For the past three years, I’ve gotten to teach an Independent Study in Calculus II. (Yes, that’s right, you should be jealous.)

We normally offer AP Calculus AB, but that’s our highest normal math. Every once in a while, though, we have a junior in Calculus AB and we have to figure out what to offer them their senior year. A few years ago, we had a student like that, and their family asked me to do an independent study with them. And thus, Calc II was born. We cover Calc BC without the AB topics (So…volume by shells, arc length, surface area, a little bit of extra diffeq like logistic and linear equations, integration methods, vectors and parametric equations, and series. Lots of series.) and then have a month or so to cover whatever we find interesting.

It’s a little bit different every year. The class is really just one – or last year, two – students, and these are students who have already been with me for two years. At that point, I know the student and can tailor the class to their interests and needs. Like a lot of proofs? Sure, we’ll prove everything? Dislike proofs, but need a lot of practice? Ok, more hand-waving, fewer proofs, more time practicing. This year, my student is asking really, really good questions: “Wait, what would happen if…?” kind of questions, questions that I’ve never really thought about too deeply. My answer is often, “I don’t know, but let’s find it. If we try this,… .” If nothing else, she’ll learn that math is about exploring and finding answers for yourself. I’d like to make sure I come back and explore some of these questions more intentionally next time around, so I’m going to make more of a point of blogging about them.

- When we covered the shell method, we derived the formula for the volume of a sphere. When we did surface area, we derived the formula for the surface area of a sphere. We compared them. We noticed their connection. We talked about why that would be true. She wondered if there are other shapes where that was true. We explored – cylinders yes, circles yes (but circumference and area), squares no, rectangular prisms no, triangular prisms no. Why is this true? We were stumped.
- When we started integrals that use trig substitutions, we set up a triangle to do a sine substitution. She asked what would happen if we set it up the other way. We tried it out (use a cosine substitution) and saw what happened. (Works just as well!)

Her questions and her obvious desire to understand and explore also make me be more careful with my explanations – ok, yes, we did a clever algebra thing with this integral; what did I see that made me do the clever algebra thing? I should take it a step further – what’s an integral that would also need this clever algebra thing?