PCMI Post 5: Ketchup Math

Today I was sitting at a lunch table of 8 people. We somehow got on the topic of grilled cheese. You may know that I’m a vegetarian, but I think a more accurate term for me is a queso-carbotarian because I love cheese and carbs together. So I’m a real fan of grilled cheese. And growing up, when my mom made me grilled cheese made with American singles inbetween two pieces of white bread, she always served it with ketchup.

I thought that was normal. That everyone ate their grilled cheese with ketchup. I mean, yes, you can dip it in tomato soup. And you can dip it in ketchup. And maybe, just maybe, you could eat it by itself. And everyone is going to have their own opinions about what’s best.

What I wasn’t prepared for is that no one at my table of 8 had ever even heard of eating grilled cheese by dipping it in ketchup. I’m not talking about not liking it, or not doing it. They hadn’t ever heard of anyone doing it.

While I was totally certain that everyone at least must have heard of it, and I’d have guessed that most people did it except for just a few. So that fact that of the 8 of us, I was the only one who heard of it led me to believe one of two things:

I was at a very weird table that was not representative of the rest of the American/Canadian population, or that I grew up with this very unique cultural tradition of eating grilled cheese with ketchup — and that was something my mom invented or maybe it’s an Indian thing — but not widely known.

Of course I decided to double down, and claim that my lunch table was TOTALLY WEIRD and that they were the outlier — and more people had to have heard of ketchup and grilled cheese. So I get up to ask another group of around 6 teachers at a neighboring table. And what happened? They all said the combination was new to them. (And later in the day, I ran into author and mathematician Jordan Ellenberg who also said he had never heard of the combination, but understood why it would likely be good.)

Now at this point I feel like my world is crashing down. Something that I thought was so commonplace in the world wasn’t. My sense of reality was being called into question. And instead of accepting it, I double down again. Everyone at my table is interested in this — maybe because I feel totally bonkers and can’t let it go. So I put up a poll on twitter, set for 3 hours, which asks the question below and gets these results from 41 people:

Another person at the table did a similar poll on facebook, and got 25-ish results, and they said about 2/3rds of the respondants said “yes.”

What to conclude?

I figure I can think of it like this. I want to know if my two lunch tables of 14 people (where 13 hadn’t heard of eating grilled cheese with ketchup) WEIRD or if I am WEIRD for doing this. I am using the two polls to be representative of the true population… and to make things easier, I’d even argue there’s some error and so maybe I could even say about 50% of the world has heard of eating grilled cheese with ketchup, and 50% of the world hasn’t heard of eating grilled cheese with ketchup.

Then in a group of 14, for only 1 person to have heard of eating grilled cheese with ketchup IS TOTALLY WEIRD. It’s like flipping a coin 14 times, and only one time you get heads (and 13 times you get tails). Hello binomial distribution.


But… I still don’t know… tonight we had an event at 7pm where a bunch of people showed up. Maybe there were 20 people in the room and the ketchup question was asked and I think only 3 or 4 people had heard of eating grilled cheese with ketchup. So I’m not sure what’s up.

Regardless, I loved this random fun math question that popped up as I was having a mental breakdown involving the internal reality of my world. And that others got into it.

P.S. For another fun condiment digression, our lunch table then started talking about the tiktok trend where people put yellow mustard on oreos and claimed it was so so good. Even Lizzo tried it. Randomly, there was a box of oreos on a table in our classroom, and we had mustard packets from lunch . So later in the day, another person and I tried the combination. I have a screenshot of the video someone took where I tried it. It wasn’t awful at all, but it wasn’t like enhancing the flavor either for me. It was just kinda fine.


PCMI 2022 Post 2: 3D printing

I was so grateful to the neat ideas that I got on twitter about 3d printing, which I included in my last post. (If you want to check the tweet and the replies, they are here.) Some of the responses that really stood out to me are here:

(1) Have students design (using their knowledge of some core functions and transformations) bubble wands using Desmos. Ashley Tewes wrote a moving blogpost about it here (and how she tied it in with empathy and a larger audience than just the students). And just look at how fun and beautiful these are!

And in a similar vein, Martin Joyce has kids use Desmos functions to create objects involving their own names! And @dandersod showed how to 3D print a polar graph from Desmos to be an ornament, which I then did on our school’s 3D printers:

… and I was going to have my kids do our polar graph contest and have the winner’s graph get 3D printed (but the designs were too intricate for that, sadly).

(2) Kids can design their own tesselations (learning the ideas behind how various constructions can build tesselations) and then create 3D printed cookie cutters for them — so they can create “cookie tiles” that tesselate! Or penrose tiles! I initially found a neat blogpost which I’ve lost, but here’s a tweet that showcases it!

