Conceptual Correlates of Counting: Children’s Spontaneous Matching and Tracking of Large Sets Reflects Their Knowledge of the Cardinal Principle

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; Gallistel & Gelman, 1992; Gelman & Gallistel, 1978). In another sense, counting is a feat of conceptual development: Children build an explicit symbolic representation that enables them to determine, track, and remember exact quantities (Carey, 2009; Carey & Sarnecka, 2006; Frank, Everett, Fedorenko, & Gibson, 2008; Gordon, 2004). A question central to this subject is how acquisition of the verbal count list interacts with non-verbal conceptual development in numerical reasoning.

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., Fuson, 1988; Gelman & Gallistel, 1978; Schaeffer, Eggleston, & Scott, 1974; Wynn, 1992). The cardinal principle is a foundational concept that serves as a building block for later mathematical thinking. Using the Give-N task, researchers have consistently shown that English-speaking children start off learning the meanings of number words in a piecemeal fashion. First, around the age of two, children develop an understanding of the meaning of “one” (“one”-knowers; i.e., they can give one object when asked for one, and avoid giving one when asked for other quantities). They then make stage-like jumps approximately every six months to become “two”-, “three”-, and then “four”-knowers—collectively termed “Subset Knowers” (henceforth SS-knowers) because during these stages children know the exact quantities referred to by numerals for a subset of the count list they are able to recite (Le Corre, Van de Walle, Brannon, & Carey, 2006; Lee & Sarnecka, 2010, 2011; Sarnecka & Lee, 2009; Wynn, 1990, 1992). Children who can accurately provide an experimenter with any requested number (usually tested up to 6) are labeled “Cardinal-Principle Knowers” (henceforth CP-knowers). Becoming a CP-Knower is often regarded as a categorical shift distinct from the subset knower levels. While SS-Knowers might understand cardinalities of small numbers, they have not generalized the cardinality principle such that they can flexibly apply it to a range of set sizes.

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., Condry & Spelke, 2008; Le Corre et al., 2006; Le Corre & Carey, 2007; Sarnecka & Carey, 2008; Slusser & Sarnecka, 2011; Wynn, 1990, 1992). For example, only CP-knowers—but not SS-knowers— have exact numerical meanings for quantities larger than three or four. On the Give-N task, only CP-knowers can generate a large set of objects upon request (e.g., Can you give me six fish?), and this ability defines their status as CP-knowers. Further, CP-knowers, but not SS-knowers, appear to understand that moving forward one number word on the count list means adding one item to the set—e.g., when an object is added to a set of five objects, the set now has six objects and not seven (Sarnecka & Carey, 2008; see Davidson, Eng, & Barner, 2012 for evidence that this generalization extends only to a limited count list). Moreover, only CP-knowers can correctly match pictures that are labeled with large number words (e.g., This picture has eight turtles. Find another picture with eight turtles.), while SS-knowers are equally likely to choose the foil that matches on continuous extent such as total surface area, or non-numerical attributes such as color (green turtles) or mood (happy turtles; Slusser & Sarnecka, 2011). Finally, only CP-knowers, but not SS-knowers, show evidence that they understand equinumerosity—if two sets have the same cardinal value, they hold the same quantity (Sarnecka & Wright, 2013). Together, these studies show that CP-knowers differ from SS-knowers with regard to their knowledge of how number words relate to cardinalities of sets of objects, particularly for sets greater than four.

