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Perceptual representations of objects and approximate magnitudes are often invoked as building blocks that children combine to acquire the positive integers. Systems of numerical perception are either assumed to contain the logical foundations of arithmetic innately, or to supply the basis for their induction. I propose an alternative to this framework, and argue that the integers are not learned from perceptual systems, but arise to explain perception. Using cross-linguistic and developmental data, I show that small (~1–4) and large (~5+) numbers arise both historically and in individual children via distinct mechanisms, constituting independent learning problems, neither of which begins with perceptual building blocks. Children first learn small numbers using the same logic that supports other linguistic number marking (e.g. singular/plural). Years later, they infer the logic of counting from the relations between large number words and their roles in blind counting procedures, only incidentally associating number words with approximate magnitudes.

How do children map number words to the numerical magnitudes they represent? Recent work in adults has shown that two distinct mechanisms—structure mapping and associative mapping—connect number words to nonlinguistic numerical representations (Sullivan & Barner, 2012). This study investigated the development of number word mappings, and the roles of inference and association in children's estimation. Fifty-eight 5- to 7-year-olds participated, and results showed that at both ages, children possess strong item-based associative mappings for numbers up to around six, but rely primarily on structure mapping—an inferential process—for larger quantities. These findings suggest that children rely primarily on an inferential mechanism to construct and deploy mappings between number words and large approximate magnitudes.

How does cross-linguistic variation in linguistic structure affect children’s acquisition of early number word meanings? We tested this question by investigating number word learning in two unrelated languages that feature a tripartite singular-dual-plural distinction: Slovenian and Saudi Arabic. We found that learning dual morphology affects children’s acquisition of the number word two in both languages, relative to English. Children who knew the meaning of two were surprisingly frequent in the dual languages, relative to English. Furthermore, Slovenian children were faster to learn two than children learning English, despite being less-competent counters. Finally, in both Slovenian and Saudi Arabic, comprehension of the dual was correlated with knowledge of two and higher number words.

How does linguistic structure affect children's acquisition of early number word meanings? Previous studies have tested this question by comparing how children learning languages with different grammatical representations of number learn the meanings of labels for small numbers, like 1, 2, and 3. For example, children who acquire a language with singular-plural marking, like English, are faster to learn the word for 1 than children learning a language that lacks the singular-plural distinction, perhaps because the word for 1 is always used in singular contexts, highlighting its meaning. These studies are problematic, however, because reported differences in number word learning may be due to unmeasured cross-cultural differences rather than specific linguistic differences. To address this problem, we investigated number word learning in four groups of children from a single culture who spoke different dialects of the same language that differed chiefly with respect to how they grammatically mark number. We found that learning a dialect which features \"dual\" morphology (marking of pairs) accelerated children's acquisition of the number word two relative to learning a \"non-dual\" dialect of the same language.

Children learn their first number words gradually over the course of many months, which is surprising given their ability to discriminate small numerosities. One potential explanation for this is that children are sensitive to the numerical features of stimuli, but don’t consider exact cardinality as a primary hypothesis for novel word meanings. To test this, we trained 144 children on a number word they hadn’t yet learned, and contrasted this with a condition in which they were merely required to attend to number to identify the word’s referent, without encoding number as its meaning. In the first condition, children were trained to find a “giraffe with three spots.” In the second condition, children were instead trained to find “Mr. Three”, which also named a giraffe with three spots. In both conditions, children had to attend to number to identify the target giraffe, but, because proper nouns refer to individuals rather than their properties, the second condition did not require children to encode number as the meaning of the expression. We found that children were significantly better at identifying the giraffe when it had been labeled with the proper noun than with the number word. This finding contrasted with a second experiment involving color words, in which children ( = 56) were equally successful with a proper noun (“Mr. Purple”) and an adjective (“the giraffe with purple spots”). Together, these findings suggest that, for number, but not for color, children’s difficulty acquiring new words cannot be solely attributed to problems with attention or perception, but instead may be due to difficulty selecting the correct meaning from their hypothesis space for learning unknown words.

How do children form beliefs about the infinity of space, time, and number? We asked whether children held similar beliefs about infinity across domains, and whether beliefs in infinity for domains like space and time might be scaffolded upon numerical knowledge (e.g., knowledge successors within the count list). To test these questions, 112 U.S. children (aged 4;0–7;11) completed an interview regarding their beliefs about infinite space, time, and number. We also measured their knowledge of counting, and other factors that might impact performance on linguistic assessments of infinity belief (e.g., working memory, ability to respond to hypothetical questions). We found that beliefs about infinity were very high across all three domains, suggesting that infinity beliefs may arise early in development for space, time, and number. Second, we found that—across all three domains—children were more likely to believe that it is always possible to add a unit than to believe that the domain is endless. Finally, we found that understanding the rules underlying counting predicted children’s belief that it is always possible to add 1 to any number, but did not predict any of the other elements of infinity belief.

We review advances in the experimental study of the mass-count distinction and highlight problems that have emerged. First, we lay out what we see to be the scientific enterprise of studying the syntax and semantics of the mass-count distinction, and the assumptions we believe must be made if additional progress is to occur, especially as the empirical facts continue to grow in number and complexity. Second, we discuss the new landscape of cross-linguistic results that has been created by widespread use of the quantity judgment task, and what these results tell us about the nature of the mass-count distinction. Finally, we discuss the relationship between the mass-count distinction and non-linguistic cognition, and in particular the object-substance distinction.  

Conditional statements often have two interpretations. For instance, the statement, “If you mow the lawn, you will receive $5”, might be understood to mean that mowing the lawn is just one possible way to earn $5 or, more strongly, that mowing the lawn is the only way one can receive $5 – an interpretation sometimes called Conditional Perfection. We investigated how people arrive at “perfected” interpretations of conditional statements: whether they initially consider a statement's literal meaning and then perfect it or begin with a perfected interpretation and revert to the weaker meaning only when necessary. Reaction time data from Experiment 1 supports the latter sequence, as evidenced by the longer time required to arrive at literal interpretations than perfected ones. Additionally, in Experiment 2, we found that participants under cognitive load were more likely to perfect conditional statements relative to participants not under load, again suggesting that people begin with a perfected meaning that is optionally canceled with effort.

Four experiments investigated 4-year-olds' understanding of adjective—noun compositionality and their sensitivity to statistics when interpreting scalar adjectives. In Experiments 1 and 2, children selected tall and short items from 9 novel objects called pimwits (1—9 in. in height) or from this array plus 4 taller or shorter distractor objects of the same kind. Changing the height distributions of the sets shifted children's tall and short judgments. However, when distractors differed in name and surface features from targets, in Experiment 3, judgments did not shift. In Experiment 4, dissimilar distractors did affect judgments when they received the same name as targets. It is concluded that 4-year-olds deploy a compositional semantics that is sensitive to statistics and mediated by linguistic labels.

Mental abacus (MA) is a technique of performing fast, accurate arithmetic using a mental image of an abacus; experts exhibit astonishing calculation abilities. Over 3 years, 204 elementary school students (age range at outset: 5–7 years old) participated in a randomized, controlled trial to test whether MA expertise (a) can be acquired in standard classroom settings, (b) improves students' mathematical abilities (beyond standard math curricula), and (c) is related to changes in basic cognitive capacities like working memory. MA students out-performed controls on arithmetic tasks, suggesting that MA expertise can be achieved by children in standard classrooms. MA training did not alter basic cognitive abilities; instead, differences in spatial working memory at the beginning of the study mediated MA learning.