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Early mathematics skills relate to later mathematics achievement and educational attainment, which in turn predict career choice, income, health and financial decision-making. Critically, large differences exist among children in early mathematics performance, with parental mathematics engagement being a key predictor. However, most prior work has examined mothers' mathematics engagement with their preschool- and school-aged children. In this Registered Report, we tested concurrent associations between mothers' and fathers' engagement in mathematics activities with their 2- to 3-year-old toddlers and children's mathematics performance. Mothers and fathers did not differ in their engagement in mathematics activities, and both parents' mathematics engagement related to toddlers' mathematics skills. Fathers' mathematics engagement was associated with toddlers' number and mathematics language skills, but not their spatial skills. Mothers' mathematics engagement was only associated with toddlers' mathematics language skills. Critically, associations may be domain-specific, as parents' literacy engagement did not relate to measures of mathematics performance above their mathematics engagement. Mothers' and fathers' mathematics activities uniquely relate to toddlers' developing mathematics skills, and future work on the nuances of these associations is needed.

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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.

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BACKGROUND: Children's spontaneous focusing on numerosity (SFON) is related to numerical skills. This study aimed to examine (1) the developmental trajectory of SFON and (2) the interrelations between SFON and early numerical skills at pre-school as well as their influence on arithmetical skills at school. METHOD: Overall, 1868 German pre-school children were repeatedly assessed until second grade. Nonverbal intelligence, visual attention, visuospatial working memory, SFON and numerical skills were assessed at age five (M = 63 months, Time 1) and age six (M = 72 months, Time 2), and arithmetic was assessed at second grade (M = 95 months, Time 3). RESULTS: SFON increased significantly during pre-school. Path analyses revealed interrelations between SFON and several numerical skills, except number knowledge. Magnitude estimation and basic calculation skills (Time 1 and Time 2), and to a small degree number knowledge (Time 2), contributed directly to arithmetic in second grade. The connection between SFON and arithmetic was fully mediated by magnitude estimation and calculation skills at pre-school. CONCLUSION: Our results indicate that SFON first and foremost influences deeper understanding of numerical concepts at pre-school and-in contrast to previous findings -affects only indirectly children's arithmetical development at school.

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Recent evidence suggests that infants and toddlers may recognize counting as numerically relevant long before they are able to count or understand the cardinal meaning of number words. The Give-N task, which asks children to produce sets of objects in different quantities, is commonly used to test children's cardinal number knowledge and understanding of exact number words but does not capture children's preliminary understanding of number words and is difficult to administer remotely. Here, we asked whether toddlers correctly map number words to the referred quantities in a two-alternative forced choice Point-to-X task (e.g., "Which has three?"). Two- to three-year-old toddlers (N = 100) completed a Give-N task and a Point-to-X task through in-person testing or online via videoconferencing software. Across number-word trials in Point-to-X, toddlers pointed to the correct image more often than predicted by chance, indicating that they had some understanding of the prompted number word that allowed them to rule out incorrect responses, despite limited understanding of exact cardinal values. No differences in Point-to-X performance were seen for children tested in-person versus remotely. Children with better understanding of exact number words as indicated on the Give-N task also answered more trials correctly in Point-to-X. Critically, in-depth analyses of Point-to-X performance for children who were identified as 1- or 2-knowers on Give-N showed that 1-knowers do not show a preliminary understanding of numbers above their knower-level, whereas 2-knowers do. As researchers move to administering assessments remotely, the Point-to-X task promises to be an easy-to-administer alternative to Give-N for measuring children's emerging number knowledge and capturing nuances in children's number-word knowledge that Give-N may miss.

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Language skills and mathematical competencies are argued to influence each other during development. While a relation between the development of vocabulary size and mathematical skills is already documented in the literature, this study further examines how children's ability to map a novel word to an unknown object as well as their ability to retain this word from memory may be related to their knowledge of number words. Twenty-five children were tested longitudinally (at 30 and at 36 months of age) using an eye-tracking-based fast mapping task, the Give-a-Number task, and standardized measures of vocabulary. The results reveal that children's ability to create and retain a mental representation of a novel word was related to number knowledge at 30 months, but not at 36 months while vocabulary size correlated with number knowledge only at 36 months. These results show that even specific mapping processes are initially related to the acquisition of number words and they speak for a parallelism between the development of lexical and number-concept knowledge despite their semantic and syntactic differences.

