We describe UMA (Unified Model of Arithmetic), a theory of children’s arithmetic implemented as a computational model. UMA extends a theory of fraction arithmetic (Braithwaite et al., 2017) to include arithmetic with whole numbers and decimals. We evaluated UMA in the domain of decimal arithmetic by training the model on problems from a math textbook series, then testing it on decimal arithmetic problems that were solved by 6th and 8th graders in a previous study. UMA’s test performance closely matched that of children, supporting three assumptions of the theory: (1) most errors reflect small deviations from standard procedures, (2) between-problem variations in error rates reflect the distribution of input that learners receive, and (3) individual differences in strategy use reflect underlying variation in learning parameters.

Conditional reasoning, or reasoning with if-then statements, depends in part on knowledge. However, the mechanisms underlying this dependence are not fully understood. We propose that example knowledge—the ability to generate and categorize examples of logical possibilities—plays a central role, and therefore hypothesize that individual differences in example knowledge contribute to differences in conditional reasoning. Two studies tested this hypothesis in the domain of algebra. In Study 1, individual differences in example knowledge predicted differences in conditional reasoning about algebra when controlling for everyday conditional reasoning and general algebra knowledge. In Study 2, training designed to improve example knowledge improved conditional reasoning about algebra. We discuss implications of the findings regarding the mechanisms underlying the knowledge-dependence of conditional reasoning and the nature of individual differences in conditional reasoning.

Study sequence can have a profound impact on learning. Previous research has often shown advantages for interleaved over blocked study, though the reverse has also been found. Learners typically prefer blocking even in situations for which interleaving is superior. The present study investigated learner regulation of study sequence, and its effects on learning in an ecologically valid context – university students using an online tutorial relevant to an exam that counted toward their course grades. The majority of participants blocked study by problem category, and this tendency was positively associated with subsequent exam performance. The results suggest that preference for blocked study may be adaptive under some circumstances, and highlight the importance of identifying task environments under which different study sequences are most effective.