A little graph theory

Basically, some graphs are the same. Basically.

Like these two:

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And if you don’t believe me, pretend that you tangled the right graph. You end up with something basically identical to the left one.

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Straighten out both of these, and you get just a straight line, or a chain. That’s another way of seeing that they’re both (basically) the same:

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Here is another pair of graphs. They’re also basically the same, i.e. isomorphic!

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I like imagining swinging around the parts of these graphs to convince myself that they really are the same.

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I took the above examples from the truly fantastic Introduction to Graph Theory by Richard Trudeau. I found it lying around the math department office and have been carrying it around since. (Though I get why they changed it, the original title was “Dots and Lines” which is awesome.)

Here are a few more of Trudeau’s puzzles. In each pair, are the graphs isomorphic (i.e. basically the same)?

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You can check yourself by playing with the diagrams digitally, trying to drag the points around to change their appearances. Here are links to all of the diagram pairs I’ve so far shared:







I love the idea of opposites in math, and there is a great way to think about what the opposite of a graph should be. The fancy term is “complement” but I like thinking of every graph as having an “anti-graph.” Here are some examples:

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If you overlay the graph and its anti-graph, the result should be a completely connected graph. Meaning, a graph’s complement should consist of just the edges that are missing from the original.

Now, here is an AWESOME question: are any graphs the same as their anti-graphs? Are any graphs their own opposites? One last way of putting the question, to maximize googleability: are any graphs self-complementary?

The answer is, definitely! Mess around with the graphs in the image above to see what I mean:


One way to start looking for self-complementary graphs is by thinking about the number of edges that a graph with n dots can have, if it is going to be (basically) the same as its anti-graph. After all, the complement can’t have more edges than the original graph…

And then it’s fun to think about how many vertices (dots) a graph can have if it’s going to evenly split its edges between the graph and its complement. For instance, if you have 6 vertices there is a maximum of 15 edges — so there’s no way any graph with 6 vertices could be self-complementary, because there’s no way for a graph and its complement to have an equal share of 15 edges.

It’s fun to look for both of the 5-vertex graphs that are self-complementary.

It’s fun to ask how many graphs that look like empty rings (i.e. a regular polygon) are self-complementary. There’s at least one…

And those are all the fun things that I know about self-complementary graphs. I know it’s not a ton, but nearly all of it can be shared with young children.


This is my post critiquing National Board Certification for Teaching


This hardly seems worth writing, except that so few people write about this stuff.

Six, maybe seven years ago, I started thinking about what it would take for me to teach in public schools. I had already been teaching for a couple years, and the idea of taking time off of teaching to get a teaching degree…I couldn’t convince myself it was financially feasible, and it seemed like it would be a bore, compared to teaching.

Somewhere along the line I tossed off a doomed application to NYC’s teaching fellows program. I remember writing something, like hey, you could use a teacher with some experience, I need a teaching degree, you scratch my back I scratch your’s. Dear applicant: no. 

For a while NY had an independent pathway towards certification that seemed possible, but then they discontinued it.

I kept on reading this bit of the certification website, making sure I wasn’t misunderstanding it: “An applicant who possesses a National Board for Professional Teaching Standards (NBPTS) certificate may obtain an Initial New York State certificate in a comparable title through the National Board Pathway.” This seemed like exactly what I needed.

So, four years ago, I started the process. They were revising the NBPTS portfolios, so I could only do it one bit at a time.

The math test was my first encounter with a Pearson Testing Center. I tried to prepare for the exam by cramming some calculus that I was rusty on. The entire test day was surreal. Went into a surprisingly small office in midtown Manhattan. I was imagining that it would be like when I took the SATs, that a whole crew of stressed out teachers would be sitting for an exam simultaneously. Nah, it’s more like a self-service gas station. Put your belongings in a cubby. Sign here. Here is your computer. Here is your sheet of plastic and a dry-erase marker. Boop. Time’s up. Have a great day.

Component 2 was my first experience with the written stuff. It was then that I learned my most important NBCT lesson: how to condense text.

How little I knew about condensing text when I began NBCT! This is from my first draft of my C2 written commentary:

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Awful, right! I mean, look at all that space. Here is what I ended up submitting, after getting feedback from a couple NBCT geniuses:

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I passed C2. The next year was C3, the video portfolio. This was annoying because you couldn’t do any preparatory work until you had the video, and the little camera that I had set up would constantly run out of battery in the middle of the lesson.

The hardest thing about the videos was that you needed them to provide evidence for exactly what NBCT was assessing you on. I felt like it was hard to capture a video that gave them exactly what I wanted. Here is the feedback I got from NBCT after I received my passing score (3.375) on the portfolio:

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OK, yes, there is irony in the quality of the feedback that NBCT gives you. Good luck parsing any of that. I just read “evidence of insight on your future instructional practices” three times to figure out if I can figure it out — not yet.

That left Component 4, which was no question the worst component. It’s sort of a mess. There are three parts, each calling for exactly the right kinds of evidence, and the three parts have very little to do with each other. It’s like three mini-portfolios glued together. I hated it, but I did it, and it’s done.

It’s done — I passed.

If a teacher tells me that they are NBCT, I think I know something about that teacher. They’re hard-working, because NBCT is a lot of work. They are likely ambitious, probably not on their way out of the profession.

All this I know because NBCT was a ton of work. I can’t imagine a teacher going through this without something pushing them — either a financial incentive or something internal.

So I know they’re hard-working and committed to teaching, but that’s pretty much all that I know. Nothing about the NBCT process gives me any confidence that it was assessing the quality of my teaching in any sense at all.

I have a couple friends who have been on the other end of things, assessing candidates. I believe them when they tell me there’s a clear difference in quality between different candidates. But having done all the work, I have trouble seeing exactly how you can tell the difference between a candidate who just didn’t understand the prompts and someone whose teaching meets the standards. Because it was really hard to figure out what the prompts were calling for — that was a lot of the work.

Maybe I’m just in a grouchy mood. Even though I love working at my school — public school is going to have to wait — I’ve been feeling a bit down lately.

It all feels sort of bad right now. Writing’s bad, I won’t even edit this piece. Bad at math. Kids hate math, though kids like class. Small apartment, we try not to flush in the AM because it might wake up the kids. Kitchen’s small, fridge is small, always catching mice.

Education can be so, so dumb so often, math education in particular. The dumb stuff is the most lucrative. Teachers seem to love this stuff, though, so what am I doing? All the people I knew teaching math six years ago are off doing other stuff.

But I got this certificate, and now I’m NBCT, and I also have a letter from NBCT saying “your voice matters,” so there’s that.


A quick shout out to proteacher.net. The people on there are the best. If you have questions about NBCT you should absolutely hop on there and make an account. If you’re starting NBCT, you should go there and make an account. The people there were just ridiculously generous with their time and it’s a lovely corner of the internet of teachers. That’s my only useful piece of advice for NBCT.

Learning is Weird


There I was, helping Samantha with some subtraction, when I hear another kid nearby — Lena — cracking up, really losing it. Lena was laughing, and though I try to ignore her, she’s laughing persistently. Lena turns and looks at me with a huge, ridiculous smile across her tiny little third grader face.

“It’s just zero!” she says.

“Yep,” I say. I force a smile.

“It’s just zero!” she says it again. I try to grin convincingly back, as my mind races. What’s so funny?

“Haha, that’s right,” I replied, hoping that I sounded sort of like a human does when they get a joke.

For context, here is what Lena was working on: a big-fat subtraction worksheet. Here is a sampling of some of the hilarious problems I’d included on the page:

120 – 30


21 – 2

Don’t forget:

110 – 60

And this classic:

8 – 3

You may also notice that this list of uproarious problems seems a bit on the easier side for third graders. For Lena (and Samantha) it was not. Subtraction has been coming exceedingly slowly for these kids — much slower than their multiplication, actually. It’s February, so we’re not anywhere near the finish line. Even so, I’m beginning to start to anticipate to realize that my time with my students is, ever so slowly, slipping away. I want these kids to have a good year next year in math, to be happy about school. I don’t want this to gnaw at me over the summer.

Anyway, Lena is cracking herself up so I have to go over and see what she’s up to. I look at her page. Suddenly, I’m in on the joke.


You see Lena subtracts digit by digit, because someone taught her to do that. I don’t know exactly what to say — it’s not wrong, and she is so shaky with so much subtraction. It gets her in trouble with problems like 17 – 8, because she brings the 1 down unnecessarily. Still, it’s something to work with.


But the thing is that she really needs to focus on each digit with all her attention. She can’t yet take that step back to see the problem as a whole. So there she is, with 251 – 251. Carefully, slowly, she considers each digit:

2 minus 2 is…0.

5 minus 5…0 as well.

1 minus 1…wait a second…

And there you go, there’s the joke, it’s just zero.


Ooh, by the way, Samantha is pretty interesting too.

