Meditations on a Cogen (No. 5) • Friday, November 12, 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 fifth post in the series.

Lunch and a debate
We usually meet on Thursdays, but because of Veteran’s Day, that couldn’t happen this week. I remembered that we all have lunch 4th period and asked my cogen students if they could do Friday during lunch. They said ok. I don’t like sandwiching the cogen between classes like this, but I had to take it over not meeting at all. I spiritedly promised to bring the students Slim Jims for their flexibility, but ended up forgetting them. I now owe them double next week.

As I began dumping all the snacks onto the table, the kids trickled in. Two forgot that we were meeting today, but someone ran down to the cafeteria to get them. After a few minutes, we were all sitting around the table again. This was our 5th session together. I was unexpectedly comforted by this.

We checked in on how everyone was feeling. As a means of breaking the ice, I like picking up on whatever energy, vibe, or idea reveals itself in a given moment. I tend to be pretty observant, so it’s not that hard. I do this in class, too. It’s my way of staying present, acknowledging one another, and warming up to dialogue. Today, one of the students brought a cup of water they got from the cafeteria. For whatever reason, something spoke to me about that cup and we spent the next few minutes airly debating water dispensers vs. water fountains. It was fun.

Collective brainpower
We opened the formal dialogue by recapping what we discussed last week. I mentioned the feedback quiz that I tried last Friday (the day after our last cogen), and how that led to Monday’s lesson on trig ratios in the coordinate plane. I used VNPS and VRG for that lesson — which is the structure the students told me they learned a lot from. We ended up spending two days this week using it as well, which the students confirmed were effective. Last Friday the cogen also persuaded me to bring out the dry erase sleeves for worked examples and practice problems in small groups. I botched the timing so it wasn’t the best lesson, but it was cool. I don’t think the cogen students even realize it, but I often use worked examples at the start of class now. I didn’t mention this last point at the cogen today, but it’s interesting how our talks are impacting class in unforeseeable ways. Our ideas are helping me rethink areas of the class that we don’t explicitly touch on in our dialogues.

At one point during our recap, one student pointed out how another student was responsible for a couple of the ideas. “That was from her…” the student said casually a couple of times. I was proud that the student vividly remembered our last talk, but commented that those ideas, though one person may have said them, were the direct result of our dialogue. Our collective brainpower, problem solving abilities, and creativity birthed them; through our dialogue, we co-created the context that enabled the solutions to emerge. The students nodded.

Making journals relevant
I left last week’s cogen desperately wanting to discuss metacognitive math journals with the group, so that’s what we did for the remainder of today’s talk. I’ve done metacognitive journaling with my students for the last several years and have even used editorial boards to peer-review them, but the journals themselves tend to be fairly dull. They need some sprucing up.

I overviewed the journal process to the cogen and asked for their suggestions. I told them I was looking for ways to make it more meaningful, but also for ways to make it fun and engaging. After some clarifying questions, they let loose. One recommendation was that students turn the journal into a written dialogue between two people. If this is the case, maybe students should be permitted to work in pairs to do the journal? This is a fascinating idea! Other ideas were allowing the journal to be a letter to self or even a short story. The journal requires a student to reflect on their solution pathway to a problem, but could they integrate drawings or other art to do this? Instead of students choosing any problem (what I have always done), one student commented that they should have to choose between a handful of challenging problems that I curate for them. In years past, I have required that the journal entries come from our standard problem set, but why can’t it be one from an exam that the entire class struggled with? Or an entirely new problem they haven’t seen yet? Our recent usage of worked examples even made me think about students using one for the journal. Why not? Given the right problem and work, it could make for juicy reflections.

An assumption throughout our dialogue and their recommendations was choice. In my three years of doing journaling, crazily enough, I never considered giving students a choice in the matter. But why not? Reflecting on one’s mathematical thinking should not be uniform. Putting their ideas in a box is probably why the journals have failed to come alive for my students these last few years. I need to throw out the box. As we wrapped up, I thanked the students and them that I would use the ideas we generated today to create an updated version of the journal assignment. I would present it to the cogen next week to get their feedback before I assigned it to all my classes.

During our discussion, I noticed one student in the group was kind of quiet and I encouraged him to share what he was thinking. He remarked that he didn’t really like writing. I sensed that he wasn’t looking forward to this assignment and withdrew from giving me feedback on it. I respected this, but told him this is precisely why I wanted him to give his opinion. What could make the journal worthwhile for him? At a minimum, what could make it less annoying for him to do? Of all the students sitting around the table, his opinion mattered a lot because of his overall disdain for writing. I even followed up with him after the cogen to reiterate this message and hear his thoughts.

I got the feeling that, for the cogen students, they felt they didn’t do much today. They were just offering up their opinion on some writing assignment. This was true, they were. But I shared that I never considered any of these types of ideas in past years. There have been hundreds of students who have written journals in the past — and all of them were confined to one way of expressing their metacognitive reflections. Our conversation today changed that. As a result of our talk today, I hope more students will find meaning in the assignment.

