This of course brings up the question of what the "next" topic is. Calculus is not linear, and at a school with absolutely no prescribed curriculum the thought of moving on often propels me into a panic-ridden tizzy. What's the best next topic - the one that will tap into the students' current understanding, keep their interest, draw on their mathematical strengths, and work on their weaknesses? (Am I asking too much of a single topic?) My two thoughts are:
- Having them discover the power rule. They're primed to appreciate it now that they've spent weeks calculating derivatives using limits. Also, my other classes have known about the power rule for awhile now and have already tried to ruin the surprise (I have to admit, part of me loves that there even is a math rumor mill!), so I'd feel a little slimy hiding it any longer.
- Having them practice graphing functions and their derivatives. This will get back to the concept of the derivative as the "rate of change," which is something I really want to hammer home. I would have them start out simply by making observations using this super-rad calculus grapher, which I just read about on Sam Shah's blog. Gosh I love the internet.
I teach three calculus classes - two honors and one regular. Teaching this non-honors class in particular is really forcing me to hone in on the essence of calculus. Somehow I'd be missing the point if I tried to push the "standard" calculus curriculum on these students, because they're still getting comfortable with so much of the algebra that successful calc students take for granted. I'm sure that we could eventually get to the point where they could apply the quotient rule and the chain rule to complicated functions, but it would be at the cost of a greater conceptual understanding. This reminds me of something my office-mate Kyle said the other day: "A year from now, I'd rather have students say that they understand what derivatives are and they used to be able to calculate them, than that they know the derivative of x^2 is 2x and they used to actually understand why."
This brings up the juicy debate over how and why calculus is taught in high school, but that's for another day...