Meditations on a Cogen • Post-School Year Reflections

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 28th and final post in the series.

As the last post in this series, here are my final reflections and takeaways on my experiences holding weekly cogens this school year. A lot happened. There is a lot to think about.

Time
A primary concern I had at the start of the year was time. Last year — my first year holding cogens — fitting cogens into my schedule was a breeze. But with the rigid agenda of a traditional school year returning, how could I make time to hold weekly cogens?

In the end, I worried about time often this year, but not in the way I originally thought. Finding time for my cogens turned out not to be an issue. I simply built them into my schedule and prioritized them over other demands. In October, we adopted Thursdays after-school as our meeting time and, other than a few exceptions, it stayed that way. Given the 23 students who were part of the cogen, it was shocking that our meeting time didn’t need to change.

The bigger issue was student attendance. I constantly racked my brain about it. Would the students show? How many would remember? Though I reminded the students about our sessions regularly, I asked myself these questions every week (mainly during the first few weeks of a cohort). Surely, the chemistry of certain cohorts made them less prone to forget about sessions. But because cogens were new for my students, I wasn’t sure how serious they would take them. Attendance each week felt like a crapshoot.

Maintaining interest
Even when some kids forgot about the cogen, however, the future of the cogen was never in doubt. What kept my students coming back week after week? Perhaps it was the 5% extra credit they got for attending, but I doubt it. On an exit survey, 86% of cogen students reported that they would “Definitely” or “Probably” still have joined without the extra credit. (Many also confirmed this during the cogens themselves.) Several students praised my having snacks for them. At the end of a long day, it was nice to have something to munch on. Many others said that being heard and helping the class were big draws for them. On why they decided to join and keep attending, here are student comments that I found to be representative:

  • Because you’re a teacher willing to ask us for help instead of the other way around.
  • I saw it as a great chance in not only helping myself but also helping my classmates.
  • Mainly just to see what it was about.
  • The manner in which you approached me about it made me feel special.
  • I liked how we all got a say in whatever we were discussing. Even for the quiet ones, you would always invite them into the convo for feedback.

Class Improvements
Cogens are just conversations if they don’t result in action to improve the class. I’m proud of the class improvements we made this year. They were diverse and met the moment. In a year where we all were searching for answers, my cogen delivered. Some improvements we made included:

  • Developing DeltaMath goals to motivate the class (link)
  • Creating DeltaMath exam reviews (link)
  • Changing exam review strategies (link)
  • Holding in-class “DeltaMath days” (link)
  • Group quizzes (link)
  • Ungraded, “feedback” quizzes (link)
  • Restructuring after-school tutoring (link)
  • Key revisions of the metacognitive journal assignment (link)

Other ideas we had came up short. For example, the “Weekly Planner” we devised back in December was good in theory, but had no staying power. I used it for a week and then ditched it. I’ve learned that if my cogen is productive, then many of the solutions that get brought up will fail or simply never get implemented.

Coteaching
About halfway through the year, the cogens began striving for more. It wasn’t enough to simply talk about improvement; the cogens themselves had to be the change. Thus, the cogens evolved to be a space where we planned and taught lessons together. We accomplished this three times during the year. The first was a game of Bingo. For the second, we created an original board game called Infinite Levels. The third was a lesson that introduced rational exponents.

Each lesson took effort and took 3-5 cogens to make a reality, but I found designing, planning, and teaching alongside students exciting. Sure, my students don’t have the experience or command that I do, but that’s what made our partnership work so well. They delivered fresh, practical, and student-forward ideas about content and pedagogy that were often in my blindspot.

Looking forward to next year, the success of our coteaching leaves me wondering: How do my current instructional routines limit my students’ potential for taking ownership of their own learning? What inherent constraints have I placed on their learning without even knowing it? How might coplanning and coteaching with my students break the glass ceiling that exists in my classroom?

The development “on-demand” cogens
In the spring, the consistency and progress of my weekly cogens inspired me to create a multi-day lesson using a series of “on-demand” cogens. Based on the lesson, which was rooted in the historical mistreatment of farmers of color, I recruited particular students who I knew could support the design of the lesson. We met three times over the course of two months. There were so many layers to the lesson that I wrote blog posts after each cogen to help me sort it all out.

