General Ideas for the Classroom

The power of the feedback loop

Note: I have some phenomenal colleagues in my school. One of them gave a powerful presentation about some changes she made in her classroom, and I asked her to write a guest post on it! The kicker: she’s not a math teacher. She teaches French. But pedagogy can transcend the subject matter at hand, and this is one of those cases. So enjoy!


When I adopted a no-homework model for my classes several years ago, my role as a teacher shifted drastically. I was no longer strictly giving instruction, but rather facilitating the movement from one activity to the next and offering on-the-spot feedback and answering questions that my students might have. The goal was to remove myself from the equation as much as possible and put the students at the center of their learning. With all of the emphasis placed on class time, it became incumbent on the student to focus completely and participate thoroughly in each activity. It also became incumbent on me to come up with a system that would allow me to objectively and accurately calculate the quality of student note-taking and participation during class.

The rubric I currently use in my French classes was designed to allow for effective and efficient use of class time, which, in turn, facilitates maximum learning. It looks like this:

  • Is punctual
  • Is ready to work at the start of class
  • Takes active notes, keeps an organized notebook
  • When speaking to the teacher, uses French only
  • Engages in activities in French
  • Engages in activities for the duration of the time indicated

Each of the six components is worth 1 point per class day, for a potential total of 36 points per cycle. I designed a page that has this rubric at the top and a box for each day of the cycle underneath, and I keep a copy of it on my clipboard at all times:



Whenever a student makes an infraction, I point it out to him or her and I write it down immediately in the box corresponding to the day of the cycle. On day 1 of cycle 3, for example, I noted that three boys were not prepared to work at the beginning of class. I also collect the students’ notebooks daily and write down any issues regarding the quality and organization of their written work in these boxes as well. You can see an example of that on day 2 of cycle 3, when two boys passed in notebooks that had missing or incomplete notes. At the end of the cycle, I calculate the points lost and keep a running tally of total points in my gradebook.


In my work this year with several colleagues regarding the importance of feedback, it became apparent to me that it would be useful for my students to have the opportunity to see and discuss the breakdown of the information from these pages. So I organized a table that allows for the student to see when and how many points were lost for each component. I also included on the page the overall GPA, as well as a list of commendations, areas for improvement, and suggested challenges. I then scheduled 10-minute individual conferences during breaks and community time to discuss the results. Below is an example of one of these reports :


Student : Jean-Paul de la Montagne

Total Notes & Participation points

  • mid-semester 1 : 139/156
  • semester 1 : 97/108
Total infractions Distribution
Is punctual 1 Cycle 2
Is ready to work at the start of class 2 Cycles 3, 6
Takes active notes, keeps an organized notebook 13 Cycles 2 – 7, 9
Speaks French only (with the teacher) 0
Concentrates on activities / Engages fully in activities / Participates for the expected duration 12 (chatting, following instructions) 3-10


  Mid-semester 1 Semester Average
GPA 89.74 94.7 92.2

 Commendations :

  • accurate accent
  • ability to properly formulate full, complex sentences
  • frequently volunteers answers/comments during large group work
  • notable increase in use of French with peers

 Areas for improvement :

  • consistency in the quality of note-taking
  • drop the habit of chatting

Suggested challenges :

  • read Daniel Pennac’s L’œil du loup
  • watch movies, listen to songs in French

 This intensive participation grading model allowed me to remove subjectivity and emotion from my participation grades. It also eliminated the potential for students or their parents to debate the grade. The final step of conferencing with each of my students was the piece I’ve been missing all these years. These conferences yielded almost 100% reduction in the behaviours that hinder productivity and learning, not to mention costing students points.

My ultimate take-away from this experience is that providing students direct feedback on the quality of their notes and class participation resulted in the kind of behaviour modifications that have made for an even more effective learning environment. In a no-homework class where every minute counts, this is key. I am so excited about what this experience has taught me, and am looking forward to refining it in the future.

Gaspable Moments

Sometime last year, I started thinking about why I love math so much. And when I kept on delving, deeper and deeper past all the adjectives (beauty, creativity, awe-inspiring, structure, …) to what really was beneath those adjectives, I came up with one fundamental, visceral answer. It is the spark of electricity you get when you think you start figuring something out, and the chase that happens, until things finally click into place, and you are so excited about your discovery that you want to share it with someone else. It’s that gaspable moment — that rush of endorphins — that feeling of sheer joy.

I’m guessing if you’re reading this blog, you’ve had that happen to you.

