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.