A striking way in which humans differ from non-human primates is in their ability to represent numerical quantity using abstract symbols and to use these 'mental tools' to perform skills such as exact calculations. How do functional brain circuits for the symbolic representation of numerical magnitude emerge? Do neural representations of numerical magnitude change as a function of development and the learning of mental arithmetic? Current theories suggest that cultural number symbols acquire their meaning by being mapped onto non-symbolic representations of numerical magnitude. This Review provides an evaluation of this contention and proposes hypotheses to guide investigations into the neural mechanisms that constrain the acquisition of cultural representations of numerical magnitude.

• NOTICE: this is the author's version of a work that was accepted for publication in Trends in Neuroscience and Education. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be

Using Developmental Trajectories to Understand Developmental DisordersPurpose: In this article, the authors present a tutorial on the use of developmental trajectories for studying language and cognitive impairments in developmental disorders and compare this method with the use of matching. Method: The authors assess the strengths, limitations, and practical implications of each method. The contrast between the methodologies is highlighted using the example of developmental delay and the criteria used to distinguish delay from atypical development. Results: The authors argue for the utility of the trajectory approach, using illustrations from studies investigating language and cognitive impairments in individuals with Williams syndrome, Down syndrome, and autism spectrum disorder. Conclusion: Two conclusions were reached: (a) An understanding of the underlying mechanism will be furthered by the richer descriptive vocabulary provided by the trajectories approach (e.g., in distinguishing different types of delay) and (b) an optimal design for studying developmental disorders is to combine initial cross-sectional designs with longitudinal follow-up.

Math relies on mastery and integration of a wide range of simpler numerical processes and concepts. Recent work has identified several numerical competencies that predict variation in math ability. We examined the unique relations between eight basic numerical skills and early arithmetic ability in a large sample (N = 1391) of children across grades 1-6. In grades 1-2, children's ability to judge the relative magnitude of numerical symbols was most predictive of early arithmetic skills. The unique contribution of children's ability to assess ordinality in numerical symbols steadily increased across grades, overtaking all other predictors by grade 6. We found no evidence that children's ability to judge the relative magnitude of approximate, nonsymbolic numbers was uniquely predictive of arithmetic ability at any grade. Overall, symbolic number processing was more predictive of arithmetic ability than nonsymbolic number processing, though the relative importance of symbolic number ability appears to shift from cardinal to ordinal processing.

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Using functional magnetic resonance imaging, we examined developmental differences in the functional neuroanatomy underlying symbolic number processing. Twelve adults and 12 children had to judge the relative magnitude of two single-digit Arabic numerals. We investigated which brain areas were significantly (P<0.005, uncorrected) more activated during processing of number pairs with small relative to large numerical distances. In the adult group, symbolic distance modulated bilateral parietal regions. In contrast, the group of children primarily engaged frontal regions. We conclude that the functional neuroanatomy underlying symbolic numerical magnitude processing undergoes an ontogenetic shift towards greater parietal engagement. This change may reflect maturation of underlying representations and increasing automaticity in mapping between numerical symbols and the magnitudes they represent.

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