For me, this internal joy is everything. Why I love math, below all the flowery adjectives, is because of the emotional impact it has on me when things click.

It’s an internal thing. A private thing. But what if it didn’t have to be. [1]

I had this idea last year, and I’m terrified and thrilled about introducing it on Tuesday. This idea is designed to specifically make this joy visible and public. 

Here it is…


A bell. I’m going to ask kids to ring the bell on their group tables when they have some sort of insight or discovery or revelation that simply causes them joy. I want their internal joy to become external. I want my kids to recognize that this joy is something they should be conscious of and recognize it when it happens. I want a class culture where moments of joy are acknowledged and celebrated.

When a bell rings, what happens? Nothing. Kids continue working. Near the end of class, I will ask kids “anyone who rang a bell, can you share a bit about that moment and insight that you had? and can you describe that feeling with one adjective?” Or maybe I’ll call on someone who rang a bell? I do know that I don’t want the bell to interrupt the flow of class and thinking.  Will I have the bells out every day for that one class? I will probably have the bells out a lot after I first introduce them, to start building the classroom culture around them, but then I can see me probably make more conscious choices about when the bells will be out based on what we are doing in class.

Will this work? I don’t know. I’m going to roll it out in only one of my classes. Why? Today I had a killer class. The kids were persevering and having so many gaspable moments. It was ridiculous. I went into the math office after just to tell the other teacher of the class to get psyched for his next class (he hadn’t taught that lesson yet). But because this happened, it is a perfect time to have a conversation with my kids about the joy of mathematics. (I’ve had the bells since school started… I just hadn’t found the right time to introduce them.)

I can see a number of things happening.

  1. I can see kids being too “scared” to ring the bell. Because it is public. And they might feel like they’re insights or feelings aren’t “valid” or “good enough” (compared to their classmates). Not ringing the bell has no risks, so why do that?
  2. I can see kids being too “bell happy.”
  3. I can see this not going well in the first few days, and then me abandoning this idea.
  4. I can see this becoming a positive and normalized part of our classroom.

What do I suspect? Truth be told, I’m super excited about this, but I think #3 is the most likely outcome. It’s hard to be consistent with something that doesn’t get off to a solid start, because then keeping it up even though it isn’t working well feels fruitless, and finding ways to fix things and change course is way tough.

However I will say that I started using, with this class, the red/yellow/green solocup strategy for groups to self-assess where they are in terms of their own progress, and it’s been amazing. So that gives me hope for this idea working with these kids. Wish me luck!



[1] Okay, sometimes it isn’t private. I love when a group high fives when they figure something out. That happens when it is something hard-earned. Something they worked for. I also remember years ago a kid getting so worked up about understanding the sum of angles formula for sine that he literally fell in the floor. So sometimes the joy is visible. But I suspect that a lot of the joy that kids feel (when given the right kind of tasks, which put them at a place where they can have those hard-earned moments) often has a momentary and fleeting nature. I hope a ding! can give voice to those fleeting moments.




Good Conversations and Nominations, Part II

This is a short continuation of the last blogpost.

In Advanced Precalculus, I start the year with kids working on a packet with a bunch of combinatorics/counting problems. There is no teaching. The kids discuss. You can hear me asking why a lot. Kids have procedures down, and they have intuition, but they can’t explain why they’re doing what they’re doing. For example, in the following questions…


…students pretty quickly write (4)(3)=12 and (4)(3)(5)=60 for the answers. But they just sort of know to multiply. And great conversations, and multiple visual representations pop up, when kids are asked “why multiply? why not add? why not do something else? convince me multiplication works.”

Now, similar to my standard Precalculus class (blogged in Nominations, Part I, inspired by Kathryn Belmonte), I had my kids critique each others’s writings. And I collected a writeup they did and gave them feedback.

But what I want to share today is a different way to use the “Nomination” structure. Last night I had kids work on the following question:


Today I had kids in a group exchange their notebooks clockwise. They read someone else’s explanations. They didn’t return the notebooks. Instead, I threw this slide up:


I was nervous. Would anyone want to give a shoutout to someone else’s work? Was this going to be a failed experiment? Instead, it was awesome. About a third of the class’s hands went in the air. These people wanted to share someone else’s work they found commendable. And so I threw four different writeups under the document projector, and had the nominator explain what they appreciated about the writeup. As we were talking through the problem, we saw similarities and differences in the solutions. And there were a-ha moments! I thought it was pretty awesome.

(Thought: I need to get candy for the classroom, and give some to the nominator and nominee!)

