Meditations on a Cogen (No. 21) • Thursday, March 31, 2022

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 21st post in the series.

Cohort 4
My cogen didn’t meet last week because of parent-teacher conferences, so this week is the first official gathering of the new cohort. This was their first meeting together, so I provided reminders yesterday. Thankfully, they all remembered. One seat from period 3 is unfilled (with a prospect in sight) and one of my period 1 students couldn’t make it today because of an appointment. She ran up to me during lunch earlier today apologizing for not being able to make it, which was super kind of her — and reassuring.

Despite the many cogens I’ve held this year, transitions are always tricky. I remember the first cogen changeover back in November and the anxiety I felt back then. The cyclical question of “Will the new students care as much as the old ones?” hasn’t gone away each time new students rotate in. In this way, new cohort kickoffs have first-day-of-school vibes. The students may have heard of the cogen from other students, but do not arrive convinced of its value. Trust and community aren’t inherent; the conditions need to be created for students to buy into this unusual space. This newness is exciting, but it takes work to help it all come alive.

I drape the tablecloth over the table and dump snacks out. I have a few items on my mind, but, on a limb, I ask the students about this week’s exam. It was yesterday. I haven’t finished grading them all yet, but I’m interested in how it went for them. This unexpectedly takes up the bulk of our time together.

Three of the students admit struggling, but the general consensus is that the exam was fair. One aspect of a problem on writing exponential functions using two points is criticized by the group because we only indirectly discussed it during class. I appreciated their honesty, because they’re right. When I probe students about their struggles, we get to talking about pre-exam review. What advice can they give me to help them feel better prepared for exams?

We first chat about about how I’ve used stations to review for the last few exams. The “review” has coincided with the end of problem set, so it hasn’t felt like review per se to me, but I think that’s how students viewed it. Each station was based on a different topic and equipped with dry erase sheet protectors for students to practice some given problems. A couple of cogen students refer to their chemistry teacher when they recommend that, instead of using stations, I create a review worksheet and dedicate the day before the exam to studying it in groups. In other words, instead of having stations with dry-erase practice, why not create a more permanent solution that students can study even after they leave class? They also recommend that I remain stationary during the review and allow students to come to me.

I’m on board with their recommendations and promise to try this strategy when our next exam rolls around. I do cringe a little inside when thinking about a review sheets and review days, however. In the name of the cogen, I’ll get over my discomfort. Perhaps there’s a way to remix the review sheet with the cogen’s help?

We shift gears and bring up the lessons that the last two cogen cohorts have taught (Bingo and Infinite Levels). What does this group want to do?

It’s an broad question and they hesitant to jump in. Sharing some personal reflections, I recommend that they teach a content-based lesson. The prior two lessons were based in a game, which is fine, but it would be interesting to try something new. Imagining I would use cogen time teach them an Algebra 2 topic first, I inform them we would then co-plan and co-teach a lesson about that topic to the whole class. The kids like the idea. They nod when I acknowledge that students have ways of explaining ideas and helping other students understand that teachers can’t match and this lesson could be a perfect illustration of that. One student sees us turning the lesson into a type of comedy show and using kids names in the problems. We end the preliminary discussion with excitement. I tell them that by next week I hope to have a topic identified.

With about 5 minutes remaining, I open up talk on another issue that may potentially be addressed by this cohort: seating arrangements. Since I can remember, I have been using a deck of cards to assign weekly, visibly-random groups. When students walk into class on Monday, I wait at the door with the deck spread out in my hands. They pick a card and each suit corresponds to a group. They can sit anywhere in the group, but have to be in that group all week. The next Monday we do it all over again. There’s a ton of reasons why I like this approach to seating, but I’ve been doing it so long that it seems natural to try something new — especially now that I have the cogen to help me think through it.

I ask the students how they feel about the random random groups, and they don’t feel strongly one way or another. They express concern about sitting with people they don’t know and I say that’s one of the chief reasons why I do it. I wonder out loud whether they could help me brainstorm a new way to seat the class. After just talking about lesson, we mention using the lesson as a vehicle to experiment with a new seating strategy. They appreciate this.

(After our cogen today, I search my blog and remember that I used to use four quadrants to assign seats.)


Cogens for Social Justice • Part 2

This is the second post of a four-part series where I explore planning and implementing a social justice-themed activity in Algebra 2. In addition to traditional collaboration with colleagues, my use of three cogenerative dialogues to develop and reflect on the activity were critical to its design and execution.

Part 2: Planning and Revision

The last several weeks have been a wild ride. After my preplanning cogen with the students a month ago, I was encouraged by the feedback they gave me, but I still didn’t know what social issue I was going to explore using compound interest. Were my students doing to study the experiences of Black farmers? Or were they going to learn about predatory lending?

