Functional data analysis : children's mathematical development
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[ACCESS RESTRICTED TO THE UNIVERSITY OF MISSOURI AT AUTHOR'S REQUEST.] Bailey et al. (2014) suggested that children's mathematical development is related more to trait characteristics than to prior mathematical development. However, their study analysis was designed to examine trait and state effects that did not vary flexibly across grades. This may lead to statistical bias in estimating the magnitude of effects, as well as educational interventions that are not targeted appropriately in time. We utilize both functional data analysis (FDA) and mixed FDA approaches to describe in detail grade-on-grade changes in the magnitude of trait and state effects. Mixed FDA methodology confirms the scientific hypotheses from Bailey et al. (2014) in a very general model setting. It acommodates both flexibly varying regression effects and testing of statistical hypotheses about them. In particular, trait terms account for 81.6% of variability in mathematical development vs. 68.2% for prior mathematical development. Additionally, of the four 'mathematical cognition' terms, the effects of simple addition and number line were not significant and these two terms were dropped out of the model. The effect of prior reading was not significant either. The effects of intelligence and prior fluency of processing the magnitudes associated with Arabic numerals were about constant over time. Skill at solving more complex addition problems became increasingly important over time, revealing an overall U-shaped pattern; central executive effect revealed an overall inverted-U pattern. With the possible exception of the central executive and complex addition, the pattern suggests that despite grade-to-grade differences in the significance of individual state and trait terms, the magnitude of the estimates of the effects of these terms is consistent across grades. Notably, in terms of relative magnitude, the regression term effects in first grade were in the same order as in 8th grade. We are thus able to answer research questions related to potential educational interventions in each grade that may improve children's mathematical development.
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