Concept Image and Concept Definition

Abstract

The “New Mathematics” of the 1960s was based on the structural approach to mathematics, assuming that if only we could formulate the mathematical definitions and deductions correctly, then this would improve the learning of mathematics. But even when this was done, difficulties continued to persist. Careful analysis suggest that these difficulties are not stupidity on the part of students, but a natural human phenomenon that is found within all of us. In the last decade, empirical research has emphasised that individuals build up their mental imagery of a concept in a way that may not always be coherent and consistent, and that previous experiences may colour the meanings of phenomena when they are met in new contexts. This is particularly in evidence in the introduction of more advanced mathematical concepts: For instance, the verbal definition of a limit “snφιs”, in the form “we can make s as close to s as we please, provided that we take n sufficiently large”, induces many individuals the belief that s n can never equal s (Schwarzenberger & Tall 1978).

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