- Email: [email protected]
Do young children acquire number words through subitizing or counting?
Cognitive Development 19 (2004) 291–307
Do young children acquire number words through subitizing or counting? Laurent Benoit a , Henri Lehalle a,b,∗ , François Jouen b a
b
Université Paul Valéry, Montpellier III (EA 3021: Mémoire et Cognition), Route de Mende, F 34199 Montpellier, Cedex 5, France 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 meaning of small-number words. The first stresses the role of counting and predicts better performance when the items are presented in succession. The second considers the role of subitizing and predicts better performance 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 strictly controlled conditions: simultaneous short presentations versus consecutive short presentations (one dot at a time), and familiar versus unfamiliar configurations. The children had to indicate how many dots they had seen. The results showed that, for small numbers, the children were better at giving the right number–word in the simultaneous task. Performance generally improved when familiar configurations were used, except for small numbers in simultaneous presentation. As a whole, subitizing appears 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 number words (Gelman & Gallistel, 1978). But we know little about how young children are able to grasp 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 both explanations take preverbal knowledge of numerosity into account, they differ as to how they 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 development (Gallistel & Gelman, 1992; Gelman & Gallistel, 1978). The main idea in this interpretation is the precociousness of counting principles, which are assumed to constitute truly implicit knowledge in infant numerical competence. Like animals (at least in some species), infants can react to magnitude differences, in accordance with the so-called accumulator model originally proposed by Meck and Church (1983). In this model, each focus on 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 whole can be evaluated in terms of how “full” the accumulator is. This mechanism is sufficient for discriminating small numbers. Moreover, according to Gallistel 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 few number words simply results, through counting, from a mapping between the verbal forms and the magnitudes already represented at the preverbal level. This explanation is supported by empirical findings showing that by the age of 3, children already behave in accordance with the counting principles (Gelman, Meck, & Merkin, 1986). But other authors (Baroody, 1991; Rittle-Johnson & Siegler, 1998) have stressed that, while counting principles are indeed embedded in effective early procedures for counting, it does not follow that these principles constitute underlying concepts that guide the acquisition of verbal counting (and number words). Moreover, Wynn (1992a) and Carey (2002) argued that if Gallistel and Gelman were right, the acquisition of the number–word sequence via the mapping of preverbal magnitudes to verbal names, would be easier than it appears in Fuson’s and Wynn’s data (Fuson, 1988; Wynn, 1992a). Furthermore, recent findings suggest that infants’ “numerical” competence is not truly numerical. When infants seem to discriminate small “numerosities,” they probably use perceptual 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 still have 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 of quantitative evaluation (Klahr & Wallace, 1976). Subitizing, is generally defined as an accurate quantitative evaluation of small sets without explicit (neither internal nor external) counting (see for instance Carey, 2002, p. 315). But, beyond this common definition, subitizing processes and the underlying competence may vary greatly from one author to the 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 particular, 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, as Tan and Bryant (2000, p. 1162) have pointed out, “There may be a special mechanism, often called subitizing, for recognizing and distinguishing small numbers independently of their perceptual appearance and without counting (e.g., Simon, 1997; Starkey & Cooper, 1995). It is quite likely that infants 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, whatever the perceptual configuration of number. Note that for Gallistel and Gelman (1992, p. 58), subitizing is in fact supported by a short preverbal counting process which produces the preverbal magnitude. (c) The third is perceptual–verbal subitizing, where number words are used to express numerosity. This is the traditional use of the word “subitizing” since subitizing was initially 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 developmental pathway 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 behind this 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 presence of 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, since it can explain the infant behavior noted in Wynn’s (1992b) paradigm (see for example Bideaud, 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 not strictly numerical. In controlled conditions, these cues may, therefore, not induce the “quasi numerical” processing usually observed (Feigenson et al., 2002). This is not really surprising because, while the object-file model can easily account for infants’ behavioral differentiation of small numbers, this does not imply that the whole is conceptualized, i.e., the magnitude itself, irrespective of spatial extent or density. Thus, the object-file model is not sufficient to account for the accurate use of small-number words. As a consequence, the second explanation of early number–word acquisition states that object indexing is combined with subitizing to account for the first appraisal of numerosity per se. Remember that Starkey and Cooper’s data (1995) clearly demonstrated subitizing in 2–4 years old children. At the theoretical level, many earlier developmental studies have stressed the importance of subitizing in number development, instead of reliance on counting alone. 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 situations within the range of subitizing. von Glasersfeld (1982) agreed with the position that lessons are drawn from the coincidence between the last counted word and “the same number word as result of a figural pattern perceived as a whole” (p. 207). According to Fischer (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
294
L. Benoit et al. / Cognitive Development 19 (2004) 291–307
not be acquired through a mere linear counting process. Considering both counting and subitizing, 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, numerous children 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 cardinal naming relies on subitizing. As for counting, success in a subitizing task can be assumed to be based on certain implicit principles, which can be drawn from the object-file model. But verbal subitizing is more than a perceptual skill, since it requires knowledge of the exact number words for small numerosities. This knowledge implies some categorical processes (Gallistel, 1993; von Glasersfeld, 1982) in addition to those proposed in the object-file model. Verbal subitizing is 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 there are 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 principle underlying general verbal subitizing may need to be constructed. In other words, this principle may not govern the early verbal subitizing efforts, even if it does underlie later general verbal subitizing. However, even with homogeneous arrays, numerosity has to be abstracted from the spatial detection of each element. 4. Combination irrelevance principle: The spatial combination of elements does not alter the perceptual outcome. This means that the scanning process (which may depend on the specific array) does not change the category file. In sum, there are two alternative explanations of how the cardinal meanings of the first few number words are acquired. One states that these number words are learned through counting, while their meaning simply results from a mapping of words to preverbal magnitudes. The other starts from indexed objects, which are grouped together through subitizing to give the cardinal referents labeled by the number words. The present study was aimed at evaluating predictions based on the two alternative theoretical accounts described above. Since recent evidence suggests that infants do not rely on numerical cues to differentiate numerosities, our general hypothesis was that subitizing is a way 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. Subitizing could be the developmental process that allows children to simultaneously grasp the elements and the whole, i.e., that enables them to go beyond separate, one by one indexing of objects, to the appraisal of the whole, but without ignoring the parts. In this experiment, small numbers of dots were briefly presented either simultaneously or consecutively. In both cases, young children had to indicate the number of the just perceived collection of elements. The dots arrangement was either familiar (dice configurations) or unfamiliar (random). Note that the elements for both types of display appeared in exactly the
