Understanding and procedural knowledge in mathematics instruction1

Abstract Using ideas developed recently in the theory of understanding language, an analysis is given of the process of understanding mathematical concepts. An analysis also is given of procedural knowledge in mathematics, based on recently developed theories of problem solving including production systems and automaton models. A suggestion is made as to how these theories might be synthesized to provide an analysis of performance of mathematical skills with understanding. Two examples of such performance are described, in the form of theories that are realized as computer programs, and there is discussion of ways in which these theories succeed and fail in representing procedural knowledge with understanding. Notes This research was sponsored by Grant MH25218 from the National Institute of Mental Health. This paper is revised from a talk given at the American Educational Research Association meetings in San Francisco, April, 1976.1 am grateful to Division C of AERA for inviting me to give that address, and to Patrick Suppes for a thoughtful and stimulating set of comments on that presentation. I have tried in this revision to be responsive to some of Suppes' remarks. The author's address is Learning Research and Development Center, University of Pittsburgh, 3939 O'Hara Street, Pittsburgh, PA 15260.