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There is evidence that early variations in the development of an approximate number system (ANS) and symbolic number understanding are both influences on the later development of formal arithmetic skills. We report a large-scale (N = 552) longitudinal study of the predictors of arithmetic spanning a critical developmental period (the first 3 years of formal education). Variations in early knowledge of symbolic representations of number and the ordinal associations between them are direct predictors of later arithmetic skills. The development of number ordering ability is in turn predicted by earlier variations in arithmetic, the ANS (numerosity judgments), and rapid automatized naming (RAN). These findings have important implications for theories of numerical and arithmetical development and potentially for the teaching of these skills.

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Previous literature has revealed that visual-spatial processing is associated with both reading and arithmetic. Yet the strength of their relations and the reasons why visual-spatial processing contributes to reading and arithmetic remain ambiguous. The current study focused on two types of visual-spatial skills that recent evidence has suggested are crucial in children's early reading and arithmetic development: visual-perceptual and spatial visualization skills. With an interval of 6 months, we assessed 104 Hong Kong kindergarten children's visual-spatial skills, word reading, arithmetic performance, and vocabulary knowledge at Wave 1; orthographic awareness, basic number knowledge, and number line estimation at Wave 2; and Chinese word reading and arithmetic performance at Wave 3. Correlational analysis showed that both visual-perceptual and spatial visualization skills were associated with later Chinese word reading and arithmetic performance. Further mediation analyses revealed that spatial visualization skills, rather than visual-perceptual skills, contributed to Chinese word reading via orthographic awareness and also predicted arithmetic performance through basic number knowledge. However, number line estimation failed to mediate any relations of visual-spatial skills with children's arithmetic abilities. The results suggest the importance of visual-spatial processing in Chinese word reading and mathematics, with spatial visualization contributing to reading and mathematics for different reasons.

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The purpose of this study was to analyze the contribution of specific executive function (EF) components to different number knowledge skills. A sample of 143 children attending the last year of preschool educational services (Mage = 65.01 months, SD = 3.57) were tested on five number knowledge tasks from the Numerical Intelligence Battery and four EF tasks assessing working memory (WM) and inhibition. First, we examined the interrelationship between different number skills; the results suggested that the relationship between basic informal skills (set comparison and number sequence) and formal skills (seriation of Arabic numerals and number comparison) was mediated by the ability to link sets to numerals. Next, we explored the contribution of WM and inhibition to different number knowledge skills. The structural equation model showed that WM and inhibition were differentially related to specific number knowledge skills. Specifically, WM predicted most components of number knowledge, including the two basic informal skills and the number comparison, whereas inhibition contributed to the seriation of Arabic numerals. The ability to link sets to numerals was predicted only by number sequence, not by EF components.

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In a preschool through first grade longitudinal study, we identified groups of children with persistently low mathematics achievement (n = 14) and children with low achievement in preschool but average achievement in first grade (n = 23). The preschool quantitative developments of these respective groups of children with mathematical learning disability (MLD) and recovered children and a group of typically achieving peers (n = 35) were contrasted, as were their intelligence, executive function, and parental education levels. The core characteristics of the children with MLD were poor executive function and delayed understanding of the cardinal value of number words throughout preschool. These compounded into even more substantive deficits in number and arithmetic at the beginning of first grade. The recovered group had poor executive function and cardinal knowledge during the first year of preschool but showed significant gains during the second year. Despite these gains and average mathematics achievement, the recovered children had subtle deficits with accessing magnitudes associated with numerals and addition combinations (e.g., 5 + 6 = ?) in first grade. The study provides unique insight into domain-general and quantitative deficits in preschool that increase risk for long-term mathematical difficulties.

