Mapping skills between symbols and quantities in preschoolers: The role of finger patterns.

Mapping skills between different codes to represent numerical information, such as number symbols (i.e., verbal number words and written digits) and non-symbolic quantities, are important in the development of the concept of number. The aim of the current study is to investigate children's mapping skills by incorporating another numerical code that emerges at early stages in development, finger patterns. Specifically, the study investigates (i) the order in which mapping skills develop and the association with young children's understanding of cardinality; and (ii) whether finger patterns are processed similarly to symbolic codes or rather as non-symbolic quantities. Preschool children (3-year-olds, N = 113, Mage = 40.8 months, SDage = 3.6 months; 4-year-olds, N = 103, Mage = 52.9 months, SDage = 3.4 months) both cardinality knowers and subset-knowers, were presented with twelve tasks that assessed the mappings between number words, Arabic digits, finger patterns, and quantities. The results showed that children's ability to map symbolic numbers precedes the understanding that such symbols reflect quantities, and that children recognize finger patterns above their cardinality knowledge, suggesting that finger patterns are symbolic in essence. RESEARCH HIGHLIGHTS: Children are more accurate in mapping between finger patterns and symbols (number words and Arabic digits) than in mapping finger patterns and quantities, indicating that fingers are processed holistically as symbolic codes. Children can map finger patterns to symbols above their corresponding cardinality level even in subset-knowers. Finger patterns may play a role in the process by which children learn to map symbols to quantities. Fingers patterns' use in the classroom context may be an adequate instructional and diagnostic tool.

The development of number line estimation in children at risk of mathematics learning difficulties: A longitudinal study.

Children with mathematics learning difficulties (MLD) show poorer performance on the number line task, but how performance on this task relates to other mathematical skills is unclear. This study examined the association between performance on the number line task and mathematical skills during the first 2 years of school for children at risk of MLD. Children (N = 100; Mage = 83.63 months) were assessed on four occasions on the number line task and other mathematical skills (math fluency, numerical operations, and mathematical reasoning). Estimation patterns were analyzed based on the representational shift and proportional judgment accounts separately. More consistent longitudinal trends and stronger evidence for differences in mathematical skills based on estimation patterns were found within the representational shift account. Latent growth curve models showed accuracy on the number line task as a predictor of growth in some mathematical skills assessed. We discuss impacts of methodological limitations on the study of estimation patterns.

Heterogeneity in children at risk of math learning difficulties.

This study recruited 428 Singaporean children at risk of math learning difficulties (MLD; Mage  = 83.9 months, SDage  = 4.35 months; 41% female). Using a factor mixture model that considered both quantitative and qualitative differences in math ability, two qualitatively different groups were identified: one with generalized difficulties across different math skills and the other with more focal difficulties in arithmetic fluency. Reading, working memory capacity, and numeracy (number line estimation skills and numerical discrimination) uniquely explained group membership. Children within each group differed in the extent of difficulties they exhibited, with numeracy variables differentially contributing to math ability in each group. Findings speak against a dimensional view of MLD and underscore the conceptual limitations of using basic numeracy performance to profile learning difficulties.

Non-symbolic Ratio Reasoning in Kindergarteners: Underlying Unidimensional Heuristics and Relations With Math Abilities.

Although it is thought that young children focus on the magnitude of the target dimension across ratio sets during binary comparison of ratios, it is unknown whether this is the default approach to ratio reasoning, or if such approach varies across representation formats (discrete entities and continuous amounts) that naturally afford different opportunities to process the dimensions in each ratio set. In the current study, 132 kindergarteners (Mage = 68 months, SD = 3.5, range = 62-75 months) performed binary comparisons of ratios with discrete and continuous representations. Results from a linear mixed model revealed that children followed an additive strategy to ratio reasoning-i.e., they focused on the magnitude of the target dimension across ratio sets as well as on the absolute magnitude of the ratio set. This approach did not vary substantially across representation formats. Results also showed an association between ratio reasoning and children's math problem-solving abilities; children with better math abilities performed better on ratio reasoning tasks and processed additional dimensions across ratio sets. Findings are discussed in terms of the processes that underlie ratio reasoning and add to the extant debate on whether true ratio reasoning is observed in young children.