Samantha also does that column-by-column thing, and it serves her well until she gets to problems like 125 – 50, since you can’t take away 5 from 2.

She started the year trying to borrow in these situations, but she really lost all sense of gravity as soon as she got permission to mess with the numbers. She’d do some of the weirdest things I’d ever seen with subtraction — I can’t remember them, they’re so weird. All I remember is that a bunch of times she would proudly shove a piece of paper in front of me and with, like, innocent puppy eyes, ask, “Is this right?”

And 100% of the time the paper would look like this:

125 – 50 = 972

Seriously! It was all over the place.

My take is that Samantha’s brain is just overloaded when she tries to keep track of all the parts of these problems. Every stage of it requires understanding and attention. She uses a strategy to compute 12 – 5, to take away 1, to realize that this leaves 0, to turn the 2 into a 12, to realize that this is, you know, subtraction so it should make things smaller, etc., etc.

I don’t think she should be going all-in on borrowing yet, not until she has a bit more knowledge to rest on.

But what do we do for her? Samantha asks for lots of help, and until recently I’ve been a bit stumped about how to help her.

I think I might have figured it out, though. The other day Samantha comes over to me, once again stumped on a problem. Her paper looks like this:


I have a false start, going into some totally different strategy for subtracting. Whatever, she gets that far-off stare, she can’t deal with all of it. It’s another way of thinking — it’s not her way of thinking which — for better or for worse — is column-by-column subtraction.

I think, and then I have an idea. She can, I know, subtract two-digit numbers — it’s laborious, but she can do it. So I write an example next to the problem on her page. How about this, I say?


OK, this actually makes sense to her! She uses it to work on the original problem. I offer to give her some more questions to practice — she completes each, surprised that she’s handling the problems correctly.

Is there more to notice here? Sure there is. She should know that the “32” in “324” means 320,  she should know how to handle 320 – 150 without drawing little lines, and down the line I sure hope that 32 – 15 doesn’t take quite so much out of her.

But has she learned something? By any fair reckoning, of course she has.


Math class should be joyous, they say, full of laughter and insight. I agree! But it seems that a lot of people in education go further, as they’re eager to point you to the source of classroom joy. See this? It’s a picture of kids smiling while studying math. Want it? You’ve got to try instructional practice X, Y and Z.

I promise, you, though, that kids and learning are weirder than that. You’ll plan for fun, and they’ll hate it. The next day you’ll run out of fresh ideas, open a new browser window, type in www dot kuta software dot com slash free dot html, print out worksheets with answer keys, sort of just push them over the desks until each kid has a sheet nearby, then mumble incoherently for a couple of minutes when all you’d really like to say is “here is this, I’m sorry, please do it” and you’ll brace for the worst…

…and that will be the day when everyone is having a blast with math, even Tobias, which is surprising because Tobias has just been sitting there quietly since October when he broke up with Julia, and like you told his mother it’s been very tricky to get him to open up, but there he is chatting about exponent rules with Harry, and he seems alive and (to be honest) happy in a way that you haven’t seen him in a long time.

(In case you missed it, we moved from third to eighth grade with that last bit.)

All of this is to say that joy and humor in a classroom can come from where you’d least expect it — depending on what you expect.

And Samantha? Well, people will also tell you that you need to listen to the ideas of students, to truly build on their thinking, not to override their thinking but to build on it.

I agree. But what does it look like to build on how your students think? What if your student thinks about a problem in a way that isn’t just wrong, but wrong in the wrong way? It’s not just that her technique is incomplete, but it feels like a trick, like a machine that was designed to perform half the job, like a car that can only turn left?

I’m not always sure that I understand the difference between procedural and conceptual knowledge, but I think Samantha’s case is clear. She has a (half-working) procedure married with a not-quite-there-yet conceptual grounding. Is this a time to accept what she knows and to develop it? Or to dismiss her approach and bring her back to square one, conceptually speaking? Is this an exception to the rule — a time when we shouldn’t build on what she knows, but should instead sort of veer around her structures and start construction on a new lot?

Learning is weird — it will surprise you. Procedures can be a start. Subtraction can be hilarious. Go ahead, come up with a theory about how all of this works, but be ready to find out that something entirely different gets the same results. Share what you’ve found, and then also have the humility to know that something quite different might work as well.

I love being able to laugh about math with kids, and learning how kids think is just about my favorite part of this job. I love that so many people in education want classrooms to be joyous places where children feel understood — I want that too. But if you find yourself setting terms on how this can happen or what this looks like, please proceed with caution: it doesn’t look just one way.

Talking About Diversity in Education with Marian Dingle

Marian Dingle is one of my favorite bloggers, though she’s only written two blog posts — I hope for many more! The most recent of the two is a couple of things at once, including an expression of the idea that people have this somewhat strange, absolutely fundamental desire to be understood by someone else. Her first post asked a simple, basic question: why should we care about diversity? That question forms the basis of this conversation. Marian has been teaching for 18 years in Maryland and Georgia and tweets thoughtfully from @DingleTeach.


I thought we could start by talking about your blog post about TMC [the conference] and diversity, and your question — why should we be more diverse? Do you feel like MTBoS [the community] or TMC is any closer to answering that question for itself than it was last summer, when you asked it?

Because TMC meets only once a year, answering the question has been difficult. I’ve participated in online communication with a subset of folks, but it’s just not the same as face-to-face discussion with a larger group.

I’ve also had several fairly deep online conversations with people one-on-one, and people are in various stages of comfort in answering the question. I do wonder how much of a concern the question is for the majority of MTBoS, though.

I don’t know if you’re counting me in your tally of people you’ve had fairly deep conversations with, but I’ve felt that you’ve pushed me to think more clearly about diversity in a lot of ways.

For example, there was a certain point in this past year when I said to myself, “Diversity matters because it’s a fancy name for affirmative action, and affirmative action is a good thing.” But when I shared that with you, I think you essentially told me, nobody wants to feel like they’re an affirmative action case — they want to feel valued for who they are.

Do you think there’s a way for a focus on diversity without making people feel like affirmative action cases?

My point was that everyone wants to feel valued. I’d much rather be sincerely invited for my value than my presence tolerated.

It’s all about the why. There are plenty of educators at TMC who do not work or teach with any people of color. It’s important to me that they, or other non-persons of color, are really clear of their reasons. It’s unfair for people of color to have to prove that they belong in a group of self-described like-minded mathematics educators, when we have to do this in nearly every other facet of our lives.

Have there been times when you’ve been in school and thought, this is something that just wouldn’t happen if diversity were better in this space?

Maybe it’s a bum question. Feel free to pass on it. Or to tell me the question you wish I had asked.

Question: Tell me about a time when you felt your ability or competence as an educator was questioned because of your color. Was it an issue of diversity?

In my ninth year of teaching, I began teaching third grade in an affluent, mostly White school and district with few teachers of color. Every January, students were tested for eligibility for the gifted program, which began in fourth grade in math. Both teachers and parents could recommend students for testing. While my competence had been questioned in different ways, it peaked when I recommended “too many” students, which included students of color — this per the gifted teacher and my teammate.

Some of those kids did end up qualifying for the program, and I later discovered that the gifted teacher had been asked to vacate her position since the students were performing poorly. Fate intervened, and I was asked to assume her position. Whispers ensued followed by interrogation about my qualifications from colleagues. Apparently, this was a coveted position.

To avoid a similar situation with parents, my principal agreed to a meeting with parents to introduce me and a brief question and answer. That summer I held a math boot camp for the fourth graders, and parent conferences for all. It was my attempt to make them comfortable with me, but also to gain insight into their expectations. The vetting process continued with the fifth graders, who also asked me if I was in a gifted program as a student, and if I’d skipped any grades. I was also given math problems to do.

Things went well, but my experience highlights what we have to go through. No one asked me to do any of the “extras”, but I knew they had to be done to be considered equal. After I proved myself, students and parents saw my worth, and the program grew. Had there been more teachers of color before me, that road may have been easier. It’s never easy being first.

I want to get back to the idea of a quota, and how that could be problematic. Suppose that this school — or TMC, or some other space in education — decided that they wanted to recruit more Black or Hispanic teachers, and so they created a quota. I hear you as saying that this would create a situation where the teachers filling the quota feel they have to prove themselves, to prove that they belong.

Did I get that right? You said earlier it’s all about “the why.” How then does “the why” fit into this?

As far as TMC, I’m saying that even before we hit the quota stage, there has to be an articulated reason for the quota in the first place. What’s the benefit? For whom? Is it for political correctness? I think some have an answer to this, but most don’t. At least not an answer that they are comfortable with. That’s the work – to really get at *why* this is a goal. Does that make sense?

If it’s about a school, then there’s a history of eliminating and limiting the number of teachers of color, and yes, a quota would seem to remedy that. Still, if it is only about filling that quota, it’s not sustainable. Experiences similar to mine and much worse will likely ensue. Some forethought should be given about retention.

