)have pointed out, “There may be a special mechanism, often called subitizing, for recog-nizing and distinguishing small numbers independently of their perceptual appearanceand without counting (e.g.,Simon, 1997; Starkey & Cooper, 1995). It is quite likely thatinfants as well as adults use this mechanism successfully to distinguish small numbers.”In this definition, verbal naming is not required, but the focus is on numerosity, whateverthe perceptual configuration of number. Note that forGallistel and Gelman (1992, p. 58),subitizing is in fact supported by a short preverbal counting process which producesthe preverbal magnitude.

(c) The third is perceptual–verbal subitizing, where number words are used to expressnumerosity. This is the traditional use of the word “subitizing” since subitizing wasinitially considered to be an explicit quantification operator (Klahr & Wallace, 1976,following Kaufman, Lord, Reese, & Volkmann, 1949).

Of course, these three kinds of subitizing can be understood as forming a developmentalpathway that ends with true awareness of the cardinal meaning of small-number words(ones which can be linked to their numerosity through subitizing). But the reasons behindthis sequence can be sought in the object-file model. This model (Kahneman et al., 1992)argues that there is a primitive competence via which the individual notices the presenceof objects in a given environment. Objects are assumed to be stored in separate “files.”This theory has repeatedly been used to account for numerical skills in infancy, sinceit can explain the infant behavior noted inWynn’s (1992b)paradigm (see for exampleBideaud, 1995, 2002; Carey, 2002; Leslie, Xu, Tremoulet, & Scholl, 1998; Uller et al.,1999; Vilette, 2002). Nonetheless the cues infants that use for indexing objects are notstrictly numerical. In controlled conditions, these cues may, therefore, not induce the “quasinumerical” processing usually observed (Feigenson et al., 2002). This is not really surprisingbecause, while the object-file model can easily account for infants’ behavioral differentiationof small numbers, this does not imply that the whole is conceptualized, i.e., the magnitudeitself, irrespective of spatial extent or density. Thus, the object-file model is not sufficientto account for the accurate use of small-number words.

As a consequence, the second explanation of early number–word acquisition states thatobject indexing is combined with subitizing to account for the first appraisal of numerosityper se. Remember thatStarkey and Cooper’s data (1995)clearly demonstrated subitizingin 2–4 years old children. At the theoretical level, many earlier developmental studies havestressed the importance of subitizing in number development, instead of reliance on countingalone.

For Klahr and Wallace (1976, p. 68)counting episodes acquire a semantic meaning(including the meaning of the last counting word) through the use of counting in situationswithin the range of subitizing.von Glasersfeld (1982)agreed with the position that lessonsare drawn from the coincidence between the last counted word and “the same number wordas result of a figural pattern perceived as a whole” (p. 207). According toFischer (1984,1991) empirical evidence reveals a discontinuity “after 3” (i.e., between ages 3 and 4),when children learn the sequence of number words. It follows that number words may


not be acquired through a mere linear counting process. Considering both counting andsubitizing,Fischer (1984)showed that, in 4- and 5 years old, subitizing implies counting(i.e., a child who can subitize a number can count that number as well). Moreover, numerouschildren at this age seem to be aware of the two alternative modes of quantitative evaluation.In several interviews, the children said that they counted whenever they could not subitize.Fischer nevertheless considered his data to be consistent with the hypothesis that cardinalnaming relies on subitizing.

As for counting, success in a subitizing task can be assumed to be based on certain implicitprinciples, which can be drawn from the object-file model. But verbal subitizing is morethan a perceptual skill, since it requires knowledge of the exact number words for smallnumerosities. This knowledge implies some categorical processes (Gallistel, 1993; vonGlasersfeld, 1982) in addition to those proposed in the object-file model. Verbal subitizingis thus supported by the following principles:

1. Distinction-of-elements principle: Each element is mapped to one and only one object-file.2. Cardinal principle: Each configuration as a whole is mapped to one and only one category

file, which is labeled by a specific number–word; we need a categorical file since thereare many possible configurations for the same number word.

3. Abstraction principle: The nature of the elements does not alter a category file. But,given the evidence that the type of element does affect number recognition (Baroody,Benson, & Lai, 2003) and equivalence judgments (Mix, 1999), the abstraction principleunderlying general verbal subitizing may need to be constructed. In other words, thisprinciple may not govern the early verbal subitizing efforts, even if it does underlie latergeneral verbal subitizing. However, even with homogeneous arrays, numerosity has tobe abstracted from the spatial detection of each element.

