Children gradually construct spatial representations of temporal events.

English-speaking adults often recruit a "mental timeline" to represent events from left-to-right (LR), but its developmental origins are debated. Here, we test whether preschoolers prefer ordered linear representations of events and whether they prefer culturally conventional directions. English-speaking adults (n = 85) and 3- to 5-year-olds (n = 513; 50% female; ~47% white, ~35% Latinx, ~18% other; tested 2016-2018) were told three-step stories and asked to choose which of two image sequences best illustrated them. We found that 3- and 4-year-olds chose ordered over unordered sequences, but preferences between directions did not emerge until at least age 5. Together, these results show that children conceptualize time linearly early in development but gradually acquire directional preferences (e.g., for LR).

Starting small: exploring the origins of successor function knowledge.

Although most U. S. children can accurately count sets by 4 years of age, many fail to understand the structural analogy between counting and number - that adding 1 to a set corresponds to counting up 1 word in the count list. While children are theorized to establish this Structure Mapping coincident with learning how counting is used to generate sets, they initially have an item-based understanding of this relationship, and can infer that, e.g, adding 1 to "five" is "six", while failing to infer that, e.g., adding 1 to "twenty-five" is "twenty-six" despite being able to recite these numbers when counting aloud. The item-specific nature of children's successes in reasoning about the relationship between changes in cardinality and the count list raises the possibility that such a Structure Mapping emerges later in development, and that this ability does not initially depend on learning to count. We test this hypothesis in two experiments and find evidence that children can perform item-based addition operations before they become competent counters. Even after children learn to count, we find that their ability to perform addition operations remains item-based and restricted to very small numbers, rather than drawing on generalized knowledge of how the count list represents number. We discuss how these early item-based associations between number words and sets might play a role in constructing a generalized Structure Mapping between counting and quantity.

What Counts? Sources of Knowledge in Children's Acquisition of the Successor Function.

Although many U.S. children can count sets by 4 years, it is not until 5½-6 years that they understand how counting relates to number-that is, that adding 1 to a set necessitates counting up one number. This study examined two knowledge sources that 3½- to 6-year-olds (N = 136) may leverage to acquire this "successor function": (a) mastery of productive rules governing count list generation; and (b) training with "+1" math facts. Both productive counting and "+1" math facts were related to understanding that adding 1 to sets entails counting up one number in the count list; however, even children with robust successor knowledge struggled with its arithmetic expression, suggesting they do not generalize the successor function from "+1" math facts.

Do children use language structure to discover the recursive rules of counting?

We test the hypothesis that children acquire knowledge of the successor function - a foundational principle stating that every natural number n has a successor n + 1 - by learning the productive linguistic rules that govern verbal counting. Previous studies report that speakers of languages with less complex count list morphology have greater counting and mathematical knowledge at earlier ages in comparison to speakers of more complex languages (e.g., Miller & Stigler, 1987). Here, we tested whether differences in count list transparency affected children's acquisition of the successor function in three languages with relatively transparent count lists (Cantonese, Slovenian, and English) and two languages with relatively opaque count lists (Hindi and Gujarati). We measured 3.5- to 6.5-year-old children's mastery of their count list's recursive structure with two tasks assessing productive counting, which we then related to a measure of successor function knowledge. While the more opaque languages were associated with lower counting proficiency and successor function task performance in comparison to the more transparent languages, a unique within-language analytic approach revealed a robust relationship between measures of productive counting and successor knowledge in almost every language. We conclude that learning productive rules of counting is a critical step in acquiring knowledge of recursive successor function across languages, and that the timeline for this learning varies as a function of count list transparency.

Children's Intergroup Attitudes: Insights From Iran.

Children generally favor individuals in their own group over others, but it is unclear which dimensions of the out-group affect this bias. This issue was investigated among 7- to 8-year-old and 11- to 12-year-old Iranian children (N = 71). Participants evaluated in-group members and three different out-groups: Iranian children from another school, Arab children, and children from the United States. Children's evaluations closely aligned with the perceived social status of the groups, with Americans viewed as positively as in-group members and Arabs viewed negatively. These patterns were evident on measures of affiliation, trust, and loyalty. These findings, which provide some of the first insights into the social cognition of Iranian children, point to the role of social status in the formation of intergroup attitudes.