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 (Frank et al., 2008; Gordon, 2004; Pica, Lemer, Izard, & Dehaene, 2004). In one study of the Pirahã, a tribe that lacks a natural number system in their language, performance on set-matching tasks was excellent for small numbers (one, two, three) but became more variable as the set size increased (Gordon, 2004). A follow-up study by Frank and colleagues (2008) replicated Gordon’s results, showing near-perfect performance on small set sizes and a linear decline in performance for quantities greater than four. In particular, they found that performance on larger sets (> 4) was especially impaired for tasks that required participants to remember particular quantities for some short period of time or to re-create an array in a different spatial orientation than the target array. This pattern of results has also been found in another population. Flaherty and Senghas (2011) showed that older deaf signers in Nicaragua who lacked a stable count list also faced difficulties re-creating exact set sizes larger than six, but they were capable of re-creating small sets. Collectively, these studies show that speakers of languages that do not have a verbal count list have difficulty representing and tracking large but not small quantities, suggesting that the acquisition of verbal number is central to solving nonverbal numerical problems that require the representation of 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 Feigenson, Dehaene, & Spelke, 2004). Using PI representations, individual objects are represented as mental symbols in working memory, but the system’s working memory capacity is limited to representations of set sizes of up to three or four items in human adults (e.g., Feigenson & Carey, 2003, 2005; Feigenson, Carey, & Hauser, 2002; Hauser & Carey, 2003; Trick & Pylyshyn, 1994). This explains how people from the Amazonian tribes and Deaf Nicaraguan signers can still represent small exact sets by deploying the PI system, despite not having a verbal count list. However, given its set size limit, the PI system cannot support the representation of large sets beyond four. In fact, much previous research has pointed towards another innate non-verbal representation of sets—the approximate number system (ANS)—for representing sets of all sizes (Dehaene, 2011; Piazza, Pinel, LeBihan, & Dehaene, 2007; Whalen, Gallistel, & Gelman, 1999; Xu & Spelke, 2000; see Feigenson et al., 2004 and Cantrell & Smith, 2013 for reviews). Unlike the PI system, the ANS represents number as continuous mental magnitudes and is limited in precision: The ability to discriminate any two numbers is determined by their ratio (i.e., Weber’s law). For example, it is easier to discriminate 5 from 6 dots than 55 from 56 dots, even though the absolute difference is one dot in both comparisons. Therefore, although the ANS can represent large sets, it cannot generate an exact representation of large quantities, leading to the less precise performance on number tasks seen amongst people from Amazonian tribes and the Deaf Nicaraguan signers. Consistent with this cognitive science literature, in this paper we refer to sets within the PI range as “small” sets and beyond this range as “large” sets.

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 (Carey, 2004, 2009). Thus, one possibility is that acquiring the cardinal principle coincides with gaining the ability to track large exact quantities. Drawing on the cross-linguistic evidence, Frank et al. (2008) argued that the verbal counting system is a “cognitive technology” that makes existing exact number representations explicit, allowing a person to note and remember exact quantities larger than four via counting. It is possible that the acquisition of verbal counting influences even more fundamental processes than memory for ephemeral quantities: Learning number language may actually change perceptual, attentional, or conceptual aspects of number representation.

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 (Xu & Spelke, 2000) and 2:3 in 9- month-old infants (Lipton & Spelke, 2003, 2004). Numerical acuity dramatically increases in children between 3 and 5 years—the same period during which children acquire verbal counting (Halberda & Feigenson, 2008)—and eventually reaches a discriminability ratio of about 7:8 in Western adults (Barth, Kanwisher, & Spelke, 2003). Developmental change is also evident in the PI system: Newborns can track only two objects (Coubart, Izard, Spelke, Marie, & Streri, 2014); 12-month-olds fail tasks requiring them to track more than three objects simultaneously (Feigenson & Carey, 2003, 2005); between the ages of 3 and 6, this limit increases to sets of four or even five (O’Hearn, Hoffman, & Landau, 2011; Ross-Sheehy, Oakes, & Luck, 2003; Starkey & Cooper, 1995).

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 (Brannon & Van de Walle, 2001; Huntley-Fenner & Cannon, 2000; Mix, 1999a, 1999b, 2008a, 2008b; Mix, Huttenlocher, & Levine, 1996; Negen & Sarnecka, 2010). To test the hypothesis that acquiring a count list allows us to represent and track large exact quantities, two criteria need to be met: First, it is necessary to compare preschoolers who have not yet acquired the cardinal principle (i.e., SS-knowers) to those who have (i.e., CP-knowers). The SS-knowers serve as a meaningful comparison group: Although they often have a verbal count list up to “eight” or higher, they only have meanings for the small numbers. Second, it is important to use tasks that require children to represent large exact sets greater than four.

