Lessons, Math, teaching

How I Teach Calculus: A Comedy (The Product Rule)

March 10, 2010 Shawn Cornally

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:

I’ve been taking some heat in the comments, so I’d like to clarify a couple of things.

1. I do not think I’m an expert. This blog is my way of saving my wife’s ears.
2. These posts on Calculus are not timely, nor are they exhaustive of my classroom activities. Some of these things happened months ago, and we do much more than I write about. I only write about the most interesting parts.
3. I teach on the block, which means I teach two year-long calculus courses every school year.

The kids now have some semblance of understanding regarding derivatives and instantaneous change. The tradition then is to introduce how to find the derivatives of functions that are much more complicated than simple polynomials. The students have graphically found the derivatives of the trigonometric functions, and the book throws a function like this at them:

This is what the book calls motivation. This is what students call, when-will-that-ever-happen? As I teach, the more it becomes blatantly apparent that kids have no idea where math fits in. I know that there’s a lot of merit to doing math for math’s sake, but only a few kids will want that. Most are just-burgeoning abstract thinkers, it really helps them to have ideas grounded somewhere.

So, I ask myself the same question I always ask myself: “When will the product rule actually need to be used.” Not can, but need. As a secondary question, I ask myself what will give the students a handle to hold on to when we delve back in to the abstraction provided by the book (and mandated by the gargoyles at CollegeBoard)? The physicist in me heads towards all sorts of fanciful functions, but in the end I settle on the damped oscillator. NOTE: I would NEVER say that to start a lesson. Good God, never say the esoteric name of the lesson first. That’s like announcing to a group of 7th grade boys that you’re going to watch “Titanic.”

The Guitar String:

A guitar being played untraditionally

So I bring in some guitars. Who cares where you get them. I asked the English teacher, the Band teacher, and I brought in a few of my own. I ask the question: How long is the sustain for these guitars? Do the more expensive ones have greater sustain than the cheap ones, as advertised?

The students then hop to it. Most start by pretending to rock out, or just strumming random notes. A few of my more timid kids just set the guitar on the table and look at it. Some start looking up prices. Some jump for the microphones. Some start timing. Who cares what they do. The point is to relinquish your stranglehold on their processes. Stop designing lessons with perfect outcomes, you know that the world doesn’t work that way. Trust the invitation to lead them to a genuine interest in the math you’re going to present. It doesn’t even have to match perfectly. Here’s some example output from a student:

Screen capture from Garageband of a single guitar note.

There’s not much of a chance to do all of the things that you’d probably want to do as a teacher: fit a function, do some fancy statistics, but hey, it’s not really about that. They can see something strange going on. Something that hints at some more interesting functions. The goal now turns from guitars to functions. What function looks like this? What is happening here? My kids were interested in the following:

• Are the little spikes equally spaced?
• Why does it waver up and down?¹
• Why does it shrink?

I then take the time to hit another standard, which is understanding graphical transformations and all that. I sit them down with Grapher or their calculator, and they get crackin’. What looks like this? The kids try all sorts of stuff. I don’t know what yours will do. Mine tried functions from all sorts of families. We eventually talked about envelopes and how a trig function must be being modified by some function the squishes as x increases.

We haven’t really talked about exponential functions in class, which is generally the mathematics used to describe a damped oscillator. We settled on:

This function doesn’t match our data perfectly, but in the end, who cares? On a graphing utility it looks like their shape, and gets at the concept that I’d like to teach about. I never said these lessons were perfect, but they have been effective.

Students that have learned something interesting about their guitar’s behavior then get their chance to report out to everyone. It’s enjoyable when you have no idea what they’ve learned, and they get to tell you. Very opposite to the traditional teacher role of giving knowledge and grading regurgitation.

The above function necessitates the use of the product rule, which is all I’ve been looking for this whole time. We now can enter the traditional math content with a little context.

1. This question also plays nicely into my hand later when we discuss Fourier Analysis.