L. Benoit et al. / Cognitive Development 19 (2004) 291–307
295
same places, so that the “configurations” were the same in the simultaneous and consecutive conditions. The following predictions were made. 1. If subitizing is the first way of understanding cardinal referents, then the performance of young children acquiring number words should be better in the simultaneous condition than 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 performance will be as good or even better in consecutive presentation—which allows the child to verbalize the number–word list—than in simultaneous presentation where too short a display time prevents explicit counting. 3. Success in quantitative evaluations by young children should be more frequent with small 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 important role, then the spatial aspects of the configuration may have an impact, even in young children, 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 three age groups: 3 years old (range: 3 years 2 months to 3 years 10 months, M = 3.5), 4 years old (range: 4 years 2 months to 4 years 9 months, M = 4.6), 5 years old (range: 5 years 3 months to 5 years 8 months, M = 5.4). There was approximately the same number of boys and girls in each group. All of the children were attending preschool. The sample was socio-economically varied, reflecting the diversity of the regional population (South of France). 2.2. Procedure The stimuli consisted of red dots 1.5 cm in diameter. Collections of 1–6 dots were presented on a white computer screen. Depending on the presentation mode, elements were displayed in different ways. In the simultaneous-presentation mode, all elements in the collection were displayed at the same time in the middle of the screen for 800 ms. In the consecutive-presentation mode, the elements in the collection were displayed one at a time for 800 ms each; after the last element in each set, a white screen with red edges appeared to indicate the request for a number word. A pause button (“Enter” key of the numeric keypad) could be pressed to stop the task at any time. A left click (correct name) and a right click (incorrect name) on the mouse was used to record the children’s verbal responses. Children were tested in a quiet dimly-lit room. In the simultaneous task, the instructions were: “You will see red spots on the computer screen. Each time, tell me how many there are.” In the consecutive task, the instructions were: “You will see red spots appearing one
296
L. Benoit et al. / Cognitive Development 19 (2004) 291–307
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.” Note that the counting task had two parts, the counting part (serial naming of each element) and the number–word part (cardinal naming of the whole). A familiarization phase was carried out with a three-element collection displayed simultaneously 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) numerical evaluation 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) (see Fig. 1 for the two configurations chosen for each number); (d) dot display direction 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 according to a repeated-measure design. Because of the necessity of controlling for order effects, the various 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 configuration (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 configuration within 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 other four 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 were displayed from left to right (LR). Two other children were presented with the second
L. Benoit et al. / Cognitive Development 19 (2004) 291–307
297
series of 24 items, and for these two children, the dots in the counting task were displayed from bottom to top (BT). As a consequence from the entire combination, there were 16 children in each age group.
3. Results The dependent variable was the number of correct responses in each experimental condition. In the simultaneous task, a correct response was defined as the verbal naming of the exact number of dots just presented. In the first part of the consecutive task, a correct response 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) when the child could verbalize the exact number of dots just presented in succession. In accordance with the experimental design, an ANOVA was computed with two betweengroup factors (age and display direction), and three repeated-measure factors (one factor for the three tasks, one for the two configurations, and one for the two number ranges). Even though the dots displayed from bottom to top seemed slightly more difficult, this effect was not significant (F1/42 = 0.41, P > 0.05). Therefore, the display direction factor was not considered further. The number of children who obtained each score in each experimental condition is presented in Fig. 2a–c. The scores varied from 0 (no correct response) to 3 (correct responses for 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 (see Figs. 3 and 4). More interesting were the two 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 a significant 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; and F1/42 = 11.98, P < 0.01, for the comparison with consecutive-cardinality), but there was no significant difference between the two parts of the consecutive task (counting and cardinality). (b) For the 4 years old, it was the consecutive-cardinality task that differed from the two easier tasks (F1/42 = 3.81, P = 0.05, for the comparison with the simultaneous task; and F1/42 = 9.87, P < 0.01, for the comparison with consecutive-counting), and the simultaneous task did not differ from consecutive-counting. (c) For the 5 years old, there were no significant differences between the three tasks. A significant Age × Number Range interaction was also observed (F2/42 = 3.93, P < 0.05). It is clear in Fig. 4 that the difference between the two number ranges (1–3, and 4–6) decreased with age. Nevertheless, planned comparisons yielded significant differences at every 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).
298
L. Benoit et al. / Cognitive Development 19 (2004) 291–307
(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 who obtained 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) on the consecutive-cardinality task, by age (3, 4, or 5 years), number range (1–3 or 4–6) and configuration (familiar FA or unfamiliar UNFA).
L. Benoit et al. / Cognitive Development 19 (2004) 291–307
(b) Fig. 2. (Continued )
299
300
L. Benoit et al. / Cognitive Development 19 (2004) 291–307
(c) Fig. 2. (Continued ).
L. Benoit et al. / Cognitive Development 19 (2004) 291–307
301
3,0
Mean Number of Correct Names
2,5
2,0
1,5
1,0 AGE 3 years
0,5
AGE 4 years
0,0 Simultaneous
Consecutive-Counting
Consecu.-Cardinality
AGE 5 years
TASK
Fig. 3. Mean number of correct names given, by age and task.
3,0
Mean Number of Correct Names
2,5
2,0
1,5
1,0
AGE 3 years 0,5
AGE 4 years
0,0 R1: from 1 to 3
R2: from 4 to 6
Number Range
Fig. 4. Mean number of correct names given, by age and number range.
AGE 5 years
302
2,6 2,4
Correct Names (Means)
2,0 1,8 1,6 1,4 1,2
TASK Simultaneous
1,0
TASK Consecutive-Counting
0,8 CONFIGURATIONS Familiar
Unfamiliar
NUMBERS: 1 to 3
CONFIGURATIONS Familiar
Unfamiliar
TASK Consecu.-Cardinality
NUMBERS: 4 to 6
Fig. 5. Mean number of correct names given, by task, number range, and configuration.
L. Benoit et al. / Cognitive Development 19 (2004) 291–307
2,2
L. Benoit et al. / Cognitive Development 19 (2004) 291–307
303
3.2. Three factors in interaction As for the age factor above, it is not very informative to consider the significant main effects of the other three factors: task (F2/84 = 4.99, P < 0.01), configuration (familiar better than 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 with age have been already presented. Fig. 5 shows how the other three factors interacted (age factor 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 how the other three factors interacted (age excluded). With small numbers (1–3), configuration familiarity 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 was always poorer and familiarity did have an effect on the simultaneous recognition of these numbers. This interaction suggests that with larger numbers children mainly recognize specific familiar patterns.
