Finding one’s meaning: A test of the relation between quantifiers and integers in language development☆

The roles of mathematical language and emergent literacy skills in the longitudinal prediction of specific early numeracy skills

Mathematical language (i.e., content-specific language used in mathematics) and emergent literacy skills predict children's broad numeracy development. However, little work has examined whether these domains predict development of individual numeracy skills (e.g., cardinality, number order). Thus, the aim of the current study was to examine longitudinal relations among mathematical language, emergent literacy skills, and specific early numeracy skills. Participants included 114 preschool children aged 3.12 to 5.26 years (M = 4.17 years, SD = 0.59). Specifically, this study examined whether mathematical language and three emergent literacy skills (print knowledge, phonological awareness, and general vocabulary) in the fall of preschool predicted 12 individual early numeracy skills in the spring, controlling for age, sex, rapid automatized naming, parent education, and autoregressors. Results indicated that mathematical language predicted development of most of the early numeracy skills (e.g., set comparison, numeral comparison, numeral identification), but findings for emergent literacy skills were not robust. Among the three emergent literacy skills, only print knowledge was a significant predictor of development in some specific numeracy skills, including verbal counting, number order, and story problems. Results highlight the important role of mathematical language in children's numeracy development and provide the foundation for future work in designing interventions to improve early numeracy skills.

Challenges with computing scalar and ad-hoc implicatures in Mandarin-speaking 4-8-year-old autistic children

INTRODUCTION

Mixed findings have been reported about the computation of scalar or/and ad-hoc implicatures in primarily school-age autistic verbal children and adolescents: while some studies reported their struggles with both implicatures, others observed their strengths in computing scalar implicatures. This study extends the previous investigation by testing the derivation of scalar (including both number and quantifier) and ad-hoc implicatures of a younger group of Mandarin-speaking autistic 4-8-year-olds; moreover, we assess the biological, linguistic, and cognitive factors affecting children's implicature acquisition.

METHODS

The participants included 22 4-8-year-old autistic verbal children (mean age = 67.64 months) and 19 typically developing (TD) children who did not significantly differ in age, receptive vocabulary, and non-verbal IQ. Both groups completed a computer-based Truth Value Judgment task, assessing their knowledge of scalar (involving the number 'three' and the quantifier 'some') and ad-hoc implicatures. We also examined whether their implicature computation was linked to age, receptive vocabulary, non-verbal IQ, and Theory of Mind (ToM).

RESULTS

Compared with the TD controls, autistic children derived significantly fewer scalar and ad-hoc implicatures. Specifically, TD children successfully computed number and ad-hoc implicatures, contrasting to the bimodal distribution of their pragmatic vs. logical responses to quantifier implicatures. Though autistic children performed better with number implicatures slightly above the chance level, they had difficulties in computing quantifier and ad-hoc implicatures. Further, autistic children's knowledge of the number and ad-hoc implicatures was linked to their ToM skills.

CONCLUSIONS

These findings underscore the overall delayed implicature knowledge of young autistic children, and their low sensitivity to the implicatures is related to the core ToM deficits. Furthermore, our data confirm the coherent pattern of the earlier acquisition of number over quantifier implicatures and illuminate the distinct mechanisms underlying the computation of scalar vs. ad-hoc implicatures.

Math items about real-world content lower test-scores of students from families with low socioeconomic status

In many countries, standardized math tests are important for achieving academic success. Here, we examine whether content of items, the story that explains a mathematical question, biases performance of low-SES students. In a large-scale cohort study of Trends in International Mathematics and Science Studies (TIMSS)-including data from 58 countries from students in grades 4 and 8 (N = 5501,165)-we examine whether item content that is more likely related to challenges for low-SES students (money, food, social relationships) improves their performance, compared with their average math performance. Results show that low-SES students scored lower on items with this specific content than expected based on an individual's average performance. The effect sizes are substantial: on average, the chance to answer correctly is 18% lower. From a hidden talents approach, these results are unexpected. However, they align with other theoretical frameworks such as scarcity mindset, providing new insights for fair testing.

