I just finished up arithmetic series, and I wanted to push my Advanced Precalculus students to think hard. I usually provide them with enough scaffolding that I know they will be able to get from Point A to Point B. But today I decided I wanted to “be less helpful.”
I told the groups to mix themselves up — for a change of pace. And then I handed this out.
I told each group that they had one opportunity to call me over so I could give them A Big Hint. Then I let them go, giving them giant whiteboards to work on if they wanted.
In my two classes, I had: one group solve the problem explicitly (they had a formula that worked) and one group come up with a recursive solution that really impressed me! The other eight groups were at varying stages of understanding. Most all the groups were gung ho about working, and most all the groups started discovering all these patterns.
Only one of the ten groups asked for The Big Hint, which means my kids have perseverance! I did give varying degrees of mini-hints to kids as I saw them progress, to nudge them this way or that way.
At the end of the second class, a few kids said how they are now braindead, because they did so much thinking. They were exhausted. As a teacher, I call that winning!
I’m still at a bit of a loss as to what I am going to do tomorrow. Since kids hadn’t really finished, I thought I would have them work a bit more. I’m not good at debriefing. Also, many kids have different observations, and I don’t have time to really dwell on this. This wasn’t supposed to even take the whole day!
My plan is to give all my groups collectively A Big Hint, give each group 12 minutes to figure out how to find the nth pentagonal number, and then after those 12 minutes are up, I will give them this:
We will go over the triangular and square numbers together as a class. And then WHAM, I leave them with the pentagonal and p-gon figures to figure out on their own.
Wish me luck that it goes well tomorrow. I can see it crashing and burning, or being a good wrapup.