Children are sensitive to reputation when giving to both ingroup and outgroup members.

Previous studies establish that reputation concerns play an important role in outgroup giving. However, it is unclear whether the same is true for ingroup giving, which by some accounts tends to be motivated by empathic concerns. To explore this question, we tested the extent to which 5 to 9-year-old children (Study 1: N = 164) and adults (Study 2: N = 80) shared resources with ingroup and outgroup members, either when being watched by an observer (where we expected reputation concerns to be salient) or in private (where we expected no effect of reputation concerns). We also assessed whether children and adults differ in their beliefs about which form of sharing (ingroup or outgroup giving) is nicer. Although we found that both children and adults exhibited an ingroup bias when sharing, there was no evidence in either group that reputation concerns were greater for outgroup members than for ingroup members. We also found that, in contrast to adults, children shared more resources when observed than in private. Additionally, children evaluated ingroup giving as nicer across different sharing scenarios, whereas adults identified outgroup giving as nicer when the two forms of giving were contrasted. These results are the first to suggest that reputational concerns influence children's sharing both with ingroup and outgroup members, and that children differ from adults in their reasoning about which form of group sharing is nicer.

Language-specific numerical estimation in bilingual children.

We tested 5- to 7-year-old bilingual learners of French and English (N = 91) to investigate how language-specific knowledge of verbal numerals affects numerical estimation. Participants made verbal estimates for rapidly presented random dot arrays in each of their two languages. Estimation accuracy differed across children's two languages, an effect that remained when controlling for children's familiarity with number words across their two languages. In addition, children's estimates were equivalently well ordered in their two languages, suggesting that differences in accuracy were due to how children represented the relative distance between number words in each language. Overall, these results suggest that bilingual children have different mappings between their verbal and nonverbal counting systems across their two languages and that those differences in mappings are likely driven by an asymmetry in their knowledge of the structure of the count list across their languages. Implications for bilingual math education are discussed.

Do children's number words begin noisy?

How do children acquire exact meanings for number words like three or forty-seven? In recent years, a lively debate has probed the cognitive systems that support learning, with some arguing that an evolutionarily ancient "approximate number system" drives early number word meanings, and others arguing that learning is supported chiefly by representations of small sets of discrete individuals. This debate has centered around the findings generated by Wynn's (, ) Give-a-Number task, which she used to categorize children into discrete "knower level" stages. Early reports confirmed Wynn's analysis, and took these stages to support the "small sets" hypothesis. However, more recent studies have disputed this analysis, and have argued that Give-a-Number data reveal a strong role for approximate number representations. In the present study, we use previously collected Give-a-Number data to replicate the analyses of these past studies, and to show that differences between past studies are due to assumptions made in analyses, rather than to differences in data themselves. We also show how Give-a-Number data violate the assumptions of parametric tests used in past studies. Based on simple non-parametric tests and model simulations, we conclude that (a) before children learn exact meanings for words like one, two, three, and four, they first acquire noisy preliminary meanings for these words, (b) there is no reliable evidence of preliminary meanings for larger meanings, and (c) Give-a-Number cannot be used to readily identify signatures of the approximate number system.

Differentiating scalar implicature from exclusion inferences in language acquisition.

During acquisition, children must learn both the meanings of words and how to interpret them in context. For example, children must learn the logical semantics of the scalar quantifier some and its pragmatically enriched meaning: 'some but not all'. Some studies have shown that 'scalar implicature' - that some implies 'some but not all' - poses a challenge even to nine-year-olds, while others find success by age three. We asked whether reports of children's successes might be due to the computation of exclusion inferences (like contrast or mutual exclusivity) rather than scalar implicatures. We found that young children (N = 214; ages 4;0-7;11) sometimes compute symmetrical exclusion inferences rather than asymmetric scalar inferences. These data suggest that a stronger burden of evidence is required in studies of implicature; before concluding that children compute implicatures, researchers should first show that children exhibit sensitivity to asymmetric entailment in the task.