I do some research. PayDay loans companies aren’t new, so looking into them turned into a rabbit hole pretty fast. John Oliver’s hilarious take provided a great start. It didn’t take long to find prepackaged ways to help students learn about their outrageous interest rates (examples: here and here). Turning to the farmers, I reread the article in the NYTimes that originally put me on to bank lobbyists’ efforts to block the $4 billion debt relief plan that’s part of the American Rescue Act. I saw this article as the launching point; I envisioned students studying farmer’s loans and interest rates and how much money the banks would lose to do right by the farmers. I hope to relate it to the promise of “40 acres and a mule” and what that land might be worth today.

After spending a couple weeks looking into both issues, I was torn. Both could evoke worthwhile discussions and expose discrimination. Both were interesting — I was learning a lot! But I couldn’t spend all my time reading articles and watching YouTube videos. I needed to make a decision and commit.

In the end, I went with the farmers. The vertical alignment with my Geoomery colleague — who also studied injustice in the farming industry — was simply irresistable. Plus, my reading allowed me to dive deep into a part of our country’s history that I knew nothing about. I wanted to know more; designing the project would satisfy my learning needs, too.

The activity
I figured the activity would span a week at most. Using student feedback, I would teach compound interest for two days up front. Then, on day 3, the farming industry would be introduced by showing students a video from ABC News and reading, as a class, the Times article about the bank lobbyists.

Students would then work through a series of problems based on seven fictional farmers of color that currently have outstanding balances on their loans. I included photos and wrote personal narratives of the farmers to humanize them. I wanted to capture their passion for farming and love for the land, but also the historical context of the racism and discrimination they and their families have faced. All of the details are based on actual farmers I came across in my readings. Included in each farmer’s profile is info about their land and loan.

A few farmer profiles that I created to illustrate their love for farming, but also the inequality experienced

Originally, I wanted the mathematics was to focus on strictly on compound interest related to the farmer’s loans. As I gained momentum in my planning, that quickly flew out the window. I created problems that brought in average rate of change, exponential functions, and profit functions, too. Here are a few.

Meeting with students
I wasn’t finished writing up the activity when I asked the students to meet with me for our 2nd cogen. I wanted to present them with what I came up with and hear their thoughts. A few students couldn’t make it, but I still was able to meet with 6 of the original 9 during a lunch cogen.

We opened with me reminding them of the feedback they gave me at our initial meeting. As I described the activity and gave them the handouts to look over, I explained how their feedback was used in my planning. One of the first questions from the group was, “Mister, how long is this going be?” After I said “probably 4-5 days,” the students sighed in relief. They were fine with the activity, but didn’t want it to drag on; they had bad memories from the Geometry project that extended past their comfort zone. That said, they saw the alignment with the Geometry project, which was good. The students also wondered about how they were going to be graded. They preferred credit for effort and group work and I told them about the need for a 1-2 problem quiz every couple of days. They recommended that I utilize an answer key for groups to check their work as they progress through the activity.

Near the end of the cogen, I admitted my skepticism about studying farmers. Reminder: we live in New York City! I’m concerned that my students’ disconnect from farms could lead to disengagement. To this point, one student playfully suggested that I take them to a farm. We laughed. I can’t do that, but I did mention that I am in communication with Shoun Hill, co-writer and director of the documentary I’m Just a Layman in Pursuit of Justice: Black Farmers Fight Against the USDA, about guest speaking with us via Zoom at the end of the activity. Before his visit I want us to watch at least part of the film in class. The students liked this.

Post-cogen revisions
After the students left, I spent the next several hours thinking deeply about how our conversation went. I couldn’t place it, but something didn’t feel right. The students engaged with me, asked questions, and offered suggestions, but there was still something missing.

Adding to my uncertainty was a comment from one of the students later in the day. We were in class and I thanked her in private for her time at the cogen. She smiled and politely said that learning about farmers wasn’t interesting to her in the least bit, she just didn’t feel comfortable mentioning it front of the whole group. “Why can’t math class just be about math?,” she asked.

Upon reflection, I think my biggest takeaway from the cogen wasn’t anything the students said. Instead, it was what they didn’t say. The students had an indifferent reaction to the activity and our cogen took on an apathetic mood as a result. That was the real feedback: their lack of excitement. Even for the young lady who admitted that she wasn’t interested in farmers, I believe her concerns were rooted in how utterly boring the activity was. All of my well-intentioned research and planning generated little anticipatory energy amongst the students. What was that something that was missing? Gholdy Muhammad would call it joy, a critical element of her five pursuits.

This was bothersome and I went to work to find some way to help my students experience joy in the activity. What could I do that would excite them? Given a foundational aspect of the activity was using math to expose systemic racism, I knew that encountering joy in the content would be a stretch. Could I instead elicit joy through the learning process?

Then it hit me: make it an immersive experience. Instead of having students study the farmers, why can’t they be the farmers? This question caused me to reimagine everything. Here’s what I came up with.

