Although children often begin to count at around age 2, they don't initially understand what number words mean, or how they are related to one another. For example, young children often begin by reciting numbers up to 5 or 10 as part of a routine, much like singing the ABCs (Fuson, 1991), although early count routines can be unstable and contain consistent inaccuracies (see Baroody & Price, 1983 for review). Crucially, these children lack an understanding of what number words mean or how they represent exact cardinalities (see Baroody & Price, 1983; Wynn, 1992). However, children ultimately learn not only the meanings of the number words in their list, but also the rules that govern the construction of the count list, including the decade+unit rule, whereby the words “one” through “nine” can be composed with decade labels like “twenty” and “thirty” to create larger numbers (e.g., Chu et al., 2020; Fuson, 1988). Understanding how to compose numbers using rules allows children to count to large numbers and to reason about numbers that they have never before encountered – such an understanding of the pattern and structure of the number system is a potentially powerful aspect of early numerical competence (Mulligan & Mitchelmore, 2009; Mulligan, Oslington, & English, 2020). In the present study, we investigated emerging counters’ understanding of how the rules of counting generalize to all numbers, including numbers that are entirely novel to them.

Substantial work has demonstrated that learning the rule structures of the number system is a critical step in early numerical competence (Moss, Bruce, & Bobis, 2015), and that children use these structures to organize their understanding of the mathematical world (Mulligan & Mitchelmore, 2009). Studies of children’s counting behaviors – which we review below – provide two specific sources of evidence that children begin to learn these rules during the preschool and early elementary school years. One source of evidence comes from children’s early counting errors. Children who have memorized their count routine should make errors that are randomly distributed across the count list, with errors becoming more likely for larger numbers and less likely for small ones. In contrast, children who use rules to combine decade labels like “twenty” and “thirty” with the units “one” through “nine” should make errors mostly at decade transitions, since in many languages (including English), decade labels like fifty are not transparently related to their component parts such as “five” or “ten” and therefore must be memorized. Compatible with the use of rules, English-speaking preschoolers are especially likely to make counting errors in the irregular teens (Baroody & Price, 1983; Fuson, 1988; Johnson et al., 2019; McMillan et al., 2024) and at decade transitions, often stopping at 29, 39, or 49, rather than stopping mid-decade at 24, 35, or 47 (Chu et al., 2020; Fuson, 1991; Gould, 2017; Siegler & Robinson, 1982; Wright, 1994). Further, children who stop at decade transitions can often continue counting if they are given the decade label (e.g., “actually, the number that comes after 39 is 40…can you keep counting?”; Chu et al., 2020). And, one longitudinal study of early counting showed that preschoolers had substantial knowledge of number words outside their count range – for example, one child, Daniela, could only count to 13 before making an error, but was able to recover and generate the numbers 16-25, showing knowledge of the structure of the count list beyond her range of comfortable production (McMillan et al., 2024). Taken together, these data are compatible with the hypothesis that these children have access to a rule for adding the units 1-9 to decades. Notably, these abilities are less often observed in children learning languages like Hindi and Gujarati, where it is difficult to notice the rules that govern counting because the counting systems are much less transparent and contain many exceptions to rules (Schneider et al., 2020). Meanwhile, children learning languages that are more transparent than English, like Cantonese, often count higher at earlier ages (Miller & Stigler, 1987; Miller et al., 2000; Miller et al., 1995).

A second source of evidence that children develop rule-governed knowledge of counting in the first years of life comes from their intuitions about infinity (Gelman, 1980). According to some accounts, children might come to believe that numbers are infinite by learning that numbers can be generated by rules, and that these rules can be applied iteratively, over and over, without end. Compatible with this, between the ages of 4 and 6, many children begin to claim that it’s possible to add 1 to any number, and that numbers never end (Cheung et al., 2017; Hartnett & Gelman, 1998). Further, children’s beliefs about infinity have been shown to be related to how high they can count (Cheung et al., 2017), whether they can continue counting up from decade transition errors (Chu et al., 2020), and how successful they are at generating the next number in the count list when given a prompt (e.g., “57….what comes next?”; Sullivan et al., 2023). Together, these studies suggest the emergence of a suite of beliefs compatible with a rule-governed understanding of numbers.