I think I see what you mean. Like, if the purpose of a quota was “we have a lot to learn from Black educators” then that diversity policy would not put those educators on the spot. Like, the whole premise of the policy would be that Black educators are valuable, so those educators wouldn’t have to prove themselves. But if the purpose was something that didn’t affirm the value of those educators, there would be a situation where people have to prove themselves. Is this sounding like your take?

In part. But even if it’s “we have a lot to learn from Black educators” then there is the question of who is benefiting. What exactly can be learned? Why can that knowledge be obtained from teachers of color? And who does that benefit? Care should be taken with that line of reasoning.

Sometimes answers to the “why diversity” question can border on exploitation, and it’s a kind of exploitation which is pretty pervasive in schools now. For example, Black male teachers are often assigned discipline duties because “they are good at it” and Black educators are given more students of color because “we are so good with them”.

So, to come to TMC and be expected to “teach how to teach Black kids” is another form of what we experience daily. No thanks.

Right. “Hey, we really want you to come to this conference so that you can teach us about how to handle diversity and racism issues.” That seems to me a close parallel to the exploitation you’re talking about in schools.

Can you imagine a “why” that would justify a demographic quota for you? The alternative seems to me for diversity to be more about a culture change that indirectly changes the demographics of a community or a conference, which feels a bit more slippery to me.

Honestly, I don’t think I’m comfortable with a quota, which is not to say that one isn’t needed. What I’d like to imagine is that it happens naturally. That is, I wish it would happen without being forced. Yes, we need to have uncomfortable conversations, but then after the smoke clears, it should get easier. Naive, perhaps.

I am still advocating for articulating a reason why we believe in diversity. And then, if it’s needed, we should take steps to make that happen, a quota being one option among others.


[Marian sends me Through Our Eyes: Perspectives and Reflections from Black Teachers.]

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This looks really cool and interesting. Does it all resonate with you?

I just skimmed it but pretty much. I’d add that I was once in an all-Black space: administration, staff, students, families. For 3 years, my children and I lived without racism directly affecting me in the workplace or my children in their schooling. I’ve missed that kind of freedom.

This reminds me of something I first learned about from Dana Goldstein’s Teacher Wars, that a lot of Black teachers left the profession after Brown v. Board of Ed, because before the decision they had been teaching in all-Black schools. Mandated integration also greatly expanded the opportunities for racism.

Right. Black teachers were not hired at White schools — explicitly told they weren’t qualified to teach White students. That began the Black teacher shortage. Black schools were closed, Black teachers were fired, and Black students entered White schools.

I know that this isn’t the same because Jews and Black people are in vastly different social situations in the US, but college was my first time out of an all-Jewish environment, and my first teaching job was in an all-Jewish school. I really like my school now, but my wife teaches at an all-Jewish school and all the time I’m like, ok, how amazing would it be to teach in a place where I don’t have to explain myself? Do you ever think about going back to work at an all-Black space?

I completely understand that feeling.

I won’t go back to that particular space, due to other reasons, but I have imagined what it would be like somewhere else. At this point in my life, I see the value in being that person in my students’ lives that they’ve not seen before. It’s so much more than breaking a stereotype. My pedagogy is most valuable in a space where there are multiple cultures represented. That’s my strength.

It seems to me that the thing you’re doing for your students — being the person they’ve never seen before, challenging stereotypes — it’s so valuable. But would it be OK for a school to ask you to play that role? I think probably not, and for the reasons we’ve talked about earlier, that this feels like a justification for diversity that comes close to exploitation: “Come to our school/conference, so that you can break our stereotypes!” It seems like the role you’re describing is so important, but it can only come from the choice of the educator. If it came from the school, conference or organization, it would feel exploitative.

And then I suppose it gets back to what you were also describing above: ideally, spaces in education should transform themselves so that educators naturally want to play these roles. But I’m left with the question, is it possible to engineer a natural development like this?

That’s the question precisely. If racism is structural and systemic, is there any manner of individual pursuits, even if coordinated, that are sufficient to tackle it? My gut says no, yet I persist anyway. It’s human to use what agency we do have, right?

I want to not have to explain/prove my humanity, yet I do. Quite the conundrum.

So you can employ quotas, but then there’s the chance that people are made to feel like they don’t belong. The alternative though seems bad too, which is to let things take their natural course. I think in the case of an organization like TMC, that means embracing non-White educators when they come along and trying to be welcoming, but not really doing much to change the fundamental demographic facts of the organization.

But being welcoming is no small thing. The work begins with attending to personal bias, and really taking a hard look at what is happening in your work space. Maybe that involves looking at equity. Do all students have access to opportunities? Do all colleagues? Or maybe if you are in a non-diverse space, analyze that. Is it by choice?

There’s a lot to be said for making people feel welcomed and considered. I think that’s a matter of being human, and not a big ask. But not ever having the conversation will never get us there. There are a few talking about these things, perhaps more than I know. I hope so.

What I hear you saying is that you actually think “welcoming” is a very good place to focus the attention of a diversity conversation.

Yes, precisely.

One last thing I’m wondering — are things simpler when there has been clear discrimination in the past? Like, suppose that TMC was 100 years old and had a policy at some point of not accepting Black educators to the conference. Would that provide a clear “why” for a quota, justifying the policy without making people wonder if they really belong?

I think we’re past having to have a historical reason we can point to (i.e. exclusion policy) in order to bring other voices in. That gets into another dialogue of assigning blame that’s not helpful. We know what we have now, so the questions become: Do we want to change it? Why do we want it changed? How will we do that? In that order.

Talking about teaching and politics with Taylor Belcher

I’m not sure how to format this thing, but it seems jarring to just drop you into a long conversation without context. But what context do you need? Taylor (@teachbarefoot) is someone who I chat with often online. As the conversation below makes clear, he’s incredibly thoughtful. I reached out to Taylor after a conversation we had on Twitter about whether teaching should be political or not. As part of that conversation he wrote “While I understand many of you see politics as a part of education, I do not. And I’m not particularly interested in the opinion that my pedagogy isn’t sufficiently political.” Read on, though, and you’ll see that there are interesting layers to his position.

By the way, this was a lot of fun and I love this format. If you’d like to chat with me about something there’s a good chance we can turn it into a post like this. Pitch me ideas! 

Maybe let’s start by talking a bit about our teaching situations. I teach at a fancy-shmancy private school in New York, and the vast majority of my students would self-identify as progressive or liberal. I’d say that my students are, on the whole, very politically engaged. My students are mostly white, mostly well-off.

Myself, my default lens for seeing the world is a liberal one, but with a lot of doubt and some deviation. On any given issue I might deviate, but I have bleeding-heart values. So I’m not exactly in tension with my students’ political views, but I’m probably less-progressive than most of them.

How about you?

I teach at a public school in South Carolina. It’s a big school (over 2,200 students) in a huge district (6 high schools, 4ish middle schools, and double digit elementary schools). The school itself has students across the whole economic spectrum from the really well off down to “Mr. Belcher I don’t have internet at home”. I have the “lower track” math classes, students who scored below a certain threshold on some test in 8th grade. They take Algebra I in two parts. The school itself is about 65%ish African American and the rest is split rougly evenly between Caucasian and Hispanic with a some Asian students, but my classes are either entirely African American or almost entirely.

I am not sure where my political views and those of my students line up, mostly because I haven’t asked them and I don’t usually share mine. But also I don’t know how much students are exposed to my brand of politics before college. I was card-carrying Libertarian for a while, and lately I have felt more sympathy for anarchist views and I don’t really know if students know about those views and I don’t usually tell them. Last year during the election students would ask me if I was voting for Trump and I gave an emphatic no, but I also told them I wasn’t voting for Hillary either. They were confused. “Who are you voting for then?”

Most of the political conversations that I have had from students have come from them being worried or confused about what is going on in our country. When they have asked me about events or expressed concern I have given them time and space in class to share their thoughts, and I may make a comment, but I mostly let the students talk to each other. (I had an English teacher in middle school who let us talk through the Iraq war when it started. She didn’t share her views but just let us talk. I have tried to emulate that.) I did express firm support for my students. It was important to me that they feel safe in my class and that they know I was for them and their well-being.

The closest I have come to sharing my political views was when the state had a mandatory constitution day. We were required by state law to incorporate the constitution into the lesson. I tried to have the students talk about whether the Bill of Rights granted rights to people or merely enshrined protections for rights that already existed.

What sorts of things make your students worried or confused?

A lot of the anti-Hispanic things that Trump said during the election worried my Hispanic students, but they responded to it in different ways. Some of them expressed concern for whether they would be allowed to stay. Others would make their usernames Trump2016 in Kahoot. I think the latter behavior was a defense mechanism or some way to take back control or comfort, but I don’t know for sure.

Shootings by police worry my students as well. I have let them talk about that and tried to navigate the weirdness that is leading a group of non-white students in a discussion on race. I find it is much easier than talking to people on twitter about race.

