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The acquisition of counting is a major milestone for children. A central question is how children’s non-verbal number concepts change as they learn to count. We assessed children’s verbal counting knowledge using the Give-N task and identified children who had acquired the cardinal principle (Cardinal Principle Knowers, or CP-knowers) and those who had not (Subset-Knowers, or SS-knowers). We compared their performance on two tests of nonverbal numerical cognition. We report comparable performance between SS- and CP-knowers for matching and tracking small sets of objects up to four, but disparate performance for sets between five and nine, with CP-knowers outperforming SS-knowers. These results indicate that the difference between CP- and SS-knowers extends beyond their knowledge of the verbal number system to their non-verbal quantitative reasoning. The findings provide support for the claim that children’s induction of cardinality represents a conceptual transition with concurrent, qualitative changes in numerical representation.

The acquisition of counting, though it may seem to be a mundane skill, is a major milestone for children. In one sense, counting is clearly a feat of word learning: Children need to learn words for highly abstract number concepts and grasp how the structure of the count list gives meaning to the number words (i.e., going up one word in the list represents adding one item to the set;

Previous studies of children’s verbal counting abilities have documented that children begin to recite a count list long before they develop stable meanings for the number words in this list. It then takes children two to three years from the time they learn the count list to acquire the cardinal principle—the idea that the last number in the count represents the cardinality of the set (e.g.,

Using a wide range of tasks, several studies have found that SS-knowers and CP-knowers differ qualitatively on their interpretations of number word meanings (e.g.,

Several studies on populations that lack a verbal count list suggest that the ability to represent large exact quantities is dependent on having exact meanings for large number words. For example, studies with two Amazonian tribes show that verbal counting may allow humans to represent, match, and track large exact quantities (

These findings are consistent with previous research showing that small sets of objects can be represented with parallel individuation (PI)—an object tracking system that is present in infants, adults, and other animal species (for review, see

If neither PI nor the ANS can support the representation of large exact quantities, how do humans represent, for example, a set of ten objects exactly? As studies on populations that lack a verbal count list suggest, the process of acquiring number word meanings may allow us to represent and track large exact quantities (

Notably, both of the nonverbal number systems undergo significant developmental change over the same time period during which most children acquire meanings for number words. The acuity of the ANS improves in a protracted fashion throughout early childhood, with a discriminability ratio of 1:2 in 6-month-old infants (

Previous studies studying the relationship between children’s counting skills and their ability to solve non-verbal numerical tasks (i.e., tasks that do not require the interpretation of numerical language like “two,” “six,” “more”) have yielded mixed findings. Importantly, many of the studies on number language and concepts do not provide convincing evidence regarding the relationship between acquiring the cardinal principle and representing large sets in typical development, because the methods used do not evaluate both of these abilities simultaneously (

Previous developmental studies exploring the relationship between counting and non-verbal numerical representation have not met these two criteria. Using a non-verbal triad task,

In another study exploring the relationship between counting and non-verbal number representation,

Finally, numerous recent studies have reported a more general correlation between symbolic number knowledge, typically performance on a math achievement test, and nonverbal numerical acuity in the ANS, in infants through adults (e.g.,

To summarize, previous developmental studies have not found a relationship between the acquisition for number words beyond four (i.e., beyond the PI range) and the representation of exact quantities above four. The studies conclude only a minimal relationship between number words and performance on non-verbal tasks beyond the onset of counting, in contrast with the data from atypical populations, which suggest a strong relationship between number language and number concepts. However, those developmental studies did not explicitly hypothesize or test for a relationship between language and thought in the representation of large numbers. Additionally, there is a need in the field to develop non-verbal tasks other than dot comparison to assess the nature of children’s cognitive representations of number. The current study addresses these challenges.

The goal of the present research is to investigate whether children who have grasped the counting system in language (i.e., CP-knowers) show better performance in matching and tracking large quantities in non-verbal numerical tasks than those who have not (i.e., SS-knowers). We hypothesized that children should be able to solve nonverbal problems involving small quantities (one, two, and three) regardless of their verbal number knowledge, drawing on the PI system (

To test this, we examined preschool children’s performance on verbal and nonverbal number tasks using both small (1-3) and large (5-20) sets. In the present research, we developed two non-verbal numerical tasks, inspired by Hannula and colleagues’ research on ‘spontaneously focusing on numerosity’ or SFON (e.g.,

