Ask Again

I've been going into my calculus classes to teach with fewer and fewer notes this semester. I've got a few examples in mind, but that's about it. And I really think it is going well. I think it helps that I started this way on the first day. I also seem to have a pretty good class - I've certainly been pleased so far. After today's class, and some others recently, I'm starting to think that posing the same question to my class a few times, either asking for a different solution, or just a differently-stated solution, is a good habit.

Last week we were talking about improper integrals in class. After some introductory discussion, I asked them to tell me which of the integrals, $\int_1^{\infty} \frac{e^{\sqrt{x}}}{\sqrt{x}}\ dx$ or $\int_1^{\infty} \frac{1}{\sqrt{x}e^{\sqrt{x}}\ dx$ could possibly converge, and why. The first answer I got was to actually compute the first integral and notice that it diverges, so that couldn't be it. A good answer, certainly, with an opportunity to talk about $u$-substitution with improper integrals. And then I asked again, for another reason. A student brought up the integral $\int_1^{\infty} \frac{1}{\sqrt{x}}\ dx$, which we already knew diverged, and this gave us the chance to talk about the comparison test. I think at this point we about ran out of time, but I know somewhere in there we also had a discussion about functions that approach 0 not necessarily having a convergent improper integral to infinity. We didn't get to L'Hospital's rule (hopefully a reminder, this being calc 2) that day, but did eventually, with this same example.

Today I was talking about parametric curves. I asked them to take a few minutes to try to sketch the parametric curve $(t^2-1,t^3)$, and then had students come to the board to draw what they had. I was happy to get two different answers - something like a sideways parabola, and then the correct graph, with a cusp. I asked them how we could tell which could possibly be the correct graph, because they both went through the first few easy points, (0,-1), (-1,0), and (0,1). A student pointed out that $y$ was changing faster than $x$, so it shouldn't be the sideways parabola. I didn't want to talk about concavity just yet, but we did get to it eventually.

Then I asked basically the same question, drawing three different curves that weren't the sideways parabola. One had a cusp at (-1,0) where the derivative approached 0 from both sides, another had a cusp where the derivative approached +/-1 (or so), and the third had a vertical asymptote at (-1,0), almost like the graph of $e^{-x^2}$ flipped sideways. I asked which of these it could possibly be, several times. I was delighted to get lots of answers, and continue to pressure the students to reformulate their answers to try to be more precise. I like to think that I waited through the quiet moments while students were working on more answers, or reformulating old answers. I like to think that's what they were doing, instead of just waiting for me to tell them the answer.

I love the feeling that nobody in the class is talking (me especially) because everybody is thinking hard about the same question. I hope that was actually the case with the quiet moments today. I think this process of asking the same question a few ways, or asking for students to answer it a few ways, and me harassing them about their answers, is a good process. We're not just going through "routine" calculations, and I like that. I also hope that I can continue to get lucky with examples that I can do this with.