PCMI 2022 Post 3: Stupid Square

One of the highlights of my day on Thursday was our first book club meeting to talk about Francis Su’s Mathematics for Human Flourishing. Just the first chapter. At my table, there were five teachers and one graduate student who got the book so he could join us (bonus: he was wearing a Harry Potter shirt and was not intimidated to be surrounded by people who all knew each other). We basically had an hour to introduce ourselves, and then informally talk about two things:

  1. Responding to one, two, or three questions that were posted on the screen, which went something like (my own paraphrase, since I have a terrible memory): “If you were at a coffeeshop and had to define what mathematics is to someone in 2 minutes, what would you say?” & “What is the connection between mathematics and being human?” and “What would you say to someone who asks what’s the point of learning math if you aren’t ever going to use it in real life?”
  2. In the book, we read a letter that someone in prison wrote to the author. It was a letter where the person was vulnerable, and in the letter talked a bit about his journey that led him to prison, but also that he had previously had a proclivity for math and so he was studying it on his own and reached out to the author for assistance. We were asked to think about what we would think and do if we had received the letter.

The looseness of the prompts (and for the first one, we had choice, and our group even modified some of the questions as we talked about them), and the lack of needing to produce something tangible at the end of our discussion, was lovely and freeing. (Over the years, I have led a lot of math book clubs with kids, and you can read some of my advice at the bottom of this article here. This structure for today worked well, and I loved it.) And since the first prompt got at the heart of what we love, why we’d spend our lives devoted to it, our passions, we all had something we could bond over and really feel connected to the other people sharing their thoughts. Or at least I felt really connected with the people at my table.

I want to first write a bit about the second prompt. It reminded me of blinders I often have. Some shared that they would not respond to the letter because of fear for their own safety and fear for people in their lives — a strange unsolicited letter coming to their home — and others shared similar thoughts. It never even occurred to me to think about that, but it reminds me of conversations I’ve had with friends who are women and have to move about the world so differently with a totally different lens (like one told me years ago that when they go into their hotel room, they have a routine where they check under the bed and in the bathroom for someone). I didn’t even consider that aspect of things.

I did think a lot about my friend Sara Rezvi who posted a few months ago about doing this exact thing — communicating with an incarcerated person about math through the prison math project. She tweeted about it here:

Now to the first set of questions, which got us going! A few of us gravitated to the third question (“What would you say to someone who asks what’s the point of learning math if you aren’t ever going to use it in real life?”) which as we discussed it, really seemed to dovetail into the second question (“What is the connection between mathematics and being human?”). One person shared the idea “why do we read Shakespeare if it doesn’t come into our daily lives?” which is often my go-to! There is something inherently captivating about the act of reading it, and analyzing it. And we see the beauty in it. As we talked, I kept on having the idea that mathematics is the act of world building, which seemed to encapsulate much that had been said. Under constraints, we invent, we use creativity to push the things we invent, we explore, we get bored and go somewhere else, we feel emotions as we construct: angst, elation, frustration, anticipation, sadness, and when we’re really lucky, love. There are also lots of other things that we said that don’t fit in here (an informed citizenry, ability to analyze, ability to draw connections, etc.), but that metaphor really resonated with me in the moment.

When we talked about the first question (“If you were at a coffeeshop and had to define what mathematics is to someone in 2 minutes, what would you say?”), we changed the question. Because normally, when we meet someone in a coffeeshop and math comes up, they get turned off. So if we said something like “math is about patterns and meaning in those patterns,” I’m pretty sure that would kinda lead to an end to that part of the conversation for most people. So instead, we changed the question to “what would you say to someone in 2 minutes to express why math is something you want to spend time with?” Like give them something to be captivated by which would let them get sparked and have a glimpse of what we glimpse.

I would have loved to brainstorm this with other teachers for hours — because I think the answers we’d come up with would be amazing for us to use in our classrooms (in addition to random hypothetical coffeeshop situations). Maybe we talk about the 30 second responses, the 2 minute responses, the 10 minute responses, and more! And we teachers have so many ways to do this! (One person said they’d bust out Pascal’s Triangle!) We didn’t get too much time to do this specific brainstorming, but one teacher said something that really encapsulated what I think what many of us teachers feel: in school, we teach under all these constraints and traditionally that amounts to students seeing math as fitting in this “stupid square” when we all see math as (waving all around the square) as this much bigger and beautiful and wondrous thing.

