Imperfect proofs

Mathematics is often presented as a very sterile subject, particularly in colleges and undergraduate courses. Exercises and theorems are accompanied by perfect solutions and proofs. In order to solve X you should do A, B, and then C. It’s easy to see the attraction of such an approach — after all maths is the pantheon of logical arguments and, usually, there is only one “correct” answer.

However, maths doesn’t work this way in “real life”. No one conjures a theorem out of thin air and then proceeds to write a perfect proof at the first attempt (except, maybe, Gauss). If a mathematician solves a new problem on the first attempt, then it really wasn’t a problem worth solving. This should be reflected in how we teach maths — problems are rarely solved by following a sterile, linear, and entirely logical path from A to B. Humans are highly illogical beasts — mathematicians included — and we solve problems in very unstructured ways. We make mistakes, backtrack, course correct, amend, adjust, and fix.

It is therefore important that we are able to identify when we have made mistakes and develop strategies capable of finding where we have erred. Moreover, there is plenty of scholarly literature (see, for example, articles by Grosse & Renkl and Rushton) which indicates that error analysis can be beneficial in the learning process. These articles focus on a very structured form of error analysis in which intentionally flawed solutions are constructed and students are typically required to identify, explain, and correct the errors.

I also like to take an informal approach by simply not having model solutions to hand when I am teaching. This means that I solve problems on the spot and live, which inevitably means that I will make mistakes and, in all likelihood, won’t know that I’ve made a mistake until I get to the end and discover that my pendulum violates energy conservation. This also highlights another important aspect of learning maths — being able to identify that you have made a mistake without knowing what the correct answer is or being told that the solution is incorrect. By watching and discussing the approach that I take to identify, (a) that the solution is wrong and, (b) where the error arose, students begin to learn and understand how to do this themselves.

Just as importantly, this approach teaches students that it’s okay to make mistakes. To try something and fail. To be wrong most of the time. We, as teachers, need to give students the confidence to be wrong. As I said yesterday, almost all first attempts at anything will almost certainly be complete garbage. If we don’t show students that 99% of maths is learning to identify and correct mistakes, then we are not really teaching students maths, we are simply teaching them a collection of rules to follow.

Day eleven of one hundred.

This post is part of my #100Days writing challenge, in which I have challenged myself to write for 10-20 minutes for 100 consecutive days.