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British Journal of Developmental Psychology (2014), 32, 163–177 © 2013 The British Psychological Society www.wileyonlinelibrary.com
Children acquire the later-greater principle after the cardinal principle Mathieu Le Corre* Centro de Investigaci on Transdisciplinar en Psicologıa, Universidad Autonoma del Estado de Morelos, Cuernavaca, Mexico Many have proposed that the acquisition of the cardinal principle (CP) is a result of the discovery of the numerical significance of the order of the number words in the count list. However, this need not be the case. Indeed, the CP does not state anything about the numerical significance of the order of the number words. It only states that the last word of a correct count denotes the numerosity of the counted set. Here, we test whether the acquisition of the CP involves the discovery of the later-greater principle – that is, that the order of the number words corresponds to the relative size of the numerosities they denote. Specifically, we tested knowledge of verbal numerical comparisons (e.g., Is ‘ten’ more than ‘six’?) in children who had recently learned the CP. We find that these children can compare number words between ‘six’ and ‘ten’ only if they have mapped them onto non-verbal representations of numerosity. We suggest that this means that the acquisition of the CP does not involve the discovery of the correspondence between the order of the number words and the relative size of the numerosities they denote.
By age 2, typical children can recite part of their language’s count list, but, to them, counting is nothing but a meaningless list of words. Then, about 1–2 years later, their understanding of verbal counting changes radically. They acquire the cardinal principle (CP): they learn that when a count has been recited in the correct order and has correctly been placed in one-to-one correspondence with elements of a set, the last word denotes the numerosity of the set (Le Corre, Van de Walle, Brannon, & Carey, 2006; Sarnecka & Carey, 2008; Schaeffer, Eggleston, & Scott, 1974; Wynn, 1990, 1992). What changes in children’s understanding of verbal counting when they acquire the CP? Many have proposed that what changes is their understanding of the meaning of the order of the number words in the count list. Initially, the order is meaningless, like the order of the letters of the alphabet. Then, children suddenly grasp at least some of the numerical implications of the order of the number words. This is, what allows them to learn the CP. Some have proposed that they learn that the order of the number words corresponds to the numerical relation successor – that is, that ‘immediately follows’ corresponds to ‘add 1’ (Carey, 2004, 2009; Piantadosi, Tenenbaum, & Goodman, 2012; Sarnecka & Carey, 2008). Others proposed that they learn how order corresponds to relative numerical size (Spelke & Tsivkin, 2001) – that is, that number words that occur
*Correspondence should be addressed to Mathieu Le Corre, Asia 2 #101, Colonia Barrio La Concepcion, Coyoacan, Mexico, D.F., Mexico (email: [email protected]). DOI:10.1111/bjdp.12029
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later in the count list denote larger numerosities. The latter correspondence is sometimes known as the ‘later-greater principle’. Alternatively, it may be that children acquire the CP without learning anything about the numerical significance of the order of the number words. Indeed, knowledge of the CP as Gelman and Gallistel (1978) first described it does not involve any representation of correspondences between the order of number words and relations between the numerosities they denote. On Gelman and Gallistel’s original proposal, children understand how counting represents numerosity when they learn that the last number word of a count denotes the numerosity (or cardinality) of a counted set if and only if the count is recited in the correct order and matches the set on one-to-one correspondence. Taken literally, this knowledge states conditions for determining what numerosity is denoted by what number word. It does not include any knowledge of correspondences between the order of the number words and relations between the numerosities they denote. Therefore, the acquisition of the CP need not involve the acquisition of knowledge of correspondences between the order of number words and relations between the numerosities they denote. This study investigates whether the acquisition of the CP involves noticing the correspondence between the order of the number words and the relative size of the numerosities they denote. In other words, we ask whether the acquisition of the CP involves the acquisition of the later-greater principle. If it does, then all children who know the cardinal principle (henceforth, ‘CP-knowers’) should be able to infer relative size relations between any pair of number words in their count list. However, on its own, such evidence is not sufficient to conclude that CP-knowers know the later-greater principle. Indeed, CP-knowers could also infer these relations from other knowledge. First, prior to becoming CP-knowers, children learn the meanings of ‘one’ to ‘four’ by mapping them onto non-verbal representations of numerosity (e.g., Le Corre & Carey, 2007; Wynn, 1990, 1992). They also take unmapped number words to denote larger numerosities than mapped number words. For example, children who have learned ‘one’ to ‘three’ (but are not CP-knowers) know that all number words beyond ‘three’ denote larger numerosities than ‘one’ to ‘three’ (Condry & Spelke, 2008). The exact source of this knowledge is still unknown. However, it is clear that it does not come from knowledge of how the order of the number words encodes relative numerical size. If children have such knowledge, they should be able to compare any two number words in their list. Yet, although children can recite the count list beyond ‘four’ prior to becoming CP-knowers, they cannot compare pairs of number words beyond ‘four’ (Condry & Spelke, 2008). Second, CP-knowers could infer relative size relations between number words beyond ‘four’ from mappings between these number words and non-verbal representations of numerosity. Multiple studies have shown that, starting in infancy, humans have access to non-verbal representations of numerosity (Feigenson, 2005; Lipton & Spelke, 2003; Xu & Spelke, 2000) and that these representations support computations of relative numerical size (Brannon, 2002; Suanda, Tompson, & Brannon, 2008). Mappings between number words and these representations could allow children to infer relative numerical size relations between the numerosities denoted by number words without using their relative positions in the count list. There is evidence that many children map number words between ‘four’ and ‘ten’ onto these non-verbal representations a few months after they become CP-knowers (Le Corre & Carey, 2007) and that some do so even earlier (Wagner & Johnson, 2011). Therefore, CP-knowers could carry out verbal numerical comparisons by relying on such mappings rather than on the later-greater principle.
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In sum, prior to becoming CP-knowers, children can compare the numerosities denoted by two number words between ‘one’ to ‘four’ or by a number word between ‘one’ and ‘four’ and a number word beyond ‘four’ without knowing the later-greater principle. Thus, it goes without saying that CP-knowers’ performance on such comparisons cannot be used to test whether they know the later-greater principle. Moreover, evidence that CP-knowers know some numerical relations between number words beyond ‘four’ is not sufficient to show that they have learned the later-greater principle. Rather, it is also necessary to show that they have not mapped these number words onto non-verbal representations of numerosity. Hence, to test whether the acquisition of the CP involves learning the later-greater principle, the present study will test whether CP-knowers can compare the numerosities denoted by number words beyond ‘four’ even if they have not mapped these number words onto non-verbal representations of numerosity. One study attempted to determine whether CP-knowers can compare the numerosities denoted by number words beyond ‘four’ even if they have not mapped them onto non-verbal representations of numerosity (Davidson, Eng, & Barner, 2012). In this study, mappings between number words and non-verbal representations of numerosity were assessed with a verbal estimation task where children had to estimate the numerosity of sets of 5–51 objects without counting. Davidson et al. found that CP-knowers performed above chance on comparisons of number words beyond ‘four’, even if they could not verbally estimate the numerosity of sets of 5–51. This could be taken to suggest that learning the later-greater principle is involved in learning the CP. However, there are some reasons to doubt this conclusion. First, as Davidson et al. point out, the CP-knowers who could not verbally estimate the numerosity of sets of 5–51 performed quite poorly on the verbal comparisons task – on average, they answered correctly about 65% of the time. Second, it also could be that some of the CP-knowers who failed to estimate the numerosity of sets of 5–51 actually had formed mappings for at least some number words beyond ‘four’. Indeed, about half of the CP-knowers in their sample could not count further than ‘twenty-nine’. Yet, the verbal estimation task included sets of up to 51 objects. Thus, the CP-knowers who could not count further than ‘twenty-nine’ may have failed to estimate the numerosity of sets of 5–51 because the task required many more number words than they knew, not because they did not have any mappings for number words beyond ‘four’. Therefore, Davidson et al.’s study does not provide conclusive evidence that CP-knowers can carry out verbal numerical comparisons without accessing non-verbal representations of numerosity. The current study presents a stronger test of CP-knowers’ knowledge of the later-greater principle. As Davidson et al., we asked whether CP-knowers can compare the numerosities denoted by number words beyond ‘four’ even if they have not mapped them onto non-verbal representations of numerosity. However, unlike Davidson et al., we focused exclusively on number words beyond ‘four’ that are well within the counting range of typical young CP-knowers – that is, ‘six’ to ‘ten’. That is, we asked whether CP-knowers can compare ‘six’ and ‘ten’ and ‘eight’ and ‘ten’, even if they cannot estimate the numerosity of sets of 6–10 objects. We were thus less likely to underestimate children’s knowledge of the mappings of the number words we targeted. Therefore, evidence that CP-knowers can compare number words between ‘six’ and ‘ten’ even if they cannot estimate the numerosity of sets of 6–10 would strongly suggest that the change in children’s understanding of counting involves learning the later-greater principle. Evidence to the contrary would suggest that learning the CP does not involve learning the later-greater principle.
