I recently read Zach Cresswell’s post on an ideal classroom. I feel lucky because I’m teaching in what we are calling a prototype room, as we think about new spaces for the math department at my school. We have begun to find furniture and lighting, but I realized after reading Zach’s post that my class was not yet feeling like the class he described, and I wished it did.

It was pure coincidence that we were about to begin an axiomatic trigonometry multi-day lesson, where students would be working together and I would not be a prominent voice in the room. These three days (especially the last two) made me so happy! On day one, the first few proofs to do were fairly straightforward, so the students were working quietly and diligently on their own. I didn’t like how it felt because I was hoping they would be talking to each other, and they weren’t doing that. However, I realized afterwards that they didn’t need that help yet; they were doing fine solo. Once they reached a harder proof, they immediately started talking. For most students, that began on Day 2 of our in-class work on these problems. Then, I started looking around (but forgot to take a few pictures) and realized I was seeing much more of the classroom that I hoped to have. Even then, I couldn’t observe it all because I was deep in conversation for about 10 minutes at a time with one student at a time. I did see students up at the board (of their own volition), talking through how to solve one problem. Others were huddled (standing) in a group of about 6 or 7 people, listening to one person explain how to think about another question. No one was using our two beanbags that day, but three girls were at the stand-up desk, and others moved around the room when they wanted to do so. I had music playing through the projector.

I was already excited before I walked in the door to Day 3 of this unit. In many ways, this was a repeat of Day 2, with new problems to consider. A few new things arose:

- My two students who were in the play and in the midst of run week apologized for not making it farther through the packet. As I had already said to them, this was an experience and not the type of assignment where every student needed to get to the last question. Whatever they could do in class would be the right amount. I have to believe that this shift of expectations helped to make their week more manageable. It also made me wonder if I could make this same statement about other sets of problems I ask kids to do. Is it ever about doing all of the problems?
- One of my freshman began to shine. I had seen his energy before, but others in the class didn’t yet know him well enough to see his “light.” As I looked over his shoulder, he definitely did not need my help. Instead, he wanted to show me the proof he wanted to type up for homework. They were going home that weekend with the assignment below:

*Using LaTeX, type up three of your Axiomatic Trig proofs. Include:*

*(1) a proof where you felt great about your final product, but you really struggled figuring out the proof*

*(2) a proof where you knew what to do, did it, and wrote a great proof*

*and *

*(3) a proof that showcases something else about this week for you. In your .tex file, be sure to explain why/how you chose the proofs that you did, and what your third proof represents for you, and why you selected it. How did this week go for you? As a student? As a learner? As a way to solidify prior trig knowledge? I really want to know what you think! This is a journal entry-type thing but your writing will be with your proofs a .tex’d document. *

This student was so proud of his work on a particular problem that he knew he wanted to use that proof as his “3” choice proof. It was fun to hear his exuberance and to be able to share that with him!

3. This same student had a chance, at my suggestion, to help two others in the class who hadn’t made it as far, and were struggling with one particular proof. I love seeing a freshmen teach a junior something, and have that be a real, fruitful conversation.

So – as I had these two great days, I want to try and help myself remember what led to this moment:

- letting go of being at the front of the room for any significant portion of class time
- encouraging students to help each other, particularly building connections across grade levels
- having students progress through material at their own pace, and only set the amount of total time spent on the unit (here, three days)
- recognizing that students working quietly and productively can also be useful

Mostly, how can I provide just enough of a structure that there is something to explore, and then get out of the way so the students can dive in?