The best part — something Kathryn Belmonte noted when presenting this idea to math teachers — is that kids now see what makes a good writeup, and what their colleagues are doing. Their colleagues are setting the bar.


I also wanted to quickly share one of my favorite combinatorics problems, because of all the great discussion it promotes. Especially with someone I did this year. This is a problem kids get before learning about combinations and permutations.


Working in groups, almost all finish part (a). The different approaches kids take, and different ways they represent/codify/record information in part (a), is great fodder for discussion. Almost inevitably, kids work on part (b). They think they get the right answer. And then I shoot them down and have them continue to think.

This year was no different.

But I did do something slightly different this year, after each group attempted part (b). I gave them three wrong solutions to part (b).


The three wrong approaches were:

And it was awesome. Kids weren’t allowed to say “you’re wrong, let me show you know to do it.” The whole goal was to really take the different wrong approaches on their own terms. And though many students immediately saw the error in part (a), many struggled to find the errors in (b) and (c) and I loved watching them grapple and come through victorious.

And with that, time to zzz.

Nominations, Part I

At TMC this past summer, Kathryn Belmonte introduced an idea about sharing student work in the classroom. Something she termed “NOMINATIONS!” I loved the idea — and wanted to use it when kids do their explore-math project. But I saw it was so flexible, and pretty early on, the time was right to test it out. So I modified it slightly and this post is about that…

In all of my precalculus classes (I teach two standard sections and one advanced section), my kids are being asked to do tons of writing. A few who have had me before in geometry are used to this, but most are not. And honestly: getting down what mathematical writing is, and how to express ideas clearly, is hard.

So what do I do? I throw them into the deep end.

On day two of class, I ask them to write an answer to a problem for a seventh grader to understand. On the third day of class, they come in, and are given the name of the student who comes after them alphabetically (and the last person is given the name of the first person alphabetically). Then they read these instructions:


Everyone moves to the desk of the name they were given. Then I project on the board:


And I give students to read through a different student’s solution. They have to make sense of it — pretending to be new to the problem. And then they critique it. Eventually, probably after 3-5 minutes, I left them return to their seats. They read over the comments. I talk about why the feedback is important. And how specific feedback is useful (so “good explanation” is less useful than “your explanation of how the groups were made was easy for me to follow”). And then we continue on with class.

Here are examples of some post-its (front of a few, then back of a few):

To follow up: that night for nightly work, I gave students a writing problem — a simple probability problem. My hope was that this would help them pay attention to their explanations. I collected the problem and read through the writeups.

They weren’t so hot. Most of them didn’t talk about why and some didn’t have any diagrams or visuals to show what was happening with the problem. So I marked them up with my comments. (They got full credit for doing it.) The next day I handed them back and shared my thoughts. I also shared a copy of a solid writeup — one that I had created — along with four or five different possible visuals they could have used. (I realized –after talking with Mattie Baker about this — that I couldn’t really get my kids from point A to point B unless they saw what point B looked like, and what my expectations were.)

At this point, I wanted to figure out if they were taking anything away from all of this. So I created a page with three questions. A formative assessment for me to see what my kids understand and what they don’t about the content. But I also asked them to take all the feedback they’ve gotten about writing and explanations, and explain the heck out of these problems. Here’s an example of one of the problems (one I’m particularly proud of):


I collected them today. I haven’t looked through them carefully yet, but from a cursory glance, I saw some thoughtful and extensive writeups. And even from this cursory glance, I can see that these two activities — plus all the conversations we’re having about explaining our thinking in class — have already made an impact.

Yes, they’ve gotten some ideas of what a good writeup looks like. They know diagrams can be helpful. They know words to explain diagrams are important. They know the answer to why is what I’m constantly looking for when reading the explanations.

But more important to me is the implicit message I’m trying to send about my values in the classroom. I think a lot about implicit messaging to communicate my values, especially at the start of the year. And I am confident my kids know with certainty that I value all of us articulating our thinking as best as we can, both when speaking but also when doing written explanations.

Notes on the Start of the Year

Today was my first day with kids. I can’t tell you how terrified I was to be back. I had about a zillion normal reasons (the standards: do i still remember to teach? so many kids names to learn and i’m terrible at it! what if I totally suck?). I also have a lot on my plate right now, a few of which are out of the ordinary, which have put me in a weird headspace. #cryptic #sorry

However I had a really good day today. I saw my advisory and two of my four classes. I even went to some of the varsity volleyball game after school!

