The contributions of domain-general abilities and domain-specific knowledge to subsequent mathematics achievement were longitudinally assessed (n = 167) through 8th grade. First grade intelligence and working memory and prior grade reading achievement indexed domain-general effects and domain-specific effects were indexed by prior grade mathematics achievement and mathematical cognition measures of prior grade number knowledge, addition skills, and fraction knowledge. Use of functional data analysis enabled grade-by-grade estimation of overall domain-general and domain-specific effects on subsequent mathematics achievement, the relative importance of individual domain-general and domain-specific variables on this achievement, and linear and non-linear across-grade estimates of these effects. The overall importance of domain-general abilities for subsequent achievement was stable across grades, with working memory emerging as the most important domain-general ability in later grades. The importance of prior mathematical competencies on subsequent mathematics achievement increased across grades, with number knowledge and arithmetic skills critical in all grades and fraction knowledge in later grades. Overall, domain-general abilities were more important than domain-specific knowledge for mathematics learning in early grades but general abilities and domain-specific knowledge were equally important in later grades.

The study assessed the relations among acuity of the inherent approximate number system (ANS), performance on measures of symbolic quantitative knowledge, and mathematics achievement for a sample of 138 (64 boys) preschoolers. The Weber fraction (a measure of ANS acuity) and associated task accuracy were significantly correlated with mathematics achievement following one year of preschool, and predicted performance on measures of children's explicit knowledge of Arabic numerals, number words, and cardinal value, controlling for age, sex, parental education, intelligence, executive control, and preliteracy knowledge. The relation between ANS acuity, as measured by the Weber fraction and task accuracy, and mathematics achievement was fully mediated by children's performance on the symbolic quantitative tasks, with knowledge of cardinal value emerging as a particularly important mediator. The overall pattern suggests that ANS acuity facilitates the early learning of symbolic quantitative knowledge and indirectly influences mathematics achievement through this knowledge.

Visuospatial competencies are related to performance in mathematical domains in adulthood, but are not consistently related to mathematics achievement in children. We confirmed the latter for first graders and demonstrated that children who show above average first-to-fifth grade gains in visuospatial memory have an advantage over other children in mathematics. The study involved the assessment of the mathematics and reading achievement of 177 children in kindergarten to fifth grade, inclusive, and their working memory capacity and processing speed in first and fifth grade. Intelligence was assessed in first grade and their second to fourth grade teachers reported on their in-class attentive behavior. Developmental gains in visuospatial memory span (d = 2.4) were larger than gains in the capacity of the central executive (d = 1.6) that in turn were larger than gains in phonological memory span (d = 1.1). First to fifth grade gains in visuospatial memory and in speed of numeral processing predicted end of fifth grade mathematics achievement, as did first grade central executive scores, intelligence, and in-class attentive behavior. The results suggest there are important individual differences in the rate of growth of visuospatial memory during childhood and that these differences become increasingly important for mathematics learning.

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