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Humans count to indefinitely large numbers by recycling words from a finite list, and combining them using rules—for example, combining sixty with unit labels to generate sixty‐one, sixty‐two, and so on. Past experimental research has focused on children learning base‐10 systems, and has reported that this rule learning process is highly protracted. This raises the possibility that rules are slow to emerge because they are not needed in order to represent smaller numbers (e.g., up to 20). Here, we investigated this possibility in adult learners by training them on a series of artificial number “languages” that manipulated the availability of rules, by varying the numerical base in each language. We found (1) that the size of a base—for example, base‐2 versus base‐5—had little effect on learning, (2) that learners struggled to acquire multiplicative rules while they learned additive rules more easily, (3) that memory for number words was greater when they were taught as part of a sequential count list, but (4) that learning numbers as part of a rote list may impair the ability to map them to magnitudes. 

The ability to communicate about exact number is critical to many modern human practices spanning science, industry, and politics. Although some early numeral systems used 1-to-1 correspondence (e.g., ‘IIII' to represent 4), most systems provide compact representations via more arbitrary conventions (e.g., ‘7’ and ‘VII'). When people are unable to rely on conventional numerals, however, what strategies do they initially use to communicate number? Across three experiments, participants used pictures to communicate about visual arrays of objects containing 1–16 items, either by producing freehand drawings or combining sets of visual tokens. We analyzed how the pictures they produced varied as a function of communicative need (Experiment 1), spatial regularities in the arrays (Experiment 2), and visual properties of tokens (Experiment 3). In Experiment 1, we found that participants often expressed number in the form of 1-to-1 representations, but sometimes also exploited the configuration of sets. In Experiment 2, this strategy of using configural cues was exaggerated when sets were especially large, and when the cues were predictably correlated with number. Finally, in Experiment 3, participants readily adopted salient numerical features of objects (e.g., four-leaf clover) and generally combined them in a cumulative-additive manner. Taken together, these findings corroborate historical evidence that humans exploit correlates of number in the external environment – such as shape, configural cues, or 1-to-1 correspondence – as the basis for innovating more abstract number representations. 

Humans are unique in their capacity to both represent number exactly and to express these representations symbolically. This correlation has prompted debate regarding whether symbolic number systems are necessary to represent large exact number. Previous work addressing this question in innumerate adults and semi-numerate children has been limited by conflicting results and differing methodologies, and has not yielded a clear answer. We address this debate by adapting methods used with innumerate populations (a “set-matching” task) for 3- to 5-year-old US children at varying stages of symbolic number acquisition. In five studies we find that children’s ability to match sets exactly is related not simply to knowing the meanings of a few number words, but also to understanding how counting is used to generate sets (i.e., the cardinal principle). However, while children were more likely to match sets after acquiring the cardinal principle, they nevertheless demonstrated failures, compatible with the hypothesis that the ability to reason about exact equality emerges sometime later. These findings provide important data on the origin of exact number concepts, and point to knowledge of a counting system, rather than number language in general, as a key ingredient in the ability to reason about large exact number. 

The distinctly human ability to both represent number exactly and develop symbolic number systems has raised the question of whether such number concepts are culturally constructed through symbolic systems. Although previous work with innumerate and semi-numerate groups has provided some evidence that understanding exact equality is related to numeracy, it is possible that previous failures were driven by pragmatic factors, rather than the absence of conceptual knowledge. Here, we test whether such factors affect performance on a test of exact equality in 3- to 5-year-old children by modifying previous methods to draw children’s attention to number. We find no effect of highlighting exact equality, either through framing the task as a “Number” game or as a “Sharing” game. Instead, we replicate previous findings showing a link between numeracy and an understanding of exact equality, strengthening the proposal that exact number concepts are facilitated by the acquisition of symbolic number systems. 

Humans make frequent and powerful use of external symbols to express number exactly, leading some to question whether exact number concepts are only available through the acquisition of symbolic number systems. Although prior work has addressed this longstanding debate on the relationship between language and thought in innumerate populations and seminumerate children, it has frequently produced conflicting results, leaving the origin of exact number concepts unclear. Here, we return to this question by replicating methods previously used to assess exact number knowledge in innumerate groups, such as the Piraha, with a large sample of semi- numerate US toddlers. We replicate previous findings from both innumerate cultures and developmental studies showing that numeracy is linked to the concept of exact number. However, we also find evidence that this knowledge is surprisingly fragile even amongst numerate children, suggesting that numeracy alone does not guarantee a full understanding of exactness.