This post isn’t about my kids or my classes. It’s going to be about some things I’ve done at the start of this year.

(1) Inclusivity. I read a book about trans teens this summer. We had a lot of conversations about pronouns last year. We as a school have taken gendered pronouns out of our mission statement. Last year I included this in my course expectations:


But this year, in my get-to-know-you google form that I give to kids, I asked for their pronouns.


Chances are, I probably am not going to get any different answers that what I expect this year. But I’m not including this question for the majority of kids. I want to be ready when that first kid gives me pronouns that differ from what I may expect. I want that kid to know they can find comfort (not just safety) in our room. And I want all kids to know things that I value. And I think this question sends that message — no matter who the kid is.

That’s the idea behind it. Who knows if my intention is how the kids will understand/interpret it?

(2) Mattie Baker and I were working at a coffeeshop before the year started, and he showed me his class webpage, which had this video (which I’d seen before) on it front and center:

I loved how *real* this video felt to me. Not like something education schmaltzy which makes me want to roll my eyes. I then went searching for a twin video that explicitly talks about the growth mindset. I had a dickens of a time finding one that I felt would be good for students to watch, but didn’t seem… well… lame. I found one:

So as part of the first set of nightly work, I’m having kids watch these videos and write a comment on them in google classroom. (So others can read their comments.) As of writing this, one class has already had two kids post their comments (even though I don’t see them until next week). I read them and my heart started singing with happiness. I have to share them:


Two videos aren’t a cure-all. But having kids realize how important having a positive can-do attitude, and how important it is to look at math as a skill to be developed (rather than something you’re innately born with and is fixed) is so important to me. I have to remember to be cognizant about how important this stuff is, and how important it is to reinforce daily.

(3) In both of my classes today, one student said something akin to “I first thought this, but then I talked with Stu (or listened to the whole class discussion) and I changed my mind.” I stopped both classes and made a big deal about how important that was for me. And how those types of statements make my heart sing. And why they make my heart sing. So they should say those sort of things aloud a lot. Okay, so I said it once in each class. How can I remember to say it a lot more? In any case, it was a teacher move I was proud of.

Oh oh another teacher move… I saw when one student was sharing their thinking with the class, but not everyone was facing the student. And I remembered Mattie Baker and Chris Luzniak’s training from this summer (on dialogue in the classroom). I told everyone they had to face the person that was speaking. I need to remember next week to make this more explicit — and talk about (or have kids articulate) what they should look like when actively listening to someone. And why it’s important to give this respect to someone. They are sharing their thinking — which is a piece of them — with us. They took a risk. We need to celebrate that. And try to learn from their thoughts. Doing anything else would be a disservice to them and to our class. (Okay, clearly you can tell I’m thinking through this in real-time right now by typing.)

(4) Robert Kaplinsky has created a movement around opening classrooms up. I personally hate being observed. Before someone comes, I freak out. Of course as soon as I start teaching, I absolutely forget that they are watching. Totally don’t even recognize them as an entity. In fact, I think I often teach better, probably because I’m subconsciously aware I’m being watched so I’m hyperaware of everything I’m doing.  But leading up to it is horrible. And I also hate the idea of “surprise visits” because… well, who likes them?

That being said, I know that getting feedback is important, and I know that in my ideal school, classrooms wouldn’t be silos. So I joined in. Not for all my classes… I need to dip my toe in gently. But I posted this next to my classroom door:


Next week or the week after, I’ll probably put this up as a “do now” and ask kids “what do they notice/wonder?” about it. Then I’ll tie it into a conversation about growth mindset and the videos they watched.

(5) For the past two years, I’ve been teaching only advanced courses. (In fact, because of that, I asked to teach a standard course… I have taught many, and it was weird to not have that on my plate for two years.) And I heard from someone that a few kids were nervous about having Mr. Shah because “he teaches the really hard courses? will he be able to teach us?”

I know that my first few classes with these kids need to show them that I am different than they expected. I also was proud of this paragraph I put on the first page of their course expectations…


(6) I met my advisory for the first time. Seniors. The thing is: we’re ramping up our advisory program to be more meaningful. Advisors are going to be with their advisees for four years. We are going to be the initial point of contact for many things. And we want to be there to support and celebrate our advisees in a way that we haven’t been able to in our previous set up.