Data from studies of counting errors and infinity beliefs are also compatible with the idea that children use rules to govern counting prior to formal schooling (see also Schneider et al., 2021). However, two considerations limit the strength of these conclusions. First, because previous studies of children’s counting abilities are generally limited to testing relatively familiar numbers, children’s behaviors are often compatible with both rules and rote memorization. For example, the finding that Cantonese-speaking children count higher than English-speaking children (Miller & Stigler, 1987; Miller et al., 2000; Miller et al., 1995) could simply reflect the possibility that Chinese children receive more, or different, numerical training than some of their English-speaking counterparts (e.g., Miller et al., 2005). And, although the clustering of counting errors at decade transitions is compatible with the use of rules, knowledge of rules can only be inferred from these errors, because counting errors do not provide direct evidence that children can apply a rule to novel examples that fall outside their training set. Similarly, although children’s beliefs about the infinity of numbers are often correlated with their counting ability (Cheung et al., 2017; Chu et al., 2020), such correlations could emerge even in the absence of the use of rules to support counting – for example, some children may simply have higher exposure both to counting and to ideas about infinity.

While previous studies are consistent with the idea that early counters have a rule-governed understanding of the count list, one strong piece of evidence for this would come from children’s ability to apply the rules that structure their native language’s count list to entirely unfamiliar contexts. The goal of the present study was therefore to test whether children could generate successors to novel numbers, and – if so – how their ability to do so relates to their understanding of counting in their native language. Specifically, we asked children to count as high as they could, and recorded their highest count prior to making an error, and their highest count when given counting feedback. This allowed us to measure their familiarity with and mastery of the English count list and their ability to recover after making counting errors. As with previous work (Cheung et al., 2017; Chu et al., 2020; Schneider et al., 2020), we reasoned these metrics would allow us to quantify not only children’s familiarity with the English count list, but their ability to combine decades and units. Next, we measured children’s ability to generate successors outside the context of the count routine. We presented children with familiar number words like 7 and 24 and asked them “What comes next?”. Critically, our final measure was the Novel Next Number task, in which we introduced children to two novel decade terms (mobi, blicky), and then asked them to generate successors to numbers containing those decade terms (e.g., “what comes after blicky-seven?”, “what comes after two hundred mobi six?”). We reasoned that if children’s ability to count in English is driven by knowledge of rules, then children’s use of rule-governed counting in English should be related to their ability to generate successors for novel number words like blicky and mobi.

In the present study, we asked four main questions: (1) Can young children generate successors for novel numbers that contain novel decade terms? (2) Do they do so as successfully as they do for English? (3) what are the predictors of their ability to generate successors to English and Novel numbers? and (4) Are their errors when generating successors consistent with the use of particular decade-unit rules for generating successors?

Understanding the structure of the number system is critically important for both early numeracy and for academic math success (Moss et al., 2015; Mulligan & Mitchelmore, 2009). We tested whether preschoolers could generate successors to novel numbers and whether their ability to do so was related to their knowledge of counting in their native language of English. To do this, we measured children’s ability to count in English, as well as their ability to generate successors to both English number words and number words that contained novel decade terms. Children were often able to generate successors for English number words both inside and outside their counting range and also for numbers containing the novel decade terms mobi and blicky. Because children had not been introduced to the terms blicky and mobi prior to the experiment, such success could not have been due to memorization, and could only have arisen by applying the rules that structure the English count list to this novel context. In other words, their ability to generate successors to novel decade terms was likely due to their ability to use the decade+unit rule that structures the English count list (Chu et al., 2020), and their knowledge of the ordering of the unit terms in English (i.e., that 1 comes before 2 comes before 3; Fuson, 1988). Our data suggest that children are able to apply their understanding of how the English count system is structured to novel contexts, using the rules that govern English counting to generate novel successors. Given that fewer than 10% of our participants could count to 100 (and the majority of participants couldn’t even count to 20 without making errors), these data suggest that the ability to generate successors for novel numbers can emerge prior to mastering the English count list.

As noted above, children were able to generate successors both for novel terms like blicky 7 and for numbers outside their English counting range. These data suggest that children can use their understanding of the rules for composing decades and units to generate successors for numbers far outside their counting range. The data from the English Next Number task are consistent with previous reports that have demonstrated that children have some knowledge of the number words outside of their productive count range (e.g., Barth et al., 2009; Chu et al., 2020; McMillan et al., 2024). We add to this that children are also able to generate novel numbers beyond their counting range. More generally, these data address the theoretical claim, made in past studies, that behaviors ostensibly compatible with the use of rules – such as errors in counting (e.g., Chu et al., 2020; Gould, 2017; Miller et al., 1995; Schneider et al., 2020), may indeed reflect a generalized induction regarding the structure of counting, including rules such as the decade+unit rule. These data therefore contribute to a larger literature arguing that the counting system becomes a critical organizing structure for mathematical reasoning early in development (Baroody & Price, 1983; Mulligan & Mitchelmore, 2009; Sullivan et al., 2023).

