Toward a unified theory of rational number arithmetic.

Abstract

To explain children's difficulties learning fraction arithmetic, Braithwaite et al. (2017) proposed FARRA, a theory of fraction arithmetic implemented as a computational model. The present study tested predictions of the theory in a new domain, decimal arithmetic, and investigated children's use of conceptual knowledge in that domain. Sixth and eighth grade children (N = 92) solved decimal arithmetic problems while thinking aloud and afterward explained solutions to decimal arithmetic problems. Consistent with the hypothesis that FARRA's theoretical assumptions would generalize to decimal arithmetic, results supported 3 predictions derived from the model: (a) accuracies on different types of problems paralleled the frequencies with which the problem types appeared in textbooks; (b) most errors involved overgeneralizafion of strategies that would be correct for problems with different operations or types of number; and (c) individual children displayed patterns of strategy use predicted by FARRA. We also hypothesized that during routine calculation, overt reliance on conceptual knowledge is most likely among children who lack confidence in their procedural knowledge. Consistent with this hypothesis, (d) many children displayed conceptual knowledge when explaining solutions but not while solving problems; (e) during problem-solving, children who more often overtly used conceptual knowledge also displayed doubt more often; and (f) problem solving accuracy was positively associated with displaying conceptual knowledge while explaining, but not with displaying conceptual knowledge while solving problems. We discuss implications of the results for rational number instruction and for the creation of a unified theory of rational number arithmetic.