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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.

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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.

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Children often display non-adult-like behaviors when reasoning with quantifiers and logical connectives in natural language. A classic example of this is the symmetrical interpretation of universally quantified statements like "Every girl is riding an elephant", which children often reject as false when they are used to describe a scene with, e.g., three girls each riding an elephant and a fourth elephant without a rider. We present evidence that children's understanding of these sentences is not attributable to syntactic, semantic, or general processing limitations. Instead, in two experiments, we argue that children's behavior stems primarily from difficulty in correctly identifying the speaker's intended "question under discussion", and that when this question is made contextually unambiguous, children's judgments are almost completely adultlike.

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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).

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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-This-Card" than the other. We discuss the theoretical and practical implications of these results.

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Whether estimating the size of a crowd or rating a restaurant on a five-star scale, humans frequently navigate between subjective sensory experiences and shared formal systems. Here we ask how people manage this in the case of estimating number. We present participants with arrays of dots and ask them to report how many dots there are. Our results produce two novel findings. First, people's estimates are best fit by a bilinear function in log space, rather than the traditional power law described in previous literature. Second, we find that people's estimates do not have a stable coefficient of variation at higher magnitudes, and that the likely cause of this is a "drift" in people's estimate calibration over many trials which has not previously been identified. Building on these results, we present a model of the mapping function from subjective numerosity to symbolic number that relies primarily on a constrained set of previous estimates and familiar numerosities, rather than the robust internal scale used in existing models. Our model is able to generate an accurate mapping with limited data and reproduce notable aspects of estimation seen in our experimental results. This suggests that human number estimation, and perhaps other domains in which we must navigate between subjective representations and formal systems, is governed by a relatively simple decision process that primarily seeks to maintain consistency with previous estimates. (PsycInfo Database Record (c) 2022 APA, all rights reserved).

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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.

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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.

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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.

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Number words allow us to describe exact quantities like sixty-three and (exactly) one. How do we derive exact interpretations? By some views, these words are lexically exact, and are therefore unlike other grammatical forms in language. Other theories, however, argue that numbers are not special and that their exact interpretation arises from pragmatic enrichment, rather than lexically. For example, the word one may gain its exact interpretation because the presence of the immediate successor two licenses the pragmatic inference that one implies "one, and not two". To investigate the possible role of pragmatic enrichment in the development of exact representations, we looked outside the test case of number to grammatical morphological markers of quantity. In particular, we asked whether children can derive an exact interpretation of singular noun phrases (e.g., "a button") when their language features an immediate "successor" that encodes sets of two. To do this, we used a series of tasks to compare English-speaking children who have only singular and plural morphology to Slovenian-speaking children who have singular and plural forms, but also dual morphology, that is used when describing sets of two. Replicating previous work, we found that English-speaking preschoolers failed to enrich their interpretation of the singular and did not treat it as exact. New to the present study, we found that 4- and 5-year-old Slovenian-speakers who comprehended the dual treated the singular form as exact, while younger Slovenian children who were still learning the dual did not, providing evidence that young children may derive exact meanings pragmatically.

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By around the age of 5½, many children in the United States judge that numbers never end, and that it is always possible to add 1 to a set. These same children also generally perform well when asked to label the quantity of a set after one object is added (e.g., judging that a set labeled "five" should now be "six"). These findings suggest that children have implicit knowledge of the "successor function": Every natural number, n, has a successor, n + 1. Here, we explored how children discover this recursive function, and whether it might be related to discovering productive morphological rules that govern language-specific counting routines (e.g., the rules in English that represent base-10 structure). We tested 4- and 5-year-old children's knowledge of counting with three tasks, which we then related to (a) children's belief that 1 can always be added to any number (the successor function) and (b) their belief that numbers never end (infinity). Children who exhibited knowledge of a productive counting rule were significantly more likely to believe that numbers are infinite (i.e., there is no largest number), though such counting knowledge was not directly linked to knowledge of the successor function, per se. Also, our findings suggest that children as young as 4 years of age are able to implement rules defined over their verbal count list to generate number words beyond their spontaneous counting range, an insight which may support reasoning over their acquired verbal count sequence to infer that numbers never end.

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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.

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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.

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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.

