More than you bargained for

This semester has been out of control since January. One of many consequences is that I haven’t had time to write about my teaching. There’s a serious backlog of stuff to document; no time to type it up.

Today’s post is a bit of a mix.

Nat Banting wrote this week about a lovely moment when a student looked at a task from a new perspective.

The look on my face must have been priceless, because she started to laugh. The scene went on for quite a while. Slowly but surely, every student had approached the desk to see what was up. The student beamed as she explained…

Explained what? Go read it.

It got me thinking about similar episodes in my classroom; those moments when I take the time to ask instead of tell, and when my students’ ideas blow me away. The moments when my students teach me some mathematics.

Calculus 2 has had many of these moments this semester. We were studying approximate integration (about which, much more in future posts). We had a motto, “When you cannot integrate, you must approximate“. We had approximated with rectangles and trapezoids. We had built up the formulas for these methods based on students’ intuitions. And then it was time to deal with Simpson’s Rule.

If it’s been a while since you studied such things (or indeed if you never have), the basic idea is in the picture below:

We’re trying to find the area enclosed between the function above, the x-axis below, x=-1 on the left and x=1 on the right. Those rectangles give a pretty good approximation of that area. Each rectangle has a bit of extra area (above the function) and leaves out a bit of area; those roughly compensate for each other and the result is a good estimate.

The reason it’s not a perfect measure is that we are using straight lines (apologies to Chris Lusto) to approximate a curvy function.

So I asked my students what a reasonable solution to this problem would be. What curvy functions should we use to approximate f(x)? We all agreed that it would be desirable for these functions to have nice calculus properties, since that’s why we’re approximating in the first place (that original function doesn’t submit to the standard set of techniques for finding this area exactly).

We were building towards Simpson’s Rule, which is based on parabolas. Use parabolas as tops on those rectangles instead of horizontal line segments and you can get a really nice fit. Plus, parabolas are easy to integrate. Plus if you do a ton of complicated algebra, you can find a really nice formula so that you don’t even have to bother integrating (the pedagogical benefits of this are debatable; the calculational benefits are massive and undeniable).

So I asked. Not just rhetorically. I asked and I listened to their answers. They wanted to use sine (or cosine). And they wanted to use exponentials.

Of course. These are the things with the simplest antiderivatives. Polynomials get more complicated when we integrate. Sine, cosine and e^{x} pretty much stay the same so they’re easy to evaluate. It’s brilliant, right? The algebra won’t work out nicely, and you won’t end up with a clean and tidy rule. But who cares? We’re trying to learn some Calculus here; some ways of thinking mathematically about relationships among functions.

I made the evaluation of this integral by a sine-based version of Simpson’s Rule into an A assignment. Several students are working on that right now.

So much more to report. It’s been a productive semester. I’ll get on that in just a few more weeks.

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