I taught a bad lesson this week. It’s not unusual. I teach hundreds of lessons every year: Some go very well, most are just fine, a few go poorly. A successful school year has its ups and downs, for students and for teachers, and you take the bad days with the good.

This week was the area of a circle. It’s always a challenge teaching a topic that students already know: It can be hard to get students to engage deeply with an essential question like “How do you conceive of the area of curved things when your notion of area is based on straight things?” when they’ve been using $A = \pi r^2$ since 6th grade.

I took a standard approach and had students explore area relationships between a circle of radius 1 and various inscribed and circumscribed polygons. The goal was for students to engage with the essential question while sharpening their area-computing skills and previewing some important arguments they’ll see again in calculus.

Usually this lesson works just fine, with occasional moments of greatness. But this year I had to trim the area unit a bit to accommodate the schedule, so prior to this lesson we just hadn’t done enough with polygon area. As a result, the conversation bogged down in the early stages of the exploration and it was hard to get unstuck. And because the remaining schedule is so tightly packed, there really wasn’t really time to stop and regroup. We just had push through.

It wasn’t a disaster: We did math together, students grappled with and argued about important ideas, connections were made. But the lesson didn’t go as I’d hoped, and that was a downer. On the other hand, it was comforting to realize that this lesson went bad for some very normal reasons. It didn’t fall flat because my wifi went out or because I couldn’t share my screen or because I had no idea how to access student thinking on a computer. The lesson didn’t work for the same reasons most lessons don’t work: I misjudged prior knowledge and tried to execute a plan that was too ambitious and too rigid. It might have been a bad lesson, but it also might have been the most normal lesson I taught all year.

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