Drawing on a tradition in the counting literature of using children’s counting errors to make inferences about their representations (e.g., Fuson, 1991; McMillan et al., 2024; Miller et al., 1995; Gould, 2017), we analyzed not only children’s overall performance, but also the types of errors that they made. To do this, we coded children’s responses for whether they provided the correct numeral in the 1s place, the 10s place, and when relevant, the 100s place. For both English and the novel count systems, errors were most prevalent in the 100s place, consistent with the fact that our sample contained relatively newer counters. Indeed, given that less than 10% of our sample could spontaneously count to 100, it’s perhaps more striking that children generated the correct hundreds place over 35% of the time on our next number tasks. However, for both the English and Novel Next number tasks, children’s failures were equally common in the tens place and in the ones place. In other words, children’s errors were not random, but rather reflected a failure to combine correct decade information with (correct) unit information. Consistent with this, performance was exceptionally low when children had to generate successors to mobi or blicky (with no ones place added). This is consistent with the idea that many preschoolers understand the rules underlying the generation of successors, but sometimes fail to correctly combine decades and units.

One of our goals was to understand the predictors of children’s ability to generate successors both in English and in a novel count system. The strongest predictor of children’s ability to provide successors on our Novel Next Number task was their performance on the English Next Number task, and this was true even when controlling for counting ability and age. These data suggest that while higher levels of fluency with the English count list confer a benefit for generating successors, the ability to generate successors in English provided a unique advantage for generating successors to items from a novel count list.

Our data suggest that early counters are sometimes able to generate successors to (a) novel number terms and to (b) numbers outside their productive count range. These data suggest that even some early counters may use rules to generate number words (e.g., by combining units and decades). However, children’s performance for unfamiliar numbers was not perfect, they frequently made errors, and performed better for familiar numbers overall. In addition, while children were generally good at generating successors to numbers like “blicky-7”, they struggled to generate successors to bare decade terms (e.g., “blicky…what comes next?”; Figure 1). These data suggest that while children may be able to combine units and decades when generating novel numbers, they benefit from being reminded how to do so -- after all, when the experimenter says something like “mobi-5, what comes next?”, the experimenter has modeled one possible rule for combining decades and units.

The present data are not without their limitations. First, we only tested English-speaking children. As noted in the Introduction, English is a moderately transparent count language, meaning that it is relatively easy to identify the pattern underlying the combination of decades and units. The present data would predict cross-linguistic variability in performance on Novel Next Number tasks, such that learners of languages where the structure of the count list is more opaque (e.g., Hindi) may be slower to abstract a rule for generating successors, while learners of transparent count systems may have an easier time learning rules for generating successors. Future research should measure performance on Novel Next Number tasks across language groups in order to assess the causal role of the transparency of count list structure in shaping the ease with which children abstract rules for combining decades and units. In addition, our novel count systems were base-ten systems modeled closely on the English count system; the present data do not speak to children’s general ability to generate successors across different count systems, or their ability to attach numerical meaning to the number words they generated -- a critical component of early numeracy (e.g., Mulligan & Mitchelmore, 2013). Instead, our study focused more specifically on children's ability to apply the rules used for generating successors in English to novel contexts. More generally, there is good evidence to suggest that children’s performance on counting tasks depends on the nature of the task (e.g., Johnson et al., 2019), and therefore the possibility that some elements of children’s performance were impacted by the nature of our task remains.

In addition, while there was a strong relation between the ability to generate successors in English and the ability to generate successors in a novel count system, the present data do not demonstrate the amount of data needed to infer the abstract rules for generating successors. Because most of our participants were newer counters with very limited mastery of the English count, it is clear that total mastery of the English count system through 100 is not required for learning some abstract rules for generating successors. That said, our current participants were neither at floor nor at ceiling, and thus future research would benefit from more explicitly mapping the relation between mastery of one’s native count list and their understanding of the rules for generating successors, perhaps adopting longitudinal methods (McMillan et al., 2024).

The present study demonstrated that novice counters can successfully generate successors to novel decade terms. Their ability to do so was predicted by their mastery of the English count list, and due to the use of a novel count system, could not be attributed to memorization alone. Instead, these data provide some of the first evidence that emerging counters have access to rules for generating successors, and that these rules are robust enough to apply to novel decade terms. Children’s ability to generate successors was strikingly similar for English numbers and novel numbers, consistent with the possibility that children routinely use rules to generate successors when counting (as opposed to simply memorizing the count list).