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Two components in Learning to Reason Using Definitions

By Lara Alcock and Adrian Simpson

Abstract

This paper discusses the transition to the use of formal definitions in mathematics, using the example of convergent sequences in Real Analysis. The central argument is that where in everyday contexts humans categorize objects in flexible ways, the introduction of mathematical definitions imposes a much more rigid structure upon the sets so defined, and hence upon the acceptability of different types of argument. The result is that, in order to have their reasoning accepted in proof-based mathematics courses, students must do two things: 1. align their notion of what mathematical objects belong to a given set with the extension of the defined set, and 2. (more fundamentally) learn to express their reasoning about such sets exclusively in terms of the definitions or other results traceable to these. The importance of these two components is illustrated using two exa mples. First, a student whose idea of what objects belong to the set of convergent sequences does not closely correspond with the definition, and whose reasoning is therefore insufficiently general. Second, a student whose set corresponds well to that given by the definition, and whose work is arguably more mathematicall

Topics: definitions, proof, advanced mathematics, imagery, AnalysisHuman categorization and mathematically-defined sets Human cultural categories are usually n, in the sense that their extension is not determined by necessary and sufficie nt conditions for memb, many have
Year: 2002
OAI identifier: oai:CiteSeerX.psu:10.1.1.333.2695
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