In Experiments 1 and 2, we adapted the Cardinality task from

We predicted that if acquiring the cardinal principle makes it easier for children to encode large quantities, CP-knowers, but not SS-knowers, should be more accurate in retrieving socks for the caterpillars. However, if any difference were found between SS- and CP-knowers’ performance on the non-verbal matching task, it could be alternately explained by children’s ability to generate verbal estimates corresponding to the number of items in the set. Therefore, we also assessed children’s estimation knowledge, and the quality of children’s mapping between verbal numerals and approximate representations of quantity in the ANS, using an estimation task—“Fast Cards”—used in previous studies (

While Experiments 1 and 2 tested children’s ability to

In Experiment 1, children were tested on two assessments of verbal number knowledge (Give-N and Fast Cards) and one assessment of nonverbal number reasoning (Caterpillar Game). In the Caterpillar Game, three small set sizes (1, 2, and 3) and three large set sizes (6, 7, and 9) were used to test the hypothesis that CP-knowers would outperform SS-knowers in the high but not in the low number range.

Forty-nine children (

Children were run in a single session on the Caterpillar Game, Elicited Counting, Give-N, and Fast Cards, in that order. The Caterpillar Game was run first so that its performance would not be affected by exposure to the explicit counting tasks.

Seven 19 in. long caterpillars were created from dark green soccer socks that were stuffed with batting and sewn shut (

Representative schematic drawings of caterpillars (from photographs) with different numbers of ‘feet’.

On each trial, children were introduced to a caterpillar and were told the following:

Sammy wants to go for a walk, but he needs socks. See the socks over there? Could you get just enough socks for Sammy? Be careful though! If you don’t bring enough socks, his feet will be cold. But if you bring too many socks, it will make a mess. Sammy’s parents really do not like a messy room, so we don’t want to have extra socks lying around. Can you go there and get

Critically, the experimenter never explicitly suggested to children that they should count the socks, and they avoided using phrases like “how many socks” or “the right number of socks.” Children were then allowed to go to the sock table, which was located between 2m and 5m away from the testing area, and bring socks, which the experimenter and the child then put onto the caterpillar’s feet. Once all the socks that the child had brought were used (or all the feet covered), the experimenter asked the child if there were “just enough socks.” Children were encouraged to retrieve more socks or return extra socks to the pile as many times as needed. If they brought too many and did not spontaneously correct their error, the experimenter pointed out that the area was now messy and asked the children to return the extra socks; if they did not bring enough, the experimenter pointed out that the caterpillar’s feet would be cold, and asked the child to retrieve additional socks. The child thus received feedback on every trial. We only present analyses of children’s responses on the first attempt to retrieve socks. This is because for most trials, the number of socks that children needed to bring was small after the first retrieval, and therefore children were very accurate with the second retrieval.

Each session began with a one-footed caterpillar to help children understand the task^{i}, and ended with a two-footed caterpillar to ensure that children were attentive throughout the session (i.e., if children were consistently correct on this trial, we assumed that they understood the constraints of the task through all trials). Counting was neither encouraged nor forbidden, but we noted whether the child counted caterpillars’ feet or socks. The one-footed caterpillar was used as the practice trial for each participant. The remaining trials (three, five, six, seven, or nine feet) were administered in one of three pseudo-random orders. A different caterpillar was used on every trial.

To establish if the child had a stable count list, the experimenter placed a line of 12 rubber ducks on a table and asked the child to count them. Each child’s highest count was recorded.

This task was adapted from

Following

Thirty-eight of the 49 participants were able to count to 10 with no errors, and all could count to 8 without error. Using the Give-N task, children were classified into knower levels. We found 19 SS-knowers (

For the practice trial, children were shown a one-footed caterpillar. We found that SS-knowers brought a mean of 2.16 socks, while CP-knowers
brought a mean of 2.47 socks. CP-knowers performed perfectly when the caterpillar had two feet, and SS-knowers were also highly accurate (

Group | Set Size ( |
||||||||
---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 5^{a} |
6 | 7 | 9 | 20 | ||

Expt. 1 | |||||||||

SS ( |
2.16 | 2.05 | 2.89 | 4.71^{a} |
5.68 | 5.37 | 5.58 | - | |

1.64 | 0.23 | 1.66 | 2.93 | 2.54 | 2.61 | 3.06 | |||

CP ( |
2.47 | 2.00 | 2.83 | 4.90^{a} |
5.80 | 6.10 | 7.40 | - | |

2.00 | 0.00 | 0.70 | 1.52 | 1.32 | 1.77 | 1.81 | |||

Expt. 2 | |||||||||

SS ( |
2.14 | 2.43 | - | 4.88 | - | - | - | 7.00 | |

1.35 | 1.13 | 2.53 | 4.36 | ||||||

CP ( |
2.50 | 2.00 | - | 5.25 | - | - | - | 9.75 | |

2.22 | 0.00 | 1.06 | 4.74 |

^{a}Only 7 SS- and 10 CP-knowers were tested with the 5-footed caterpillar in Experiment 1.