The cogens proved to be invaluable not only to the lesson itself, but also to my understanding of cogens and how to employ them. For the last two years, my cogens only happened weekly and were standard issue. While weekly cogens are now a mainstay for me, this new, “on-demand” variety of cogen helped me see how to use cogens to address specific needs in the classroom in a more spontaneous and nimble way.

Next Year
For my weekly cogens, there are two changes I’m looking to implement next year. First, I’m going to make them slightly longer. On the exit survey, many students said they wished the cogens were 45 minutes instead of 30. Increasing our meeting time would enable us to talk about more, hear more perspectives, and get more done. I wasn’t pinched for time this year, but still felt guilty asking the students for more than a half-hour each week. But given their feedback, it seems they desired a longer cogen too.

Second, I’m going run the cogens without offering extra credit. With the cogen culture that I know is possible, I want to try developing it without holding a carrot in front of my students. By offering them extra credit, not only am I inflating their grades (however little it may be), but I may also be assuming that they won’t attend without it. Besides, both last year and this year, most students told me in-person and reported on the exit survey that they would have attended the cogens without extra credit. Removing the carrot will be a big jump for me, as I use extra credit as a the cherry on top when selling the cogen to students, but I’m confident it‘ll work out. I hope this doesn’t blow up in my face.

What if…
The lasting impact of my cogens on my practice lies in how they systematically marry my knowledge of teaching to the learning profiles of my students. Cogens help me tailor my pedagogy to my students, empower them, and keep them engaged in learning. I do wonder, however, how much more impactful my cogens would be if I weren’t the only teacher doing them at my school. Whenever the makeup of my cogens changed, which was about every six weeks, onboarding the new kids took considerable thought and energy. As the year went on, this took away from the cogen’s effectiveness.

But what if cogens were normalized across all (or most) subjects and classes at my school? What if students were routinely a part of cogens with other teachers? This would mean that wherever students go in my school, they would already understand the purpose of cogens. They would know the different forms cogens can take (they need not only happen after class, like mine do). They would be well-versed in how to support their teachers in building community and making learning meaningful.

I don’t think this is a lot to ask of my school. A belief in cogenerative dialogues and a willingness to pilot them in our professional development plan is all it would take. This would give cogens the opportunity to take hold in others’ classrooms, give students the opportunity to be heard, and ultimately have a say in what happens in the classroom. By normalizing these nontraditional academic spaces for students, it would also make everyone’s cogens stronger.

The posts
Since this is the last post in the Meditations series, commenting on all these posts seems fitting. Despite a recent ChalkBeat article that references them, I’m pretty sure these posts were boring to the average reader. They were long-winded, unpolished, and sloppy. Yet, despite being off the cuff, I thoroughly enjoyed firing up WordPress after each week’s cogen and pouring myself into a new Meditations post. It was stimulating and reflective. Long after the students left, the posts helped me replay the cogens and make sense of what we discussed. Many weeks, I actually listened to the audio recording of a given cogen as I wrote.

In what was a long, tiring school year, I’m glad I followed through on my commitment to these posts. They will no doubt be a resource for me in the years ahead.

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The Pre-student

Any good teacher understands the importance of getting to know their students. It’s a vital part of what we do. It allows us to craft our instruction to meet student needs, form bonds that improve learning, and make teaching personal.

While getting to know students who are on my roster is essential to successful teaching, I often wonder about the student who isn’t in my class, but will be in the coming school years. This is the kid I see around school that I don’t know, but who I will eventually teach. What about them?

I consider them the Pre-student.

The Pre-student is any student who is not in my class…yet. They might be that student who walks past me at the same time every day as I greet my students in the classroom doorway. Or perhaps we paths cross regularly at school events or extracurriculars. They may tag along with their friend during my after-school tutoring sessions. The Pre-student could even be the kid who I meet and chat with once while covering for an absent colleague. These are just a few ways that I come across the Pre-student, but at a small school like mine, these kinds of encounters happen a lot. Truth is, any student at my school who I don’t know has the potential of being a Pre-student.

I view the relationship I nurture with each student as a story, a personal narrative that writes itself over the course of a school year. For the Pre-student, the introduction to their story happens before they even sit down in my room. In this way, the Pre-student offers me something invaluable: not having to wait until they’re in my class to establish a relationship with them. Our relationship is a proverbial ice-breaker. It’s a low-stakes opportunity to establish common ground before they enter my class and there are suddenly so many other things to worry about. Through these students is where I catch glimpses of my instructional future.

