A unified model of arithmetic with whole numbers, fractions, and decimals.

Abstract

This article describes UMA (Unified Model of Arithmetic), a theory of children’s arithmetic implemented as a computational model. UMA builds on FARRA (Fraction Arithmetic Reflects Rules and Associations; Braithwaite et al., 2017), a model of children’s fraction arithmetic. Whereas FARRA—like all previous models of arithmetic—focused on arithmetic with only one type of number, UMA simulates arithmetic with whole numbers, fractions, and decimals. The model was trained on arithmetic problems from the first to sixth grade volumes of a math textbook series; its performance on tests administered at the end of each grade was compared to the performance of children in prior empirical research. In whole number arithmetic (Study 1), fraction arithmetic (Study 2), and decimal arithmetic (Study 3), UMA displayed types of errors, effects of problem features on error rates, and individual differences in strategy use that resembled those documented in the previous studies of children. Further, UMA generated correlations between individual differences in basic and advanced arithmetic skills similar to those observed in longitudinal studies of arithmetic development (Study 4). The results support UMA’s main theoretical assumptions regarding arithmetic development: (a) most errors reflect small deviations from standard procedures via two mechanisms, overgeneralization and omission; (b) between-problem variations in error rates reflect effects of intrinsic difficulty and differential amounts of practice; and (c) individual differences in strategy use reflect underlying variation in parameters governing learning and decision making.