To examine if SS-knowers and CP-knowers performed differently on the Caterpillar Game, we first analyzed the number of socks children retrieved for the caterpillars, followed by an analysis of the mean absolute errors on each retrieval.

On low-number trials, SS-knowers and CP-knowers showed similar performance: On their first attempt, both groups brought more socks for the three-footed than the two-footed caterpillar, SS: ^{ii} Additionally, for the nine-footed caterpillar, CP-knowers brought significantly more socks than did the SS-knowers, _{slope}_{slope}

SS-knowers and CP-knowers also differed in the magnitude of their errors in retrieving socks. Error rate was defined as the absolute difference between the number of feet on the caterpillar (target) and the number of socks brought on the first attempt (response). For example, for a seven-footed caterpillar, children who brought back either five socks or nine socks would have an error of 2.

We conducted a 2x2 mixed ANOVA with Knower-Level (SS vs. CP) as a between-subjects factor and Set Size (Small vs. Large) as a within-subjects factor. The dependent variable is summed errors across trials. The analysis revealed an overall main effect of Knower Level, ^{2}_{p} = .241, showing that the magnitude of errors of CP-knowers (^{2}_{p} = .662, showing that the magnitude of errors on large number trials (^{2}_{p} = .200. For the small-number trials, the magnitude of errors was similar for SS-knowers,

(a) SS- and CP-knowers’ mean number of socks retrieved at each set size. (b) SS- and CP-knowers’ mean error on small and large number trials. Error bars represent 1 SEM.

Although the sample size did not permit analysis by each knower-level, we were interested in whether children’s performance increased across knower-levels prior to the CP transition. We separated the SS-knower group into two groups, 1/2-knowers (^{2}_{p} = .005; error: ^{2}_{p} = .018; slope: ^{2}_{p} = .046). Finally, to confirm that the difference between SS-Knowers and CP-Knowers would hold for both groups of SS-Knowers, not just the younger or less knowledgeable ones, we ran a series of One-Way ANOVAs with Group as the independent variable (3 levels: 1/2-Knowers, 3/4-Knowers, and CP-Knowers) and Total Error for the large sets, Slope for the large sets, and Mean Retrieval for the large sets as the dependent variables. All three ANOVAs were statistically significant, ^{2}_{p} > .22. Importantly, post-hoc LSD tests indicated that the differences between the two groups of SS-Knowers were not statistically significant, but the differences between 3/4-Knowers and CP-Knowers were (Total Error: 1/2-Knowers (

One explanation for why CP-knowers outperformed SS-knowers is that CP-knowers more readily engaged counting as a problem-solving strategy. To address this, we analyzed effects of counting on task performance. We loosely defined children as “Counting” if they showed evidence of engaging in overt counting on any trial (^{2}(1) = 3.28,

Of the 26 children who counted, only four, all CP-Knowers, brought the correct number of socks for each trial on which they counted, implying that counting does not guarantee success on this task. However, on large number trials (6, 7, and 9), children who counted made fewer total errors (3.27) on average than children who did not count (7.91),

Next we explored whether Counting and Knower-Level contributed independently to success on the task, or whether a propensity to count was behind the better performance seen in the CP-knowers. An ANOVA with total error on large sets as the dependent measure, counting behavior (Counting, Not Counting) and Knower-Level (SS, CP) as fixed factors, and age in months as a covariate, showed main effects of counting, ^{2}_{p} = .215, and knower-level, ^{2}_{p} = .137, no interaction, and no effect of nor interaction with age. Looking only at the sub-group of children who did not count, CP-Knowers still made fewer total errors in the large-number range than did SS-knowers,

We also tested the possibility that CP-knowers outperformed SS-knowers because of superior abilities in generating verbal estimates. The Caterpillar Game is, in some sense, a non-verbal estimation task. Our group of CP-knowers included both Mappers, who could generate fairly accurate verbal estimates for set sizes between 6 and 10, and Non-Mappers, whose estimates were far less accurate. We reasoned that if verbal estimation ability was beneficial to performance on this task, then CP-Mappers should outperform CP-Non-Mappers. Contrary to this prediction, CP-Mappers and CP-Non-Mappers made an equal number of total errors on the large-number trials (mean total error 3.94 vs. 3.97, respectively,

