An exam is coming up soon, and the students are beginning to panic a bit about quizzes. I keep trying to tell them that the quizzes are a pretty low key affair (they don’t make up a huge portion of the final grade, and I tend to grade leniently and give lots of feedback), but I have enough type-A overachievers that it seems to be a constant concern. Don’t get me wrong—I prefer to have really motivated students, I just with that they were motivated by something other than grades.
What I Taught
Product rule and quotient rule.
That was pretty much it.
Okay, okay—the narrative version: I started the class by finishing the discussion of exponential functions. We started from where we left off earlier, namely the identity
\[
\frac{d}{dx}a^x = a^x\lim_{h\to 0}\frac{a^h-1}{h}.
\]
I noted that the limit is some well defined number (with some mild assumptions on \(a\)), but that we don’t yet have the tools to do much more with it. I then suggested that it would be kind of convenient if we could find a value for \(a\) such that the limit were equal to 1, as, in that case, we would have
\[
\frac{d}{dx}a^x = a^x
\]
Using the magic of poofing entities into existence by fiat, I said that we might call such a number \(e\) (for Euler’s number). That is, we can define Euler’s number \(e\) to be the magical constant such that
\[
\lim_{h\to 0} \frac{e^h-1}{h} = 1
\]
and such that
\[
\frac{d}{dx} e^x = e^x.
\]
While this is not my favorite way of defining \(e\) (because I find the aesthetics so charming, I prefer a definition in terms of the integral of the hyperbola \(1/x\)), it is certainly functional, and gives the students something to work with.
After finishing up the left-over bits from the previous lecture, I administered a quiz, and then spent the next 55 minutes proving the product rule and the quotient rule. And I didn’t even do everything that I wanted to do. Specifically, I presented proofs that work straight from the definition of the derivative, given by
\[
f'(x) = \lim_{h\to 0}\frac{f(x+h)-f(x)}{h}.
\]
This works, but relies on an algebra trick that seems really unmotivated and magical (I called it pulling a rabbit out of a hat). In the derivations of both the product rule and the quotient rule, there expressions that look like
\[
ab – \hat{a}\hat{b}.
\]
We can’t do much with this, but by introducing an intermediate term of the form \(ab’\), some factoring can be done:
\[
ab – \hat{a}\hat{b} – a\hat{b} + a\hat{b} = a(b-\hat{b}) + \hat{b}(a-\hat{a})
\]
The differences turn out to be just precisely the terms required to make some derivatives pop out, but he motivation for tossing in those intermediate terms is unintuitive.
The textbook presents a rather nice proof of the product rule using Leibniz’s notation and some geometry, though many of the details are left to the imagination of the reader. I had really wanted to go over that argument, as it is both elegant and short, but just didn’t have time. I think that I will type it up and give it to them as a bonus…
What Worked
I teach from 5:30 until 7:05 in the evening, and then have a mile long walk home, which generally allows me time to decompress and say high to the llamas (really). I started judging how well my class has gone based on how I feel when I walk in the door—the more exhausted I am, the better class went. By that metric, the product and quotient rules went quite well.
Since I wasn’t entirely sure how I wanted to present the proofs, I spent much of my prep time going over various alternatives. By the time I got into the classroom to teach, I had gone over both derivations several times in a variety of ways. I felt really well prepared, and I think that the crispness of the presentation bore that out.
As the presentation went well, I suppose that this is an appropriate place to go a little bit meta about the class. I believe that mathematics is an important part of a well-rounded, liberal education not because it is a tool for performing computations, but because mathematics can be used to teach people how to think rationally in a sterile setting devoid of emotional content. This can be difficult in other fields (physics, history, art, &c.), as what is capital-T “True” is nebulous, ill-defined, and may be relative to the viewer. In mathematics, there are solid notions of what is true or not. If you can prove it from the axioms, it is True. If not, then it isn’t.1Yes, I know. This is an oversimplification. The axioms are only true by fiat, and there are true statements that can’t be proved, a la Gödel. With respect to a liberal education, these issues from the deepest depths of mathematical philosophy probably aren’t relevant. This allows us to employ reason to answer questions that have definite answers, which models how we can think about questions that don’t have definite answers.
As such, my lectures tend to focus more on the theory of calculus. I like to present proofs, and so I present a fair number of them. This takes time, since every step needs to be explained, and I prefer that the students do some (if not all) of the work. Sometimes, I wonder if this is a good idea, since most of my students (all of my students?) are not math majors. In fact, most of them are engineering majors.
Thus I had a nice moment of vindication last night when a student commented that she really like to see all of the proofs. She felt that it helped her to understand what was going on, and gave her insight into the computations that she had to do in other classes that memorizing formulae could never provide. So, there’s something that worked. If nothing else, it made me quite happy.
What Didn’t Work
This is a topic that I need to think more about in the future, and my thoughts are not entirely formed at the moment, but since the previous section is a bit meta, I’ll steer in that direction here, as well.
A truly Socratic method involves asking questions in order to illuminate an idea. In a real Socratic dialog, the participants should act as equals, whether or not they are really equals in a particular context (i.e. there is a power dynamic between a teacher and students, and equality is uncommon). An honest Socratic dialog is difficult (if not impossible) for an instructor to hold.
Instead, in many classrooms, teachers ask questions (because the Socractic method is all about asking questions) with the intention of eliciting specific replies from students. As a student, I find this technique annoying, and I try like hell to avoid it while teaching. And yet, I do it quite a bit. That is, I ask questions to which I want a very specific reply. Why do I do this?
I hadn’t really thought about it until walking home after the last lecture—there were a couple of times that I tried to get incredibly specific responses during the lecture, and it was bugging me. As I said, I don’t have a fully formed answer yet, but here is an attempt:
I teach calculus because I think that it is an important way to help students learn to reason. When I am presenting a proof, I am presenting a certain kind of reasoning. However, much of the reasoning is done “off the board,” and has to do with the kinds of questions that I am asking myself (e.g. “I would love to have a term that looks like \(f(x)g(x+h)\) in this expression, because then I could pull out a derivative. How can I make such a term appear?”). Thus when I ask the class such a question, I shouldn’t expect a specific answer, or any answer at all! Really, the goal is to demonstrate what I would be thinking if I were trying to solve the problem for the first time. This philosophy of questioning needs to be expanded some (there is probably some publications that I need to read), so I won’t comment on it more, but I feel like I have found a workable answer to a problem that has been bothering me for years.