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Method Participants Thirty 3- and 4-year-olds participated in this study (Mage = 3 years, 10 months, range: 3 years, 2 months to 4 years, 6 months).1 All were fluent English speakers recruited in the Greater Boston area. Participants were initially recruited by letter and phone calls through commercially available lists or by letters sent home by the day care centres. They were tested at a university child development laboratory or at local day care centres. A caregiver accompanied all children tested at the laboratory. Children received a small gift for their participation. Caregivers who came to the laboratory received reimbursement for their travel expenses; day cares were given children’s books. The majority of the children were from middle-class backgrounds, and most were Caucasian although a small number of Asian, African American, and Hispanic children participated. Another 22 children also participated, but are not reported here because they had not yet become CP-knowers (n = 21), or because they did not meet the criterion for inclusion in the verbal numerical comparisons task (n = 1).
Materials and procedures All children were tested on four tasks: a verbal counting task, give a number, Le Corre and Carey’s (2007) verbal numerical estimation task (Fast Cards), and a verbal numerical comparisons task. The tasks were administered over two sessions. In the first session, children were tested on fast cards, give a number, and the verbal counting task, always in that order, and they were tested on verbal numerical comparisons in the second session. The two sessions usually occurred within 10 days of each other and were never separated by more than 3 weeks.
Verbal counting Children were presented with a row of 10 small plastic toys. The toys were identical to each other. Children were asked to count the toys. If children’s first attempt at counting the toys was grossly wrong, the experimenter asked them to recount them and assisted their counting by pointing to each toy as they counted.
Give a number Children were provided with a container filled with 12–15 small plastic toys. Children were always asked for ‘one’ toy first. If they correctly gave one toy, the experimenter replaced it in the container and moved on to ask for ‘two’ toys. If children did not give the correct number, they were given a chance to count the objects they had given and to fix their answer. The experimenter kept asking for the next highest number as long as children could give the number of toys she had requested. If a child gave the requested number of toys up to ‘six’, the experimenter tested ‘five’ and ‘six’ once more each. If a child failed before reaching ‘six’, the experimenter asked for the next smallest number
1 The give a number and verbal numerical estimation (Fast Cards) data for 26 of these children were reported in Le Corre and Carey (2007). The verbal numerical comparisons data and their analysis in terms of the other two tasks are reported here for the first time for all children.
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and kept moving up or down in steps of one to establish the highest number the child could give correctly two out of three times.
Verbal numerical estimation This task assessed children’s knowledge of mappings between number words and non-verbal representations of numerosity. It was first modelled by an experimenter. The experimenter presented the child with cards with sets of circles printed on them and then told the child how many circles were on each card. Modelling included sets of 1–12 dots and a set of 15. The total surface area of the sets of circles (i.e., the sum of the individual areas of the circles comprising a set) used to model the task was negatively correlated with numerosity, so that circle size decreased as numerosity increased.2 The experimenter initiated the test phase by saying ‘Now it’s your turn!’ At the beginning of each test trial, the experimenter held the card with the set facing towards her and attracted the child’s attention by saying ‘Ready?’ Then, the experimenter said, ‘Go!’ and flipped the card over so the set faced the child. After approximately 1 s, the experimenter removed the set from the child’s view. If the child had not already guessed the numerosity, she or he was asked to do so. If the child still refused to produce an answer, the experimenter presented the set again and encouraged the child to make a guess. The set was left in sight until the child made a guess. If the child still did not answer, the experimenter told the child how many circles were on the card. Trials where cards were re-presented were only used to incite children to answer. They were not included in any of the analyses. The test numerosities were 1, 2, 3, 4, 6, 8, and 10. Each numerosity was presented four times, each time in a different configuration of circles. Sets for large numerosities (i.e., 6, 8, or 10) were configured so that they could not be easily broken into smaller perceptual groups (e.g., none of the sets of six consisted of two parallel rows of three circles). The sets were presented in four series. Each series included one trial of each numerosity. The order of presentation of the sets was randomized within each series. In two of the series, the summed area of the circles remained constant, so that the size of individual circles decreased as numerosity increased. In the other two series, summed area and individual circle area both decreased as numerosity increased. Therefore, children could not rely on total area nor on individual element size to estimate numerosity.
Verbal numerical comparisons Familiarization. The experimenter introduced the child to a stuffed toy animal who loved to eat fish and was very hungry. The experimenter then placed two open tin boxes – one uniformly painted in bright orange and the other in purple – on the table between them. One box contained one small plastic fish and the other contained two. The experimenter told the child that the stuffed animal could only eat the fish in one of the two boxes. The animal looked into each open box in turn saying, ‘This box has two fish. This box has one fish. I want this box (referring to the box with two fish) because it has more fish. Two fish is more than one fish!’ After the animal had indicated its preference, it ate the fish in the selected box. 2
The stimuli used here were the same as were used in Le Corre & Carey (2007). See their paper for exact dot sizes.
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Next, children were presented with three ‘one’ versus ‘two’ comparisons and two ‘two’ versus ‘four’ comparisons, with the boxes left open. These comparisons were presented in alternation, with ‘one’ versus ‘two’ always presented first. On each trial, the experimenter pointed to each box and prompted children to look into the boxes as she described their contents (‘Look! The orange box has one fish in it. The purple box has two fish in it’). On the last familiarization trial, the experimenter said how many fish were in each box, but then put lids on the boxes. After the boxes were closed, children were reminded of how many fish were in each box and were asked to find the box that the animal wanted.
Test. At test, the boxes were already closed when they experimenter placed them on the table – children only saw the contents of the boxes after they chose which box contained more objects. Therefore, children had to base their choices on their understanding of the meaning of the number words that were paired with the boxes. The test comparisons were ‘one’ versus ‘eight’, ‘two’ versus ‘three’, ‘six’ versus ‘ten’, and ‘eight’ versus ‘ten’. The experimenter pointed to each box and named its colour as she described its contents (e.g., ‘The orange box has 10 fish in it. The purple box has eight fish in it’). Then, she asked ‘Which box has more fish?’ and/or ‘Which box does Mr. Bear want?’ After the child made a choice, she or he was given feedback, and the box with the largest number of fish so she or he could see the contents. Each comparison was presented three times for a total of 12 test trials. The comparisons were presented in one of two orders. Throughout the task, the boxes were always placed in the same position, with the orange one on the left of the child and the purple one on the right. The largest set was never in the same box on more than two consecutive trials. For each comparison type, the largest numerosity was in one box twice and in the other box once. Moreover, one of the numerosities in each pair was introduced first twice and was introduced last once (and vice versa for the other numerosity in the pair). Over all trials, the larger numerosity was in the orange box on half of the trials and was introduced first on half of the trials.
Memory checks. On every third test trial (for a total of four trials), the experimenter checked whether the child remembered how many objects were in each box prior to asking him/her to choose the largest set. We only included children who showed evidence that they could remember the number of objects in each box long enough to answer the questions. To show such evidence, children had to do one of two things: (1) answer all four memory questions correctly or (2) answer correctly on at least five of the six comparisons including one number word between ‘one’ and ‘four’. Twenty-one children answered all four memory questions correctly. Nine others made either one (n = 8) or two (n = 1) errors on the memory questions, but were included because they answered correctly on at least five of the six comparisons including one small number word.3 Only one of the children who participated did not meet either of the criteria.
3 All of the analyses of the results of the verbal numerical comparisons task were also performed without the children who made one or more memory errors. There were no differences in the results, but for one exception: the interaction between mapping and order was significant, F(1, 17) = 5.9, p < .05.
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Results Screening analyses: Verbal counting, give a number, and verbal numerical estimation Verbal counting: Screening for knowledge of the counting sequence up to ‘ten’ Children were considered to be able to recite the count list to ‘ten’ if they recited the number words in the correct order up to ‘ten’ as they tried to count the objects in the verbal counting task, with or without assistance from the experimenter. One-to-one correspondence errors (e.g., skipping or double-counting items) were not taken into account in this analysis. All children who were included in the study could recite the count list to ‘ten’ in the correct order.
Give a number: Screening for knowledge of the CP As in many previous studies of children’s performance on the give-a-number task (Ansari et al., 2003; Barner, Chow, & Yang, 2009; Barner, Libenson, Cheung, & Takasaki, 2009; Halberda, Taing & Lidz, 2008; Le Corre et al., 2006; Sarnecka & Carey, 2008; Sarnecka & Gelman, 2004; Wynn, 1990, 1992), children were categorized as CP-knowers if they met the following three criteria: 1. Correctly gave n objects when asked for ‘n’ (e.g., gave 3 when asked for ‘three’) at least twice of a maximum of three trials.4 2. Gave n objects when asked for numbers other than ‘n’ no more than half as often as they did when they were asked for ‘n’ (e.g., if they always correctly gave three objects when asked for ‘three’, they gave three objects on less than half of the trials where they were asked for other numbers). 3. Met criteria 1 and 2 for all number words up to ‘six’.