4. Discussion The results suggest that for the exact and explicit appraisal of numerosity, subitizing is a more primitive tool than counting. Accordingly, for small numbers, the 3 years old performed better on short, simultaneous presentations than on short, one by one presentations of the same items. It follows that subitizing is unlikely to be based on a preverbal counting process. If it were, performance on a verbal task would be at least as good in a simultaneous display task as in a consecutive task, since there is no reason for verbalization to disturb the basic relations between subitizing and counting, no matter what mechanisms support the preverbal differentiation of numerosities. Of course, we could have compared our simultaneous verbal task with a simultaneous counting task, i.e., a task involving the enumeration of simultaneously presented items instead of consecutively presented ones. In fact, the rather difficult consecutive counting task seems to “stack the deck” in favor of subitizing. But a simultaneous counting task implies longer viewing of the dots, which for small numbers, leaves enough time to subitize first and then count (Fischer, 1984) in order to 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, young children may know the number–word sequence used in counting without knowing the cardinal property of the last word said (Fuson, 1988, 1991; Baroody, 1991). At the age of 5, there were no differences between the two presentation modes; but of course, this result is limited to the numbers 1–6 used in this experiment. If subitizing is a primitive tool for quantitative evaluations, it is not surprising that small numbers are recognized better than larger ones, since subitizing is usually limited to the number 4. Fig. 5 summarizes the main findings: greater success with small numbers, especially when displayed simultaneously, and delayed acquisition for larger numbers, with a substantial drop for subitizing, especially on unfamiliar configurations.
304
L. Benoit et al. / Cognitive Development 19 (2004) 291–307
Based on these results, and in accordance with previous authors (Fischer, 1984, 1991; Klahr & Wallace, 1976; Vilette, 1994, 2002; von Glasersfeld, 1982;), subitizing may be regarded as the necessary developmental pathway for understanding the significance of the first few number words. Let us imagine a plausible scenario based on the object-file model (Kahneman et al., 1992). As we have seen above, this model suffices to account for numerical behavior in infants, even if infants do not focus on numerosity per se. According to the object-file model, when several objects are presented (but not many), they are stored in separate files, provided that they can be differentiated on some spatiotemporal cues. But naming the whole (cardinality) implies considering the whole as a single object. So if the whole is stored as a single object, the elements should not be represented one by one. On the other hand, knowing the exact meaning of a given number word requires maintaining awareness of the distinct elements that can ensure exact quantification. Especially if infant performance is not strictly numerical, verbal subitizing appears to be the only tool that allows young children to grasp the whole and the elements at the same time, and in this way, it guides the acquisition of the first true numerical categories (with the constraint of accuracy). As von Glasersfeld (1982) said some time ago, it is a matter of “experiencing units of units” (p. 205). From these results and considerations, it follows that the global, simultaneous aspects of numerosity play a major role in children’s understanding of what the number words mean. Remember that for numerical-equivalence judgments, Mix (1999) showed that children recognized numerical equivalence on static sets earlier than on sequential sets. Thus, the type of configuration (familiar or unfamiliar) should have an impact on quantitative evaluation, even in young children. In this experiment, familiar configurations were generally named better than unfamiliar ones (see Fig. 5). This configuration effect appears to refute the fourth subitizing principle (combination irrelevance principle) stated above. But we must consider the following: (a) with small numbers, there was no configuration effect in the simultaneous task, which means that the fourth principle was in effect when the subitizing strategy was possible; (b) by contrast, with larger numbers where subitizing is more difficult, simultaneous short presentation became configuration-sensitive, as if the reason for success was the recognition of specific patterns; and (c) global, simultaneous cues are generally used by young children, even if the subitizing capacity is overloaded (see classic conservation tasks). The reasons for this early configuration effect are not clear. Mandler and Shebo (1982) suggested that “familiar” configurations are learned, since all configurations can potentially be learned. But the precociousness of the configuration effect makes this interpretation doubtful. Moreover, in line with traditional Gestalt Theory, a regular figure may be easier to scan perceptually. So when children use subitizing, the coordination between the elements and the whole should be easier with “familiar” configurations. Furthermore, note that for small 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 and counting. From a developmental standpoint, subitizing appears to be a more primitive and necessary skill for understanding what the counting process means (Vilette, 1994; von Glasersfeld, 1982). On this topic, neuropsychological data are not yet entirely clear. Some arguments based on clinical reports would lead one to believe that counting and subitizing are grounded on two different processes (Dehaene & Cohen, 1994). But a PET scan study
L. Benoit et al. / Cognitive Development 19 (2004) 291–307
305
by Piazza, Mechelli, Butterworth, and Price (2002) showed that counting and subitizing implicated a common neuronal area, whereas counting activated an additional neuronal area that was not activated by subitizing (but subitizing did not activate a specific area of its own). Thus, counting appears to be “something more” than subitizing, as if subitizing were a more primitive, prerequisite skill. Yet in a recent study using functional magnetic resonance imaging, Piazza, Giacomini, Le Bihan, and Dehaene (2003) confirmed the parallel-serial dichotomy of subitizing and counting in adults. The apparent primitive nature of subitizing, in its strictly-perceptual, preverbal and verbal forms, raises the question of the potentially innate abilities in which it is rooted. Using electroencephalographic techniques, Grice et al. (2001) showed that the integration of perceptual features (as in human-face perception) appears to be both primitive and variable when developmental disorders are compared (autism and Williams syndrome). It is likely, then, that subitizing stems from a primitive skill that binds spatially separate elements into a whole. However, even if a primitive form of perceptual integration (spatial binding of separate elements into a whole) is in effect early on, quite a long process of development and abstraction is necessary for the child to focus on the relationship between the elements and 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 is entirely ready-made from the beginning either.
Acknowledgments The authors wish to thank Jacqueline Bideaud and David I. Anderson for their valuable advice on earlier drafts of this article. They also thank Peter Bryant and the reviewers at Cognitive Development for providing many helpful comments.
References Baroody, A. J. (1991). Procédures et principes de comptage: Leur développement avant l’école [Procedures and counting principles: Their development in preschool]. In J. Bideaud, C. Meljac, & J. -P. Fischer (Eds.), Les chemins du nombre [Pathways to number] (pp. 133–158). Lille: Presses Universitaires de Lille. Baroody, A. J., Benson, A. P., & Lai, M. L. (2003, April). Early number and arithmetic sense: A summary of three studies. Paper presented at the annual meeting of the Society for Research in Child Development, Tampa, FL. Bideaud, J. (1995). From Wynn’s infants “calculations” to cardinality: What develops? Current Psychology of Cognition, 6, 685–694. Bideaud, J. (2002). Les animaux et les bébés comptent-ils ? [Do animals and babies count?]. In J. Bideaud, & H. Lehalle (Eds.), Le développement des activités numériques chez l’enfant [The development of numerical skills in children] (pp. 55–80). Paris: Editions Hermès Sciences. Carey, S. (2002). Les discontinuités au cours du développement conceptuel sont-elles possibles? [On the very possibility of discontinuities in conceptual development]. In E. Dupoux (Ed.), Les langages du cerveau [Language, brain and cognitive development: Essays in honor of Jacques Mehler] (pp. 305–325). Paris: Editions Odile Jacob. Dehaene, S., & Cohen, L. (1994). Dissociable mechanisms of subitizing and counting: Neuropsychological evidence from simultanagnosis patients. Journal of experimental Psychology Human Perception and Performance, 20(5), 958–975.
