Against the background of neuroimaging studies on how the brain processes numbers, there is now converging evidence that numerical magnitude representations are crucial for successful mathematics achievement. One major drawback of this research is that it mainly investigated mathematics performance as measured through general standardized achievement tests. We extended this research by investigating the association between these numerical magnitude representations and children's strategy use during single-digit arithmetic. Our findings reveal that children's symbolic but not nonsymbolic numerical magnitude processing skills are associated with individual differences in arithmetic. Children with better access to magnitude representations from symbolic digits, retrieve more facts from their memory and are faster in executing fact retrieval as well as procedural strategies. These associations remain even when intellectual ability, digit naming and general mathematics achievement were additionally controlled for. This all indicates that particularly the access to numerical meaning from Arabic symbols is key for children's arithmetic strategy development, which suggests that educators and remedial teachers should focus on connecting Arabic symbols to the quantities they represent.

Keywords:longitudinal design numerical processing arithmetic development fact retrieval The present longitudinal study investigated whether children's numerical processing skills at the start of formal schooling predict subsequent development in single-digit arithmetic and fact retrieval. At the start of first grade, we administered measures of numerical processing (digit naming, symbolic numerical magnitude comparison, nonsymbolic numerical magnitude comparison) as well as measures of intellectual ability, preschool mathematical abilities, working memory and processing speed. Our longitudinal data indicate that children's numerical processing skills at primary school entrance were predictively related to their future competence in singledigit arithmetic and their reliance on arithmetic fact retrieval. This association was not explained by children's intellectual ability, preschool mathematical abilities, verbal working memory, visual-spatial short-term memory and processing speed. These findings indicate that numerical processing skills precede children's development in single-digit arithmetic.

Disabilities in reading and arithmetic often co-occur, but (dis)abilities in reading and arithmetic have mostly been studied in isolation from each other. This study explicitly focused on the co-development of early reading and early arithmetic before primary education. The Multiple Deficit Model was used as theoretical framework (Pennington, 2006). According to this model, the overlap between early reading and early arithmetic is due to a constellation of shared and unique cognitive correlates. Therefore, we investigated whether key cognitive correlates of one academic ability also correlate with the other. Participants were 188 five-year-old kindergartners who had not yet been formally instructed in reading and arithmetic. Phonological awareness was selected as reading-specific cognitive correlate and (non)symbolic numerical magnitude processing and numeral recognition were considered as arithmetic-specific cognitive correlates. We administered a productive letter knowledge task as a proxy of early reading. Early arithmetic was assessed with simple problems such as 2 + 3 =?. Regression analyses and Bayesian hypothesis testing revealed significant correlations between early reading and early arithmetic before children start primary education. Phonological awareness predicted not only early reading but also, early arithmetic, even when controlling for early reading and arithmetic-specific cognitive correlates. Likewise, numeral recognition predicted not only early arithmetic, but also early reading, even when controlling for early arithmetic and phonological awareness. Phonological awareness and numeral recognition can be considered shared cognitive correlates of both academic domains. In contrast, non-symbolic and symbolic numerical magnitude processing skills were specifically correlated to early arithmetic, and not to early reading, indicating that they are unique to only one academic domain. In line with the Multiple Deficit Model, our data suggest that early reading and early arithmetic have a shared as well as unique underlying cognitive basis. Further unravelling what these academic abilities have in common can be of high value for detecting children at risk already before their transition to formal primary education.

The current longitudinal study tried to capture profiles of individual differences in children's arithmetic fact development. We used a model-based clustering approach to delineate profiles of arithmetic fact development based on empirically derived differences in parameters of arithmetic fact mastery repeatedly assessed at the start of three subsequent school years: third, fourth, and fifth grades. This cluster analysis revealed three profiles in a random sample-slow and variable (n = 8), average (n = 24), and efficient (n = 20)-that were marked by differences in children's development in arithmetic fact mastery from third grade to fifth grade. These profiles did not differ in terms of age, sex, socioeconomic status, or intellectual ability. In addition, we explored whether these profiles varied in cognitive skills that have been associated with individual differences in single-digit arithmetic. The three profiles differed in nonsymbolic and symbolic numerical magnitude processing as well as phonological processing, but not in digit naming or working memory. After also controlling for cluster differences in general mathematics achievement and reading ability, only differences in symbolic numerical magnitude processing remained significant. Taken together, our longitudinal data reveal that symbolic numerical magnitude processing represents an important variable that contributes to individual variability in children's acquisition of arithmetic facts.

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