The process by which adults develop competence in symbolic mathematics tasks is poorly understood. Nonhuman animals, human infants, and human adults all form nonverbal representations of the approximate numerosity of arrays of dots and are capable of using these representations to perform basic mathematical operations. Several researchers have speculated that individual differences in the acuity of such nonverbal number representations provide the basis for individual differences in symbolic mathematical competence. Specifically, prior research has found that 14-year-old children's ability to rapidly compare the numerosities of two sets of colored dots is correlated with their mathematics achievements at ages 5-11. In the present study, we demonstrated that although when measured concurrently the same relationship holds in children, it does not hold in adults. We conclude that the association between nonverbal number acuity and mathematics achievement changes with age and that nonverbal number representations do not hold the key to explaining the wide variety of mathematical performance levels in adults.

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This paper reports on a collaborative exercise designed to generate a coherent agenda for research on mathematical cognition. Following an established method, the exercise brought together 16 mathematical cognition researchers from across the fields of mathematics education, psychology and neuroscience. These participants engaged in a process in which they generated an initial list of research questions with the potential to significantly advance understanding of mathematical cognition, winnowed this list to a smaller set of priority questions, and refined the eventual questions to meet criteria related to clarity, specificity and practicability. The resulting list comprises 26 questions divided into six broad topic areas: elucidating the nature of mathematical thinking, mapping predictors and processes of competence development, charting developmental trajectories and their interactions, fostering conceptual understanding and procedural skill, designing effective interventions, and developing valid and reliable measures. In presenting these questions in this paper, we intend to support greater coherence in both investigation and reporting, to build a stronger base of information for consideration by policymakers, and to encourage researchers to take a consilient approach to addressing important challenges in 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...

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Mathematics-related affect comprises an individual's attitudes, beliefs, emotions and motivations towards mathematics. These affective constructs have been widely studied in mathematics education and cognitive psychology and have been shown to be related to cognitive outcomes such as performance on a range of mathematical tasks. However, it is not yet clear how these constructs develop, or how they relate to cognitive factors in young children who are in the early stages of learning mathematics. As such, the current special issue focuses on mathematics-related affect in primary school children aged 4 to 10 years. It brings together five recent empirical studies and two discussant articles looking at the development of attitudes, beliefs, emotions and motivations towards mathematics, and the relations between affective and cognitive factors in these young age groups. In this introductory paper we provide some brief historical context, followed by a rationale for the special issue, and an overview of its structure and scope.

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