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Little is known about the development of number knowledge (NK) and the antecedents of low-persistent NK profiles in early childhood. We documented the developmental trajectories of NK across the transition from preschool to elementary school, their predictive validity with respect to later math achievement, and the child and family early-life factors associated with low NK profiles. Children's NK was assessed four times at regular intervals between the ages 4 and 7 years in a large, representative population-based sample. Developmental trajectories of NK were established for 1597 children. These children were also assessed with respect to several features of their family environment at 5, 17, and 29 months, as well as their cognitive skills at age 41 months. Analyses revealed a best-fitting 4-trajectory model, characterized by Low-Increasing (10% of the children), Moderate-Increasing (39%), Moderate-Fast Increasing (32%) and High-Increasing (19%) groups. Children of these trajectory groups differed significantly with respect to math achievement at ages 8 and 10 years, with the Low-Increasing group persistently scoring lower than the other groups throughout these years. Children of Low-Increasing NK group were from household of lower income and father with low educational background, poorer early cognitive development, and more importantly, reduced visual-spatial skills and memory-span. Children displaying reduced cognitive abilities and impoverished living conditions early in life are at greater risk of low NK throughout late preschool and school entry, with ensuing difficulties in math achievement. They deserve early preventive attention to help alleviate later mathematic difficulties.

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BACKGROUND: Prior studies on fraction magnitude understanding focused mainly on students with relatively sufficient formal instruction on fractions whose fraction magnitude understanding is relatively mature. AIM: This study fills a research gap by investigating fraction magnitude understanding in the early stages of fraction instruction. It extends previous findings to children with limited and primary formal fraction instruction. SAMPLE(S): Thirty-five fourth graders with limited fraction instruction and forty fourth graders with primary fraction instruction were recruited from a Chinese primary school. METHODS: Children's fraction magnitude understanding was assessed with a fraction number line estimation task. Approximate number system (ANS) acuity was assessed with a dot discrimination task. Whole number knowledge was assessed with a whole number line estimation task. General reading and mathematics achievements were collected concurrently and 1 year later. RESULTS: In children with limited fraction instruction, fraction representation was linear and fraction magnitude understanding was concurrently related to both ANS and whole number knowledge. In children with primary fraction instruction, fraction magnitude understanding appeared to (marginally) significantly predict general mathematics achievement 1 year later. CONCLUSIONS: Fraction magnitude understanding emerged early during formal instruction of fractions. ANS and whole number knowledge were related to fraction magnitude understanding when children first began to learn about fractions in school. The predictive value of fraction magnitude understanding is likely constrained by its sophistication level.

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Preschool children vary tremendously in their numerical knowledge, and these individual differences strongly predict later mathematics achievement. To better understand the sources of these individual differences, we measured a variety of cognitive and linguistic abilities motivated by previous literature to be important and then analyzed which combination of these variables best explained individual differences in actual number knowledge. Through various data-driven Bayesian model comparison and selection strategies on competing multiple regression models, our analyses identified five variables of unique importance to explaining individual differences in preschool children's symbolic number knowledge: knowledge of the count list, nonverbal approximate numerical ability, working memory, executive conflict processing, and knowledge of letters and words. Furthermore, our analyses revealed that knowledge of the count list, likely a proxy for explicit practice or experience with numbers, and nonverbal approximate numerical ability were much more important to explaining individual differences in number knowledge than general cognitive and language abilities. These findings suggest that children use a diverse set of number-specific, general cognitive, and language abilities to learn about symbolic numbers, but the contribution of number-specific abilities may overshadow that of more general cognitive abilities in the learning process.

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This study investigated the stable and transient genetic and environmental contributions to individual differences in number knowledge in the transition from preschool (age 5) to Grade 1 (age 7) and to the predictive association between early number knowledge and later math achievement (age 10-12). We conducted genetic simplex modeling across these three time points. Genetic variance was transmitted from preschool number knowledge to late-elementary math achievement; in addition, significant genetic innovation (i.e., new influence) occurred at ages 10 through 12 years. The shared and nonshared environmental contributions decreased during the transition from preschool to school entry, but shared and nonshared environment contributed to the continuity across time from preschool number knowledge to subsequent number knowledge and math achievement. There was no new environmental contribution at time points subsequent to preschool. Results are discussed in light of their practical implications for children who have difficulties with mathematics, as well as for preventive intervention.

