for the learning of mathematics

an international journal of mathematics education

Paul Christian Dawkins - Vol. 34 Num. 2 (2014)
 When proofs reflect more on assumptions than conclusions


This paper demonstrates how questions of 'provability' can help students engaged in reinvention of mathematical theory to understand the axiomatic game. While proof demonstrates how conclusions follow from assumptions, 'provability' characterizes the dual relation that assumptions are "justified" when they afford proof of appropriate results. I provide examples of students’ learning in teaching experiments connected to an advanced, undergraduate, neutral axiomatic geometry course. Students used 'provability' relations to choose definitions, develop the notion of independence, and understand the relationship between axioms and theorems. I argue that provability promotes cognitive need for systematization, allaying concerns that geometric proof lacks illuminative value.