A model of the relations among cognitive precursors, early numeracy skill, and mathematical outcomes was tested for 182 children from 4.5 to 7.5 years of age. The model integrates research from neuroimaging, clinical populations, and normal development in children and adults. It includes 3 precursor pathways: quantitative, linguistic, and spatial attention. These pathways (a) contributed independently to early numeracy skills during preschool and kindergarten and (b) related differentially to performance on a variety of mathematical outcomes 2 years later. The success of the model in accounting for performance highlights the need to understand the fundamental underlying skills that contribute to diverse forms of mathematical competence.

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Adults solved simple subtraction problems (e.g., 16 - 9). Half of the 32 participants provided immediate self-reports of their solution processes on each problem. Performance was analyzed using both traditional descriptive statistics (i.e., means, standard deviations, and percentage of errors) and with statistics derived from fitting the ex-Gaussian distributions to latencies (i.e., mu and tau). The results support the view that ex-Gaussian analyses can be useful in exploring patterns of procedure selection that relate both to characteristics of the stimuli (e.g., problem size) and to characteristics of the participants (e.g., arithmetic skill). More generally, the results provide further evidence that adults use a variety of procedures to solve simple subtraction problems and that these choices are related to patterns of performance on more complex problems that require calculation.

Is the locus of the problem-size effect in mental arithmetic different across cultures? In a novel approach to this question, the ex-Gaussian distributional model was applied to response times for large (e.g., 8 3 9) and small (e.g., 2 3 3) problems obtained from Chinese and Canadian graduate students in a multiplication production task (LeFevre & Liu, 1997). The problem-size effect for the Chinese group occurred in m (the mean of the normal component), whereas the problem-size effect for the Canadian group occurred in both m and t (the mean of the exponential component). The results support the position that the problem-size effect for the Chinese group is purely a memory-retrieval effect, whereas for the Canadian group, it is an effect of both retrieval and the use of nonretrieval solution procedures.

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