Brad DeLong asks a question that frets me, too:

For those who have difficulty learning to speak the language that is mathematics like a native, how to teach them science in a world where it is a fact that the underlying bones of reality are profoundly mathematical--for that is the conclusion Eugen Wigner's "The Unreasonable Effectiveness of Mathematics in the Natural Sciences" leads us to? Call this the "Friends of Wigner" problem.

Stated in another way, we could call it the Calvin problem:

I think the problem is more than just difficulty with mathematics. Look at the creationist's confusion I wrote about yesterday—it wasn't a failure of mathematics that lead to his foolish errors, unless you think it was an inability to count. Other disciplines have their own problems: historians have students who want history to be just the memorization of events that actually happened, rather than a difficult exercise in thinking and learning and evaluating. When I was attending the meetings to discuss the Minnesota state science standards a while back, that was the theme I kept hearing when the history standards came up: that rote memorization of dates and dogma is not history.

The real problem isn't math, it's epistemology. What we want from our students is that they understand how they know what they know. In the sciences, that often distills down into some properly applied mathematics and that common injunction on exams to "show your work." It's what we do in those peer-reviewed papers, which are all step-by-step explanations of how we got a particular answer. I suspect that one common thread among academics in all disciplines is that what we really like in a good paper is the logic and the story and the clever details that lead up to the conclusion, that what counts is the process.

The real problem is that so many people want the shortcut to the "right" answer (although students will change their tune when it's a matter of me going blind this weekend trying to decipher chicken scratches in blue books to give partial credit for applying the right method to a genetics problem, even if the final answer was off.) It's Bronowski's conflict between knowledge and certainty: most people prefer certainty, especially when knowledge might give them an answer they don't like. And they especially favor certainty when it requires nothing more than learning a single datum, rather than the work it takes to do a calculation or derivation or document a chain of evidence.

I don't know exactly what the answer is, but the root of it has to lie in teaching kids to enjoy figuring things out. One geeky personal example: I got introduced to model rocketry when I was in fifth grade, and I was a member of the model rocket club at my school up through junior high. I think, though, that I built precisely two rockets and launched them just once. The first time I'd watched these things, the instructor had handed me some gadget that I looked through and measured the angle to the rocket at the top of it's flight, and showed me how to calculate how far it went. That was it for me. Who cared about balsa wood and cardboard when there was geometry and trigonometry to do? I thought Calvin's problem was the fun part!

Our students aren't buying a finished product, they're getting a toolbox (with math at the heart of it) and instructions in how to use it. When they don't realize that central fact, that's when mutual disappointment occurs.