The mental timeline is gradually constructed in childhood.

When reasoning about time, English-speaking adults often invoke a "mental timeline" stretching from left to right. Although the direction of the timeline varies across cultures, the tendency to represent time as a line has been argued to be ubiquitous and primitive. On this hypothesis, we might predict that children also spontaneously invoke a spatial timeline when reasoning about time. However, little is known about how and when the mental timeline develops, or to what extent it is variable and malleable in childhood. Here, we used a sticker placement task to test whether preschoolers and kindergarteners spontaneously map temporal events (breakfast, lunch, and dinner) and deictic time words (yesterday, today, tomorrow) onto lines, and to what degree their representations of time are adult-like. We found that, at age 4, preschoolers were able to arrange temporal items in lines with minimal spatial priming. However, unlike kindergarteners and adults, most preschoolers did not represent time as a line spontaneously, in the absence of priming, and did not prefer left-to-right over right-to-left lines. Furthermore, unlike most adults, children of all ages could be easily primed to adopt an unconventional vertical timeline. Our findings suggest that mappings between time and space in children are initially flexible, and become increasingly automatic and conventionalized in the early school years.

To infinity and beyond: Children generalize the successor function to all possible numbers years after learning to count.

Recent accounts of number word learning posit that when children learn to accurately count sets (i.e., become "cardinal principle" or "CP" knowers), they have a conceptual insight about how the count list implements the successor function - i.e., that every natural number n has a successor defined as n+1 (Carey, 2004, 2009; Sarnecka & Carey, 2008). However, recent studies suggest that knowledge of the successor function emerges sometime after children learn to accurately count, though it remains unknown when this occurs, and what causes this developmental transition. We tested knowledge of the successor function in 100 children aged 4 through 7 and asked how age and counting ability are related to: (1) children's ability to infer the successors of all numbers in their count list and (2) knowledge that all numbers have a successor. We found that children do not acquire these two facets of the successor function until they are about 5½ or 6years of age - roughly 2years after they learn to accurately count sets and become CP-knowers. These findings show that acquisition of the successor function is highly protracted, providing the strongest evidence yet that it cannot drive the cardinal principle induction. We suggest that counting experience, as well as knowledge of recursive counting structures, may instead drive the learning of the successor function.

Today is tomorrow's yesterday: Children's acquisition of deictic time words.

Deictic time words like "yesterday" and "tomorrow" pose a challenge to children not only because they are abstract, and label periods in time, but also because their denotations vary according to the time at which they are uttered: Monday's "tomorrow" is different than Thursday's. Although children produce these words as early as age 2 or 3, they do not use them in adult-like ways for several subsequent years. Here, we explored whether children have partial but systematic meanings for these words during the long delay before adult-like usage. We asked 3- to 8-year-olds to represent these words on a bidirectional, left-to-right timeline that extended from the past (infancy) to the future (adulthood). This method allowed us to independently probe knowledge of these words' deictic status (e.g., "yesterday" is in the past), relative ordering (e.g., "last week" was before "yesterday"), and remoteness from the present (e.g., "last week" was about 7 times longer ago than "yesterday"). We found that adult-like knowledge of deictic status and order emerge in synchrony, between ages 4 and 6, but that knowledge of remoteness emerges later, after age 7. Our findings suggest that children's early use of deictic time words is not random, but instead reflects the gradual construction of a structured lexical domain.

Language, procedures, and the non-perceptual origin of number word meanings.

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.

Discourse bootstrapping: preschoolers use linguistic discourse to learn new words.

