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Humans and nonhuman animals share the capacity to estimate, without counting, the number of objects in a set by relying on an approximate number system (ANS). Only humans, however, learn the concepts and operations of symbolic mathematics. Despite vast differences between these two systems of quantification, neural and behavioral findings suggest functional connections. Another line of research suggests that the ANS is part of a larger, more general system of magnitude representation. Reports of cognitive interactions and common neural coding for number and other magnitudes such as spatial extent led us to ask whether, and how, nonnumerical magnitude interfaces with mathematical competence. On two magnitude comparison tasks, college students estimated (without counting or explicit calculation) which of two arrays was greater in number or cumulative area. They also completed a battery of standardized math tests. Individual differences in both number and cumulative area precision (measured by accuracy on the magnitude comparison tasks) correlated with interindividual variability in math competence, particularly advanced arithmetic and geometry, even after accounting for general aspects of intelligence. Moreover, analyses revealed that whereas number precision contributed unique variance to advanced arithmetic, cumulative area precision contributed unique variance to geometry. Taken together, these results provide evidence for shared and unique contributions of nonsymbolic number and cumulative area representations to formally taught mathematics. More broadly, they suggest that uniquely human branches of mathematics interface with an evolutionarily primitive general magnitude system, which includes partially overlapping representations of numerical and nonnumerical magnitude.analog magnitude | Weber's law | estimation | nonsymbolic magnitude precision | mathematical cognition H ow do humans come to understand mathematics? A common view is that the mental capacity for formal mathwhich includes access to symbolic notations of number, knowledge of quantitative concepts, and the implementation of computational operations-builds on a set of core abilities such as an intuitive, nonverbal sense of numerosity (1-5). Also known as the approximate number system (ANS), this nonsymbolic sense of numerical magnitude is shared with nonhuman animals (6) and is widespread across cultures (7). Unlike the acquisition of symbolic number (e.g., Arabic digits) and formal math concepts, which are learned via explicit instruction and allow for exact quantification, the ANS may be innate (8) and is characteristically "noisy," with variance increasing linearly as a function of the absolute numerical value (9). This imprecision can be modeled as overlapping Gaussian distributions along an internal continuum (10) and is captured by Weber's law, which holds that subjective differences in intensity are proportional to the objective ratios between values. When people compare numerical values under conditions that prevent counting or that do not...

In this chapter we provide an overview of what is known and unknown about the relationships between executive functions (EF) and written expression in children and adolescents, with an emphasis on evidence-based developmental associations and dissociations between various EF and writing skills in children and adolescents. We provide an overview of several key models of EF that are applicable to written expression, highlighting critical aspects of these models with respect to their importance for writing skills. What is known and unknown about the relationships between selected EF and writing skills for children and adolescents also is described. Based on these evidence-based findings, important associations and disassociations are underscored—when either suspected or supported by the available empirical literature. Finally, the chapter concludes with targeted directions for future research into advancing our understanding of these relationships across the paediatric developmental spectrum.

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Students with specific learning disabilities (SLDs) are more likely to exhibit internalizing and externalizing behaviors than typically developing peers. Virtually none of the literature, however, reports on the behaviors of students at-risk for writing disabilities (AR-WD). We compared the behaviors of writers who are AR-WD and typically developing writers (TDW) from first through fourth grade (N = 138). We found that students who are AR-WD were only significantly different from TDW on Externalizing and Total Behaviors at Grade 2. These findings illustrate the benefits of studying behavior across different forms of SLD, as it appears that students ARWD do not consistently manifest significant behaviors, although interventions that simultaneously target writing and behavior may be warranted and mutually beneficial.