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Recent research examining children's early mathematical abilities has focused primarily on number and operations (e.g., counting, addition) with considerably less attention directed to the role of other possible dimensions of early mathematical abilities, such as, measurement, geometry, and patterning. The current study examined the dimensionality of informal mathematical abilities by conducting categorical confirmatory factor analysis (CCFA) using data from a large sample of preschool children from low-income families (N=1630; Mean age = 4.46 years, SD = .37) using the Child Math Assessment (CMA; . The best fitting model consisted of four factors of Number and Operations, Measurement, Geometry, and Patterning, with the Number and Operations factor explaining common variance in three first-order factors of Numbering, Operations, and Relations. These findings support the view that informal mathematical knowledge is a multi-dimensional construct comprised of each of these separable dimensions. Additionally, a Multiple Indicators Multiple Causes model was used to determine if mathematical ability differed for male and female preschoolers on each of the four factors or on each of the 35 items of the CMA. Results showed no differences for mathematical abilities between males and females at this age. Future research and curricular implications are discussed.

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Policy makers face dilemmas when choosing a policy, program, or practice to implement. Researchers in education, public health, and other fields have proposed a sequential approach to identifying interventions worthy of broader adoption, involving pilot, efficacy, effectiveness, and scale-up studies. In this article, we examine a scale-up of an early math intervention to the state level, using a cluster randomized controlled trial. The intervention, Pre-K Mathematics, has produced robust positive effects on children's math ability in prior pilot, efficacy, and effectiveness studies. In the current study, we ask if it remains effective at a larger scale in a heterogeneous collection of pre-K programs that plausibly represent all low-income families with a child of pre-K age who live in California. We find that Pre-K Mathematics remains effective at the state level, with positive and statistically significant effects (effect size on the Early Childhood Longitudinal Study, Birth Cohort Mathematics Assessment = .30, p < .01). In addition, we develop a framework of the dimensions of scale-up to explain why effect sizes might decrease as scale increases. Using this framework, we compare the causal estimates from the present study to those from earlier, smaller studies. Consistent with our framework, we find that effect sizes have decreased over time. We conclude with a discussion of the implications of our study for how we think about the external validity of causal relationships.

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