The Process of Proving

Proving things is an undervalued life skill.

Catherine O’Neil recently described a refreshing approach to teaching proofs in undergraduate math. Rather than emphasizing reaching a solution fast, she emphasizes the process.

If I recall correctly, the first time I learned proofs in a math course was in 7th grade geometry. I regarded Mr. Cannon as a great teacher and still found the fourth quarter very difficult. From my years as an undergrad, my courses with Serge Lang in real analysis, and with Shelly Kagan in ethics stand out as exceptional experiences. In all cases, the memorable courses were both challenging and rewarding.

Professor Lang emphasized speed and memory, often telling us, “You should know this three days after you’re dead.” Professor Kagan demanded similar rigor in prose, with a more measured approach; I recall writing one paper three times before my grader was satisfied with my argument, and before I was satisfied with my grade.

Schools should teach the process of proving in addition to the proofs themselves. Of course, speed still matters. In a school setting, time is bounded by the end of the semester. In life, time is bounded by death. All the more important to learn life skills early. I see no reason why the opportunity of learning with Catherine’s measured approach should wait until college. I hope we see K-12 schools teaching the process of proving even earlier.