Proving as problem solving: The role of cognitive decoupling

Abstract

This paper discusses the process of proving from a novel theoretical perspective, imported from cognitive psychology research. This perspective highlights the role of hypothetical thinking, mental representations and working memory capacity in proving, in particular the effortful mechanism of cognitive decoupling: problem solvers need to form in their working memory two closely related models of the problem situation – the so-called primary and secondary representations – and to keep the two models decoupled, that is, keep the first fixed while performing various transformations on the second, while constantly struggling to protect the primary representation from being “contaminated” by the secondary one. We first illustrate the framework by analyzing a common scenario of introducing complex numbers to college-level students. The main part of the paper consists of re-analyzing, from the perspective of cognitive decoupling, previously published data of students searching for a non-trivial proof of a theorem in geometry. We suggest alternative (or additional) explanations for some well-documented phenomena, such as the appearance of cycles in repeated proving attempts, and the use of multiple drawings.