Verbal numerical estimation: Screening for mappings of number words between ‘six’ and ‘ten’ onto non-verbal representations of numerosity Minimally, if children use mappings to non-verbal representations of numerosity to estimate the numerosity of sets of 6–10, then, on average, they should produce larger number words for larger numerosities in that range. Numerically, this is captured by the slope of the best linear fit of estimates of the numerosity of sets of 6–10 – or ‘6–10 slope’ for short. If children produce larger number words for larger numerosities, then the 6–10 slope is >0. If they do not produce larger number words for larger numerosities, then the 6–10 slope is near 0 or .07. Finally, the two groups did not differ in age, t(28) = 1.0, p = ns. In theory, to produce larger number words for sets of 10 than for sets of 6 or 8, minimal mappers need not have used number words between ‘six’ and ‘ten’. They could have used any number words they wished as long as they used larger ones for larger numerosities. Therefore, to determine whether minimal mappers had genuinely mapped at least some of the number words between ‘six’ and ‘ten’, we asked (1) whether they mostly produced number words between ‘six’ and ‘ten’ as estimates of 6–10; and (2) whether their estimates were sensitive to numerosity (e.g., whether they used ‘six’ more often for sets of 6 than for sets of 8 or 10). We also used this analysis to confirm that the non-mappers had not mapped any number words beyond ‘four’. To answer these questions, we determined the relative frequency of use of each number word between ‘six’ and ‘ten’ for each large numerosity. That is, for each participant, we calculated the total number of numerical estimates produced for each large numerosity (max = 4) and then determined what percentage of these estimates were ‘six’, ‘seven’, etc. up to ‘ten’. For example, a relative frequency of 50% for the number word ‘six’ for numerosity 6 means that the number word ‘six’ represented 50% of the estimates of sets of six objects. 12 11 10 9 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 10 Numerosity
Figure 1. Verbal numerical estimation functions for non-mappers (dashed line) and minimal mappers (solid line).
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To test for evidence of mappings, separate one-way repeated measures ANOVAs with set size (6, 8, 10) as the repeated measure and relative frequency as the dependent variable were performed for each number word between ‘six’ and ‘ten’. For non-mappers, none of the ANOVAs revealed significant effects of set size (all Fs < 1.87, all ps > .18), and there were no significant linear (all Fs < 2.18, all ps > .16) or quadratic trends (all Fs < 1.68, all ps > .22). In sharp contrast, minimal mappers’ relative frequency score for ‘ten’ showed a significant effect of numerosity, F(2, 30) = 5.66, p < .01, and a significant linear trend, F (1, 15) = 8, p < .05. Figure 2 suggests that these effects were due to the fact that ‘ten’ was used more often for 8 and 10 than for 6. Moreover, minimal mappers’ relative frequency score for ‘six’ showed a marginally significant linear trend, F(1, 15) = 4.36, p = .054, and that for ‘seven’ showed a marginally significant quadratic trend, F(1, 15) = 3.81, p = .07. Figure 2 suggests that these marginal effects were due to the fact that ‘six’ was used most often on sets of 6 and least often on 10 and that ‘seven’ was used more often on sets of 8 than on sets of 6 or 10. ‘Eight’ and ‘nine’ were not used any more than 5% of the time on any numerosity. These analyses confirm that non-mappers had not mapped any number words beyond ‘four’ onto non-verbal numerical representations. Most importantly, they provide evidence that minimal mappers estimated the numerosities of sets of 6–10 by relying on mappings for number words between ‘six’ and ‘ten’. We now turn to asking whether the availability of mappings for these number words correlated with CP-knowers’ performance on verbal numerical comparisons involving them – that is, ‘six’ versus ‘ten’ and ‘eight’ versus ‘ten’.
Verbal numerical comparisons Performance on the verbal numerical comparisons task was measured as average accuracy (in percentage correct) of children’s judgements on each comparison (‘one’ vs. ‘eight’,
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‘two’ vs. ‘three’, ‘six’ vs. ‘ten’, and ‘eight’ vs. ‘ten’). Figure 3 reports average accuracy on each comparison for non-mappers and for minimal mappers. As expected, one-sample t-tests revealed that both groups of CP-knowers performed above chance (50% correct) on the ‘one’ versus ‘eight’ and on the ‘two’ versus ‘three’ comparisons, all ts > 3.2, all ps < .001. As for comparisons of ‘six’ versus ‘ten’ and ‘eight’ versus ‘ten’, non-mappers failed to carry out these comparisons, both ts(13) < 0.9, p = ns. More specifically, only two of 14 non-mappers answered correctly on at least five of the six comparisons of large number words (p = .47). In sharp contrast, minimal mappers performed above chance on both comparisons, both ts(15) > 4.9, both ps < .001. Twelve of 16 answered correctly on at least five of the six comparisons of large number words (p < .0001). More generally, a 2 9 2 9 4 repeated measures ANOVA with mapping (non-mapper, minimal mapper) and order (Order 1 and Order 2) as between-subjects factors and comparison (‘one’ vs. ‘eight’, ‘two’ vs. ‘three’, ‘six’ vs. ‘ten’, and ‘eight’ vs. ‘ten’) as a within-subjects factor revealed main effects of mapping, F(1, 26) = 20.3, p < .001, and of comparison, F(3, 78) = 12.1, p < .001, as well as a marginally significant interaction between mapping and comparison, F(3, 78) = 2.7, p = .05. None of the effects involving order were significant, all ps > .1. Bonferroni-corrected independent-samples t-tests5 (a = .0125) showed that minimal mappers were not significantly more accurate than non-mappers on comparisons of ‘one’ versus ‘eight’, t(13) = 1.0, p = ns, and of ‘two’ versus ‘three’, t(20) = 1.9, p = .075. However, minimal mappers were significantly more accurate than non-mappers on ‘six’ versus ‘ten’, t(28) = 3.3, p < .001, and on ‘eight’ versus ‘ten’, t(28) = 2.8, p < .01. To determine whether the results were an artefact of the criterion for dividing children into non-mappers and minimal mappers, we asked whether performance on the verbal numerical estimation task correlated with performance on the verbal numerical comparisons task. We plotted the sum of each CP-knowers’ accuracy score on verbal numerical comparisons of ‘six’ versus ‘ten’ and ‘eight’ versus ‘ten’ (max score = 6) as a
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Figure 3. Average accuracy on verbal numerical comparisons for non-mappers (left) and minimal mappers (right). 5 By Levene’s test of equality of variances, the variance of the accuracy scores of non-mappers and minimal mappers were not equal on ‘one’ versus ‘eight’ and on ‘two’ versus ‘three’. Therefore, for these comparisons, we report the results of t-tests that do not assume equal variances.
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function of their mean 6–10 slope for the verbal numerical estimation task (see Figure 4). Children’s summed accuracy score for the ‘six’ versus ‘ten’ and ‘eight’ versus ‘ten’ comparisons was significantly correlated with their 6–10 slope, even when age was partialled out, r = .49, p < .01. By Cohen’s (1988) standards, this is a large correlation. This suggests that the relationship between the two tasks was not an artefact of our criterion for dividing children into non-mappers and minimal mappers. In sum, the above analyses suggest that, while non-mappers and minimal mappers performed similarly on verbal numerical comparisons containing at least one number word between ‘one’ and ‘three’, their performance differed qualitatively on comparisons of number words between ‘six’ and ‘ten’ – that is, whereas the majority of minimal mappers succeeded on these comparisons, nearly all non-mappers failed. This, in turn, suggests that, although they all were CP-knowers, children could not use the later-greater principle to solve verbal numerical comparisons. Rather, they had to rely on mappings to non-verbal representations of numerosity. Alternatively, it could be that all children knew the later-greater principle, but that some failed to use it to solve the comparisons of large number words because these made heavier processing demands than the comparisons of small number words. To use the later-greater principle, children must use some strategy that involves counting to determine the relative positions of number words. Arguably, this makes heavier demands on working memory processes whenever the child must count more – that is, when the number words are further away from ‘one’, or when they are further apart from each other. By necessity, the comparisons of large number words involved number words that were further away from ‘one’ than the comparisons of small number words. Moreover, the pairs of large number words were further apart than the one pair of small number words
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that did not involve ‘one’ (i.e., ‘two’ vs. ‘three’ compared to ‘six’ vs. ‘ten’ and ‘eight’ vs. ‘ten’). Therefore, it could be that although all children knew the later-greater principle and attempted to use it to solve verbal numerical comparisons, only those with better ability to maintain and manipulate information in working memory were able to use it to solve the large number word comparisons. To test this alternative, we asked whether children’s performance on comparisons of large number words was correlated with their counting rate. Counting rate correlates highly with scores on standardized measures of the ability to manipulate and maintain information in working memory (e.g., counting span) in children (Case, Kurland, & Goldberg, 1982) and adults (Tuholski, Engle, & Baylis, 2001). Therefore, if children’s performance on comparisons of large number words was a function of their ability to manipulate and maintain information in working memory, it should correlate with their counting rate. Counting rates were measured from video records of the counting task. Specifically, we determined the time that elapsed between the beginning of children’s utterance of the word ‘one’ and the end of their utterance of the word ‘ten’ when they counted a set of 10 objects. Children were included only if they counted the entire set of objects on their own, without making any counting errors. Data were available for 15 children. The average and standard deviation of the large number word comparison scores for children included in this analysis (M = 71%, SD = 25.7) were nearly identical to those for the entire sample of children (M = 69%, SD = 25.9). This suggests that the group of children included in the analysis was representative of the sample as a whole. Yet, accuracy on comparisons of large number words did not correlate with counting rate, r = .02, p = .95.6 This suggests that it is not the case that the difference between children who could solve the large number comparisons and those who could not was that the former had a greater ability to manipulate and maintain information in working memory.
