An Integrated Model of Skill in Solving Elementary Word Problems

Abstract

Solving arithmetic word problems links the ability to execute arithmetic operations and the ability to apply these operations in real-word situations. An appropriate recent interest in word problems has produced much data on the relative difficulty of different kinds of word problems, and some data on the strategies children use to solve these problems together with the kinds of errors they make. We also have accounts of problem difficulty based on descriptions of the problem, that is, on their structural variables or on their semantic structure. We present here and account of problem difficulty based on a description of the psychological processes of the child. The purpose of this article is to outline a model of these processes, made explicit in the form of a computer-implemented model. The model solves many common word problems by acting them out with representations of physical counters. More difficult problems require augmenting this procedure first with knowledge that one object is a member of both a set and its superset, and second with knowledge that processes can be "undone" and that subsets can be exchanged. We characterize problems by the kind of knowledge the model uses to solve them. A comparison of these knowledge types with data on problem difficulty allows us to estimate how much the need to use each kind of knowledge contributes to a problem's difficulty. We also observe and relate to children's performance the sequence of steps the program produces a solution, and the errors the program makes when it lacks certain knowledge. Finally, we contrast the knowledge in the program with the knowledge we believe is acquired by children in school.