An excerpt from an essay

I received an email from a former student (R.L.) who I taught a few years ago. She’s a senior now in college and is taking an education class. She wrote a paper that she wanted to share with me, because half of it centered on her time in our Advanced Precalculus class during her junior year. I know I’m a warm-demander teacher (or at least that’s what I strive for). I try to make my classes a little bit harder than kids think they can do (but exactly at the level I know they can do). Reading her essay made me feel a lot of things, but I love how in so many ways she captured some of the things I strive for and do in class. The fact that she noticed them and remembered them years later means a lot.

She said I could share that part of her essay here on my blog when I asked. I like to archive good things in teaching, and this is something I’d like to archive. So here it is.


I’ve been thinking about ways that coaching, questioning, and telling played out in my education at Packer. Packer was the ideal setting for these methods of learning, as we had small, seminar-style classes and teachers with the capacity to work with students one-on-one and develop individual relationships. Upon reflection, my eleventh grade math class, Advanced Pre-Calculus with Mr. Shah, exemplified coaching, student-telling, and questioning, and also included exhibition-style projects.

This was the hardest class I took in high school. The content was very challenging, and Mr. Shah’s approach to math drove me crazy. Mr. Shah sees math as a creative field, one that demands critical thinking and a deep conceptual understanding of topics that many see as surface-level and robotic. Mr. Shah’s packets used broad questions as the benchmarks of understanding, pushing students to explain concepts using their own words. His problems had a playful tone, and we spent most of our class time working through material in small groups while he played music in the background and buzzed around answering questions and challenging students to think more deeply. Why are conics important? What is the meaning behind this geometric sequence? Why did you choose to solve this combinatorics problem this way? At the time, these questions made the class a nightmare for me – the content was already challenging enough (it was in this class that I failed the only test I’ve ever failed), and the way he forced us to think about it made it even harder. But, after reading Sizer and thinking about coaching, telling, and questioning, I see what Mr. Shah was doing.

In addition to trying to make math more fun and meaningful, he was pushing us to develop mathematical skills that built upon each other. Sizer writes that the “subject matter chosen should lead somewhere, in the eyes and mind of the student” (Sizer, 111). The curriculum progressed when we made “mathematical discoveries,” and those discoveries led us to mastery of complicated skills and a deeper understanding of concepts. Mr. Shah never talked at us. Discoveries came through telling, but, it was table-mate to table-mate telling. As the teacher, Mr. Shah’s role was to encourage our discoveries and offer support as we worked through our own questions and explained concepts to one another.

Through his approach, we were also practicing broader skills like critical thinking, creativity, perseverance, and thoughtful reflection. He centered the class around group work and we spent a significant amount of time reflecting on our individual contributions to the group and the strengths and weaknesses of our team. I honed these skills in my other classes at Packer, and I am sure they are part of my academic success at Tufts. Mr. Shah’s class was enormously challenging for me, but in writing this paper I have found an appreciation for his approach, and I know I owe him a thank-you.

Lastly, Mr. Shah also incorporated exhibition-style projects into our curriculum. He called them “Math Explorations,” and we had to do four of them throughout the year. They were not as big as these example exhibitions and they did not center around a presentation [like someone mentioned earlier in the paper], but they provided an opportunity for individual exploration in a subject area of our choosing. Being the English-lover I am, for one of my Math Explorations I wrote a series of math poems – a sonnet, some haikus, an ode, and a free-form poem. I was proud of these poems. It was exciting to take ownership in a class in which I often felt overwhelmed, and pursue something that made the content relevant to my interests. These projects were empowering, and they helped me feel connected to the material and confident in the class. If this is the power of exhibition-style learning, then I’m in full support, because the takeaways made a difference in my learning. If only more schools had teachers like Mr. Shah and the resources and the capacity to make classes like his more widely available.


At the end…

This is an archive of my day yesterday.

We’re nearing the close of classes with online schooling. It isn’t the end for me, with narrative comments and college letters of recommendations looming. But in terms of meeting with kids, there’s not much left, and I had my plans in order — at least well outlined in my head for some, ready to execute for others. But I felt anxious. On Thursday afternoon, I finished my last meeting, had finished my prep work for Friday, and was in a good place. But my heart was beating, racing, and I didn’t know what it was. That anxiety usually happens to me when I have something impending looming (like a stack of tests I need to grade, or a challenging email I need to write or conversation I need to have) and I am stressed about it. But I had none of that. But my heart was racing. I was a little short of breath.

I tried to relax, but I couldn’t focus on reading. My mind quickly wandered. TV worked.

A few different things had been thrown at me in the past few days. I wrote to teaching friends, as I processed and as a way to vent: “The problem about keeping boundaries in teaching is that boundaries I draw to protect myself often involves saying no to a kid who I can prop up by saying yes. I recently wanted to say no to four requests — some entitled, some respectful. I don’t have much left of me to give. I couldn’t. When it comes to me or them, I pick them. Again and again. As I slowly burn out. Teaching is hard. It will never not be hard. Because we give so much of our selves, emotionally, mentally. I don’t always want to do that. But I (honestly) don’t know how not to.” The same thing happened two weeks ago with college recommendations. I teach two junior classes. All but three juniors asked me for a recommendation. I felt utterly deflated. And honored. If someone could be in my head as I try to craft these, you’d understand the deflation. They not only take me forever, but they are emotionally draining because I want to capture my kids with integrity. I told my classes I had to limit them and I hoped they understood. I tried to be transparent and vulnerable with them about what doing this meant for me, and why I had to limit them, because they deserved that. But when I read their reflections, I couldn’t say no. I thought about the Giving Tree. I’m not that self-destructive. But for a moment I martyred myself.