Discussion The present study asked whether the acquisition of the CP involves learning how the order of the number words corresponds to relative size relations between the numerosities they denote. In other words, we asked whether the acquisition of the CP involves learning the later-greater principle. To answer this question, we tested whether children who had recently learned the CP could carry out verbal numerical comparisons of number words, even if they had not mapped them onto non-verbal representations of numerosity. We found a nearly perfect correspondence between the availability of mappings and the ability to compare the relative size of numerosities denoted by number words. That is, just like children who have not yet learned the CP (Condry & Spelke, 2008), non-mappers could compare pairs of number words only if one of them was between ‘one’ and ‘four’. Non-mappers failed to compare pairs of number words between ‘six’ and ‘ten’. Minimal mappers were the only ones who succeeded on these pairs. Several explanations of these results can be ruled out. First, all CP-knowers could recite the count list at least up ‘ten’. Therefore, it cannot be that non-mappers failed to estimate sets of 6–10 because they did not know enough number words to do so. Similarly, it
6 There was a wide range of counting rates – from 4.5 to 10 s (M = 7.3, SD = 1.6) – and a wide range of scores on comparisons of large number words – four of the children had scores below 40%, seven had scores between 67% and 83%, and four had perfect scores. Therefore, the lack of a correlation was not due to a lack of variability in either of the variables.
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cannot be that they could not compare number words between ‘six’ and ‘ten’ because they could not count up to them. Second, non-mappers performed significantly above chance when at least one of the number words in the comparisons was between ‘one’ and ‘four’. Therefore, it cannot be that they failed on comparisons of number words beyond ‘four’ because they did not understand the task. Third, we obtained the same pattern of results if we only included the children who answered all of the memory probes correctly. Therefore, there is no evidence that non-mappers’ failure to compare pairs of number words between ‘six’ and ‘ten’ was due to failure to remember the number words in the comparisons. Fourth, Le Corre and Carey (2007) showed that minimal mappers7 and non-mappers are equally good at perceptual comparisons of sets of 6–10 objects. This suggests that non-mappers’ failure to estimate the numerosity of sets of more than four objects was genuinely due to a lack of mappings, and not to a limitation of the precision of their non-verbal representations of numerosity, or of their capacity to use such representations. Finally, performance on comparisons of number words beyond ‘four’ did not correlate with counting rate – a measure that has been shown to correlate highly with standardized measures of the ability to manipulate and maintain information in working memory in children (Case et al., 1982) and adults (Tuholski et al., 2001). Thus, it is unlikely that all CP-knowers knew the later-greater principle, but that only those with a greater ability to manipulate and maintain information in working memory could use it to compare the pairs of large number words. We suggest that the best explanation of our results is that the acquisition of the CP does not involve learning the later-greater principle. Rather, the later-greater principle is something that children learn separately, after they have acquired the CP. Thus, we suggest that CP-knowers’ ability to compare the relative size of the numerosities denoted by number words strongly correlated with the availability of mappings to non-verbal numerical representations because they had to draw on such representations to carry out the comparisons. Incidentally, we also suggest that Davidson et al. (2012) found what seemed like evidence that CP-knowers can compare number words beyond ‘four’ without mappings for these number words because their assessment of mappings for number words beyond ‘four’ underestimated the availability of these mappings. Our results converge with evidence obtained by Fuson and Hall (1983) with a different set of tasks. Fuson and Hall found that children aged between 4.5 and 6.5 are better at comparing the numerosities denoted by number words between ‘one’ and ‘ten’ than at determining their relative positions in the count list. They tentatively took this to suggest that young children do not use the later-greater principle to compare number words between ‘one’ and ‘ten’ but rather rely on what they called ‘a magnitude process’ (p. 98; non-verbal representations of numerosity are sometimes known as representations of magnitude, e.g., Feigenson, Dehaene, & Spelke, 2004; Moyer & Landauer, 1967). More generally, the present results also converge with other evidence that the acquisition of the CP does not involve the discovery of the numerical significance of the order of the number words. Davidson et al. (2012) asked whether all CP-knowers know that ‘immediately follows’ means ‘add 1’. They found that many did not know the successor relations between any number words beyond ‘four’. Others knew successor relations between low number words in their list (e.g., between ‘five’ and ‘six’), but did not know the successor relations between higher number words in their list (e.g., between ‘fifteen’ and ‘sixteen’). Thus, the latter most probably knew a
7
Le Corre and Carey referred to minimal mappers as ‘CP mappers’.
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few isolated addition facts (e.g., ‘five + one = six’) rather than a general principle linking the order of number words to the successor relation (but see Sarnecka & Carey, 2008). In sum, taken together with studies of young CP-knowers’ knowledge of the correspondence between order and the successor relation, our results suggest that the acquisition of the CP does not involve or require the acquisition of any knowledge of the numerical significance of the order of the number words in the list. Rather, we suggest that knowledge of the CP is exactly as Gelman and Gallistel (1978) first described it: that is, knowledge that the last number word of a count denotes the numerosity (or cardinality) of the counted set when the count is correct. Taken literally, this knowledge states conditions for determining what numerosity is denoted by what number word. It does not include any knowledge of correspondences between the order of the number words and relations between the numerosities they denote. This suggests that proposals that children acquire the CP by noticing a correspondence between the meaning of number words and their order in the count list (e.g., Carey, 2004, 2009; Piantadosi et al., 2012; Spelke & Tsivkin, 2001) are unlikely to be right. Instead, we propose that children acquire the CP by noticing that the last number word of a count is the very number word that is used to denote the numerosity of the counted set only when the count is correct (see Dehaene, 1997, for a similar proposal). This learning process leads to the acquisition of the CP without requiring learners to notice any correspondences between the order of number words in the list and relations between numerosities.
Acknowledgements This paper owes much to multiple discussions with David Barner, Susan Carey, Pierina Cheung, Ori Friedman, Justin Halberda, and Katherine White. Thanks to Pierina Cheung, Caryn Harris, Mindy Hsu, Bridjet Lee, Paul Muentener, and Dora Thalwitz for their help with the collection and analysis of data. Finally, we thank the many parents, children, and day care personnel who generously contributed to this project. This research was supported by NIH Grant no. RO1-HD038338 and NSF ROLE Grant no. REC-0196471 to Susan Carey and by an NSERC Discovery grant to Mathieu Le Corre.
References Ansari, D., Donlan, C., Thomas, M. S., Ewing, S. A., Peen, T., & Karmiloff-Smith, A. (2003). What makes counting count? Verbal and visuo-spatial contributions to typical and atypical number development. Journal of Experimental Child Psychology, 85 (1), 50–62. Barner, D., Chow, K., & Yang, S.-J. (2009). Finding one’s meaning: A test of the relation between quantifiers and integers in language development. Cognitive Psychology, 58, 195–219. Barner, D., Libenson, A., Cheung, P., & Takasaki, M. (2009). Cross-linguistic relations between quantifiers and numerals in language acquisition: Evidence from Japanese. Journal of Experimental Child Psychology, 103, 421–440. Brannon, E. M. (2002). The development of ordinal numerical knowledge in infancy. Cognition, 83, 223–240. Carey, S. (2004). Bootstrapping and the origin of concepts. Daedalus, Winter, 59–68. Carey, S. (2009). The origin of concepts. New York, NY: Oxford. Case, R., Kurland, D. M., & Goldberg, J. (1982). Operational efficiency and the growth of short-term memory span. Journal of Experimental Child Psychology, 33, 386–404. Cohen, J. (1988). Statistical power analysis for the behavioral sciences. Hillsdale, NJ: L. Erlbaum.
