Latest piece in the philosophy

So, there's a thread on the AP-Calculus mailing list right now, about rationalizing the denominator (that is, if you have a fraction with, say, a square root in the bottom, you multiply the top and bottom both by that square root, which effective puts the root in the top instead of the bottom). A couple of people defend the practice, for reasons I couldn't much understand. A few others slam it, for reasons that I do very much understand. One person, who is maybe in the middle ground, pointed out that it's really a pretty neat hack, and so might be taught just because it's kinda cool (see also most of the "neat" series, Pascal's Triangle, etc).

Here's what I just wrote to the email list. I'm posting it here, because I think it sums up a piece of my current teaching philosophy. It's a long way from two years ago, when I was dying to find a science teaching job...


Taught well, rationalizing the denominator can make for an intriguing half hour for a student who is at the right place in his math-student career to spend that half hour contemplating it. Excellent teachers will help put that student in that right place, by routinely, consistently inspiring students to be interrogative, and to see the beauty in mathematical patterns.

Without those routine elements to drive the exercise, rationalizing the denominator is a dry and meaningless exercise, with no context or meaning for the student-- just another "math rule" that they do because they were told to.

As you point out, there are LOTS of beautiful mathematical patterns that a good teacher can rely on to create those intriguing half-hours. Rationalizing the denominator isn't one of the ones that I choose to rely on, usually. Sometimes it comes up on its own, and then we spend that half-hour, but it's never something I plan to teach, on grounds that an interrogative student, in the exceptionally unlikely event that she ever needs to know it, will figure it out for herself in that same half hour...

To me, THAT'S what math education is about: to develop core problem-solving skills, and enough of a sense of pattern to allow students to understand patterns that I DIDN'T teach them, for themselves.