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, Mix (1999a, 1999b, 2008a, 2008b) showed children a target set of objects (2, 3, or 4), and asked them to match it from two to three alternative arrays that differed from the target set in various dimensions (e.g., the number of objects, density between individual objects, length, and object type). Mix tested children with a range of counting abilities (i.e., ages consistent with SS- and CP-knowers) and found that children with minimal counting proficiency (i.e., those who could give at least two objects in Give-N) could recognize numerical equivalence between sets despite surface dissimilarities, but children who completely lacked counting ability could not (see Negen & Sarnecka, 2010, for similar findings). These findings reveal that verbal counting knowledge is correlated with non-verbal numerical cognition, in that minimal counting ability corresponded with above-chance performance on a matching task. In these studies, further counting development, including performance on the Give-N task consistent with understanding cardinality, was not related to performance on the non-verbal tasks. However, the stimuli used in the study entailed small numbers only. If cardinality is specifically related to non-verbal representation of large exact sets, the relationship could not have been observed in this study.

In another study exploring the relationship between counting and non-verbal number representation, Brannon and Van de Walle (2001) used a non-verbal numerical comparison task and found an overall effect of verbal counting knowledge similar to Mix’s (1999a, 1999b, 2008a, 2008b) studies. Specifically, children who knew at least some number word meanings were better at numerical comparison than those who did not know any number words. However, while this study used both small and large sets, the children were 2- and 3-year-olds and all in the pre-counting or SS-knower range. Thus, these findings again indicate a relationship between the onset of verbal counting and non-verbal numerical cognition, but the question remains whether there is an additional relationship between acquiring cardinality and non-verbal representation of large sets.

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., Libertus, Feigenson, & Halberda, 2011; Libertus, Odic, & Halberda, 2012; Mazzocco, Feigenson, & Halberda, 2011; Sasanguie, Göbel, Moll, Smets, & Reynvoet, 2013; Starr, Libertus, & Brannon, 2013). Most of the studies in this area do not focus on how counting ability and the acquisition of the cardinal principle relates to the development of nonverbal numerical representations. Several recent studies explicitly test for and find a relationship between cardinality and ANS acuity (Chu & Geary, 2015; Shusterman, Slusser, Halberda, & Odic, 2016; Wagner & Johnson, 2011), indicating a qualitative change in magnitude representations when children acquire cardinality. These studies used a dot comparison task in which participants were shown two sets of dots and asked to select the more numerous set (e.g., “Point to the side with more dots”). While this task is often used to investigate the ANS, this paradigm reveals how finely participants perceive more-less relations, but not how accurately participants represent any particular quantity. Thus, the dot-comparison studies do not explicitly test for a difference between CP- and SS-knowers in the exact representation of quantities specifically larger than four.

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.

Experiment 1

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.

Experiment 2

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.

Experiment 3

While Experiments 1 and 2 revealed an advantage in CP-knowers in their use of approximate representations of large sets, Experiment 3 explored whether CP-knowers show advantages in the representation and use of exact numerical quantities. To this end, we created a new non-verbal task that required children to track an exact number of objects.

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 x objects from x – 1 objects by answering a yes-or-no question; thus, on a given trial, children’s responses were always correct or incorrect.

General Discussion

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., Huntley-Fenner & Cannon, 2000; Mix, 1999a, 1999b; Rousselle, Palmers, & Noël, 2004). Just like adults whose language lacks an integer list (Flaherty & Senghas, 2011; Frank et al., 2008; Spaepen, Coppola, Spelke, Carey, & Goldin-Meadow, 2011), children who do not yet understand cardinality exhibit relatively coarse representations of numerical quantity when the quantity to be represented is greater than four—beyond the limit of the parallel individuation system. On small-number trials with set sizes of one, two, and three, SS-knowers performed no differently from CP-knowers, consistent with evidence for a primitive, language-independent cognitive system for representing small exact quantities (Agrillo, Piffer, Bisazza, & Butterworth, 2012; Feigenson, Carey, & Hauser, 2002; Starr, Libertus, & Brannon, 2013). However, on large-number trials with set sizes between five and nine, CP-knowers were more sensitive to the differences between quantities: In the Caterpillar Game, they brought more socks for larger sets of feet, and the number of socks they brought was closer to the target, and in the Mr. Elephant game, they more accurately identified whether the exact number of hidden objects re-appeared. Thus, mastery of verbal counting was correlated with performance on large, but not small, set sizes.