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This study focused on the relative contributions of the acuity of the approximate number system (ANS) and knowledge of quantitative symbols to young children's early mathematical learning. At the beginning of preschool, 191 children (Mage=46 months) were administered tasks that assessed ANS acuity and explicit knowledge of the cardinal values represented by number words, and their mathematics achievement was assessed at the end of the school year. Children's executive functions, intelligence, and preliteracy skills and their parents' educational levels were also assessed and served as covariates. Both the ANS and cardinality tasks were significant predictors of end-of-year mathematics achievement with and without control of the covariates. As simultaneous predictors and with control of the covariates, cardinality remained significantly related to mathematics achievement, but ANS acuity did not. Mediation analyses revealed that the relation between ANS acuity and mathematics achievement was fully mediated by cardinality, suggesting that the ANS may facilitate children's explicit understanding of cardinal value and in this way may indirectly influence early mathematical learning.

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Although everyone perceives approximate numerosities, some people make more accurate estimates than others. The accuracy of this estimation is called approximate number system (ANS) acuity. Recently, several studies have reported that individual differences in young children's ANS acuity are correlated with their knowledge of exact numbers such as the word 'six' (Mussolin et al., 2012, Trends Neurosci. Educ., 1, 21; Shusterman et al., 2011, Connecting early number word knowledge and approximate number system acuity; Wagner & Johnson, 2011, Cognition, 119, 10; see also Abreu-Mendoza et al., 2013, Front. Psychol., 4, 1). This study argues that this correlation should not be trusted. It seems to be an artefact of the procedure used to assess ANS acuity in children. The correlation arises because (1) some experimental designs inadvertently allow children to answer correctly based on the size (rather than the number) of dots in the display and/or (2) young children with little exact-number knowledge may not understand the phrase 'more dots' to mean numerically more. When the task is modified to make sure that children respond on the basis of numerosity, the correlation between ANS acuity and exact-number knowledge in normally developing children disappears.

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A growing amount of empirical data is showing that the ability to manipulate quantities in a precise and efficient fashion is rooted in cognitive mechanisms devoted to specific aspects of numbers processing. The analog number system (ANS) has a reasonable representation of quantities up to about 4, and represents larger quantities on the basis of a numerical ratio between quantities. In order to represent the precise cardinality of a number, the ANS may be supported by external algorithms such as language, leading to a "precise number system". In the setting of limited language, other number-related systems can appear. For example the parallel individuation system (PIS) supports a "chunking mechanism" that clusters units of larger numerosities into smaller subsets. In the present study we investigated number processing in non-aphasic patients with corticobasal syndrome (CBS) and posterior cortical atrophy (PCA), two neurodegenerative conditions that are associated with progressive parietal atrophy. The present study investigated these number systems in CBS and PCA by assessing the property of the ANS associated with smaller and larger numerosities, and the chunking property of the PIS. The results revealed that CBS/PCA patients are impaired in simple calculations (e.g., addition and subtraction) and that their performance strongly correlates with the size of the numbers involved in these calculations, revealing a clear magnitude effect. This magnitude effect was correlated with gray matter atrophy in parietal regions. Moreover, a numeral-dots transcoding task showed that CBS/PCA patients were able to take advantage of clustering in the spatial distribution of the dots of the array. The relative advantage associated with chunking compared to a random spatial distribution correlated with both parietal and prefrontal regions. These results shed light on the properties of systems for representing number knowledge in non-aphasic patients with CBS and PCA.