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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.

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Do children learn number words by associating them with perceptual magnitudes? Recent studies argue that approximate numerical magnitudes play a foundational role in the development of integer concepts. Against this, we argue that approximate number representations fail both empirically and in principle to provide the content required of integer concepts. Instead, we suggest that children's understanding of integer concepts proceeds in two phases. In the first phase, children learn small exact number word meanings by associating words with small sets. In the second phase, children learn the meanings of larger number words by mastering the logic of exact counting algorithms, which implement the successor function and Hume's principle (that one-to-one correspondence guarantees exact equality). In neither phase do approximate number representations play a foundational role.

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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.

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To interpret an interlocutor's use of a novel word (e.g., "give me the papaya"), children typically exclude referents that they already have labels for (like an "apple"), and expect the word to refer to something they do not have a label for (like the papaya). The goal of the present studies was to test whether such mutual exclusivity inferences require children to reason about the words their interlocutors know and could have chosen to say: e.g., If she had wanted the "apple" she would have asked for it (since she knows the word "apple"), so she must want the papaya. Across four studies, we document that both children and adults will make mutual exclusivity inferences even when they believe that their interlocutor does not share their knowledge of relevant, alternative words, suggesting that such inferences do not require reasoning about an interlocutor's epistemic states. Instead, our findings suggest that children's own knowledge of an object's label, together with their belief that this is the conventional label for the object in their language, and that this convention applies to their interlocutor, is sufficient to support their mutual exclusivity inferences. Additionally, and contrary to the claims of previous studies that have used mutual exclusivity as a proxy for children's beliefs that others share their knowledge, we found that children - especially those with stronger theory of mind ability - are quite conservative about attributing their knowledge of object labels to others. Together, our findings hold implications for theories of word learning, and for how children learn about the scope of shared conventional knowledge.

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Word learning depends critically on the use of linguistic context to constrain the likely meanings of words. However, the mechanisms by which children infer word meaning from linguistic context are still poorly understood. In this study, we asked whether adults (n = 58) and 2- to 6-year-old children (n = 180) use discourse coherence relations (i.e., the meaningful relationships between elements within a discourse) to constrain their interpretation of novel words. Specifically, we showed participants videos of novel animals exchanging objects. These videos were accompanied by a linguistic description of the events in which we manipulated a single word within a sentence (and vs. because) in order to alter the causal and temporal relations between the events in the discourse (e.g., "One animal handed the baby to the other animal [and/because] the baby started crying in the talfa's arms"). We then asked participants which animal (the giver or the receiver) was the referent of the novel word. Across two experiments, we found evidence that young children used the causal and temporal relations in each discourse to constrain their interpretations of novel words.

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Do children understand how different numbers are related before they associate them with specific cardinalities? We explored how children rely on two abstract relations - contrast and entailment - to reason about the meanings of 'unknown' number words. Previous studies argue that, because children give variable amounts when asked to give an unknown number, all unknown numbers begin with an existential meaning akin to some. In Experiment 1, we tested an alternative hypothesis, that because numbers belong to a scale of contrasting alternatives, children assign them a meaning distinct from some. In the "Don't Give-a-Number task", children were shown three kinds of fruit (apples, bananas, strawberries), and asked to not give either some or a number of one kind (e.g. Give everything, but not [some/five] bananas). While children tended to give zero bananas when asked to not give some, they gave positive amounts when asked to not give numbers. This suggests that contrast - plus knowledge of a number's membership in a count list - enables children to differentiate the meanings of unknown number words from the meaning of some. Experiment 2 tested whether children's interpretation of unknown numbers is further constrained by understanding numerical entailment relations - that if someone, e.g. has three, they thereby also have two, but if they do not have three, they also do not have four. On critical trials, children saw two characters with different quantities of fish, two apart (e.g. 2 vs. 4), and were asked about the number in-between - who either has or doesn't have, e.g. three. Children picked the larger quantity for the affirmative, and the smaller for the negative prompts even when all the numbers were unknown, suggesting that they understood that, whatever three means, a larger quantity is more likely to contain that many, and a smaller quantity is more likely not to. We conclude by discussing how contrast and entailment could help children scaffold the exact meanings of unknown number words.

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