306
L. Benoit et al. / Cognitive Development 19 (2004) 291–307
Feigenson, L., Carey, S., & Spelke, E. (2002). Infants’discrimination of number vs. continuous extent. Cognitive Psychology, 44, 33–66. Fischer, J.-P. (1984). L’appréhension du nombre par le jeune enfant [Young children’s appraisal of number]. Enfance, 36(2), 167–187. Fischer, J. -P. (1991). Le subitizing et la discontinuité après 3 [Subitizing and the discontinuity after 3]. In J. Bideaud, C. Meljac, & J. -P. Fischer (Eds.), Les chemins du nombre [Pathways to number] (pp. 235–258). Lille: Presses Universitaires de Lille. Fuson, K. C. (1988). Children’s counting and concepts of number. New-York: Springer-Verlag. Fuson, K. C. (1991). Relations entre comptage et cardinalité chez les enfants de 2 à 8 ans [Relations between counting and cardinality in children from 2 to 8 years old]. In J. Bideaud, C. Meljac, & J. -P. Fischer (Eds.), Les chemins du nombre [Pathways to number] (pp. 159–179). Lille: Presses Universitaires de Lille. Gallistel, C. R. (1993). A conceptual framework for the study of numerical estimation and arithmetic reasoning in animals. In S. T. Boysen, & E. J. Copaldi (Eds.), The development of numerical competence. Animal and human models (pp. 211–223). Hillsdale, NJ: Lawrence Erlbaum Associates. Gallistel, C. R., & Gelman, R. (1992). Preverbal and verbal computation. Cognition, 44, 43–74. Gelman, R., & Gallistel, C. (1978). The child’s understanding of number. Cambridge, MA: Harvard University Press. Gelman, R., Meck, E., & Merkin, S. (1986). Young children’s numerical competence. Cognitive Development, 1, 1–29. Grice, S. J., Spratling, M. W., Karmiloff-Smith, A., Halit, H., Csibra, G., de Haan, M., & Johnson, M. H. (2001). Disordered visual processing and oscillatory brain activity in autism and Williams syndrome. Neuroreport, 12, 2697–2700. Kahneman, D., Treisman, A., & Gibbs, B. (1992). The reviewing of object files: Object-specific integration of information. Cognitive Psychology, 24, 175–219. Karmiloff-Smith, A. (1992). Beyond modularity: A developmental perspective on cognitive science. Cambridge, MA: MIT Press. Kaufman, E. L., Lord, M. W., Reese, T. W., & Volkmann, J. (1949). The discrimination of visual number. American Journal of Psychology, 62, 498–525. Klahr, D., & Wallace, J. G. (1976). Cognitive development: An information-processing view. Hillsdale, NJ: Lawrence Erlbaum Associates. Lehalle, H. (2002). Connaissances numériques et modèles de développement [Numerical knowledge and developmental models]. In J. Bideaud, & H. Lehalle (Eds.), Le développement des activités numériques chez l’enfant [The development of numerical skills in children] (pp. 29–54). Paris: Editions Hermès Sciences. Leslie, A., Xu, F., Tremoulet, P., & Scholl, B. (1998). Indexing and the object concept: Developing ‘what’ and ‘where’ systems. Trends in Cognitive Sciences, 2, 10–18. Mandler, G., & Shebo, B. J. (1982). Subitizing: An analysis of its component processes. Journal of Experimental Psychology: General, 111, 1–22. Meck, W. H., & Church, R. M. (1983). A mode control model of counting and timing processes. Journal of Experimental Psychology: Animal Behavior Processes, 9, 320–334. Mix, K. S. (1999). Preschoolers’ recognition of numerical equivalence: Sequential sets. Journal of Experimental Child Psychology, 74, 309–332. Mix, K. S., Huttenlocher, J., & Levine, S. C. (2002). Multiple cues for quantification in infancy: Is number one of them? Psychological Bulletin, 128, 278–294. Piazza, M., Giacomini, E., Le Bihan, D., & Dehaene, S. (2003). Single-trial classification of parallel pre-attentive and serial attentive processes using functional magnetic resonance imaging. Proceedings of the Royal Society of London, 270, 1237–1245. Piazza, M., Mechelli, A., Butterworth, B., & Price, C. J. (2002). Are subitizing and counting implemented as separate or functionally overlapping processes. NeuroImage, 15, 435–446. Rittle-Johnson, B., & Siegler, R. S. (1998). The relation between conceptual and procedural knowledge in learning mathematics: A review. In C. Donlan (Ed.), The development of mathematical skills (pp. 75–110). Hove, UK: Psychology Press. Simon, T. J. (1997). Reconceptualizing the origins of number knowledge: A “non-numerical” account. Cognitive Development, 12, 349–372.
L. Benoit et al. / Cognitive Development 19 (2004) 291–307
307
Starkey, P., & Cooper, R. (1995). The development of subitizing in young children. British Journal of Developmental Psychology, 19, 399–420. Tan, L. S. C., & Bryant, P. (2000). The cues that infants use to distinguish discontinuous quantities: Evidence using a shift-rate recovery paradigm. Child Development, 71, 1162–1178. Uller, C., Carey, S., Huntley-Fenner, G., & Klatt, L. (1999). What representations might underlie infant numerical knowledge? Cognitive Development, 14, 1–36. Vilette, B. (1994). Des processus de quantification à la cardinality [From quantification processes to cardinality]. L’Année Psychologique, 94, 25–44. Vilette, B. (2002). Processus de quantification chez l’enfant: Peut-on parler d’une arithmétique précoce? [Quantification processes in children: Is it possible to speak of a precocious arithmetic]. In J. Bideaud, & H. Lehalle (Eds.), Le développement des activités numériques chez l’enfant [The development of numerical skills in children] (pp. 81–101). Paris: Editions Hermès Sciences. von Glasersfeld, E. (1982). Subitizing: The role of figural patterns in the development of numerical concepts. Archives de Psychologie, 50, 191–218. Wakeley, A., Rivera, S., & Langer, J. (2000). Can young infants add and subtract? Child Development, 71(6), 1525–1534. Wynn, K. (1992a). Children’s acquisition of the number words and the counting system. Cognitive Psychology, 24, 220–251. Wynn, K. (1992b). Addition and subtraction by human infants. Nature, 358, 749–750.