4. Combination irrelevance principle: The spatial combination of elements does not alterthe perceptual outcome. This means that the scanning process (which may depend onthe specific array) does not change the category file.

In sum, there are two alternative explanations of how the cardinal meanings of the firstfew number words are acquired. One states that these number words are learned throughcounting, while their meaning simply results from a mapping of words to preverbal magni-tudes. The other starts from indexed objects, which are grouped together through subitizingto give the cardinal referents labeled by the number words.

The present study was aimed at evaluating predictions based on the two alternative theo-retical accounts described above. Since recent evidence suggests that infants do not rely onnumerical cues to differentiate numerosities, our general hypothesis was that subitizing is away of progressing from infant indexing of objects, one by one, to an appraisal of the whole,and then on to an understanding of the cardinal referents of small-number words. Subitiz-ing could be the developmental process that allows children to simultaneously grasp theelements and the whole, i.e., that enables them to go beyond separate, one by one indexingof objects, to the appraisal of the whole, but without ignoring the parts.

In this experiment, small numbers of dots were briefly presented either simultaneously orconsecutively. In both cases, young children had to indicate the number of the just perceivedcollection of elements. The dots arrangement was either familiar (dice configurations) orunfamiliar (random). Note that the elements for both types of display appeared in exactly the


same places, so that the “configurations” were the same in the simultaneous and consecutiveconditions.

The following predictions were made.

1. If subitizing is the first way of understanding cardinal referents, then the performance ofyoung children acquiring number words should be better in the simultaneous conditionthan in the consecutive condition, at least for small numbers (less than 4).

2. Alternatively, if the meaning of small-number words relies on counting, then perfor-mance will be as good or even better in consecutive presentation—which allows thechild to verbalize the number–word list—than in simultaneous presentation where tooshort a display time prevents explicit counting.

3. Success in quantitative evaluations by young children should be more frequent withsmall numbers, but mainly in the simultaneous condition where they can subitize.

4. If a perceptual appraisal of the whole (while still noting the elements) plays an importantrole, then the spatial aspects of the configuration may have an impact, even in youngchildren, and evaluations of familiar configurations should be better.

2. Method

2.1. Participants

Forty-eight children participated in the study. There were 16 children in each of threeage groups: 3 years old (range: 3 years 2 months to 3 years 10 months,M = 3.5), 4 yearsold (range: 4 years 2 months to 4 years 9 months,M = 4.6), 5 years old (range: 5 years3 months to 5 years 8 months,M = 5.4). There was approximately the same number ofboys and girls in each group. All of the children were attending preschool. The samplewas socio-economically varied, reflecting the diversity of the regional population (South ofFrance).

2.2. Procedure

The stimuli consisted of red dots 1.5 cm in diameter. Collections of 1–6 dots were pre-sented on a white computer screen. Depending on the presentation mode, elements weredisplayed in different ways. In the simultaneous-presentation mode, all elements in thecollection were displayed at the same time in the middle of the screen for 800 ms. In theconsecutive-presentation mode, the elements in the collection were displayed one at a timefor 800 ms each; after the last element in each set, a white screen with red edges appearedto indicate the request for a number word.

A pause button (“Enter” key of the numeric keypad) could be pressed to stop the task atany time. A left click (correct name) and a right click (incorrect name) on the mouse wasused to record the children’s verbal responses.

Children were tested in a quiet dimly-lit room. In the simultaneous task, the instructionswere: “You will see red spots on the computer screen. Each time, tell me how many thereare.” In the consecutive task, the instructions were: “You will see red spots appearing one


Fig. 1. The two types of dot configurations: familiar and unfamiliar.

after another. Count them. And then when I ask you, tell me how many there were.” Notethat the counting task had two parts, the counting part (serial naming of each element) andthe number–word part (cardinal naming of the whole).

A familiarization phase was carried out with a three-element collection displayed simul-taneously or consecutively. A break was provided after the familiarization tasks.

2.3. Design

The five factors manipulated as independent variables were (a) age (3–5); (b) numericalevaluation task: simultaneous, consecutive-counting (the first part of the consecutive task:serial counting of dots), consecutive-cardinality (the second part of the consecutive task:cardinal naming after serial counting); (c) type of configuration: familiar (FA) or unfamiliar(UNFA) (seeFig. 1 for the two configurations chosen for each number); (d) dot displaydirection in the consecutive task: LR (from left to right) or BT (from bottom to top); and(e) number of dots, divided into two ranges: R1 (1–3) and R2 (4–6).

