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.

I don’t have an answer, Joseph, but you have given me a great post to share with my teen. He is an atypical non-linear learner, but learn he does. Personally, I perceive myself as a lifelong learner so I lean toward moving on from difficult material with the hope of many opportunities to revisit it later, perhaps with more experience (wisdom, patience, etc.)

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I hope he’s able to draw something useful. You are probably right that there’s a difference between lifelong learners — well, let’s call them recreational learners — and people who learn because they want to get a specific skill or block of knowledge. A recreational learner probably has an easier time of letting a difficult point go and coming back to it later, not least because there isn’t a deadline demanding knowledge of this particular thing by this particular time.

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Exactly. One challenge younger students can face is when deadlines come before developmental readiness and/or interest. My goal is to keep my kids in the learning game, to see themselves as learners, until hopefully they get past all the deadlines and can have space to creatively tackle topics that were more difficult when they were younger. Hope your week is off to a great start, Joseph!

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Thank you; I hope yours is turning out well. I’ve been away from teaching for a couple of years so have been getting rusty in working out how to keep students interested and somewhere near deadline.

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When I think back to learning literary theory, in particular deconstruction back in the day when Jaques Derrida was still alive, there was a need for patience, as well as trust that it would make sense one day. Everything was cutting edge and so few people really understand this highly abstract philosophy, so it took time to absorb even the basic structures. To answer your question, I agree that learning requires a lot of suspension. In a couple weeks I start university again, this time studying mathematics, and it helps to remember that I already know my learning process, and it is not so different from the way you’ve described yours. (One less thing to sweat about.)

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I hadn’t thought about the parallels in other fields but certainly a similar process has to apply.

New or cutting-edge material has to be more difficult to learn, it seems to me. For one even the founding terms have to be more in flux. What the core concepts are, and what their meanings should be, requires experience and practice and a new field isn’t going to have that to draw on. And there’s less time for the development of good introductory exposition, and for alternate lines of presentation, in something new.

I confess I forget the details now but I remember in differential equations hearing of a cornerstone paper from about 1950 that, allegedly, no one had ever understood from reading it. What knowledge people had of it came from people who’d had it explained by the original author, who went on to explain to their students with only passing reference to the actual paper, and so on. I’m dubious of any claims about an unreadable paper, but the

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I think “not understanding the math course until you take its successor” is very true.

Von Neumann said something like, “One never really understands mathematics, one just gets used to it”.

Good article!

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Was that mathematics? I’d thought it was quantum mechanics, although there is a point where the difference doesn’t matter. The quote and the sentiment are meaningful either way.

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My advice to anyone embarking on a new topic in math is to read just the text on first, and maybe second, reading. You may get a clue as to what it’s all about that way. Then go back and try figuring out the symbolic stuff.(a bit at a time, and don’t worry about skipping chunks)

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I hadn’t thought of it, but realize that I do tend to read a mathematics text that way. That is, I’ll do a pass where I just read the text and don’t worry about the equations. Later on I do the equations with only glancing reference to the text. At some point I try them together. This is why it’s very slow reading mathematics, I suppose.

I’m a bit surprised that everyone who’s commented seems to have a similar, or at least compatible, learning style to me. I’m curious whether that’s really widespread, or if I just give off vibes that attract people with similar styles to mine.

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I remember learning physics as quite ‘patchy’. You learn about some things as if they were fundamental axioms, only to discover later you could derive them from something else; sometimes you were not so sure after a while ‘what came first’ actually. Then, suddenly, everything fell in place.

For example, it is hard to pin down for me how and when torque and angular momentum have been explained really the first time. In the first introduction it felt like a parallel world to force / momentum, introduced by using analogues … but not the full-blown derivation from Newton’s laws of course, using Euler’s angles. I think it is also related to the way and order how math and physics are taught. In my physics degree programme, we had two math lectures in the first year – Linear Algebra and Real Analysis (I think I mentioned before that we don’t had any ‘Generad Ed’ or ‘calculus for everybody’ lectures, these were lectures specifically for math and physics ‘majors’ only).

Theoretically, these were the perfect preparation for dealing with, say, tensors, and we were given a rigid prove of those theorems in vector analysis. But in parallel to the advanced math lectures we had an ‘intro to physics’ lecture using simpler math, and ‘theoretical physics’ started only one year later – and at that time it was hard to relate the physicist’s way of introducing Gauss and Stokes theorem to the much more rigid way you had learned it already. In Real Analysis the focus was on proofs for those aspects that were done more, well, hand-waving, in physics. For example, in ‘theoretical physics’ it was tacitly assumed that functions in the physical world, like distribution of charges, are sort of well-behaved, without kinks and singularities – so you did not have to make such a fuss over functions being continuous or differentiable :-)

Feynman once said, in relation to his second volume of lectures (electromagnetism) that he would do a ‘mathematical methods for physicists’ lecture, if he would have to do it again, instead of introducing those theorems together with Maxwell’s equations as it is usually done (and obviously also instead of teaching them in a way unrelated to physics).

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In thinking about it I realize my introduction to physics was also a patchy thing. I originally attributed that to starting out in an advanced program in high school, and so trying to rush physics and calculus into the heads of high school students who probably weren’t really quite up to it. And then Advanced Placement credit meant I was able to skip the college-level introductory courses that might have been a little more harmonious, or at least might have gone back over the basic ground with a little more system.

Now, Gauss’s theorem I remember getting in (high school) physics as part of the magic of working out electrostatic problems. I don’t remember there being any attempt to link it to what was going on in calculus. I’m not sure my high school calculus course got that far into multivariable integrals either. We did get to it in vector calculus in college, although there again I’m not sure it was ever specifically linked to what was going on physics. There, though, my skipped courses might have made things worse because I

thinkif I’d been taking electromagnetism at the same time as vector calculus the links between the two would have been more obvious. As it was, I had a bunch of vector calculus tools sitting around for a semester (and summer break) without any use.Somehow, I never got into a course that gave me enough to do with tensors to really get them straight. They did get a little mention in Classical Mechanics, although the professor wasn’t very interested in the problems they required so we got

justenough to get on to what he really liked. (Mind, he did some great work, including an exam problem I thought, then and now, was brilliant: given this completely different pairwise interaction law, derive the equivalent of Kepler’s laws for point masses, and work out whether solid bodies and point masses orbit in different ways.)While Linear Algebra did serve my physics courses I don’t think the links were very clear. It’s something I recognize in hindsight, probably because as physics gets more advanced it turns into many more linear algebra or linear algebra-like problems. Real analysis I don’t think served anything in my physics courses, although I realize it was the foundation on which the quantum chemistry course I taught one semester was built. (They needed anybody who could do it, and I was nearby.)

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