The activity is no longer an exploration of the discrimination farmers of color have faced; it’s now the process of completing a debt-forgiveness application with the USDA. As my students walk in the door on day 1 of the activity, they’re going to be transported to Washington D.C. I will welcome them wearing a suit and tie and give them a nametag with a farmer’s name on it (3-4 students will “be” the same farmer, i.e., be in the same group). I’m going to be Tom Vilsack, the U.S. Secretary of Agriculture, and also will wear a nametag.

In accordance with the American Rescue Act, I’ve invited them here to pay off their farm-related debts — given they complete the application correctly. The students/farmers, of course, will be highly confused, and that’s when I’ll start providing the historical context and significance of our gathering. And I will be a gracious host: there will be light refreshments and snacks out on the tables when the students/farmers arrive. I’m going all in!

The inside cover of the “application” my students will complete

And what will the application consist of? The original problems, of course! It’ll take some reconfiguring, but I’ll use the problems to distinguish the different “sections” of the application. Each farmer’s application will be personalized with their info.

An example “application” cover page. It displays the personal narrative and farm/loan info.

As the students/farmers complete their applications over 2-3 days, I will check work and hand out checks in the amount of their outstanding balances.

This all sounds way more fun and engaging than what I had before. Will it spark joy in my students? I hope so!


Meditations on a Cogen (No. 20) • Thursday, March 17, 2022

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 20th post in the series.

New members
I’m starting to get used to the cyclical nature of my cogens. It’s been a privilege to have had this so many sessions with so many willing students this year. Today I welcome my 5th cohort and four new members. Four kids from my outgoing cohort join us for their last official session. Another cogen student from the fall also shows up and brings a friend with her from period 1. That makes nine students. The table — meant for six — bulges. The chairs sprawl outwards in all directions.

As the snacks start disappearing from the table and into the hands and mouths of hungry teenagers, I eye the newcomers and give them an overview of the space and formally reveal its name: “cogenerative dialogue” or simply “cogen.” At this point, my introductory speel seems routine for me. I find this comforting.

Infinite Levels debrief
We played our board game, Infinite Levels, for two days this week. We reflect on it for about 10 minutes, hearing from my coteachers and students who were players. I wrote a separate post about the experience and our debrief today.

Seeking variety
I had a few talking points for the crew today, but they were all thrown out the window when — out of nowhere — a student remarked, “Mister, exams and quizzes make up most of our grade. I think we should have more variety to our grade than just those two things.” She says this politely, but still catches me off guard. What’s even wilder is that she says it smack in the middle of our discussion of Infinite Levels. Talk about a curveball.

To be fair, there are other assignments that contribute to students’ grades in our class. I would also like to add that, despite the thoughtfulness and honesty of the remark, context matters. The student who made it — someone who was part of a previous cohort — hasn’t done much of what I’ve asked of her recently and is struggling to pass. That said, it’s clear that it’s been on her mind for a while and I want to honor that. I ask her to elaborate and listen with open ears. Her status in class notwithstanding, she makes a good point; I sense a worthwhile conversation looming and advance towards it. I ask the rest of the cogen what they think.

Groupthink could be at play — especially at such a large cogen — but every student agrees. They would appreciate more diversity in the grade book. I explain the purpose of exams and quizzes being weighted so heavily: their grade must be reflective of their knowledge of Algebra 2, and not biased judgments of abstract things like participation or “classwork.”

As our dialogue matures, we start transitioning to possible solutions. One student’s suggestion makes me think of the “Turn In” assignments I did last year during remote learning. These weekly assignments were distributed on Monday, due Friday, and were formally graded like an exam. They could also be edited and resubmitted throughout the week after receiving feedback from me. The students think this is a good idea.

Another idea that’s brought up is to allow retakes for quizzes. Right now, retakes are available only for exams. Quizzes are only one question and are based on what we learn on the day it’s administered. If retakes are allowed for quizzes, I share a concern about students bombing their initial attempt because they know they can retake it for a higher grade. The students quickly shoot this down. Why would someone voluntarily mess up on a quiz only to sacrifice time and energy to come after school and attend tutoring to fix it? My crew also points out that if students wanted to do this, I would have witnessed it on our exams already (which I haven’t).

As we reach time, I go around and ask, as of now, which of the two options students prefer. Retake quizzes earn about 70% of the vote. In the end, perhaps students don’t want another assignment; they just want another avenue to improve their grades. We don’t reach a final answer on the issue, but I vow to take action on it in the coming weeks. I make eye contact with my new cogen members; this will probably be the first issue we address together.

