How I Teach Calculus: A Comedy (Front-loading Infinitesimals)

My heart doesn’t shrivel nor my soul distend because I hate parabolas, not even because I’m morally opposed to plurals that end in “i,” no, it’s the lack of a grand narrative that causes me unrest.

I’m now enjoying the true luxury of having a blog; I can review my own posts about lessons from previous iterations and revise them for lessons that I’m currently teaching. This post extends the post previously written about the limit definition of the derivative.

We started with one of these:

Slides.

<Socratic Dialogue>

How does it work? Yea, but how does it know how fast they’re going? What math does it do? Did some archaeologist find this in the dirt and reverse engineer it for all other radar gun manufacturers? Who programmed the math?

We then attempted to make it work. A running human? Not fast enough. A wavering hand? Too jumpy.

Interesting.

We took this to the streets. They radar-ed cars, trucks, bikes, birds, & each other. Turns out 7 mph is really as low as the thing goes. Hmm, it must not be able to do the math below that number.

Interesting.

</Socratic Dialogue>

Let’s rewind. Why did I bring a radar gun to school? Well, last year I just didn’t feel like my students had a handle to hang their understandings of infinitesimals on. The idea that something can come as close to zero without being zero is central to understanding calculus, and I was doing a poor job of teaching it. Sleep has been lost.

I firmly believe in context before content, whether that context is something from the “real world,” or just an infuriatingly curious problem. The words “Ok, so, today we’re moving on to parabolas and their foci. I know yesterday we were talking about proofs, but trust me, you’ll need to know this for some class later” make me die inside like the rose from Beauty and the Beast. (Yikes, I google-imaged “Dying Rose” for a picture. Mistake!)

My heart doesn’t shrivel nor my soul distend because I hate parabolas, not even because I’m morally opposed to plurals that end in “i,” no, it’s the lack of a grand narrative that causes me unrest. I want my students to feel immersed. I want them to love the characters. I want them to care when we can’t do something, because dividing by zero is keeping us down.

<Socratic Dialogue>

The radar gun sends a wave packet out at some microwave frequency. When that wave bounces off of an object (say, your car’s bumper) that wave returns. If it returns as a replica of itself, the gun interprets that as rest. If the car was moving, the wave will be smooshed or stretched as the car bounces pieces of the wave back. This is the Doppler effect, holmes.

We stopped just short of dissecting the gun — which I had just received — and hooking it up to an oscilloscope to see what was happening. My legal and accounting team wouldn’t authorize that one.

Here’s my Doppler slide (I’m not very good at Keynote, sorry)

A problem arises when the car changes speed while the wave packet is in mid bounce. The wave comes back all wonky, stretched here and smooshed there. This is no good.

Let’s attack this from a different angle. We then get to graphing. What does a car look like on  graph of d vs. t when standing still? moving? Any of these following situations. The kids graphed these, and made up some of their own.

It’s the changes in speed that give the gun fits. It’s the curvy parts of the graphs. Ah-ha. I win. Calculus wins. Society wins. The hulk has been abated.

In short, the makers of the gun need to create a wave packet so short that it almost has zero length. Luckily, this is easily obtainable with the kind of GHz radiation used in these guns, which can provide thousands of oscillations in even the shortest of wave packets.1 The motivation for that sentence is what I want from my kids. It’s the motivation for the limit (There’s a sweet slide with Isaac on it, if you want to see the all the slides. Link at the top.)

The lesson at large then begins. A night separated these two sessions, and I had all sorts of fantastic questions come in from students via email and conversation. They had looked things up on their own. I knew I had them, and that’s all it takes for some direct instruction to be meaningful.

See: the Limit Definition of the Derivative


1. Yes, I know, there’s a lot more going on inside of even the chintziest radar gun, relax, it’s the go-to-zero I wanted out of them.