When we first start learning mathematics, the problems we are asked to solve have been carefully chosen to suit the method we’ve been taught. But as we advance, we accumulate an ever-growing toolkit of methods, which can be used to solve an increasing variety of problems. Getting the answer is no longer just a matter of following a standard recipe. Instead, we have to decide which method — or which combination of methods — will do the trick.
At that stage, the subject starts to resemble what happens at the research frontiers. There, the range of possible methods is vast, and often no existing methods are suitable, so you have to invent new ones. Worse: it’s not always clear what the right question should be. And even when you’ve sorted out the question, there might be several answers. Or none.
Trying to solve a simple exercise, or tackling a research problem, may seem very different, but they have important features in common. Next time you’re stuck on a mathematical question, tell yourself that professional mathematicians often get stuck too. At the very least, that will help you to calm down, take a deep breath, and apply a few simple tactics.
Mathematics is like a maze: a complicated network of interconnected paths, going every which way, dense and apparently impenetrable. That’s more than a metaphor—it’s a fairly literal description of the logical structure of mathematics. The paths in the maze are simple logical steps, deducing a new fact from ones you already know to be true. Your problem, like that of the research mathematician, is to explore the maze and reach your goal.
How do you explore a maze? Try a few tentative steps, see if they lead anywhere that looks familiar, or seems closer to the goal. Search systematically around the places you recognise, and see whether some of them link up. Work backwards from the goal, and forwards from the start, and try to get the paths meeting up. You can’t do that in a real maze, but you can in a mathematical one.
There are some extra tricks, where the maze metaphor doesn’t really apply. Try a simpler version of the same problem. Once you’ve understood that one, the important new features of the harder problem start to stand out from the background. Look for a route into the problem, a place where you can get some leverage. If you can turn it into an algebraic equation, for example, a battery of standard techniques can be used to solve that equation.
If all else fails—and provided you’re not working to a fixed timetable—sit back and relax. Tell yourself that there has to be an answer, and you’re not going to give up until you find it… but that there’s no rush. Let your subconscious mind play around with ideas. If it finds anything useful, it will let you know. Then the little light bulb goes off and the breakthrough happens.
I don’t guarantee that any of these tactics will work, but at the very least they’ll give you something constructive to try. And it’s amazing how often they get you past the place where you’re stuck.
- Ian Stewart is Emeritus Professor of Mathematics at Warwick University. In 1995 he was awarded the Royal Society’s Faraday Medal for contributions to the public understanding of science. He was elected a Fellow of the Royal Society in 2001. He won the gold medal of the Institute of Mathematics and Its Applications in 2000. He delivered the televised Christmas Lectures in 1997, bringing a live tiger into the lecture room. He has made over 400 radio broadcasts and 80 television appearances. He has published over 80 books including Professor Stewart’s Cabinet of Mathematical Curiosities. He collaborated with Terry Pratchett and Jack Cohen on the best-selling Science of Discworld series. He is active in research with over 180 published papers.
- Professor Stewart’s website: http://freespace.virgin.net/ianstewart.joat/
- Twitter: @JoatStewart
- Ian Stewart’s new book: Professor Stewart’s Casebook of Mathematical Mysteries