I also forgot to mention that my school is the school where the police officer flipped a student out of her desk and it made national news. I was not working at this school at the time but the administration still asks us not to comment on it.


So there are a lot of people in education who say that teachers need to bring politics into their classrooms, and I think they’d really applaud what you’re describing. You let your kids talk about the political things that are worrying them, you don’t shy away when they ask you about who you’re voting for, you know what worries your kids.

What would you say are your limits? For example, would you ever teach a lesson that analyzes data about police misconduct?

Yes I probably would if I was teaching a stats course, but I would want them to draw their own conclusions from the data.

I had students who were complaining about the dress code at school. “Mr Belcher why do we have a dress code. They are being racist about enforcing the codes.”

I told them if they believed that was the case, then they should request demographics data on dress-code write-ups from the administration and I would help them crunch the data. They never followed up on it.

I’ve also tried to get some of my students to stop smoking weed by showing them incarceration rates by race for marijuana. (That didn’t work. One student got suspended and another is in juvenile detention. Both for drugs.)

When/if I bring politics in the classroom I want students to discuss / share / defend ideas rather than arrive at a certain conclusion. I don’t want to indoctrinate my students politically anymore than I want to indoctrinate them religiously. I don’t believe it is appropriate for me to do that as their teacher and not their parent. As my friend, @chrisexpthenews would say, I don’t want to act in loco parentis.

What are your limits / what have you talked about with your students in the classroom?

One thing that I think gets lost too much in these discussions is that kids don’t really care what we think about politics. Or, at least, I’ve never gotten the sense that I could really influence my kids too much by telling them what I think. If political indoctrination was easy, there would be a lot more of it.

I think my limits are about the kids and making them feel bad, hurt, or like there’s a pile-on. Once, when doing a lesson about systems of equations and the minimum wage, one kid started talking about how there’s a trade-off between minimum wages and unemployment. A bunch of other kids started piling on this kid. These kids were a group that usually had no trouble disagreeing respectfully, but politics was different, and this kid started feeling lousy and he withdrew.

I don’t regret doing the lesson, but I hate to think what that kid would have thought if I had sided with his opponents, or if the lesson had taken a clearer pro-minimum wage take. (As it happens, I think controversies about minimum wage are genuine, and you look at some of the things that happened in Seattle when they raised wages and economists seem split on whether it’s a disaster or not.)

There was another time when there was a big protest happening outside a courthouse right by our school. It was a protest against police misconduct, and so many kids wanted to go that the school decided to let kids miss class to go to the protest. Kids came back in the middle of my class period — there was no way we were going to learn any math that day, and kids were having an increasingly heated argument with one kid from the class.

I decided to make it a conversation, and I tried to really control who was talking, whether you were attacking someone, etc. Again, it was a pile-on, with one kid who saw herself as anti-Anti-police (I thought she was quite wrong) getting whacked around by the other kids in the class.

I’m not sure about whether I made the right move that day, letting kids talk about the protest. The school had already given them the chance to engage politically during the school day. Their politics had been honored, and all I think I accomplished was make a righter-wing kid feel ostracized by her peers.

We live in super-liberal Brooklyn, and the kids aren’t on the losing side of the demographic lottery, for the most part. School is already a place that feels like it honors who they are, mostly, I think, both politically and in other ways.

I guess my big limit is that I don’t want kids who disagree with the dominant view in the room to feel badly. I care about those kids too, you know? They’re just kids

I completely agree with wanting to protect the kids. Adults can’t handle political discussion a lot of the time either and I think you can feel humiliation and social exclusion more intensely as a teenager.

And I agree with the kids probably not caring either. When I was a student in high school I didn’t particularly know or care what my teacher’s politics were. I wasn’t exposed to regular political views of my teacher until college and even then that was rare.


I was thinking about what you said about how teachers who want to bring politics into classrooms would applaud what I was describing.

I don’t know if they would or not, but I really think that if a teacher chose not to do the things that I described, they aren’t doing anything wrong. I feel that way sometimes when edu-twitter talks about teacher responsibility when it comes to white privilege too. I may or may not already be doing things that are “approved” when it comes to my role in race relations in our country but I am not interested in whether they think I am doing enough or the right thing or not.

It frustrates me when I see teachers on twitter tell colleagues to leave the profession because those colleagues don’t check off all the boxes on some list.

Thanks for bringing this up — let’s spend a second talking about this.

So it makes sense to me that you’d want to say that a teacher who doesn’t do these things isn’t doing anything wrong, in the same way you’d probably be OK with a teacher who structured their lessons in a different way than you do. Like, I’d be perfectly OK with a teacher who approached congruent triangles differently than I do, because in teaching there are a lot of different ways to go about things. This is a limitation of the “best practices” approach to teaching that people often criticize.

But just as there actually is some understanding or skill having to do with congruent triangles that we’re all mostly aiming for, there’s probably some non-negotiable having to do with classroom environment that you share. Meaning, OK, maybe you would be OK with other teachers not doing what you do, but that’s because it’s not really at the heart of things, you know?

Maybe you don’t — I’m being super unclear, I know — but is there some other value, or practice, or core whatever that you think is really the non-negotiable? Like, a level deeper than talking politics with your kids, or sharing data about arrest rates. Does that make any sense?

Yeah, that’s pretty much what I was getting at and I think it answers your second question too. I don’t have a political learning objective in my goals for my students. If there aren’t best practices and only different effective practices, then there may be effective practices for helping students become more politically educated, but that’s not one of my goals. If I taught a course where we never talked about anything except abstract nonsense and my students learned the abstract nonsense well, I would be happy.

This may be my pure math background / bent a little bit though. I am also not interested in whether my class prepares students for a job or “real-world applications”. I only want them to be able to do math and hopefully enjoy math. And I understand the argument that “everything is political and choosing not to include politics in class is a political choice”. I am fine with that charge. The political choice I am making is to just have fun doing abstract nonsense with children. I think there has to be a space for that. (And there should be a space for politics too. Just because everything is political doesn’t mean we have to politicize everything, if that makes sense at all. Like, “Yes, I understand. But let’s talk about that in another setting.”) And at the risk of repeating myself, I don’t mind (necessarily) if other teachers feel the need to explicitly politicize their math classroom. It’s their math classroom and since politics is a learning target for them, they are using their effective practices.

So I guess to me the deeper value when it comes to math class is just doing and enjoying math.

My beef with Sunil Singh’s piece about math and math education


If you’re just joining us: I wrote this, some people liked it, others did not. Sunil Singh, whose post I was critiquing, understandably didn’t like what I had to say.

(He also didn’t like how I said it. Admittedly, I was a bit obnoxious. But Sunil’s rhetoric was not kind either and, dammit, sometimes two wrongs do make a right. Happy to deescalate the rhetoric, though.)

As I see it, Sunil’s core argument in his post goes like this:

  • People, in general, hate math
  • It’s math education’s fault
  • The problem is that math education has deviated from the purposes and values of math (as identified by Francis Su)
  • Math needs to take back math education — in particular, mathematicians need to direct math education

I wanted to make sure I was understanding the argument correctly — it seems to check out with Sunil.

Now, the goal here is to reconstruct Sunil’s argument in a way that he would recognize — maybe, if we do a really good job, in a way that he would recognize as even clearer than his own version.

Towards that, let’s try to clarify: what exactly would it mean for mathematicians to “take back” math education? This gets clearer from Sunil’s examples of where math education has gone wrong. He calls out homework as a major problem (“homework is on life support”). He wants to get rid of grades. He wants to radically change assessment. He thinks math education has been infected by financial and political forces. He says math education is not operating in the best interest of children.

The point being that it’s not only classroom teaching that is making children hate math — it’s everything, the whole system. Mathematicians need to be in charge of all this.

And who are these “mathematicians”? Am a mathematician? The problem, as Matt Enlow points out, is that while the term “mathematician” can mean “someone who gets paid to do mathematics” or “someone who has received an unusual amount of training in mathematics,” it can also mean “a lover of mathematics, someone committed to the discipline.” (The exact same difficulty surrounds words like “artist” or “educator.”)

Sunil Singh isn’t a professional working mathematician, so let’s assume that he means to use “mathematician” broadly.

His argument, all together, therefore goes like this: If mathematicians — people who love and truly get mathematics — had control of math education, there’s no way it would look the way it does. There would be no homework. There would be none of this testing to decide who is worthy or unworthy of more math. There would be no grades. The universal values of mathematics would be the focus of education: play, justice, truth, beauty and love. Students would have a chance to learn math that truly interests them, not the garbage we throw at them in algebra classes. They’d learn the “dream team” of beautiful math, not a hierarchy of topics on this inevitable, dreary march to nowhere. 

This, as I understand it, is Sunil’s argument.


I disagree with basically all of it.