As a teacher, my Pre-student relations have a nontrivial and underrated impact on the start of the year. By establishing them, it means that, come September, we’re not starting from ground zero. At a time when so much is brand new, it means that we won’t be strangers. Building relationships and getting to know students takes so much effort, so doing some of that work beforehand gives my beginning of the year dealings with students a leg up. Ironically, although they happen outside my classroom, these early relationships also help form the basis of my efforts to build community in my classroom.

What’s also special about these relationships is how they create a context that deemphasizes traditional teacher and student roles. Unlike that of current students, there are no strings attached when it comes to my Pre-students. No grades, no assignments, no nothing. This blissful state of detachment is liberating and frees us to simply connect. This has been especially valuable coming back from remote learning since interacting with Pre-students wasn’t possible or, if they were, happened a lot less frequently last year.

Interestingly, the same dynamic that makes interacting with the Pre-student so freeing is the same one that can make relationships with them feel strange. For me, most of teaching and learning is transactional. Everything I do with my students has some prescribed end goal, some tangible output. If there isn’t a deliverable that accompanies an exchange between me and a student, as is the case with the Pre-student, then it can feel odd establishing such a relationship.

It’s worth noting that when I say “relationship” or “relations,” I don’t mean anything other than putting a face to a name along with knowing a personal detail or two about them. Usually, these long-distance relationships don’t foster anything more than that. For many of my Pre-students this year, I only know their name, an experience we’ve shared, and a salient detail about their life. Initiating such a relationship takes minimal energy; it requires me to simply identify a student who I don’t know and ask them their name. Maintaining it is just as simple because small talk is a necessary part of it. In a day and age when teachers never seem to have enough time for anything, caring for a Pre-student relationship is not a big ask.

As the sun sets on this school year and I look forward to the next, several Pre-students come to mind. I look forward to seeing them again and acknowledging them by name. In a sea of unacquainted faces that will be waiting for me in September, my Pre-students will offer me a needed sense of familiarity. In them, the future will become the present. Our preestablished kinship will not only make getting to know my students more manageable, but also make the start of the year friendlier, brighter, and more hopeful.

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Mathematical Voices, Volume 3

My earliest memory of math is very vague but what I do remember is I was in elementary school, and we would use these dots. They were yellow and red, each side was a different color and we would use that to add and subtract.

-E__, Class of ’24


I know you picked this problem because it seems to be the easiest out of a bunch, but just remember, despite it looking easy, there are still many mistakes you can make. So don’t get overconfident.

-J__, Class of ’24

Everyone uses math in their life and I think it is overlooked and underrated when considering how much we use it in our daily lives. Math will always be around us no matter where we are – not literally solving an equation, but involving what is around you.

I__, Class of ’24

Graphs are like us
With different lengths and widths
Overall, you can see the difference
When you look from within

J__, Class of ’23

Those are excerpts from my students’ writing in our book Mathematical Voices, Volume 3.

Image of Mathematical Voices Volume 3

This is the third consecutive year that I’ve been able to compile my students’ math writing, bound it, and turn it into a physical book. Volume 1 happened in 2020 — the year everything changed. Volume 2 dropped last year — the year of remote learning.

When compared to the previous two, I’m glad to reveal that Volume 3 arrives on a much brighter note. The pandemic still rages on, but there is much to be thankful for. Schools reopened, in-person learning resumed, and so much of what I’ve longed for during the last two school years returned. With my faith in teaching and learning restored, my career experienced a rebirth. There were challenges, but I savored every mask-covered, hand-sanitized minute of being back.

Given the joy I rediscovered by returning to the classroom, it may come as no surprise that I deeply enjoyed my students’ writing experiences this year. Given the historic circumstances that kept us apart, there was so much to catch up on, create, and look forward to. In this way, writing helped us to reclaim both learning and mathematics. The celebratory spirit and honorary tone the book took on opened the door for even more student voice: Despite fewer students on my roster this year, Volume 3 contains more writing than either of the previous two.