Because of the marked differences between SS-and CP-knowers on the high number set sizes, it was of interest to know whether SS-knowers’ ability to solve the task was poor for any set size outside the PI limit, or alternatively, whether their ability declined incrementally with increasing set size. Consequently, the last seventeen children tested received an additional trial with a five-footed caterpillar intermixed with the other trials. SS- and CP-knowers brought similar numbers of socks, 4.71 (SS) vs. 4.90 (CP), _{slope}

As a final test, separate ANCOVAs were run with Knower-Level as the independent factor and total error as the dependent variable; counting and one other factor (age, sex, or spring vs. fall testing) were included as covariates in each analysis. Time of testing was identified as a possible covariate because children who were tested in the spring had several more months of schooling than those tested in the fall, and thus may have demonstrated better performance on the Caterpillar Game. Knower-Level was significantly associated with total error in all three analyses. None of the three covariates yielded significant effects (Age: ^{2}_{p} = .051; Sex: ^{2}_{p} = .037; Time of testing: ^{2}_{p} = .096). These results indicate that having exact meanings for number words higher than four is robustly related to the ability to solve problems with quantities higher than three in a non-verbal task, even after controlling for the effects of spontaneous counting, age, sex, and amount of schooling.

Consistent with our hypothesis, Experiment 1 demonstrated comparable performance between SS- and CP-knowers for small numbers up to four, and contrasting performance between SS- and CP-knowers for large numbers between six and nine. CP-knowers gave increasing responses for larger targets in the high-number range, while SS-knowers did not show differential responses for caterpillars that had a large number of feet. These differences were statistically significant and robust with large effect sizes.

Although there were not significant differences for two of the large set sizes (six and seven), this is likely because SS-knowers brought roughly 5.5 socks no matter how many were required. Thus, while their responses appeared accurate for set sizes 6 and 7, this was likely just an artifact of 5.5 being SS-knowers’ typical response for all set sizes. In the more informative comparisons on set size nine, error rates for 6-9, and slopes in the 6-9 range, SS- and CP-knowers’ responses were clearly different.

These effects could not be explained by overt counting or by estimation skills. Rather, the results suggested that children’s knowledge of verbal counting and cardinality were related to more refined approximate performance on the non-verbal numerical task. The results implied a sharp cutoff in performance beyond the PI range, but this needed to be tested further as Experiment 1 did not systematically include, for all children, a set size of 5, just beyond the PI boundary. Experiment 1 also showed a stark contrast between small and large sets for subset-knowers, in that they were sensitive to quantities within the small-number range, but completely insensitive to quantities beyond it. Given that even subset-knowers have access to approximate number representations, which should support reasoning about larger quantities, this result raises the question of why such representations were not engaged on trials with quantities above four. Experiment 2 was designed to explore these two findings further by introducing a 5-footed caterpillar, just beyond the PI range, and a 20-footed caterpillar, to test whether subset-knowers would express more sensitivity to a more extreme numerical difference.

Results from Experiment 1 indicated a clear distinction between SS-knowers and CP-knowers on their performance on large-number trials. Specifically, SS-knowers appeared to treat all large sets (between 5 and 9) similarly in the Caterpillar Game, while CP-knowers differentiated among the large set sizes. Experiment 2 was conducted to address two questions that were raised by the Experiment 1 results. First, it was not clear whether children’s representation of set size ‘five’ followed the pattern of exact responses to small numbers, or poorly differentiated responses to large numbers, or an intermediate pattern. Many SS-knowers brought approximately five socks for a variety of large number trials, making it difficult to tell whether responses of five socks to five feet was an accurate response based on an exact representation, or a typical response to large numbers. The five-footed caterpillar trials in Experiment 1 provided preliminary evidence that children handled sets of five as a “large” number, suggesting a sharp drop-off in performance beyond the PI system’s limit. However, because this set size was added partway into the study, only a small group of children was tested on ‘five’ and the comparison of error between SS-knowers and CP-knowers did not reach statistical significance. Experiment 2 therefore aimed to replicate the patterns on set size five with a different sample of children.