The next day I woke up at 6:40. to an alarm. I quickly showered so I could check my kiddos nightly work. With online learning, I go through what my kids worked on each day before class to give individual feedback, but also know what we need to talk about together as a class. Although my heart wasn’t racing, I still felt… tight. Wound up. I was off. My kids were doing really strong thinking with hyperbolas and that made me happy. Some kids really struggled applying the quadratic formula for the first time and that made me wonder about my approach this year. I tried to address that by breaking it into two pieces and having them work on those separately — but maybe there was a better way? I woke up earlier than normal because I had an optional meeting I wanted to attend at 8:15am about the finances of our school. Our new head of school wanted to share where we were at — trying to be transparent about the decisions being made by the board and the school. I appreciated that, and her openness to talk about it.

Yesterday evening all advisors received an email saying that instead of meeting in our advisories, we would be meeting as an entire high school to give some time to discuss the recent tragedies involving Ahmaud Arbery  Christian Cooper  George Floyd  Nina Pop. I hadn’t heard of Nina Pop. Her name was misspelled in the email we received. Both of those facts together speaks to something about me, and something bigger than me. I entered the zoom meeting and scroll through 16 pages of faces. More, maybe. Tired, yawning. And then the meeting begins. It was planned to be a 20 minute meeting. It starts with Feel. Reflect. Act. Grow. Repeat. 

Two hours later, we’ve heard the voices of students and faculty. Anger and frustration about teachers not bringing up these current events, and in general black/POC issues, in classes. About the ability to check in and opt out depending on skin color, and whether things should be optional. Calls for people to go to march at Foley Square at 4pm. Kids requested breakout rooms early on, and I was assigned Breakout Room 22 with eight students, seven of whom I had never met, nervous about facilitating a conversation about race that I hadn’t prepared with strangers. My heart sang as my colleague and friend Monika was also randomly placed in the room. We launched the conversation with a focused question, but that didn’t work, so we went broader to “how are you feeling?” Kids started talking. At times, they were discussing with themselves. At times they were asking the opinions of the two adults in the room. One asked our thoughts on ACAB. I had never heard of that and with slight trepidation (should I know?) asked. “All cops are bad” two students responded. It’s a thing, they said. Reductionist. Monika was more eloquent and nuanced in her response. We talked about multiple narratives (it’s complex). The reasons for violence in riots/protests, and if it pushes things forwards or not (it depends). There are no pat answers and easy fixes, everything is layered, structural, cultural, historical. This is one slice, one slide, of a much longer movie. We end with each saying a word, phrase, or something we’re thinking about. I say “over-intellectualizing.” We move back to the main discussion room and I hear our high school principal talking. We’ve gone past first period. I’ve already emailed my kids in my second period class saying we’re not having class and to stay in the meeting. I look and see over 300 students and faculty are in the room. Faculty and students speak. I don’t know how they avoid talking over one another with 300 people, but somehow they do. Outrage. Questioning of the school curricula. Frustration with the statistics around black lives. Calling out hypocrisy. Asking questions about whose responsibility it is to educate kids on race — some say kids are responsible for themselves, some say its the school. Discussions about cultural appropriation. Sharing of feelings. The importance of being called out and of calling out. About the difference between being a liberal and an anti-racist, and being a liberal doesn’t make you an anti-racist. About the conflation between being liberal and being “good.” About being called out and accepting that as a gift. I think to myself that some of these kids must have read White Fragility. A concurrent conversation happens in the chat, twisting and turning with the vocal conversation. I keep silent. I like spaces for kids to talk. I don’t often know what they think about. This isn’t a “conversation” in the sense that we’re talking to find a goal. Right now, we’re talking because people need to talk because they have a lot bubbling up inside. We all have a lot of bubbling up inside. That… that was when I knew why my heart racing yesterday.

Our high school principal ended the meeting before our third period class. It was two hours from when we started. I open my zoom meeting for our Algebra 2 class. Obviously we aren’t doing math today. As the large meeting was winding down, I brainstormed what would make sense and gave my kids some options. They chose to continue the conversation. Unlike in the first breakout room I was in, this conversation was mainly had by kids, where I tried to give them space to speak and I listened. I took notes. They were in the same grade and class and knew each other. They didn’t turn to me for answers. I don’t have answers. I interjected sometimes, and then stepped away. When class was officially over, they were still talking, so I told them they could leave if they wanted or continue talking. Most wanted to stay and so we held court for 20 more minutes.

I was late to a book club I had scheduled during lunch. I emailed the student leader to start it up without me. We were reading Hannah Fry’s Hello World: Being Human in the Age of Algorithms. We were two adults and five kids. The conversation at the beginning stayed close to the book — which also had some themes on race — but very occasionally would go back to something from the discussions the school was having. It was fun and a lighthearted discussion with smiles. We had finished the book, and the scheduled time we had together was done, but we all continued chatting, and the conversation turned into questioning about what teachers think about lots of things. It morphed into an unguarded and vibrant conversation, one that I love having with my friends when they ask a lot of questions showing interest in what I do. I felt close to these kids, and appreciative for them. I don’t know much about what students think about, and looking back I wished I had asked students questions too. It was 12:50 and I had a class at 1:15 so we bid each other farewell. My eyes were literally vibrating at this point. I had been emotionally run raw and I had been staring at my computer screen since 8:15. All I had ingested was a cup of coffee. I heated up some frozen leftover macaroni and cheese and wolfed it down. I emailed a student I was supposed to meet at 2:20pm — an important meeting so I hated doing this — to reschedule.

