How I Teach Calculus: A Comedy (Power Rule Families)

This is going to be awfully subtle, sorry if this is useless to you.

Today we read the first part of A Mathematician’s Lament as I usually have kids do at the start of any math-based course. We ended up having a long discussion about what creativity is, and how we can inject some while we deal with the competing influences put on a math class.

We had just finished finding derivatives using limits, and the whole time these little effervescent bubbles of “we learned an easier way last semester” kept popping up.

They’re talking about the power rule of course, but what they’re tacitly talking about is the underlying battle that math education is fighting right now.

The Power Rule:

[latex] \frac{d}{dx}x^n = nx^{n-1}[/latex]

…as opposed to the much more general, complicated, but motivated:

[latex] f'(x) = \lim_{d \to 0} \frac{f(x+d) – f(x)}{d} [/latex]

Most students see math as a thing to be done with; a bodiless insatiable monster that consumes correct answers that stuents must produce as tribute.

Why else would someone propose what is an algorithmic, baseless, and unmotivated solution to something so pretty as baby’s first infinitesimal. It’s not that the kids are wrong; they’re conditioned.

So, in light of our discussion of Lockhart, and my obsession with meeting kids where they happen to be, we decided to whiteboard up some questions about the power rule and how it relates back to the “hard way.”

The most important question that arose was, “when does it work? Like, what kinds of functions work with the power rule and which ones don’t?”

This led directly to students creating the following families of functions:

[latex] x^n \: where \: n>0 \: \mathbb{Z} [/latex]


[latex] x^n \: where \: n<0 \: \mathbb{Z} [/latex]


[latex] x^{m/n} [/latex]


[latex] x^{in} [/latex]


[latex] x^{m/n} \: where m/n < 0 [/latex]


[latex] n^x [/latex]

You can see how they have some idea for where the power rule will work, but it’s more guilt-by-association than really understanding how the limit process can be summed up algorithmically.

  1. They graphed the function (using modern graphing software is new to a lot of them)
  2. They then blindly applied the “easy way” to their function and graphed it
  3. They then attempted to reason whether the graph of f’ matched the slopes of f, this is a difficult thing to do.

This is where it gets really subtle. Where I have to ride the governor and put in just enough teacher-ism to keep the ball rolling without losing the organic anti-establishment vibe.

We could really easily at this point have this collapse back into the seeking of correct book answers.

We consensus build. Some families are out, some are in.

We argue.

We get freaked out by [latex]x^{in}[/latex]

We see that everyone is perfectly capable to evaluating the scope and validity of “random boxed math” as they so lovingly call the theorems in the book.