Mathematical Problem Solving: The Roles of Exemplar, Schema, and Relational Representations

Human mathematical abilities are a deep and somewhat underappreciated mystery in cognitive science. How can we explain the apparent fact that humans develop a set of relational representations that are completely portable and abstract? Much of the power of mathematics stems from the fact that it can be divorced from context and applied across an unlimited range of situations. However, our experiences, mathematical or otherwise, are contextually bound; everything occurs in some rich and textured environment. It is not clear how abstract, relational representations could arise from context-rich experience or what the nature of those representations might be. Therefore, how mathematical representations might come into being poses a serious and important theoretical puzzle. Viewed this way, mathematical representations constitute a prime instance of this classic problem in cognitive science, the problem of abstraction.