But for all this to happen, I need to form relationships with my advisees. Relationships that go beyond pleasantries. In our training for our new advisories (amazing training… I think I should write a post to archive that thinking before I forget it… done by the Stanley King Institute) we did an exercise. We found someone we didn’t know (I found a new teacher at our school). We had to think about something meaningful to us, and something real (not something like our favorite sports teams… sorry sports fans)… and then talk about it with our partner for EIGHT MINUTES. Anyone who has been a teacher knows that speaking about anything for eight minutes straight is tough. It feels like eternity. While that person is speaking, the other person is actively listening. They can say a few words here and there, like “oh yeah…” or “totally,” but it wasn’t about having a conversation.

Normally I’d roll my eyes at something like this. But at the end, I felt like I got to know this new person at our school pretty well… actually, considering we only had eight minutes, amazingly well… and we bypassed all the initial superficial stuffs. That stuff, like movies and books and stuff, we’ll get to later. Yes, it was awkward. But yes, it worked.

So here’s how I adopted it for my advisory. I met with them today to do a bunch of logistics, and then I took them to a different room. I had cookies, goldfish, crackers, and a cold drink for them. And I explained this exercise. And I said: “I want to do this with you. I want to get to know you.” And so I took out the notecards I prepared, and I shared stuff about my life with them. And they were rapt. I told them about stuff going on in my family that was exciting and stuff going on that was tough, I told them “things I wish my students knew” (this is such a great way to flip “things I wish my teacher knew”). I told them my total anxiety for the start of the school year and why I had it, and I told them my total excitement for the start of the school year and why I had it. I even said: “I never feel like I’m a good enough teacher.” When I was saying that, I wondered how many kids think “I never feel like I’m good enough.”

A photo of my index cards are here… but I only used them as launching points. I didn’t want to be rehearsed.


I’m a guarded person, and I made sure never to cross the line between personal and professional, but when I finished, I sensed some (all?) of them were processing that a teacher opened up to them in this way. A few thanked me for sharing with them.

I wanted to set up an initial connection, and send the message I want you to know that I’m not an advisor in name only… I’m opening up to you because I want you to believe that when you’re ready, you can open up to me. They’re seniors. They have a lot figured out. But I hope they know I’m here for the stuff they don’t have figured out.

In the next week and a half, I have 10 minute meetings with all of my advisees individually. I told my kids they are going to talk to me about what’s meaningful to them for 8 minutes. I acknowledged it would feel awkward. I told them they didn’t need to open up in any way that made them feel uncomfortable. But I wanted them to speak about whatever is meaningful to them. We’ll do favorite books later. Now I’ll get to know them on a more personal level. [1]

(6) As you might have noticed from #4 above, I’m trying to be better about formative assessments. I want to make sure I know what kids are thinking, and where they are at, and use that to refine or alter future classes. I haven’t tried this out yet (today was just our first day!!! I only saw two classes!!!), but I made a google form for exit tickets.


This is a #MTBoS sample version, so feel free to click on it, and fill out fake feedback to get an idea of the form.

Pretty awesome idea, right? I didn’t want to have a bunch of pre-printed slips (something I knew I wouldn’t actually do).

(7) I took a page from Sara Van DerWarf’s playbook. I didn’t do this on the back of name tents, but I have a separate sheet that they’re filling out. For my two classes today, I asked them to share something about themselves that would help me get to know them as something other than a kid in our math class. Some kids gave a lot, some a little, but I learned something about each one of my kids. As I’ve mentioned, I’m terrible with names. But what’s nice is on this sheet I created, I put photos (they’re in a school database for us to use) and knowing something about them is helping me remember their names. It’s odd and unexpected and lovely. Kids interested in arts/photography/social justice/sports/debating-arguing/nature/etc. I liked writing that little note back to my kids. I don’t know what question I’ll ask next. I may ask them “Math is like…” (like James did). For the penultimate one, I should definitely take a cue from Sara and ask them to ask ME a question.

I have a few more ideas for posts percolating. I hope that I get the time and motivation to write them. But it’s nice to be back!!! SCHOOL IS IN SESSION!


[1] This is what I included in my email to my advisees:

I know it may feel awkward, but when you meet with me during this meeting, you’re going to speak for 8 minutes about things that are meaningful to you. So something more than a listing of your favorite books/movies. If you need help thinking about this: what makes you tick? what makes you gasp? what are your thoughts about senior year and the future? what could you not imagine doing? what are you feeling? what keeps you up at night? These are all questions that might help you find things that are truly meaningful to you. I found it really helpful to have an index card of things when I was talking with you, because I was nervous. I suggest doing that!