Not because the status quo in math education is ideal to me. I don’t like grades, I think they’re overall bad for learning. I think the math curriculum is over-stuffed and deserves a healthy pruning. Our current testing regime in the US is nuts, and my experience with NY’s Regents exams have all been frustrating.

But here’s the thing about mathematicians and math education: you don’t have to sit and wonder what a math education designed by people who love and truly get math would look like. It’s not some mystery. University math departments are designed and filled exclusively with people who love math so much that they’ve signed up for a lifetime of studying and teaching it.

You know what’s coming next, right? Because university math courses are, on the whole, taught far more poorly than k-12 math courses. Where were all the mathematicians when they were designing entrance exams to decide which Calculus section you get to sign up for? Where is the play and exploration in Abstract Algebra classes? Why is the dominant pedagogy notes and lecture? What on Earth is the deal with a class like Math 55?

And it’s also not some sort of mystery what professional lovers of math advocate for when it comes to k-12 education. They call for more rigor, they worry that their students are coming with weaker skills than they used to, they criticize textbooks for having ever-so-fuzzy definitions, and, not to put too fine a point on it, they aren’t exactly lining outside of the statehouse asking k-12 educators to ditch all that algebra.

The way I see things, Singh’s mistake is in thinking that math education is some deviation from the desires of mathematicians and lovers of mathematics. I mean, yes, k-12 math education absolutely is a deviation from a particular vision of math that Singh articulates. But if you look at university teaching and if you look at the rigor, precision, and gate-keeping that mathematicians frequently call for in k-12 education, you can see that this isn’t a deviation; it’s a reflection of what a major segment of the math-loving population wants out of math education.

Singh’s piece was written as a polemic against math education. This is entirely unfair, though, as math education and math culture are codependent. The issues with math education are equally issues facing the broader mathematical community.

Mathematicians and lovers of mathematics love to hate on math education and its deviations. But it’s the lovers of mathematics who have set up the system that we have. They protect it and extend it into higher education. It’s painful to see that some of the problems we have in math education can even spring from that love — from a desire to protect mathematics, or a desire to spread mathematics.

And realizing all of this is a way to realize that Singh’s diagnosis is incorrect. The problems with math education come from the competing desires that non-mathematicians along with mathematicians want from k-12 classrooms. Mathematicians may want students to be exposed to the beauty of math, but they equally want to find the gifted few who can enroll in their college classes, and they want those students to be well-prepared. Lovers of mathematics want to make sure that the discipline — which they love — is well-understood and used by the populace. And they want to make sure that engineers, doctors, accountants, NSA hackers, software designers, everyone is given a solid grounding in math. And, like the rest of us, they want to think that everyone gets a fair shot at any of those jobs.

You can’t improve math education without understanding what exactly is going on. Singh’s vision of mathematics isn’t universal among lovers of math, even among those who really know and get it. (Unless we say no true lover of math would disagree with Singh’s vision, which is totally cheating.)

We’ll never improve math education in our current system by trying to blow it up, and I think that would only make things worse. I have a great respect for those who operate outside of the world of math education who try to spread their love of the field more widely. But here in the world of math education, we’re all trying to figure out how to help kids deal with the mess that mathematicians and everyone else have left us.

And, actually, we’re making some progress. So ease up on the attack on math education.

Four answers to “Should teaching be political?”

[This one sort of ran away from me. It started with just taking some notes about the historical sources and kind of just exploded from there. This all is very provisional for me. I don’t trust the commentary. Mostly you should read it for the sources cited. OK, caveat lector etc.]


George Counts was a fire-breathing advocate for using school to influence students’ political views. Specifically, Counts comes down in favor of the political indoctrination of students:

You will say, no doubt, that I am flirting with the idea of indoctrination. And my answer is again in the affirmative. Or, at least, I should say that the word does not frighten me. We may all rest assured that the younger generation in any society will be thoroughly imposed upon by its elders and by the culture into which it is born. For the school to work in a somewhat different direction with all the power at its disposal could do no great harm. At the most, unless the superiority of its outlook is unquestioned, it can serve as a counterpoise to check and challenge the power of less enlightened or more selfish purposes.

This is from Counts’ 1932 talk, “Dare Progressive Education Be Progressive?” It was delivered in front of the Progressive Education Association, and caused quite a stir:

The challenge of Dr. Counts was easily the high point of the program. Following the dinner meeting at which he spoke, small groups gathered in lobbies and private rooms to discuss, until far into the night, the issues raised in Dr. Counts’ sharp challenge. These discussions were marked by a general willingness to accept the viewpoint of Dr. Counts that the schools have a real responsibility for effective social change. There was, however, a considerable difference of opinion as to how this was to accomplished. The method of indoctrination, advocated by Dr. Counts, was widely questioned.

I took this quote from Richard Niece and Karen Viechnicki’s very interesting article on Counts’ talk. They cite many contemporary educators’ reactions to Counts’ talk. They ran the gamut, from complete agreement to shock at his frank talk of power and imposition on the child. For his progressive audience, it was the talk of indoctrination of the child — as opposed to commitment to the child’s own natural flourishing — which was truly shocking. Counts was trying to steer his audience away from children and towards socialism.

Niece and Viechnicki think that Counts’ speech split the progressive movement in a way that ultimately led to its dissolution. “The chasm between child-centered supporters and social welfare advocates became too vast to bridge.”

In 1935, three of Counts’ talks — including his live-wire PEA lecture — were reprinted as Dare the School Build a New Social Order? Intriguingly, the section on indoctrination was rewritten to include an anecdote about Counts’ discussion with the Commisar of the Soviet school system:

The advocates of extreme freedom have been so successful in championing what they call the rights of the child that even the most skillful practitioners of the art of converting others to their opinions disclaim all intention of molding the learner. And when the word indoctrination is coupled with education there is scarcely one among us possessing the hardihood to refuse to be horrified. This feeling is so widespread that even Mr. Luncharsky, Commissar of Education in the Russian Republic until 1929, assured me on one occasion that the Soviet educational leaders do not believe in the indoctrination of children in the ideas and principles of communism. When I asked him whether their children become good communists while attending the schools, he replied that the great majority do. On seeking from him an explanation of this remarkable phenomenon he said that Soviet teachers merely tell their children the truth about human history. As a consequence, so he asserted, practically all of the more intelligent boys and girls adopt the philosophy of communism. I recall also that the Methodist sect in which I was reared always confined its teachings to the truth!

Counts is just absolutely delicious in the way he stares power in the face. He’s just so quotable:

That the teachers should deliberately reach for power and then make the most of their conquest is my firm conviction…In doing this they should resort to no subterfuge or false modesty. They should say neither that they are merely teaching the truth nor that they are unwilling to wield power in their own right. The first position is false and the second is a confession of incompetence. It is my observation that the men and women who have affected the course of human events are those who have not hesitated to use the power that has come to them.

People just don’t talk like that anymore, unless they’re part of antifa or something. It’s a blast.


Is Counts right? He is absolutely insistent that teaching involves imposition and indoctrination. Another juicy quote:

There is the fallacy that the school should be impartial in its emphases, that no bias should be given instruction. We have already observed how the individual is inevitably molded by the culture into which he is born. In the case of the school a similar process operates and presumably is subject to a degree of conscious direction. My thesis is that complete impartiality is utterly impossible, that the school must shape attitudes, develop tastes, and even impose ideas.

I think it’s clear that Counts, at some level, is right. School is an imposition; teaching is an imposition. So is parenting.

But there are impositions, and there are impositions. Counts correctly notes that all education is an imposition on a child. Then he says teachers should unilaterally take control of the curriculum and train children in the truths of socialism in the hopes of creating a class of socialist revolutionaries. Woah, Counts!

“Any time you watch a TV show it changes your mood, how you think,” George Counts might’ve said, “so the question isn’t whether or not to use TV for mind control but how.” George Counts is the sort of fellow responsible for Facebook’s emotion control experiment. He’s really into controlling people, but then again he’d be happy to tell you that himself.

It’s true: all education is an imposition on the student, parents and communities, but should we or should we not try to minimize this imposition? That’s the question that George Counts doesn’t ask.

Personally, I like to think of our education system in terms of tensions and equilibria. It’s not a question of the amount of imposition but the direction of those impositions. Some parts of society want us to impose professional training on children. Others want political training. We also want to use our powers of imposition to impose a safe environment, where kids are happy and safe.

And I think this is more-or-less how schools should be. They should be places that find points of balance between the competing needs of students and communities. At school, we use our powers of imposition towards contradictory ends so that a broad range of students can get something out of these institutions.

It’s true — there’s no neutrality in school — but we shouldn’t mistake the slurry, messy non-neutrality of schools for an institution with a particular aim and purpose. Counts’ vision would be a disaster for public education, I have no doubt.


W.E.B. Du Bois was not a fan of George Counts’ take on education. In a 1935 address to black teachers in Georgia — a few things I’ve read say that he was addressing the Counts controversy in this talk, but I can’t see where exactly — he began by deemphasizing the ability of schools to effect direct social change. “The public school,” he says, “is an institution for certain definite ends. It is not an institution intended or adapted to settle social problems of every kind.”