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Condry, K. F., & Spelke, E. S. (2008). The development of language and abstract concepts: The case of natural number. Journal of Experimental Psychology: General, 137 (1), 22. Davidson, K., Eng, K., & Barner, D. (2012). Does learning to count involve a semantic induction? Cognition, 123 (1), 162–173. Dehaene, S. (1997). The number sense: How the mind creates mathematics. New York, NY: Oxford University Press. Feigenson, L. (2005). A double-dissociation in infants’ representations of object arrays. Cognition, 95 (3), B37–B48. Feigenson, L., Dehaene, S., & Spelke, E. (2004). Core systems of number. Trends in Cognitive Sciences, 8, 307–314. Fuson, K. C., & Hall, J. W. (1983). The acquisition of early number word meanings: A conceptual analysis and review. In H. Ginsburg (Ed.), The development of children’s mathematical thinking (pp. 49–107). New York, NY: Academic Press. Gelman, R., & Gallistel, C. R. (1978). The child’s understanding of number. Cambridge, MA: Harvard University Press. Halberda, J., Taing, L., & Lidz, J. (2008). The development of most comprehension and its potential dependence on counting ability in preschoolers. Language Learning and Development, 4 (2), 99–121. Le Corre, M., & Carey, S. (2007). One, two, three, four, nothing more: An investigation of the conceptual sources of the verbal counting principles. Cognition, 105, 395–438. Le Corre, M., Van de Walle, G., Brannon, E. M., & Carey, S. (2006). Re-visiting the competence/ performance debate in the acquisition of the counting principles. Cognitive Psychology, 52, 130–169. Lipton, J. S., & Spelke, E. S. (2003). Origins of number sense large-number discrimination in human infants. Psychological Science, 14, 396–401. Moyer, R. S., & Landauer, T. K. (1967). Time required for judgments of numerical inequality. Nature, 215, 1519–1520. Piantadosi, S. T., Tenenbaum, J. B., & Goodman, N. D. (2012). Bootstrapping in a language of thought: A formal model of numerical concept learning. Cognition, 123, 199–217. Sarnecka, B. W., & Carey, S. (2008). How counting represents number: What children must learn and when they learn it. Cognition, 108, 662–674. Sarnecka, B. W., & Gelman, S. A. (2004). "Six" does not just mean "a lot": Preschoolers see number words as specific. Cognition, 92 (3), 329–352. Schaeffer, B., Eggleston, V. H., & Scott, J. L. (1974). Number development in young children. Cognitive Psychology, 6, 357–379. Spelke, E. S., & Tsivkin, S. (2001). Initial knowledge and conceptual change: Space and number. In S. Bowerman & S. Levinson (Eds.), Language acquisition and conceptual development (pp. 70– 97). Cambridge, UK: Cambridge University Press. Suanda, S., Tompson, W., & Brannon, E. (2008). Changes in the ability to detect ordinal numerical relationships between 9 and 11 months of age. Infancy, 13, 308. Tuholski, S. W., Engle, R. W., & Baylis, G. C. (2001). Individual differences in working memory capacity and enumeration. Memory & Cognition, 29, 484–492. Wagner, J. B., & Johnson, S. C. (2011). An association between understanding cardinality and analog magnitude representations in preschoolers. Cognition, 119 (1), 10–22. Wynn, K. (1990). Children’s understanding of counting. Cognition, 36, 155–193. Wynn, K. (1992). Children’s acquisition of the number words and the counting system. Cognitive Psychology, 24, 220–251. Xu, F., & Spelke, E. S. (2000). Large number discrimination in 6-month-old infants. Cognition, 74 (1), B1–B11. Received 2 November 2013; revised version received 20 November 2013
British Journal of Developmental Psychology (2014), 32, 163–177 © 2013 The British Psychological Society www.wileyonlinelibrary.com
Children acquire the later-greater principle after the cardinal principle Mathieu Le Corre* Centro de Investigaci on Transdisciplinar en Psicologıa, Universidad Autonoma del Estado de Morelos, Cuernavaca, Mexico Many have proposed that the acquisition of the cardinal principle (CP) is a result of the discovery of the numerical significance of the order of the number words in the count list. However, this need not be the case. Indeed, the CP does not state anything about the numerical significance of the order of the number words. It only states that the last word of a correct count denotes the numerosity of the counted set. Here, we test whether the acquisition of the CP involves the discovery of the later-greater principle – that is, that the order of the number words corresponds to the relative size of the numerosities they denote. Specifically, we tested knowledge of verbal numerical comparisons (e.g., Is ‘ten’ more than ‘six’?) in children who had recently learned the CP. We find that these children can compare number words between ‘six’ and ‘ten’ only if they have mapped them onto non-verbal representations of numerosity. We suggest that this means that the acquisition of the CP does not involve the discovery of the correspondence between the order of the number words and the relative size of the numerosities they denote.
By age 2, typical children can recite part of their language’s count list, but, to them, counting is nothing but a meaningless list of words. Then, about 1–2 years later, their understanding of verbal counting changes radically. They acquire the cardinal principle (CP): they learn that when a count has been recited in the correct order and has correctly been placed in one-to-one correspondence with elements of a set, the last word denotes the numerosity of the set (Le Corre, Van de Walle, Brannon, & Carey, 2006; Sarnecka & Carey, 2008; Schaeffer, Eggleston, & Scott, 1974; Wynn, 1990, 1992). What changes in children’s understanding of verbal counting when they acquire the CP? Many have proposed that what changes is their understanding of the meaning of the order of the number words in the count list. Initially, the order is meaningless, like the order of the letters of the alphabet. Then, children suddenly grasp at least some of the numerical implications of the order of the number words. This is, what allows them to learn the CP. Some have proposed that they learn that the order of the number words corresponds to the numerical relation successor – that is, that ‘immediately follows’ corresponds to ‘add 1’ (Carey, 2004, 2009; Piantadosi, Tenenbaum, & Goodman, 2012; Sarnecka & Carey, 2008). Others proposed that they learn how order corresponds to relative numerical size (Spelke & Tsivkin, 2001) – that is, that number words that occur
*Correspondence should be addressed to Mathieu Le Corre, Asia 2 #101, Colonia Barrio La Concepcion, Coyoacan, Mexico, D.F., Mexico (email: [email protected]). DOI:10.1111/bjdp.12029
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later in the count list denote larger numerosities. The latter correspondence is sometimes known as the ‘later-greater principle’. Alternatively, it may be that children acquire the CP without learning anything about the numerical significance of the order of the number words. Indeed, knowledge of the CP as Gelman and Gallistel (1978) first described it does not involve any representation of correspondences between the order of number words and relations between the numerosities they denote. On Gelman and Gallistel’s original proposal, children understand how counting represents numerosity when they learn that the last number word of a count denotes the numerosity (or cardinality) of a counted set if and only if the count is recited in the correct order and matches the set on one-to-one correspondence. Taken literally, this knowledge states conditions for determining what numerosity is denoted by what number word. It does not include any knowledge of correspondences between the order of the number words and relations between the numerosities they denote. Therefore, the acquisition of the CP need not involve the acquisition of knowledge of correspondences between the order of number words and relations between the numerosities they denote. This study investigates whether the acquisition of the CP involves noticing the correspondence between the order of the number words and the relative size of the numerosities they denote. In other words, we ask whether the acquisition of the CP involves the acquisition of the later-greater principle. If it does, then all children who know the cardinal principle (henceforth, ‘CP-knowers’) should be able to infer relative size relations between any pair of number words in their count list. However, on its own, such evidence is not sufficient to conclude that CP-knowers know the later-greater principle. Indeed, CP-knowers could also infer these relations from other knowledge. First, prior to becoming CP-knowers, children learn the meanings of ‘one’ to ‘four’ by mapping them onto non-verbal representations of numerosity (e.g., Le Corre & Carey, 2007; Wynn, 1990, 1992). They also take unmapped number words to denote larger numerosities than mapped number words. For example, children who have learned ‘one’ to ‘three’ (but are not CP-knowers) know that all number words beyond ‘three’ denote larger numerosities than ‘one’ to ‘three’ (Condry & Spelke, 2008). The exact source of this knowledge is still unknown. However, it is clear that it does not come from knowledge of how the order of the number words encodes relative numerical size. If children have such knowledge, they should be able to compare any two number words in their list. Yet, although children can recite the count list beyond ‘four’ prior to becoming CP-knowers, they cannot compare pairs of number words beyond ‘four’ (Condry & Spelke, 2008). Second, CP-knowers could infer relative size relations between number words beyond ‘four’ from mappings between these number words and non-verbal representations of numerosity. Multiple studies have shown that, starting in infancy, humans have access to non-verbal representations of numerosity (Feigenson, 2005; Lipton & Spelke, 2003; Xu & Spelke, 2000) and that these representations support computations of relative numerical size (Brannon, 2002; Suanda, Tompson, & Brannon, 2008). Mappings between number words and these representations could allow children to infer relative numerical size relations between the numerosities denoted by number words without using their relative positions in the count list. There is evidence that many children map number words between ‘four’ and ‘ten’ onto these non-verbal representations a few months after they become CP-knowers (Le Corre & Carey, 2007) and that some do so even earlier (Wagner & Johnson, 2011). Therefore, CP-knowers could carry out verbal numerical comparisons by relying on such mappings rather than on the later-greater principle.