Second, it was not clear whether SS-knowers had entirely failed to notice the differences between the six, seven, and nine-footed caterpillar, or, alternatively, whether they had difficulty discriminating these quantities in order to solve a number-relevant problem. To address this, we included a caterpillar with 5 feet and a caterpillar with 20 feet, as children should be more likely to notice the difference between 5 and 20 feet than the difference between 6 and 9 feet. If SS-knowers used this very marked difference to retrieve more socks for the 20-footed caterpillar, it is likely that they simply did not notice the differences between the high-number caterpillars in Experiment 1. If, however, SS-knowers did not bring more socks for the 20-footed caterpillar, perhaps they noticed the difference but were unable to apply this information to solve the numerical problem.

Twenty-three children (

The Caterpillar game consisted of four trials, with set sizes 1, 2, 5, and 20. As in Experiment 1, the first trial involved a one-footed caterpillar to make sure children understood the task and the two-footed caterpillar came last. The order of the two middle trials (five- and twenty-footed caterpillars) was counterbalanced across children. Two additional children, both subset-knowers, did not complete the Caterpillar Game and were therefore not analyzed.

We identified 7 SS-knowers (1 one-knower, 4 two-knowers, 1 three-knower, and 1 4-knower) and 16 CP-knowers using Give-N. CP-knowers were further sorted using Fast Cards into 8 Non-Mappers and 6 Mappers; 2 CP-knowers did not complete the task. A preliminary analysis revealed no differences at all between CP-Mappers and CP-Non-Mappers on both 5- and 20-foot trials (p > .4), so these groups were combined for analysis as CP-knowers.

We first analyzed children’s performance on the five-footed and twenty-footed caterpillars separately to better understand children’s responses to quantities outside the PI range (see

For the twenty-footed caterpillar, both SS- and CP-knowers noticed and commented that there were “a lot” of feet, and indeed retrieved more socks, on average, for this caterpillar than children had in Experiment 1 for any other set size (7.00 for SS, 9.75 for CP), including the 9-footed caterpillar in Experiment 1; this difference between 9 and 20 was statistically significant only in CP-knowers,

Next, we asked whether children, particularly the SS-knowers, were sensitive to the difference between the 5-footed caterpillar and the 20-footed caterpillar. CP-knowers retrieved significantly more socks for the 20- than the 5-footed caterpillar,

Experiment 2 extended the findings from Experiment 1 by examining children’s performance with sets of five (just beyond the PI range) and sets of twenty (more than double the highest set size from Experiment 1). For the five-footed caterpillar, the variance in the responses was much higher in SS- than CP-knowers, just as it was for the six-, seven-, and nine-footed caterpillars in Experiment 1. This pattern suggests that five is treated as a ‘large’ number by subset-knowers, corroborating the conclusion from Experiment 1 that there is a divergence in the quality of SS- and CP-knowers’ performance on this task with set sizes outside the PI range.

For the twenty-footed caterpillar, the pattern of responses did not exactly mirror those observed for large sets under ten, in two ways. First, SS-knowers showed some sensitivity (by Wilcoxon test, above) to the difference between five and twenty, indicating that with sufficiently different target quantities, they could differentiate their responses. Second, even CP-knowers underestimated how many socks to bring back, and the differences between SS-and CP-knowers were not as strong for the 20-footed caterpillar as for other set sizes. If CP-knowers were generally more attuned to quantity–for instance, if their performance on twenty could be predicted by extrapolating from their responses for six, seven, and nine–then the 20-footed caterpillar provided them with the best chance to demonstrate a large difference from SS-knowers, but they did not. Notably, “twenty” was beyond the comfortable estimation range for most of the children. Rather than revealing a robust difference between SS- and CP-knowers in their representation of twenty, it appears that the difference between groups diminished when children were presented with a larger quantity that was less familiar to both groups.

In short, Experiment 1 and 2 indicated that CP-knowers had a more refined response to a quantity-matching task, with more accurate and less noisy estimates of socks for a given number of feet. We note two results that could have occurred but did not: CP-knowers could have solved the task precisely, simply by counting the feet and the socks; or CP-knowers could have exhibited better exact matching of the numbers of socks and feet with a different non-verbal representation (e.g., chunking). However, the observed pattern suggests that CP-knowers had a more refined

While Experiments 1 and 2 revealed an advantage in CP-knowers in their use of

We created a different non-verbal paradigm–the Mr. Elephant Game–to test children’s ability to track large exact quantities beyond the PI range. In the Mr. Elephant Game, the experimenter placed balls inside a box named “Mr. Elephant.” On half of the trials, the experimenter surreptitiously stopped one ball from coming out of Mr. Elephant (by toggling a small plastic disc), and on the other half of the trials, all of the balls came out. At the end of each trial, children were asked whether there were balls left in the box. While both the Mr. Elephant Game and the Caterpillar Game were non-verbal numerical tasks, one fundamental difference between these two tasks is the nature of the responses elicited from the children. The Caterpillar Game requires children to retrieve “just enough socks” from a very large pile, posing an essentially open-ended problem of how many socks to retrieve. In contrast, the Mr. Elephant Game asks children to distinguish

Nineteen children (

Fifteen of the 19 children were run in a single testing session on Give-N and the Mr. Elephant game, in that order. The remaining four participants were run on Give-N in one testing session and on Mr. Elephant in a second testing session two days later.

