Making mathematical claims

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.