The Chasm between Thought Experiment and Formal Proof

Abstract

This presentation will address the conceptual demands placed on students attempting to deal with formal proof for the first time and present empirical evidence that reveals the subtlety of this transition. It transpires that there is more than one route to move from informal experience of proof to formal proof. Informal proof often occurs in the style of a thought experiment, using a variety of imagery to infer that, when a certain situation occurs, then another must also occur as a consequence of the first Formal proof on the other hand, is based on verbal/symbolic definitions and focuses only on those results that can be deduced logically from the definitions. The presentation will show that there are (at least) two cognitively different routes from informal to formal. One builds on imagery and constantly reconstructs it to fit new formalisms. Another starts from the definitions and develops only those properties that can be built by formal deduction. Empirical evidence will be given, collected in longitudinal studies from students in their first year of university mathematics, to demonstrate how both of these routes can lead to success, but that each involves a different array of cognitive difficulties that can lead to failure.