At least around here school is starting up again and that’s got me thinking about learning mathematics. Particularly, it’s got me on the question: what should you do if you get stuck?
You will get stuck. Much of mathematics is learning a series series of arguments. They won’t all make sense, at least not at first. The arguments are almost certainly correct. If you’re reading something from a textbook, especially a textbook with a name like “Introductory” and that’s got into its seventh edition, the arguments can be counted on. (On the cutting edge of new mathematical discovery arguments might yet be uncertain.) But just because the arguments are right doesn’t mean you’ll see why they’re right, or even how they work at all.
So is it all right, if you’re stuck on a point, to just accept that this is something you don’t get, and move on, maybe coming back later?
Some will say no. Charles Dodgson — Lewis Carroll — took a rather hard line on this, insisting that one must study the argument until it makes sense. There are good reasons for this attitude. One is that while mathematics is made up of lots of arguments, it’s also made up of lots of very similar arguments. If you don’t understand the proof for (say) Green’s Theorem, it’s rather likely you won’t understand Stokes’s Theorem. And that’s coming in a couple of pages. Nor will you get a number of other theorems built on similar setups and using similar arguments. If you want to progress you have to get this.
Another strong argument is that much of mathematics is cumulative. Green’s Theorem is used as a building block to many other theorems. If you haven’t got an understanding of why that theorem works, then you probably also don’t have a clear idea why its follow-up theorems work. Before long the entire chapter is an indistinct mass of the not-quite-understood.
I’m less hard-line about this. I’m sure that shocks everyone who has never heard me express an opinion on anything, ever. But I have to judge the way I learn stuff to be the best possible way to learn stuff. And that includes, after a certain while of beating my head against the wall, moving on and coming back around later.
Why do I think that’s justified? Well, for one, because I’m not in school anymore. What mathematics I learn is because I find it beautiful or fun, and if I’m making myself miserable then I’m missing the point. This is a good attitude when all mathematics is recreational. It’s not so applicable when the exam is Monday, 9:50 am.
But sometimes it’s easier to understand something when you have experience using it. A simple statement of Green’s Theorem can make it sound too intimidating to be useful. When you see it in use, the “why” and “how” can be clearer. The motivation for the theorem can be compelling. The slightly grim joke we shared as majors was that we never really understood a course until we took its successor. This had dire implications for understanding what we would take senior year.
What about the cumulative nature of mathematical knowledge? That’s so and it’s not disputable. But it seems to me possible to accept “this statement is true, even if I’m not quite sure why” on the way to something that requires it. We always have to depend on things that are true that we can’t quite justify. I don’t even mean the axioms or the assumptions going into a theorem. I’m not sure how to characterize the kind of thing I mean.
I can give examples, though. When I was learning simple harmonic motion, the study of pendulums, I was hung up on a particular point. In describing how the pendulum swings, there’s a point where we substitute the sine of the angle of the pendulum for the measure of the angle of the pendulum. If the angle is small enough these numbers are just about the same. But … why? What justifies going from the exact sine of the angle to the approximation of the angle? Why then and not somewhere else? How do you know to do it there and not somewhere else?
I couldn’t get satisfying answers as a student. If I had refused to move on until I understood the process? Well, I might have earlier had an understanding that these sorts of approximations defy rigor. They’re about judgement, when to approximate and when to not. And they come from experience. You learn that approximating this will give you a solvable interesting problem. But approximating that leaves you too simple a problem to be worth studying. But I would have been quite delayed in understanding simple harmonic motion, which is at least as important. Maybe more important if you’re studying physics problems. There have to be priorities.
Is that right, though? I did get to what I thought was more important at the time. But the making of approximations is important, and I didn’t really learn it then. I’d accepted that we would do this and move on, and I did fill in that gap later. But it is so easy to never get back to the gap.
There’s hope if you’re studying something well-developed. By “well-developed” I mean something like “there are several good textbooks someone teaching this might choose from”. If a subject gets several good textbooks it usually has several independent proofs of anything interesting. If you’re stuck on one point, you usually can find it explained by a different chain of reasoning.
Sometimes even just a different author will help. I survived Introduction to Real Analysis (the study of why calculus works) by accepting that I just didn’t speak the textbook’s language. I borrowed an intro to real analysis textbook that was written in French. I don’t speak or really read French, though I had a couple years of it in middle and high school. But the straightforward grammar of mathematical French, and the common vocabulary, meant I was able to work through at least the harder things to understand. Of course, the difference might have been that I had to slowly consider every sentence to turn it from French text to English reading.
Probably there can’t be a universally right answer. We learn by different methods, for different goals, at different times. Whether it’s all right to skip the difficult part and come back later will depend. But I’d like to know what other people think, and more, what they do.