Some mathematical proofs are essentially magic, aren’t they? “OK, there’s nothing up my sleeve, right? Now watch what happens when I multiply by !” “Stay with me now: we’re going to take logs for reasons that will become clear later, then substitute for .”
Students say “I understand what you did, but how did you know it would work? I would never be able to come up with that.” This is a big problem with the way we teach maths, especially at the more advanced level. We want students to develop into resilient problem solvers who learn from their mistakes, but most of the time all they see modelled from their teachers is carefully crafted examples, pruned and polished until they are as efficient and sanitized as we can make them, with no hint of the rough-n-ready manner in which they were born.
When’s the last time you stood up in front of a class of students and tried to tackle a problem that you didn’t already know how to solve? Not just through inadequate subject knowledge or bad planning, but in a deliberate attempt to model failure and how to deal with it for our students. They already know what success looks like, and half the time we make it look so effortless they get even more demoralised when their own experience doesn’t match up. Don’t they deserve to watch us flounder, sweat, get stuck and be confused once in a while, too?
Even if we occasionally take a punt at an unfamiliar question in class, I’m guessing for most topics we can see how the problem will play out before we start (or at the very least, its general trajectory), which doesn’t quite fit the bill. To get well and truly stuck, you either have to be testing the boundary conditions of your subject knowledge comfort zone (Further Maths A-level has a few areas like this for me, still), or tackle something like a UKMT maths challenge problem, or something from the excellent Integral Maths Ritangle competition.
Solving a maths problem to me feels like the film Groundhog Day. The first time you tackle it, it’s a mess. You may not get the solution at all, or even close to it. Perhaps when you come back to it, having reflected on the dead-ends and false trails, you can do better. Define your variables, because that tripped you up last time. Clarify your assumptions, and tidy up your notation because those simultaneous equations get messy fast. A couple more attempts, though, and you can solve it in your sleep. You even spot shortcuts that would never have occurred to you the first time around. And when you miraculously pull the perfect substitution out of your hat the next day in class, and they whisper “It worked! But how did he know to do that??” it’ll look like magic when really it was good old-fashioned hind-sight. A colleague shared a geometrical problem with me that he said could be done in your head “after you know the answer”. Which makes perfect sense to mathematicians: solving a problem doesn’t just give you a solution, but an insight into the entire scenario.
When I’m feeling especially reckless, I bring my selected unsolved questions into the classroom. Narrating my thought processes, and letting the whiteboard fill up in the half-structured half-chaotic manner that back-of-the-envelope problems do, I can show students what it looks like for me to struggle with an unfamiliar problem. They see the wild goose chases, the uncertainty, the dubious assumptions and second-guessing. They see me get stuck, and take stock of what I’ve tried and how it helped, if at all. They see me back-track, and start again from scratch, and get it wrong, and hunt for that dropped minus sign, and look for ways to verify my solution. And with any luck, in the post-mortem, we’ll spot that elusive, elegant pathway through the maze which is often only visible from the end.