Simple and complex addition problems were presented for true/false verification to 22 subjects across two times of measurement to test the general model for simple and complex addition proposed by Widaman, Cormier, & Geary (1985). Models fit to average RT data revealed that subjects were processing complex problems columnwise, beginning with the units column. Column sums seemed to be obtained through an incrementing process, and subjects exited problems as soon as a colimn error was encountered. Group level models were the same across complex problem types and for both times of measurement. However, individual level analyses suggested that nearly half of the subjects used a different processing strategy to obtain column sums for the second time of measurement. Results support the multi-staged model proposed by Widaman et al. (1985), but individual level results suggest that information processing models developed from group data may not represent the processing strategies used by all subjects, or the same subjects at different times.

Substantial longitudinal relations between children's early mathematics achievement and their much later mathematics achievement are firmly established. These findings are seemingly at odds with studies showing that early educational interventions have diminishing effects on children's mathematics achievement across time. We hypothesized that individual differences in children's later mathematical knowledge are more an indicator of stable, underlying characteristics related to mathematics learning throughout development than of direct effects of early mathematical competency on later mathematical competency. We tested this hypothesis in two longitudinal data sets, by simultaneously modeling effects of latent traits (stable characteristics that influence learning across time) and states (e.g., prior knowledge) on children's mathematics achievement over time. Latent trait effects on children's mathematical development were substantially larger than state effects. Approximately 60% of the variance in trait mathematics achievement was accounted for by commonly used control variables, such as working memory, but residual trait effects remained larger than state effects. Implications for research and practice are discussed.

Children's understanding of the quantities represented by number words (i.e., cardinality) is a surprisingly protracted but foundational step in their learning of formal mathematics. The development of cardinal knowledge is related to one or two core, inherent systems - the approximate number system (ANS) and the object tracking system (OTS) - but whether these systems act alone, in concert, or antagonistically is debated. Longitudinal assessments of 198 preschool children on OTS, ANS, and cardinality tasks enabled testing of two single-mechanism (ANS-only and OTS-only) and two dual-mechanism models, controlling for intelligence, executive functions, preliteracy skills, and demographic factors. Measures of both OTS and ANS predicted cardinal knowledge in concert early in the school year, inconsistent with single-mechanism models. The ANS but not the OTS predicted cardinal knowledge later in the school year as well the acquisition of the cardinal principle, a critical shift in cardinal understanding. The results support a Merge model, whereby both systems initially contribute to children's early mapping of number words to cardinal value, but the role of the OTS diminishes over time while that of the ANS continues to support cardinal knowledge as children come to understand the counting principles.