Do young children acquire number words through subitizing or counting? Laurent Benoit a , Henri Lehalle a,b,∗ , François Jouen b a
b
Université Paul Valéry, Montpellier III (EA 3021: Mémoire et Cognition), Route de Mende, F 34199 Montpellier, Cedex 5, France 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 meaning of small-number words. The first stresses the role of counting and predicts better performance when the items are presented in succession. The second considers the role of subitizing and predicts better performance 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 strictly controlled conditions: simultaneous short presentations versus consecutive short presentations (one dot at a time), and familiar versus unfamiliar configurations. The children had to indicate how many dots they had seen. The results showed that, for small numbers, the children were better at giving the right number–word in the simultaneous task. Performance generally improved when familiar configurations were used, except for small numbers in simultaneous presentation. As a whole, subitizing appears 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 number words (Gelman & Gallistel, 1978). But we know little about how young children are able to grasp 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 both explanations take preverbal knowledge of numerosity into account, they differ as to how they 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 development (Gallistel & Gelman, 1992; Gelman & Gallistel, 1978). The main idea in this interpretation is the precociousness of counting principles, which are assumed to constitute truly implicit knowledge in infant numerical competence. Like animals (at least in some species), infants can react to magnitude differences, in accordance with the so-called accumulator model originally proposed by Meck and Church (1983). In this model, each focus on 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 whole can be evaluated in terms of how “full” the accumulator is. This mechanism is sufficient for discriminating small numbers. Moreover, according to Gallistel 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 few number words simply results, through counting, from a mapping between the verbal forms and the magnitudes already represented at the preverbal level. This explanation is supported by empirical findings showing that by the age of 3, children already behave in accordance with the counting principles (Gelman, Meck, & Merkin, 1986). But other authors (Baroody, 1991; Rittle-Johnson & Siegler, 1998) have stressed that, while counting principles are indeed embedded in effective early procedures for counting, it does not follow that these principles constitute underlying concepts that guide the acquisition of verbal counting (and number words). Moreover, Wynn (1992a) and Carey (2002) argued that if Gallistel and Gelman were right, the acquisition of the number–word sequence via the mapping of preverbal magnitudes to verbal names, would be easier than it appears in Fuson’s and Wynn’s data (Fuson, 1988; Wynn, 1992a). Furthermore, recent findings suggest that infants’ “numerical” competence is not truly numerical. When infants seem to discriminate small “numerosities,” they probably use perceptual 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 still have 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 of quantitative evaluation (Klahr & Wallace, 1976). Subitizing, is generally defined as an accurate quantitative evaluation of small sets without explicit (neither internal nor external) counting (see for instance Carey, 2002, p. 315). But, beyond this common definition, subitizing processes and the underlying competence may vary greatly from one author to the 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 particular, 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, as Tan and Bryant (2000, p. 1162) have pointed out, “There may be a special mechanism, often called subitizing, for recognizing and distinguishing small numbers independently of their perceptual appearance and without counting (e.g., Simon, 1997; Starkey & Cooper, 1995). It is quite likely that infants 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, whatever the perceptual configuration of number. Note that for Gallistel and Gelman (1992, p. 58), subitizing is in fact supported by a short preverbal counting process which produces the preverbal magnitude. (c) The third is perceptual–verbal subitizing, where number words are used to express numerosity. This is the traditional use of the word “subitizing” since subitizing was initially 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 developmental pathway 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 behind this 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 presence of 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, since it can explain the infant behavior noted in Wynn’s (1992b) paradigm (see for example Bideaud, 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 not strictly numerical. In controlled conditions, these cues may, therefore, not induce the “quasi numerical” processing usually observed (Feigenson et al., 2002). This is not really surprising because, while the object-file model can easily account for infants’ behavioral differentiation of small numbers, this does not imply that the whole is conceptualized, i.e., the magnitude itself, irrespective of spatial extent or density. Thus, the object-file model is not sufficient to account for the accurate use of small-number words. As a consequence, the second explanation of early number–word acquisition states that object indexing is combined with subitizing to account for the first appraisal of numerosity per se. Remember that Starkey and Cooper’s data (1995) clearly demonstrated subitizing in 2–4 years old children. At the theoretical level, many earlier developmental studies have stressed the importance of subitizing in number development, instead of reliance on counting alone. 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 situations within the range of subitizing. von Glasersfeld (1982) agreed with the position that lessons are drawn from the coincidence between the last counted word and “the same number word as result of a figural pattern perceived as a whole” (p. 207). According to Fischer (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
294
L. Benoit et al. / Cognitive Development 19 (2004) 291–307
not be acquired through a mere linear counting process. Considering both counting and subitizing, 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, numerous children 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 cardinal naming relies on subitizing. As for counting, success in a subitizing task can be assumed to be based on certain implicit principles, which can be drawn from the object-file model. But verbal subitizing is more than a perceptual skill, since it requires knowledge of the exact number words for small numerosities. This knowledge implies some categorical processes (Gallistel, 1993; von Glasersfeld, 1982) in addition to those proposed in the object-file model. Verbal subitizing is 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 there are 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 principle underlying general verbal subitizing may need to be constructed. In other words, this principle may not govern the early verbal subitizing efforts, even if it does underlie later general verbal subitizing. However, even with homogeneous arrays, numerosity has to be abstracted from the spatial detection of each element. 4. Combination irrelevance principle: The spatial combination of elements does not alter the perceptual outcome. This means that the scanning process (which may depend on the specific array) does not change the category file. In sum, there are two alternative explanations of how the cardinal meanings of the first few number words are acquired. One states that these number words are learned through counting, while their meaning simply results from a mapping of words to preverbal magnitudes. The other starts from indexed objects, which are grouped together through subitizing to give the cardinal referents labeled by the number words. The present study was aimed at evaluating predictions based on the two alternative theoretical accounts described above. Since recent evidence suggests that infants do not rely on numerical cues to differentiate numerosities, our general hypothesis was that subitizing is a way 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. Subitizing could be the developmental process that allows children to simultaneously grasp the elements and the whole, i.e., that enables them to go beyond separate, one by one indexing of objects, to the appraisal of the whole, but without ignoring the parts. In this experiment, small numbers of dots were briefly presented either simultaneously or consecutively. In both cases, young children had to indicate the number of the just perceived collection of elements. The dots arrangement was either familiar (dice configurations) or unfamiliar (random). Note that the elements for both types of display appeared in exactly the