That reminded me of something I heard years ago… In physics, in high school, kids learn about quantum mechanics. They learn about the wave-particle duality, they learn about the probabilistic weirdness, they learn the world is so strange. Learning about that has the ability to captivate the minds of kids (it did me, anyway). But kids in high school can’t do the mathematical parts of quantum mechanics, the wave function. But that doesn’t stop the physics teachers from teaching it. We should be doing stuff like that with modern mathematics in schools, to capture the developing wonder and imagination of kids.

Jordan Ellenberg is at PCMI and I was too nervous to go up to him to say “thank you.” You see, he wrote How Not To Be Wrong: The Power of Mathematical Thinking, which I read when it first came out and loved. (I am super critical about popular math books, in general.) And since then, I’ve done two math book clubs with How Not To Be Wrong with kids at my school — holding a few sessions, each, talking about what we read. It’s part of my trying to get out of the “stupid square.” So instead of talking to him, I send him an email thanking him for giving me another avenue to do this! The day after I sent the email, Prof. Ellenberg gave a talk about “Outward-facing Mathematics” (books, blogs, popular articles, tiktoks, etc.) and why and how to get involved with it. There are so many ways teachers are doing this now… Sidewalk Math, #MathGals, Math on a Stick, Play with your Math, Playful Math Education BlogCarnival, …

When I left his talk, I remembered I actually had done some “Outward-facing Mathematics” with the Big Internet Math Off 2019 (where I came in second place of sixteen competitors!). In case you want to see
my attempts at learning or explaining math for a slightly more general audience, to captivate, they are here:

Entry 1: a confounding conundrum: https://aperiodical.com/2019/07/the-big-internet-math-off-2019-group-2-jorge-nuno-silva-vs-sameer-shah/

Entry 2: a card trick: https://aperiodical.com/2019/07/the-big-internet-math-off-2019-group-2-vincent-pantaloni-vs-sameer-shah/

Entry 3: a magical property of circles: https://aperiodical.com/2019/07/the-big-internet-math-off-2019-group-1-marianne-and-rachel-vs-sameer-shah/

Entry 4: an unexpected break in a mathematical pattern: https://aperiodical.com/2019/07/the-big-internet-math-off-2019-semi-final-1-lucy-rycroft-smith-vs-sameer-shah/

Entry 5: two beautiful squares: https://aperiodical.com/2019/07/the-big-internet-math-off-the-final-sameer-shah-vs-sophie-carr/

(And here are all entries to all the Big Internet Math Offs throughout the years, run by the Aperiodical.)

With that, I’m getting tired so I shall bid this blogpost adieu.



  1. I like what you said here,
    “We teach under all these constraints and traditionally that amounts to students seeing math as fitting in this “stupid square” when we all see math as (waving all around the square) as this much bigger and beautiful and wondrous.”
    I also think it’s tragic that someone left behind by society making a prosocial inquiry, perhaps for even the first time with something they can get behind and genuinely love, receives no reply because they are afraid of their safety. I am sure that kind of unansweredness when they are being “good” is devasating, and reaffirms why they must violate social norms and get themselves. Though some human spirits do indeed take it too far, and they know who they are, many of them are victims of a less than intelligent society that could not provide them with the interventions or aid it should have. Then, they are stuck in a small concrete box, a true stupid square. It is easy to see why the term “square” comes with such vitriol. To put a victim in a concrete box is probably one of the stupidest things imaginable! For that reason, I love love love your citation of the Prison Math Project, and I hope people continue to become awre intelligent on the matter of injustice, social determinants of health, signs of true will to change, and investing very generously in those signs instead of citing a previous history that likely was a product of grief, rage, pain, and systemic failure. From an abolitionist perspective, after all, any civilization that still has prisons is inherently failing.

    1. Wow, thank you for sharing your thoughts, and drawing a connection to the “stupid square” and literal prison. It gets me thinking about the larger structures and constraints that lead many of us teachers to teach in the stupid square, and also the larger structures and constraints that lead to many to actually ending up in the literal stupid square (prison). And maybe there are some real overlaps in those two sets of larger structures and constraints.

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