Yesterday I started a lesson on right triangle similarity and the geometric mean by throwing a task to students and seeing what they would come up with after 5 minutes of independent work time.
In all classes, I had a similar (get it, geometry teachers?) breakdown on students’ ability to approach the problem: Some were stuck on where to even begin, some tried to use yesterday’s skill, and some made connections to earlier geometry units.
What really varied were the conversations and debates we had after the independent task work.
First period: My opening question was, “What are you thinking about this problem?” I got three hands (out of 33 students). The first two to respond were pretty timid, though student #2 did build on student #1. Student #3 then confidently gave the answer and explained the problem. I wasn’t satisfied with this at all because the whole idea was that it was a totally new problem on something they hadn’t seen before, and I had wanted students to bring up different ideas, respond to each other, and then argue until we settled on the new idea that all three triangles in the diagram are similar.
I realized that my question had to change. By asking, “What are you thinking about this problem?” which to me, seemed like a pretty low-stakes question, I think I was unintentionally suggesting that they needed to give me a right answer– no matter what.
Second period: This time I asked, “What are you thinking about this image that may or may not help us solve it?” More hands this time, from students of varying skill levels, but again, not a lot of debate and/or responding to each other.
Third period: The new question was, “What do you notice about the picture?” Participation improved, as did the follow-up conversation.
Fourth period: I finally tried out “What do you notice? What do you wonder?” I’ll admit, I hadn’t used this word choice before because it sounded a little corny to me, but wow, students really responded to it! Participation was the best of the day, and what I was really happy about was that many of the contributions came from students who typically don’t participate and who are nervous about giving an incorrect answer when cold-called.
Rather than prefacing their responses with “So I’m not really sure, but…,” they just automatically started with “I notice… and I wonder …” They totally got that the idea was to just squeeze whatever mathematical juice they could out of the problem– whether or not it was relevant. By the end of the conversation, I had a rich list of discussion notes on the white board, and the class had basically generated all of the key points for the mini-lesson.
“What do you notice? What do you wonder?” is definitely here to stick.