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 (Flaherty & Senghas, 2011; Frank et al., 2008; Gordon, 2004; Spaepen et al., 2011).

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 (Frank et al., 2008), but this does not seem to be the right explanation for CP-knowers’ more precise performance in this study. In the Caterpillar Game, even CP-knowers did not use exact representations of quantity to match numbers of socks to feet; rather, they tended to retrieve an approximately correct number of socks. Furthermore, both groups’ near-perfect performance on the small set sizes indicates that they understood the task and its specifically numerical demands. Therefore, if CP-knowers paid more attention to numerosity, this was not a binary switch between attending or not attending to number: Both groups attended to numerosity in the small number trials, and even CP-knowers failed to engage exact number in the large number trials. Furthermore, when comparing adults from numerate cultures to adults from innumerate cultures, the reason for differences in performance seems clear: The adults in numerate cultures can count, and thus form a stable representation to assist memory for ephemeral events. The children in our study who understood counting, however, did not necessarily deploy it in the service of solving the non-verbal problems. Moreover, the advantage of CP-knowers over SS-knowers held even after taking counting behavior into account. Therefore, our findings suggest a different role of language than a “tool for thought”: Acquiring language for numbers might be more deeply related to the cognitive representation of numerical quantity, or the use of these representations in computation, beyond providing an efficient tool for memory.

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., Chu & Geary, 2015; Shusterman et al., 2016; Wagner & Johnson, 2011), which may help them more accurately encode and reason about the larger set sizes in the non-verbal tasks. This explanation accords with the data in some ways. CP-knowers’ responses in the Caterpillar Game demonstrated two characteristics of representations in the ANS: They were approximately but not exactly correct, and their responses exhibited scalar variability (i.e., increasing standard deviations for larger targets). Thus, change in numerical acuity in the ANS could underlie changes in non-verbal numerical problem-solving in the Caterpillar Game.

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., Halberda & Feigenson, 2008). It would therefore be surprising if differences in ANS acuity between CP- and SS-knowers drove differential performance on that task.

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 either by noting the total set size or by noting the set size on each side (e.g., four on one side and five on the other on the nine-footed caterpillar). CP-knowers might have been more likely than SS-knowers to use a chunking strategy, noting the set on each side and combining them in their final response.

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 (Starkey & Cooper, 1995). This increased capacity could plausibly support children’s performance with the seven- and nine-footed caterpillar, where children would have to remember a set of four or five on each side; a less mature system, with a limit of only three objects, would limit children’s performance with these larger set sizes. Thus, a small increase in the set-size limit of the PI system could support the observed pattern in the Caterpillar Game.

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 (Rathé et al., 2016). Children who intuitively pay more attention to numerosity in SFON tasks might enter more quickly into skillful counting or might be more motivated to practice counting (Hannula, Räsänen, & Lehtinen, 2007). Individual differences in SFON are consistently related to children’s counting skills such as subitizing in the small number range and rote counting in the large number range (Batchelor, Inglis, & Gilmore, 2015; Edens & Potter, 2013; Hannula & Lehtinen, 2005; Hannula et al., 2007); furthermore, SFON in 3.5-year-old children is related to higher subsequent mathematical knowledge at ages 5 and 12 (Hannula & Lehtinen, 2005; Hannula-Sormunen, Lehtinen, & Räsänen, 2015). However, the current study differs from previous SFON studies in several important ways. Previous SFON studies have not distinguished performance between children who fully understand cardinality past the number “four” from those who do not. Counting ability is typically evaluated in SFON studies by procedural knowledge (e.g., Hannula et al., 2007) rather than a generalized understanding of cardinality. Additionally, the SFON tasks themselves typically focus on smaller set sizes (2, 3, or 4) than the ones used here (e.g., Hannula et al., 2007). In some studies, the participants are slightly older and may have already acquired cardinality (e.g., Batchelor et al., 2015). Finally, few SFON studies take place in the cultural context of the U.S., where children’s early number experiences may meaningfully differ. In short, development of theory related to SFON has enabled novel ways of thinking about the relationship between children’s non-verbal enumeration and their verbal counting skills. The current study is the first to explore this relationship between language and thought using the SFON approach with numbers beyond the subitizing range, and to emphasize the conceptual rather than procedural development of counting and cardinality.