L. Benoit et al. / Cognitive Development 19 (2004) 291–307
295
same places, so that the “configurations” were the same in the simultaneous and consecutive conditions. The following predictions were made. 1. If subitizing is the first way of understanding cardinal referents, then the performance of young children acquiring number words should be better in the simultaneous condition than 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 performance will be as good or even better in consecutive presentation—which allows the child to verbalize the number–word list—than in simultaneous presentation where too short a display time prevents explicit counting. 3. Success in quantitative evaluations by young children should be more frequent with small 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 important role, then the spatial aspects of the configuration may have an impact, even in young children, 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 three age groups: 3 years old (range: 3 years 2 months to 3 years 10 months, M = 3.5), 4 years old (range: 4 years 2 months to 4 years 9 months, M = 4.6), 5 years old (range: 5 years 3 months to 5 years 8 months, M = 5.4). There was approximately the same number of boys and girls in each group. All of the children were attending preschool. The sample was socio-economically varied, reflecting the diversity of the regional population (South of France). 2.2. Procedure The stimuli consisted of red dots 1.5 cm in diameter. Collections of 1–6 dots were presented on a white computer screen. Depending on the presentation mode, elements were displayed in different ways. In the simultaneous-presentation mode, all elements in the collection were displayed at the same time in the middle of the screen for 800 ms. In the consecutive-presentation mode, the elements in the collection were displayed one at a time for 800 ms each; after the last element in each set, a white screen with red edges appeared to indicate the request for a number word. A pause button (“Enter” key of the numeric keypad) could be pressed to stop the task at any time. A left click (correct name) and a right click (incorrect name) on the mouse was used to record the children’s verbal responses. Children were tested in a quiet dimly-lit room. In the simultaneous task, the instructions were: “You will see red spots on the computer screen. Each time, tell me how many there are.” In the consecutive task, the instructions were: “You will see red spots appearing one
296
L. Benoit et al. / Cognitive Development 19 (2004) 291–307
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.” Note that the counting task had two parts, the counting part (serial naming of each element) and the number–word part (cardinal naming of the whole). A familiarization phase was carried out with a three-element collection displayed simultaneously 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) numerical evaluation 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) (see Fig. 1 for the two configurations chosen for each number); (d) dot display direction 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 according to a repeated-measure design. Because of the necessity of controlling for order effects, the various 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 configuration (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 configuration within 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 other four 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 were displayed from left to right (LR). Two other children were presented with the second
L. Benoit et al. / Cognitive Development 19 (2004) 291–307
297
series of 24 items, and for these two children, the dots in the counting task were displayed from bottom to top (BT). As a consequence from the entire combination, there were 16 children in each age group.
3. Results The dependent variable was the number of correct responses in each experimental condition. In the simultaneous task, a correct response was defined as the verbal naming of the exact number of dots just presented. In the first part of the consecutive task, a correct response 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) when the child could verbalize the exact number of dots just presented in succession. In accordance with the experimental design, an ANOVA was computed with two betweengroup factors (age and display direction), and three repeated-measure factors (one factor for the three tasks, one for the two configurations, and one for the two number ranges). Even though the dots displayed from bottom to top seemed slightly more difficult, this effect was not significant (F1/42 = 0.41, P > 0.05). Therefore, the display direction factor was not considered further. The number of children who obtained each score in each experimental condition is presented in Fig. 2a–c. The scores varied from 0 (no correct response) to 3 (correct responses for 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 (see Figs. 3 and 4). More interesting were the two 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 a significant 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; and F1/42 = 11.98, P < 0.01, for the comparison with consecutive-cardinality), but there was no significant difference between the two parts of the consecutive task (counting and cardinality). (b) For the 4 years old, it was the consecutive-cardinality task that differed from the two easier tasks (F1/42 = 3.81, P = 0.05, for the comparison with the simultaneous task; and F1/42 = 9.87, P < 0.01, for the comparison with consecutive-counting), and the simultaneous task did not differ from consecutive-counting. (c) For the 5 years old, there were no significant differences between the three tasks. A significant Age × Number Range interaction was also observed (F2/42 = 3.93, P < 0.05). It is clear in Fig. 4 that the difference between the two number ranges (1–3, and 4–6) decreased with age. Nevertheless, planned comparisons yielded significant differences at every 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).
298
L. Benoit et al. / Cognitive Development 19 (2004) 291–307
(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 who obtained 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) on the consecutive-cardinality task, by age (3, 4, or 5 years), number range (1–3 or 4–6) and configuration (familiar FA or unfamiliar UNFA).
L. Benoit et al. / Cognitive Development 19 (2004) 291–307
(b) Fig. 2. (Continued )
299
300
L. Benoit et al. / Cognitive Development 19 (2004) 291–307
(c) Fig. 2. (Continued ).
L. Benoit et al. / Cognitive Development 19 (2004) 291–307
301
3,0
Mean Number of Correct Names
2,5
2,0
1,5
1,0 AGE 3 years
0,5
AGE 4 years
0,0 Simultaneous
Consecutive-Counting
Consecu.-Cardinality
AGE 5 years
TASK
Fig. 3. Mean number of correct names given, by age and task.
3,0
Mean Number of Correct Names
2,5
2,0
1,5
1,0
AGE 3 years 0,5
AGE 4 years
0,0 R1: from 1 to 3
R2: from 4 to 6
Number Range
Fig. 4. Mean number of correct names given, by age and number range.
AGE 5 years
302
2,6 2,4
Correct Names (Means)
2,0 1,8 1,6 1,4 1,2
TASK Simultaneous
1,0
TASK Consecutive-Counting
0,8 CONFIGURATIONS Familiar
Unfamiliar
NUMBERS: 1 to 3
CONFIGURATIONS Familiar
Unfamiliar
TASK Consecu.-Cardinality
NUMBERS: 4 to 6
Fig. 5. Mean number of correct names given, by task, number range, and configuration.
L. Benoit et al. / Cognitive Development 19 (2004) 291–307
2,2
L. Benoit et al. / Cognitive Development 19 (2004) 291–307
303
3.2. Three factors in interaction As for the age factor above, it is not very informative to consider the significant main effects of the other three factors: task (F2/84 = 4.99, P < 0.01), configuration (familiar better than 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 with age have been already presented. Fig. 5 shows how the other three factors interacted (age factor 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 how the other three factors interacted (age excluded). With small numbers (1–3), configuration familiarity 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 was always poorer and familiarity did have an effect on the simultaneous recognition of these numbers. This interaction suggests that with larger numbers children mainly recognize specific familiar patterns.