I entered my last class. At this point I had very little more to give. I had very little left I could discuss. I told kids I had been on zoom all day, and I had been engaging all day, and I needed a 10 minute break to do math and recenter myself. So I did math with them for 10 minutes. And then instead of giving my kids a choice like I did for an earlier class, I made the decision that we’d read something together. I pulled up Francis Su’s keynote address to the Mathematical Association of America — his farewell address as president. A section of this address is on “Justice.” It is centered around this quotation by Simone Weil:

“Justice. To be ever ready to admit that another person is something quite different from what we read when he is there (or when we think about him). Or rather, to read in him that he is certainly something different, perhaps something completely different from what we read in him.  Every being cries out silently to be read differently.”

We read it popcorn style. Someone would read a paragraph or two and then stop. Someone else would jump in. I hadn’t read this in a while. I have the book based on this speech that Prof. Su wrote afterward, after his speech went viral in the math community, trapped at school, unread, sitting on my desk. As kids started reading, I remember hoping it would speak to the events of the day at school. I hadn’t time to review it to see if it did. Everything I was doing today was on the fly. The section did, at least to me. It was longer than I remembered. It took a while to get through, or maybe I just lost sense of time because I was exhausted. When I read, every so often, I forgot words or mixed them around. My eyes were jumping around the text, unable to focus. At the end, I told kids if they wanted space to talk about the events of the day or the piece we just read, we could have it. Or we could call it an end. I consciously tried not to make it sound, when giving the options, like I really wanted to end, so they knew they truly had a choice. We called it an end, and I was grateful. I was empty.

I posted the nightly work for my classes from today. I shared with them some articles, including  this piece on Edray Goins, which I remember having my classes read last year and gave us a lot to talk about. And this keynote speech by Marian Dingle about “centering.” I apologized in my google classroom post for not bringing in more conversations about race, gender, and other -isms into class, and said it wasn’t just the jobs of the history and English teachers (something kids had brought up in the conversations).

I had 45 minutes to lie on my couch and close my eyes. I thought about how much the world has changed since I was in high school. I thought about how much the kids we teach are aware and know and care… and I wondered if the non-POC kids and faculty really cared in their hearts or if they cared because they were supposed to care. I wondered if I cared in my heart or if I cared because I am supposed to care. I felt the weight of guilt — trying to parse what I’ve done to help my kids know they’re “being read” — especially the POC, especially the girls and non-binary kids. I found myself thinking of when I had kids read about a trans mathematician and a queer mathematician and we talked about that in class. Or when I had kids read about a black mathematician and we discussed. And then I thought: wait, maybe we did that last year and not this year? About the ways I tried to encourage the bright girls in my precalculus classes who were outnumbered by boys — to know they were valuable and their ideas were valuable. But maybe it was too subtle, or not enough, or nothing at all. About my intention to have kids listen to a podcast of latinx mathematicians and how I never made space and time for it. As I thought about these things to assuage and exacerbate my own guilt, I clung to certain things as I was hoping to justify I was “good.” Though I’m not white, this was definitely white fragility. And I thought about if and how today would change anything. For me. For our community. Because I’ve been around, and these sorts of days have happened before. And we as a school community get all riled up and passionate. And then a week later, we’re back to the status quo. Or have we? Have small changes been made that add up over time? I remember thinking, maybe one day the fire will catch and maybe we’ll have something sustainable. Days like these are the sparks. You can’t have fire without a spark.

At 3pm, we had a department meeting. It felt incongruous to the day because we had no mention of the happenings of the day. We toasted to colleagues (friends) who were moving on — one to another school, one to blissful retirement. And we did some work together. When the meeting ended at 3:45pm, I moved to my couch and laid down.

A little over an hour later, I hopped onto a zoom call with three colleagues and friends. We have this tradition every Friday. It’s nice to have this normalcy and camaraderie. These are people I need to vent to, but also laugh with, because otherwise I would feel very alone as a teacher in isolation. We talked about the day, processed, a few tears were shed early on, and laughed. We lasted two hours. And we imagined ourselves on a beach adventure together, in person, and committed to making that happen when we could. It dawns on me that this trip is literally the only actual thing I have to look forward to in my life right now. A hypothetical trip to the beach.

This is an archive of my day yesterday.

Advisory Games/Activities

A friend asked me what sorts of games might be playable over zoom.


I was going to write an email with some games I’ve played with my advisory, but I figured I might as well just blog about it in case it’s useful to anyone else.

Some important context:  I have an awesome advisory group of 8 kids. They’re juniors now but I’ve been with them since they were ninth graders. So we’re comfortable with each other and they know a bit about each other. In quarantine, we always start our distance learning school day with a 15 minute zoom advisory. Honestly, it almost always is one of my favorite parts of the day because … well, if you met my kiddos you’d know why. Sometimes I lead advisory with an activity, sometimes I assign them to come to advisory with a plan. Here are some things we’ve done and things we may do in the future. Some of them are games. The ones with * are things they’ve suggested but we haven’t done or I’ve been thinking about doing it but we haven’t done.

