A theory of expertise has developed over the last few years on the basis of research into expert-novice differences. This research has shown an impressive degree of consistency in a variety of fields of knowledge and is beginning now to have direct implications for the teaching of mathematics and related disciplines. Research data, generated in part by current theories of expertise, call into question many long-held assumptions and procedures. We suggest that some views of mathematics and the way it should be taught owe more to tradition than to our current knowledge of cognitive processes. The purpose of this paper is to outline the theory, describe some of the associated experimental findings, and indicate the consequences of those findings for educational practice in mathematics. The work on expert-novice differences can be divided into three sets of interrelated findings: the memory of relevant problem details, modes of categorizing problems, and problem-solving strategies. There is now strong evidence that experts have a better memory than novices for realistic problem states or configurations. De Groot (1966) and Chase and Simon (1973a, 1973b), using realistic chess configurations; Egan and Schwartz (1979), using electronic circuit diagrams; Jeffries, Turner, Polson, and Atwood (1981), using computer programs; and Sweller and Cooper (1985), using algebra, all found a superior memory by experts for relevant items. With respect to modes of categorizing problems, Chi, Glaser, and Rees (1982) found, when studying the differences between experts and novices in solving physics problems, not only that the experts possess a greater body of knowledge, as one might expect, but more significantly that experts classified problems according to the underlying physics principle (e.g., the conservation of energy) that must be invoked to solve the problem. Novices, however, classified on the basis of surface features mentioned in the problem statement (e.g., inclined planes or pulleys) regardless of the principle underlying the solution method. Novices and experts also differ dramatically in their use of problem-solving strategies. A great many routine mathematics and mathematics-related problems
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