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In sum, prior to becoming CP-knowers, children can compare the numerosities denoted by two number words between ‘one’ to ‘four’ or by a number word between ‘one’ and ‘four’ and a number word beyond ‘four’ without knowing the later-greater principle. Thus, it goes without saying that CP-knowers’ performance on such comparisons cannot be used to test whether they know the later-greater principle. Moreover, evidence that CP-knowers know some numerical relations between number words beyond ‘four’ is not sufficient to show that they have learned the later-greater principle. Rather, it is also necessary to show that they have not mapped these number words onto non-verbal representations of numerosity. Hence, to test whether the acquisition of the CP involves learning the later-greater principle, the present study will test whether CP-knowers can compare the numerosities denoted by number words beyond ‘four’ even if they have not mapped these number words onto non-verbal representations of numerosity. One study attempted to determine whether CP-knowers can compare the numerosities denoted by number words beyond ‘four’ even if they have not mapped them onto non-verbal representations of numerosity (Davidson, Eng, & Barner, 2012). In this study, mappings between number words and non-verbal representations of numerosity were assessed with a verbal estimation task where children had to estimate the numerosity of sets of 5–51 objects without counting. Davidson et al. found that CP-knowers performed above chance on comparisons of number words beyond ‘four’, even if they could not verbally estimate the numerosity of sets of 5–51. This could be taken to suggest that learning the later-greater principle is involved in learning the CP. However, there are some reasons to doubt this conclusion. First, as Davidson et al. point out, the CP-knowers who could not verbally estimate the numerosity of sets of 5–51 performed quite poorly on the verbal comparisons task – on average, they answered correctly about 65% of the time. Second, it also could be that some of the CP-knowers who failed to estimate the numerosity of sets of 5–51 actually had formed mappings for at least some number words beyond ‘four’. Indeed, about half of the CP-knowers in their sample could not count further than ‘twenty-nine’. Yet, the verbal estimation task included sets of up to 51 objects. Thus, the CP-knowers who could not count further than ‘twenty-nine’ may have failed to estimate the numerosity of sets of 5–51 because the task required many more number words than they knew, not because they did not have any mappings for number words beyond ‘four’. Therefore, Davidson et al.’s study does not provide conclusive evidence that CP-knowers can carry out verbal numerical comparisons without accessing non-verbal representations of numerosity. The current study presents a stronger test of CP-knowers’ knowledge of the later-greater principle. As Davidson et al., we asked whether CP-knowers can compare the numerosities denoted by number words beyond ‘four’ even if they have not mapped them onto non-verbal representations of numerosity. However, unlike Davidson et al., we focused exclusively on number words beyond ‘four’ that are well within the counting range of typical young CP-knowers – that is, ‘six’ to ‘ten’. That is, we asked whether CP-knowers can compare ‘six’ and ‘ten’ and ‘eight’ and ‘ten’, even if they cannot estimate the numerosity of sets of 6–10 objects. We were thus less likely to underestimate children’s knowledge of the mappings of the number words we targeted. Therefore, evidence that CP-knowers can compare number words between ‘six’ and ‘ten’ even if they cannot estimate the numerosity of sets of 6–10 would strongly suggest that the change in children’s understanding of counting involves learning the later-greater principle. Evidence to the contrary would suggest that learning the CP does not involve learning the later-greater principle.
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Method Participants Thirty 3- and 4-year-olds participated in this study (Mage = 3 years, 10 months, range: 3 years, 2 months to 4 years, 6 months).1 All were fluent English speakers recruited in the Greater Boston area. Participants were initially recruited by letter and phone calls through commercially available lists or by letters sent home by the day care centres. They were tested at a university child development laboratory or at local day care centres. A caregiver accompanied all children tested at the laboratory. Children received a small gift for their participation. Caregivers who came to the laboratory received reimbursement for their travel expenses; day cares were given children’s books. The majority of the children were from middle-class backgrounds, and most were Caucasian although a small number of Asian, African American, and Hispanic children participated. Another 22 children also participated, but are not reported here because they had not yet become CP-knowers (n = 21), or because they did not meet the criterion for inclusion in the verbal numerical comparisons task (n = 1).
Materials and procedures All children were tested on four tasks: a verbal counting task, give a number, Le Corre and Carey’s (2007) verbal numerical estimation task (Fast Cards), and a verbal numerical comparisons task. The tasks were administered over two sessions. In the first session, children were tested on fast cards, give a number, and the verbal counting task, always in that order, and they were tested on verbal numerical comparisons in the second session. The two sessions usually occurred within 10 days of each other and were never separated by more than 3 weeks.
Verbal counting Children were presented with a row of 10 small plastic toys. The toys were identical to each other. Children were asked to count the toys. If children’s first attempt at counting the toys was grossly wrong, the experimenter asked them to recount them and assisted their counting by pointing to each toy as they counted.
Give a number Children were provided with a container filled with 12–15 small plastic toys. Children were always asked for ‘one’ toy first. If they correctly gave one toy, the experimenter replaced it in the container and moved on to ask for ‘two’ toys. If children did not give the correct number, they were given a chance to count the objects they had given and to fix their answer. The experimenter kept asking for the next highest number as long as children could give the number of toys she had requested. If a child gave the requested number of toys up to ‘six’, the experimenter tested ‘five’ and ‘six’ once more each. If a child failed before reaching ‘six’, the experimenter asked for the next smallest number
1 The give a number and verbal numerical estimation (Fast Cards) data for 26 of these children were reported in Le Corre and Carey (2007). The verbal numerical comparisons data and their analysis in terms of the other two tasks are reported here for the first time for all children.
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and kept moving up or down in steps of one to establish the highest number the child could give correctly two out of three times.
Verbal numerical estimation This task assessed children’s knowledge of mappings between number words and non-verbal representations of numerosity. It was first modelled by an experimenter. The experimenter presented the child with cards with sets of circles printed on them and then told the child how many circles were on each card. Modelling included sets of 1–12 dots and a set of 15. The total surface area of the sets of circles (i.e., the sum of the individual areas of the circles comprising a set) used to model the task was negatively correlated with numerosity, so that circle size decreased as numerosity increased.2 The experimenter initiated the test phase by saying ‘Now it’s your turn!’ At the beginning of each test trial, the experimenter held the card with the set facing towards her and attracted the child’s attention by saying ‘Ready?’ Then, the experimenter said, ‘Go!’ and flipped the card over so the set faced the child. After approximately 1 s, the experimenter removed the set from the child’s view. If the child had not already guessed the numerosity, she or he was asked to do so. If the child still refused to produce an answer, the experimenter presented the set again and encouraged the child to make a guess. The set was left in sight until the child made a guess. If the child still did not answer, the experimenter told the child how many circles were on the card. Trials where cards were re-presented were only used to incite children to answer. They were not included in any of the analyses. The test numerosities were 1, 2, 3, 4, 6, 8, and 10. Each numerosity was presented four times, each time in a different configuration of circles. Sets for large numerosities (i.e., 6, 8, or 10) were configured so that they could not be easily broken into smaller perceptual groups (e.g., none of the sets of six consisted of two parallel rows of three circles). The sets were presented in four series. Each series included one trial of each numerosity. The order of presentation of the sets was randomized within each series. In two of the series, the summed area of the circles remained constant, so that the size of individual circles decreased as numerosity increased. In the other two series, summed area and individual circle area both decreased as numerosity increased. Therefore, children could not rely on total area nor on individual element size to estimate numerosity.
Verbal numerical comparisons Familiarization. The experimenter introduced the child to a stuffed toy animal who loved to eat fish and was very hungry. The experimenter then placed two open tin boxes – one uniformly painted in bright orange and the other in purple – on the table between them. One box contained one small plastic fish and the other contained two. The experimenter told the child that the stuffed animal could only eat the fish in one of the two boxes. The animal looked into each open box in turn saying, ‘This box has two fish. This box has one fish. I want this box (referring to the box with two fish) because it has more fish. Two fish is more than one fish!’ After the animal had indicated its preference, it ate the fish in the selected box. 2
The stimuli used here were the same as were used in Le Corre & Carey (2007). See their paper for exact dot sizes.
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Next, children were presented with three ‘one’ versus ‘two’ comparisons and two ‘two’ versus ‘four’ comparisons, with the boxes left open. These comparisons were presented in alternation, with ‘one’ versus ‘two’ always presented first. On each trial, the experimenter pointed to each box and prompted children to look into the boxes as she described their contents (‘Look! The orange box has one fish in it. The purple box has two fish in it’). On the last familiarization trial, the experimenter said how many fish were in each box, but then put lids on the boxes. After the boxes were closed, children were reminded of how many fish were in each box and were asked to find the box that the animal wanted.
Test. At test, the boxes were already closed when they experimenter placed them on the table – children only saw the contents of the boxes after they chose which box contained more objects. Therefore, children had to base their choices on their understanding of the meaning of the number words that were paired with the boxes. The test comparisons were ‘one’ versus ‘eight’, ‘two’ versus ‘three’, ‘six’ versus ‘ten’, and ‘eight’ versus ‘ten’. The experimenter pointed to each box and named its colour as she described its contents (e.g., ‘The orange box has 10 fish in it. The purple box has eight fish in it’). Then, she asked ‘Which box has more fish?’ and/or ‘Which box does Mr. Bear want?’ After the child made a choice, she or he was given feedback, and the box with the largest number of fish so she or he could see the contents. Each comparison was presented three times for a total of 12 test trials. The comparisons were presented in one of two orders. Throughout the task, the boxes were always placed in the same position, with the orange one on the left of the child and the purple one on the right. The largest set was never in the same box on more than two consecutive trials. For each comparison type, the largest numerosity was in one box twice and in the other box once. Moreover, one of the numerosities in each pair was introduced first twice and was introduced last once (and vice versa for the other numerosity in the pair). Over all trials, the larger numerosity was in the orange box on half of the trials and was introduced first on half of the trials.
