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Meditations on a Cogen (No. 19) • Thursday, March 10, 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 19th post in the series.

Pre-cogen recruitment
With my current cohort of cogen students transitioning out, I went hard this week on my recruitment efforts. Last week, four prospective replacements were identified by my current group. The cogen students did their part and reached out to them. They told me that most of the students they identified were hesitant about joining. That was my cue. Time for some encouragement and cogen advocacy!

I found most of the prospective students during lunch. There I pulled them out for a few minutes and gave them my elevator pitch. I managed to convince all of them that the cogen was worth their while. Two said that they would make it today and the others said they would come next week. (After today’s cogen I learned that one newbie from period 3 — who said he coming today — now cannot do the cogen because of a prior commitment. I’ll be on the hunt again next week.)

Prep
We finalized the directions and logistics of our board game last week, so today has a singular focus: prep the materials. We’re playing tomorrow and Monday. The two major tasks are: (1) create four gameboards on chart paper based on a revised blueprint and (2) label 34 problems with their respective levels (1-4). I dig through the mess inside my desk to find markers to draw the game boards and bright paper for the problems.

As we begin, I realize that I’m short two members that should be here. I find out later that both are attending an Emerging Leaders Program meeting. I have no problem at all about this, but only one of the students informs me. I’m slightly annoyed because of how much we have to do in just 30 minutes. We’ve been planning this for weeks, I say to myself. With so much prep to do for tomorrow’s kick-off, how could one of them bail on us today?

I quickly shake off my negativity and focus on who’s here. I see the new member and quickly (and poorly) orient him to cogen. Other than the snacks and water, I mention that today’s meeting will look nothing like it usually does. We will not be sitting around the table and there will be no formal dialogue. He nods willingly and unassumingly. It’s clear that he’s grateful to be here. I’m grateful he is, too. He’s dependable and eager and will be a wonderful asset to the next cohort.

We divide up the work. There’s a total of six of us. Half create the boards and the other half (which includes me) label the problems. After some scrambling to find additional staplers and staples, my half finds our rhythm. The students who are creating the gameboards work together nicely, too. Unknownst to me, they even coordinate colors on the board with the colors we’ve chosen for the different levels of problems. There’s synergy in the room.

As 3:15pm approaches and we are completing everything, a few members have to leave. I bid them farewell and thank them vigorously for their leadership. They should be proud of what they’ve created — not just today, but in the weeks leading up to today. As they pack up, I’m placing game pieces in makeshift envelopes for the tables when suddenly remember that we don’t have a name for the game. Ha! Weeks of playing and we forget the most obvious of things. The kids quickly throw out one name after another. One sticks: Infinite Levels. It’s fitting. We run with it.

Upon finalizing the name, one of the students who missed today due to Emerging Leaders flies into the room. He apologizes for his absence and asks if there’s anything he can do to help. He was actually the one who came up with the idea of playing a board game in the first place. When I see him come in, it’s clear that he feels horrible about not being here to help us today. I thank him for coming back and checking in and then ask him to look over the slides he created with the directions. The cogen students are going to use his slides to introduce the game tomorrow. Now that he sees the game in the flesh, is there anything he wants to edit? We make a few additions. I give him a huge high five. He heads home.

Let’s play
After all the students leave, I tidy up and get the tables in order for tomorrow. I place the gameboards, problems, and game pieces out. I drop some whiteboards on each group and call it a day. Infinite Levels is born.

I leave happy.

Meditations on a Cogen (No. 18) • Thursday, March 3, 2022

A new tradition
We’re coming off mid-winter break and at some point during the last two weeks, I found inspiration — I forget from where — for a new tradition at the cogen. It’s pretty simple: I want to start each session with everyone answering the question, What’s something you have done for someone else in the last week? It’s a way of starting every cogen with generosity and kindness and also helps me get to know the students better.

Everyone shows up again today with no in-person reminders. I’m not surprised; this cohort has been so consistent these last several weeks. They’re great. I ask everyone how they’re feeling and then introduce our new tradition. We all go around and share our acts of generosity. The standout: earlier today, someone gave a friend a snack because they were hungry.

Finalizing Plans
I long to touch on several issues with the team today (seating arrangements, journal 2 reflections, problem presenters), but I have to push them aside and prioritize our board game. Our planning is basically complete, and we were originally scheduled to play tomorrow and Monday, but we decided to wait one more week. There are details that need to be ironed out. I’ve been super flexible on timing — there’s no need to rush.

The first item up for discussion is the gameboard. The student who was responsible for creating it emailed it to me earlier this week. I present it to the rest of the cogen. Everyone loves it.

