A Theory of Algebra-Word-Problem Comprehension and Its Implications for the Design of Learning Environments

Abstract

A tutoring approach is derived from a model of problem comprehension, based on the van Dijk and Kintsch (1983; Kintsch, 1988) theory of discourse processing. A problem statement is regarded as a text from which the student must glean propositional and situational information and make critical inferences. The competent student must coordinate this information with known problem models so that formal (i.e., algebraic) operations can be applied and exact solutions can be obtained. We argue that this task is a highly reading-oriented one in which poor text comprehension and an inability to access relevant long-term knowledge lead to serious errors. In particular, poor students often omit from their solutions or misspecify necessary mathematical constraints that are based on reading inferences needed to describe fully the problem situation. Furthermore, formal algebraic expressions are so abstract that their meaning is often elusive; this contributes to mistranslations and misinterpretations. The competent approach is teachable, however. We describe experimental results with ANIMATE, a learning environment that knows nothing of the problem at hand or of the student's actions. Subjects encouraged to reason explicitly about the situations described in typical word problems consistently performed as well as or better than those who were not, in both training and transfer task. We conclude that, by using an environment that gives equations situation-based meaning through computer animation, students learn to relate formal expressions to the referent situations. This enhances problem comprehension and gives a stronger representational base to the problem-solving process. A call for evaluation methods beyond just algebra problem-solving performance is made. The implications of this work for the design of future computer-based tutors and other learning environments are also discussed.