Du Bois, however, was a big believer in the ability of knowledge to erode social barriers, eventually:

Now, of course, indirectly, and in the long run, all men must believe that human wrong is going to be greatly ameliorated by a spread of intelligence; that the spread of such intelligence beyond the confines of a narrow aristocracy and the pale of race, down to the masses of men, is going to open great and inexhaustible reservoirs of ability and genius. But, mind you, this intelligence is the essential and inescapable step between the school and the social program, which cannot be omitted without disaster. The school cannot attack social problems directly. It can and must attack them indirectly by training intelligent men, and these intelligent men through social institutions other than the school will work for a better organization of industry, a juster distribution of income, a saner treatment of crime, a more effective prevention of disease, a higher and more beautiful ideal of life without race, prejudice and war.

Du Bois goes on to criticize the ability of progressive education trends — “new methods of teaching” — to address the needs of black children. Du Bois takes aim at the entire progressive education enterprise, preferring a sort of early version of the “back to basics” approach:

What we need, then, and what the public school, college and university must supply, is intelligence concerning history, natural science and economics; and the essential key to this intelligence is a thorough, long-disciplined knowledge of the three “R’s.” To assume that instead of this, and be allowing the curriculum of the public school to encroach upon thorough work in reading, writing and arithmetic, we can cure the ills of the present depression by training children directly as artisans, workers and farmers without making them intelligent men, is absolutely false.

Du Bois, though, is clear that this sort of training can’t possibly help the impoverished and oppressed break through all on its own. School can educate, but the rest of the work happens outside the institution.

There’s a fascinating moment when Du Bois talks about his disappointment of his daughter’s education in a progressive school:

My daughter attended as a child a first-class kindergarten and a progressive grade school. But there was so much to study and to do, so much education rampant, that when she went to a real school and entered the fifth grade, she had to stop everything and learn the multiplication table. The learning of the multiplication table cannot be done by inspiration or exortation. It is a matter of blunt, hard, exercise of memory, done so repeatedly and for so many years, that it becomes second nature so that it cannot be forgotten.

I remember a parent-teacher conference with the mother of a student of mine who was black. I couldn’t tell if she was particularly concerned about my class, but she was very clear about her concerns: “My son is a black boy, and he’s going to leave school a black man in a society where it’s not easy to be that. I want to make sure his math won’t hold him back.” I hear that parental concern in Du Bois.

Du Bois, like Counts, is just so direct and quotable. (“The pressure upon Negroes is to increase our income. That is the main and central Negro problem.”) They also shared a love of communism. (Du Bois would suffer terribly at the hands of McCarthyism, an episode I know basically nothing about.)

While reading Du Bois’ address, I found myself thinking of James Baldwin’s remarkable Talk to Teachers, and Baldwin’s closing message:

Now if I were a teacher in this school, or any Negro school, and I was dealing with Negro children, who were in my care only a few hours of every day and would then return to their homes and to the streets, children who have an apprehension of their future which with every hour grows grimmer and darker, I would try to teach them –  I would try to make them know – that those streets, those houses, those dangers, those agonies by which they are surrounded, are criminal.  I would try to make each child know that these things are the result of a criminal conspiracy to destroy him.  I would teach him that if he intends to get to be a man, he must at once decide that his is stronger than this conspiracy and they he must never make his peace with it.  And that one of his weapons for refusing to make his peace with it and for destroying it depends on what he decides he is worth.

Du Bois would probably have little patience for this project, and he’d probably also take issue with Baldwin’s characterization of the purpose of schooling:

The purpose of education, finally, is to create in a person the ability to look at the world for himself, to make his own decisions, to say to himself this is black or this is white, to decide for himself whether there is a God in heaven or not.  To ask questions of the universe, and then learn to live with those questions, is the way he achieves his own identity.

This comes much closer to Counts’ progressive view on the role of school, but Baldwin has no interest in indoctrination. He is interested in self-sufficiency, the ability to decide. Baldwin just wants school to tell the truth about the world, so that students don’t blindly make their choices.

I do think that something like a mish-mash of Baldwin and Du Bois is possible in our schools today. Baldwin was not a teacher, and (as far as I know) didn’t have any concrete views on curriculum, not like Counts did. Where do those lessons about America come? How are they learned? Is Baldwin talking about the curriculum, skills and knowledge, or is he talking about the space between all that stuff?

So much happens in the empty space of school. Kids are formed by the ephemeral qualities of school just because school is where they are for (if all goes well) more than a decade of their lives — everyone’s influenced by their environments, no? Kids will make friends, find adults to admire and despise, and learn a way of moving through their world. Kids really do take something from all that other stuff when they leave — though what any particular student takes with them may vary. Tricky, maybe impossible to engineer, yes, but there are chances in school to help children really understand their place in the world, and we shouldn’t lie.


Eric Gutstein is a professor of math education at University of Illinois. Following Paulo Freire, he talks about math education for reading and writing the world. In other words, math education should help you understand social injustices and also lead to you battling them.

Gutstein’s pedagogy is nicely encapsulated in his article Teaching and Learning Mathematics for Social JusticeThe piece is his account of the two years he spent teaching social justice math at an “urban, Latino high school.” His approach is a version of real-world math, and he is a big believer that engagement with personally relevant mathematics leads to better learning and more positive feelings about math.

Much to discuss in Gustein’s work. I’m interested, mostly, in the Counts’ question: are we OK with indoctrination?

There is no clear and direct statement, ala Counts. We do get a very measured statement that seems to address the concern of imposition of political views:

I did not try to have my students answer questions so much as raise them. Questions such as why females, students of color, and low-income students score lower on SAT and ACT exams are not easily answerable–and students did raise that question in one of our projects…And I did not want my students to accept any view without questioning it. I did share my own opinions with my students because I agreed with Freire’s contention that progressive educators need to take the responsibility to dispel the notion that education can be the inactive transfer of inert knowledge and instead to promote the idea that all practice (including teaching) is inherently political I take Freire to mean that educators need to be explicit in their views while at the same time to respect the space of others to develop their own.

There is a lot of “yes, but…” going on here. Yes, I share my political views with my students, but I don’t ask them to accept them on my authority. Yes, I show my students examples of social injustice, but I don’t offer simplistic answers. (Though Gutstein in his paper describes prompting his students to explain why they thought a given injustice arose.  I think that this inevitably supports students in forming definite answers to the questions he raises.) For Gutstein, we are to be explicit in our political views and design math lessons around them, but we are to also give students space to develop their own views. (Where?)

Counts might have caused a fissure in progressive education by placing a wedge between child-centered and social justice progressives; Gutstein represents an attempt to reconcile these forces in his own pedagogy. For that matter, it aims to satisfy Du Bois by claiming rigor and mathematical sophistication for his students, and Baldwin too by aiming for student independence in deciding what to believe, as long as they’re armed with the truth.

And what would Counts say? I imagine this would bother him. After all, how can you claim that you’re leaving room for students to develop their own views when you’re hand-picking topics to support your political views? (Counts on curriculum: “the dice must always be weighted in favor of this or that.”) He would see in Gutstein a contradiction, an attempt to erase himself from the role of political influencer even though he totally echoes Counts’ line in saying teaching is inherently political.

If teaching is inherently political, then why not own up to the attempt to politically influence your students? Own up, man!

For better or for worse, though, we aren’t living in the 1930s. (Come to think of it — of course it’s better that we aren’t living in the 1930s.) Back in those days communism was a semi-respectable political view, fascism wasn’t funny, an educator like Counts could earnestly go around talking about power grabs, and everyone had the distinct feeling that the world was about to fall apart at the seams — they were correct. School was up for grabs, along with everything else, and advocates for social reform wanted a piece.

This is not the world we live in now. We’ve been through the 20th century. Schools are now places of compromise, as reflected in Gutstein’s rhetoric. George Counts, W.E.B. Du Bois, James Baldwin all tug at school, along with like seven other totally different tuggers.

And because of all this — because schools are places of compromise, and because they should be — I am generally not excited by teachers trying to help their students come to take one side in a political controversy.

It’s not that I want politics out of schools. There’s no reason for that, and politics is part of the story that we need to tell — Baldwin is right. Du Bois is also right: schools have enough trouble teaching the curriculum, and can only create social change indirectly, by educating students who go on to create institutions for good.

Counts, however, is wrong, and to the extent Gutstein echoes him I think he is wrong as well. True: there’s nothing “natural” or apolitical about education. But I think we do our best job when schools accurately reflect the incompatible desires our society places on schools, rather than taking a particular social desire and running with it. So while schools will never be politically neutral, I think in our lessons we should try.

Should I see my son’s misconceptions?


Yosef turns three tomorrow — happy birthday, kid! My sister got him some new puzzles for his birthday, and that’s how we spent a big chunk of the afternoon.