4. Discussion The results suggest that for the exact and explicit appraisal of numerosity, subitizing is a more primitive tool than counting. Accordingly, for small numbers, the 3 years old performed better on short, simultaneous presentations than on short, one by one presentations of the same items. It follows that subitizing is unlikely to be based on a preverbal counting process. If it were, performance on a verbal task would be at least as good in a simultaneous display task as in a consecutive task, since there is no reason for verbalization to disturb the basic relations between subitizing and counting, no matter what mechanisms support the preverbal differentiation of numerosities. Of course, we could have compared our simultaneous verbal task with a simultaneous counting task, i.e., a task involving the enumeration of simultaneously presented items instead of consecutively presented ones. In fact, the rather difficult consecutive counting task seems to “stack the deck” in favor of subitizing. But a simultaneous counting task implies longer viewing of the dots, which for small numbers, leaves enough time to subitize first and then count (Fischer, 1984) in order to 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, young children may know the number–word sequence used in counting without knowing the cardinal property of the last word said (Fuson, 1988, 1991; Baroody, 1991). At the age of 5, there were no differences between the two presentation modes; but of course, this result is limited to the numbers 1–6 used in this experiment. If subitizing is a primitive tool for quantitative evaluations, it is not surprising that small numbers are recognized better than larger ones, since subitizing is usually limited to the number 4. Fig. 5 summarizes the main findings: greater success with small numbers, especially when displayed simultaneously, and delayed acquisition for larger numbers, with a substantial drop for subitizing, especially on unfamiliar configurations.
304
L. Benoit et al. / Cognitive Development 19 (2004) 291–307
Based on these results, and in accordance with previous authors (Fischer, 1984, 1991; Klahr & Wallace, 1976; Vilette, 1994, 2002; von Glasersfeld, 1982;), subitizing may be regarded as the necessary developmental pathway for understanding the significance of the first few number words. Let us imagine a plausible scenario based on the object-file model (Kahneman et al., 1992). As we have seen above, this model suffices to account for numerical behavior in infants, even if infants do not focus on numerosity per se. According to the object-file model, when several objects are presented (but not many), they are stored in separate files, provided that they can be differentiated on some spatiotemporal cues. But naming the whole (cardinality) implies considering the whole as a single object. So if the whole is stored as a single object, the elements should not be represented one by one. On the other hand, knowing the exact meaning of a given number word requires maintaining awareness of the distinct elements that can ensure exact quantification. Especially if infant performance is not strictly numerical, verbal subitizing appears to be the only tool that allows young children to grasp the whole and the elements at the same time, and in this way, it guides the acquisition of the first true numerical categories (with the constraint of accuracy). As von Glasersfeld (1982) said some time ago, it is a matter of “experiencing units of units” (p. 205). From these results and considerations, it follows that the global, simultaneous aspects of numerosity play a major role in children’s understanding of what the number words mean. Remember that for numerical-equivalence judgments, Mix (1999) showed that children recognized numerical equivalence on static sets earlier than on sequential sets. Thus, the type of configuration (familiar or unfamiliar) should have an impact on quantitative evaluation, even in young children. In this experiment, familiar configurations were generally named better than unfamiliar ones (see Fig. 5). This configuration effect appears to refute the fourth subitizing principle (combination irrelevance principle) stated above. But we must consider the following: (a) with small numbers, there was no configuration effect in the simultaneous task, which means that the fourth principle was in effect when the subitizing strategy was possible; (b) by contrast, with larger numbers where subitizing is more difficult, simultaneous short presentation became configuration-sensitive, as if the reason for success was the recognition of specific patterns; and (c) global, simultaneous cues are generally used by young children, even if the subitizing capacity is overloaded (see classic conservation tasks). The reasons for this early configuration effect are not clear. Mandler and Shebo (1982) suggested that “familiar” configurations are learned, since all configurations can potentially be learned. But the precociousness of the configuration effect makes this interpretation doubtful. Moreover, in line with traditional Gestalt Theory, a regular figure may be easier to scan perceptually. So when children use subitizing, the coordination between the elements and the whole should be easier with “familiar” configurations. Furthermore, note that for small 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 and counting. From a developmental standpoint, subitizing appears to be a more primitive and necessary skill for understanding what the counting process means (Vilette, 1994; von Glasersfeld, 1982). On this topic, neuropsychological data are not yet entirely clear. Some arguments based on clinical reports would lead one to believe that counting and subitizing are grounded on two different processes (Dehaene & Cohen, 1994). But a PET scan study
L. Benoit et al. / Cognitive Development 19 (2004) 291–307
305
by Piazza, Mechelli, Butterworth, and Price (2002) showed that counting and subitizing implicated a common neuronal area, whereas counting activated an additional neuronal area that was not activated by subitizing (but subitizing did not activate a specific area of its own). Thus, counting appears to be “something more” than subitizing, as if subitizing were a more primitive, prerequisite skill. Yet in a recent study using functional magnetic resonance imaging, Piazza, Giacomini, Le Bihan, and Dehaene (2003) confirmed the parallel-serial dichotomy of subitizing and counting in adults. The apparent primitive nature of subitizing, in its strictly-perceptual, preverbal and verbal forms, raises the question of the potentially innate abilities in which it is rooted. Using electroencephalographic techniques, Grice et al. (2001) showed that the integration of perceptual features (as in human-face perception) appears to be both primitive and variable when developmental disorders are compared (autism and Williams syndrome). It is likely, then, that subitizing stems from a primitive skill that binds spatially separate elements into a whole. However, even if a primitive form of perceptual integration (spatial binding of separate elements into a whole) is in effect early on, quite a long process of development and abstraction is necessary for the child to focus on the relationship between the elements and 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 is entirely ready-made from the beginning either.
Acknowledgments The authors wish to thank Jacqueline Bideaud and David I. Anderson for their valuable advice on earlier drafts of this article. They also thank Peter Bryant and the reviewers at Cognitive Development for providing many helpful comments.
References Baroody, A. J. (1991). Procédures et principes de comptage: Leur développement avant l’école [Procedures and counting principles: Their development in preschool]. In J. Bideaud, C. Meljac, & J. -P. Fischer (Eds.), Les chemins du nombre [Pathways to number] (pp. 133–158). Lille: Presses Universitaires de Lille. Baroody, A. J., Benson, A. P., & Lai, M. L. (2003, April). Early number and arithmetic sense: A summary of three studies. Paper presented at the annual meeting of the Society for Research in Child Development, Tampa, FL. Bideaud, J. (1995). From Wynn’s infants “calculations” to cardinality: What develops? Current Psychology of Cognition, 6, 685–694. Bideaud, J. (2002). Les animaux et les bébés comptent-ils ? [Do animals and babies count?]. In J. Bideaud, & H. Lehalle (Eds.), Le développement des activités numériques chez l’enfant [The development of numerical skills in children] (pp. 55–80). Paris: Editions Hermès Sciences. Carey, S. (2002). Les discontinuités au cours du développement conceptuel sont-elles possibles? [On the very possibility of discontinuities in conceptual development]. In E. Dupoux (Ed.), Les langages du cerveau [Language, brain and cognitive development: Essays in honor of Jacques Mehler] (pp. 305–325). Paris: Editions Odile Jacob. Dehaene, S., & Cohen, L. (1994). Dissociable mechanisms of subitizing and counting: Neuropsychological evidence from simultanagnosis patients. Journal of experimental Psychology Human Perception and Performance, 20(5), 958–975.
