Making mathematical claims

TL;DR

This paper reflects on the instructional routine of engaging students with conjectures and proofs to teach mathematical processes, questioning its effectiveness in conveying the beauty and joy of mathematics.

Contribution

It critically examines the traditional teaching routine of conjecture and proof, suggesting the need for diverse approaches to better convey mathematical beauty.

Findings

Students may develop an impoverished view of mathematics if only exposed to conjecture-proof routines.

The routine may not fully capture the beauty and joy inherent in mathematical discovery.

Alternative instructional methods could enhance students' appreciation of mathematics.

Abstract

Good problems grab us. They invite us to find patterns, make conjectures, and prove-or perhaps disprove-a conjecture. When I first taught, I saw my work as tantalizing students with structures just beyond their reach, so that I could elicit conjectures from promising half-phrases. With a community conjecture crystallized on the board, "we" proved the statement. "We" anointed the conjecture a community theorem, and "we" moved on. I hoped that, through repeated exposure to this routine, students would absorb a mathematical process from discovery to proof. But I've since wondered: What does this routine teach students? I've concluded that if this is the only instructional routine that students experience, they may leave with an impoverished image of the beauty and joy that doing math can offer.