After the students left, I noticed a colleague in the room. I believe she was there the entire time. I explained to her what it was. She remarked that she really liked the idea and wished she was “at the point” where she could do something like it in the future. We chatted briefly about the difference between giving kids a survey to gather their feedback and being in dialogue with them. One provides real-time opportunities for follow-up questions, divergent thinking, and adjacent ideas, and the other one doesn’t. Our exchange reminded of me last year when I asked a few colleagues to join the cogen to provide me with an outsider’s point of view.

Last two things. Our meeting today felt a tad rushed and it caused me to forget to inform the students that they need to choose their replacement next week. I emailed them afterward about this. All good things must come to an end. Secondly, I realized that we had an exam this week and didn’t revisit the DeltaMath review/exam score correlation idea that was brought up a couple weeks back. Today would have been the perfect time to discuss it, but I had to prioritize the journals. So much to discuss, so little time.


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.


Meditations on a Cogen (No. 3) • Thursday, October 28, 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 third post in the series.

Because of parent-teacher conferences, all after-school activities were canceled today. As a result, my cogen met during lunch. It was an impromptu change that I made last minute to build early continuity. I didn’t want to miss a week. I also wanted to show the students that our dialogue was vitally important to me. Even if after-school was canceled, our weekly exchange needed to happen. We had to find a way.

Thankfully, I lucked out because all the members of the cogen have lunch 4th period. After running down to retrieve two kids from the cafeteria (they forgot), we ended up spending 30 minutes together over snacks. Heeding their suggestions from last week, I brought bottles of water for us to sip while we chatted. We had bevs.

We opened the dialogue by checking in. How was everyone doing? We had a funny conversation about Slim Jims and Pringles. Who knew there was “mechanically separated chicken” in those things? Ouch.

After more small banter that warmed us up, I asked if they would prefer to meet during lunch (since we all share it) or after school like we have been doing. They were all indifferent, so we decided to keep holding our meetings on Thursday afternoons.

Next, we reviewed last week. We debriefed the roundtable I had with each of the classes last week on collaboration. It happened Friday and the cogen helped prep me for it last week Thursday. At the roundtable, the classes and I co-created a rubric to help us improve things like communication, asking questions, and presenting problems — all key elements of collaboration. The cogen felt that it went OK, but, outside of period 3, they hadn’t noticed much of a change in the levels of collaboration this week. This was hard for me to hear, but I owned it. I had to keep enforcing the rubric. To continue to improve collaboration, one suggestion from the cogen was to turn it into an in-class competition. We didn’t know what the competition would look like and how we might measure it, but agreed that it could motivate students to collaborate more. I loved the idea. As time passes, I also hope that our chemistry improves. This would help us build a more cohesive classroom community, foster collaboration, and help us feel responsible for each other’s learning.

The second item on the agenda — and what I wanted to focus heavily on — was tutoring. After I led a good-sized tutoring session this week for students retaking our most recent exam, I asked the cogen about it. How was it? Three members of the cogen were present at tutoring. But even if they didn’t attend, I ask them to think about what would make a good tutoring session. How could we ensure that attendees get the most of these sessions, I wondered. Is it better that it be teacher-led (i.e. whole group direct instruction) or can students be organized in small groups with me floating?

Their feedback was really helpful. After brainstorming, we decided that I would provide an answer key of the original exam (it seems basic, but I have never done this) and additional practice problems that mirror those from the exam. This way, the students could see correct solutions for the exam, discuss them together and with me, and then be able to spend time actually doing math. The tutoring session earlier this week consisted mainly of me reviewing the solutions on a whiteboard (i.e. direct instruction) and fielding their questions, but included no time for students to actually do mathematics. In the moment, I totally overlooked this. It’s no wonder why several students did poorly on the retake.

Towards the end, we spent a few minutes discussing the “cheat sheets” that I allow students to use on exams and — surprisingly — how a good amount of students don’t actually use them. This led to a bigger conversation about note-taking and studying for exams. I mentioned the idea of my future plans for “notebook quizzes” and how this will help them be more intentional about the notes they take this year. There was a mixed reaction from the group, some professing their dedication to taking noes while others dismissed it as tedious and confusing. I appreciated their honesty.

As the session concluded, my vibe was that the students seemed to feel genuinely comfortable expressing their ideas this week. They seemed less rehearsed and more natural in their gestures and commentary. Nonexistent was the awkwardness and silence that lurked during our first two meetings. This was reassuring and proof that we’re making progress as a group. There was even some back and forth between students, which was a weak point of my cogens last year. They were exceedingly brief, but during those moments I caught myself listening in to my students being critical of the class without me needing to keep the dialogue afloat. I appreciated that.