  1. We’ve done a pet share. Kids brought their pets (or some sort of stuffed animal or other object) and introduced their pet to the advisory.
  2. We’ve shown our workspaces during digital learning to each other.
  3. We shared things that were difficult about online learning and problem solved together.
  4. I posted a list of 8 random items (a balloon, a jar of peanut butter, the playing card the 7 of hearts, etc.). I assigned point values to the items based on their obscurity.  Students took a photo of the list and had 10 minutes to gather everything and come back at a certain time. Then they tallied up their points, and we declared a winner. (I hope a student does this when leading their advisory… I wanted to play!)
  5. Played Kahoot (with pre-made Kahoots)
  6. Each of us bring baby pictures of ourselves and share them with each other
  7. We break up into two teams on zoom. We pick a word (e.g. “love” or “red”). Each team has 30 seconds to come up with a song lyric or title with that word in it. We go back and forth until a team can’t come up with another one.
  8. The same thing as the previous one, but we changed it from song lyrics to song titles, movie titles, book titles, magazine titles, etc. (media titles).
  9. Played Mad Gab. I projected a card and kids tried to figure out what the card was saying. We played this fine with kids just sounding things out on their own, but I was thinking they could mute themselves and then they could unmute themselves when they think they got it.
  10. Played drawphone. This is like telephone, but with words/phrases and drawings. Initially we started out with pre-written words from telestrations/pictionary, but recently we’ve done it where we ourselves write the initial words/phrases. This is one of my favorites! Caution: there is a cards against humanity version that you can play on this site — so you should always initiate the game, and ask kids the join (and not let kids create the game).
  11. Played pictionary/skribbl.io. Initially, we did this using the pictionary words built-in, but then one of the students (when they were leading the advisory) came with words associated with our school (people, places, terms, etc.). That was so sweet.
  12. Picked a Sporcle quiz, I projected it, and we collectively tried to see how good we could do working together. I typed what they told me to type. We did this today and we did a Pixar quiz (3 minutes) and a Harry Potter quiz (8 minutes). Also, one of my favorites!
  13. Projected some “would you rather” questions I found online and had kids discuss them.
  14. Since kids are doing a lot of cooking, I told them to take photos of their cooking process and their final foodstuffs… and then we shared those photos.
  15. *One person gets put in a breakout room alone. The rest of the people get assigned some weird feature (e.g. they can only speak in questions, all their sentences have to start with the letter T, they have to cough each time they speak, they have to mention a color each time they speak, they have to address each person they speak to by their name, they have to do a strange hand movement at the camera each time they speak, etc.). Then the person has to come out of the breakout room and talk with everyone and try to determine what the person’s quirk is.
  16. *Play Spyfall.
  17. *Play Evil Hangman.
  18. Update: We did this! (But I just found it online too!) We can do it with the annotation/whiteboard feature on zoom. Me (not playing) secretly emails everyone but one person the same object (e.g. A teddy bear) and one person gets an email saying they don’t get told what the object is. We go around and each person draws one line/shape on the board. A line is considered anything you draw without picking your pencil/pen up. The group has to decide — after two rounds of drawing — who wasn’t told the object being drawn. For example, let’s say everyone but one person was told the object was “teddy bear.” So the person who wasn’t told “teddy bear” has to be careful about what they draw… they’re always taking a risk when drawing… but the rest of the people can’t be too obvious either, because if it’s clear early on it’s a creature of some sort, the person who wasn’t told could draw an eyeball or something. If the group correctly guesses what the item is (watch the video to determine how the group guesses) then the fake person can still win if they can identify what the object was supposed to be.
    (This is like Spyfall, but with drawings!)
    Update: We did this today! It’s based on a game called A Fake Artist Goes to New York [video tutorial]
  19. Update: My advisory played scattegories online today! I admit to not really understanding how it worked at the beginning, but by the end I think I had the hang of it.
  20. Update: Today one of my advisees created a playlist on spotify. She shared her desktop with audio but had a random chrome page shared so we couldn’t see her spotify. She then controlled her spotify from her phone and played the music. The first person to identify the song won a point. I kinda loved this even though I definitely didn’t know many songs… and then someone else suggested next time we all rotate who shares a song so we can get lots of different genres and get to know other peoples’ musical tastes also. Without screensharing with audio, I tried just playing a song from my computer to see if kids could hear it, and they said they could hear it fine… so it seems like that would work pretty seamlessly.
  21. Update from an advisor colleague at school (Carla K): Go visit a museum tour virtually together!
  22. Update: We played “Charades” today! We started by each person just acting out whatever they wanted to and everyone else trying to guess. And then when a student had trouble coming up with an idea, I private messaged them on zoom to give them 3 choices to choose from. And then we played that way! No teams. Nothing serious. Super fun!

If anyone has anything they’ve done that’s fun and low-key like this with their kiddos in advisory or class, please throw down ideas in the comments! I’d love to add to our list!

Update: John Golden shared, in a recent culling of neat math/teaching things, this google doc which not only has games but also digital escape rooms!

Update: My friend Mattie Baker sent me this image which has lots of boardgames online! It includes some that I’ve listed above, and a lot more!
online board games

Update: A friend sent me this spreadsheet with virtual games to play.

Going off the beaten path…

I find myself always pressed for time in class. Doing inquiry-based work, and going at the pace of the kids’ understanding, means that inevitably there are stretches of time when I’m feeling like “ARGH! We need to go faster! We have to be further along to cover the content!” Every teacher I know feels this, and in my office we talk about this, so that makes me feel better. I’m not alone. I also know in the back of my mind that somehow, each year, we manage to get things done. That helps.

That being said, this past week I had one of the very few times that one class was about half a day in front of the other and I didn’t feel the need to forge forward as acutely as I usually do. And in that short exciting window, I saw a student’s shirt which I thought was mathematically beautiful…


… which of course I told to the student. “Math?” He didn’t see it, nor did others in the class. It was an 90 minute class, and when having kids work on some problems together, my mind started thinking… I want to show kids how I see this shirt. The glasses I use to see the world are different from theirs. We take a break in the middle of class for kids to get water or quickly grab a snack, and so 5 minutes before the break I stop everyone. I have them look up. And I say something like: “I know many of you don’t see math in [Stu]’s shirt, but I think if you start looking at the world with math in mind, something like that shirt will pop out as beautiful mathematics. So grant me 5 minutes where I’m on stage, you’re the audience, and I live give you my thought process for creating a version of that shirt on desmos. It may not work, but I think it will be neat to try.”