Pitching college math courses

Ooops. This turned out to be a post with no images. So here’s a TL;DR to whet your appetite: I wanted to expose my seniors to what college mathematics is, but instead of lecturing, I had them “pitch” a college course to the rest of the class.

My multivariable calculus courses was coming to an end, and I got some questions about what college courses in math are about. It reminded me of a comic strip I read years ago, which I frustratingly can’t find again. It has an undergraduate going to meet with his math professor adviser, saying something like “I want to major in triple integrals.” Which is crazy-sounding — but maybe not to a high school student who has only ever seen math as a path that culminates in calculus. What more is out there? What is higher level math about? (These questions are related to this post I wrote.)

So here’s what I told my students to do. They were asked to go onto their future college math department websites (or course catalog), scour the course offerings, and find 3-4 courses that looked interesting and throw these courses down on a google doc.

It was awesome, and made me jealous that they had the opportunities to take all these awesome classes. Some examples?


After looking through all the courses, I highlighted one per student that seemed like it involved topics that other students had also chosen — but so that all the courses were different branches/types of math. I told each student to spend 10-15 minutes researching their highlighted course — looking up what the words meant, what the big ideas were, finding interesting videos that might illustrate the ideas — so they can “pitch the course to the class” (read: explain what cool math is involved to make others want to take the course).

I’m fairly certain my kids spent more than 10-15 minutes researching the courses (I’m glad!). Each day, I reserved time for 2-3 students to “pitch” their courses. And since some of the ideas were beyond them, after the pitches, I would spend 5 or so minutes giving examples or elaborating on some of the ideas they covered.

If you want to see the research they did for their pitches, the google doc they chucked their information into is here.

Some fun things we did during the pitches?

(1) We watched a short clip of a video about how to solve the heat equation (that was for a course in partial differential equations)

(2) I showed students how to turn a communication network into a matrix, and explained the meaning of squaring or cubing the matrix (this was for a course on network theory)

(3) A student had us play games on a torus (a maze, tic tac toe) (this was for a course on topology)

(4) I had students store x=0.3 on their calculators. Then I had each student store a different “r” value (carefully chosen by me) and then type r*x*(1-x)->x in their calculators. They then pressed enter a lot of times. (In other words, they were iterating x_{n+1}=rx_n(1-x_n) with the same initial conditions but slightly different systems. Some students, depending on their r value, saw after a while their x values settle down. Some had x values that bounced between two values. Some had x values that bounced between four values. And one had x values that never seemed to settle down. In other words, I introduced them to a simple system with wacky wacky outcomes! (If you don’t know about it, try it!) (This was for a course on chaos theory)

(5) A student introduced us to Godel’s incompleteness theorem and the halting problem (through a youtube video)

It was good fun. It was an “on the spot” idea that turned out to work. I think it was because students were genuinely interested in the courses they chose! If I taught a course like AP Calculus, I could see myself doing something similar. I’m not sure how I would adapt this for other classes… I’m thinking of my 9th grade Advanced Geometry class… I could see doing something similar with them. In fact, it would be a great idea because then they could start getting a sense of some of the big ideas in non-high school mathematics. Kay, my brain is whirring. Must stop now.

If anyone knows of a great and fun introduction to the branches of college level math (or big questions of research/investigation), I’d love to know about it. Something like this is fine, but it doesn’t get me excited about the math. I want something that makes me ooh and ahh and say “These are great avenues of inquiry! I want to do all of them!” I think those things that elicit oohs and ahhs might be the paradoxes, the unintuitive results, the beautiful images, the powerful applications, the open questions… If none exists, maybe we can crowdsource a google doc which can do this…

Reading? For math class?

This year, our school adopted this weird rotating schedule where we see our classes 5 times out of every 7 days. And four of those times are 50 minute classes and one of those times is a 90 minute class.

I didn’t have a clear idea of what to do in multivariable calculus for the block. I still had to cover content, but I wanted it to be “different” also. After many hours of brainstorming, I came up with a solution that has worked out pretty well this year.

We had a book club.

The 90 minute block was divided into 50 minutes of traditional class, and 40 minutes of book club. (Or 60 minutes of class, and 30 minutes of book club.)

Now, to be clear, this is a class of seven seniors who are highly motivated and interested in mathematics. I can see ways to adapt it in a more limited way to other courses, with more students, but this post is about my class this year.


We started out reading Edwin Abbott’s Flatland.