Meditations on a Cogen (No. 1) • Thursday, October 15, 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 first post in the series.

Come October, I knew I wanted to start my cogens. I needed a month to let the dust settle on the start of the year and to get the know students. I also needed them to get to know me before I asked them to be critical of our class. Interestingly, last year, my first cogen was on October 9, 2020 — almost a year ago today.

Despite my eagerness to do cogens in person this year, with so much going on, identifying my initial students for the cogen kept escaping me. “I’ll get to it, I’ll get to it…” is what I kept telling myself. Drowned in a million To Dos, my cogen would have been swallowed up whole if I didn’t force myself to drop everything and write down the names of prospective students who would make up the initial group. I did this seemingly at random one day. I had to.

And so, last week, I sat down and identified six students from my three Regents-bound Algebra 2 classes (two from each class). They were diverse (ability, gender, race) and with whom I had formed an early connection in some small way. I wanted to two from each class because (a) six is a good number and (b) if any student from a class can’t make it on a given week, there’s still another student who can represent that class.

After I had my kids, I used my free period to traverse the school and visit them in their other classes to formally ask them if they would like to participate. I gave my best elevator pitch and framed it as my group of “advisors” who give me ideas and feedback in order to make the class better. I explained the time commitment (~6 weeks) and the accompanying perks (snacks, extra credit). Thankfully — and surprisingly — they all felt like it would be a good use of their time. I wanted to meet weekly and the consensus was to meet every Thursday from 2:45-3:15pm. We would convene in my classroom, room 227.

This week, leading up to today, I half-thought the kids would all forget about the cogen so I found myself reminding them almost every day. I’m pretty sure they found it annoying, but I could help it. In preparation, I scribbled down some loose talking points, but mainly wanted to see what came up naturally, and then spend time following those threads. I bought a variety of snacks from the local supermarket.

I was anxious — the good kind — when the kids started trickling in. They remembered! I hid my excitement about their arrival by wiping down the tables with disinfectant wipes and putting out the snacks. After an entire year of remote cogens, I was elated to be on the brink of having an in-person one.

In the end, three of the students I spoke with couldn’t make it today for various reasons. They still wanted to be part of the group, but couldn’t do it today. Nonetheless, this meant I was down to three kids. Despite feeling slightly discouraged, I managed to find a late addition right before we started. This made four students, plus me. Importantly, all of my class periods had at least one representative.

I opened the dialogue by profusely thanking them for being there. After that, I grabbed a snack, sat back, and asked them how they were doing. We did a soft whip around and I sprinkled in some follow-up questions about each of their days to help them feel comfortable. I asked them to introduce themselves to the group. We were off and running.

I offered up some house rules: one mic, not privileging one voice over any other, and being action-oriented. The kids were ok with these. I reminded them of the extra credit that they’ll be receiving, the time commitment, and overall expectations. I reexplained the purpose of the cogen and the need for it. For the betterment of the class, I asked them to try their best to be earnest and honest while also focusing on solutions. I vowed the same.

Ten minutes in, we transitioned into a conversation about our class. I asked them how everything was going. What’s working? What’s not? I didn’t realize it at the time, but this was far too idealistic and broad. To the students, everything was fine. They liked the vibe of the class, the whiteboards, the energy. As we went around the table, there were questions — like how newer, more complicated topics were going to be introduced — but no real critiques or problems of practice surfaced.

Our talk was pretty bland until a student commented on the most recent DeltaMath assignment. He had completed the entire assignment, but his grade online didn’t reflect this. I assured him that I would double-check later, but his question got me thinking about DeltaMath more generally and its role in our class. Seeing an opening, I threw it out there to the cogen. What did they think of the weekly DeltaMath assignments I give? Are they too long? Too hard? Do they align well with what is learned in class?

The students told me that they find value in the DeltaMath assignments and felt they were challenging, but not overbearing. This was good to hear, but it wasn’t until a student mentioned the possibility of using DeltaMath to review for exams that I began to move to the edge of my seat. The student said how her teacher — my colleague — did this last year and how they found it helpful in prepping for exams. The other members of the cogen agreed. I haven’t given/done review before an exam in years, but thought this was a great opportunity to try it again. With their suggestion, I promised the students that I will post a DeltaMath review assignment for our next exam, which is next week. We agreed the review would be ungraded and optional. We will debrief how it went next week.

Unlike what I tried to do earlier, asking them about DeltaMath targeted a specific aspect of our class. It allowed us to dig into something definitive, which ultimately made for better, more opinionated discussion. It proved to be a turning point in the dialogue.

I was concerned at first, but I’m glad that we walked away from the initial cogen with a concrete action that can improve our class. This won’t always happen, but in our early meetings, I need my students to feel that their ideas are being used to shape our class (I also need to feel this). I want to reassure them — through action — that their voice matters. This will hopefully keep them coming back and keep our train of improvement moving.