(2) Mike Lawler has almost a hundred posts where he and his two sons (who do math together for fun) have used 3D printing. And they are all pretty dang fabulous — an amazing resource. He even chose his favorite ten 3D printed projects here if you don’t want to scroll through all of his posts. The last one he listed in his ten was a model that illustrates Archimedes’ method for deriving the volume of a sphere (without calculus)! I remember learning this in high school and was blown away (so unexpected! so beautiful!), but in all my years teaching, I had never seen this particular manipulative. You can see and download the manipulative here, but I’ll throw down a screenshot of it:

(3) When I taught Multivariable Calculus, we had talked about mappings and coordinate systems, and so one year a student 3D printed this stereographic projection (among other things) and then wrote a paper which analyzed how this all worked:

And I remember showing my multivariable calculus students, in another year, a bunch of optical illusions made by Kokichi Sugihara. They blew my mind, and the kids were smitten. One read some papers on the math behind how you can design these and wrote up a cogent explanation of how this worked using a neat analysis of vector-valued functions.

And goodness knows 3D printing is so cool for surfaces in multivariable calculus, and so much in regular calculus.

But I have to say: after doing a lot of sleuthing, getting things sent to me by others, and just trying to wrack my brain, I’m honestly pretty disappointed with what I think I can do with it in the classroom. It might just be me, but all these schools a decade or so ago were like “WE NEED THESE 3D PRINTERS BECAUSE THEY ARE GOING TO REVOLUTIONIZE STEM EDUCATION.” Maybe so. But after doing an initial foray into them, my current thoughts are: pfft. Maybe I’ll change my mind, but right now: pfft.

Right now, for me, I see the value in 3D printing in two main domains:

MANIPULATIVES: So as I noted, in my last post, there are tons of cool manipulatives a teacher can find and 3D print to illustrate an idea. Like the Archimedes’ proof for the volume of a sphere, or the optical illusion, or creating penrose tiles or printing many of the 15 pentagons that tile (so kids can fit them together and play!), kids will learn. They may be captivated. But kids are learning just from the manipulative, not from the process of 3D printing. That’s just the point of the manipulative — and the 3D printing is one way of getting the manipulative. So great. It isn’t the process of 3D printing that drives student understanding, it is just the manipulative that the teacher finds to illustrate the idea, that happens to be a 3D manipulative. And that’s cool. There’s some value. But in the same value that you can open any math teacher catalog and find lots of hands-on things for kids to play with. This is just a 3D printer printing them, instead of ordering them.

OBJECTS TO SPARK JOY, BUT DON’T HEIGHTEN MATHEMATICAL UNDERSTANDING: Then there are things that I think kids would love doing with the 3D printer in a math class… building bubble wands by using Desmos and function transforemations… developing cookie cutters by learning about transformations… creating polar ornaments by designing creative and beautiful polar graphs. Kids will be able to hold their creations, feel an ownership of mathematics, be proud! So I think there’s a lot to be said for these types of activities. I want to do them! But at the same point, I also truly feel like all the conceptual mathematical learning is happening before the 3D print. The 3D print doesn’t do anything to build on that understanding. What does printing the polar graph ornament from the 2D Desmos polar graph actually teach kids in terms of math? Nothing. I’d argue a kid who printed their bubble wand and a kid who didn’t probably learned the same things. Yes, these things are dang cool, so there’s something to be said for that, but I would argue they don’t build student understanding.

I posited in my last post that there might be a third domain where 3D printing is powerful: where the act of kids actually doing the building in tinkercad or whatever software builds conceptual mathematical understanding. This has been my unicorn, the thing I’ve been really trying to think about or find in the past few days. Because if I’m going to have kids spend time learning new software and troubleshooting finicky 3D printers, there better be a big learning payoff. But at least for Geometry, Algebra 2, and Precalculus, I have yet to anything that really fits the bill.

So for now, as a teacher, I say “hey, 3D printing is cool, but overall, pfft.”

(You might feel differently about this and that’s cool. And I might change my mind. But since I’ve been sending a lot of time trying to think about this and look stuff up, I have just felt a lot of disappointment when I was hoping there was a lot of untapped promise.)

PCMI 2022 Post 1

Haha, I was going to write PCMI 2022 Day 1, but I know I absolutely wouldn’t be able to keep up blogging every day. It’s 8:22pm, and I’m just getting back to my room after leaving it at 7:15am, and golly gee, I’m spent. For those not in the know, PCMI stands for Park City Math Institute and it’s a 3 week summer professional development, and the teacher program is called TLP (Teacher Leadership Program). The information for the teacher program is here, and the constantly updating webpage with the materials we’re using this year is here. I did it two times in the past, in 2010 and 2011, and both were transformative for my teaching, and allowed me to meet and make some amazing math teacher friends around the country! So if you haven’t heard of it, and you want a recommendation from someone who is super critical about most professional development, this is me saying that it’s worth it.