Why? Because after they read this, they understand why I can’t help them visualize the fourth (spatial) dimension! But it convinces them that they can still understand what it is (by analogy) and makes them agree: if we can believe in the first, second, and third spatial dimensions, why wouldn’t we believe in higher spatial dimensions too? It’s more ludicrous not to believe they exist than to believe they don’t exist! A perfect entree into multivariable calculus, wouldn’t you say?

After reading this, we read the article “The Paradox of Proof” by Caroline Chen on the proposed solution to the ABC conjecture.


This led us to the notion of “modern mathematics” (mathematics is not just done by dead white guys) and raised interesting questions of fairness, and what it means to be part of a profession. Does being a mathematician come with responsibilities? What does clear writing have to do with mathematics? (Which helps me justify all the writing I ask for on their problem sets!) It also started to raise deep philosophical questions about mathematical Truth and whether it exists external to the human mind. (If someone claims a proof but no one verifies it, is it True? If someone claims a proof and fifty people verify it, is it True? When do we get Truth? Is it ever attainable? Are we certain that 2+2=4?)

At this point, I wanted us to read a book that continued on with the themes of the course – implicitly, if not explicitly. So we read Steve Strogatz’s The Calculus of Friendship:


What was extra cool is that Steve agreed to sign and inscribe the book to my kids! The book involves a decades long correspondence between Steve and one of his high school math teachers. There are wonderful calculus tricks and beautiful problems with explanations intertwined with a very human story about a young man who was finding his way. Struggling with choosing a major in college. Feelings of pride and inadequacy. The kids found a lot to latch onto both emotionally and mathematically. Two things: we learned and practiced “differentiating under the integral sign” (a Feynman trick) and talked about the complex relationship that exists between teachers and students.

After students finished this book, I had each student write a letter to the author. I gave very little guidelines, but I figured the book is all about letters, so it would be fitting to have my kids write letters to Steve! (And I mailed the letters to Steve, of course, who graciously wrote the class a letter back in return.)

Our penultimate reading was G.H. Hardy’s A Mathematician’s Apology:


I went back and forth about this reading, but I figured it is such a classic, why not? It turned out to be a perfect foil to Strogatz’s book — especially in terms of the authorial voice. (Hardy often sounds like a pompous jerk.)  It even brought up some of the ideas in the “Paradox of Proof” article. What is a mathematician’s purpose? What are the responsibilities of a mathematician? Why does one do mathematics? And for kids, it really raised questions about how math can be “beautiful.” How can we talk about something that is seen as Objective and Distant to be “beautiful”? What does beauty even mean? Every section in this essay raises points of discussion, whether it be clarification or points that students are ready to debate.

What is perfect about this reading is at the same time we were doing it, the movie about G.H. Hardy and S. Ramanujan was released: The Man Who Knew Infinity (based on the book of the same name).

Finally, we read half of Edward Frenkel’s Love and Math:


Why? Because I wanted my students to see what a modern mathematician does. That the landscape of modern mathematics isn’t what they have seen in high school, but so much bigger, with grand questions. And through Frenkel’s engaging telling of his life starting in the oppressive Russia and ending up in the United States, and his desire to describe the Langland’s program understandably to the reader, I figured we’d get doses of both what modern mathematics looks like, and simultaneously, how the pursuit of mathematics is a fully human endeavor, constrained by social circumstances, with ups and downs. Theorems do not come out of nowhere.Mathematicians aren’t the blurbs we read in the textbooks. They are so much more. (Sadly, we didn’t read the whole thing because the year came to a close too quickly.)


I broke the books into smaller chunks and assigned only them. For Flatland, it might have been 20-30 pages. For Love and Math or A Mathematician’s Apology, it might have been 30-50 pages. We have our long block every 7 school days, so that’s how much time they had to read the text.

At the start, with Flatland, students were simply asked to do the reading. Two students were assigned to be “leaders” who were to come in with a set of discussions ready, maybe an activity based on something they read. And they led, while I intervened as necessary.

For every book club, students who weren’t leading were asked to bring food and drink for the class, and we had a nice and relaxing time. On that note,  never did I mention anything about grades. Or that they were being graded during book club. (And they weren’t.) It was done purely for fun.

Later in the year, I had students each come to class with 3-4 discussion questions prepared, and one person was asked to lead after everyone read their questions aloud.

The discussions were usually moderated by students, but I — depending on how the moderation was going — would jump in. There were numerous times I had to hold back sharing my thoughts even though I desperately wanted to concur or disagree with a statement a student had made. And to be fair, there were numerous times when I should have held back before throwing my two cents in. But my main intervention was getting kids to go back to the texts. If they made a claim that was textually based, I would have them find where and we’d all turn there.