One last option — which never reaches the ears of the students because I think about it after the cogen — is to take a mastery-based (or standards-based) grading approach to the quizzes. Quizzes usually show up in batches of 3-4 in the days leading up to an exam. Each quiz is based on a single concept, matches that of the exam, and helps assess where students are on the unit objectives. Here’s my idea: What if, after the unit exam, I retroactively adjust students’ quiz grades to reflect their understanding of the concepts based on their performance on the exam? This way, if a student demonstrates a higher level of understanding of a concept on the exam (i.e. they learn after they take a quiz), their grade doesn’t suffer as a result of a low quiz grade on that same concept. I’ve been meaning to test drive this idea for months now. This cogen and our conversation today might be the push I need to finally make it happen. It would be especially good to try it this spring in preparation for next year.

A colleague
On my out of the building today, I run into the English 9 teacher at the copy machine. We’ve spoken on and off about my cogen work. She even attended a couple of my cogens last year during remote learning. She’s been telling me for awhile that she’s been wanting to do one with her students.

Well, today she did! She met with her students for 45 minutes after school and was thrilled about how it went. We hang around for a few minutes talking about how her students were selected, how the cogen aided her planning, and what’s next for them as a group.

Little did I know it, but our respective cogens were running in parallel today. I’m so happy for her and her kids. And, if I’m being honest, I’m happy for me too: now I have someone to talk to about all my cogening!


The Infinite Levels Board Game

Last month, as a result of my ongoing cogenerative dialogues, me and my students coplanned and cotaught a lesson. It was nothing fancy, but I was so excited about how it went that I asked the next cohort of cogen students if they wanted to plan and teach a lesson with me. They were game! And I say that literally because we ended up designing an actual board game to play with the class. We called it Infinite Levels.

“Infinite Levels” board game

Gameplay consists of players rolling a die to make their way around a large infinity sign. Players land on spaces marked with a 1, 2, 3, or 4, which denote the level of problem they must answer to earn points (5, 10, 15, or 20 points, respectively). There are also “blackout” spaces (no points can be earned if landed on them) and “Second Chance” spaces (earns the player a card that is akin to a “Get out of Jail Free” in Monopoly; they get another try at answering the problem that they initially answered wrong). Players can steal problems from other players, too. The goal of Infinite Levels is simple: earn the most points. Each of four the tables in the room ran its own stand-alone game; my cogen students prepped all the materials the day before.

I documented the design process of the game with my most recent Meditations posts. Let me tell you — what a ride! Having never created anything like it before from scratch, I underestimated how much work and how many decisions we would need to make. From the game board, to player movement, to content, to implementation, there was a lot for us to consider and then reconsider.

We decided to play over the course of two days. This would give us enough time to really settle into the game. To explain the rules to the class, one of my coteachers created this slide:

Rules for Infinite Levels, as created by a cogen student

The extra time we spent on the details of the game paid off because, after some initial hiccups, it didn’t take long for the class to get into the flow of the game. As my coteachers and I floated to check solutions, there was a noticeable buzz in the air. Several students got so into it that they even stood up while they played. Some groups were laid back and others were fiercely competitive. It was all good fun. All the problems were selected by me and based on recent concepts we learned, so some groups naturally needed more scaffolding than others in solving them. My coteachers and I had the answer key and provided hints when necessary.

Cogen students introducing the game

On the exit tickets, we asked the classes to rate playing Infinite Levels on a scale of 1-10 (1 = horrible, 10 = amazing). The average rating was 8.1. Not bad! Here was some specific feedback the students gave us:

  • I liked the fairness everyone gets when tryng to solve the problems
  • I liked the extra turn we got if the original team got it wrong
  • I really liked having fun with a little competition
  • I liked how it refreshed my mind on older topics
  • There should be a time limit on answerng the problems, like 3-5 minutes
  • There should be drawings on the board, it was just plain
  • The game should have more variety of problems

A few days later, at our next cogen, I asked my coteachers and a few other students about how they thought the game went. They echoed much of what was on the exit tickets. We agreed that timers were needed for each table to keep the game moving along. When we were designing Infinite Levels, we talked about having a space where a player would get points from other players (similar to how the “Collect $50 from each player” card works in Monopoly), but forgot to include it in the final version of the game. That would have been a nice twist. When I asked whether they would like to play again, everyone nodded without hesitation. One of my coteachers recommended that I get the boards laminated for future play, which was a great idea. I can totally see us playing again once Regents prep starts.

I must say, it was only the second time, but coteaching with students is kind of catching on. Students are starting to see that it’s a thing and I’m definitely enjoying it. Given these last two lessons have been based on a game and not tied to any particular Algebra 2 topic, I would like to coplan a lesson or series of lessons around a specific topic with my cogen students. Logarithms and polynomial long division immediately come to mind, both of which we’ll be studying in a few weeks. Focusing on a particular topic would require more content-specific planning from us, but it would provide a nice challenge for the cogen. We could even co-author a lesson plan for it, too, which would add depth to our planning.