Except for age and display direction (LR or BT), the factors were presented accordingto a repeated-measure design. Because of the necessity of controlling for order effects, thevarious conditions were counterbalanced in the following way.

- Items: Each child received 24 items.- Blocks: The 24 items were divided into four blocks of six items, each of a single type. A

given block was characterized by the task (simultaneous or consecutive) and the config-uration (familiar or unfamiliar). The six items in all blocks were the six numbers (1–6).

- Block order: Four block orders were used by simply reversing the order of the tasks(simultaneous-consecutive or consecutive–simultaneous) and the type of configurationwithin the task order (familiar–unfamiliar or unfamiliar–familiar).

- Item order: Eight random item orders were defined for the six numbers (i.e., the six items).Four were used for half of the participants (4 orders× 6 numbers= 24 items); the otherfour orders were used for the other half of the participants.

- Children: Two children in each age group were presented with one series of the 24 items(as specified above), and for these two children, the dots in the consecutive task weredisplayed from left to right (LR). Two other children were presented with the second


series of 24 items, and for these two children, the dots in the counting task were displayedfrom bottom to top (BT). As a consequence from the entire combination, there were 16children in each age group.

3. Results

The dependent variable was the number of correct responses in each experimental con-dition. In the simultaneous task, a correct response was defined as the verbal naming of theexact number of dots just presented. In the first part of the consecutive task, a correct re-sponse implied indicating the exact sequence of number words (e.g., “one,” “two,” “three”)as the dots were displayed. In the second part, a response was correct (cardinality) whenthe child could verbalize the exact number of dots just presented in succession.

In accordance with the experimental design, an ANOVA was computed with two between-group factors (age and display direction), and three repeated-measure factors (one factor forthe three tasks, one for the two configurations, and one for the two number ranges). Eventhough the dots displayed from bottom to top seemed slightly more difficult, this effect wasnot significant (F1/42 = 0.41,P > 0.05). Therefore, the display direction factor was notconsidered further.

The number of children who obtained each score in each experimental condition ispresented inFig. 2a–c. The scores varied from 0 (no correct response) to 3 (correct responsesfor all three numbers in that experimental condition).

3.1. Age effect and interactions

It was not surprising to find a significant main effect of age (F2/42 = 17.65,P < 0.0001).Of course, performance improved with age (seeFigs. 3 and 4). More interesting were thetwo significant interactions described below.

As Fig. 3 shows, there was a significant Age× Task interaction (F4/84 = 2.83, P <

0.05). Planned comparisons indicated the following: (a) for the 3 years old, there was asignificant difference between the simultaneous task and the other two, more difficult tasks(F1/42 = 5.78,P < 0.05, for the comparison with consecutive-counting; andF1/42 = 11.98,P < 0.01, for the comparison with consecutive-cardinality), but there was no significantdifference between the two parts of the consecutive task (counting and cardinality). (b) Forthe 4 years old, it was the consecutive-cardinality task that differed from the two easiertasks (F1/42 = 3.81,P = 0.05, for the comparison with the simultaneous task; andF1/42= 9.87,P < 0.01, for the comparison with consecutive-counting), and the simultaneous taskdid not differ from consecutive-counting. (c) For the 5 years old, there were no significantdifferences between the three tasks.

A significant Age× Number Range interaction was also observed (F2/42 = 3.93,P <

0.05). It is clear inFig. 4that the difference between the two number ranges (1–3, and 4–6)decreased with age. Nevertheless, planned comparisons yielded significant differences atevery age (F1/42 = 33.39,P < 0.0001 for age 3;F1/42 = 26.53,P < 0.0001 for age 4;F1/42 = 4.30,P < 0.05 for age 5). The Age× Configuration interaction was not significant(F2/42 = 1.24,P > 0.05).


(a)

Fig. 2. (a) Number of participants who obtained each score (0–3) on the simultaneous task, by age (3–5 years),number range (1–3 or 4–6) and configuration (familiar FA or unfamiliar UNFA). (b). Number of participants whoobtained each score (0–3) on the consecutive-counting task, by age (3, 4, or 5 years), number range (1–3 or 4–6)and configuration (familiar FA or unfamiliar UNFA). (c) Number of participants who obtained each score (0–3) onthe consecutive-cardinality task, by age (3, 4, or 5 years), number range (1–3 or 4–6) and configuration (familiarFA or unfamiliar UNFA).


(b)

Fig. 2. (Continued )


(c)

Fig. 2. (Continued ).


