JMB JOURNAL OF MATHEMATICAL BEHAVIOR, 17 (2), 137-l 65 ISSN 0364-0213. Copyright 8 1998 Ablex Publishing Corp. All rights of reproduction in any form reserved. Representational Systems, Learning, and Problem Solving in Mathematics GERALD A. GOLDIN Rutgers University This article explores aspects of a unified psychological model for mathematical learning and prob- lem solving, based on several different types of representational systems and their stages of devel- opment. The goal is to arrive at a scientifically adequate theoretical framework, complex enough to account for diverse empirical results but sufficiently simple to be accessible and useful in math- ematics education practice. Some perspectives on representational systems are discussed, and components of the model are described in relation to these ideas-including constructs related to imagistic thinking, heuristics and strategies, affect, and the fundamental role of ambiguity. 1. KEY CONSTRUCTS IN A UNIFIED MODEL As the study of learning and problem solving in mathematics advances, and the complexity of these processes is recognized and better understood, the need grows for theoretical bases adequate to guide exploratory investigation, verifiable scientifically through empirical studies, and helpful in informing teaching practices. The psychological theory of mathe- matics education has evolved greatly in the past forty years. However the avenues of this evolution appear to be (and their proponents have claimed them to be) mutually incompat- ible. Partly as a consequence, we lack the model that serves the growing needs adequately. This article, synthesizing and expanding on some earlier ideas, describes how a model based on several different types of representational systems unifies ideas from diverse the- oretical perspectives, and permits the description and analysis of a variety of key con- structs.’ Let us consider first some of those constructs. 1.1. Broad Theoretical Perspectives One strand of thought in the study of mathematical learning developed from behaviorist ideas. Adherents of this school emphasized empirically observable behaviors and environ- ments as the most fundamental entities in a scientific theory, and even took them to be the only admissible entities. Theorists moved from stimulus-response theory, operant condi- tioning, and rule-governed learning, into neobehaviorist perspectives that accepted the pos- sibility of internal responses by learners, chained responses, and so forth (Gag&, 1970; Skinner, 1953, 1974). The study of algorithmic learning and debugging, and the consider- Direct all correspondence to: Gerald A. Goldin, Center for Mathematics, Science, and Computer Education. Rut- gers University, Piscataway, NJ 08854 <gagoldin@dimacs.rutgers.edu>. 137