Demystifying Proofs Through Structured Interaction: A Case Study of One Instructor’s Teaching in an Undergraduate Analysis Course

Abstract

The concept of proof is central to every undergraduate mathematics curriculum, but it is also a difficult concept for university students to understand and for university instructors to teach. Prior studies have contributed useful research knowledge about the nature and sources of undergraduate students' difficulties with proof, but they have paid less attention to the development of effective instructional practices to enhance student learning of proof. As a step towards this direction, this paper makes a contribution to the limited body of case studies of promising or potentially effective teaching practices in undergraduate mathematics by reporting a case study of an instructor's teaching, locally characterized as "effective," in an undergraduate analysis course at a leading Indian university. The instructor did not deviate from the so called "Definition-Theorem-Proof" (DTP) format that is followed in most prooforiented university mathematics courses, but her teaching presented a set of features that, collectively, form a rather innovative teaching practice at the undergraduate level. Specifically, our case study shows that, even in the context of a rather crowded university classroom, proof can be demystified for students through structured interaction between the instructor and the students, that is, an interactive, conversational style of proof instruction invoking the participation of students. This is based on a solid foundation in symbolic logic at the very outset and on a significant time investment in definitions being explained in depth using informal language, visual aids, and real-life analogies. In addition to contributing to existing images of potentially effective teaching practices in undergraduate mathematics, this paper draws attention to the Indian educational context that has had little representation in international forums of mathematics education research thus far. ' This is obviously equivalent to |𝑎 " -a|< ! '* and |𝑏 " -b| < ! '+ Now as <𝑎 " > → 𝑎, so corr. to ! '* > 0, ∃ 𝑛 & 𝜖𝑵 such that |𝑎 " -a|< ! '* ∀ 𝑛 ≥ 𝑛 & Also as < 𝑏 " > → b, so corr. to ! '+ >0, ∃ 𝑛 ' 𝜖𝑵 such that |𝑏 " − 𝑏| < ! '+ ∀ 𝑛 ≥ 𝑛 '