CONNECTING LOGIC AND PROOF TECHNIQUES: IDENTIFYING LEARNING IN AN INTRODUCTION TO PROOFS COURSE

Abstract

Proof and proving are a core component of the discipline of mathematics and proving is a required exercise in many upper-level courses in mathematics at the undergraduate level. Writing proofs remains difficult for many students (Moore, 1994; Stylianides et al., 2017). To address this difficulty many universities have began offering Introduction to Proof courses. These courses typically cover three main areas, logic, proof techniques, and sets and functions (David & Zazkis, 2020). With this course’s importance in students’ transition to upper-level mathematics, it is worthwhile to investigate the connections that students make between the subcomponents of such a course. As such, in this dissertation study I sought to understand the connections that students make between logic and the techniques of proof in and Introduction to proofs course. In the first chapter I state the broad issues related to students learning of logic and proof techniques, to set the stage for the remainder of the manuscript. In the second chapter I present a research study on the connections that students make between logic, direct, and indirect modes of proof. In the third chapter I present a research study on the struggles that students face as they learn to write proofs with mathematical induction. In the fourth chapter I present a practitioner-minded piece where I highlight the typical issues that students face throughout an Introduction to Proofs course. Finally, in the fifth chapter I share some broad conclusions across these three manuscripts and reflect on students’ learning throughout an Introduction to Proofs course.