Numeral order and the operationalization of the numerical system.

Recent years have witnessed an increase in research on how numeral ordering skills relate to children's and adults' mathematics achievement both cross-sectionally and longitudinally. Nonetheless, it remains unknown which core competency numeral ordering tasks measure, which cognitive mechanisms underlie performance on these tasks, and why numeral ordering skills relate to arithmetic and math achievement. In the current study, we focused on the processes underlying decision-making in the numeral order judgement task with triplets to investigate these questions. A drift-diffusion model for two-choice decisions was fit to data from 97 undergraduates. Findings aligned with the hypothesis that numeral ordering skills reflected the operationalization of the numerical system, where small numbers provide more evidence of an ordered response than large numbers. Furthermore, the pattern of findings suggested that arithmetic achievement was associated with the accuracy of the ordinal representations of numbers.

Maternal education and siblings: Agents of cognitive development in kindergarten.

In this study (n = 1000, Mage at K1entry  = 53.4 months, SD = 3.4; 53% females), we investigated the contributions of the family socioeconomic status (SES; maternal education and an income-related measure) and number and age of siblings to the development of children's math, reading, and working memory (WM) updating skills over the kindergarten years. Results from a multivariate multilevel growth curve model showed that children from more disadvantaged SES backgrounds already had a multifaceted developmental lag at kindergarten entry. Maternal education was the aspect of SES that more clearly affected the child's cognitive development; the mother's education predicted children's math, reading, and WM-updating skills at kindergarten entry as well as the rate of development of reading skills over the kindergarten years. Independently of SES status, children with more siblings also showed poorer reading and math skills than those in one-child families at kindergarten entry. We also found that both older and younger siblings affected, negatively, children's reading skills before they attended kindergarten-which suggests that the development of reading skills is more responsive to environmental factors during the first years than other aspects of the child development. The findings underscore the independent role of siblings upon entry to kindergarten, and the enduring role of maternal education even after children are exposed to formal schooling.

Cross- and Within-Domain Associations of Early Reading and Mathematical Skills: Changes Across the Preschool Years.

Existing research has mainly examined the role of cognitive correlates of early reading and mathematics from a stationary perspective that does not consider how these skills unfold and interact over time. This approach constraints the interpretation of cross-domain associations and the specificity of domain-specific covariates. In this study, we disentangle the role of these predictors and investigate cross-domain associations between reading, math, and two related domain-specific predictors (phonological awareness and fluency with number sets) over the kindergarten years (n=512, Mage=54months, SDage=3.5, 52% females). Results reveal that the overlap between reading and math skills changes over development. Reciprocal associations between reading and math abilities are observed at earlier stages; then, reading abilities become the lead force. Findings also show that phonological awareness and fluency with number sets are domain-specific predictors that do not contribute to cross-domain gains in academic skills. Indeed, there is a trend for domain-specific skills to be more strongly related to achievement at the beginning of formal education than at the beginning of kindergarten, which suggests an increasing differentiation of domains over the kindergarten years. Such findings have implications for the timing and nature of interventions that aim to support children's reading and mathematical development.

Socioeconomic status, home mathematics environment and math achievement in kindergarten: A mediation analysis.

Growing evidence suggests that parents' practices contribute to their children's cognitive development and that such practices may reflect SES disparities. This study investigated longitudinal interrelations between home mathematics environment (HME), children's math achievement, and two facets of SES (mother's educational attainment and household income-subsidy status) during the first year in kindergarten (n = 500 children; Mage at T1 = 57.3 months, SD = 3.8). Results revealed that these facets of SES operated through different mechanisms in kindergarten-the association between mothers' education and math growth at the end of K1 is fully mediated by HME and children's baseline math knowledge. Furthermore, only home math activities that explicitly supported the understanding of addition and subtraction contributed to children's math growth independently of SES background. The pattern of longitudinal associations suggests that the provision of home math activities may reflect children's mathematical abilities rather than SES disparities.