Memory checks. On every third test trial (for a total of four trials), the experimenter checked whether the child remembered how many objects were in each box prior to asking him/her to choose the largest set. We only included children who showed evidence that they could remember the number of objects in each box long enough to answer the questions. To show such evidence, children had to do one of two things: (1) answer all four memory questions correctly or (2) answer correctly on at least five of the six comparisons including one number word between ‘one’ and ‘four’. Twenty-one children answered all four memory questions correctly. Nine others made either one (n = 8) or two (n = 1) errors on the memory questions, but were included because they answered correctly on at least five of the six comparisons including one small number word.3 Only one of the children who participated did not meet either of the criteria.
3 All of the analyses of the results of the verbal numerical comparisons task were also performed without the children who made one or more memory errors. There were no differences in the results, but for one exception: the interaction between mapping and order was significant, F(1, 17) = 5.9, p < .05.
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Results Screening analyses: Verbal counting, give a number, and verbal numerical estimation Verbal counting: Screening for knowledge of the counting sequence up to ‘ten’ Children were considered to be able to recite the count list to ‘ten’ if they recited the number words in the correct order up to ‘ten’ as they tried to count the objects in the verbal counting task, with or without assistance from the experimenter. One-to-one correspondence errors (e.g., skipping or double-counting items) were not taken into account in this analysis. All children who were included in the study could recite the count list to ‘ten’ in the correct order.
Give a number: Screening for knowledge of the CP As in many previous studies of children’s performance on the give-a-number task (Ansari et al., 2003; Barner, Chow, & Yang, 2009; Barner, Libenson, Cheung, & Takasaki, 2009; Halberda, Taing & Lidz, 2008; Le Corre et al., 2006; Sarnecka & Carey, 2008; Sarnecka & Gelman, 2004; Wynn, 1990, 1992), children were categorized as CP-knowers if they met the following three criteria: 1. Correctly gave n objects when asked for ‘n’ (e.g., gave 3 when asked for ‘three’) at least twice of a maximum of three trials.4 2. Gave n objects when asked for numbers other than ‘n’ no more than half as often as they did when they were asked for ‘n’ (e.g., if they always correctly gave three objects when asked for ‘three’, they gave three objects on less than half of the trials where they were asked for other numbers). 3. Met criteria 1 and 2 for all number words up to ‘six’.
Verbal numerical estimation: Screening for mappings of number words between ‘six’ and ‘ten’ onto non-verbal representations of numerosity Minimally, if children use mappings to non-verbal representations of numerosity to estimate the numerosity of sets of 6–10, then, on average, they should produce larger number words for larger numerosities in that range. Numerically, this is captured by the slope of the best linear fit of estimates of the numerosity of sets of 6–10 – or ‘6–10 slope’ for short. If children produce larger number words for larger numerosities, then the 6–10 slope is >0. If they do not produce larger number words for larger numerosities, then the 6–10 slope is near 0 or .07. Finally, the two groups did not differ in age, t(28) = 1.0, p = ns. In theory, to produce larger number words for sets of 10 than for sets of 6 or 8, minimal mappers need not have used number words between ‘six’ and ‘ten’. They could have used any number words they wished as long as they used larger ones for larger numerosities. Therefore, to determine whether minimal mappers had genuinely mapped at least some of the number words between ‘six’ and ‘ten’, we asked (1) whether they mostly produced number words between ‘six’ and ‘ten’ as estimates of 6–10; and (2) whether their estimates were sensitive to numerosity (e.g., whether they used ‘six’ more often for sets of 6 than for sets of 8 or 10). We also used this analysis to confirm that the non-mappers had not mapped any number words beyond ‘four’. To answer these questions, we determined the relative frequency of use of each number word between ‘six’ and ‘ten’ for each large numerosity. That is, for each participant, we calculated the total number of numerical estimates produced for each large numerosity (max = 4) and then determined what percentage of these estimates were ‘six’, ‘seven’, etc. up to ‘ten’. For example, a relative frequency of 50% for the number word ‘six’ for numerosity 6 means that the number word ‘six’ represented 50% of the estimates of sets of six objects. 12 11 10 9 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 10 Numerosity
Figure 1. Verbal numerical estimation functions for non-mappers (dashed line) and minimal mappers (solid line).
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To test for evidence of mappings, separate one-way repeated measures ANOVAs with set size (6, 8, 10) as the repeated measure and relative frequency as the dependent variable were performed for each number word between ‘six’ and ‘ten’. For non-mappers, none of the ANOVAs revealed significant effects of set size (all Fs < 1.87, all ps > .18), and there were no significant linear (all Fs < 2.18, all ps > .16) or quadratic trends (all Fs < 1.68, all ps > .22). In sharp contrast, minimal mappers’ relative frequency score for ‘ten’ showed a significant effect of numerosity, F(2, 30) = 5.66, p < .01, and a significant linear trend, F (1, 15) = 8, p < .05. Figure 2 suggests that these effects were due to the fact that ‘ten’ was used more often for 8 and 10 than for 6. Moreover, minimal mappers’ relative frequency score for ‘six’ showed a marginally significant linear trend, F(1, 15) = 4.36, p = .054, and that for ‘seven’ showed a marginally significant quadratic trend, F(1, 15) = 3.81, p = .07. Figure 2 suggests that these marginal effects were due to the fact that ‘six’ was used most often on sets of 6 and least often on 10 and that ‘seven’ was used more often on sets of 8 than on sets of 6 or 10. ‘Eight’ and ‘nine’ were not used any more than 5% of the time on any numerosity. These analyses confirm that non-mappers had not mapped any number words beyond ‘four’ onto non-verbal numerical representations. Most importantly, they provide evidence that minimal mappers estimated the numerosities of sets of 6–10 by relying on mappings for number words between ‘six’ and ‘ten’. We now turn to asking whether the availability of mappings for these number words correlated with CP-knowers’ performance on verbal numerical comparisons involving them – that is, ‘six’ versus ‘ten’ and ‘eight’ versus ‘ten’.
Verbal numerical comparisons Performance on the verbal numerical comparisons task was measured as average accuracy (in percentage correct) of children’s judgements on each comparison (‘one’ vs. ‘eight’,
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‘two’ vs. ‘three’, ‘six’ vs. ‘ten’, and ‘eight’ vs. ‘ten’). Figure 3 reports average accuracy on each comparison for non-mappers and for minimal mappers. As expected, one-sample t-tests revealed that both groups of CP-knowers performed above chance (50% correct) on the ‘one’ versus ‘eight’ and on the ‘two’ versus ‘three’ comparisons, all ts > 3.2, all ps < .001. As for comparisons of ‘six’ versus ‘ten’ and ‘eight’ versus ‘ten’, non-mappers failed to carry out these comparisons, both ts(13) < 0.9, p = ns. More specifically, only two of 14 non-mappers answered correctly on at least five of the six comparisons of large number words (p = .47). In sharp contrast, minimal mappers performed above chance on both comparisons, both ts(15) > 4.9, both ps < .001. Twelve of 16 answered correctly on at least five of the six comparisons of large number words (p < .0001). More generally, a 2 9 2 9 4 repeated measures ANOVA with mapping (non-mapper, minimal mapper) and order (Order 1 and Order 2) as between-subjects factors and comparison (‘one’ vs. ‘eight’, ‘two’ vs. ‘three’, ‘six’ vs. ‘ten’, and ‘eight’ vs. ‘ten’) as a within-subjects factor revealed main effects of mapping, F(1, 26) = 20.3, p < .001, and of comparison, F(3, 78) = 12.1, p < .001, as well as a marginally significant interaction between mapping and comparison, F(3, 78) = 2.7, p = .05. None of the effects involving order were significant, all ps > .1. Bonferroni-corrected independent-samples t-tests5 (a = .0125) showed that minimal mappers were not significantly more accurate than non-mappers on comparisons of ‘one’ versus ‘eight’, t(13) = 1.0, p = ns, and of ‘two’ versus ‘three’, t(20) = 1.9, p = .075. However, minimal mappers were significantly more accurate than non-mappers on ‘six’ versus ‘ten’, t(28) = 3.3, p < .001, and on ‘eight’ versus ‘ten’, t(28) = 2.8, p < .01. To determine whether the results were an artefact of the criterion for dividing children into non-mappers and minimal mappers, we asked whether performance on the verbal numerical estimation task correlated with performance on the verbal numerical comparisons task. We plotted the sum of each CP-knowers’ accuracy score on verbal numerical comparisons of ‘six’ versus ‘ten’ and ‘eight’ versus ‘ten’ (max score = 6) as a
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Figure 3. Average accuracy on verbal numerical comparisons for non-mappers (left) and minimal mappers (right). 5 By Levene’s test of equality of variances, the variance of the accuracy scores of non-mappers and minimal mappers were not equal on ‘one’ versus ‘eight’ and on ‘two’ versus ‘three’. Therefore, for these comparisons, we report the results of t-tests that do not assume equal variances.
