Today I gave my students the very excellent activity from Jo Boaler’s YouCubed on Pascal’s Triangle.

I modified it for my own purposes, but essentially the heart of the activity was the same. My students had to complete a half written triangle and then answer some questions. Before we began I gave a quick introduction by talking about how mathematicians don’t usually do what we sometimes do in math class, which is copy the procedure that has been shown to us. Like Jo writes in the activity, I told the students that mathematicians look for patterns. I also told my students that the triangle is named after one of my “dowgs” (my dead, old, white guys). I always get some nice eye-rollers for that joke.

I was really pleased with how well this activity went. There were so many light bulb moments for the students, with plenty of verbal exclamations.

“OH my goodness, y’all.”

“Oh SNAP I see it!”

*hand smacking desk noise* “DUDE no way”

“Ooooohhhhhh”

It really was a pleasure to facilitate the lesson. (Well, I did have some off-task behavior in my final block that was draining, but for the most part it all went really well. Some new seat assignments are happening for that block tomorrow.) I kept wishing I could simultaneously take notes and facilitate. This was a lesson that I should have recorded to play back and analyze because I feel like my “end-of-the-day what do I remember” list in my head just doesn’t cut it.

And my students really surprised me. They saw patterns that I did not expect them to see and sometimes even patterns I had not yet noticed. In particular I was really excited because one of the questions is about triangular numbers and I got two different definitions of triangular numbers from a group of students and BOTH of them were legitimate definitions.

The sheet from YouCubed gave the first two triangular numbers (3 and 6) and then asked the students to find the next two, writing down both the number and a representation. One student in the group saw the “traditional” definition of triangular numbers and drew pictures for 10 and 15, the “correct” next numbers. But another student saw 3 and 6, assumed the pattern was to add 3 every time and said 9 and 12. But this was great because she drew triangles with dots that were simply missing the middle. She still had legitimate triangular numbers. They were at odds about who was correct when I stepped near their group to listen. I asked the first student why he had the triangles the way he did and he answered, “Gravity, it’s like stacking cups. You have to have middle dots.” Which I thought was a spectacular answer. I drew both types on the board and then had the whole class debate. (I couldn’t get them to debate as much as I wanted to, but they were able to recognize and extend the pattern in both cases.)

It really was a nice day. My take-aways for the day:

- I need to figure out how to integrate this style of lesson into other tasks, even when they are not as open ended as this one.
- Sometimes it is the class and not the task. My final block did not respond well at all and the issue was classroom behavior rather than the task itself.

Thanks for reading.