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Quantifiers, like "some" or "few," are frequent in daily language. Linguists posit at least three distinct classes of quantifiers: cardinal quantifiers that rely on numerosity, majority quantifiers that additionally depend on executive resources, and logical quantifiers that rely on perceptual attention. We used BOLD fMRI to investigate the roles of frontal and parietal regions in quantifier comprehension. Participants performed a sentence-picture verification task to determine whether a sentence containing a quantifier accurately describes a picture. A whole-brain analysis identified a network involved in quantifier comprehension: This implicated bilateral inferior parietal, superior parietal and dorsolateral prefrontal cortices, and right inferior frontal cortex. We then performed region-of-interest analyses to assess the relative contribution of each region for each quantifier class. Inferior parietal cortex was equally activated across all quantifier classes, consistent with prior studies implicating the region for quantifier comprehension due in part to its role in the representation of number knowledge. Right superior parietal cortex was up-regulated in comparison to frontal regions for cardinal and logical quantifiers, but parietal and frontal regions were equally activated for majority quantifiers and each frontal region is most highly activated for majority quantifiers. This finding is consistent with the hypothesis that majority quantifiers rely on numerosity mechanisms in parietal cortex and executive mechanisms in frontal cortex. Also, right inferior frontal cortex was up-regulated for logical compared to cardinal quantifiers, which may be related to selection demands associated with logical quantifier comprehension. We conclude that distinct components of a large-scale fronto-parietal network contribute to specific aspects of quantifier comprehension, and that this biologically defined network is consistent with cognitive theories of quantifier meaning.

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BACKGROUND: Serious difficulties in formal mathematical skills have been identified in preterm children. By contrast, basic-level numerical skills like magnitude judgments have not yet been tested in these children. AIMS: The aim of the present research was to investigate whether preterm birth also affects these basic numerical abilities, with particular attention to the transition from preschool to formal education. METHOD: One hundred-forty very preterm children and 60 age-matched controls were recruited in a cross-sectional study at 6 and 8years of age. Magnitude comparison tasks with non-symbolic dot displays or symbolic Arabic-number stimuli, measuring accuracy and reaction time, were administered to participants. We also investigated explicit number knowledge, as well as general cognitive developmental levels, to gain a broader picture of preterm abilities. RESULTS: Despite no general cognitive delay, the more simple approximate non-symbolic representation of numerical magnitude was affected by preterm birth, with slower reaction times at both ages compared to controls. Additionally, clear difficulties in the construction of the symbolic representation of numerical magnitude and in explicit number knowledge emerged in the 6-year-old preterm children, with a recovery from this serious delay finally by 8years. CONCLUSIONS: The serious delays identified here in basic numerical abilities in preterm children, despite normal IQ, point to the need for further studies in order to elucidate the relationship between basic numerical abilities and subsequent difficulties in formal mathematic achievement at school.

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How does symbolic number knowledge performance help identify young children at risk for poor mathematics achievement outcomes? In research and practice, classification of mathematics learning disability (MLD, or dyscalculia) is typically based on composite scores from broad measures of mathematics achievement. These scores do predict later math achievement levels, but do not specify the nature of math difficulties likely to emerge among students at greatest risk for long-term mathematics failure. Here we report that gaps in 2nd and 3rd graders' number knowledge predict specific types of errors made on math assessments at Grade 8. Specifically, we show that early whole number misconceptions predict slower and less accurate performance, and atypical computational errors, on Grade 8 arithmetic tests. We demonstrate that basic number misconceptions can be detected by idiosyncratic responses to number knowledge items, and that when such misconceptions are evident during primary school they persist throughout the school age years, with variable manifestation throughout development. We conclude that including specific qualitative assessments of symbolic number knowledge in primary school may provide greater specificity of the types of difficulties likely to emerge among students at risk for poor mathematics outcomes.

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This paper examines Piantadosi, Tenenbaum, and Goodman's (2012) model for how children learn the relation between number words ("one" through "ten") and cardinalities (sizes of sets with one through ten elements). This model shows how statistical learning can induce this relation, reorganizing its procedures as it does so in roughly the way children do. We question, however, Piantadosi et al.'s claim that the model performs "Quinian bootstrapping," in the sense of Carey (2009). Unlike bootstrapping, the concept it learns is not discontinuous with the concepts it starts with. Instead, the model learns by recombining its primitives into hypotheses and confirming them statistically. As such, it accords better with earlier claims (Fodor, 1975, 1981) that learning does not increase expressive power. We also question the relevance of the simulation for children's learning. The model starts with a preselected set of15 primitives, and the procedure it learns differs from children's method. Finally, the partial knowledge of the positive integers that the model attains is consistent with an infinite number of nonstandard meanings-for example, that the integers stop after ten or loop from ten back to one.

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