Who’s There? A Study of Students’ Reasoning about a Proof of Existence

Abstract

Drawing on prior research on indirect proof, this paper reports on a series of exploratory studies that examine the extent to which findings on students' ways of reasoning about contradiction and contraposition characterize students' views of indirect existence proofs. Specifically, Study 1 documents students' comparative selections and selection rationales when asked to choose the Bmost convincing^proof, given a constructive and nonconstructive existence proof. Study 2 further examines findings from Study 1 by documenting novices' levels of conviction and interpretations of a nonconstructive existence proof. Findings show that when presented with a nonconstructive proof, students tended to not only find the proof convincing but also interpreted the proof constructively. Moreover, the data indicate students who exhibit an awareness of the nonconstructive structure were divided in terms of their views of which form -constructive or non-constructive -was the most convincing. The discussion considers students' reactions to the disjunctive structure of nonconstructive existence proofs and use of pragmatic and theoretical modes of thought.

A Trio of Courses to Support Mathematics Transfer Students’ Development of Professional Vision

“…Mathematics departments and scholars have designed and assessed the effectiveness of introduction to higher mathematics classes, mathematics seminars (Levine & Shanfelder, 2000), and transition to proof courses (David & Zazkis, 2020). Still, difficulties surrounding learning how to construct and write proofs persist (e.g., Brown, 2017). The literature notes two specific ways of supporting mathematics students through this transition to proof-bridge programs (Gordon & Nicholas, 2013) and seminars (Polhill, 2007).…”

A Hypothesis Framework for Students’ Difficulties with Proof By Contradiction

“…(p. 44) Mancosu (1991) catalogued additional historical examples noting the "lower epistemological status" (p. 26) of IP and the failure to engender the feeling of causality that DP often does. More recently, Brown (2017) cited modern resistance to indirect methods, including opposition to Hilbert's non-constructive proof of the Hilbert Basis Theorem and a recent critique of Cantor's non-constructive proof establishing the existence of transcendental numbers. As one might expect, these historical themes gained traction in pedagogical spaces (see comments under "Training Hypotheses").…”