This is his first foray into “big kid” puzzles. We had no idea he was ready for them, and he can do a lot of it on his own, though he always asks for help. (Like every three year-old, he likes attention from grown-ups.)

While he’s been playing, I’ve been watching and trying to make sense of how he’s thinking. As far as I can tell, his main strategy is to match the pictures of pieces: fish goes with fish, yellow with yellow, etc. He also has an eye for missing pieces — meaning, he matches holes with pieces that are congruent to the holes.

One thing that’s fascinated me: he doesn’t really notice the difference between edge pieces and interior pieces. Here he is, trying to stick an edge piece into the middle of the puzzle.

IMG_0760 (1)

I got curious, so I started asking him about potential fits. Could this piece go here?


He never mentions the shape of the piece, or the way that it would partly stick out. When I asked him about this piece he only mentioned the color. When I swapped out the yellow piece would another green-sea colored one, he would try to smoosh it into the hole. Only then would he tell me why it wouldn’t work — “It doesn’t go in the hole.”

Now, I honestly don’t care how well or poorly he solves puzzles. But learning stuff is fun, and I was curious whether I could help him see the difference between edge and inside pieces.

“Look Yosef,” I said. Just like in that picture, one of those inside pieces was along the top row of pieces, protruding out of the frame. I put my finger at the top of the puzzle on the top left side of the puzzle, and I slowly dragged my finger to the right. “My finger can just keep going, going, going…until it bumps into this. Bonk! This piece doesn’t belong!”

Yosef laughed. “Bonk!” he said. “Bonk!”

“But look Daddy. If my finger goes like this” — he loops down into the interior of the puzzle, far below the top row of pieces, slowly meanders up until it reaches the false piece, right under where my finger had bumped — “if it goes like this, then it doesn’t bump.”

Which was true! Had to cop to that.

He returned to the puzzle. He matched pictures — dolphin into dolphin, clownfish to clownfish — and every so often mystified me by quickly intuiting where a piece went. He also continued to shove edge pieces all along the inside of the puzzle.

I wasn’t lying when I said that I didn’t care how he plays with these puzzles…but doesn’t it just kill you to watch someone painstakingly — delicately with care — try like seventeen different ways of putting an inside piece into the side of a puzzle?

I mostly keep my mouth shut and let him have fun. He’s clearly not seeing edge pieces yet, which is interesting, but obviously fine.

Every once in a while though, I nudge at his understanding. “Pass me an edge piece,” I’ll say, hoping that he starts thinking of edge pieces as a distinctive category. If he asks me to fill in part of the puzzle I’ll talk aloud about my thinking: “This piece couldn’t go here because it doesn’t have a straight side.”

I have no idea if this stuff is connecting with him. Learning to see shapes in different ways is messy and slow. My little two-second nudges won’t make much of a difference to him — right up until he becomes ready for them, and then they might.


It’s pretty clear to me that there are things about shapes and puzzle pieces that Yosef doesn’t yet understand. He’s three. Of course there are. But how should I think about his understanding? In some quarters of the math education world, the answer is contentious.

Brian Lawler is someone who has been incredibly patient with me on Twitter, as we’ve gone back and forth discussing his positions on the nature of mathematical knowledge, teaching and learning. He passionately believes that any talk of misconception is not only wrong-headed, but also the act of labeling someone as holding a misconception is harmful to that person. Ditto for a smattering of other terms that imply that the other person’s thinking is worse than your’s, or on the way to some better understanding — this includes talk of alternate conceptions, early conceptions, preconceptions.

Rochelle Gutierrez likewise asks us to refuse to talk of misconceptions.

These scholars aren’t talking about me doing math with my kid — they’re talking about the ways math teaching can beat down kids in a lot of school situations. Still, their arguments are that thinking in terms of misconceptions or even not-there-yet conceptions is harmful — even violent — to a math learner. Their framework should apply to me doing a puzzle with my son too, I think.

Now, it doesn’t seem entirely accurate to me to say that Yosef has misconceptions about puzzles or shapes. It’s not like he actively thinks that edges don’t matter — he just doesn’t see the difference between edge and interior pieces yet. Yet he has so many amazing things in his little-kid brain that help him put pieces together. He absolutely has a conception of shape that is letting him have a blast with puzzles, and he loves doing them with me. I love playing puzzles with him. I love him.

Is it still harmful for me to think and talk about the things he doesn’t yet see?

I try to be a good father to my children. This is not always easy with a three-year old, but I really do try. I think I’m getting more patient — hopefully in time for the baby’s turn at toddlerhood — and I try hard to give Yosef room to play with toys the way he wants to play with them. I certainly don’t want to visit “intellectual violence” (as the phrase goes) on him by getting all up in his face about the right way to do a freaking 48-piece puzzle. I’d rather save our conflict for when he’s dropping a block on his baby sister’s head.

At the same time, part of our play is learning. The kid wants to put the puzzle pieces in on his own. He wants me to help. He likes learning new things — he’s a kid, he’s deeply curious about the world. The world includes mathematical language. Every time we put on his shoes we end up doing a whole routine about left/right: No, that’s not left. It’s right. No, not right, right. Right. Left. That’s right. His conception of left/right is relatively weak — it’ll get stronger.

Kids love improving their conceptions of the world, if they really get a chance to feel like it’s their own conceptions that are improving. Humans are curious creatures, and we like improving. There are a million ways for this to go wrong and to turn into abuse — in a lot of schools, this is happening.

In a lot of schools and homes, though, it isn’t. I don’t think it’s inherently abusive to see your child’s misconceptions or to help them see things in a new and richer way. It can be, of course, and that abuse needs to be detailed and discussed.


Some people might disagree with the above, but not many. The real question is a linguistic, or even a strategic one:

Does refusing to talk of ‘misconceptions’ cut down on the abuse?

Pretty much everyone I talk to online seems to think that this is a good way to chip away at the problem of abusive teaching practices. The first way this might chip away at the problem: the word “misconception” itself connotes the bad teaching practices. You can hardly use the word without being further nudged towards abuse — or you might nudge a colleague to abuse. If you eliminate the word, you eliminate the subconscious priming of yourself and of colleagues.

The second way: Changing your pedagogy is hard, and it’s easy to forget your principles. The refusal to talk of “misconceptions” is a relatively easy change to make, but it’s like a red string around your finger. It will remind you of your commitment to the proper pedagogy, and you’ll continuously improve as a result.

I actually think there really is something to that second thing, but I also think it’s incredibly risky for the cause of good pedagogy to tie it to refusing to use the word “misconception.”

It’s because my kid’s conception of shape really does have room to improve.

We see misconceptions in children because it really is true that there’s stuff that they don’t yet know. Noticing this doesn’t have to be an act of violence — in fact, I don’t think that it usually is. Usually it’s like me playing with my son and noticing there’s stuff he doesn’t yet know how to do, even as my mind is blown because oh my god my son is into puzzles! When did our baby turn into a kid?

Is it good pedagogy to ask people who don’t already see their pedagogy as abusive to forswear from using words that they use all the time? Isn’t this exactly the sort of “intellectual violence” that we’re being urged to refrain from? Shouldn’t we start with the way people actually see the world, rather than asking them to use language that is not their own?

Seriously: imagine what a teacher whose well-meaning administrator announces that they don’t want to hear any talk of misconceptions any longer, that this is now school policy. Is that good pedagogy?

There is real injustice and unkindness in this world, and I have no clue how to fix it. I think a focus on which words are allowed or not is a tactical mistake. Take any word that you associate with fear, abuse, pain; there are others out there who associate those same words with love, play and growth. To bridge those gaps we need to talk with each other and find a common language. That can only happen if we agree to use each others’ words.

Weird, Sloppy Rant about Giftedness


People who know me or my work in the goyishe world sometimes ask me how my traditional Jewish education — which mostly involves learning to carefully analyze texts —  influences what I do in math education. “You’re super-duper pedantic but you read things closely,” they say. “Isn’t that a result of a life studying Talmud?”

The answer I tend to offer is something like: Who knows? I have a lot of friends from yeshiva, but most of them aren’t nearly as annoying as I am. So, really, who’s to say? Besides, I also studied philosophy, and philosophers aren’t exactly the most easy-going people of all-time either. Maybe philosophy is why I’m such a pain in the ass.

Some people have stories about how their upbringing or education has made them who they are today. But memory is a funny thing; it’s hardly a reliable witness. If we’re honest, how sure can we be about what made us who we have become? All this sort of backwards-speculation is just guessing, and shouldn’t be taken too seriously.


OK, throat cleared, let’s speculate.

I was a good student, but I wasn’t a quote-unquote genius. That’s partly a matter of personality but it also accurately reflects the fact that nobody was ever, like, dude, Michael is breaking the system!

