Teaching problem solving as viewed through a theory of models

Summary

This paper presents a small amount of theory including an analysis of the task of teaching problem solving and the act of problem solving derived from a theory of models of operations. It also presents the results of an exploratory investigation which tends to add credence to elements of the analysis.

Two of the general problem solving strategies which are suggested are (1) classification of word problems as instances of models previously learned (taught) and (2) a variety of techniques involving mapping individually word problems onto instances of familiar models. A third problem solving strategy which has not been mentioned is indicated by the theory: In this strategy, from the given problematical situation (word problem), a model is generalized which subsumes the given situation, and search for an isomorphism between the model and various operations (i.e., multiplication) is conducted. If an isomorphism is observed, then the selected operation and attending computational algorithms can be used to solve the problem. Of course, it is unlikely that this type of activity would be appropriate for primary grade children, but perhaps it could be taught at a later stage of the curriculum.

The extensive mathematics education literature devoted to the study of teaching problem solving in the elementary school includes numerous theoretical and methodological approaches to investigating problem solving. This paper, the author believes, presents a unique, systematic, and reasonable perspective on the matter. Both a systematic view of the act of solving one-step word problems and a two stage general level analysis of practices found in teaching problem solving have been presented as they were derived from a theory of models.