I show them a die that players will roll to move along the board. We agree that the center space should be a “challenge problem” and decide that it will be a Level 4. We have an interesting debate over which direction players will move and whether it matters. Can players change direction after passing through the center? This gets messy and we determine a uniform direction for all players. We figure that we will remove the word “space” should be removed in favor of an empty black space (if a player lands on it, they get no problem to answer and, hence, can earn no points on that round). A student proposes an “entry” square to the immediate left of the board where players will start play. Chess and checkers pieces are to be used for players’ game pieces.

If a player gets a problem wrong on their turn, the other players will have the opportunity to “steal” — to answer the problem correctly and earn the points. There’s going to be 3-4 teams per table, so how will the “stealing team” be determined? I’m baffled, as I usually am when it comes to simple things, when a student says, “Let them roll the die for it. Highest number wins.” It’s perfect. Because of this, we decide that the problem cards (that I’ll be creating this week) should be printed on 8.5″ x 11″ paper. This way, all teams can see and work on the problem at the same time. This will not only keep everyone engaged even when it’s not their turn, but speed up gameplay during stealing because stealing teams will have their answer ready.

We do a mini walkthrough of the game and feel good about it. I remind the students that I will be creating the answer key for the problems. They offer some pointers on its structure that will allow for easy checking as players call them over. I take note. At the end of our meeting, someone volunteers to create a couple of slides with game rules and instructions. The cogen students will use them to introduce the game to the class.

There’s nothing left to do now but play! We’ll use next week’s cogen to recreate the gameboards on four large chart papers that will be placed at the center of each table and make any other final touches. I forget to ask the group about a name for the game — but we decide that next week, too. I promise to have the problem cards and answer key ready for them to review. I tell the crew that it’s been so much fun planning the game with them that even though we haven’t even played yet, I see myself using it again next year. We’ve put a ton of thought and effort into it. Why not reuse it? One student remarks that I ask next year’s cogen to modify the game for their class — to make it even better. It’s a brilliant idea.

We depart, eager to play.

Transitions
This is sixth week for the current cohort. We’ve been talking about their replacements for a few weeks now — including at the start of today’s session. All of them have identified at least one potential successor, but most of their choices have expressed little interest in joining the cogen or need to be convinced. I’ve been pretty good at helping students network cogen replacements this year, but I’ve been so absorbed with planning our game these last few weeks that I haven’t done any advocacy work for the next cohort. That needs to change next week. Time to recruit!

Cogens for Social Justice • Part 1

This is the first 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 1: Preplanning

For years, I’ve been in search of meaningful ways to bring social justice into my Algebra 2 class. I haven’t had much success, but there is hope. It comes in the form of the math department at my school as well as my students — namely my cogenerative dialogue.

Last year, the Geometry teacher at my school designed a project based on the 1619 Project podcast. It was a highly creative endeavor that focused on the historic discrimination in the farming industry. He used excerpts from the last two episodes which profile a Black family, the Provosts, and their experiences with their farm.

For his project, he presented students with fictional data of many plots of land. The data included info on plot sizes, bank loan information, and the demographics of the loan recipients. The goal was for the students to determine if discrimination was involved in handing out the bank loans. It was part of his triangle congruency unit (all plots of land were triangular). If two plots were equal in size and location, for example, are the two people of equal qualifications applying for loans given the same amount? Are there patterns of discrimination? Could students prove it in court?

He first ran the project last year during remote learning. He did it again this year. Intrigued by his creativity and willingness to embrace the intersection of social justice and mathematics, I’ve decided to build on what he started. More specifically, I see a throughline between his project’s focus on farm loans and how compound interest is explored in Algebra 2. Could I continue what he started — but instead of exploring plot sizes, could I anchor my activity on interest rates? Or extend it to discuss something like predatory lending and its impact on urban communities?

I brought this idea to the rest of the math department. We brainstormed. Our wealth of experience fueled my idea with momentum. In our talks, however, I wondered: what do my students think? If there’s anything that I’ve learned these last few years, it’s to make space to listen to students. Talking my head off with fellow teachers absent student voice is not something I want to do anymore. I pump the brakes on my planning to seek out some student perspectives. I want to hear what they think.

To do this, I begin planning for a series of three cogenerative dialogues (or cogens). The first cogen will be used to gather baseline info from students that will shape my planning. I’ll use the second cogen to present students with the activity itself, make space to receive their feedback, and then modify the activity based on what they say. The third and final cogen will occur after the activity’s conclusion and will be used to hear from the students how it went.

I identify students for the cogens by asking my Geometry colleague for recommendations. Who was active last year and can supply me insights into his project? He drops five students’ names. I also pinpoint four other students who are currently in Geometry and just experienced his project this year (they’re doubling up on Geo and Alg 2). I check all of their schedules and determine that lunch is a viable meeting time. I find them a few days later and explain that I want their feedback on an upcoming class activity. They agree to meet during lunch this past week.