Disentangling the Mechanisms of Symbolic Number Processing in Adults' Mathematics and Arithmetic Achievement.

A growing body of research has shown that symbolic number processing relates to individual differences in mathematics. However, it remains unclear which mechanisms of symbolic number processing are crucial-accessing underlying magnitude representation of symbols (i.e., symbol-magnitude associations), processing relative order of symbols (i.e., symbol-symbol associations), or processing of symbols per se. To address this question, in this study adult participants performed a dots-number word matching task-thought to be a measure of symbol-magnitude associations (numerical magnitude processing)-a numeral-ordering task that focuses on symbol-symbol associations (numerical order processing), and a digit-number word matching task targeting symbolic processing per se. Results showed that both numerical magnitude and order processing were uniquely related to arithmetic achievement, beyond the effects of domain-general factors (intellectual ability, working memory, inhibitory control, and non-numerical ordering). Importantly, results were different when a general measure of mathematics achievement was considered. Those mechanisms of symbolic number processing did not contribute to math achievement. Furthermore, a path analysis revealed that numerical magnitude and order processing might draw on a common mechanism. Each process explained a portion of the relation of the other with arithmetic (but not with a general measure of math achievement). These findings are consistent with the notion that adults' arithmetic skills build upon symbol-magnitude associations, and they highlight the effects that different math measures have in the study of numerical cognition.

Processing of situational information in story problem texts. An analysis from on-line measures.

In three experiments, we investigated the extent to which readers process information related to the construction of a situation model when they are confronted with solving word problems. Considering that generation of inferences to match actions with particular goals is part of constructing of the situation model, we constructed "rich story problems", that is, word problems included in the context of a story, in which the characters propose goals, and then these goals are followed by actions to achieve it. In Experiments 1 and 2 the story problems were designed so that the character's goal was related to the activation of a problem schema, either explicitly (Experiment 1) or implicitly (Experiment 2). In Experiment 3 the problem schema activation was clearly separated from the goal information. In all three experiments, goal information availability was assessed by on-line measures. The results showed that participants processed situational information by keeping track of characters' goals. These results fit nicely with those studies that emphasize the role of situation model construction in word problem solving.

Arithmetic word problem solving: evidence for a magnitude-based mental representation.

Previous findings have suggested that number processing involves a mental representation of numerical magnitude. Other research has shown that sensory experiences are part and parcel of the mental representation (or "simulation") that individuals construct during reading. We aimed at exploring whether arithmetic word-problem solving entails the construction of a mental simulation based on a representation of numerical magnitude. Participants were required to solve word problems and to perform an intermediate figure discrimination task that matched or mismatched, in terms of magnitude comparison, the mental representations that individuals constructed during problem solving. Our results showed that participants were faster in the discrimination task and performed better in the solving task when the figures matched the mental representations. These findings provide evidence that an analog magnitude-based mental representation is routinely activated during word-problem solving, and they add to a growing body of literature that emphasizes the experiential view of language comprehension.

Inverse reference in subtraction performance: an analysis from arithmetic word problems.

Studies of elementary calculation have shown that adults solve basic subtraction problems faster with problems presented in addition format (e.g., 6 ± = 13) than in standard subtraction format (e.g., 13 - 6 = ). Therefore, it is considered that adults solve subtraction problems by reference to the inverse operation (e.g., for 13 - 6 = 7, "I know that 13 is 6 + 7") because presenting the subtraction problem in addition format does not require the mental rearrangement of the problem elements into the addition format. In two experiments, we examine whether adults' use of addition to solve subtractions is modulated by the arrangement of minuend and subtrahend, regardless of format. To this end, we used arithmetic word problems since single-digit problems in subtraction format would not allow the subtrahend to appear before the minuend. In Experiment 1, subtractions were presented by arranging minuend and subtrahend according to previous research. In Experiment 2, operands were reversed. The overall results showed that participants benefited from word problems where the subtrahend appears before the minuend, including subtractions in standard subtraction format. These findings add to a growing body of literature that emphasizes the role of inverse reference in adults' performance on subtractions.