He doesn’t learn algebra
in the algebra course;
he learns it in calculus.
I have been catching up on my Bloglines reading [procrastinating blogger at work — I should be going over the MathCounts lesson for Friday’s homeschool co-op class], and found the following quotation at Mathematics under the Microscope [old blog posts are no longer archived].
…a phenomenon that everybody who teaches mathematics has observed: the students always have to be taught what they should have learned in the preceding course. (We, the teachers, were of course exceptions; it is consequently hard for us to understand the deficiencies of our students.)
The average student does not really learn to add fractions in an arithmetic class; but by the time he has survived a course in algebra he can add numerical fractions. He does not learn algebra in the algebra course; he learns it in calculus, when he is forced to use it. He does not learn calculus in a calculus class either; but if he goes on to differential equations he may have a pretty good grasp of elementary calculus when he gets through. And so on throughout the hierarchy of courses; the most advanced course, naturally, is learned only by teaching it. This is not just because each previous teacher did such a rotten job. It is because there is not time for enough practice on each new topic; and even it there were, it would be insufferably dull.
— Ralph P. Boas
[Scroll down a bit for Boas’ essay.]
Unfortunately, the quote is too long for my blackboard. I’m not sure my students would understand it anyway, but it sure rings true to me.
Boas was one of the authors behind the famous (but unfortunately unavailable on the Web) 1938 paper, A Contribution to the Mathematical Theory of Big Game Hunting, published in the American Mathematical Monthly under the pseudonym H. Pétard. You can read that article and much more in Boas’s book: