There was a postdoc in my research group in grad school who had a sister in college. She called him once to ask for help with a math assignment dealing with series expansions. He checked a book to refresh his memory, and then told her how to generate the various series needed for her homework assignment.
A week or so later, he asked how she’d done. “Terrible,” she said. It seems that he had just plunged ahead with generating series terms without doing the convergence tests and other proofs that a mathematician would do for the same problems. She told him, “My professor said I answered all the questions like a physicist!”
I was reminded of this in the comments to a recent post, where “alkali” suggested getting around the problem of algebra-based physics by “teaching a really degenerate form of single-variable calculus (no derivations, basically “here it is”) and then go[ing] on to teach an actually useful kind of physics.” David responds, saying:
If you do this, you will get students whose knowledge of calculus has analagous flaws to the physical knowledge Chad describes above. And this will hurt them in the end. When I took introduction to electricity and magnetism, I watched my fellow students destroyed because they didn’t know what div, grad and curl actually meant. In mechanics, there were similar problems with respect to single variable calculus–why do you integrate to get an average pressure, why is acceleration a double derivative–although I don’t remember it being so bad.
There’s always a certain tension regarding the degree to which we need to emphasize mathematical correctness in teaching physics. In an ideal world, of course, we would do everything precisely correctly. But then, if you approach physics derivations like a mathematician, you end up spending so much time proving theorems that you never get anywhere…
Physicists, particularly experimental ones, tend to be a little cavalier about mathematics. We have a tendency to neglect strict formalism, and fall back on physical intuition to justify steps in our derivations. In lecture last week, I used a couple of tricks that tend to give math majors hives– things like taking a “dx” dividing it by a “dt,” and calling it a velocity. The steps are all correct, but they’re justified not by first-principles mathematical reasoning and proof, but by a sort of hand-waving reference to basic physics concepts. When you’ve got an infinitesimal distance divided by an infinitesimal time, of course that’s a velocity. What else would it be?
We get away with this sort of thing because we’ve always got comparisons to physical reality to fall back on (at least in the low-energy physics world). As my colleague pointed out regarding his sister’s homework, we don’t usually bother checking the validity of series expansions because they work very well for the vast majority of physically interesting cases. And when a series expansion fails, it usually shows up as a physically impossible prediction, in which case you know to go back and check your assumptions. You don’t have that luxury in pure math, so you need to be a lot more careful about following formal procedure. (More after the cut.)
Personally, I’m perfectly happy with this state of affairs. I’m not all that mathematically inclined, and I have a much easier time understanding things when I can construct a concrete physical picture of what’s going on. This is why my understanding of solid state physics is absolutely abysmal– two chapters in, they started doing everything in terms of “reciprocal lattice vectors,” and lost me completely. I couldn’t see where the atoms were, and what the electrons were doing, and I never really recovered.
As a result, I tend to emphasize intuition over formalism when I’m teaching. I spend a good deal of time working on physically appealing justifications for how things work, and try to gloss over the mathematical details as much as I can. That’s how I learn best, and that’s what I’m comfortable with teaching.
I was amused to learn, when this subject came up at a New Year’s party thrown by a math professor (we’re wild and crazy in academia– earlier in the party, I had spent about half an hour talking about the BEC-BCS crossover…), that there are people who have the opposite problem. One of the math faculty said that she had bombed introductory physics back in college, because of exactly the dilemma suggested by my commenters. Her professor was trying to teach a bastard version of calculus on the fly, and she struggled with it because she wanted to see proofs and theorems, and was very disturbed by the frequent use of “proof by physical plausibility” and “proof by dimensional analysis.”
I’ve got a couple of students this term who might be in that boat, and I’ll do my best to accomodate them (even if I have to fire up the Mathematica Integrator to fill in some of the details). Still, part of my job is to teach students what it means to “think like a physicist,” so you give me a length and a time, and I’ll give you a velocity, and theorems be damned…