Sometimes the conversations veered away from the texts. Often. But it was because students were wondering about something, or had a larger philosophical point to make (“Is math created or discovered?”) which was prompted by something they read. And most of the times, to keep the relaxed atmosphere and let student interest to guide the conversation, I allowed it. But every so often I would jump in because we had strayed so far that I felt we weren’t doing the text we had read justice (and we needed to honor that) or we were just getting to vague/general/abstract to say anything useful.


I mentioned students generated discussion questions on their own. Here are some, randomly chosen, to share:

  •  Strogatz talks about how math is a very social activity. We see this exemplified in the letters between Steve and Mr. Joffray, but where else do we see this exemplified in math? (papers, etc.) How do you think Strogatz might have felt about Shinichi Mochizuki’s unwillingness to explain his paper and proof to the math community?
  • What do you think about Strogatz and Joff using computer programs to give answers to their problems? Are computers props, and their answers unsatisfying? Or are they just another method, like Feynman’s differentiating under the integral?
  • Do you like A Square? In what ways is he a product of his society? Does he earn any redeeming qualities by the end of the book?
  • Can you draw any connections between things in Flatland and religion? Do you think Abbott is religious? Why/why not?
  • When we first read about Mochizuki’s ABC Conjecture, we debated whether or not math is a “social” subject. Perhaps many mathematicians do much of the “grind” work on their own, however, throughout everything we’ve read this year, there has been one common link when it comes to the social aspects of math: mentorship. It appears to me that all of the great mathematicians we know about have been mentored by, or were mentors others. In what ways have Frenkel’s mentors – he’s had a few – had an influence on the path of his mathematical career? Do you think he would/could be where he is today without all of those people along the way? Can you think of any mentors that have had a profound influence on your life? (The last one can just be a thought, not a share.)
  • Frenkel talks about the way in which math, particularly interpretations of space and higher dimensions, began to influence other sectors of society, specifically the cubist movement in modern art. This movement was certainly not the first time math and science influenced art and culture – think about the advent of perspective in the Renaissance and the use of technology on modern art now – however math and art are often thought as opposites and highly incompatible. Why do you think that people rarely associate the two subjects? Would you agree that the two are incompatible? Can you think of other examples of math/science influence art/culture/society?


In many ways, I felt like this was a perfect way to use 30 minutes of the long block. After doing it for the year, there are a few things that stood out to me, that I want to record before summer hits and I forget:

(a) I think students really enjoyed. It isn’t only a vague impression, but when I gave a written survey to the class to take the temperature of things, quite a few kids noted how much they are enjoying the book clubs.

(b) For the post-Flatland book club meetings, I need to come up with multiple “structures” to vary what the meetings look like. Right now they are: everyone reads their discussion questions, the leader looks for where to start the discussion, the discussion happens. But I wonder if there aren’t other ways to go about things.

One example  I was thinking was students write (beforehand) their discussion questions beforehand on posterpaper and bring it to class. We hang them up, and students silently walk around the room writing responses and thoughts on the whiteboard. Then we start having a discussion.

Or we break into smaller groups and have specific discussions (that I or students have preplanned) and then present the main points of the discussion to the entire class.

Clearly, I need to get some ideas from English teachers. :)

(c) I love close readings of texts. I think it shows focus, and calls on tough critical thinking skills. At the same time, I need to remember that this is not what the book clubs are fundamentally about. They are — at the heart, for me — inspiration for kids. So although for Flatland I need to keep the critical thinking skills and close readings happening, I need to remember (like I did this year) to keep things informal.

(d) Fairly frequently, I will know something that is relevant to the conversation. For example, I might talk about of the math ideas that were going over their heads, or about fin de siecle Vienna, or branches of math that might show how the line between “theoretical” and “applied” math is blurry at best. I have to remember to be judicious about what I talk about, when, and why. We only have limited time in book club, so a five minute tangent is significant. And one thing I could try out is jot down notes each time I want to talk about something, and then at the end of the book club (or the beginning of the next class), I could say them all at once.

(e) I usually reserve 30 minutes for book club. But truthfully, for most, 40 minutes turned out to be necessary. So I have to keep that in mind next year when planning class.

(f) Should we come up with collaborative book club norms? Should I have formal training on how to be a book club leader? Should we give feedback to the leaders after each book club? Can we get the space to feel “safe” where feedback could actually work?

And… that’s all!