Priming scalar and ad hoc enrichment in children

Sentences can be enriched by considering what the speaker does not say but could have done. Children, however, struggle to derive one type of such enrichments, scalar implicatures. A popular explanation for this, the lexical alternatives account, is that they do not have lexical knowledge of the appropriate alternatives to generate the implicature. Namely, children are unaware of the scalar relationship between some and all. We conducted a priming study with N = 72 children, aged 5;1 years, and an adult sample, N = 51, to test this hypothesis. Participants were exposed to prime trials of strong, alternative, or weak sentences involving scalar or ad hoc expressions, and then saw a target trial that could be interpreted in either way. Consistent with previous studies, children were reluctant to derive scalar implicatures. However, there were two novel findings. (1) Children responded with twice the rate of ad hoc implicature responses than adults, suggesting that the implicature was the developmentally prior interpretation for ad hoc expressions. (2) Children showed robust priming effects, suggesting that children are aware of the scalar relationship between some and all, even if they choose not to derive the implicature. This suggests that the root cause of the scalar implicature deficit is not due to the absence of lexical knowledge of the relationship between some and all.

Questions About Quantifiers: Symbolic and Nonsymbolic Quantity Processing by the Brain

One approach to understanding how the human cognitive system stores and operates with quantifiers such as "some," "many," and "all" is to investigate their interaction with the cognitive mechanisms for estimating and comparing quantities from perceptual input (i.e., nonsymbolic quantities). While a potential link between quantifier processing and nonsymbolic quantity processing has been considered in the past, it has never been discussed extensively. Simultaneously, there is a long line of research within the field of numerical cognition on the relationship between processing exact number symbols (such as "3" or "three") and nonsymbolic quantity. This accumulated knowledge can potentially be harvested for research on quantifiers since quantifiers and number symbols are two different ways of referring to quantity information symbolically. The goal of the present review is to survey the research on the relationship between quantifiers and nonsymbolic quantity processing mechanisms and provide a set of research directions and specific questions for the investigation of quantifier processing.

Different impacts of long-term abacus training on symbolic and non-symbolic numerical magnitude processing in children

Abacus-based mental calculation (AMC) has been shown to be effective in promoting math ability in children. Given that AMC relies on a visuospatial strategy to perform rapid and precise arithmetic, previous studies mostly focused on the promotion of AMC training on arithmetic ability and mathematical visual-spatial ability, as well as its transfer of advanced cognitive ability. However, little attention has been given to its impact on basic numerical comparison ability. Here, we aim to examine whether and how long-term AMC training impacts symbolic and non-symbolic numerical comparisons. The distance effect (DE) was utilized as a marker, indicating that the comparison between two numbers becomes faster as their numerical distance enlarges. In the current study, forty-one children matched for age and sex were recruited at primary school entry and randomly assigned to the AMC group and the control group. After three years of training, the event-related potential (ERP) recording technique was used to explore the temporal dynamics of number comparison, of which tasks were given in symbolic (Arabic number) or non-symbolic (dot array) format. In the symbolic task, the children in the AMC group showed a smaller DE than those in the control group. Two ERP components, N1 and P2p, located in parietal areas (PO7, PO8) were selected as neural markers of numerical processing. Both groups showed DE in the P2p component in both tasks, but only the children in the AMC group showed DE in the N1 component in the non-symbolic task. In addition, the DE size calculated from reaction times and ERP amplitudes was correlated with higher cognitive capacities, such as coding ability. Taken together, the present results provide evidence that long-term AMC training may be beneficial for numerical processing in children, which may be associated with neurocognitive indices of parietal brain regions.