But, yes, ok, I was smart in school and made to feel that way by grades, peers, teachers, etc. I knew there were things I could do that others could not. The things people tell me I’m good at now are things that people were telling me then: that I ask good questions, that I read carefully, that I write clearly (if not quite, you know, beautifully).

Sarah HANNAH Gómez, in her tweets, says there’s a problem with gifted education. She was gifted, told she was smart, but never told to engage with classwork, to push herself, to really do anything at all. She says this is endemic to education and that teachers need to figure out ways to engage their most gifted students.

Here’s what I’m trying to say: in my yeshiva we were taught that we had an obligation to learn, and that obligations were a big deal. “Obligation to learn” means there’s optional Talmud class on Wednesday nights, and also on Sunday afternoons, and also on Thursday nights, and aren’t you going to stick around for it? Aren’t you a serious student?

There were silly parts of this culture, as there are of any culture. Kids trying to out-macho themselves by staying up late, attempting pious acts of learning into the early morning. For some kids it caused a lot of stress, when they were tracked into a middle shiur or out-shined by their classmates. There were stupid competitions about who could learn the most, and there was also a ridiculous award ceremony every year to honor the school’s top students.

(Though, I should add, being a “top student” didn’t mean you were a genius. It mostly meant that you took your studies seriously, logged a lot of hours, and also were a moral exemplar in the eyes of your teachers.)

I can’t imagine a gifted student at my high school somehow getting the message that he didn’t have to learn. That you had an obligation to learn was pretty much the whole point of the school.

You might wonder what our non-Jewish studies classes looked like, whether the same verve was applied to these other classes.

Based on what I saw, nah.

A lot of my other high school classes were a joke. There was not an obligation to e.g. know the Roman emperors or get really good at balancing chemical reactions. I remember reading a lot of textbook chapters during my free period, right before a 30-question multiple choice test.

(Many teachers used the same tests each year, and there was a shady tradition of kids saving the answers and inputting them into TI-83s, which they might get away with using on some test. This worked best for Mr. Rosenbaum’s AP Biology class, since you could often legitimately ask to use a calculator to help with genetic probabilities. Mr. Rosenbaum told us he was mystified why so many of us nailed the multiple choice but put no effort into the essay.)

Looking at my life since yeshiva, though, I think somehow I got bonked with the weird idea that there is an obligation to learn anything, especially if you can. I think I can thank my yeshiva for that idea, and I think that’s the sort of education that Sarah HANNAH Gómez wished she had received.


The yeshiva is an educational institution rooted in scarcity. Your towns and cities needed scholars and rabbis, but how many people could the community support? You need wealth to support equality of opportunity. Post-Holocaust, though, many have found that wealth.

Yeshivas today hold together two ideas side-by-side: the historical belief that some students really are iluys, savants, and are destined for greatness, and the more democratic belief that each student has an obligation to reach their own potential.

My read of the culture is that the drive for equity is subservient to that towards serving giftedness. The reason for equality of opportunity is because everyone has an obligation to explore their own giftedness — the difference between local and global maxima.

(A story that has become popular: Reb Zusha lies on his deathbed, shaking in fear of the conversation he’ll have after he dies. “When I get to Heaven they won’t ask why I wasn’t like Moses; they’ll ask why I wasn’t more like Zusha.” One must imagine himself like Zusha, terrified.)

American schools, as Gómez points out, are largely not like this at all. In fact, they’re sort of the other way around, which makes sense for an institution borne out of plenty, not scarcity. American public schools exist for the least among us. (Initially, out of concern that left unschooled they would rip society to shreds.)

American public schools are the mirror image of yeshivas. The drive to teach gifted students comes out of a drive for equity, the belief that schools should teach everybody.


So, which is a better system? Should giftedness be subservient to equity? Should equity be subservient to giftedness?

The popular answer is that schools can achieve both, that neither concern has to be subservient to the other.

The way that plays out in yeshiva is that there’s a universal obligation to study — and therefore teach — each student to their ability. But no such obligation exists in mainstream culture.

I don’t really know how teachers, in general, think about the needs of the few vs. the needs of the many, the majority of class.

I know, for me personally, I experience this as a tension in my classrooms. I both want to help every student (I really do believe in an obligation to learn) while also making sure that gifted kids get to develop their gifts.

When I say I experience this as a tension, I mean that my efforts in one direction get in the way with my efforts in the other. There is no synthesis, no one way to teach that gives each student what, ideally, they would get.

I think Rochelle Gutierrez describes this well as the “inherent contradictions of teaching mathematics from an equity stance”:

Although teachers must recognize they are teaching more than just mathematics, they also have to reconcile that fact with the idea that, ultimately, they are responsible for helping students learn mathematics. Teachers who are committed to equity cannot concern themselves with their students’ self-esteem and negotiated identities to the exclusion of the mathematics that the students will be held responsible for in later years. Yet preparation for the next level of mathematics must also not be the overriding feature of a teacher’s practice. In answer to which of the two foci are important (teaching students or teaching mathematics), I would answer “neither and both.” It is in embracing the tension…”

That tension I feel as a teacher is the same tension I feel about myself as somebody with gifts. (Trite but true: we all have some.) You have to know that your gifts really are gifts — you really are gifted — also, nobody gives a shit about your gifts. You have an obligation to learn, and everyone has that same obligation. The more time you spend wondering if maybe you really are special, the less likely you are to do anything of value. This is the old growth mindset mantra, and it’s true, but it should only be concerning if you actually do want to do something of value.

So I think there’s maybe no way to solve this cleanly in mainstream US schools. The main thrust of classroom teaching is the need to reach everyone; gifted students are just another everyone. At the same time, there really are gifted students and they really do have different needs. And every inch in one direction takes away an inch in the other. As Labaree puts it, from the perspective of schools and teachers someone has to fail,

The tension is real, but I do think there’s something that would have helped a student like Gómez. Parents, teach your children: there is an obligation to learn.


Doodling the Axioms of Set Theory

Our son is almost three, and he’s starting to really like to draw. He’s also getting to the age where, if he doesn’t have something to do, he’ll tear our apartment to shreds, so most Sunday afternoons we dump a lunchbox of crayons and whatever paper we have on the floor. It’s arts and crafts time, Yosef!

Now, don’t get me wrong, I like watching my kid draw as much as the next parent. (“It’s a fish? No, a dog? Oh, wow, that’s terrific.”) But, what can I say, I get a bit bored.

More to the point, drawing with crayons is so much fun. Arts and crafts time is great. So I draw along with him. And very often I find myself trying to doodle some math.

Lately I’ve been studying a book called Classic Set Theory. It’s been really working for me. It has great exercises, clear organization, oodles of historical context. It’s subtitled “For Guided Independent Study,” and it’s making me wonder why every math textbook isn’t for guided independent study too.

Set theory and logic was my way back into math. I was a philosophy major and had all sorts of worries about whether I could handle a college math class. Then I took a philosophy class about Frege, Russell and Wittgenstein, figures who stand at the creation of mathematical logic and set theory.

In that class, I was finally starting to understand how I had to study this stuff — line by line, ask myself lots of questions, don’t leave anything fuzzy. (Which is more of less how I know to study Talmud.) And I was realizing that if I put in this sort of effort, and if I was honest with myself about what I got and what I didn’t, I could understand some (if I do say so) ridiculously abstract stuff at at least a basic level.

“At some level,” because there was still a lot of stuff that I didn’t get. Since college, set theory and logic has been some of the math that I’ve read the most on. But I still haven’t felt like I really understood set theory, as I hadn’t been able to make much headway in any set theory text.

This is part of why Classic Set Theory is so much fun for me to read. Finally, I feel like I’m starting to get this stuff. Finally!

Here’s the question I found myself asking while drawing with my son today: what could the axioms of set theory* look like if you drew them?

The axioms of Zermelo-Fraenkel set theory with the Axiom of Choice, I mean. I’d love to understand some of the other set theories better. During that class I took in college we studied type theory as Russell’s attempt to patch up the contradiction he discovered in Frege’s system, but I think I only ever had a surface-level understanding of what this meant.

There are nine axioms in all. I know that explanations work better when the objects to be explained are doing stuff, so I tried to make the axioms as active as possible in the drawings. In practice, this means I interpreted the axioms as little machines, rather than as rules or laws. (Axioms have always been presented to me as rules, and until the last couple of days that’s always how I’d always thought of them.)

The toughest one to visually represent was Replacement. The ones I’m most worried about having misrepresented are the Axiom of Infinity and of Foundation. Honestly, all of them are probably flawed in some way. There might be mathematical errors or misinterpretations — as long as you’re nice about it, would you point those out to me?

But I’m not sharing these as resources or to make some point about teaching. I’m sharing these out of celebration, and a bit of relief, that I might be finally figuring out how to learn some math on my own.

Extensionality, Empty Set, and the Axiom of Pairs
Axiom of Separation
Power Set Axiom
Union Axiom
Axiom of Replacement
Axiom of Infinity
Axiom of Foundation
Axiom of Choice