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Entering last school year, I knew it was going to be a roller coaster. So much of what happened was unpredictable and unclear and unstable. To help chronicle my adventures as a teacher during this time, I decided to write two humble sentences each school day and publish them here on my blog. The goal for these sentences was to capture some of the impressions, feelings, experiences, or thoughts I had each day. These sentences were small, but mighty. Discrete, but representative. Ordinary, but special. Most of these reflections were composed in a haste and often forget about until days later. I posted them each week and called the series My Two Cents.

Today I went back and reread my sentences and felt like I was reliving an unforgettable year. Here are some standouts. There is at least two sentences from each week. They are organized sequentially.


September 18, 2020
Midday, I took a walk in the park as a result my extreme disappointment/anxiety with my school’s “you must turn on your camera” policy.

September 21, 2020
Teachers teaching students in empty classrooms throughout the school. By far the most bizarre first day of school ever.

September 24, 2020
Feeling isolated in both body and spirit. I have no idea who I’m teaching; the future seems bleak.

October 5, 2020
They have free breakfast and lunch for staff now. I had both, yum!

October 15, 2020
I fell into a discussion today with my eighth period class which let me know that, despite our long distance relationship, we are in fact making progress. Whether it is pumpkin spice, the Lakers, oranges, velvet red cake, or Ninja, I sense that the glue is forming.

October 16, 2020
I had anxiety about my cogen not showing up today, but they did, and it was great.

October 26, 2020
Responding to email is more taxing than I could ever imagine. 

October 27, 2020
Out of sheer mental exhaustion and frustration, I find myself bypassing meetings and forgoing previous commitments. This is disappointing, but strangely satisfying; what must be done, must be done.

November 4, 2020
Anxiously refreshing both the CNN a NY Times homepages frequently throughout the day. At a loss for for motivation, I did pushups in 9th period after a student gave a thoughtful response to question.

November 9, 2020
The beautiful weather enticed me to teach three of my classes from a bench near my apartment building; having leaves fall on my face as we discussed the unit circle was refreshing, but wiping bird poop off my keyboard was not.

November 10, 2020
After a student said “Happy 50th Birthday” to me, I ran into his room (which was across the hall) to give him a playful piece of my mind; you deserved it A! Ninth period absolutely made my day when they sang happy birthday to me.

November 18, 2020
I somberly learned today that, because the city’s 7-day infection rate rose above 3%, the schools were closing. I get why, but I can’t deny how disheartening and demoralizing it is to be ripped from the comforting walls of my classroom yet again — even when there are no students present.

November 24, 2020
Another roller coaster. I barely taught in eighth period today before realizing that they needed a mental health day; I wish I had the answers they needed.

December 1, 2020
I shaved my head in 5th period. That was random and fun.

December 9, 2020
I added five new call-and-response, virtual handshakes to my collection; one was connected to the first snow of the year. My 5th period class and I created an impromptu masterpiece.

December 21, 2020
My crusade for student engagement resulted in many minutes of silence today in both 1st and 9th periods. I get frustrated as hell, but, right now, who can blame them for wanting to hide in the shadows of vulnerability and learning?

January 5, 2021
In a random act of engagement, I had first period vote for the shirt that I was going to wear for the entire day; they chose a denim button-up.

January 6, 2021
Feeling defeated and isolated, I still find myself leaving my camera off during certain staff meetings. Played an uplifting game of Ultimate Tic-Tac-Toe with two students during office hours.

January 12, 2021
A rough day; students weren’t learning, I struggled to teach, nothing worked. By the end of it, I found myself questioning so much of what means to be a teacher.

January 26, 2021
In helping a student with makeup work, I realized how uncomfortable it made me to ask him to complete missing assignments…given that I knew next to nothing about him. I told him this and asked him to tell me something meaningful about himself; he told me he liked soccer.

February 5, 2021
Reflecting on my week, I held back tears in the afternoon as I wrote several of my students emails expressing gratitude for their hard work, willingness to participate, and the connections we’ve created during remote learning this year. Having finally made time to do this, it was extremely cathartic; when some of the students replied with kind sentiments of their own, I felt the tremendous weight of the year land on my shoulders and couldn’t but get emotional.