Early number word learning: Associations with domain-general and domain-specific quantitative abilities

Cardinal number knowledge-understanding “two” refers to sets of two entities-is a critical piece of knowledge that predicts later mathematics achievement. Recent studies have shown that domain-general and domain-specific skills can influence children’s cardinal number learning. However, there has not yet been research investigating the influence of domain-specific quantifier knowledge on children’s cardinal number learning. The present study aimed to investigate the influence of domain-general and domain-specific skills on Mandarin Chinese-speaking children’s cardinal number learning after controlling for a number of family background factors. Particular interest was paid to the question whether domain-specific quantifier knowledge was associated with cardinal number development. Specifically, we investigated 2–5-year-old Mandarin Chinese-speaking children’s understanding of cardinal number words as well as their general language, intelligence, approximate number system (ANS) acuity, and knowledge of quantifiers. Children’s age, gender, parental education, and family income were also assessed and used as covariates. We found that domain-general abilities, including general language and intelligence, did not account for significant additional variance of cardinal number knowledge after controlling for the aforementioned covariates. We also found that domain-specific quantifier knowledge did not account for significant additional variance of cardinal number knowledge, whereas domain-specific ANS acuity accounted for significant additional variance of cardinal number knowledge, after controlling for the aforementioned covariates. In sum, the results suggest that domain-specific numerical skills seem to be more important for children’s development of cardinal number words than the more proximal domain-general abilities such as language abilities and intelligence. The results also highlight the significance of ANS acuity on children’s cardinal number word development.

Children's understanding of most is dependent on context

Children struggle with the quantifier "most". Often, this difficulty is attributed to an inability to interpret most proportionally, with children instead relying on absolute quantity comparisons. However, recent research in proportional reasoning more generally has provided new insight into children's apparent difficulties, revealing that their overreliance on absolute amount is unique to contexts in which the absolute amount can be counted and interferes with proportional information. Across two experiments, we test whether 4- to 6-year-old children's interpretation of most is similarly dependent on the discreteness of the stimuli when comparing two different quantities (e.g., who ate most of their chocolate?) and when verifying whether a single amount can be described with the term most (e.g., is most of the butterfly colored in?). We find that children's interpretation of most does depend on the stimulus format. When choosing between absolutely more vs. proportionally more as depicting most, children showed stronger absolute-based errors with discrete stimuli than continuous stimuli, and by 6-years-old were able to reason proportionally with continuous stimuli, despite still demonstrating strong absolute interference with discrete stimuli. In contrast, children's yes/no judgements of single amounts, where conflicting absolute information is not a factor, showed a weaker understanding of most for continuous stimuli than for discrete stimuli. Together, these results suggest that children's difficulty with most is more nuanced than previously understood: it depends on the format and availability of proportional vs. absolute amounts and develops substantially from 4- to 6-years-old.

Relational language influences young children's number relation skills

Relational language is thought to influence mathematical skills. This study examines the association between relational language and number relation skills-knowledge of cardinal, ordinal, and spatial principles-among 104 U.S. kindergartners (5.9 years; 44% boys; 37% White, 25% Black, 14% Asian, 24% other) in the 2017-2018 academic year. Controlling for general verbal knowledge, executive function, and counting and number identification skills, relational language predicted later number relation skills, specifically number line estimation, β = .30. Relational language did not differentially predict number line estimation performance in children with low or high number relation skills, likely due to the restricted ranges of data within subgroups. Number relation skills, specifically number line estimation and number ordering, may be a pathway between relational language and mathematical skills.

Assessing the knower-level framework: How reliable is the Give-a-Number task?

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.

Early ERP Evidence for Children's and Adult's Sensitivity to Scalar Implicatures Triggered by Existential Quantifiers (Some)

How quickly do children and adults interpret scalar lexical items in speech processing? The current study examined interpretation of the scalar terms some vs. all in contexts where either the stronger (some = not all) or the weaker interpretation was permissible (some allows all). Children and adults showed increased negative deflections in brain activity following the word some in some-infelicitous versus some-felicitous contexts. This effect was found as early as 100 ms across central electrode sites (in children), and 300–500 ms across left frontal, fronto-central, and centro-parietal electrode sites (in children and adults). These results strongly suggest that young children (aged between 3 and 4 years) as well as adults quickly have access to the contextually appropriate interpretation of scalar terms.