AGE 3 years

AGE 4 years

AGE 5 years

TASK

Mea

n N

umbe

r of

Cor

rect

Nam

es

0,0

0,5

1,0

1,5

2,0

2,5

3,0

Simultaneous Consecutive-Counting Consecu.-Cardinality

Fig. 3. Mean number of correct names given, by age and task.

AGE 3 years

AGE 4 years

AGE 5 years

Number Range

Mea

n N

umbe

r of

Cor

rect

Nam

es

0,0

0,5

1,0

1,5

2,0

2,5

3,0

R1: from 1 to 3 R2: from 4 to 6

Fig. 4. Mean number of correct names given, by age and number range.

302L

.Benoitetal./C

ognitiveD

evelopment19

(2004)291–307

TASKSimultaneous

TASKConsecutive-Counting

TASKConsecu.-Cardinality

NUMBERS: 1 to 3

Cor

rect

Nam

es (

Mea

ns)

0,8

1,0

1,2

1,4

1,6

1,8

2,0

2,2

2,4

2,6

CONFIGURATIONSFamiliar

Unfamiliar

NUMBERS: 4 to 6

CONFIGURATIONSFamiliar

Unfamiliar

Fig. 5. Mean number of correct names given, by task, number range, and configuration.


3.2. Three factors in interaction

As for the age factor above, it is not very informative to consider the significant maineffects of the other three factors: task (F2/84= 4.99,P < 0.01), configuration (familiar betterthan unfamiliar:F1/42= 7.64,P < 0.01), number range (1–3 better than 4–6:F1/42= 56.35,P < 0.0001). The pertinent information is found in the interactions. The interactions withage have been already presented.Fig. 5 shows how the other three factors interacted (agefactor excluded). This three-way interaction was significant (F2/84 = 5.303,P < 0.01),whereas the four-way interaction (age factor included) was not.

Keeping in mind the above age effects,Fig. 5 nevertheless helps us understand howthe other three factors interacted (age excluded). With small numbers (1–3), configurationfamiliarity had no effect in the simultaneous task but did affect the two consecutive tasks(counting and cardinality). On the other hand, with larger numbers (4–6), performance wasalways poorer and familiarity did have an effect on the simultaneous recognition of thesenumbers. This interaction suggests that with larger numbers children mainly recognizespecific familiar patterns.

4. Discussion

The results suggest that for the exact and explicit appraisal of numerosity, subitizingis a more primitive tool than counting. Accordingly, for small numbers, the 3 years oldperformed better on short, simultaneous presentations than on short, one by one presen-tations of the same items. It follows that subitizing is unlikely to be based on a preverbalcounting process. If it were, performance on a verbal task would be at least as good in asimultaneous display task as in a consecutive task, since there is no reason for verbalizationto disturb the basic relations between subitizing and counting, no matter what mechanismssupport the preverbal differentiation of numerosities. Of course, we could have comparedour simultaneous verbal task with a simultaneous counting task, i.e., a task involving theenumeration of simultaneously presented items instead of consecutively presented ones.In fact, the rather difficult consecutive counting task seems to “stack the deck” in favor ofsubitizing. But a simultaneous counting task implies longer viewing of the dots, which forsmall numbers, leaves enough time to subitize first and then count (Fischer, 1984) in orderto satisfy the experimenter’s demands. In other words, with a simultaneous counting task,one cannot disentangle subitizing and counting, especially for small numbers.

The other results obtained here are in line with previous findings. At the age of 4, youngchildren may know the number–word sequence used in counting without knowing thecardinal property of the last word said (Fuson, 1988, 1991; Baroody, 1991). At the age of5, there were no differences between the two presentation modes; but of course, this resultis limited to the numbers 1–6 used in this experiment.

If subitizing is a primitive tool for quantitative evaluations, it is not surprising that smallnumbers are recognized better than larger ones, since subitizing is usually limited to thenumber 4.Fig. 5summarizes the main findings: greater success with small numbers, espe-cially when displayed simultaneously, and delayed acquisition for larger numbers, with asubstantial drop for subitizing, especially on unfamiliar configurations.


