Today I flew across the nation, farting. On the plane, having gorged far beyond satiety on my Dave Eggers anthology, I turned my attention to trying to clarify for myself some points I had been considering in linear algebra.
Later that night, in catching up after a year apart, my older sister asked about my flight. Jul had once considered becoming a math teacher. She took the same AP math classes I did in high school, studied a few technical topics here and there on her way to a linguistics degree. She has some background.
So I told her briefly about my attempts to understand dual spaces. I don’t think it got through, really. The point wasn’t that a vector space is isomorphic to the dual of its dual. The point was that, yes, I had a nice flight, because I sat there with a fresh notebook and step by step watched the algebra of this thing grow out of blank space. The results I had heard about from one source here and another there were materializing right in front of me. It was sloppy. I’m no mathematician. But it was getting increasingly better as I cleaned up a point here and there.
Soon I saw how we could go about associating vectors in the dual spaces, and in one sudden flash of insight, saw that a certain freedom in this choice could lead to Euclidean spaces, or Minkowski spaces, or Hilbert spaces, or, although I can’t claim I actually understand what these are, more complicated Reimann geometries. It all depends on a “metric”, I had been told. But here I was on a bumpy, dry sky-bullet, with stewardesses slamming carts of full of orange juice and assorted Pepsi products against my knee every twenty minutes, serendipitously discovering what the hell a “metric” could be. It was a nice flight.
“Linear algebra”, said Jul, “was a pretty dull class.” Dull? Are we talking about the same linear algebra? And then I realized – no, of course we were not. “So your linear algebra class,” I asked, “was mostly about matrices, and multiplying them and finding determinants and stuff?”
“Yes, that’s right.”
Dammit! Because see, she didn’t take a class on linear algebra. She took a class on formulas. Which is a shame. My sister deserves a lot better than that. She’s smart. Really smart. She was the captain/president/founder of her high school robotics team. Scored like a bajillion points on the SAT (I got a bajillion and one. Sorry, sis.) She taught me how to multiply numbers by 11 when we were this high.
She has a little baby who is grasping after a new syllable or two every day now, and tentatively standing a few momentous seconds at a time on wobbly little legs. Will he sit up straight in his chair at lunch one day and declare through a mouth full of PB&J that it’s obvious a circle is the shortest possible line to enclose a given area, and then laugh and ask to go play Explorers with the kid next door? And if he does, who will notice?
ZapperZ at Physics and Physicists links to a recent paper on physics education. The authors tried to quantify the problem physics teachers are constantly battling – the wide gap in the way they and their students view the nature of the subject.
It’s inevitable that physicists will be more enthralled by their material than physics students on average. If they weren’t enthralled to begin with, the professors would never have gone to grad school. Still, it’s a somewhat saddening that so many students think of physics as a collection of formulas handed down from on high. That’s essentially what the survey shows.
Even at Caltech, I hear the constant complaint, “The problems on the test weren’t the same as the ones we did in class or on the homework.” Or, “the book doesn’t have any worked out examples.” I opened the book. I couldn’t understand, for a while, what they meant. The book definitely did have worked out examples. They were in the paragraphs that began “for example…” and then carried out a calculation. What they meant was, “the book doesn’t do everything for me.”
The other complaint, which I hear more often from younger students, is “I understand the concepts. I just don’t know how to solve the problems.” This has a variant for younger kids, which comes from the parents’ mouth, and is “He understands the math, he just has trouble with the word problems.” Then there is a long, expectant pause, “Can you just help him a bit with the word problems?”
No, not like that. It works the opposite way. I can normally solve the problems well before I understand the concepts. Occasionally I do understand the stuff but not the problem, if there’s some sort of sneaky trick to find. But the mantra of “I understand, but just can’t quite apply,” is some sort of warped refrain that echoes back and forth between students across the nation the way all meaningless idioms of speech do. It’s just something to say about a problem so arcane you aren’t really even sure what it is, or where to look for it.
I want so much to do something. To show them just a bit here or there, to get them started. I don’t know how. I think maybe the best thing to do is to take care of understanding more of this stuff for myself, first.
There are millions of people who really do get it, and can enjoy math on an airplane. Of course I know many of them in person, from school. Over the last few months, as I’ve spent more time on the sorts of places around the internet these people frequent, I see that they’re actually an incredibly strong and interconnected community. Interconnected, but disconnected. Floating in isolation through a nation of anti-intellectualism.