February 11, 2021
In ninth period, it made me happy to drop two close friends in the same breakout so they could talk to one another; given how foreign my students have been to me this year, the fact that I knew they were close friends and was able to act on it was an unusual feeling that I appreciated so much.

February 25, 2021
For Black History Month, I had my students research a Black mathematician of their choosing and write a short profile of them. Today, after one of my students emailed their mathematician to see if she would be a guest speaker in our class, one of these mathematicians (Dr. Lauren L. Thomas) visited my fifth period class and shared her story with us.

March 10, 2021
Found myself slowing down several times in my classes to express deliberate amounts of patience; my students weren’t readily offering up responses, but I tried to remain empathetic. In the past, I would have done this in a way that was passive aggressive, but today I was more understanding — I genuinely waited for them to be ready to engage.

March 15, 2021
Our collective struggle was on full display in our grade team meeting after school; an extended moment of silence swept over the group as we contemplated next steps in helping our most struggling students.

March 18, 2021
Today was the first day back in the building since late November; getting ready in the morning triggered some “first day of school” vibes; seeing colleagues in the flesh filled me with a renewed sense of hope and positivity.

March 22, 2021
Today was the first day back at school with students present. Out of nowhere, access to a whiteboard and dry erase markers launched me into a dizzying state of excitement while introducing my students to complex numbers; at the end of one of my classes, a student remarked sarcastically, “Mister, this was one of the best lessons ever!”

March 24, 2021
Because of in-person scheduling headaches and space constraints, I’m now teaching my ninth period class in the gym. Today, after HR (who hasn’t given one answer over the mic all year) voiced several correct responses in a row, I ecstatically grabbed a basketball from the utility closet and made a layup at nearby basket in her honor (not going to front: I was so happy that it took me two attempts).

April 7, 2021
Had a pre-observation meeting with my colleague whose doing the case study on me for her grad class. I admitted to her that I’ve felt a bit exposed and insecure from all the attention she’s given me these last few weeks; my thoughtless, mediocre teaching has been on full display to someone that I deeply respect.

April 14, 2021
I held back tears in conversation with a colleague about the end of the school year; I realized that I’m concealing a lot of dismay that will probably come back to bite me in June.

April 16, 2021
Had a thought-provoking conversation with two colleagues and a student about The Glass Castle this afternoon…interestingly, the student informally enticed all of us teachers to read it individually (both last year and this year) and then got us all together to chat about it for 45 minutes; it was an incredibly unique way of connecting during these crazy times.

April 19, 2021
Thoroughly enjoyed my lunch banter with some students in the courtyard (SHEESHHH!). My morning walks with BD are routine and therapeutic; I see them as an investment in self-awareness and self-care.

April 22, 2021
Still feeling defeated by remote learning, I dragged myself through my classes, and my students sensed it. I’m coming to terms with what it means to have sustained this level of professional stress for this long; today was really hard, this week has been horrible.

April 26, 2021
Gave a public apology to my students at the beginning of class today. Owning up to my weaknesses when it comes to remote learning, I apologized for my sour, emotionally-absent attitude at the end of last week; I’m not sure how it was received, but it had to be done — they deserve better.

April 29, 2021
Thrilled when several students and I got to play basketball in the gym during lunch. It was a socially distant game of 500 with everyone wearing masks and surgical gloves, but I had the time of my life.

May 10, 2021
Out of no where, smack in the middle of 5th period, I picked up my laptop, sat next to one of my students in an adjacent room, and co-taught the class with him via Zoom.

May 17, 2021
In 8th period, we shared a cool moment of connection when a student, who is remote but was in school for one day, visited the smears I created in the hallway which represent our figurative children. It’s crazy to think about the levels I’ve gone to to engage my students this year.

May 19, 2021
Myself along with six other staff members went to visit the grave of Malcolm X after school at Ferncliff Cemetery.

May 27, 2021
Disappointed at the end of the day when I removed a few students who were non-responsive to my attempts to engage them. I was frustrated based on the lack of engagement from the previous class and didn’t lead with compassion.

June 1, 2021
The first day of the Math Olympics, an event that we’ll use to close the year which was co-designed by me and my cogen. To add an official feel to it, we held the opening ceremonies; we played the Olympic music, lit the Olympic torch, had a parade of participants, read-aloud some math writing from the year.

