16 August, 2008

Bet You Never Thought of Mathematics as Emotional Before

Efrique, whose couch I may someday temporarily have to beg as it's located far away from McCain, has a glorious post up exploring the emotions elicited by mathematics.

Before I got older and wiser, I used to see mathematicians as cold, passionless logic machines. I couldn't conceive of an emotional connection to all of those rigid numbers. It took a lot of reading in science before I realized that math can do exactly what Efrique describes:

A really clever manipulation (I can't help but think of them as "tricks") or an inspired substitution that makes a difficult problem easy can produce a tingling sensation up the back of my neck and head. A particularly beautiful piece of mathematics can, on occasion, move me almost to tears.

Then there's joy and delight. On occasion I have had the fortune to look at some neat, if modest, just-derived result and wonder if perhaps I am the first to have ever seen it (it is, obviously, rarely the case that I am - it is not unusual to find that my result has been tucked away in some mathematical corner for many decades ... on one occasion I found I had been beaten by Gauss - but the thrill of discovery is there all the same).

Mathematics can be intensely emotional. I've read mathematicians talking about math with the same passion and thrill that I experience at discovering a tremendously well-written sentence. When I understood enough physics, I finally caught of echo of the excitement and awe E = mc 2 elicits. It truly is dramatic, and beautiful.

I think that's what's missing from so much science and math education: emotion. Grammar suffers from the same disease to a lesser degree. We get so caught up in teaching kids the foundations that we forget to keep them excited about the edifice that could eventually arise.

If you're learning by rote and told there's only one possible right answer, you're not likely to understand that strong, rewarding emotion is possible. When I tutored English, I invariably discovered that all the joy'd been sucked out of it for the struggling students. They were so beaten down by rules they couldn't feel a damned thing. That had killed their motivation to master those rules to the point where the rules vanished and the beauty began. I'd usually spend a few sessions pumping them up: English is easy, it's exciting, it's really really awesome!! Once they could feel, they could punctuate. And when they could do that, I'd show them how to transcend the rules, which really got 'em going.

We need something like that with science and math. We need teachers who can make it seem simplicity itself, too exciting to stop even when it's tough, and so dramatic that you're determined to keep slogging right through to the breathtaking vistas at the top. We need drama. We need passion. We need blood, sweat, toil and tears. We need, in a word, to make it emotional.

Rationally emotional, o' course. Let's don't get carried away. But you can be utterly rational and beside yourself with emotion at the same time. The two states aren't mutually exclusive. Ask Efrique.

When people understand that, I don't think they'll see science and math as esoteric arts for emotionless experts anymore.


Efrique said...

Hey Dana, thanks!

I typed a longer comment before but managed to lose it (and yeah, I didn't take my own advice about how to avoid that...). So this will be shorter.

I couldn't conceive of an emotional connection to all of those rigid numbers.

The thing is, of course, mathematicians don't deal with numbers themselves very much. They talk about properties of numbers (well, the number theorists do), but most mathematicians don't deal with ordinary arithmetic much. Indeed, a professor of my acquaintance (quite a good one, as it happens) is unable to do mental arithmetic at all. I watched him pull out a calculator when in the course of a conversation about the high school curriculum, he needed to know 14+9. Algebra, sure - even someone working in knot theory is mostly doing stuff with polynomials. But actual arithmetic - most mathematicians aren't dealing with that. Depending on which area of mathematics we're talking about, it could be a bit like a writer worrying whether the words are spelled right (which is kind of important), but for others it would be more like worrying about the font their book would be printed in, or even what the texture of the paper was like on the pages their novel was printed on. For a rare few, it's more like worrying about the motivations of their characters (and so of relatively central importance), but for most mathematicians numbers aren't central at all.

I'm a statistician, so you'd think numbers would be critical to me. But actually, most of the mathematics I do is algebra, calculus and a few other things. I'm pretty good with numbers, but actual calculation is mostly not what I do as a statistician. It's not like I am going to perform Markov Chain Monte Carlo or a bootstrap or even fit a GLM in my head, or on paper. Computers do that stuff. I might design the algorithm. I may even write the code. I'll decide which models to consider. I will use the computer to draw some pictures of the results. I probably won't do any arithmetic myself. I will probably mention a few numbers in discussing my results (such as estimates or intervals for parameters), but they won't be the results of calculations I actually performed.

That's one of the tragedies of mathematics - the early stages that puts people off mathematics, like arithmetic, isn't actually all that much like mathematics that a mathematician does (and I include a lot of high school mathematics in this). It's like only learning the alphabet and then wondering why people think it's interesting to write.

I think one of the problems at school is too much drilling, too much routine homework, too much testing - at least that's the case here. It's boring, and that's counter-productive. There's obviously a need to learn skills, but I think that can be done largely in the context of other things. The thing is, I don't think it's the fault of the teachers, I think it's the other parents pushing the teachers into more and more homework. Fortunately the level of homework has dropped a little more recently, down to a level that I can accept as reasonable, but it was getting ridiculous there for a while.

We play a lot of games at home - boardgames and cardgames. They're full of mathematical skills. Something like Blokus Trigon, for example, covers a whole pile of different skills in the course of a game. Monopoly is full of arithmetic. Even just adding up die rolls and working out what you need to get to do something or avoid something is exercising skills. Even when the kids want to buy some Pokemon cards, we're doing mathematics, as they work out what they can afford ... "I can get a starter and a booster, or four boosters, or that nifty tin full of cards with the special Level X card in it". Then they check their change. Sharing out food gets you into fractions. The trick is not to be too obvious while you're doing it - after all, we're just playing board games, buying cards, sharing food and fun stuff like that, right?

With a lot of the basic skill practice tucked away into everyday activities**, that leaves room to do some mathematics that's interesting or fun.

**(sometimes it's a bit hard to practice certain skills in everyday activities without putting in a lot of effort, but that's okay - because a little bit of homework is not a big hassle if mathematics isn't boring yet.)

Efrique said...

meant to add - cooking is another fun activity that's full of basic skills.