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:

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That’s so true! I have always known that I learned algebra in physics & chemistry, despite having really good math teachers. I thought it was just me!

That is very true. It drives my students crazy, too. They can’t see why they should have to learn a skill, and I explain they’ll need to master it now to succeed easily at the next level of math. They always think I’m kidding.

It is important, though, not to let this turn into an excuse. It’s not ok if they don’t learn it this year because they will learn it three years from now. It is understandable if they don’t master it the first time through. Very different ideas.

Perhaps it is a difference between learning and mastery. We can learn the topic the first time through, in the sense that we can perform it on tests, but we don’t understand and internalize it until we have used it repeatedly in other contexts. And if we learn it only to the “successful test performance” level, it will fade from memory quickly.

@ kirylin: Did you click through and read the article by Boas? Here is the part that comes right before what I quoted:

Our students will need to master whatever we are teaching so that they can learn more math in the future. But they also need it so that they can practice and retain what they have learned in the past, which is what Boas talks about. And I think, too, that the process of learning and doing math gives the mind a workout and makes it to some degree more capable of thinking clearly in general.

One of my favorites:

The art of doing mathematics consists in finding that special case which contains all the germs of generality.

David Hilbert

There are much more at http://www.brainyquote.com

A couple more quotes:

Althogh it’s usually simpler to to prove a general fact than to prove numerous special cases of it, for a student the content of a mathematical theory is never larger than the set of examples that are thoroughly understood.

Vladimir Arnold.

Don’t give a definition, give an example!

Michael Atiyah