When children acquire language, they often learn words in the absence of direct instruction (e.g. 'This is a ball!') or even social cues to reference (e.g. eye gaze, pointing). However, there are few accounts of how children do this, especially in cases where the referent of a new word is ambiguous. Across two experiments, we test whether preschoolers (2- to 4-year-olds; n = 239) can learn new words by inferring the referent of a new word from the surrounding linguistic discourse. Across two experiments, we show that children as young as 2 can learn a new word from the linguistic discourse in which it appears. This suggests that children use the linguistic discourse in which a word appears to learn new words.

Do attitudes toward societal structure predict beliefs about free will and achievement? Evidence from the Indian caste system.

Intuitive theories about the malleability of intellectual ability affect our motivation and achievement in life. But how are such theories shaped by the culture in which an individual is raised? We addressed this question by exploring how Indian children's and adults' attitudes toward the Hindu caste system--and its deterministic worldview--are related to differences in their intuitive theories. Strikingly, we found that, beginning at least in middle school and continuing into adulthood, individuals who placed more importance on caste were more likely to adopt deterministic intuitive theories. We also found a developmental change in the scope of this relationship, such that in children, caste attitudes were linked only to abstract beliefs about personal freedom, but that by adulthood, caste attitudes were also linked to beliefs about the potential achievement of members of different castes, personal intellectual ability, and personality attributes. These results are the first to directly relate the societal structure in which a person is raised to the specific intuitive theories they adopt.

Learning Mathematics in a Visuospatial Format: A Randomized, Controlled Trial of Mental Abacus Instruction.

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 outperformed 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.

Children's evaluation of public and private generosity and its relation to behavior: Evidence from China.

This research examined children's evaluation of public and private prosocial giving and whether such evaluation would predict actual behavior. We tested children between 6 and 12 years old (N=192) in China, where children are socialized not to call positive attention to themselves. In Study 1, a significant age-related change was found; younger children evaluated public acts of prosocial giving more favorably than private acts, whereas older children showed the opposite pattern. Study 2 not only replicated the findings of Study 1 but also showed that children's evaluation of public versus private giving predicted their actual behavior in communicating about their own prosocial giving. These findings are the first to show that age-related changes in children's understanding of generosity predict reputation management behavior.

Grammatical morphology as a source of early number word meanings.

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.

Learning the language of time: Children's acquisition of duration words.

Children use time words like minute and hour early in development, but take years to acquire their precise meanings. Here we investigate whether children assign meaning to these early usages, and if so, how. To do this, we test their interpretation of seven time words: second, minute, hour, day, week, month, and year. We find that preschoolers infer the orderings of time words (e.g., hour>minute), but have little to no knowledge of the absolute durations they encode. Knowledge of absolute duration is learned much later in development - many years after children first start using time words in speech - and in many children does not emerge until they have acquired formal definitions for the words. We conclude that associating words with the perception of duration does not come naturally to children, and that early intuitive meanings of time words are instead rooted in relative orderings, which children may infer from their use in speech.

Why is number word learning hard? Evidence from bilingual learners.

Young children typically take between 18 months and 2 years to learn the meanings of number words. In the present study, we investigated this developmental trajectory in bilingual preschoolers to examine the relative contributions of two factors in number word learning: (1) the construction of numerical concepts, and (2) the mapping of language specific words onto these concepts. We found that children learn the meanings of small number words (i.e., one, two, and three) independently in each language, indicating that observed delays in learning these words are attributable to difficulties in mapping words to concepts. In contrast, children generally learned to accurately count larger sets (i.e., five or greater) simultaneously in their two languages, suggesting that the difficulty in learning to count is not tied to a specific language. We also replicated previous studies that found that children learn the counting procedure before they learn its logic - i.e., that for any natural number, n, the successor of n in the count list denotes the cardinality n+1. Consistent with past studies, we found that children's knowledge of successors is first acquired incrementally. In bilinguals, we found that this knowledge exhibits item-specific transfer between languages, suggesting that the logic of the positive integers may not be stored in a language-specific format. We conclude that delays in learning the meanings of small number words are mainly due to language-specific processes of mapping words to concepts, whereas the logic and procedures of counting appear to be learned in a format that is independent of a particular language and thus transfers rapidly from one language to the other in development.