How I Teach Calculus: A Comedy (Concavity)
This post is a part a larger series documenting the changes I am making to my calculus course. My goals are to implement standards-based grading and to introduce genuine applications of the concepts being taught. I’m not suffering any delusions that any of this is all that ground-breaking, I just want to log the comedy that ensues:
The students and I have spent some considerable time working with optimization. This technique really shows them the value of differential calculus, and I don’t mean wal*mart “value.” By doing optimization out of “order” I’ve also introduced the concepts of critical points and increasing/decreasing behavior in context. We spent a few days working — fairly traditionally I might add — on these two concepts, but always within the context of optimization.
The next concept that naturally follows is concavity, or the behavior of the second derivative. Here’s what I thought to myself: “Concavity really shows a kind of acceleration. The slopes may be positive or negative, but they might be getting more or less positive or negative.” I’m also very concerned with obfuscation. I hate the idea of real-world examples for the sake of real-world examples. As a commenter pointed out, there’s even a little push back in the research world against real-world examples.1 As much as one, vaguely-written study synopsis can show, I think we can all agree that students need to be attacked from multiple angles. I do mean attacked, by the way.
I gave them a toy situation to play with, which modeled the behavior of a website based on how many times you ran your ad during the super bowl. The kids then plotted the data given by the program in Excel. This is what I would consider very structured. I have pre-programmed this site, I have decided what they will do. Despite looking student-centered, this activity really was teacher-centered, at best it’s “hands on.” It also worked famously. See “attacked” above. Here are some student generated graphs:
Discussion ensues. Why are the curves shaped the way they are? Hypotheses fly. We start talking about some basic marketing psychology. Funny’s only funny a few times. Serious just gets your name out.
“How much money would you recommend spending?” I ask. At this point I haven’t said the words “marginal cost,” “infelction point,” or event “derivative.” The kids come up with all sorts of ideas. They start hinting at the idea that you’d want to get the most new hits per dollar; after this point you still gain hits, but not as many per extra millions of dollars. They identify the inflection points. I’m not kidding; I was floored. They derived the concept of marginal cost on their own. I start to wonder if they even need me.
I ask, “What’s happening to the slope that this point?” A student explodes, “It’s the fastest it ever gets!!” Holy crap, yes Timmy. We call this an inflection point. The lesson begins. I pull up the source code and show them how the program worked. (You can too, just go to View->Source, the functions are on lines 26 and 31) They like seeing code. I haven’t taught them much yet, but we will later.
So the lesson on the second derivative test begins. Pretty standard, but I feel that their brains have been primed with the concept first. I’ve never had something this contrived go so well.
1. Education Research. Where do I begin? I feel like we’re all just sophists when it comes to ed research. We read what we like if it supports our strategies. We marginalize the research we don’t like saying it’s just done by quacks in Europe. I’m also pretty young, and probably offending all of my higher ed readers. Here’s what I do: I try stuff on my kids, and when they can generalize it to unique situations, I keep that strategy.