306
L. Benoit et al. / Cognitive Development 19 (2004) 291–307
Feigenson, L., Carey, S., & Spelke, E. (2002). Infants’discrimination of number vs. continuous extent. Cognitive Psychology, 44, 33–66. Fischer, J.-P. (1984). L’appréhension du nombre par le jeune enfant [Young children’s appraisal of number]. Enfance, 36(2), 167–187. Fischer, J. -P. (1991). Le subitizing et la discontinuité après 3 [Subitizing and the discontinuity after 3]. In J. Bideaud, C. Meljac, & J. -P. Fischer (Eds.), Les chemins du nombre [Pathways to number] (pp. 235–258). Lille: Presses Universitaires de Lille. Fuson, K. C. (1988). Children’s counting and concepts of number. New-York: Springer-Verlag. Fuson, K. C. (1991). Relations entre comptage et cardinalité chez les enfants de 2 à 8 ans [Relations between counting and cardinality in children from 2 to 8 years old]. In J. Bideaud, C. Meljac, & J. -P. Fischer (Eds.), Les chemins du nombre [Pathways to number] (pp. 159–179). Lille: Presses Universitaires de Lille. Gallistel, C. R. (1993). A conceptual framework for the study of numerical estimation and arithmetic reasoning in animals. In S. T. Boysen, & E. J. Copaldi (Eds.), The development of numerical competence. Animal and human models (pp. 211–223). Hillsdale, NJ: Lawrence Erlbaum Associates. Gallistel, C. R., & Gelman, R. (1992). Preverbal and verbal computation. Cognition, 44, 43–74. Gelman, R., & Gallistel, C. (1978). The child’s understanding of number. Cambridge, MA: Harvard University Press. Gelman, R., Meck, E., & Merkin, S. (1986). Young children’s numerical competence. Cognitive Development, 1, 1–29. Grice, S. J., Spratling, M. W., Karmiloff-Smith, A., Halit, H., Csibra, G., de Haan, M., & Johnson, M. H. (2001). Disordered visual processing and oscillatory brain activity in autism and Williams syndrome. Neuroreport, 12, 2697–2700. Kahneman, D., Treisman, A., & Gibbs, B. (1992). The reviewing of object files: Object-specific integration of information. Cognitive Psychology, 24, 175–219. Karmiloff-Smith, A. (1992). Beyond modularity: A developmental perspective on cognitive science. Cambridge, MA: MIT Press. Kaufman, E. L., Lord, M. W., Reese, T. W., & Volkmann, J. (1949). The discrimination of visual number. American Journal of Psychology, 62, 498–525. Klahr, D., & Wallace, J. G. (1976). Cognitive development: An information-processing view. Hillsdale, NJ: Lawrence Erlbaum Associates. Lehalle, H. (2002). Connaissances numériques et modèles de développement [Numerical knowledge and developmental models]. In J. Bideaud, & H. Lehalle (Eds.), Le développement des activités numériques chez l’enfant [The development of numerical skills in children] (pp. 29–54). Paris: Editions Hermès Sciences. Leslie, A., Xu, F., Tremoulet, P., & Scholl, B. (1998). Indexing and the object concept: Developing ‘what’ and ‘where’ systems. Trends in Cognitive Sciences, 2, 10–18. Mandler, G., & Shebo, B. J. (1982). Subitizing: An analysis of its component processes. Journal of Experimental Psychology: General, 111, 1–22. Meck, W. H., & Church, R. M. (1983). A mode control model of counting and timing processes. Journal of Experimental Psychology: Animal Behavior Processes, 9, 320–334. Mix, K. S. (1999). Preschoolers’ recognition of numerical equivalence: Sequential sets. Journal of Experimental Child Psychology, 74, 309–332. Mix, K. S., Huttenlocher, J., & Levine, S. C. (2002). Multiple cues for quantification in infancy: Is number one of them? Psychological Bulletin, 128, 278–294. Piazza, M., Giacomini, E., Le Bihan, D., & Dehaene, S. (2003). Single-trial classification of parallel pre-attentive and serial attentive processes using functional magnetic resonance imaging. Proceedings of the Royal Society of London, 270, 1237–1245. Piazza, M., Mechelli, A., Butterworth, B., & Price, C. J. (2002). Are subitizing and counting implemented as separate or functionally overlapping processes. NeuroImage, 15, 435–446. Rittle-Johnson, B., & Siegler, R. S. (1998). The relation between conceptual and procedural knowledge in learning mathematics: A review. In C. Donlan (Ed.), The development of mathematical skills (pp. 75–110). Hove, UK: Psychology Press. Simon, T. J. (1997). Reconceptualizing the origins of number knowledge: A “non-numerical” account. Cognitive Development, 12, 349–372.
L. Benoit et al. / Cognitive Development 19 (2004) 291–307
307
Starkey, P., & Cooper, R. (1995). The development of subitizing in young children. British Journal of Developmental Psychology, 19, 399–420. Tan, L. S. C., & Bryant, P. (2000). The cues that infants use to distinguish discontinuous quantities: Evidence using a shift-rate recovery paradigm. Child Development, 71, 1162–1178. Uller, C., Carey, S., Huntley-Fenner, G., & Klatt, L. (1999). What representations might underlie infant numerical knowledge? Cognitive Development, 14, 1–36. Vilette, B. (1994). Des processus de quantification à la cardinality [From quantification processes to cardinality]. L’Année Psychologique, 94, 25–44. Vilette, B. (2002). Processus de quantification chez l’enfant: Peut-on parler d’une arithmétique précoce? [Quantification processes in children: Is it possible to speak of a precocious arithmetic]. In J. Bideaud, & H. Lehalle (Eds.), Le développement des activités numériques chez l’enfant [The development of numerical skills in children] (pp. 81–101). Paris: Editions Hermès Sciences. von Glasersfeld, E. (1982). Subitizing: The role of figural patterns in the development of numerical concepts. Archives de Psychologie, 50, 191–218. Wakeley, A., Rivera, S., & Langer, J. (2000). Can young infants add and subtract? Child Development, 71(6), 1525–1534. Wynn, K. (1992a). Children’s acquisition of the number words and the counting system. Cognitive Psychology, 24, 220–251. Wynn, K. (1992b). Addition and subtraction by human infants. Nature, 358, 749–750.