Children Change Their Answers in Response to Neutral Follow-Up Questions by a Knowledgeable Asker

Burgeoning evidence suggests that when children observe data, they use knowledge of the demonstrator's intent to augment learning. We propose that the effects of social learning may go beyond cases where children observe data, to cases where they receive no new information at all. We present a model of how simply asking a question a second time may lead to belief revision, when the questioner is expected to know the correct answer. We provide an analysis of the CHILDES corpus to show that these neutral follow‐up questions are used in parent–child conversations. We then present three experiments investigating 4‐ and 5‐year‐old children's reactions to neutral follow‐up questions posed by ignorant or knowledgeable questioners. Children were more likely to change their answers in response to a neutral follow‐up question from a knowledgeable questioner than an ignorant one. We discuss the implications of these results in the context of common practices in legal, educational, and experimental psychological settings.

Do children derive exact meanings pragmatically? Evidence from a dual morphology language

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.

Longitudinally adaptive assessment and instruction increase numerical skills of preschool children

Social inequality in mathematical skill is apparent at kindergarten entry and persists during elementary school. To level the playing field, we trained teachers to assess children's numerical and spatial skills every 10 wk. Each assessment provided teachers with information about a child's growth trajectory on each skill, information designed to help them evaluate their students' progress, reflect on past instruction, and strategize for the next phase of instruction. A key constraint is that teachers have limited time to assess individual students. To maximize the information provided by an assessment, we adapted the difficulty of each assessment based on each child's age and accumulated evidence about the child's skills. Children in classrooms of 24 trained teachers scored 0.29 SD higher on numerical skills at posttest than children in 25 randomly assigned control classrooms (P = 0.005). We observed no effect on spatial skills. The intervention also positively influenced children's verbal comprehension skills (0.28 SD higher at posttest, P < 0.001), but did not affect their print-literacy skills. We consider the potential contribution of this approach, in combination with similar regimes of assessment and instruction in elementary schools, to the reduction of social inequality in numerical skill and discuss possible explanations for the absence of an effect on spatial skills.

Ontogenetic Origins of Human Integer Representations

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.

The role of salience in young children's processing of ad hoc implicatures

Language comprehension often requires making implicatures. For example, inferring that "I ate some of the cookies" implicates that the speaker ate some but not all (scalar implicatures), and "I ate the chocolate chip cookies" where there are both chocolate chip cookies and raisin cookies in the context implicates that the speaker ate the chocolate chip cookies but not both the chocolate chip and raisin cookies (ad hoc implicatures). Children's ability to make scalar implicatures develops at around 5 years of age, with ad hoc implicatures emerging somewhat earlier. In the current work, using a time-sensitive tablet paradigm, we examined developmental gains in children's ad hoc implicature processing and found evidence for successful pragmatic inferences by children as young as 3 years in a supportive context and substantial developmental gains in inference computation from 2 to 5 years. We also tested whether one cause of younger children's (2-year-olds) consistent failure to make pragmatic inferences is their difficulty in inhibiting an alternative interpretation that is more salient than the target meaning (the salience hypothesis). Our findings support this hypothesis; younger children's failures with pragmatic inferences were related to effects of the salience mismatch between possible interpretations.

Contrast and entailment: Abstract logical relations constrain how 2- and 3-year-old children interpret unknown numbers

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.

Some Pieces Are Missing: Implicature Production in Children

Until at least 4 years of age, children, unlike adults, interpret some as compatible with all. The inability to draw the pragmatic inference leading to interpret some as not all, could be taken to indicate a delay in pragmatic abilities, despite evidence of other early pragmatic skills. However, little is known about how the production of these implicature develops. We conducted a corpus study on early production and perception of the scalar term some in British English. Children's utterances containing some were extracted from the dense corpora of five children aged 2;00 to 5;01 (N = 5,276), and analysed alongside a portion of their caregivers' utterances with some (N = 9,030). These were coded into structural and contextual categories allowing for judgments on the probability of a scalar implicature being intended. The findings indicate that children begin producing and interpreting implicatures in a pragmatic way during their third year of life, shortly after they first produce some. Their production of some implicatures is low but matches their parents' input in frequency. Interestingly, the mothers' production of implicatures also increases as a function of the children's age. The data suggest that as soon as they acquire some, children are fully competent in its production and mirror adult production. The contrast between the very early implicature production we find and the relatively late implicature comprehension established in the literature calls for an explanation; possibly in terms of the processing cost of implicature derivation. Additionally, some is multifaceted, and thus, implicatures are infrequent, and structurally and contextually constrained in both populations.

