Transitions and Proof and Proving at Tertiary Level

Abstract

This chapter discusses implications of the requirement that students become autonomous in understanding and constructing rigorous proofs when they transition to tertiary level mathematics. As they struggle with this requirement, students encounter various difficulties including: the proper use of logic; the necessity to employ formal definitions; the need for a repertoire of examples, counterexamples, and nonexamples; the requirement for a deep understanding of the concepts and theorems involved; the need for strategic knowledge of which theorems are important, and the importance of being able to read and check arguments for correctness. Beyond conveying proofs linearly in lectures, university teachers can employ generic proofs or structured proofs. Additional ways to teach include: transition-to-proof courses, communities of practice, the Moore Method, the co-construction of proofs, and the method of scientific debate. Resources developed to help university mathematics teachers include videos, DVDs, and written materials.