This happened a while ago.

Last year, I was never really satisfied with the Calculus students’ understandings of the nuances of integration. Did they really know the difference between a definite and indefinite integral? Which one involved a +C, and why? What exactly was an antiderivative? They did ok on the problems, but there were some mistakes, and I just wasn’t happy about their conceptual understanding.

Fast forward to this past January.

I teach definite integrals as the limit of the Riemann sum in November (finals are the first week of December). We talk a lot about rectangles. Then we come back in January and talk about antiderivatives. “Hey, we took derivatives. Let’s do this cool thing where we go backwards!” We practice that for a day and talk about why +C. Then we hit the FTC and tie them all together nicely before we hit u-substitution and all that jazz.

The day after the FTC, we spend some time talking. Last year, I’d given them discussion questions, but dealing with the nuances of vocabulary words that you just learned using nothing but those vocabulary words doesn’t help you learn very well. So, this year, I told them to make concept maps.

I’m assuming that at some point, you learned what a concept map was. Maybe you called them mind maps, or idea webs, or something like that, but you stuck your main idea in a circle in the middle and connected it to sub ideas around it, then connected those to other ones. (Go ahead and google “concept map” if you’re still lost. It’s ok. I’ll wait.)

See? You knew what they were. The kids didn’t, though, so first I spent two minutes explaining what they were and giving them an example with polynomials. (How did they get to be seniors and not know about concept maps? When did you learn about them? “What do they teach in schools these days?”) Then they got in groups and connected ideas. I made them explain their connections, too – sure, definite and indefinite integrals are connected, but *how*?

Finally, I asked them to share. I didn’t get any volunteers at first, so I picked on a group that I thought could handle it. They did ok, with a little prompting. This is what they came up with.

For those of you wondering, Tol is the god of math that they’ve invented. Story for another day, but they’ve essentially created a cult in my classroom. They’re so going to get me fired.

The best part of their presentation, though, was when I told the kids that they could heckle the presenters. “That doesn’t look like a circle. It’s not very round. More of a corn-flake.” “Your line isn’t straight.” “You misspelled derivative.” These kids have been together for a long, long time, and while I normally hate teasing, I knew that they could have fun with it and do it in love. Especially since the recipients were the ones who usually dish it out. Maybe that makes me a bad teacher…

But then things got better. While they were working on some subtle nuance, I got a volunteer! Two kids who are normally pretty quiet *wanted to share with the class!* I was over the moon. This is what they came up with.

So, did it help? I think so, largely. Last year, maybe half the class struggled with when to include +C. This year, it was more like 20%. I’ll take that. Plus, they learned a study skill. I’ll take that, too. Now that we’re reviewing for the AP exam, I should probably remind them of concept maps. A big, end-of-the-year concept map connecting limits, derivatives, and integrals might be good. Maybe after their final review project…