Creating a Theory of Decimal Arithmetic Learning

Florida State University, Tallahassee FL

Investigators

Abstract

Proficiency with rational numbers-fractions, decimals, and percentages-is essential for success in more advanced mathematics such as algebra. It is also important for occupational success; majorities of both white- and blue-collar workers report using rational numbers in their jobs. Yet, many children struggle with rational numbers even after years of instruction. The goal of this project is to create a theory of children's learning in one area of rational numbers: decimal arithmetic. The project will identify types of knowledge that help children to learn decimal arithmetic more easily, clarify the mechanisms by which this facilitation occurs, and develop a computational model that simulates the process of learning decimal arithmetic. Based on the results, recommendations will be generated for improving children's learning of decimal arithmetic including recommendations (1) to focus classroom and practice time on conceptual approaches that are used by successful learners, (2) to place special emphasis on types of problem that pose difficulty for children, (3) to devote classroom time to illustrating common errors and explaining why they are incorrect, and (4) to use discussion of common errors as an opportunity to illustrate general concepts. These recommendations are anticipated to have implications for improving mathematics instruction in general. Learning mathematics involves learning both concepts and procedures. Concepts include principles and relations; procedures are step-by-step action sequences for solving problems. Understanding of concepts is believed to help children learn procedures, but how this occurs is not known. This project aims to create a theory of how conceptual understanding - when present - facilitates learning of procedures within a particularly difficult and important area of math: decimal arithmetic. To accomplish this goal, the project will adopt a three-pronged approach including longitudinal, microgenetic, and computational modeling methods. Longitudinal methods will identify specific types of conceptual knowledge that predict success in learning decimal arithmetic procedures; microgenetic methods will provide evidence for specific mechanisms by which these types of conceptual knowledge facilitate learning; computational modeling will be used to describe these mechanisms precisely and to simulate the empirical phenomena observed using the previous two methods. The computational model will build on and extend a modeling architecture previously employed in a model of fraction arithmetic learning, FARRA; its success will be assessed based on its ability to generate levels of accuracy, patterns of errors, and correlations between conceptual and procedural knowledge similar to those observed among children. The proposed research will advance scientific knowledge in three ways: by connecting individual differences in learning outcomes with a theory of learning processes, by advancing understanding of the relations between conceptual and procedural knowledge, and by extending theories of numerical development into a new domain, decimal arithmetic. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.