I’m slowly getting edged out of my math department, so I only get to teach one section of calculus this whole year. I’m greatful for the opportunity, because I love math: back to teaching math! Yippeeeee!

This semester of calculus is brought to you by the concepts creativity and proof. I shudder a bit when I type the word “creativity,” because, like so many fine pillars of the English language, “creativity” has been perverted by countless inspirational posters and efforts of similar ilk.

This is deep creativity I’m after. This is my need for our class time to engage me beyond simple management and looking for dropped negative signs. Engagement is not just students working diligently with pencils wiggling like a 3rd-grade rendition of the Nutcracker ballet, no no no.

I have my students convinced that there are no lesson plans; they, as much as me, are in control of where this “course” in calculus is headed.

I don’t want this to come across as some whelp trying to come out of the gate shooting from the hip. There’s a plan; and it’s nominally to learn calculus. However, something snapped in me this summer.

I’m not an alcoholic, so I don’t want to call it a moment of clarity. These students aren’t depending on me to learn calculus. There’s no way I can teach them everything that’s in those silly blue text books. There’s no way I can prepare them for every cheap trick some petty professor may throw at them. There’s no way I can teach them calculus procedurally and pretend I’m doing a service to humanity.

I have to teach them to follow the story. That’s the only way humans remember anything. They replay the story in their minds. If there aren’t characters, it’s lost. If there isn’t a plot, it’s lost. Getting the students to write the story is even better.

Here are some of my first attempts:

# The Product Rule:

The students have a firm grasp of limits and how they help create derivatives. This conversation in and of itself was enjoyable, but I’m sure you all do it. (Let the sliver of time go to almost nothing!)

The call for creativity this day started with a request for functions whose derivatives we can’t possibly find. Here are the responses:

[latex size=3]f(x) = i[/latex]

[latex size=3]f(x) = x^{\infty}[/latex]

[latex size=3]f(x) = \frac{\sqrt{x^3+1}+\frac{x^{10}}{9x+9}}{tanx}[/latex]

The first two are interesting, and we spent a large time discussing what they and their derivatives might look like. The third is the obvious attempt of a teenager to string things together that they recognize into something that to them seems impossible. This is the only reasonable function in the list, and it is differentiable.

The conversation turns to attempting to solve this problem. They suggest breaking it down, and I know exactly what they mean. Multiplication, division, everything else. Let’s break everything apart. They want to do something like this:

[latex size=3]\frac{d}{dx}x^2sinx=2xcosx[/latex]

Which seems totally reasonable, until you graph them. My students did, and they realized something was amiss.

So, how should we handle products? I’ve never done it this way before, and I’m totally surprised by where they took this. We tried to do products for which we already knew the answers, like:

[latex size=3]f(x) = x*x[/latex]

and

[latex size=3]h(x) = x*x^2[/latex]

They had a hunch that the derivative of each individual piece was going to show up, but we couldn’t put our fingers on why. So, for the second one they wrote

[latex size=3]h'(x) = 1 + 2x[/latex]

They know this isn’t correct. So, they started to play around with patterns. I’ve got them. I think you see where this is headed, and I hope you can imagine what fun it was to do with students as they whiteboarded their patterns out. Eventually, this was reached:

[latex size=3]h'(x)=1*x^2+2x*x[/latex]

[latex size=3]Product \: Rule = f’g+g’f[/latex]

To my students, every single piece has a motivation. Each part is connected to the original problem. This held power for them. They tried it with crazy-silly products (as they called them), like:

[latex size=3]f(x) = \sqrt{x}\sqrt{x}[/latex]

Is this an exhaustive proof? No. Are my students ready for that? No. When will they be? When they bring it up; eventually someone is going to realize that showing something is true for one case doesn’t mean it’s true for all cases. I can’t wait for that day. Until then, the product rule has been motivated. They even named it “The Easy Way for Products.”

Had I planned the product rule for that day? Maybe. I’m just saying that perhaps this kind of we-have-a-problem-let’s-solve-it-no-matter-how-long-it-takes is a better story than, “Tomorrow, we’ll do section 3.4.”

Good enough, for me anyways.