Differential Development of Children's Understanding of the Cardinality of Small Numbers and Zero

Counting and the understanding of cardinality are important steps in children’s numerical development. Recent studies have indicated that language and visuospatial abilities play an important role in the development of children’s cardinal knowledge of small numbers. However, predictors for the knowledge about zero were usually not considered in these studies. Therefore, the present study investigated whether the acquisition of cardinality knowledge on small numbers and the concept of zero share cross-domain and domain-specific numerical predictors. Particular interest was paid to the question whether visuospatial abilities – in addition to language abilities – were associated with children’s understanding of small numbers and zero. Accordingly, we assessed kindergarteners aged 4 to 5 years in terms of their understanding of small numbers and zero as well as their visuospatial, general language, counting, Arabic number identification abilities, and their finger number knowledge. We observed significant zero-order correlations of vocabulary, number identification, finger knowledge, and counting abilities with children’s knowledge about zero as well as understanding of the cardinality of small numbers. Subsequent regression analyses substantiated the influences of counting abilities on knowledge about zero and the influences of both counting abilities and finger knowledge on children’s understanding of the cardinality of small numbers. No significant influences of cross-domain predictors were observed. In sum, these results indicate that domain-specific numerical precursor skills seem to be more important for children’s development of an understanding of the cardinality of small numbers as well as of the concept of zero than the more proximal cross-domain abilities such as language and visuospatial abilities.

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.

Quantifier comprehension is linked to linguistic rather than to numerical skills. Evidence from children with Down syndrome and Williams syndrome

Comprehending natural language quantifiers (like many, all, or some) involves linguistic and numerical abilities. However, the extent to which both factors play a role is controversial. In order to determine the specific contributions of linguistic and number skills in quantifier comprehension, we examined two groups of participants that differ in their language abilities while their number skills appear to be similar: Participants with Down syndrome (DS) and participants with Williams syndrome (WS). Compared to rather poor linguistic skills of individuals with DS, individuals with WS display relatively advanced language abilities. Participants with WS also outperformed participants with DS in a quantifier comprehension task while number knowledge did not differ between the two groups. When compared to typically developing (TD) children of the same mental age, participants with WS displayed similar levels regarding quantifier abilities, but participants with DS performed worse than the control group. Language abilities but not number skills also significantly predicted quantifier knowledge in a linear regression analysis, stressing the importance of linguistic abilities for quantifier comprehension. In addition to determining the skills that are relevant for comprehending quantifiers, our findings provide the first demonstration of how quantifiers are acquired by individuals with DS and WS, an issue not investigated so far.

The development of linguistic prediction: Predictions of sound and meaning in 2- to 5-year-olds

Language processing in adults is facilitated by an expert ability to generate detailed predictions about upcoming words. This may seem like an acquired skill, but some models of language acquisition assume that the ability to predict is a prerequisite for learning. This raises a question: Do children learn to predict, or do they predict to learn? We tested whether children, like adults, can generate expectations about not just the meanings of upcoming words but also their sounds, which would be critical for using prediction to learn about language. In two looking-while-listening experiments, we show that 2-year-olds can generate expectations about meaning based on a determiner (Can you see one…ball/two…ice creams?) but that even children as old as 5 years do not show an adult-like ability to predict the phonology of upcoming words based on a determiner (Can you see a…ball/an…ice cream?). Our results, therefore, suggest that the ability to generate detailed predictions is a late-acquired skill. We argue that prediction might not be the key mechanism driving children's learning, but that the ability to generate accurate semantic predictions may nevertheless have facilitative effects on language development.

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.

