Mastering New Mathematical Concepts

Bernd Buldt, Indiana University - Purdue University Fort Wayne

Abstract

Frank Quinn proposed an analysis of mathematical practice that is informed by both an analysis of contemporary mathematics and its pedagogy (see [1]). Taking this account as our starting point, we can characterize the current mathematical practice to acquire and work with new concepts as a cognitive adaptation strategy that, first, emerged to meet the challenges posed by the growing abstractness of its objects, and which, second, proceeds according to the following three-pronged approach:

(1) sever as many ties to ordinary language as possible and limit ordinary language explanations to an absolute minimum;

(2) introduce axiomatic definitions and bundle them up with a sufficient number of examples, lemmata, propositions, etc. into small cognitive packages;

(3) practice hard with one new cognitive package at a time.

Drawing on research in cognitive science, and especially in mathematics education, I will argue that results from both areas provide supporting evidence for the effectiveness of this mathematical practice.

[1] Quinn, Frank. Contributions to a Science of Mathematics, manuscript (October 2011), 98pp.