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.

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.


A Broken Chair, A Keepsake

A couple of weeks ago, after an ordinary lesson, a student in first period came up to me at the end of class. “Mister, your chair just broke,” he said. In disbelief, I paused. Huh? What? When?

I gather myself and he casually described what happened and proceeded to hand me the flat, top portion of the chair — the seat. Then he showed me the rest:

He smiled and walked out of the room matter-of-factly as if a dismembered chair is an everyday occurrence. Teenagers kill me.

In my illustrious 16 years as a teacher, I can’t recall ever having a chair fall apart during class. This was a first. Instead of laughing it off and leaving it to the mercy of our custodian, I was pulled to hold on to the fragmented piece of classroom furniture. Something interesting could come of it, I thought.

The next day, after relishing in the uniqueness of my newfound treasure with another class, it hit me: the seat could be a memento for the year. Why not keep it? A chair is a defining physical element of any classroom, but is woefully taken for granted and unseen. After being removed from the classroom for so long, the seat could celebrate our return. It was the perfect way to memorialize our repossessing of the physical space we were forced to vacate.

In the subsequent days, I passed the seat around the room with some Sharpies and asked my students to sign it. The result was a keepsake that ensures I’ll never forget this extraordinary school year and how it restored my faith in teaching and learning.

I’m not sure if I want to hang it or simply lean it up against a wall, but it’s going to be around for a while. When I look at it, perhaps I’ll think about all the students who it supported throughout its lifetime in Room 227. Many have come and gone through the years, and this seat afforded their bodies a space to sit, think, and feel. The seat lived a purposeful, yet overlooked life.

Now, thanks to impeccable timing, it has earned itself a second life as a token of our unapologetic reinhabiting of the classroom. Before the pandemic and remote learning, we thought nothing of the everyday support it provided. After a revival year of service to many, the seat made the ultimate sacrifice and will be overlooked no more. Instead, my students and I have immortalized it and shoved it into the spotlight where it will live its remaining days as a cheerful representation of our comeback.


Shaving Day

Since 2018, I haven’t shaved once during a school year. What started as a fun pact with a student has since turned into a tradition that I’ve come to be known for. I’m the guy whose face gradually turns into that of a caveman as the weeks and months pass by.

My Paleolithic look notwithstanding, the abnormal growth of facial hair that I call a beard is symbolic. I view it as a physical representation of the growth my students experience in my class throughout the year. Given that my beard is an unmistakable part of me, it’s also a physical expression of the personal attachment I feel being their teacher. My beard travels with me wherever I go and, as it matures, becomes a more significant part of who I am, just like my students. I often extend individual hairs from my face during class and jokingly comment how they resemble my students. “This long one is Tatiana,” I’ll say while tugging on a hair underneath my chin. “This one that’s growing sideways on my left cheek? That’s Raul.” By the end of the year, many students playfully ask which hair is theirs. I always lose track.

What’s really special about my beard tradition is that I allow students to shave it off in June. They’re leaving me, and so must my beard. Thus, I bring in clippers, sit in the middle of a crowd of brazen teenagers, and get shaved by my students. It’s a pretty vulnerable state to place myself in, but also terribly fun. There’s nothing quite like getting my facial hair removed by young people whose grades I sign off on every six weeks. This unique and unforgettable experience has become known as Shaving Day. It marks the informal end of the school year for me and my students.

For the last two years, because of the ongoing pandemic, Shaving Day has been held remotely. The spirit of the annual event has been there, but the vibe has suffered greatly. Every ounce of anticipation and exuberance that it accomplishes is lost on Zoom.

Thankfully, that changed today. Shaving Day was back in Room 227. So was the crowd.

A selfie I took just before my students shaved off my beard. The crowd swelled after we began.

The return of Shaving Day was the cherry on top of what has been a beautiful school year for me. The return of in-person teaching brought with it real challenges, but these have been subdued by the magic that came with being back together. As a teacher, it’s moments like Shaving Day that I live for. The community and love that arise from students passing around hair clippers, rooting for each other to take off every last hair, is special. It’s important to our collective mental health, our sense of purpose and togetherness, and can only be produced by sharing space in the same room.