Does Grammatical Structure Accelerate Number Word Learning? Evidence from Learners of Dual and Non-Dual Dialects of Slovenian

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.

Cross-linguistic patterns in the acquisition of quantifiers

Learners of most languages are faced with the task of acquiring words to talk about number and quantity. Much is known about the order of acquisition of number words as well as the cognitive and perceptual systems and cultural practices that shape it. Substantially less is known about the acquisition of quantifiers. Here, we consider the extent to which systems and practices that support number word acquisition can be applied to quantifier acquisition and conclude that the two domains are largely distinct in this respect. Consequently, we hypothesize that the acquisition of quantifiers is constrained by a set of factors related to each quantifier's specific meaning. We investigate competence with the expressions for "all," "none," "some," "some…not," and "most" in 31 languages, representing 11 language types, by testing 768 5-y-old children and 536 adults. We found a cross-linguistically similar order of acquisition of quantifiers, explicable in terms of four factors relating to their meaning and use. In addition, exploratory analyses reveal that language- and learner-specific factors, such as negative concord and gender, are significant predictors of variation.

Numerical morphology supports early number word learning: Evidence from a comparison of young Mandarin and English learners

Previous studies showed that children learning a language with an obligatory singular/plural distinction (Russian and English) learn the meaning of the number word for one earlier than children learning Japanese, a language without obligatory number morphology (Barner, Libenson, Cheung, & Takasaki, 2009; Sarnecka, Kamenskaya, Yamana, Ogura, & Yudovina, 2007). This can be explained by differences in number morphology, but it can also be explained by many other differences between the languages and the environments of the children who were compared. The present study tests the hypothesis that the morphological singular/plural distinction supports the early acquisition of the meaning of the number word for one by comparing young English learners to age and SES matched young Mandarin Chinese learners. Mandarin does not have obligatory number morphology but is more similar to English than Japanese in many crucial respects. Corpus analyses show that, compared to English learners, Mandarin learners hear number words more frequently, are more likely to hear number words followed by a noun, and are more likely to hear number words in contexts where they denote a cardinal value. Two tasks show that, despite these advantages, Mandarin learners learn the meaning of the number word for one three to six months later than do English learners. These results provide the strongest evidence to date that prior knowledge of the numerical meaning of the distinction between singular and plural supports the acquisition of the meaning of the number word for one.

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.

Children's interpretations of general quantifiers, specific quantifiers, and generics

Recently, several scholars have hypothesized that generics are a default mode of generalization, and thus that young children may at first treat quantifiers as if they were generic in meaning. To address this issue, the present experiment provides the first in-depth, controlled examination of the interpretation of generics compared to both general quantifiers ("all Xs", "some Xs") and specific quantifiers ("all of these Xs", "some of these Xs"). We provided children (3 and 5 years) and adults with explicit frequency information regarding properties of novel categories, to chart when "some", "all", and generics are deemed appropriate. The data reveal three main findings. First, even 3-year-olds distinguish generics from quantifiers. Second, when children make errors, they tend to be in the direction of treating quantifiers like generics. Third, children were more accurate when interpreting specific versus general quantifiers. We interpret these data as providing evidence for the position that generics are a default mode of generalization, especially when reasoning about kinds.

On the relation between grammatical number and cardinal numbers in development

This mini-review focuses on the question of how the grammatical number system of a child’s language may help the child learn the meanings of cardinal number words (e.g., “one” and “two”). Evidence from young children learning English, Russian, Japanese, Mandarin, Slovenian, or Saudi Arabic suggests that trajectories of number-word learning differ for children learning different languages. Children learning English, which distinguishes between singular and plural, seem to learn the meaning of the cardinal number “one” earlier than children learning Japanese or Mandarin, which have very little singular/plural marking. Similarly, children whose languages have a singular/dual/plural system (Slovenian and Saudi Arabic) learn the meaning of “two” earlier than English-speaking children. This relation between grammatical and cardinal number may shed light on how humans acquire cardinal-number concepts. There is an ongoing debate about whether mental symbols for small cardinalities (concepts for “oneness,” “twoness,” etc.) are innate or learned. Although an effect of grammatical number on number-word learning does not rule out nativist accounts, it seems more consistent with constructivist accounts, which portray the number-learning process as one that requires significant conceptual change.

