16 October 2008

Another shot at the "theory vs. practice" question

Michael Alison Chandler, an education reporter for the Washington Post, is currently taking a high school "Algebra II" class. As she said in her first post, she's "not a math person" and yet math increasingly comes up in what she's reporting on, so she wanted to see what math education in the schools is like these days.

In today's post, she writes:
It's difficult to describe how or why math works. It's easier to just write the formula and say, "Do this." Several readers have commented on this blog that what's often missing from math education is more of a focus on why certain applications work. I agree. It's harder to remember what to do, if you don't have some sense of why it works.
This is something that many mathematicians should remember. But there's a balance to be struck; if you spend too much time on the theory ("why" it works) and never apply it that's silly too.

As you may have guesed, I primarily view mathematics as a tool for solving various interesting problems; I'm often not interested in the theory for its own sake, but I do like that knowing "theoretical" things makes lots of problems more tractable. Often such problems involve lots of calculation, and as many of you know it's easy to get lost in a calculation if you don't understand why you're doing it. But if you never calculate, and you only prove general results, I feel like you're ignoring why this subject exists in the first place. (Mathematicians of a more theoretical inclination may, of course, disagree.)

3 comments:

Calculus Dave said...

I think I do disagree a little. I mean, as for myself, I am with you and am often looking for an application. But there are two reasons I support "math for math's sake." One is the purist approach and I think it's cool to plumb the depths of logic and understanding in some situations. Much like some kinds of modern art or climbing Mt. Everst...you do it because it's there.

The other, though, dovetails with your viewpoint. I've heard a number of examples where more "theoretical" maths have become applicable much later than the original theory. From my own understanding, an example is knot theory. Originally Kelvin started it trying to describe the elements as knots in the ether. As that theory left physics/chemistry so did the original application. But not with knot theory showing up in biology and in creation of cyclic molecules, etc, it's become applicable again.

Anyhow, back on the Algebra II front, a lot of the "real life" examples we use in there is so contrived to use only two variables or to result in integer solutions that some of the true application gets lost. One of the other teachers at our high school likes to day, "True, you may never use this in your life. In fact, probably 90% of people do not do this in their lives and live very rich and fulfilling lives. BUT, the people who run your life use these all the time."

jd2718 said...

So, real question, why are they solving little systems with matrices?

I am concerned, in general, that the "doing" has recently been allowed to get way beyond the "why."

In New York, state standards now call for kids to do regression, linear, exponential, etc, using a calculator, not really knowing what they are doing, or why. I think that is unacceptable.

On the other hand, all theory and no examples... I don't think that happens very often in American high schools.

What I'm aiming at is more application than theory, but no application (or almost no application) that is unsupported.

My line, and I take it fairly seriously, is that students do not have to be able to explain why everything they do works, but I have an obligation to show them wy something works before they start applying it.

Jonathan

Todd Trimble said...

I don't think there's much to disagree with here, but I personally don't know any examples of the hypothetical theoretical type who doesn't do calculations. My own area of research (category theory) is considered by many to be extremely theoretical, but most of the best exemplars in the field (Lawvere, Joyal, Freyd, ...) love to calculate. (I do too.) But they also get off on the tremendous power and economy of thought that theory can contribute toward the carrying out of calculations, as you were suggesting.