June 10, 2021
The last day of classes and an emotional day for me. For me, saying goodbye to my students — my comrades, my companions, my battlemates — is an act I take seriously that honors the space they’ve taken up in my life for the last 10 months; this year I found it liberating, but sad.

June 11, 2021
Today was the annual cutting of my beard. I fully expected to do it alone on Zoom, but, in a twist, two students walked in the classroom just as I was getting started and ended up doing it all.

June 15, 2021
This afternoon, with remote learning knowing that its days are numbered, it lashed out at me one last time; with the sun in my eyes, I smoothly dodged its noxious thrust and extended grace to a student in need — I consider it my watered-down version of a federal pardon.

June 25, 2021
Alone with the echo of an empty classroom, I packed up my crap for the summer and said farewell; what a year this was. Tried my best to consolidate my gear into a few regions of room 227 and clumsily posted “Belongs to Mr. Palacios – Please do Not Remove” signs on them.


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Yesterday was the last day of school. Thinking back on the last 10 months, I have a confession. Despite limping though most of the school year, I’m going to miss something about it. It took me a long time to realize this, but it sunk in one day while in conversation with my first period class back in May. It was an uneasy realization. It made me uncomfortable at first because I thought nothing about this stressful year was missable. I questioned my feelings. I didn’t want to believe them.

What will I miss? It’s definitely not remote learning. That experience was terrible. There’s no need to unleash my contempt for it now. It has felt my wrath all year. Now is a time for letting go, a time for healing.

I will say this though. Remote learning was a war. It consisted of many battles fought by me and my students every day. Small battles like clicking a link, larger battles like turning in a project. These battles were waged at home, on the bus, at the park, in school, and wherever else we accessed Zoom. It wasn’t merely academic blood that was shed during these battles either. There was social and emotional blood too, blood that is far thicker than any curriculum or lesson or breakout room. Through these battles, through this loss of blood, we fought to learn and make meaning. We fought to understand and persevere. If I’m being real, we fought because we had to.

This warring wasn’t me vs. them. It was never me fighting with students to get them to turn in assignments or show up for class or turn on their camera. Some teachers have felt this way, but I never did. In my eyes, there was only one enemy: remote learning. We were all struggling against it. We were side-by-side in the trenches, fighting together. We all have scars that will leave us changed. No one was exempt from this. Our objectives may have been different based on our roles in the war, but we all had a shared goal of survival.

And that is what I’ll miss. It won’t be the war itself, but instead the companionship and sense of purpose that I shared with my students as we survived it. I’ll miss our collective struggle. I’ll miss our uncompromising laugher in the midst of fatigue, our fierce efforts in the face of hardship, our inexplicable togetherness during a year of academic isolation. I’ll miss being comrades with my students during this once-in-a-lifetime event. When my teaching career is over, it is this subset of students who will hold a special place in my heart. I may not know what they look like, but I will remember them and what we accomplished this year.

Together, we made something out of nothing. Our shared struggle was a light in a year of so much darkness. I’ll miss it.



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You’re lucky to overcome your weaknesses, especially one so fundamental. Mine is so hard to overcome despite how easy it seems. Little mistakes or a lack of understanding can shift a problem entirely. It’s been my issue for a long time and basically became a habit. Anyways, one of the weaknesses I’ve successfully overcome is fractions. Besides the occasional mix up, I’m golden.

Some of my favorite topics in math would be the classics. I’m thinking of things like basic arithmetic. Why? Because they are not as complicated as the present lessons. Despite the lessons taking more of time, it takes up a lot of space in my precious books. Those pages can be filled with stories or fan-fiction, not factoring or distributing. Still, I’ll take distribution any day, factoring may seem easy but there’s so many different types that I can’t keep up. No. Just no.

One of my students (in a letter to her math penpal)

Last August, before the school year began, I wrote a blogpost on the notion of mathematical penpals. It was essentially a public plea to see if anyone was interested in trying out the idea with me. My thinking was that we’d come together, have our students write mathematically-themed letters to each other throughout the year, and possibly have them meet on Zoom at the end of the year. If you’ve read wonderful book Mathematics for Human Flourishing by Francis Su, I imagined our the students’ letters would be, in spirit, like the ones between Su and Christopher Jackson.