So they are sitting watching me sit at a laptop. I start by graphing f(x)=e^{-x^2}. They ooh. Maybe they don’t. But in my mind’s recreation of the event, I hear them ooh. I explain that this isn’t a random thing I created. I ask who is taking statistics. I mention the “normal distribution.” A few nod knowingly, and for those who don’t, I say “this isn’t genius, this is me seeing [Stu]’s shirt and recognizing one of the most famous equation shapes in the world.” Then I graph the reflection over the x-axis. They understood that. But then I said I need more lines.


So I say: “I am now going to try to make the same curve with a higher peak. I think I should do a slight vertical stretch.”


But then I note that it isn’t just that each curve gets slightly stretched, but also the width of the bump gets slightly widened too. I go to the board and explain how I’m going to do a horizontal stretch too, and write up how I’m going to alter the x-variable in the equation to do that.


I flipped that over the x-axis and then manually entered a bunch more equations that did the same thing — slightly higher peaks, slightly wider bumps. Kids asked me to add in the two circles in the middle, so I did. It looked meh because I only had 6 or 8 curves. I sent them on break and promised them I’d get it to look a bit better when they returned. And that I would do this with just one equation.

During break, I whipped this up using lists.


When they returned, I explained how the list worked in Desmos — so one equation actually plotted a bunch of equations.

I didn’t know what to make of doing this. I wanted them to see how I saw things, how I thought about things. That math is in lots of places if you just look for it. That playing with math can be fun and what they already know mathematically are quite powerful tools. If for just a second one of the kids was like “Oh, yeah, wait, math is pretty neat,” I’d be happy. It might have happened because the next day I was talking with a science teacher who was telling me that my kids who she also taught were talking about it in her class.

Also, you know, I always find that when I deviate from my plans for something I’m excited about, I always feel so good about doing what I’m doing with my life. I have to keep this in mind and try to go off the beaten path more…

PS. Of course when I saw the shirt, I didn’t initially “see” the normal distribution. I saw fluid flow around a cylinder:


But I forgot everything mathematical I know about that. :) So normal distribution it is!


Alone with “Starry Night”


So I wasn’t actually alone with Van Gogh’s Starry Night. But I went to MoMA this morning and got to tour the museum with other math teachers before the museum opened. Our sherpa? George Hart, mathematical artist. A few months ago, I got an email from two different teacher friends letting me know about this opportunity to take a master class on Geometric Sculpture put together by the Academy for Teachers. What an opportunity indeed!

I show up at 8:30 am and me and a gaggle of math teachers (a gaggle is eighteen, right?) are raring to go. We have fancy namecards and everything. (Note to self: at the book club I’m hosting in a bit over a month, create fancy namecards.)


Beforehand, we were assigned a tiny bit of homework. We were asked to go onto the Bridges website (it’s an international annual math-art conference, organized by our sherpa), look at submitted papers for their conference proceedings, select three papers, and then read and reflect on them.

My Paper Choices and Thoughts

1. Prime Portraits, Zachary Abel

This mathematician was able to construct portraits using the digits of prime numbers. The digit 0 was black and the digit 9 was white, and the other digits were various shades pic1of gray. The digits of a number were put in order in a rectangular array (e.g. 222555777 would be put into 3×3 array, where 222 is the top row, 555 is the middle row and 777 is the bottom row) and an image results. For most numbers, the image will look like noise. But this author was able to use prime numbers put into a rectangular array to create images of Mersenne, Optimus Prime, Sophie Germain (using Sophie Germain primes), Gauss (using Gaussian primes), and others. I wpic2.pngas blown away. This intersection of math and art doesn’t quite fall neatly into any of the categories that George provided us, but it is close to “mathematics used in calculating construction details necessary for constructing an artwork.” In this case, the portraits themselves are the “art” and the author was using numbers to reconstruct that art. What makes it interesting is that the math version of these portraits feel unbelievable. Senses of awe and wonder and curiosity filled me when seeing the portraits for the first time because how could it be? It was like a magic trick, because nature couldn’t have embedded those portraits into those numbers. And before reading the paper on how these were constructed, I had a nice few moments thinking to myself how this could have been done.

(If you’re curious, the answer is to start backwards. First take an image, pixelate it, and then turn those pixels into a number. Take that number and check if it’s prime on a computer. If it isn’t prime (which is likely), slightly alter the image by the colors by +1% or -1% (some imperceptible noise), repixelate it, and turn those pixels into a number. And again, check if that number is prime on a computer. If it isn’t, do this again. It turns out that you’re going to need to do this about 2.3n times [where n is the number of pixels]. With a computer, this can go quickly.)



(a) Math: I recall faintly from college classes that the distribution of primes is related to the natural logarithm. Which explains why the 2.3n comes from something involving a natural log. But what is this relationship precisely, and how does it yield the 2.3n?

(b) Content: I think prime numbers are very rarely taught in high school math in a meaningful way. Number theory is ignored for the “race to calculus.” However there is so much beauty and investigation in this ignored branch of math. Where could I fit in conversations of prime numbers in an existing high school curriculum? Could ideas from this paper be used to captivate student interest (by letting them choose their own image), while showcasing what various types of prime numbers are?

(c) Extension: Are there other things that we teach that have visualizations that look impossible/unbelievable, but actually are possible? Can we exploit that in our teaching? I’m thinking that often numbers in combinatorics are crazy huge and defy imagination… Perhaps a visualization of the answer to some simple combinatorial problem?

(d) In order to fully appreciate this work, the viewer needs to have an understanding of prime numbers. Without that understanding, this is just a pixelated image with some numbers superimposed. All wonderment of these pieces is lost!