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function of their mean 6–10 slope for the verbal numerical estimation task (see Figure 4). Children’s summed accuracy score for the ‘six’ versus ‘ten’ and ‘eight’ versus ‘ten’ comparisons was significantly correlated with their 6–10 slope, even when age was partialled out, r = .49, p < .01. By Cohen’s (1988) standards, this is a large correlation. This suggests that the relationship between the two tasks was not an artefact of our criterion for dividing children into non-mappers and minimal mappers. In sum, the above analyses suggest that, while non-mappers and minimal mappers performed similarly on verbal numerical comparisons containing at least one number word between ‘one’ and ‘three’, their performance differed qualitatively on comparisons of number words between ‘six’ and ‘ten’ – that is, whereas the majority of minimal mappers succeeded on these comparisons, nearly all non-mappers failed. This, in turn, suggests that, although they all were CP-knowers, children could not use the later-greater principle to solve verbal numerical comparisons. Rather, they had to rely on mappings to non-verbal representations of numerosity. Alternatively, it could be that all children knew the later-greater principle, but that some failed to use it to solve the comparisons of large number words because these made heavier processing demands than the comparisons of small number words. To use the later-greater principle, children must use some strategy that involves counting to determine the relative positions of number words. Arguably, this makes heavier demands on working memory processes whenever the child must count more – that is, when the number words are further away from ‘one’, or when they are further apart from each other. By necessity, the comparisons of large number words involved number words that were further away from ‘one’ than the comparisons of small number words. Moreover, the pairs of large number words were further apart than the one pair of small number words
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that did not involve ‘one’ (i.e., ‘two’ vs. ‘three’ compared to ‘six’ vs. ‘ten’ and ‘eight’ vs. ‘ten’). Therefore, it could be that although all children knew the later-greater principle and attempted to use it to solve verbal numerical comparisons, only those with better ability to maintain and manipulate information in working memory were able to use it to solve the large number word comparisons. To test this alternative, we asked whether children’s performance on comparisons of large number words was correlated with their counting rate. Counting rate correlates highly with scores on standardized measures of the ability to manipulate and maintain information in working memory (e.g., counting span) in children (Case, Kurland, & Goldberg, 1982) and adults (Tuholski, Engle, & Baylis, 2001). Therefore, if children’s performance on comparisons of large number words was a function of their ability to manipulate and maintain information in working memory, it should correlate with their counting rate. Counting rates were measured from video records of the counting task. Specifically, we determined the time that elapsed between the beginning of children’s utterance of the word ‘one’ and the end of their utterance of the word ‘ten’ when they counted a set of 10 objects. Children were included only if they counted the entire set of objects on their own, without making any counting errors. Data were available for 15 children. The average and standard deviation of the large number word comparison scores for children included in this analysis (M = 71%, SD = 25.7) were nearly identical to those for the entire sample of children (M = 69%, SD = 25.9). This suggests that the group of children included in the analysis was representative of the sample as a whole. Yet, accuracy on comparisons of large number words did not correlate with counting rate, r = .02, p = .95.6 This suggests that it is not the case that the difference between children who could solve the large number comparisons and those who could not was that the former had a greater ability to manipulate and maintain information in working memory.
Discussion The present study asked whether the acquisition of the CP involves learning how the order of the number words corresponds to relative size relations between the numerosities they denote. In other words, we asked whether the acquisition of the CP involves learning the later-greater principle. To answer this question, we tested whether children who had recently learned the CP could carry out verbal numerical comparisons of number words, even if they had not mapped them onto non-verbal representations of numerosity. We found a nearly perfect correspondence between the availability of mappings and the ability to compare the relative size of numerosities denoted by number words. That is, just like children who have not yet learned the CP (Condry & Spelke, 2008), non-mappers could compare pairs of number words only if one of them was between ‘one’ and ‘four’. Non-mappers failed to compare pairs of number words between ‘six’ and ‘ten’. Minimal mappers were the only ones who succeeded on these pairs. Several explanations of these results can be ruled out. First, all CP-knowers could recite the count list at least up ‘ten’. Therefore, it cannot be that non-mappers failed to estimate sets of 6–10 because they did not know enough number words to do so. Similarly, it
6 There was a wide range of counting rates – from 4.5 to 10 s (M = 7.3, SD = 1.6) – and a wide range of scores on comparisons of large number words – four of the children had scores below 40%, seven had scores between 67% and 83%, and four had perfect scores. Therefore, the lack of a correlation was not due to a lack of variability in either of the variables.
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cannot be that they could not compare number words between ‘six’ and ‘ten’ because they could not count up to them. Second, non-mappers performed significantly above chance when at least one of the number words in the comparisons was between ‘one’ and ‘four’. Therefore, it cannot be that they failed on comparisons of number words beyond ‘four’ because they did not understand the task. Third, we obtained the same pattern of results if we only included the children who answered all of the memory probes correctly. Therefore, there is no evidence that non-mappers’ failure to compare pairs of number words between ‘six’ and ‘ten’ was due to failure to remember the number words in the comparisons. Fourth, Le Corre and Carey (2007) showed that minimal mappers7 and non-mappers are equally good at perceptual comparisons of sets of 6–10 objects. This suggests that non-mappers’ failure to estimate the numerosity of sets of more than four objects was genuinely due to a lack of mappings, and not to a limitation of the precision of their non-verbal representations of numerosity, or of their capacity to use such representations. Finally, performance on comparisons of number words beyond ‘four’ did not correlate with counting rate – a measure that has been shown to correlate highly with standardized measures of the ability to manipulate and maintain information in working memory in children (Case et al., 1982) and adults (Tuholski et al., 2001). Thus, it is unlikely that all CP-knowers knew the later-greater principle, but that only those with a greater ability to manipulate and maintain information in working memory could use it to compare the pairs of large number words. We suggest that the best explanation of our results is that the acquisition of the CP does not involve learning the later-greater principle. Rather, the later-greater principle is something that children learn separately, after they have acquired the CP. Thus, we suggest that CP-knowers’ ability to compare the relative size of the numerosities denoted by number words strongly correlated with the availability of mappings to non-verbal numerical representations because they had to draw on such representations to carry out the comparisons. Incidentally, we also suggest that Davidson et al. (2012) found what seemed like evidence that CP-knowers can compare number words beyond ‘four’ without mappings for these number words because their assessment of mappings for number words beyond ‘four’ underestimated the availability of these mappings. Our results converge with evidence obtained by Fuson and Hall (1983) with a different set of tasks. Fuson and Hall found that children aged between 4.5 and 6.5 are better at comparing the numerosities denoted by number words between ‘one’ and ‘ten’ than at determining their relative positions in the count list. They tentatively took this to suggest that young children do not use the later-greater principle to compare number words between ‘one’ and ‘ten’ but rather rely on what they called ‘a magnitude process’ (p. 98; non-verbal representations of numerosity are sometimes known as representations of magnitude, e.g., Feigenson, Dehaene, & Spelke, 2004; Moyer & Landauer, 1967). More generally, the present results also converge with other evidence that the acquisition of the CP does not involve the discovery of the numerical significance of the order of the number words. Davidson et al. (2012) asked whether all CP-knowers know that ‘immediately follows’ means ‘add 1’. They found that many did not know the successor relations between any number words beyond ‘four’. Others knew successor relations between low number words in their list (e.g., between ‘five’ and ‘six’), but did not know the successor relations between higher number words in their list (e.g., between ‘fifteen’ and ‘sixteen’). Thus, the latter most probably knew a
7
Le Corre and Carey referred to minimal mappers as ‘CP mappers’.
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few isolated addition facts (e.g., ‘five + one = six’) rather than a general principle linking the order of number words to the successor relation (but see Sarnecka & Carey, 2008). In sum, taken together with studies of young CP-knowers’ knowledge of the correspondence between order and the successor relation, our results suggest that the acquisition of the CP does not involve or require the acquisition of any knowledge of the numerical significance of the order of the number words in the list. Rather, we suggest that knowledge of the CP is exactly as Gelman and Gallistel (1978) first described it: that is, knowledge that the last number word of a count denotes the numerosity (or cardinality) of the counted set when the count is correct. Taken literally, this knowledge states conditions for determining what numerosity is denoted by what number word. It does not include any knowledge of correspondences between the order of the number words and relations between the numerosities they denote. This suggests that proposals that children acquire the CP by noticing a correspondence between the meaning of number words and their order in the count list (e.g., Carey, 2004, 2009; Piantadosi et al., 2012; Spelke & Tsivkin, 2001) are unlikely to be right. Instead, we propose that children acquire the CP by noticing that the last number word of a count is the very number word that is used to denote the numerosity of the counted set only when the count is correct (see Dehaene, 1997, for a similar proposal). This learning process leads to the acquisition of the CP without requiring learners to notice any correspondences between the order of number words in the list and relations between numerosities.
Acknowledgements This paper owes much to multiple discussions with David Barner, Susan Carey, Pierina Cheung, Ori Friedman, Justin Halberda, and Katherine White. Thanks to Pierina Cheung, Caryn Harris, Mindy Hsu, Bridjet Lee, Paul Muentener, and Dora Thalwitz for their help with the collection and analysis of data. Finally, we thank the many parents, children, and day care personnel who generously contributed to this project. This research was supported by NIH Grant no. RO1-HD038338 and NSF ROLE Grant no. REC-0196471 to Susan Carey and by an NSERC Discovery grant to Mathieu Le Corre.
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