# How I Teach Calculus: A Comedy (Diff Eq)

*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:*

When’s the last time you tried to build something and all but one board was already cut for you? This is the brand of real-world that has been sold to high school math teachers, and it’s about as real as Velveeta is Gruyere.

First and foremost, it is important to note that “Diff Eq” sounds like an awesome handle for a rapper. If I were going to launch a career into socially conscious hip hop — which I might — this will be my name. So, there’s my official statement of legally binding dibs. Here’s a sample of some real hip hop, if you still “hate rap” after listening to this, there is no hope for you:

The differential equations covered in my book are simple. Pretty much just separation of variables. This is fine with me, because diff eq’s are a vessel for me to teach the finer abstractions of integration. Yes, I find the world of nonlinear dynamics as buzz wordy as the rest of you, I even worked in a plasma lab studying nonlinear behavior in fusion plasmas, but hey, these kids are 17 and more worried about prom than nuclear fusion (today, anyway), and I’m more worried about their ability to become un-addicted to the letter *x* as a variable.

# My School is Near a Lake = Sailboats

So, the Cornally train gets rolling. How can a I make diff eq’s real enough without watering down the math or declaring a state of activity mania? Sailboats, I think. I actually stole this from a question in the back of my book (Larson et al.).

A word about book problems: All of us have seen the Dan Meyer TED talk by now (You haven’t?!), and the idea here is to create problems that require thinking. The book problem that I stole this idea from gives all of the necessary information for solving the “problem.” This is the functional equivalent of cutting your students’ Achilles tendons and then asking them to run a 40. To further plagiarize Dan, giving students problems with all the info in them is fake and ridiculous. When’s the last time you tried to build something and all but one board was already cut for you? This is the brand of real-world that has been sold to high school math teachers, and it’s about as real as Velveeta is Gruyere.

So, I have to cut the problem down to its core and do my best to present that in an interesting and engaging way. A story usually does the trick, because oration is one of my stronger suits, the other being of the leisure variety that I inherited from my Dad:

You may want to show a video of a sailboat competition, or do whatever else it is that invites your kids. Many of my students live on/near a lake, so this isn’t that big of a stretch.

Here’s what I want: I want the students to derive the equations of motion [a(t), v(t), and x(t)] from first principles. I want them to think about how the wind pushes a sailboat, and I want them to use their basic understanding of physics (*F=ma*) to go from there. I can’t say that aloud, though, or it has become my investigation they then have to do. I have to let it grow, otherwise all I’ll get is a picture of how well they follow my directions, not how well their mathematical intuition is developing.^{1}

So, I need to introduce a good question, one that is clear about what we’re doing, but not so clear that it maps the whole process out artificially. This process is the math. By using only the asinine problems from the book you are relegating math to a status of recipe following. Do you really want your kids to be the kind of people that won’t attempt to make a pizza from scratch for lack of the 1 tsp of anise seed that has little to do with the overall success of the dish? That’s the kind of math that is predominantly taught. Barf.

# Ferdinand M.F. Magellan:

I go all Magellan on them. Let’s talk sailing and world history for a minute. Let’s talk about using wind to get you where you want to go. Let’s talk about the prevailing west winds of the Antarctic circumpolar region that help Magellan get around Africa. This is the perfect time for a dual lesson with my Western Civ teacher. She teaches the social effects of Ferdinand’s magical mystery tour, I/we hash out the logistics of such a trip. Collaboration is a beautiful thing.

How could Magellan have had the cajones to do what he did? How could he have known what kind of time commitment a trip like his would require? These questions are sufficient enough to start us down the correct mathematical path. I cannot stress enough how hard the next half an hour will be for you as a teacher. They must be assisted but not led. They must be supported but not undercut.