2. Modular Origami Halftoning: Theme and Variations (Zhifu Xiao, Robert Bosch, Craig Kaplan, Robert Lang)

I chose my articles on different days, and I didn’t even notice that this article is very similar to the first article! I chose it because I love the idea of a gigantic public art project in a school (I tried once and failed to make a giant cellular automata that students filled in). But this article basically shows how to fold orgami paper (white on one side, colored on the other side) in five different ways to make squares where all of the square is colored, ¼ of the square is colored, ½ the square is colored, ¾ of the square is colored, and none of the square is colored. A number of each of these origami pieces are constructed.

Then an image is converted to grayscale and scaled down to the number of origami pieces you want to use. Then the image is scaled-down image is pixelated with “origami piece” size pixels, and each pixel is given a number based on brightness [0, ¼, ½, ¾, 1].

Then this origami image can be created by putting these five different origami pieces in the correct order based on the brightness of the pixelated image!


Just like with the previous paper, this intersection of math and art doesn’t quite fall neatly into any of the categories that George Hart provided us, but it is close to “mathematics used in calculating construction details necessary for constructing an artwork.” In this case, the portraits themselves are the “art” and the author was using numbers to reconstruct a variation of that art.


(a) Math Classroom: I really love the idea of having kids take an image with a particular area (w by h) and figure out how to “scale down” the image to use a particular number of origami pieces. It is an interesting question that will also involve square roots! It seems like a great Algebra I or Algebra II question.

(b) Extensions: How could this project be extended to the third dimension? 3D “halftone” origami balloons? Unlike a photograph which can be easily pixelated, can we find a way to easily pixelate the “outside”/”visible part” of a 3D object and create a balloon version of this? Similarly

(c) This is not just a low-fidelity copy of an existing piece of art. If we took a random non-professional Instagram photograph, we might call it “pretty but not art.” But if someone made this Instagram photograph out of origami sheets, we would be more likely to call it art. But why? Just one thought, but there is something about the intentionality of the artist (and the craftsmanship that goes into creating the origami piece) that isn’t in the original photograph. It also is likely to evoke something different in a viewer – a viewer will instantly wonder “how was that done” when seeing the origami piece (so the art piece evokes process) while a random photograph might not do the same (they just pressed a button on their phone and got a cool photo).

3. A Pattern Tracing System for Generating Paper Sliceform Artwork, Yongquan Lu and Erik Demaine

I chose this paper because of the beautiful sliceform image on the first and last page. I had only seen them once before, but forgot what they were called! I wanted to learn howpic4 to make them. In this paper, the authors share that most existing sliceforms are created in separate pieces (e.g. the image on the first page, a bunch of hexagons created separately) and then pieced together afterwards. The authors wanted to instead thread the paper slices together so they could create the same intricate patterns—but with the paper slices interconnected. So instead of individual hexagons placed together, a giant connected sliceform was created (e.g. the image on the last page). The authors came up with a way to do this for designed created in polygonal tiles, like in many Islamic star patterns, and then created a program to “print” the strips of paper needed – with red lines indicating where folds are, and blue notches indicating where cuts need to be made so the paper slices can be fit into each other.

They accomplished this in two steps. First, they came up with a way to notate the internal structure of a paper slice within one polygon. One notation captured lengths (where slices of paper intersected other slices of paper and where slices of paper needed to be bent/folded), and another notation (not provided) recorded angles that needed to be folded. The second step was more tricky. An algorithm was created that looked at the edge of a polygon (where a paper strip initially ended), and looked to see if it could be extended into another polygon. In that way, one strip could start in one polygon and then enter another, and then another, etc. This is the threading that the authors wanted to get. The authors created a three-step algorithm for deciding if a paper strip could enter another polygon at all, and if there were multiple possible paths for this strip to take, which one it should choose.

After doing all of this, the authors then created a program that could take in an image, calculate out the different strips of paper needed to create the sliceform, and with the notation they created, print out the appropriate slice (see image on page 370 for an example).


(a) There were two big things I didn’t totally understand when reading this paper. First, how were angles recorded/notated? Second, where did the 3-step algorithm for extending paper slices come from? How do we know if we follow it that all segments in the figure will be created by the paper slices, and no segment will be repeated?

(b) Besides just being “cool,” is there an application to this in a high school math class? What higher level research does this connect up to? (Just like origami was simply beautiful but then it also was exploited to create new and interesting questions for mathematicians, what does this bring up for us?)

Note: When I went to research these, it turns out that Lu and Demaine created a website to help amateurs out: https://www.sliceformstudio.com/app.html

(c) I was wondering what a 3D version of this might look like, but it turns out that this exists! https://www.sliceformstudio.com/gallery.html

Back to the Master Class

After getting coffee and pastries, and introducing ourselves to each other in small groups, we all were taken on a tour of MoMA, where George led us to certain pieces to spoke to him as he looked at them through mathematical lenses. There was one sculpture in particular that George stopped us at — a sculpture he remembered seeing as a kid visiting MoMA — that I would have walked right by. It was a figure cast in bronze (?), that had a lightness and movement despite it’s medium. To me, it screamed that it was a figure in tension. Rooms later, I was still thinking about how it was a collection of oppositions, form and formlessness, fluidity and stability. For George, describing what drew him to it was ineffable.


Here are more photos of George taking us around.


The whole walkthrough, George kept on saying “I’m not an art historian, but this is what I see in terms of my perspective as a mathematician…” which was just what I needed to hear. I know so little about art history and contemporary art, but hearing that let me feel a bit more “free” in looking at something and thinking about it with my own lens, instead of me passively waiting to hear what the piece is “supposed” to convey or what philosophical/conceptual trend it is a part of. In general, I feel ill-equipped to make statements/ judgments about art in museums that go beyond “I like this” or “I didn’t really like this.” But listening to George talk about what he sees as a mathematician and mathematical artist was liberating. Because I can see mathematical ideas/principles (intentional and unintentional) in some of the art too! This walk and talk reminded me a lot of what I imagine Ron Lancaster’s math walk around MoMA would be like!