Notably, one recent study provides compelling evidence that the sequential-enumeration tasks used in some SFON studies likely draw on ANS representations (Sella, Berteletti, Lucangeli, & Zorzi, 2016). Nevertheless, SFON tasks should not be taken as a measure of ANS acuity: when guided, ‘low-SFON’ children can perform these non-verbal tasks at the same level as ‘high-SFON’ children (Hannula & Lehtinen, 2005), showing that SFON reflects children’s self-guided attention to numerosity, not their underlying competence in numerical discrimination.

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 (Carey, 2010; Sarnecka & Carey, 2008; Slusser & Sarnecka, 2011; Wynn, 1990, 1992). Our findings corroborate the qualitative change by showing that the acquisition of meaningful counting and cardinality is accompanied by additional changes in non-verbal reasoning.

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 (Davidson, Eng, & Barner, 2012). They posit that these limitations in CP-knowers’ knowledge mean that there is no conceptual change, and no radical discontinuity between numerical representations in SS-knowers and those in CP-knowers. The current data argue against this perspective because they highlight a clear discontinuity in non-verbal representations that correlates with the discontinuity in number language. Additional evidence for a discontinuity between SS- and CP-knowers comes from Sarnecka and Wright (2013), who demonstrated that CP-knowers, but not SS-knowers, understand the principle of equinumerosity, and from Shusterman et al. (2016), who demonstrated in a longitudinal study that children’s numerical acuity on a non-verbal dot-discrimination task increases right around the moment when they become CP-knowers. Drawing on all of these findings, we conclude that the transition to CP-knower status does indeed represent a major conceptual shift, and that the induction of cardinality is related to a broad suite of changes in children’s representation of quantity, including a dramatic and abrupt increase in its precision.

Of course, there are limits to what children learn when they acquire the cardinal principle. Davidson et al. (2012) convincingly demonstrate that CP-knowers do not have stable number meanings ‘as high as they can count,’ contrary to the original suggestions of Carey, Sarnecka and others (Sarnecka & Carey, 2008) that children ‘bootstrap’ the meanings of all of the numbers within their count list when they become CP-knowers. Furthermore, children slowly refine the mappings between numerals and representations of quantities in the ANS (Le Corre & Carey, 2007). The acquisition of cardinality, then, might best be characterized as a ‘limited semantic induction’ but nevertheless a robust conceptual change. The CP-induction is important because it is in this moment that children acquire their first meanings for higher number words beyond the range of parallel individuation; because they recognize in a new way (i.e., in a way not available to SS-knowers) how the structure of the count list imbues numbers with their meanings; and because, as we show here, they exhibit a corresponding shift in the precision of non-verbal representations for those quantities.

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 (Shusterman & Berkowitz, 2011), and they often cannot generate sets of more than 10 objects in Give-N (Shusterman, Cheung, Sarbh, & Taggart, 2015). CP- and SS-knowers’ limited verbal meaning for 20 (perhaps as something vague like a lot) may be related to their worse performance with this set size on the non-verbal task. Lower familiarity, distinctiveness, or acquired meaning of large number words like “twenty” (and like “five” for SS-knowers) therefore appears to be associated with less accurate performance on non-verbal tasks in which these quantities need to be represented in memory. Even after becoming CP-knowers, children clearly need to learn more about and become more familiar with higher number words. An open question is whether further development of number language, beyond cardinality, correlates with more precise numerical representations in non-verbal tasks.

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 (Libertus et al., 2011; Shusterman et al., 2016) and adults (Frank et al., 2008), and extend this conclusion by demonstrating the same result with two engaging tasks with low task demands.

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.