Because of scheduling, I’m unable to meet with all of the students in a single session, so I hold two separate cogens. The first cogen has 7 students and the second has 3. I sketch out a few questions ahead of time to ask the students. I also consult my Geometry colleague for his input on what to ask. My questioning went a lot of different places during the cogen based on student responses, but here’s what I entered with:

What are your first impressions of the farming project you did in Geometry?
Was the project worth your while?
What advice can you give me as I plan a similar activity?
What did you think of the social justice aspect of the project? Is there social issue you want to explore?
Does me being a white man affect how comfortable you are exploring issues such as racism and sexism?

On the day of the first cogen, two students whom I didn’t invite were curiously hanging around the room. Noticing their inquisitiveness, I explained what we were doing. They seemed loosely interested and, before I know it, they are sitting down with us. Right place, right time. (One of them was part of my weekly cogen at one point this year, so she kind of knew what was happening.)

What follows is a synthesis of my students’ feedback from our cogen, which I will use to begin designing the activity.

Find balance. Most students commented that I need to strike the right balance between learning math and exploring social justice. This means possibly frontloading the mathematics and rolling out the social justice aspect of the project slower than I would have previously anticipated. There is genuine concern in the group about not understanding the mathematics before being asked to apply it.
Choose different launching point. Because of remote learning, last year’s students struggled on the Geometry project. They found it confusing and frustrating and struggled to say it was worth the time investment. For this reason, they recommend that I don’t use the Geometry project as a launcing point for our activity. If I do, it’ll probably cause dred and angst to sweep over our class on day 1 (many of these students rolled their eyes when I first mentioned the project to them). Instead, the kids suggest that I don’t even bring up the Geometry project — at least not initally. This was valuable insight.
Interest rates are relevent. From what I gauged, there was interest amongst the students in studying interest rates and how they could be used in things like predatory lending. The students offered up no other particular topics that they’d like to see us study. To their defense, I did put them on the spot with this question.
Don’t overcomplicated it. Students suggested that I be careful not to make the activity too complex. Social issues are inherently thorny and multilayered. I need to honor their complexity, but making it digestable for students is also important. Students don’t want their heads to be spinning because there’s so many factors to take into account.
Presentations should focus on different issues. If I ask the students to present their findings at the end of the activity, each group should have a different focus. This way, students can learn something new from each presenting group.
Be mindful of my teacher moves. The Geometry teacher is a white man — so is 60% of the math department (we’re 80% white overall). Almost all students in the cogen expressed no discomfort with exploring racism or sexism with him in the context of the 1619 project. They appreciated the need for social change and saw the project as a means to that end. That said, one young lady did wonder whether another teacher (of color?) could have led her through the activity, but did understand its purpose. I’m glad she said that because it spoke to the tension I feel exploring these sensative isssues given that I will never experience them firsthand. Another student said plainly, “Mister, it’s what you do that makes a difference.” His words were pointed and would stay with me for a long time.

As I move out of my information-gathering phase and begin planning, Gholdy Muhammad is in my head. In addition to her four pillars of identity, skills, intellect, and criticality — which I will try to address in my activity — one of her guiding principles is to start with joy. It’s terribly easy to see injustice and center it completely in activities involving social issues. Besides, what joy is there to find in predatory lending and payday loans and how they eat away at one’s finances? That said, I’ll need to ensure I make room to cultivate joy and love in whatever my activity asks students to do.

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

A full house with no reminders
This was the first cogen of the year that wasn’t preceded with an in-person reminder from me. I didn’t pull up to any of my cogen students at any this week before, during, or after class and ask them if they could make it today (also: there were no classes yesterday). The way things have been going these last several weeks, I was confident that they didn’t need me to do that. The one move I did make, however, was to email them yesterday to remind them about bringing in a blueprint for our board game. No one responded, so I couldn’t be sure if they read it or not.

It’s no matter because my students don’t disappoint today: everyone shows up. Even the newest member who couldn’t make it last week because of tutoring surprises me by coming through. One cogen alumnus also comes — he actually reminded the newest member about today. I shout him out both during and after the cogen for his act of community.

As we settle in around the table, my cogen family has me on a high. Though they’re no different than usual, the snacks taste better today. The air is light. I breathe in deeply and smile.

Successors
This week we have a singular focus: planning our board game lesson. It will be the second coteaching experience for the cogen. We left last week with plans to move forward with the idea and have a lot of work to do.

Before we start planning, I remember something a student asked me about this week: cogen replacements. I’ve been so wonderfully engrossed in the cogen’s progress these last few weeks that I didn’t even realize that the current cohort’s six-week tenure is coming to an end. Because of this, a hint of melancholy is present in my voice when I ask the students about their successors today. They have been so productive and willing, it is bittersweet for them to leave. Half of the group has a replacement in mind and drops names. I ask them to reach out to these folks and offer my support in breaking the ice.