Despite my excitement for the idea, I dampened my expectations. We were in the midst of a pandemic with no vaccine in sight. Teachers everywhere were fastening their seatbelts for a school year unlike any other. The idea math penpaling would have been outlandish even in regular circumstances, let alone the one we found ourselves in. I hit publish on the post and waited.

How many people responded to my plea? A heartwarming — and surprising — 17.

I emailed everyone to see if we could establish partnerships. Most teachers were from the northeast, but there were a few from places like Illinois and Minnesota. Thankfully, Sarah Furman and I found common ground in terms of what and who we teach and decided to give it a go. Sarah teaches Algebra 2 in upstate Michigan and the differences between our students (e.g. race, geography) was a huge draw for us. She’s a 21-year vet who is thoughtful, reflective, and eager.

We had plans for the students to write each other every 4-6 weeks. That didn’t happen. The logistics of an unpredictable school year kept us guessing and caused longer-than-expected gaps between letters. We were in-person one week and remote the next. Our students were all over the place. So were we. In all, our students ended up writing two letters to their penpal. I’m very proud of this.

And the actual exchange of the letters? How did that work? After pairing the students up, we hoped that the letters could be handwritten and mailed to each other, just like old-school penpal letters are. That also didn’t happen. Who was I kidding?

The letters turned out to be a combination of handwritten and typed correspondences. Students who hand-wrote their letters scanned them and sent them to us. Students who typed it shared it with us. Because of the remote learning mess we found ourselves in, we didn’t even attempt to mail them. Instead, we just dropped them in a shared Google folder and called it a day. The other person retrieved the letters and distributed them to their students. It worked out well.

Outside of logistics, one of the more interesting challenges about the experience was figuring out engaging and worthwhile prompts for the students to respond to in their letters — especially when it comes to mathematics. Some of the math-themed talking points I threw at the kids included:

• What is your earliest memory of math or learning math?
• How do you learn math best?
• What parts of math are challenging for you?
• What is/was your favorite topic to learn in math? Why?
• Tell your penpal about our A Mathematician and Me assignment. Share your mathematician and why you chose them.

The main goal of this project was to build community between students through letter writing and mathematics. I’m not sure to what extent Sarah and I achieved this, as students may have viewed it as one of the many random things that happened in a crazy year, but it was a lot of fun trying. For what it’s worth, Sarah and I both observed the genuine interest and enthusiasm amongst the students when the letters “arrived” and when it was time to reply. That counts for something, right?

We both agreed to try this again next year. We’re hopeful that the experience will be more meaningful for our students, allow them to write more letters, and just run a lot smoother. I’m looking forward to this and have many wonderings. Like, how might we safely connect our students over social issues and current events? And how might mathematics be a vehicle for this? What about problem solving and problem posing? Could we have students jointly solve problems through their letters? How might students debate mathematics through their writing? Could their letters make it into a future volume of Mathematical Voices?

So many juicy questions to sit on.

A fascinating subplot of this experience is that, in all of our planning and organizing for our kids’ math penpaling this year, I’ve still never seen or spoken to Sarah. It’s strange, but I kind of like it that way. It’s as if she and I have been penpaling about penpaling. It’s fitting. And beautiful.


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Last year my Algebra 2 students and I created a book using their math writings from the year. We called it Mathematical Voices, Volume 1. I wrote a blog post about the experience, which was birthed at a math conference I attended several summers ago. The goal of the book was to lift up my students’ mathematical ideas, perspectives, reflections, and creative writings in a way that was purposeful and public. I saw it as their contribution to mathematics.

Because of the pandemic and my students being ripped away from me in March, there was a serious lack of closure to the school year. This was compounded by the fact that I wasn’t able to give the book to its authors and rightful owners: my students. Despite my disdain and incertitude about how the year ended, the book excited me because of what it represented.

My delight for the book lingered all summer. I even boasted about it to my new students on Zoom on the first day of class this year. Although it was one-of-a-kind, it felt like something bigger. So, despite being remote, I implored that we were going to create a Volume 2 and it was going to be amazing. Frankly, I knew no better — I was simply riding the wave of a new school year.

