Reasoning by equivalence:The potential contribution of an automatic proof checker

Abstract

Reasoning by equivalence, a form of line-by-line algebraic reasoning, is the most important single form of reasoning in school mathematics. In this chapter I define reasoning by equivalence and examine the role of reasoning by equivalence in mathematical proof. I base the discussion on an examination of the extent to which students are currently asked to "prove", "show" or "justify" in high-stakes national examinations. I then report research into how students go about solving such problems on paper. These results inform the design of anautomatic proof checker within the STACK software which assesses students' responses. I report on the use of this software with students. Finally I discuss the implications of this work for what constitutes mathematical "proof" at school level, and how this might be taught and learned online