Creating a board game
When we left last week, I asked the students to make a prototype of our gameboard. (I took on the responsibility of creating some makeshift problem cards.) This was really interesting because it was the first time all year that the cogen students left our meeting with “homework.” I was unsure of what to expect. But if coteaching is the next phase of the cogen, then it’s natural to presume students will have some work to do outside of the allotted 30 minutes we meet each week. Besides, the buy-in was strong last week from the kids. It felt right.

Did I mention already that the students have me on a high today? Well, they do. They bring three different hand-drawn gameboards! I do my best to contain my excitement in front of them. I’m thrilled. The boards varied in shape and orientation but were similar in that they all had a starting point and ending point. Think Shoots and Ladders. I ask the students to share what went into making their boards. One student shares her vision for using mathematicians during the game. Another has neat ideas for special spaces on the board. Their ideas are clever.

After I share my improvised problem cards, we get into the nitty-gritty of the game. There’s a lot of back-and-forth action for the rest of our time together as we hash out details of the board and gameplay. Here are some highlights and key takeaways of our dialogue:

Player movement. We originally decided (last week) that the problem cards would determine how players would move on the board. Each player would get to choose the problem level they wanted (by picking a card) and, if they got the problem on the card correct, they would move that number of spaces. Well, it doesn’t take long for the students to squash this idea. Instead, players will roll a die to move. The problem levels will be scattered randomly all over the board. Whatever level a player/team lands on, they pick up that card and solve the problem. We agree that this is fairer, more equitable, and more balanced.
Using points. By changing how how players move (random), we conclude that our game cannot be won by reaching a specific location on the board. There can be no “end” space. Instead, the game should be point-based. When players get problems correct (whose levels are also random), they will earn points based on the difficulty of the problem. Whoever has the most points at the end of the game, wins.
Infinity-shaped gameboard. Because the game is no longer based on a players’ positions, a student quickly suggests the board be made into the shape of an infinity symbol (∞). Everyone loves it.

an infinity-shaped game we sketch out during the cogen

Group gameplay. Throughout our talk, we flip-flop on whether to play the game synchronously (whole class) or asynchonrounsly (small groups). There are pros and cons to each, both for gameplay dynamics and logistics, but in the end agree to play the game in small groups. This means that each group plays its own game I have 4 groups in my classroom with 5-8 students each, so each player in the game will consist of 2-3 students.
Table vs. Table. In addtion to having individual winners from each group, we also want to have a winning group. To do this, at the end of gameplay we can have all players in each group sum their points. Whichever group has the largest sum wins that award. This way, players have an incentive to support other players in their group.
Coteachers checking answers. As players move around the gameboard solving problems, the cogen students (and myself) will float around the room with an answer key. After a player solves a problem, they will signal one of us by raising their hand. A coteacher will come over and check their work and final answer (all problems will be numbered for quick reference). This provides immediate feedback to the players. If the player gets it correct, they will earn the points for that problem. Incorrect solutions do not result in a penalty. (When I proposed that a key be left in each group so players/tables can self-check, the cogen said outright: “Mister, we’re going to cheat if you do that.” This was surprising and hilarious.)
Lockers. I have lots of empty lockers in my classroom (we used to be an elemenatry school). While we ultimately didn’t adopt it, there was a fascinating suggestion to use the lockers during our game. This spoke to me because it uses the physical environment to enhance the game. The idea to use them proved unwieldy for our game, but I pocket it for future use.
Point stealing. When a player lands on a “Level 2” space and begins working on the problem, all other players will work on the same problem. If the player who chose the card gets the problem wrong, another player will have a chance to steal the points for that problem. We didn’t determine yet who gets to steal.
Special Spaces. The majority of spaces on the board will have a problem level (1-3) on them. To add to the fun, we also want to have “special” spaces. These will be different and add some spice to gameplay. Some possibilities include: (a) challenge space (hard problem worth more points), (b) nothing space (no problem, no points), (c) steal points from other players (all players give another player 1 or 2 points), (d) donate points to others (player “gives” all other players 1 or 2 points), (e) earn a second chance card (if a player gets a problem wrong, they can exchange this card for a second chance at getting it correct before another player can steal)
Homework. At the end of our discussion, I ask if someone can make a sketch of the board that includes what we talked about today. Next week (or the week after, if we need more time to finialize), we will use our cogen to redraw the boards on large chart paper for each group. The chart papers will be placed at the center of each group for gameplay. The quietest student of the cogen agrees to take on this task. I promise to email her — and the rest of the team — our notes.
Game name. After we leave, it hits me that we don’t have a name for the game. We have to think of somethign catchy.

Excitement all around
A lot happened today. The energy was high, ideas were flowing, and everyone was present in body and mind. There was plenty of debate about our game/lesson. It’s really taking shape. Everyone contributed and is feeling good about it. I even overheard a student say at the end, “Today was really productive.” I couldn’t agree more.