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.

Chemistry
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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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.


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Dear R, (Student Letter #9)

To help me be more critical and mindful of the bonds I’m forging in and out of the classroom, I write anonymous letters to some of my current and former students. This is the ninth post in the series.

Dear R,

Hey. Long time, no talk. Kidding, of course — I just saw you yesterday, but it was fitting I write you. As one of the first students I developed a genuine connection with this year, you have been a delight to have in class. Your humble, unassuming nature was refreshing from the start. I remember that during the first week of school you had your head down and I pulled up next to you and asked if everything was alright. You assured me that you were fine, just tired. It was our first real interaction.

One day, a week or two after, you showed up late to class. I was standing outside the door. I’m not sure why, but I held up my arm in the doorway, not letting you pass, and asked you for the “password.” I wasn’t going to let you into the classroom until you figured it out. You guessed three or four times before you got it correct. We laughed. It was so random and fun that, although you arrived on time, I asked you for another password the next day. And the next. And the next. Our funny, made-up “passwords” soon developed into a quirky daily routine that was a needed high note for me just before our class began. Crazily enough, on days that I wasn’t at the door, you began holding yourself accountable to the password and not walking in until I walked over and confirmed that your password was correct. Hilarious.

As I write this, I’m also left appreciating your Friday Letters. They’ve been heartwarming. I’m not sure why you decided to write one out of the blue, but I’m glad you did. You even started numbering your letters, which was unique. Through them, I enjoyed reading about your indecisiveness, struggles with AP Euro, and trip to DR. I wish I could have seen your 80 year-old grandmother’s face during her not-so-surprise birthday party that you and your family threw her. How fun.

This is why, this week, given all my excitement around teaching you and getting to know you, hearing about your transfer was so defeating for me. From your letters, I knew you and your mom were considering it, but if it was going to happen, I didn’t think it would happen so fast. I thought it would have taken a few more weeks, at least. But after I made a light-hearted reference to something next week, you told me that you wouldn’t be here for it. I asked why. You said Friday (yesterday) would be your last day. Your transfer went through. The last day of the marking period was going to be your last day as my student.

The moment was sudden and far heavier than I would have expected. Shocked, I forgot about whatever it was we were discussing. The joy left my face. The news saddened the rest of my day and woefully disrupted what has been a wonderful school year. I was dumbfounded and frustrated.

After months of heartache teaching students I never saw, this wasn’t supposed to happen. I wasn’t supposed to experience sincere teacher-student connection like ours and then have it ripped away from me after only six weeks. I paid my dues to remote learning. I’ve earned my right to bond with my students again. This all feels so wrong. Somewhere, the spirit of remote learning is laughing at me.

(Sidenote: It’s interesting how I have begun thinking about and referring to remote learning as if it’s a living thing that was here for while and then moved on. I find a strange comfort in this.)

In time, my distress over you transferring will pass, but that doesn’t mean that I’ll forget you. I won’t. You gave me the gift of renewal and revealed the beauty in new beginnings. In the wake of emptiness and counterfeit classrooms, you offered up kindness and connection. Without trying, you were my first proof that the humanity in teaching and learning has indeed returned. Truth is, you were the sign that I was longing for. You let me know that it’s OK to lean in again. Thank you.

I wish you the absolute best at your new school and beyond. Please stay in touch.


Will miss you and our passwords,
Mr. P


P.S. Thanks again for the tortoise. Ironically enough, it’s one of my favorite animals. How fitting. It’ll rest on my desk for a long time and remind me of you and the rebirth of my teaching this year.

P.P.S. I’m so glad I could give you the Token of Appreciation. I was waiting a long time to find the right person — and you were it. Appreciate you, R.

Meditations on a Cogen (No. 2) • Thursday, October 21, 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 second post in the series.

Before this meeting, I made sure to touch base with all of the students throughout the week to remind them about it. All four students from the first meeting were able to attend. One of the students who couldn’t make it last week was also able to attend. It looks doubtful that the remaining two students that I originally spoke to are going to be able to participate because of availability and attendance issues. In the coming weeks, I need to find replacements.

