How I Teach Calculus: A Comedy (Optimization)
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:
What does this map of Middle-Earth do for me? Other than indulging my current obsessions with Tolkien, it gets at the inherent inelegance and drudgery of guess and check.
At this point my kids have been working with related rates. They’ve been doing things with water jugs, as well as working the traditional book style problems. Kids tend to be frustrated by “word” problems. This is because they can’t memorize and repeat a process. Most people love to be told exactly what to do — whether they’ll admit that or not — if the process has enough complicated steps, but not too many, then they’ll feel like they’re getting something done, but it was easy because the task had already been planned out with an explicit end in mind. Word problems piss kids off because they have to try and piece together information and frame it mathematically in a form that will allow them to apply the appropriate technique. That is not a fun process for someone who has no idea what that technique actual does or means. This is why you must generate context before content. The changes to my calculus class have brought me a lot of work, but they’ve also brought on a whole lot more student conceptual understanding. Cornally is happier.
So, in a pretty large breach of calculus etiquette, I skip most of the material about critical points, extrema, 1st and 2nd derivative tests, and what have you, and I move straight to optimization. How can they do optimization, if they don’t know all that other stuff?! Easily, optimization is really only about using your noggin and finding critical points. So, what better way to introduce a whole chapter of “Applications of the Derivative,” as Larson so deftly puts it, than by starting with the most useful application itself!
A word about text book structure. Why, oh WHY, do we always put “applications” after abstract content? Don’t you get it? The only thing I’m trying to do here is switch that order. You have to be very very judicious in how you do this so as not to burn the children, but why would anyone give two shakes about critical points until they know why they’d want to find them? So, optimization we must do first. Aside of an aside: do not pretend that the “real world applications” your book gives you are sufficient. They just put those in there as a buzz word to sell more books. I blame Texas.
I start off with a map:
So, I spent a lot of this winter reading Tolkien’s Lord of the Rings. I’ve never gotten through all three in a row, and I felt it was about time. I didn’t anticipate how engrossing they would be, and when they started showing up in my lessons, I knew I had a problem. See above.
I presented the kids this map, and asked them what’s the shortest distance to Mordor? I accompanied this question with a little set-up from the book. I won’t reproduce it here, but I read an excerpt from Two Towers Book 4 Chapter 3. I also showed the complementary part from the movie (Two Towers DVD Chapter 15 The Black Gate is Closed). The story is that Gollum leads Sam and Frodo to the gates of Mordor, but they are shut. Gollum then leads the two hobbits on a crazy hike through the mountains north of the gate.
This is an exercise in implicit differentiation, orthogonal trajectories, and — nominally — optimization; however, the task becomes very daunting very quickly. The kids start drawing lines. Some quickly realize that each step down requires use of the Pythagorean theorem to find the actual distances covered. Many students eventually raise the important question, “Aren’t there an infinite number of ways to check?” Yes. They become sad at the thought that I might make them do this. Of course, we don’t.
The point has been made; we need to develop some mathematics that gets at optimization.1
So where from here? We now need to learn how to optimize. A discussion of critical points occurs. I don’t say those words; vocab always comes last. We draw some graphs, what’s common about all the highest points? My slope-minded students easily point out that the slope there is nothing. So, if only we could get a function, find its derivative, and then set it to zero, perhaps then we’ve built a process here? Yup.
We then launch into the very classic open-topped-box-from-sheet-of-paper example. I tell a story about camping and chili and having limited tin foil with which to make a chili holding vessel. This kind of contrived problem really rubs me the wrong way, but I’m OK with it now that the kids have context, we’re using it to learn process.
I have the kids build boxes the boxes by cutting out the edges and taping them up. They then measure the length, width, and height of the box and find its volume. Many students realize that there was really no point in making the box, but hey a lot don’t.
They then graph their data on the board making a graph of Volume vs. Cut-out length. It comes out a little shaky but pretty much looks like this plot: (from 0 up to 4.25 anyway…)
The 8.5 is the dimension of our paper. A great discussion of domain ensues, I’m sure you already see it. In fact, I am suffering absolutely no delusions that this is at all unique. In fact this is about as traditional as I get. Sometimes you just do what works best, even if it smacks of pretense.
All that’s left to do is develop the math for this model. We get to work out a fairly simple example to show how to combine two equations that relate the same idea:
An equation in one variable for single-variable calculus, it must be Christmas! Derivatives ensue, zeroes get thrown around, and optimum values get found. In case you’re trying to learn from this:2 we need to use the product then chain rule (or you could distribute first)
1. I don’t usually give impossible/extremely difficult tasks, but I planned this map bit in response to a conversation about the method of guess and check: A student told me that he prefers guess and check because he can usually get the right answer to a problem, and sometimes it’s even faster. He likes the method mostly because he doesn’t have to learn anything new. This should strike some of you, I know it hit me hard. He recognized the inherent ridiculousness of many math lessons; the techniques taught aren’t always the best way to do it. SO WHY ARE YOU TEACHING IT? What does this map of Middle-Earth do for me? Other than indulging my current obsessions with Tolkien, it gets at the inherent inelegance and drudgery of guess and check.
2. I never intended for this blog to teach content, but I’ve noticed a that a lot of the Google searches that lead to my site are from students looking for help. If you want to learn calculus, you can attend my high school and take my course. Otherwise there are about six trillion other sites that attempt to teach you calculus.