And as the title of this blogpost suggested, there was something so special and magical about being able to have the run of the museum before the general public was let in. And a random fun tidbit: I also learned that there is no simple mathematical equation for an egg. I (of course) had to google that when I got home, and came up with this webpage.

We Become Card Sculptors

We get back to the room that was our home base, and some people share out interesting things from the articles they read. I was going to share mine, but I noticed that even though the ratio of men to women was low, more men were taking up airtime than women proportionally. So I kept my hand down.

George gives us a set of 13 cards with notches in them. We only needed 12 but you know how we math teachers really like prime numbers… (Okay, that wasn’t the reason for the 13th card, but I want to pretend it was.) We were asked to crease them like so:


And then… we were asked to put them together somehow, into a freestanding sculpture. No glue, scissors, tape, etc. We were given a hint that you can start with three cards. So I figured we needed to create 4 sets of objects that each take three cards. So with my desk partner we made this:


This was the core object we needed to build the final thing together. It was interesting how it took different pairs different amounts of time to get these three things together. Without instructions, it was a logical guessing game, but it felt so good once we hit upon it.

Then came the tough part. Putting these four building blocks together. That took a long time and some frustration, but the good kind. It was one of those problems that you know is within your grasp, and you know that you can come out on the other side successfully, but you don’t quite know how much time and how much angst the journey will cause you. It’s that sweet spot in problem-solving that I love so much. And lo and behold:


Many people got it! I would post a picture of mine, but all my photos look terrible. You can’t see or appreciate the symmetry and freestanding nature of this beast. But it was a moment of such pride when we got the last card to slide in the last notch! (And of course when my partner and I tried building hers after finishing mine, it went much faster and we had a better sense of things.)

Oh yeah, this card sculpture is isomorphic to a cube. I was blown away by that. It was hard for me to see at first, but realized that to get my kids to see it, I would give ’em purple circular stickers to have them put on the “corners” and blue circular stickers to have them put on the “faces” and green circular stickers to have them put on the “edges.” It would help me not only count the different things (maybe put the numbers 1-8 on the purple stickers, 1-6 on the blue stickers, and 1-12 on the green stickers?), but also “see” how they are in relationship to each other. (And George told the class he liked the suggestion and would think about trying it out!) George asked the class what the “fold angle” is for each card (what angle the card was bent at in the sculpture). I loved the question because it’s so obvious when you look at the sculpture from just the perfect angle! (The answer: 60 degrees.)

We See Art and We Build More Art

Lunch was delivered from Dig Inn, and we ate and briefly chatted. And then George took us on a picture tour of his sculptures and their construction. Some choice quotes:

“Kids need to have an emotional connection to math.”

“Math and art are both about creating new things.”

Finally, we ended our day building our own mathematical sculpture. We had 60 pieces of wood that we set up in trios. And we combined those to create a hanging sculpture.


What’s neat is that this hanging sculpture is going to travel to all the schools of the teachers who were at this session for two week periods. It will come to us disassembled and we’re going to get a group of kids (or teachers!) each to build it up and hang it. And then after two weeks, send it on! I love the idea of this same set of 60 pieces being in the hands of young elementary school kids and my eleventh-grade kids.

Takeaways and Random Thoughts

I have recently been into math art. Last year, I helped organize a math-art exhibit in our school’s gallery. I get excited when kids make math-art for their math explorations that I assign in my precalculus class. (In fact, years ago I had two kids make some sculptures and now I know they came from directions George provided on his website.) For me, it isn’t about “art” per se, but about seeing math as more expansive than kids might initially think, and seeing math as a creative and emotional endeavor. That’s why this resonated with me.

At the start of the year, I had intentions of starting a math-art club. Because my mother was sick and I was not taking on any new responsibilities, I decided to put that idea on hold. But now I’m feeling more excited about trying this out. To do this, I want to create 5 pieces on my own based on things I have found online. Things that will kids to say “oooooooooh.” Heck, things that will get me to say “ooooooooh.” (Like the origami image I saw in the second paper I wrote about above.) And then show them to students and get a core group of 4-5 who want to just build stuff with me on a regular basis. Maybe as a stress reliever.

What can we make? Who knows! Maybe stuff out of office supplies? Maybe some of the zillion awesome project ideas that George and his partner Elizabeth have put together. Maybe something inspired by the awesome tweets with hashtag #mathart that I’ve been following (and sites like John Golden’s). Maybe something on geogebra or desmos? Maybe something else? The idea of a large visible public sculpture appeals to me. One that random people walking by can add to also appeals to me. (I tried last year to get a giant cellular automaton poster going at my school, with two students in the art club, but it didn’t quite work as planned.)

Maybe this happens. Maybe it doesn’t. I hope I can muster the energy to start thinking this summer and making this a reality next year.

Random thought: Based on all the photos that George posted showing him bringing his math art to little kids in public spaces, I wonder if he’s talked to Christopher Danielson who organizes Math-On-A-Stick? Or if he knows Malke Rosenfeld (we had talked about math and dance earlier in the day)? I’m hoping yes to both!

Random note: George said that among his favorite mathematical artists were Helaman Ferguson, Henry Segerman, and John Edmark. Bookmarking those names to check out later.

Random thing: At MoMA in an exhibit about the emergence of computers to help create art was fabric that was created by the artist to hold information in it. What was pointed out to me, 20180208_100531.jpgwhich made me go HOLY COW, is that the punch card idea for the first computers came out of the Jacquard loom. So loom –> computer –> loom. What a clever idea. I wish I knew what information was encoded in the fabric I saw! Additionally, this reminded me of one of the artists we had exhibited at the math-art show I helped organize: the deeply hypnotic and mathematical lace of Veronika Irvine. And that of course got me thinking about this kickstarter that I’m so sad I didn’t know about until after it was done: cellular automata scarves!

