Learning about fractions is a critical step on the path to high school mathematics, yet many children never masterbasic knowledge such as fraction arithmetic procedures. To better understand these difficulties, the present study describes acomputational model of fraction arithmetic problem solving. The model demonstrates that the majority of empirically observederrors over all four arithmetic operations can be explained by only two error-generating mechanisms – overgeneralization andrepair. Further, by assuming probabilistic selection of solution procedures using associative strengths learned from experience,the model predicts two other empirical phenomena: (1) variation in error rates and relative frequencies of specific errorsas a function of problem features, and (2) variable strategy selection within and between problems and individuals. Beyondproviding a formal account of errors, the model was used to simulate the effects of variation of instructional parameters, leadingto novel predictions regarding potentially effective instructional designs.

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.

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.

There is a norm in psychology to use causally ambiguous statistical language, rather than straightforward causal language, when describing methods and results of nonexperimental studies. However, causally ambiguous language may inhibit a critical examination of the studys causal assumptions and lead to a greater acceptance of policy recommendations that rely on causal interpretations of nonexperimental findings. In a preregistered experiment, 142 psychology faculty, postdocs, and doctoral students (54% female), ages 22-67 (M = 33.20, SD = 8.96), rated the design and analysis from hypothetical studies with causally ambiguous statistical language as of higher quality (by .34-.80 SD) and as similarly or more supportive (by .16-.27 SD) of policy recommendations than studies described in straightforward causal language. Thus, using statistical rather than causal language to describe nonexperimental findings did not decrease, and may have increased, perceived support for implicitly causal conclusions.