Math lagging (or leading) physics

Senior year of high school, I took AP Physics C and AP Calculus BC concurrently: allowable, but not advisable, the admins said. For a good part of the year, I struggled with physics because I hadn't `really' learned the math yet. Reading the textbook, I'd just skip over derivations that I didn't understand, and hope that the end result was something I could memorize, if not comprehend. Mr. T held a session or two of `physics math', which helped a little, but didn't make me feel less intimidated.

Last semester, I was in pretty much the same situation, only with line and surface integrals instead of all integrals. Oh, we never had to evaluate any -- they were always of the simple kind that reduce to multiplication problems -- but it took several tries to be able to recognize those situations, and remember how to reduce them to multiplication problems. Didn't solve the intimidation problem, either. On the final exam, even though I knew what to do and how the problems worked, I still felt like a trespasser for writing down those fancy-schmancy closed-surface double integral signs, knowing I didn't really know how they worked.

At one point Prof. Hudson said something (can't recall the exact words) that gave the impression this math-lagging-physics business was a dreadful intractable plague, oh me oh my, whatever shall we do. And at the time, struggling to understand everything, I was rather inclined to agree with him. But since then I've somewhat reversed position, or at least become neutral.

Back in high school, after I'd learned to fudge my way through integration, I had a much, much easier time with it when we `officially' learned it in math class. This was especially noticeable in the case of integrating (still in a single variable, though!) over 3D objects. We'd been doing this routinely in physics, to find the moments of inertia of various shapes. So while the rest of the calculus class was struggling to put together the relatively new concept of integration with the relatively new concept of bizarre-shaped objects in 3-space, I could relax.

Recently (within the past two weeks), the same thing happened for me in multivariable calculus. I'm in 18.01A/18.02A, which is a blend of two courses: a quick six-week review of single-variable, and then the regular multivariable curriculum, finishing over the January term. Because of the way things are timed, I never saw a line integral in 18.02A until just now, well after the end of my physics class. And the same thing happened: I'd learned to fudge my way through easy line integrals, and that gave me a leg up on people who'd never seen them before. I had a well-developed intuition about what line integrals meant in terms of the real world, which kept me from getting confused. That in turn allowed me to focus on the complicated aspects that we hadn't worked with in physics.

Given these experiences, I don't think it's a very bad thing at all that math education tends to lag physics education. It can often be a good thing: physics is leading math, to frame it positively. When you tackle something complicated and abstract, having `live-fire' experience with simple cases and workarounds can hardly do anything but help.

This comes with some caveats. It's a classic example of delayed gratification: I sure didn't feel happy about math lagging physics when I was actually struggling with the physics, and it was only a month later that I was able to grin to myself in calculus recitation, watching the people around me struggle with a problem I'd found obvious. (Schadenfreude, too, is not necessarily a good thing. In my defense, I do help my peers if I can, and if they're not already being helped.) And naturally, it's a tradeoff; who can say if knowing the math beforehand would have brought my physics grade up a little?

But most importantly, if physics is going to lead math, physics has to be taught right. I approve very strongly of Prof. Hudson's philosophy here: conceptually difficult problems with easy math = win. If you absolutely must be bashing your head against the wall at midnight the night before the problem set is due, it shouldn't be because you can't solve a differential equation. It should be because you can't figure out which way the induced current goes. Physics can be taught adequately without using too much complicated math (though of course the definition of `complicated' will vary with the physics). You learn just as much integrating (read: multiplying) over a sphere as you do integrating (read: flailing) over the surface of some crazy-shaped thing.

You wouldn't expect it, but knowing too much math can be a problem. Last semester's physics class was taught in a rather unusual format, about which more later. The relevant thing is that I spent the whole time in a group of three. There was me, there was S, and there was L. S had excellent intuition, and tended to approach problems by thinking them through qualitatively first, and then seeing if the math bore him out. This was really a very effective strategy -- a lot of the time the math looked right, but his intuition cried foul, and then we found the sign error twenty steps back. L, on the other hand, was advanced in math, and already pretty far beyond what the physics course used. She approached problems very mathematically, instead of with intuition like S. Even when something made complete physical sense, she wasn't satisfied until she justified it mathematically -- even if it was a qualitative problem and didn't have any equations in it to begin with. and because of how every other thing is backwards in E&M, she tied herself into an awful lot of knots. (Not that she wasn't brilliant; I think she outscored both S and me on every test.) L and S complemented each other, and they were a great group to work with. But I have to say that S's intuitive method was the more effective. To the same question, S might answer "because the charge is over here so the potential is higher here, and then this happens"; L might answer "because the potential is [insert equation here] and the electric field is the negative gradient of the potential."

I admit that this is based on my own experience, and what's better for different students will vary. I said that it's easier to learn complicated abstract math if you've first grounded it in physics; you could just as well say that it's easier to learn physics if you're well practiced with the mathematical tools already. But I think the first argument is more valid than the second. It's easier to abstract from the concrete than it is to learn the abstract first and only then get concrete examples. You see the same thing in computer science classes: pick up a copy of the blue wizard book and on every other page you'll find a discussion of how to abstract some instance, not how to instantiate some abstraction. Even though physics was made genuinely more difficult by not knowing the math, I'd still rather have done it this way around.