Predicting Parameters of Strategy Choice in Rational Number Arithmetic

• In Event: Development of Number Sense

Abstract

Strategy choice is a critical aspect of math problem solving, and understanding the mechanisms of strategy choice is a correspondingly important goal for research. This study aimed to advance understanding of children’s strategy choices in rational number arithmetic. Strategy choice is central in this domain, in that most errors involve using incorrect strategies. Further, children’s strategy choices display substantial individual differences. The goal of this study was to characterize these individual differences and identify predictors of them.

Our investigation relied on a recent theory of children’s arithmetic development, UMA (Unified Model of Arithmetic; Author, submitted). According to this theory, differences in strategy choice reflect variations in global bias favoring a particular strategy and in effects of relevant and irrelevant problem features on strategy choices. (A feature is relevant if it affects the appropriate solution strategy and irrelevant otherwise; for example, in “3/5*1/5”, the arithmetic operation is relevant but the presence of equal denominators is irrelevant.) We hypothesized that individual differences in global bias and effects of relevant features would be associated with differences in conceptual knowledge, whereas differences in effects of irrelevant features would be associated with differences in inhibitory control.

To test these hypotheses, children (N = 120 fifth to ninth graders) chose between correct and incorrect strategies for solving addition and multiplication problems involving fractions and decimals. Hierarchical Bayesian logistic regression was used to estimate, for each child, parameters representing global bias and effects of relevant and irrelevant features. Finally, linear regression was used to analyze how children’s parameter values related to individual differences in (1) conceptual knowledge, indexed by accuracy on a rational number magnitude comparison task; and (2) inhibition, as assessed by a numerical Stroop task.

As predicted, individual differences in magnitude comparison predicted differences in the relevant feature parameters for both fraction and decimal arithmetic. Unexpectedly, inhibition predicted differences in global bias for fraction arithmetic: Children with weaker inhibition displayed greater global bias. Neither variable predicted differences in global bias for decimal arithmetic, or differences in the irrelevant feature parameters for either fraction or decimal arithmetic. Finally, the relevant feature parameter for fraction arithmetic was correlated with the relevant feature parameter for decimal arithmetic, but neither global bias nor the irrelevant feature parameters were correlated across notations.

The findings have several implications for understanding individual differences in children’s strategy choices in fraction and decimal arithmetic. First, the relation between magnitude comparison and the relevant feature parameters may reflect conceptual understanding of fractions and decimals helping children to know which problem features are relevant and why. Second, the association between weak inhibition and increased global bias in fraction arithmetic may reflect inhibition being required to resist the impulse to overuse a frequently practiced strategy. Third, the reason why inhibition did not predict the irrelevant feature parameters may be that many children incorrectly believe irrelevant features to be relevant, and so do not inhibit effects of such features on their strategy choices.

Authors

• David Braithwaite, Florida State University
  Presenting Author

• Anna Rafferty, Carleton College