In the end, my students did a masterful job cleaning me up. Other than an excited few, their hands were calm and steady. They weren’t as smooth as a barber, but they took care of me.

Looking in the mirror, it’s no doubt refreshing to be able to see my neck again, but also sad that I can no longer extend a hair from my face and give it the name of one of my students. Being back in the classroom added weight to my students’ development this year and made Shaving Day 2022 particularly meaningful. My students’ growth this year was like no other. So was that of my beard.


Meditations on a Cogen (No. 27) • Thursday, June 8, 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 27th post in the series.

After my last cogen on May 19, it didn’t feel right to onboard another cohort. There were just four weeks remaining in school. At the same time, it seemed unnatural to finish there. Thus, right after May 19, I began planning an end-of-year cogen reunion. I wanted to bring back all the cogen students from the entire year for one last cogen. Over the course of the year, our half-hour cogens resulted in over 13 hours of talking and generating ideas to improve the class. If for no one else but myself, the reunion was a way to bring closure to this unforgettable experience.

Of course, the last cogen of the year had to be celebratory. These students stepped up for me in major ways. While most of my colleagues have cursed this year up and down, I found myself in a great mental space all year. Undoubtedly, my cogen had a lot to do with this and I’m grateful to them. To show my cogen students the love, I made party bags, ordered pizza, and set up a raffle for our last gathering. Two weeks before, I gave out invitations.

Pizza, trivia, and a raffle
As the students arrive, they walk into the room and find a large table with seating for 23 (the number of cogen students this year). They also see the party bags, pizza, and drinks. Music fills the room with rhythm and melody. I welcome each student individually and hand them a raffle ticket.

After everyone gets settled around the table, I utter some opening remarks, but mainly just thank them for coming. I spent a lot of time this year scared to death that my students would forget to come (or come back) to our cogen each week. With 21 smiling faces in front of me today, I’m relieved that I’m only missing 2, and 1 of them is a senior who isn’t in school because of prom. I don’t talk long because I see their eyes eagerly shift from me to the unopened pizza boxes in the center of the room. We eat.

In addition to eating pizza and enjoying party bags filled with candy, I want our time to be reflective. It should afford us a chance to think back on the year through our cogens and remember how far we’ve come — in a fun way. I’m the king of boring and almost make today’s reunion dull and uninspiring, but at the last minute I come up with an idea to do trivia and a raffle.

Everyone earned one raffle ticket simply by showing up. I decide that additional tickets can be earned by answering trivia questions based on the cogen. In re-reading my cogen summaries, here are some of the questions I tapped out one day during lunch:

  • What is “cogen” short for?
  • Who can name all members of their cogen?
  • Who were the only cogen students to continue to attend after their six-week tenure was complete?
  • Which group of cogen students was the only one to meet on Zoom?
  • Who first mentioned the idea for bingo?
  • Who attended basketball practice and the cogen on the same day?
  • Who first mentioned the idea for DeltaMath review for exams?
  • Who created the game board for Infinite Levels?
  • Who designed the opening slide for the lesson on fractional exponents?
  • What fruit did I leave in the snack box over-ripened during a school break to make everything in the box smell like it?

I’m generous with the raffle tickets and dish them out to anyone who comes close to a correct answer. It’s lighthearted and fun. Good memories and laughter abound.

For the raffle prizes, I picked up several inexpensive party favors from Five Below. I also raffle off two cans of Pringles and a bag of Takis. We even raffle off the last three slices of pizza. I’ve learned that it doesn’t take much to make a raffle exciting; the rush that comes with winning something is usually enough. This proves true with my cogen students. Ticket after ticket, we have a blast.

It’s a wrap
We call it a day and depart from the table one final time. The students leave enthused and fulfilled. So do I.

A shot of me and all of my cogens students around the table one last time


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