09 | April | 2009 | Continuous Everywhere but Differentiable Nowhere

In Calculus, I sound like a broken record. Each time we learn something new, I say “take what you don’t know and turn it into what you do know.” I say that at least three times a week. I said it last week when doing integrals like:

$\int \frac{1}{4x^2+1} dx$

We don’t know how to deal with that, but we do know how to deal with

$\int \frac{1}{x^2+1}dx$

So let’s try to turn what we don’t know how to do into something we do know how to do. For those who haven’t taken calculus for a while, the integral above is $\tan^{-1}(x)+C$ . So to do the original problem, we want to somehow get the original integral to look like $\int \frac{1}{(something)^2+1}d(something)$ — the integral of 1 over something squared plus 1. So we rewrite the integral as $\int \frac{1}{(2x)^2+1}dx$ . That’s much closer to what we want to get — it looks more like something we know how to deal with. Next we use $u$ -substitution to finish this beast off ( $u=2x$ ) to get $\frac{1}{2} \int \frac{1}{u^2+1}du$ . Now we have something we know how to deal with, from something we didn’t.

Again today, I showed my students how to solve $\int_0^1 \sqrt{1-x^2}dx$ , and told them to solve: $\int_0^1 5-3\sqrt{1-x^2}dx$ . At first sight, they recoiled, but again, we used the mantra of “take what you don’t know and turn it into what you do know” to solve it. If it looks scary, fine, have a moment of panic, but then ask yourself “what does this look like” and “can I turn it into that with some simple manipulation”?

I was thinking today how this actually could be my refrain in Algebra II also. Example: I could frame quadratics in that way. Students know — or quickly learn — how to solve equations like $(x+1)^2=5$ (hopefully). But what about something like $x^2+6x+1=0$ ? It’s not nearly as easy. But then we can talk about if there is a way to that what we don’t know (that equation) and turn it into something we do know how to solve ( $(x+3)^2=8$ ). It’s not that I don’t do this already, but I am not always explicit about it. It is not my mantra.

But it should be. It’s how we solve math problems. We have something we don’t initially know how to do. And we have to figure out if we can simplify/rewrite/re-envision it to bring it to a place where we know how to do it.It seems stupid and simple and obvious, so much so, that I don’t say all the time. But if I started saying that as my refrain, if students really saw that math is simply this simple process, it might stop seeming like a huge bag of tricks that never fall together. They might see it as the art that it is — where there is creativity in deciding how to get from point A (hard problem) to point B (simple problem they know how to do). And all the specifics that we do in class are giving them the tools which they can use to chisel out a path from A to B. It might finally be us always trying to work out the puzzle: what does this look like that we know how to do, and can we get it to that place?

In other words, we’re now talking processes instead of methods. We’re talking problem solving instead of rote memorization. And whenever a student is stumped on a problem, you can stimulate his/her thought process by saying “we’ve always taken what we don’t know how to do and turned it into something we do know how to do… what similar things does this beast remind you of?”

So yeah, it’s not a huge revelation or anything. But I’m thinking that it might be a really amazing experiment to frame my Algebra II and Calculus classes with this mantra next year. Heck, maybe even in the next few weeks when I’m teaching exponential and logarithmic functions! I mean, yeah $\log(2x+1)+\log(x-1)=2$ may look ugly. But is there a way to turn it into something we do know how to do? Namely something of the form $\log(something)=2$ ? Obvi.