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Unformatted text preview: Surprises from Mathematics Education Research: Student (Mis)use of Mathematical Definitions Barbara S. Edwards and Michael B. Ward 1. INTRODUCTION. The authors of this paper met at a summer institute sponsored by the Oregon Collaborative for Excellence in the Preparation of Teachers (OCEPT). Edwards is a researcher in undergraduate mathematics education. Ward, a pure math- ematician teaching at an undergraduate institution, had had little exposure to math- ematics education research prior to the OCEPT program. At the institute, Edwards described to Ward the results of her Ph.D. dissertation [ 5 ] on student understanding and use of definitions in undergraduate real analysis. In that study, tasks involving the definitions of limit and continuity, for example, were problematic for some of the students. Wards intuitive reaction was that those words were loaded with conno- tations from their nonmathematical use and from their less than completely rigorous use in elementary calculus. He said, Ill bet students have less difficulty or, at least, different difficulties with definitions in abstract algebra. The words there, like group and coset, are not so loaded. Eventually, with OCEPT support, the authors studied student understanding and use of definitions in an introductory abstract algebra course populated by undergradu- ate mathematics majors and taught by Ward. The surprises in the title are outcomes that surprised Ward, among others. He was surprised to see his algebra students having difficulties very similar to those of Edwardss analysis students. (So he lost his bet.) In particular, he was surprised to see difficulties arising from the students understanding of the very nature of mathematical definitions, not just from the content of the defini- tions. Upon hearing of Edwardss dissertation work, some other mathematicians who teach undergraduates found those difficulties surprising even when restricted to real analysis. Hereafter, we present a simple two-part theoretical framework borrowed from phi- losophy and from mathematics education literature. Although it is not our intent to give an extensive report of either study, we next indicate the methodology used in Edwardss dissertation and in our joint abstract algebra study so that the reader may know the context from which our observations are drawn. We then list the surpris- ing difficulties of the two groups of students, documenting them with examples from the studies and using the framework to provide a possible explanation for them. We conclude with what we see as the implications for undergraduate teaching, along with some specific classroom activities that the studies and our experience as teachers sug- gest might be of value....
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