Random last thing: totally unrelated to this workshop, last night someone posted on twitter that Seattle’s Center on Contemporary Art is about to open a math-art exhibition, and my friend Edmund Harriss is one of the artists in that show! Along with the work of father-son duo Eric and Martin Demaine who both do amazing paperwork (and amazing mathematics). So awesome. Wish I were there so I could go see it.

Exploding Dots! Global Math Week 2017!

Hi all,

Life is getting away from me with some tough personal stuff. So I haven’t been as active with the online math teacher community/twitter/blogging/etc. for a while, and I sadly probably I won’t be for a while.

That being said, I really wish I could participate in this initiative that Raj Shah (no relation!) shared with me a while ago. But because of life stuff I might not be able to. But one of the biggest things I want to do is bring joy into the math classroom as a core value, and this does that. And I love the idea of a collective joyful math moment for students and teachers all around the world! I’ve done a bit of exploration with this initiative — exploding dots — and I think it’s fabulous and full of wonderment. What it takes? At minimum, 15 minutes of classtime! I highly recommend you reading the guest post I asked Raj to write (below), and joining in this worldwide effort to celebrate the interestingness of mathematics!




The Global Math Project is an invitation to students, teachers, and communities everywhere to actively foster their sense of wonder and to enjoy truly uplifting mathematics. Math is a human endeavor: It’s about thinking creatively, exploring patterns, explaining structure, and solving real problems. The Global Math Project will share a unifying, joyful experience of mathematics with people all across the world.
Our aim is to thrill 1 million students, teachers, and adults with an engaging piece of mathematics and to initiate a fundamental paradigm shift in how the world perceives and enjoys mathematics during one special week each year. We are calling it Global Math Week.
This year, Global Math Week will be held from October 10–17. The focus of Global Math Week 2017 is the story of Exploding Dots™ which was developed by Global Math Project founding team member James Tanton, Ph.D.
Exploding Dots is an “astounding mathematical story that starts at the very beginning of mathematics — it assumes nothing — and swiftly takes you on a wondrous journey through grade school arithmetic, polynomial algebra, and infinite sums to unsolved problems baffling mathematicians to this day.”
The Exploding Dots story will work in any classroom, with a variety of learning styles. It’s an easy to understand mathematical model that brings context and understanding to a wide array of mathematical concepts from K-12 including:
  • place value
  • standard algorithms for addition, subtraction, multiplication, and long division
  • integers
  • algebra
  • polynomial division
  • infinite sums
  • and more!
Teachers routinely call Exploding Dots “mind-blowing”!
“I am still amazed by this. Exploding Dots has changed my fifth grade class forever!” – Jo Anna F.
“This makes me WANT to teach algebra!” – Kristin K.
“YES!” Hands up in the air in triumph! Decades of believing I couldn’t do math—poof! Exploded!”  – Jennifer P.
Join us for Global Math Week, October 10 – 17, 2017!
During Global Math Week, teachers and other math leaders are asked to commit to spending from 15-minutes to one class period on Exploding Dots and to share their students’ experience with the Global Math Project community through social media.
You can join the movement in four easy steps:
1) See Exploding Dots for yourself
Here’s a brief overview: https://youtu.be/KWJVAjONqJM
2) Register to Participate at globalmathproject.org
3) Conduct an introductory Exploding Dots experience with your students during Global Math Week
All videos, lesson guides, handouts are available for free at globalmathproject.org. Since everything is available online, inspired students (and teachers) can continue to explore on their own.
4) Share your experience on Twitter during Global Math Week using #gmw2017
That’s it!
The power of the global math education community is truly astounding. To date, over 4,000 teachers have registered to participate in Global Math Week (#gmw2017) and they have pledged to share Exploding Dots with over 560,000 kids from over 100 countries! We already over half-way to our goal
Help us reach and thrill a one million students!
The Global Math Project is a collaboration among math professionals from around the world. Spearheaded by popular speaker, author, and mathematician James Tanton, partner organizations include the American Institute of Mathematics, GDayMath.com, Math Plus Academy, and the National Museum of Mathematics.

Bragging about my school

This is a milestone for me. I have been at my school for ten years, and this is the start of my eleventh. It’s the only school I’ve worked at. That’s a testament to my school, but more specifically, to my colleagues.

Last year, my school’s awesome director of communications contacted the math department to let us know that the one issue of the magazine she publishes four times a year was going to focus on math. And she wasn’t kidding! The cover of the magazine had most of my multivariable calculus kids on it (thinking deeply at the math-art show I helped put on last year)!


One of my favorite things is that the feature article with an alliterative title, Making Math Meaningful, was simply the transcript of a roundtable discussion we had. A bunch of math teachers got in a room around a big table, and we were led by our director of communications who had done her research and come with some questions. There was a digital recorder in the center of the table. And through talking with carefully crafted prompts, we got to think deeply and collectively about our own practice. I can’t even tell you how interesting it was to listen to my colleagues during that facilitated conversation, and how proud I was to be in a school with such like-minded folks that I have the opportunity to learn from. (If you’re a department chair or academic dean, consider doing this!)

I wish I could just post a PDF of the article for you to read, but alas, the whole magazine is online but can’t be downloaded. Here are two quotations to whet your appetite:

quote 0

quote 1.PNG

So if you want to read about a department that is doing strong work moving towards inquiry-based learning, and read the words of real teachers having a real conversation playing off of each other, I highly recommend you:

  1. Go to this site
  2. Make the magazine full screen
  3. Read pages 18 to 29

That is all!