Children's Developing Intuitions About the Truth Conditions and Implications of Novel Generics Versus Quantified Statements

Generic statements express generalizations about categories and present a unique semantic profile that is distinct from quantified statements. This paper reports two studies examining the development of children's intuitions about the semantics of generics and how they differ from statements quantified by all, most, and some. Results reveal that, like adults, preschoolers (a) recognize that generics have flexible truth conditions and are capable of representing a wide range of prevalence levels; and (b) interpret novel generics as having near-universal prevalence implications. Results further show that by age 4, children are beginning to differentiate the meaning of generics and quantified statements; however, even 7- to 11-year-olds are not adultlike in their intuitions about the meaning of most-quantified statements. Overall, these studies suggest that by preschool, children interpret generics in much the same way that adults do; however, mastery of the semantics of quantified statements follows a more protracted course.

Children acquire the later-greater principle after the cardinal principle

Many have proposed that the acquisition of the cardinal principle (CP) is a result of the discovery of the numerical significance of the order of the number words in the count list. However, this need not be the case. Indeed, the CP does not state anything about the numerical significance of the order of the number words. It only states that the last word of a correct count denotes the numerosity of the counted set. Here, we test whether the acquisition of the CP involves the discovery of the later-greater principle - that is, that the order of the number words corresponds to the relative size of the numerosities they denote. Specifically, we tested knowledge of verbal numerical comparisons (e.g., Is 'ten' more than 'six'?) in children who had recently learned the CP. We find that these children can compare number words between 'six' and 'ten' only if they have mapped them onto non-verbal representations of numerosity. We suggest that this means that the acquisition of the CP does not involve the discovery of the correspondence between the order of the number words and the relative size of the numerosities they denote.

Area vs. density: influence of visual variables and cardinality knowledge in early number comparison

Current research in the number development field has focused in individual differences regarding the acuity of children's approximate number system (ANS). The most common task to evaluate children's acuity is through non-symbolic numerical comparison. Efforts have been made to prevent children from using perceptual cues by controlling the visual properties of the stimuli (e.g., density, contour length, and area); nevertheless, researchers have used these visual controls interchangeably. Studies have also tried to understand the relation between children's cardinality knowledge and their performance in a number comparison task; divergent results may in fact be rooted in the use of different visual controls. The main goal of the present study is to explore how the usage of different visual controls (density, total filled area, and correlated and anti-correlated area) affects children's performance in a number comparison task, and its relationship to children's cardinality knowledge. For that purpose, 77 preschoolers participated in three tasks: (1) counting list elicitation to test whether children could recite the counting list up to ten, (2) give a number to evaluate children's cardinality knowledge, and (3) number comparison to evaluate their ability to compare two quantities. During this last task, children were asked to point at the set with more geometric figures when two sets were displayed on a screen. Children were exposed only to one of the three visual controls. Results showed that overall, children performed above chance in the number comparison task; nonetheless, density was the easiest control, while correlated and anti-correlated area was the most difficult in most cases. Only total filled area was sensitive to discriminate cardinal principal knowers from non-cardinal principal knowers. How this finding helps to explain conflicting evidence from previous research, and how the present outcome relates to children's number word knowledge is discussed.

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.

Connecting numbers to discrete quantification: a step in the child's construction of integer concepts

The present study asks when young children understand that number words quantify over sets of discrete individuals. For this study, 2- to 4-year-old children were asked to extend the number word five or six either to a cup containing discrete objects (e.g., blocks) or to a cup containing a continuous substance (e.g., water). In Experiment 1, only children who knew the exact meanings of the words one, two and three extended higher number words (five or six) to sets of discrete objects. In Experiment 2, children who only knew the exact meaning of one extended higher number words to discrete objects under the right conditions (i.e., when the problem was first presented with the number words one and two). These results show that children have some understanding that number words pertain to discrete quantification from very early on, but that this knowledge becomes more robust as children learn the exact, cardinal meanings of individual number words.

