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Mathematical discovery: a covariance analysis.

Article Excerpt
Abstract

A covariance analysis of constructs related to the process of mathematical discovery is presented. The constructs are derived from Polya's ideas on problem solving and include sophistication, traits that predict affect, and emotion. The relative importance of these constructs is investigated in specific problem solving situations. How the model is refined and validated is discussed. The analysis provides a foundation for exploring the nature of mathematical thinking during problem solving.

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The importance of affect and emotion during mathematical discovery and problem solving has long been recognized (Dreger & Aiken, 1957; Higbee & Thomas, 1999). Emotion can organize, focus, disrupt, distract or energize problem solving (Abella & Heslin, 1989), and the influence of emotion can be immediate or delayed (Lazarus, 1991). Further, emotion has a representational function in problem solving (Bebellis & Goldin, 1997) and forms enduring affective pathways that contribute to individuals' mathematical power (Goldin, 2000).

If the student had no opportunity in school to familiarize himself with the varying emotions of the struggle for the solution, his mathematical education failed in the most vital point. (Polya, 1985, p. 94).

This paper develops a model of the process of mathematical discovery. The model is built from Polya's ideas on mathematical discovery and problem solving, and augmented with Bloom's (1956) concept of sophistication, Anderson's (1981) traits that predict affect, and Mandler's (1989) theory of emotion during problem solving. After identifying variables involved in the mathematical discovery process, the model is refined and validated to provide a foundation for exploring the nature of mathematical thinking during problem solving (Schoenfeld, 2000).

In influential work on the cognitive activities that occur during mathematical problem solving and discovery, Polya (1981, 1985) describes ongoing cognitive evaluations that lead to emotion. Polya contends that every step in problem solving is accompanied by the partially correlated evaluations of relevancy, proximity, and quality. These evaluations are judgements about the problem solving process and performance. Relevancy is an estimate of either how important the problem is to the problem solver, or how important a particular idea or guess is in obtaining the solution. Proximity is an estimate of how close or far away the solution is. Described by Polya, but not specifically named as such, quality is an estimate of how good an idea or guess is, or how good the problem solving performance is.

Polya claims that more sophisticated people experience more differentiated emotion during problem solving. Polya does not define sophistication but in the context of problem solving, Polya is probably referring to individuals who have greater experience and expertise in problem solving, who perform better at problem solving, and who have more acute evaluations of their problem solving process (Bloom, 1956). Thus, sophistication can be...

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