Achievement in mathematics is predicted by an individual's domain-specific factual knowledge, procedural skill and conceptual understanding as well as domain-general executive function skills. In this study we investigated the extent to which executive function skills contribute to these three components of mathematical knowledge, whether this mediates the relationship between executive functions and overall mathematics achievement, and if these relationships change with age. Two hundred and ninety-three participants aged between 8 and 25years completed a large battery of mathematics and executive function tests. Domain-specific skills partially mediated the relationship between executive functions and mathematics achievement: Inhibitory control within the numerical domain was associated with factual knowledge and procedural skill, which in turn was associated with mathematical achievement. Working memory contributed to mathematics achievement indirectly through factual knowledge, procedural skill and, to a lesser extent, conceptual understanding. There remained a substantial direct pathway between working memory and mathematics achievement however, which may reflect the role of working memory in identifying and constructing problem representations. These relationships were remarkably stable from 8years through to young adulthood. Our findings help to refine existing multi-component frameworks of mathematics and understand the mechanisms by which executive functions support mathematics achievement.

Despite educational initiatives to improve mathematics achievement there has been a disappointing lack of improvement in mathematics outcomes in many Western societies (Vorderman et al. 2011). Given this lack of progress, researchers have attempted to better understand the cognitive processes that underlie mathematics performance. An improved theoretical understanding of the factors involved in mathematics processing can provide the starting point from which to develop pedagogy to support mathematics learning in all young people. Over the past few decades, researchers have identified two classes of cognitive skills that are important for mathematics achievement. The first of these concerns domainspecific skills such as symbol knowledge, counting skill, and underlying numerical representations. Alongside these, researchers have identified domain-general skills which are involved in learning in many areas but which are particularly important for mathematics (e.g. language, IQ and spatial ability). Particular attention has been paid to executive functions-the skills required to monitor and control thought and action-and the role they play in learning and performing mathematics (see reviews by Cragg and Gilmore 2014; Bull and Lee 2014). Three types of executive functions have been identified: monitoring and manipulating information in mind (working memory), suppressing distracting information and unwanted responses (inhibition), and flexible thinking (shifting). To date, few models of mathematical cognition have considered the role of executive function skills, particularly inhibition. LeFevre et al. (2010) identified a role for attentional processes in their Pathways Model. This model proposed that attentional skills have a direct impact on mathematical performance in a variety of domains independent of linguistic or quantitative skills. However, the specific role of inhibition skill was not specified in this model.

Our ability to perform arithmetic relies heavily on working memory, the manipulation and maintenance of information in mind. Previous research has found that in adults, procedural strategies, particularly counting, rely on working memory to a greater extent than retrieval strategies. During childhood there are changes in the types of strategies employed, as well as an increase in the accuracy and efficiency of strategy execution. As such it seems likely that the role of working memory in arithmetic may also change, however children and adults have never been directly compared. This study used traditional dual-task methodology, with the addition of a control load condition, to investigate the extent to which working memory requirements for different arithmetic strategies change with age between 9–11 years, 12–14 years and young adulthood. We showed that both children and adults employ working memory when solving arithmetic problems, no matter what strategy they choose. This study highlights the importance of considering working memory in understanding the difficulties that some children and adults have with mathematics, as well as the need to include working memory in theoretical models of mathematical cognition.

When children learn to count, they map newly acquired symbolic representations of number onto preexisting nonsymbolic representations. The nature and timing of this mapping is currently unclear. Some researchers have suggested this mapping process helps children understand the cardinal principle of counting, while other evidence suggests that this mapping only occurs once children have cardinality understanding. One difficulty with the current literature is that studies have employed tasks that only indirectly assess children's nonsymbolic-symbolic mappings. We introduce a task in which preschoolers made magnitude comparisons across representation formats (e.g., dot arrays vs. verbal number), allowing a direct assessment of mapping. We gave this task to 60 children aged 2;7 -4;10, together with counting and Give-a-Number tasks. We found that some children could map between nonsymbolic quantities and the number words they understood the cardinal meaning of, even if they had yet to grasp the general cardinality principle of counting.Keywords: counting, magnitude comparison, cardinality, preschool children, number Running head: PRESCHOOL MAGNITUDE REPRESENTATIONS 3 Magnitude Representations and Counting Skills in Preschool ChildrenWe know from more than a decade's worth of research that infants, children and adults can represent and manipulate numerical information nonsymbolically, without number words or digits. These nonsymbolic representations are robust across multiple modalities and set sizes. Children and adults can compare, add and subtract small and large quantities in visual arrays (Barth, Kanwisher, & Spelke, 2003;McCrink & Wynn, 2004), auditory sequences (Barth et al., 2003;Barth, La Mont, Lipton, & Spelke, 2005) and moving displays of actions (e.g., puppet jumps) (Wood & Spelke, 2005;Wynn, 1996;Wynn, Bloom, & Chiang, 2002).The nonsymbolic representations employed in these tasks are approximate and in an analogue format. They are inherently noisy and the variance associated with them increases with the absolute size of the magnitudes represented. As a result, success on these tasks depends on the ratio (or numerical distance) between the numerosities to be compared 1 . As the quantities get closer together, discrimination becomes more effortful and less precise.Importantly, the precision of these representations varies across individuals and increases over development. Infants can discriminate numerosities with ratios as small as 2:3, whilst preschool children show a ratio-limit of 3:4, and adults, 7:8 (Barth et al., 2003;Feigenson, Dehaene, & Spelke, 2004).When children begin to count they learn to use external symbols to represent number.These symbolic representations enable exact number comparison and manipulation. There is evidence that when children acquire this symbolic system, the preexisting nonsymbolic system is not overridden; rather, nonsymbolic representations become mapped onto the newly acquired symbolic representations. The evidence for this is at least threefold. Firstly, children...

Large individual differences in children's mathematics achievement are observed from the start of schooling. Previous research has identified three cognitive skills that are independent predictors of mathematics achievement: procedural skill, conceptual understanding and working memory. However, most studies have only tested independent effects of these factors and failed to consider moderating effects. We explored the procedural skill, conceptual understanding and working memory capacity of 75 children aged 5 to 6 years as well as their overall mathematical achievement. We found that, not only were all three skills independently associated with mathematics achievement, but there was also a significant interaction between them. We found that levels of conceptual understanding and working memory moderated the relationship between procedural skill and mathematics achievement such that there was a greater benefit of good procedural skill when associated with good conceptual understanding and working memory. Cluster analysis also revealed that children with equivalent levels of overall mathematical achievement had differing strengths and weaknesses across these skills. This highlights the importance of considering children's skill profile, rather than simply their overall achievement.