A hollow, wooden cube (length, width, and height = 27 cm) was painted dark blue, and paper eyes and felt ears were pasted on the front and sides of the box to create “Mr. Elephant” (

Schematic drawing of (a) exterior and (b) interior of Mr. Elephant apparatus with levers used by the experimenter to control the path of the balls through the tubes.

At the beginning of the experiment, the child was shown a bowl containing 7 green Styrofoam balls 4 cm in diameter. The experimenter explained to the child that Mr. Elephant liked to eat the “green peanuts” and then blow them out of his trunk. But sometimes, the child was told, the peanuts got stuck in his trunk, so Mr. Elephant needed the child’s help to make sure all the peanuts came out.

On each trial, the experimenter placed either 2, 3, 5, or 7 balls on top of the box in a fixed pseudorandom order. Each number of balls was presented to the child twice—one trial releasing N balls, and the other trial releasing N-1 balls—yielding a total of eight trials per testing session. Each testing session began with the easier 2-ball trials, which introduced the procedure to the child, and ended with the 3-ball trials, to ensure that children understood the task and were attentive throughout the session. The remaining trials (5 and 7) were presented in one pseudo-random order (5 in/4 out, 7-in/7-out, 5-in/5-out, 7-in/6-out). Feedback was available on every trial since one more ball either came out or did not on each trial.

The experimenter circumscribed the balls with her finger and said, “Look! I'm going to feed Mr. Elephant these peanuts!” At the beginning of each new trial, the experimenter said “Remember, let me know if you think a peanut is stuck.” The experimenter then dropped the balls into the top chute one by one. The balls were blocked from immediately coming out of the front chute by the small plastic door near the trunk. On one of the two trials for each number, the experimenter surreptitiously toggled the second plastic disc to block the final ball from going down the top chute.

The experimenter then told the child that Mr. Elephant was going to blow out the ‘peanuts,’ lifted the disc blocking the front chute, and allowed the balls to come out. The child was then asked, “Did they all come out?” An affirmative response was correct for the 50% of the trials when all of the balls came out, while a negative response was correct for the 50% of trials when all but one of the balls came out. Once all of the balls were out, the experimenter said “Good job!” if the child was correct. If the child was incorrect, she would say “Let’s check! Uh oh! I think a peanut is stuck! Can you make it come out? Thank you!” or “Oops! It doesn’t look like any peanuts were stuck.”

We identified 9 SS-knowers (4 one-knowers, 1 two-knower, 3 three-knowers, 1 four-knower) and 8 CP-knowers using Give-N. Two five-knowers were also identified, but were excluded from subsequent analyses.^{iii}

We tested whether CP-knowers would respond more accurately than SS-knowers on the large number trials in the Mr. Elephant Game, as they had on the Caterpillar Game. A 2x2 mixed ANCOVA was performed on the percentage of correct responses, with Set Size (Small [2- and 3- ball trials] vs. Large [5- and 7-ball trials]) as a within-subjects factor, Knower-level (SS vs. CP) as a between-subjects factor, and age (in months) was entered as a covariate. This analysis revealed a significant main effect of Knower-Level, ^{2}_{p} = .46, with CP-knowers (^{2}_{p} = .38 (see ^{2}_{p} = .52). Furthermore, both SS- and CP-knowers performed well above chance (50% accuracy) for small set sizes (

CP- and SS-knowers’ performance on small (2 and 3 balls) and large (5 and 7 balls) set sizes in the Mr. Elephant task.

Looking at

Trial Type | SS-Knowers | CP-knowers |
---|---|---|

Empty | .111 (.220) | .500 (.378) |

One Inside | .611 (.417) | .813 (.372) |

^{1} (d-prime) |

Counting behavior was not recorded for all participants, but was available for 4 of the 8 CP-knowers and 7 of the 9 SS-knowers in our sample. No participant counted on every trial, 4 children (2 SS-knowers and 2 CP-knowers) counted on at least one trial, and 7 children never counted. Of the two SS-knowers who counted, both counted on two trials, but responded inaccurately. Of the two CP-knowers who counted, one counted on one trial and responded accurately. The other CP-knower counted on two trials, and was accurate on one trial but inaccurate on the other trial.

