Some math questions have long complicated answers. This is a problem for teachers, because students get lost in the middle of complex stuff. They end up not learning and feeling bad about it. This is not good.

This past year I taught calculus for the first time. There’s no way around it: some of those solutions take a lot of steps. Moreover, my class was a sort of intro to calc for people who aren’t especially enthusiastic about algebra. This was a class where kids apologized for being bad at math on the first day of school.

What could I, an award-winning* worked example enthusiast, do to help these students?

* *Correction 6/15/2021: I have not won any awards.*

One approach, one I don’t favor, is to break a complex problem up into many mini-problems. That way, students only have to think about a smaller part at any one moment. This is called “scaffolding.” It’s the approach Active Calculus takes for some of their problems:

Look, that’s still a lot of words and questions. And you can’t forget part (c) once you solve it — you haven’t learned anything until you take all five parts and make sense of them together. I do not like this, it is bad.

What I’ve started to think is that good teaching sometimes involves turning long ideas into a series of shorter ones. But in a way so that after each relatively short task you have learned something **complete**, gotten a full meal. The pieces add up to something greater, but each part is a meaningfully whole thought. You can breathe in between each piece without the entire structure falling apart.

For longer calculus problems this year, here was my routine for doing that.

First, I would warm up students to the lesson with any skills they’d need for the day’s learning. I was just hoping to help them notice and remember anything they’d need for what was coming next. For this related rates lesson, I knew that the chain rule was going to be important, so we started with that.

Then, I would slowly develop the problem, making sure everyone understood the question. “If we’re inflating a spherical balloon, what happens to the volume as it inflates? What happens to the radius?”

“Imagine we inflate the balloon with a constant flow of air, maybe using a pump or something? Will the volume change constantly? What about the radius?”

If I want to make sure students get it, I’ll ask them to take a guess as to what the answer will be. “Will the radius grow at a constant rate? Will it grow faster at first and then slow down? Or slower, then fast? What’s your guess?” (There is a lovely animation somebody made to go along with this problem. I projected it while collecting guesses.)

When it was time to dive into the solution, I would first give a “headline news” version of it. I’d project it on the board, and then talk through it.

“Here’s the plan of attack for this problem. We can write a function that connects the radius to the volume. But we’re going to turn that into an equation that connects *the rate of change* of volume to the *rate of change *of the radius. How? The derivative! We’ll treat volume and the radius as if they are not constants but *functions of time*, and we’ll use the chain rule to differentiate in terms of time. Then we’ll sub in all the given info from the problem — which rate of change do we already know? that’s right! — and solve for dr/dt.”

This approach is cribbed directly from Richard Catrambone’s research on “subgoal learning.”

Then I start revealing steps in the solution, filling in the empty spaces. What function do we have that connects the volume of a sphere to the radius? How do we differentiate? etc.

I would give students time to study this, time to explain it to a partner, and time to answer some self-explanation questions about it. I had previously explained in the abstract why we’d need to use the chain rule in the second step — I’d ask students to articulate that principle on their own. And I would ask students what if the diameter were 16 inches instead of 12 — what would the answer be then?

Their job now is to understand the details of this particular solution and connect it to the generalized outline I had already presented.

And once we’re comfortable with this solution, we’ll develop *another *question and ask students to solve just a bit of it on their own. In doing this we’re giving them something short to think about in the context of a much larger problem. These are “completion problems,” a type of task identified and studied by van Merrienboer (for example, here).

And it keeps going:

And so on:

Anyway, I think a lot of what I do as a teacher is I make shorter cycles of learning compared to a lot of other people. I’ll do inquiry, but for a few minutes. I’ll do worked examples, but I structure it so that it’s made of several self-contained tasks that (mostly) stand on their own. Is that just what we in the biz call “scaffolding”? I guess, but it’s not “scaffolding” in the sense of giving kids lots of help to climb up a very tall structure. More like just building a bunch of smaller structures that are just as valuable as the big tall one.

I don’t know what to call it, but it seemed to work well for some of my highly anxious, algebraphobic calculus students.