Mastering New Mathematical Concepts
Frank Quinn proposed an analysis of mathematical practice that is informed by both an analysis of contemporary mathematics and its pedagogy (see ). 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.
 Quinn, Frank. Contributions to a Science of Mathematics, manuscript (October 2011), 98pp.