Meditations on a Cogen (No. 4) • Thursday, November 4, 2021

During the 2021-22 school year I’m having weekly co-generative dialogues (or cogens) with my students. In an effort to help me process these talks and document progress, I summarize and write reflections after each cogen. This is the fourth post in the series.

An Experiment
There was a small twist behind the start of this week’s cogen. Because it was our 4th meeting, I was curious how the students view our time together. I like to think that our cogen has established itself in their hearts and minds as a thing. My hope is that they have taken some degree of ownership of the space we share together every Thursday.

To test this theory, I didn’t remind my cogen students of our meeting today. I was tempted to say something to them in class — like I’ve done every week up to this point — but stopped myself. Do they value it enough to remember on their own — without me? Has our cogen become part of their Thursday routine?

The results were mixed. Of my four cogen students in school today (there’s five, but one was absent), two remembered on their own to come. Of the two that didn’t, I saw one just before we were about to start and reminded her. She admitted that she forgot, but expressed no hesitancy in staying back for the allotted 30 minutes. As for the last student, he told me the following day that he felt bad for missing our talk and didn’t remember until he was on his way home. I told him not to worry about it.

In addition to my regular crew of five, last week I decided to invite another student from period 7 to join. We only had one representative from 7th period in the cogen. Each of my other periods had two. I asked the invitee to say after class one day last week and gave her my elevator pitch. She heard me out and said it was down. Yesterday, after going up to her in class to remind her about her first cogen today, she nodded and confirmed without blinking. She was eager to take part.

Thinking back on my recruitment of students, I’m a little surprised at how ready and willing they have been to be part of the cogen. Reflecting on this makes me uncomfortable. I thought that I would’ve had to bribe them to come to our sessions, but I haven’t. They’re all about improving our class and transforming it into a more student-facing environment. Does my discomfort stem from the fact that I didn’t expect my students to want to be stewards of their learning as much as they have? If so, what does this say about my internalized beliefs about students and their role in the classroom?

Anywho, back to today’s meeting. I had four students. Two were from 7th period (one was the newcomer) and the other two were from 1st. This meant that I didn’t have anyone from 3rd period today (one forgot and one was ill and not in school). I made mental note to connect with these students tomorrow about today’s session. Although they weren’t present, it’s still important that they be briefed on what we discussed and decide to act on this week.

Worked examples — in and out of class
After checking in with everyone, I introduced the new member of our cogen. I had the other three members overview for her the purpose of the space. We grabbed snacks, sipped water, and settled in.

I opened by updating the kids about the after-school tutoring session from this week. It was for students retaking the most recent exam. Taking the cogen’s suggestion, I provided worked-out solutions to the exam along with practice problems related to those on the exam. The idea was to get them to analyze worked examples first, learn from them, and then allow them to test their learning on new problems in small groups. Only one student at the cogen attended tutoring, so it was hard to gauge its success with the cogen. That student did say, however, that she liked it better than the teacher-led version of two weeks ago. I also thought it went better because the students were actually doing math, but I also feel that some were just waiting around on me to come to their group and help them. I want to grade the retakes before I pass further judgment.

The conversation about tutoring quickly transitioned into talk of in-class learning when I mentioned that I tried the tutoring strategy (worked example + practice problems) with 7th period earlier this week. The students from 7th period confirmed and commented that they really liked it. Instead of the exam, it was based on the homework problems. As opposed to our routine of using stations to discuss the homework problems, the cogen students liked it because it afforded them the opportunity engaged more deeply with the problems. They had a partner, a worked-out homework problem, and a related problem to try. They weren’t depending on another student to write on the board. They could just go. They also felt that, in contrast to the station work, they weren’t merely copying notes from the boards — they were trying problems. Of course, if they attempted the homework problems the night before, the students wouldn’t be copying notes — they would be editing and verifying their notes — but that’s another story. I openly shared this with the cogen. I promised them that I would continue to find more ways to build the “worked example + practice problem” routine into our lessons.

With all this talk of worked examples, I’m making a note to read Michael Pershan’s Teaching Math with Examples in the coming weeks. It’s been on my shelf for a while. Now’s the time.

The Thinking Classroom
At some point during our talk, the students commented on the recursive sequences lesson from this week. I used visually random groups and our large whiteboards. After reading Building Thinking Classrooms in Mathematics, I’ve been pressing myself to bring in more of Peter Liljedahl’s work into my classroom. This lesson was a reflection of my efforts. I’ve used his principles of a thinking classroom a lot in past years, but not usually this early. All of the cogen students loved the lesson and asked that I use the structure more often. I promised them that I would find more opportunities to bring it in. It really was great.

Quizzes, entry tickets, and polls
I can’t remember how we ended up discussing quizzes, but that’s where we found ourselves next. I like to assess content in small chunks so my quizzes are simple: one problem at the end of the class that’s based on what we learned that day. I shared that I use quizzes to help me determine how students are learning and also to give students a way of doing the same. Someone remarked — it may have been me, I’m not sure — about using ungraded quizzes to do this. Why must the quizzes be for a grade? Won’t ungraded quizzes serve the same purpose? It was an interesting idea and I told the team that I would try it out next week. They added I should use the ungraded quizzes to format the next day’s class. In teaching for so long, this idea wasn’t new to me, but hearing from the students felt different somehow. I gave my word that I would try an ungraded quiz next week and see how it goes.

They also suggested using a poll to measure how much time we spend discussing the homework problems each day. More class time should be dedicated to diving deep into problems that the class finds challenging; a poll could reveal what these problems are. And perhaps a poll would help students see that our work each day is highly dependent on their doing the problems each night. Maybe this implores more of the students to do their homework? In the past, I have done polls via a tally that students mark when they walk in the room. I may revisit this to see if it helps us have more productive discussions in class. Another idea I shared with the group was the prospect of using an “entry ticket” to diagnose the class’s needs at the start of any given class. This would be something completed on paper or on a whiteboard that would help me determine how the class looks that day. I’ve done this sparingly throughout my career in content-dependent ways, but again, hearing it from the students today felt different. I told the cogen that I would try and experiment with it soon.

The half-hour went by fast! When we closed, I found myself wondering where the time went. Our dialogue was energizing and probably the best one yet. Chemistry is forming between us and it’s enriching our discussion. The students were opinionated, thoughtful, and honest. There was a 4-6 minute stretch today where I was struggling to understand the students’ point of view on the ungraded quizzes. They slowed the discussion down in order to rephrase each other’s ideas. They were resolute about me getting it. I eventually did, but their attachment to teaching me was pretty cool. In other instances, I found myself instinctually calling on students who were quiet for a few minutes to get their opinion. I applied no pressure, they felt none (at least I don’t think so), and they shared their thoughts naturally. Any awkwardness or hesitancy that was present in previous sessions is gone. I commended them for this and more — especially the new addition from 7th period, who fit right in.

In the end, there were so many more ideas bubbling up inside me that I wanted to run by the group. A few that I wrote down for our next session — and beyond: homework completion (why they don’t do it), metacognitive journals (how to make them more fun and engaging), and “class” quizzes (what they think about them).

Afternote: The next day, I connected with the two students who didn’t make it to today’s cogen. I had a 5-minute, one-on-one with each of them to overview what the cogen discussed. Outside of keeping them in the loop, it felt like I was informing a teammate of the gameplan. In retrospect, I wonder if I could have had another student from the cogen brief them on what we discussed? Note to future self: do this. It will help promote ownership of and community within the cogen.


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