We opened by sharing how our day went and grabbing some snacks. The goods are plentiful, but the kids noticed we had no beverages. I promised to bring water next week. I was so glad that they actually came back a second time that I told them this. There are many things that young people are asked to do these days. Requesting for 30 minutes of their time after school may not seem like a huge ask for some, but to me it is. I showered them with gratitude.

After checking in, I followed up on how we ended our last cogen. How did the DeltaMath review assignment go? We agreed that I would create it to help students prepare for the exam this week. It was optional and no one — including cogen students — was obligated to do it. Last Friday, I paid a visit to the teacher who was mentioned at last week’s cogen. He was the one who gave DeltaMath reviews that the students praised for helping them on exams. He and I talk often. I simply asked him how he did his reviews and if he had any tips. After our chat, I posted the review assignment on Monday. The exam was Tuesday.

In class, when I announced that I created a review assignment, I nodded to the cogen students as a sign of respect and appreciation. They were the reason I created the assignment and I wanted them to understand that — at least on a personal level. I haven’t announced the cogen in any of my classes, but that may change down the road.

Anyway, about the assignment, half of the cogen students said they didn’t have time to do it. The other half started it but didn’t complete it because they got stuck on a problem and didn’t push through. One student complete it and said that it significantly helped her. Having all the topics in one place was what she liked best. Her exam score reflected this; she did far better on this exam than she did on the previous one.

After looking over the data, I noticed that about 20% of students completed the review assignment, with around 50% of students trying some part of it. I presented the cogen with this data and probed them. I was curious, what could encourage more students to do it? Given all the demands placed on them, what could make the review assignment a priority?

They threw out the idea of giving extra credit for doing the assignment. The extra credit could be awarded on a sliding scale and added to the final score of the exam. I expressed my hesitancy about giving extra credit but remained open to it. I got some nods from some of the students who understood the optional nature of the assignment. Interestingly, one student also asked whether we could track the students who completed the assignment with their exam performance to see if there’s a correlation. I nearly hugged him when he said this! Ultimately, we decided to try the assignment again sans extra credit and see what happens. It was only the first time, so maybe engagement with it will improve once students see its value. Given that the next exam is 2-3 weeks away, we tabled the discussion and made plans to revisit it.

Other than revisiting the review assignment, I wanted to ask the cogen about our class discussions. I’ve noticed many more of my students struggling to collaborate in effective ways these last couple of weeks. In some ways, I think the honeymoon of being back at school is over. As a result, their willingness to reach across the table for help — or to help– is starting to diminish. At the same time, I think I overlooked how intentional I needed in helping students to communicate, present solutions, and seek help from others. To address this, I’m planning a whole-class roundtable for tomorrow to discuss these issues. I want to use the time to talk about what good and bad collaboration looks like and co-create a collaboration rubric with my classes. This rubric would be used to help us measure our levels of collaboration and improve them over time.

After saying this, I asked the cogen what they thought. They offered up reasons that might explain our current struggles. Some felt uncomfortable critiquing other students’ work. Some felt that the work on the boards could be neater and the presenters could be louder. Others didn’t feel responsible for other students’ learning and instead subconsciously viewed themselves as islands. This last point struck me as something to pursue further as a possible yearlong theme/goal with my classes — to the responsible for each others’ learning. Near the end, there was a suggestion about me being more deliberate about identifying “problem leaders.” I try this when I select presenters, but this proposal felt different. It made me think of small pockets of students forming around certain boards which are led by students giving explanations. These would be students that would lead other students in understanding the problems we study each day.

As the meeting came to an end, I felt really good about how things went today. The discourse was productive and less awkward than last week. It helped that we had a change — a collective decision — to reflect on. Where and how we ended our talk really prepped me for having the whole class discussion tomorrow around collaboration.


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