Two Research Traditions Separated by a Common Subject: Mathematics and Mathematics Education. Technical Report. No. 2002-2.

There are no proofs in mathematics education." While this is true, claims are made in mathematics education research and evidence is provided for them. In this talk, I will explore the nature of such research, the kinds of claims and evidence, and what such research might have to offer teachers of mathematics, especially at the undergraduate level. Along the way, I will point out differences between the ways research is done in the two fields. The above title is meant to be both provocative and descriptive. While I will talk about how research in the two fields -mathematics and mathematics education -differs, I would like to note, at the outset, something that mathematicians and mathematics education researchers have in common -a love of mathematics and a desire that more people (especially our students) learn to love, appreciate, and work with mathematical ideas flexibly. That said, there is also much that separates the two fields that could possibly lead to some misunderstandings of the aims and methods of those engaged in mathematics education research. Having obtained a Ph.D. in mathematics and published several papers in my field (semigroups), I became increasingly interested in the problems my students were having. As a result, some fifteen years ago, I decided to take mathematics education at the undergraduate level as a serious research commitment. Consequently, I feel I have a "foot in both camps" and can understand and empathize with both. Indeed, many mathematicians I know seem to consider me a "math ed person" and a number of mathematics education researchers consider me a mathematician. While much of what I have to say will be about research in mathematics education generally (i.e., at all levels, K-16+), many of the examples, and some of my remarks, will apply specifically to research in mathematics education at the undergraduate level, sometimes abbreviated as RUME. 1 This paper was written in preparation for the author's AWM-MAA Invited Address at the MathFest in Burlington, Vermont, August 3, 2002. This address is given annually by the recipient of AWM's Louise Hay Award for Contributions to Mathematics Education. For more information on this award, see http://www.awm-math.org/hayaward.html. 2 This statement has been attributed to Henry Pollak by Schoenfeld (2000, 2001).

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