One of the things that happened in the pandemic is that I stopped truly engaging online with other math teachers. Partly, was drowning with trying to figure out how to teach in a totally new way (remote and hybrid), partly I recognized there was no real way for me to teach effectively and I was constantly comparing my teaching to what I had done in the past, and partly my soul was crushed. It was hard to go online and see all the positivity and innovative things that people were implementing when I was just barely keeping my head above water. So I just stopped engaging by blogging which was my form of reflecting (what, a blogpost about how I started doing a bit more lecturing and became less engaging while teaching online? how depressing)… and stopped reading other math teacher tweets.

I’m hoping at PCMI to start blogging again here and there, even if each blogpost is just a mishmash of stuff. I even started engaging with some math teachers again online, and it reminded me why the online math teacher community can be so powerful and why I loved it so much. Here’s an example… one of my two working groups is on how to 3D print. My school has 3D printers. But I have never had any great ideas about how to use them in my classes (except for calculus, which I haven’t taught for a while, but I see lots of connections there). So in my working group, I was learning the basics of tinkercad (which is how you can create basic 3D things to print). But I realized as I was learning the tool, I didn’t know what the motivation was for learning it. In Geometry/Algebra 2/Precalculus, what is a concept that students learn where 3D printing would actually enhance student learning of mathematics — like they would understand the concept better because they learned to 3D print it? And so I threw the question out on twitter:

It was amazing to see people reply! And share links, and ideas! I haven’t yet gotten to look through all of them, but it seems to me like there are probably two or three classes of things: (a) things that kids will make and be proud of and appreciate but didn’t actually enhance their understanding of the math (but would bring them math joy), (b) manipulatives or demonstration thingies that a teacher can make to illustrate or play with ideas (but the making of the manipulatives wouldn’t be so powerful), and (c) the thing where the actual building of the 3D model develops and enhances a kid’s sense of an underlying mathematical concept or idea (where the time spent doing the building is actually worth the payoff in understanding, rather than just using pre-created manipulatives).

So that was fun to re-engage with twitter! And when we saw that Eli Luberoff (founder of Desmos) was coming, it was fun to be able to tweet him to say everyone seemed excited!!!

I also really enjoyed starting to get to know the teachers here in person. I was actually pretty nervous about coming because I honestly think I’m going to just be alone and no one is going to want to hang out with me. But of course that’s never true, but it doesn’t make that fear any less real! (I grew up with no friends when I was younger, so I think that has just scarred me in this respect!) And to find people who want to share groan-worthy math jokes they make in their classroom while we’re eating dinner, or to talk about why someone took the leap to go from 20 year veteran classroom teacher to principal over our breakfast oatmeal, or (fill in any number of conversations here), reminded me how much I love hanging out with math teachers and geeking out.

In the morning, we do fun collaborative math problem-solving. My table (SHOUT OUT TABLE 3) spent a ton of time on this opener:

We found patterns, codified them, refined them, checked them, broke them, fixed them, posed our own questions about them, etc. It was cool because we all had different approaches and styles, but that also made it challenging. I have my kids reflect a lot about how they work in their groups — what they bring to it, what they think they can work on, etc. And so I think that in some ways we were very strong: there was a lot of idea sharing and excitement and conjecturing. But this is a wondering: I wonder if everyone felt like we were all working together to create collective knowledge. Something I tell my kids in my class is that when they’re working in groups, the goal isn’t individual understanding (that they can build at home)… it’s to build collective understanding. And so everyone has to be as invested in the thinking of others as they are in their own thinking. And I’d say that on that metric, there was a lot of interest in the thinking of others, but not for us to build some sort of group collective understanding, but rather to fill it bits and pieces of our own individual understanding. My whole thing about individual understanding versus collective understanding in my classroom… I don’t think this is actually part of the PCMI morning group philosophy, but I think it would be interesting to see if a group I was in all agreed to go in that direction and what we could accomplish both socially and mathematically. I think it would generate some really rich question-asking that would refine our own thinking and understand other people’s thinking, but also help us sortwhat we were having a surface level understanding (more pattern recognition) of to deeper conceptual understanding of (an ability to say why something worked). 

Okay now I’m really lagging, but I want to briefly talk about our “Reflecting on Practice” session. Our focus is going to be on assessments, and I think it’s going to tie into many conversations we’ve been having at my school about grade inflation. Because one of the things we’ve been talking about at our school is “what does a grade mean”, and it’s clearly an artificial construct that flattens a multidimensional thing but is super important in the larger scheme of things because grades matter (at least to my students, for a variety of reasons). And so it was interesting to think about what is “knowledge” and “how do we know that we know something” — because our leader said — we can’t really think critically about assessments until we delve into some of the philosophical underpinnings. We got some reading on assessments from NCTM’s Principles to Action which reminded me if we as teachers reframe and expand our definition and purpose of assessments, they can be much more useful in our teaching practice. 

I also went to a lecture on cryptography which was beyond my level of understanding, but the speaker was excellent and though I needed time to understand the details, I could see the larger argument and zoomed out bigger picture view of what she was sharing. 

With that, it’s 9:10pm, and I am flagging. So tired. So night night I go!

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!