My enthusiasm helped steer Volume 3, but it was also powered by the understated value of writing during moments of transition and growth. With the return of in-person learning, I believe that writing not only buttressed my students’ development as learners of mathematics, but also helped us all use mathematics to bear witness to our changing world. So much of who we are is different now. Writing helped us make sense of our evolution not by declaring blind allegiance to content or curriculum, but by instead doubling down on the commitment we have to ourselves and each other. Although Zoom was gone and we were in front of each other once more, it was writing that enabled us to actually see each other again. The details of our lives and the mathematics that runs though our hearts and minds were made discernible to each other through writing. Thus, Volume 3 is a testament to the ongoing humanization of teaching and learning in mathematics, especially during a transitional year such as this one.

An important aspect of Volume 3 that emerged from our return to the classroom is the student editors (I blocked out their names from the above image). This is a major development since, from its inception, Mathematical Voices has sought to magnify students’ mathematical perspectives and the gifts they hold. Each edition strives to upend conventional expectations for the role students need to play in math class while elevating their status from learner and consumer to author and producer. Though I was thrilled to have achieved this and for it to be so well received by readers in the first two volumes, being the sole person compiling and editing my students’ writing didn’t feel natural. After the first two editions, my exclusive rights as editor in a work that challenges teacher-student hierarchies did not sit well with me.

In addition to having their mathematical voices published, I knew that my students should also help determine how they were published. Besides, it is their writing. Thus, for Volume 3 I stepped aside and made space for my students to take greater ownership over their intellectual property and mathematical creativity. Instead of merely filling up the pages of our book, their names are now proudly displayed on its cover, as editors. As their teacher, it’s a privilege to share the role of editor with them and I couldn’t be prouder of what we produced together. To accompany mine, each of my co-editors wrote a Preface to share their own editorial journey.

In addition to my co-editors, the cover this year also has special meaning: It was 100% designed by one of my students. As a stunning creation from one of our school’s most prominent artists, the cover is her artistic vision of her classmates’ writing. Somehow, she found a way to do justice to what readers will discover inside. Looking back at the first two volumes, I realize now that only a student could achieve this. Only a student could capture visually what my students have accomplished in written form. She even inspired me to publish the book in color this year. The accent color inside was chosen as a direct result of the teal/green that she used for the cover.



As my students’ reflections, analyses, and creativity harmoniously collide for a third time, they converge in what is the most complete and student-facing version of Mathematical Voices yet. Inside are their Mathographies, Metacognitive Journals, Poetry, and Math Raps, each special in its own right. Taken collectively, they fill the pages with a brightness that could only be expressed in a year like the one we’ve had. The finishing touch was added by my colleague and good friend Adhim DeVeaux, who wrote the Foreword. Given his journey and our ongoing brotherhood, he was the first and last person that I thought of to open the book on behalf of my students. He did a marvelous job.

Because we finished in-person this year (unlike the last two), this was the first time I was able to actually hand students their copy of our book while they were still on my roster. The exchange happened in class on the last day of instruction. It was an unforgettable feeling to kneel next to each one of my students, thank them for their efforts this year, and hand them a book that they helped co-author. It’s a transaction that I’ve been looking forward to for three years. I took advantage of the opportunity by writing a personalized dedication to each student.

One of the many dedications I wrote in my students’ books

Despite all that’s happened these last few years, I’m proud that Mathematical Voices has lived on and evolved the way it has. It’s no doubt been a beacon of hope and inspiration that has helped carry my battered pedagogy to the finish line each June. I know it’s cliché, but the books have also taught me to dream big. They’ve helped me see that despite restrictive state exams, being a cog in the wheel of a bureaucracy, and a global health crisis, beautiful things are indeed possible. Mathematical Voices removes self-perceived limitations and grants my teacher imagination the permission to reach further. I’m so thankful for it.

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Haiku #11

As an alternative means of capturing my thoughts and reflections, I write haiku about my teaching practice. This is the 11th post in the series.

My students sit for their state test today, the fabled Algebra 2 Regents exam. The pandemic freed me from this calamity for two years, but today, like many things this year, it makes its return. With its reemergence comes the heart-racing build up, anxiety, and obsession on results. There is no bounce back, no retakes. It’s all or nothing. The thirty-seven problems my students will read and respond to in solitude this morning contradicts much of what I work to accomplish as a teacher. This is disheartening.

At the same time, if the length of a school year was mapped onto a football field, the three hours that the Regents exam takes up wouldn’t even amount to a yard. After so many varied experiences with my students, I refuse to give today more attention than it deserves.

This haiku attempts to carry what I’m feeling today.

Return to normal
Alone with thirty-seven
A mere three hours

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