What exactly do numbers mean?

Number words are generally used to refer to the exact cardinal value of a set, but cognitive scientists disagree about their meanings. Although most psychological analyses presuppose that numbers have exact semantics (two means EXACTLY TWO), many linguistic accounts propose that numbers have lower-bounded semantics (AT LEAST TWO), and that speakers restrict their reference through a pragmatic inference (scalar implicature). We address this debate through studies of children who are in the process of acquiring the meanings of numbers. Adults and 2- and 3-year-olds were tested in a novel paradigm that teases apart semantic and pragmatic aspects of interpretation (the covered box task). Our findings establish that when scalar implicatures are cancelled in the critical trials of this task, both adults and children consistently give exact interpretations for number words. These results, in concert with recent work on real-time processing, provide the first unambiguous evidence that number words have exact semantics.

Collectivity, Distributivity, and the Interpretation of Plural Numerical Expressions in Child and Adult Language

Sentences containing plural numerical expressions (e.g., two boys) can give rise to two interpretations (collective and distributive), arising from the fact that their representation admits of a part-whole structure. We present the results of a series of experiments designed to explore children's understanding of this distinction and its implications for the acquisition of linguistic expressions with number words. We show that preschoolers access both interpretations, indicating that they have the requisite linguistic and conceptual machinery to generate the corresponding representations. Furthermore, they can shift their interpretation in response to structural and lexical manipulations. However, they are not fully adult-like: unlike adults, they are drawn to the distributive interpretation, and are not yet aware of the lexical semantics of each and together, which should favor one or another interpretation. This research bridges a gap between a well-established body of work in cognitive psychology on the acquisition of number words and more recent work investigating children's knowledge of the syntactic and semantic properties of sentences featuring numerical expressions.

An Excel sheet for inferring children's number-knower levels from give-N data

Number-knower levels are a series of stages of number concept development in early childhood. A child's number-knower level is typically assessed using the give-N task. Although the task procedure has been highly refined, the standard ways of analyzing give-N data remain somewhat crude. Lee and Sarnecka (Cogn Sci 34:51-67, 2010, in press) have developed a Bayesian model of children's performance on the give-N task that allows knower level to be inferred in a more principled way. However, this model requires considerable expertise and computational effort to implement and apply to data. Here, we present an approximation to the model's inference that can be computed with Microsoft Excel. We demonstrate the accuracy of the approximation and provide instructions for its use. This makes the powerful inferential capabilities of the Bayesian model accessible to developmental researchers interested in estimating knower levels from give-N data.

Number-concept acquisition and general vocabulary development

How is number-concept acquisition related to overall language development? Experiments 1 and 2 measured number-word knowledge and general vocabulary in a total of 59 children, ages 30-60 months. A strong correlation was found between number-word knowledge and vocabulary, independent of the child's age, contrary to previous results (D. Ansari et al., 2003). This result calls into question arguments that (a) the number-concept creation process is scaffolded mainly by visuo-spatial development and (b) that language only becomes integrated after the concepts are created (D. Ansari et al., 2003). Instead, this may suggest that having a larger nominal vocabulary helps children learn number words. Experiment 3 shows that the differences with previous results are likely due to changes in how the data were analyzed.

Quantified statements are recalled as generics: evidence from preschool children and adults

Generics are sentences such as "ravens are black" and "tigers are striped", which express generalizations concerning kinds. Quantified statements such as "all tigers are striped" or "most ravens are black" also express generalizations, but unlike generics, they specify how many members of the kind have the property in question. Recently, some theorists have proposed that generics express cognitively fundamental/default generalizations, and that quantified statements in contrast express cognitively more sophisticated generalizations (Gelman, 2010; Leslie, 2008). If this hypothesis is correct, then quantified statements may be remembered as generics. This paper presents four studies with 136 preschool children and 118 adults, demonstrating that adults and preschoolers alike tend to recall quantified statements as generics, thus supporting the hypothesis that generics express cognitively default generalizations.