Impediments to mathematical creativity: Fixation and flexibility in proof validation

Abstract

Mathematical techniques in proof writing can be narrowed down to specific proof styles. Simply put, proofs can be direct or indirect- the latter using the Law of the Excluded Middle from logic as well as the axiom of Choice, to prove existence of mathematical objects. However, the thinking skills involved in writing indirect proofs are prone to errors, especially from novice proof writers such as prospective teachers. Creativity in mathematics entails the use of both direct and indirect approaches to determine the validity of a statement. In this article, I shed some light on this relationship, by reporting on some findings from a study on how students comprehend and validate direct and indirect proofs. Furthermore, I use the constructs of fixation and flexibility from creativity research to examine student approaches to direct and indirect proofs.