Including only the children for whom counting behavior was recorded, an ANCOVA with SS/CP knower level as the independent variable, age and counting behavior as covariates, and proportion correct on small and large trials as a repeated measure, revealed a large and significant main effect of SS/CP, ^{2}_{p} = 2.26, and a marginal interaction between set size and SS/CP, ^{2}_{p} = .40, and no other significant effects or interactions. Specifically, there was no hint that counting behavior explained the SS/CP difference, because no effect of counting was detected, ^{2}_{p} = .005. Thus, knower-level, not counting behavior or age, predicted success on the Mr. Elephant task.

Experiment 3 replicated and extended the pattern of results from Experiment 1 using a different task. SS- and CP-knowers performed similarly and were highly accurate on a non-verbal number task for smaller numbers (1 to 3), but CP-knowers significantly outperformed SS-knowers for larger numbers (> 4). Using a non-verbal tracking task, we found evidence that understanding the cardinal principle is related to better tracking and memory for large quantities. Specifically, in the Mr. Elephant Game, children were asked to

The current experiments test the relationship between preschool children’s knowledge of cardinality and their responses on two non-verbal tasks: numerical matching (Caterpillar Game) and object-tracking (Mr. Elephant Game). The results provide strong support for the hypothesis that verbal counting and nonverbal quantitative reasoning are related in children, and help bring clarity to a conflicted body of findings about the relationship between these skills during development (e.g.,

Three verbal mathematical capacities other than cardinality notably did not relate to performance on the non-verbal tasks. First, counting behavior during the non-verbal Caterpillar Game improved children’s performance on the large number trials, but it did not explain the difference in success between SS- and CP-knowers. Second, verbal estimation skill within the CP-knower groups was also unrelated to performance on the Caterpillar Game (note that children were not tested on Fast Cards in Experiment 3). Third, no differences were observed across different levels of SS-knowers (e.g., “one”-knowers, “two”-knowers, etc.), with the caveat that the samples were small; as a group, SS-knowers’ performance differed substantially from that of CP-knowers. Thus, after controlling for age, children’s mastery of verbal counting—specifically, their understanding of cardinality—was the key predictor of accuracy on a non-verbal numerical matching task.

The current data establish the relationship between the acquisition of meanings for large numbers and non-verbal processing for quantities and numerals larger than “four.” We provide developmental evidence for the cross-cultural finding that knowledge of a meaningful count list is related to more precise non-verbal representation of large numbers (

These findings raise two important questions: First, which cognitive systems underlie the observed change in performance on non-verbal tasks, and second, what is the causal mechanism underlying the relationship between verbal and non-verbal numerical knowledge?

Some researchers have argued that the role of number language is to provide a concept of exact number or a “tool for thought” enabling exact number representations (

The observed data are consistent with several possible cognitive systems for non-verbal numerical representations that could support these patterns of performance. One explanation is that CP-knowers have better numerical acuity (e.g.,

However, ANS acuity is an unlikely explanation of the CP-knowers’ superior performance on the Mr. Elephant Game: The most difficult trials required children to distinguish between outcomes of six and seven balls coming out on the seven-ball trials. This ratio of 1.16 is considered a very difficult discrimination ratio on other tests of numerical acuity in this age group (e.g.,

Differences in PI representations could also plausibly underlie the different performance between CP- and SS-knowers. In the Caterpillar Game, because the “feet” were distributed across the two sides of the caterpillar “body,” children could solve the problem

The distribution of the feet on either side of the caterpillar might have also affected SS-knowers ability to represent the total numerosity presented. Importantly, it is not the case that SS-knowers simply attended to one side and ignored the other; if they had ignored one side, their responses on the large number sets would have ranged from three (one side of the 6-footed caterpillar) to five (the largest number on one side on any trial, on the 9-footed caterpillar). This was not the case: SS-knowers responded with sets larger than five socks on many of the large number trials, suggesting that they were oriented to the totality of the presented set of feet.