We eventually settle on creating our own little sailboat system. They had fun and learned alot from the construction, and I secretly enjoy this part more than the math itself. This took about 20 minutes of solid thinking, a Gatorade bottle, 2 spray paint caps, office supply miscellany, and an air track blower:

The kids then played for a bit. I hope I don’t give the impression that our class time is spent perfectly on task moving from one beautiful realization to another in quick succession. Far from it. A lot of time is spent frustrated, sitting, thinking, and otherwise not knowing what to do. If you’re not committed, this is when you’ll quit and just go back to the white board. Stick with it; email me, and I will be your cheerleader.

Playing with the sailboat has yielded us a common experience to talk about. How does the force from the air change the speed of the boat? Can the boat go faster than the wind speed? These are all student questions that we endeavor to answer. Eventually we come up with the idea that the slower you are compared to the wind speed, the more force you’ll feel. Or:

This is acceleration. If we want to know anything else about the boat, its speed or position, we’re going to have to integrate. The introduction of *k* was my idea. Why? Units. The kids bought that pretty well. *v dub-ya* is the wind speed and *vb* is the speed of the boat. This is just a hop and a skip (jump and you’ll overshoot) from:

The lesson on diff eq can now begin. I can teach separation of variables from this point. This tends to make students uncomfortable because of the lack of an *x*. Hence getting into the deeper abstractions of integration. Here’s the work for those of you that are here for content (teachers, skip to the end):

Separate the variables from each other. *vb* and *t* are the only variables, *vw* and *k* are numbers.

This is now totally separated, and we can integrate over *vb* on the left and with respect to *t* on the right:

The integral on the left requires substitution (Yay!):

This yields us a function for the speed of the boat *[vb(t)*], assuming a constant wind speed (*vw*). Talking about these assumptions is always nice.

Working in angles and variable wind speed could definitely be an open investigation, if a kid wanted to do it. I wouldn’t necessarily make my kids, because my standard is “Diff Eqs by Separation” not “OMG Sailboats Textbook!” The sailboats are just a tool to create a handle for those kids that are not dominantly abstract thinkers yet. I know some of you scoff at all this extra time not spent on practice problems, homework, sharpening pencils and grind-stoning noses, but I’m telling you, some kids just don’t have the necessary life experiences to buy into the abstraction of math. Creating these experiences in a meaningful way will not only help that kid grab a hold of something, it will also help them communicate with their peers better due to commonality.

The next step is to check to see who understood what. I say, alright kiddos, find the x(t) function. This is my check for understanding, this is may be the first SBG moment of the lesson. Where are they? What do they know? Can they connect kinematics to differential equations? Can they handle *e* with tact? This is a certainly a formative moment. I know exactly what kind of board time I need to do after I watch them try this exercise. For those of you here for content, the answer is:

Just graphing *x(t) *and* v(t)* with your kids when you’re learning about exponentials is a rich experience. Let alone deriving them.

Is this new? Of course not, I stole it from my book. It’s the approach that I want to share with all of you. It’s avoiding disabling your kids. It’s showing them that math has an actual purpose than just being some class they have to take. It makes me genuinely sad to imagine that most of America’s children think math is an affliction with no purpose other than to scare them away from STEM careers. Why? Because they can’t give a shit about parabolas until they either buy into the beauty of math (a few kids), or they know what parabolas are good for.

You can make a stink about *h*, *k*, *x*, and *y* all you want. You know what your kids are thinking? “Why *h* and *k*? Where’d those come from? I have no idea, crap, what random thing should I try and memorize” This gets old, and they don’t want you to point at the board again where you’ve already circled the same equation twenty times. They want motivation, history, humanization, beauty; let the grind stone come naturally, please.

1. Some of you hate it when I say stuff like that. You place extreme value in following directions, because, as you see it, kids can’t. You want to teach them how to be good little students and members of society, which is nice, but in your frustration you’ve forgotten how effing boring school is when you have no idea what the motivation for a lesson is. Or you just think I’m a twit, and that’s, like, your opinion, man.