The other day I did a workshop with students from Advanced Mathematical Economics III, which is more or less a pure maths course for economics students. It covers such things as mathematical logic, analysis and topology — all a bit intimidating for students who started out the degree with almost no mathematical background!
We had just spent an hour looking at relations: the definition as a subset of the set of pairs, the common usage as statements in natural language, various ways to visualise them, examples and non-examples of various properties relations might have, and proofs involving those properties. After all this point, one of the students said, with some surprise, “It really feels like it’s just playing around here.”
The student was right, of course, and I told him so! Pure maths is play. You have some ideas and you fiddle around with them to see how they fit together, how they’re similar or different, what other things are really the same things in disguise. Most of the time there’s no particular goal in mind, but even when there is, you often get distracted by something cool and end up somewhere totally different. This is exactly what play is. Watch any child in a sandpit and you’ll see the same behaviour.
It suddenly occurred to me that I have always seem maths as play, and have liked it the least when it looked like work. More than this, the maths courses I have had the most academic success in were the ones where I allowed myself to see it as play. These were the courses where I took the concepts I was learning and pulled them apart and put them together in new ways, where I tried to do things that may or may not be possible just to give it a go, and where I drew lots of pictures in vivid colour just because. And importantly, where I didn’t question what the point of any of it was but just ran headlong into whatever crazy idea the lecturer presented next to see what happened with it.
So perhaps the best way to approach a pure maths course is to see it as play! Surrender to “just playing around” and see where it goes. That’s the advice I gave to these students, and I’m hoping it helps them to have the freedom to learn.