In addition to potentially using the two sides to help break down the task into two ‘chunks’, CP-knowers might also have been better able to maintain exact representations of the set size on each side. Some research suggests that the set-size limit on PI increases through the preschool years from a set size limit of three in 3-year-olds to four or five in 5-year olds (

The results from the Mr. Elephant Game are also consistent with this explanation: CP-knowers were much more likely to correctly assess both five-ball trials, demonstrating a true understanding of five discrete objects. SS-knowers, on the other hand, performed at chance for the sets of five. These observations strengthen the possibility that there is a relationship between an increase in the set-size limit of the PI system (to five) and the timing of children’s acquisition of the cardinality principal. With much attention in recent years on the relationship between ANS acuity and symbolic mathematics, developmental change in the PI system has received little attention in recent years but may be important for children’s emerging number concepts.

Although it is not considered a cognitive ‘system’ per se, another possible explanation for the CP-knowers’ advantage is enhanced spontaneous focus on numerosity or SFON, a child’s tendency to engage with quantities and number in her environment (

Notably, one recent study provides compelling evidence that the sequential-enumeration tasks used in some SFON studies likely draw on ANS representations (

The current findings lend support to arguments for a qualitative shift, or conceptual change, between SS-Knowers and CP-Knowers. Previous research on this transition has focused on children’s conceptual change in terms of their construction of a novel representation for numbers—namely, a meaningful count list (

Some previous researchers have argued that there is no semantic induction when children become CP-knowers based on the fact that CP-knowers fail to answer many questions about higher numbers within their count list (

Of course, there are limits to what children learn when they acquire the cardinal principle.

An additional parallel between verbal and non-verbal number concepts comes from the 20-footed caterpillar in Experiment 2. On the 20-footed caterpillar, neither group performed very accurately, and CP-knowers responded only marginally more accurately than SS-knowers. This pattern on the non-verbal task parallels the low verbal knowledge of “twenty” in both groups: “Twenty” is essentially an ‘unknown’ number for some CP-knowers and most SS-knowers: Although they may produce it, they often cannot reliably count to it using stable order (

Finally, we note that this study is correlational, and therefore cannot in itself address causality: Change in core non-verbal number representations may support number language; number language may induce change in cognitive systems involved in non-verbal numerical reasoning; or these changes may co-occur as part of an inter-related set of conceptual shifts. The cross-cultural findings with innumerate people imply that language causally changes the way in which quantities can be represented, and that in the absence of acquiring such language, conceptual representations are not pushed to change. Studies on the development of number concepts under conditions of accelerated language (e.g., with number training) or delayed language (e.g., due to limited access to a native language) will help to tease apart these possibilities.

To summarize, children’s acquisition of exact cardinal meanings for large numbers (beyond the PI range) correlates with their performance on numerical problem-solving tasks that require remembering and matching large, exact quantities. In particular, responses to target sets between five and nine were less accurate and more variable in children who had not induced cardinality than in those who had. In contrast, smaller set sizes were handled easily by all children, regardless of knower-level. Performance on the non-verbal tasks did not vary for children at different SS-knower levels, as a function of developing better verbal estimation skills, nor as a function of counting behavior. Our findings accord with reports of close links between symbolic and non-symbolic mathematical competence in children (

These findings thus bring some clarity to a previously conflicted body of literature, by showing a distinct relationship between the acquisition of number meanings larger than “four” and non-verbal problem-solving with quantities larger than four. This pattern of results helps to explain why previous studies, which lacked the specific comparison between SS- and CP-knowers on large numbers, did not find hypothesized relationships between verbal and non-verbal number knowledge. These findings open up a new set of questions regarding which non-verbal cognitive skills and causal mechanisms underlie the tight link observed between number language and number thought.

We are indebted to the many members of the Wesleyan Cognitive Development Lab who contributed to portions of this project, including Emily Compton, Lisa Drennan, Barry Finder, Kathryn Grogan, Cory Savereid, Andrew Smith, Meghan Duberek, Adèle Borden, Emma Zoloth, and Sarah Edelman. Thank you to Tom Castelli of the Wesleyan Machine Shop for helping to design and build the Mr. Elephant apparatus. We are also grateful to the preschools and families that participated in the studies. We note that some of the data in Experiment 1 was previously reported in a student journal (

Many children brought two socks on the first trial, even though the caterpillar had just one foot, presumably because socks typically come in pairs. They were then asked to return the extra sock, which helped to reinforce the rules of the game.

Given that all children could count up to 8 but not 9 on the Elicited Counting task (

Given that we had trials using ‘5’ in the Mr. Elephant game, we took a conservative approach and did not analyze performance of the two five-knowers.

This work was supported by NSF CAREER 0845966 to A.S., NSF 1420196 to A.S, and Wesleyan University.

The authors have declared that no competing interests exist.