Sadly, that bright feeling got buried once the darkness of this school year revealed itself. As we trudged our minds and bodies through the fall and winter months, my lofty plans for compiling Volume 2 got pushed waaaaaay down on the list of priorities. While their writing was therapeutic for me, a soothing balm for the wound created by remote learning, a coping mechanism for interpersonal gulf created between my students and me, I found myself merely surviving for a good part of the year.

Though the idea for Volume 2 got buried and lay dormant for most of the year, it never died. When spring arrived, bringing a COVID vaccine and hope along with it, I caught my second wind. In early April, I slowly began combing through the kids’ writing from the year and drafted a manuscript. I started this process a month later than I did last year and figured it would be a crunch to finish it by June, especially with the cumulative stress of 15 months of remote learning weighing me down. I ended up spending the entire month of May (and part of June) hunched over my computer in order to complete it. The result was beautiful.

The overall structure of the book is the same as last year. Much of the sections are the same; included are the students’ mathographies, metacognituve journals, and math poetry. To spice it up, though, I made some changes.

First, with the guidance of my co-generative dialogue, I scaled up our math poetry assignment to include Fibonacci poetry, free verse poetry, and math raps/spoken word poetry. Last year the only option was haiku. This was a major upgrade because of the ingenuity it fostered amongst the students. I also included a new section for the A Mathematician and Me task from February. I attribute this task to my dear colleagues Stephanie Murdock and Brother DeVeaux, who came up with it early in the year. And after writing many open, antiracist letters to each other last summer and fall, I asked Stephanie to pen the foreword, which she supportively did. The last change was unrelated to writing and instead came in the form of drawings. They were based on an assignment near the end of the year where I asked my students to make a hand-drawn sketch or collage representing their year in math class. Their sketches are sprinkled throughout the book. Despite all the writing they did, sometimes art can say things that words cannot.

In addition to what physically appears in the book, what I came to appreciate about this anthology in its second iteration is the collaborative nature of it. It is a joint effort between my students and me. I compile it, yes, but my students do the heavy lifting with their writing. All but a few of my Regents-bound students are included in it and, this year, one student even helped design the cover. In my eyes, this all amounts to us producing something together. It’s communal, symbiotic. It’s their voices, but our song. In this way, Mathematical Voices embodies the decentering of authority and the flattening of hierarchies in academic contexts, which includes the pages of a book. It’s a co-created solution that bows emphatically to our collective humanity and shared responsibility we have to learn from each other and grow. I am proud of this.

In a school year filled with relationships built around Zoom, virtual handshakes, and all things Google Classroom, creating a physical book seems strange. All the work that my students submitted this year was digital, but yet here is a physical manifestation of their mathematical selves. Unlike everything else from the year, you can hold it. You can touch it. You can pick it up and give it to a friend. You can read it without straining your eyes. Viewed from the perspective of a mathematical function, this year’s inputs should not have yielded this particular output. There is great deal of symbolism in this paradox that I have yet to fully wrap my head around, but do greatly appreciate. Thankfully, unlike last year, I was able to mail each of my students a copy of our book.

When thinking about last year and Volume 1, what I realize is that this year it wasn’t just about lifting up, validating, and publicly circulating my students’ math writing, it was also about finding a way to document this once-in-a-lifetime school year. The book strives to capture our persistence and refusal to let remote learning win. In many ways, the reflections and other writings function as a mathematical time capsule for me and my students. By bounding their thinking and printing their perspective again, I hope this edition serves to chronicle a most unique school year in a most unique way: through mathematics.

In the end, despite not being able to share a physical space with my students and feeling estranged from teaching and learning, my only hope is that Mathematical Voices, Volume 2 can serve as a humble reminder – at least to myself – that my students were not mere screen names or profile pictures this year. They were humans. Behind the screen and on the other side of that email was a young person. A young person with a story. A young person with thoughts, feelings, beliefs, ideas, struggles, and triumphs. A young person, who, in many ways, I’ll never know. By the time they get to see their name on those pages, our paths, both mathematical and otherwise, will have diverged. With our humble book, which represents our final attempt to repossess a school year co-opted by COVID-19, my only wish is that it honors them in a way that I was never able to on Zoom.

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