Numerate adults know that when two sets are equal, they should be labeled by the same number word. We explored the development of this principle—sometimes called 'cardinal extension'—and how it relates to children’s other numerical abilities. Experiment 1 revealed that 2- to 5-year-old children who could accurately count large sets often inferred that two equal sets should be labeled with the same number word, unlike children who could not accurately count large sets. However, not all counters made this inference, suggesting that learning to construct and label large sets may be a necessary but not sufficient step in learning how numbers represent exact quantities. Experiment 2 found that children who extended labels to equal sets were not actually sensitive to exact equality and that they often assigned two sets the same label when they were approximately equal, but differed by just one item (violating one-to-one correspondence). These results suggest a gradual, stagelike, process in which children learn to accurately count, learn to extend labels to perceptually similar sets, and then eventually restrict cardinal extension to sets that are exactly equal. (PsycInfo Database Record (c) 2025 APA, all rights reserved)

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.

How do children learn to connect expressions (e.g., “that red apple”) to the real-world objects they refer to? The dominant view in developmental psychology is that children rely on descriptive information, “red” and “apple.” In contrast, linguistic theories of the adult language attribute primacy to grammatical elements: words such as “that” or “another” first establish the status of potential referents within the discourse context (old or new) before descriptions can factor in. These theories predict that reference can succeed even when the description does not match the referent. We explored this novel prediction in adults and children. Over four experiments, we found that (a) adults relied on the articles to identify the referent, even when the description did not fit, consistent with grammar-first accounts; (b) consistent with description-first accounts, and unlike adults, 3- to 5-year-old children prioritized the descriptions provided by nouns and adjectives, despite being sensitive to grammatical information. This suggests that children connect expressions to referents differently from adults. © 2024 American Psychological Association

De Neys is right to criticize the exclusivity assumption in dual-process theories, but he misses the original sin underlying this assumption, which his working model continues to share. Conflict paradigms, in which experimenters measure how one cognitive process interferes (or does not interfere) with another, license few inferences about how the interfered-with process works on its own.

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 Give-a-Number task has become a gold standard of children's number word comprehension in developmental psychology. Recently, researchers have begun to use the task as a predictor of other developmental milestones. This raises the question of how reliable the task is, since test-retest reliability of any measure places an upper bound on the size of reliable correlations that can be found between it and other measures. In Experiment 1, we presented 81 2-to 5-year-old children with Wynn (1992) titrated version of the Give-a-Number task twice within a single session. We found that the reliability of this version of the task was high overall, but varied importantly across different assigned knower levels, and was very low for some knower levels. In Experiment 2, we assessed the test-retest reliability of the non-titrated version of the Give-a-Number task with another group of 81 children and found a similar pattern of results. Finally, in Experiment 3, we asked whether the two versions of Give-a-Number generated different knower levels within-subjects, by testing 75 children with both tasks. Also, we asked how both tasks relate to another commonly used test of number knowledge, the What's-On-This-Card task. We found that overall, the titrated and non-titrated versions of Give-a-Number yielded similar knower levels, though the non-titrated version was slightly more conservative than the titrated version, which produced modestly higher knower levels. Neither was more closely related to What's-On-ThisCard than the other. We discuss the theoretical and practical implications of these results.