Based on these results, and in accordance with previous authors (Fischer, 1984, 1991;Klahr & Wallace, 1976; Vilette, 1994, 2002; von Glasersfeld, 1982;), subitizing may beregarded as the necessary developmental pathway for understanding the significance ofthe first few number words. Let us imagine a plausible scenario based on the object-filemodel (Kahneman et al., 1992). As we have seen above, this model suffices to account fornumerical behavior in infants, even if infants do not focus on numerosity per se. Accordingto the object-file model, when several objects are presented (but not many), they are storedin separate files, provided that they can be differentiated on some spatiotemporal cues. Butnaming the whole (cardinality) implies considering the whole as a single object. So if thewhole is stored as a single object, the elements should not be represented one by one. Onthe other hand, knowing the exact meaning of a given number word requires maintainingawareness of the distinct elements that can ensure exact quantification. Especially if infantperformance is not strictly numerical, verbal subitizing appears to be the only tool thatallows young children to grasp the whole and the elements at the same time, and in thisway, it guides the acquisition of the first true numerical categories (with the constraint ofaccuracy). Asvon Glasersfeld (1982)said some time ago, it is a matter of “experiencingunits of units” (p. 205).

From these results and considerations, it follows that the global, simultaneous aspects ofnumerosity play a major role in children’s understanding of what the number words mean.Remember that for numerical-equivalence judgments,Mix (1999) showed that childrenrecognized numerical equivalence on static sets earlier than on sequential sets. Thus, the typeof configuration (familiar or unfamiliar) should have an impact on quantitative evaluation,even in young children. In this experiment, familiar configurations were generally namedbetter than unfamiliar ones (seeFig. 5). This configuration effect appears to refute thefourth subitizing principle (combination irrelevance principle) stated above. But we mustconsider the following: (a) with small numbers, there was no configuration effect in thesimultaneous task, which means that the fourth principle was in effect when the subitizingstrategy was possible; (b) by contrast, with larger numbers where subitizing is more difficult,simultaneous short presentation became configuration-sensitive, as if the reason for successwas the recognition of specific patterns; and (c) global, simultaneous cues are generally usedby young children, even if the subitizing capacity is overloaded (see classic conservationtasks). The reasons for this early configuration effect are not clear.Mandler and Shebo (1982)suggested that “familiar” configurations are learned, since all configurations can potentiallybe learned. But the precociousness of the configuration effect makes this interpretationdoubtful. Moreover, in line with traditional Gestalt Theory, a regular figure may be easier toscan perceptually. So when children use subitizing, the coordination between the elementsand the whole should be easier with “familiar” configurations. Furthermore, note that forsmall numbers, the number of possible displays is limited and the figures are always regular(two points make a line, three points make a triangle).

In addition, these results lead to the question of the relationship between subitizing andcounting. From a developmental standpoint, subitizing appears to be a more primitive andnecessary skill for understanding what the counting process means (Vilette, 1994; vonGlasersfeld, 1982). On this topic, neuropsychological data are not yet entirely clear. Somearguments based on clinical reports would lead one to believe that counting and subitizingare grounded on two different processes (Dehaene & Cohen, 1994). But a PET scan study


by Piazza, Mechelli, Butterworth, and Price (2002)showed that counting and subitizingimplicated a common neuronal area, whereas counting activated an additional neuronalarea that was not activated by subitizing (but subitizing did not activate a specific area of itsown). Thus, counting appears to be “something more” than subitizing, as if subitizing were amore primitive, prerequisite skill. Yet in a recent study using functional magnetic resonanceimaging,Piazza, Giacomini, Le Bihan, and Dehaene (2003)confirmed the parallel-serialdichotomy of subitizing and counting in adults.

The apparent primitive nature of subitizing, in its strictly-perceptual, preverbal and ver-bal forms, raises the question of the potentially innate abilities in which it is rooted. Usingelectroencephalographic techniques,Grice et al. (2001)showed that the integration of per-ceptual features (as in human-face perception) appears to be both primitive and variablewhen developmental disorders are compared (autism and Williams syndrome). It is likely,then, that subitizing stems from a primitive skill that binds spatially separate elements intoa whole. However, even if a primitive form of perceptual integration (spatial binding ofseparate elements into a whole) is in effect early on, quite a long process of developmentand abstraction is necessary for the child to focus on the relationship between the elementsand the whole, i.e., to coordinate them and then to extend that coordination, as a principle,to numbers lying beyond the limits of subitizing (Karmiloff-Smith, 1992; Lehalle, 2002).In other words, as far as development is concerned, nothing is totally new, but nothing isentirely ready-made from the beginning either.

Acknowledgments

The authors wish to thank Jacqueline Bideaud and David I. Anderson for their valuableadvice on earlier drafts of this article. They also thank Peter Bryant and the reviewers atCognitive Development for providing many helpful comments.

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do young children acquire number words through subitizing or

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