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

Study an example, see the world

I’ve been a math teacher in New York City since 2010, a few months after I graduated from college. It’s the only job I’ve ever had, besides for little things over the summer when I was a teen. (In order: babysitter, camp counselor, Pepsi vendor at Wrigley Field, tutor. All kind of relevant to teaching, come to think of it.)

Though I teach math, math didn’t feel easy for me as a student. It was never where I shined. An exception was geometry, with its heavy emphasis on proof. Proof felt natural for me in a way that algebra didn’t.

When I began teaching, I realized that for many students the situation is reversed — it’s proof that feels unnatural and cumbersome. Writing a proof involves combining statements in ways that seemed to mystify many students. This was especially true early in my career.

After a few years of hitting my head against the wall, I started to understand what made this such a difficult skill to teach. Proof is the closest that mathematics comes to writing, and writing itself is impossible without reading. How can a student who has never read an essay possibly write one? I concluded that my students needed to read more proofs.

It took me a few more years to understand how exactly to pull this off in class. My big frustration was that my students wouldn’t devote enough attention to the proof examples I shared. I would distribute a completed proof and ask the class to read it with care. Very often, it seemed that they missed the whole point of the proof. They couldn’t read it carefully yet — they didn’t know how.

Now, things go better when I share proofs in class. One big difference is I have a much better understanding of all the subtle conceptual understandings that go into a proof, many which were invisible to me at first. (In teaching, it can be trouble when a topic comes naturally to you.) There are many aspects of a proof that I need to help them uncover.

Besides for a better understanding of the subtleties of proof, I’ve learned to structure my activities in sturdier ways. I’ve learned to design these activities so that they have three parts:

  • The proof example
  • Comprehension questions about the example
  • Proof-writing practice, with the example as a model

I didn’t come to this structure on my own, by the way. I came to it through reading about Cognitive Load Theory (where these are sometimes called “example-problem pairs”) and especially from seeing it in some especially well-designed curricular materials:

Screenshot 2017-11-30 at 6.49.21 PM.png


Screenshot 2017-11-30 at 6.50.45 PM.png


(In fact, I didn’t really understand how to make my own example activities until I saw many models in these curricular materials. I needed examples, myself.)

So, for instance, I created this proof example for my students this year:

DM8ANAWWsAACjSC.jpgLooking back, the example isn’t perfect. It ended up being a bit visually crowded, and it might have been better to eliminate some of the letter-abbreviations. In class, I actually covered up each stage of the proof to focus their attention on each part.

In any event, this activity shows a lot of what I’ve learned about teaching proof. I knew I wanted to make explicit the complicated two-stage structure of some congruence arguments, so I worked hard to create a pretty clear example for my students. I then called on students to answer a trio of analysis questions about the proof — there’s a lot to notice, and students don’t yet know how to notice the underlying structure of this kind of proof all on their own. Finally, I ask students to use what they’ve noticed on a related pair of problems, so that students see that there’s something here that’s generalizable to many different kinds of diagrams.

Even when my proof activities aren’t structured so rigidly, I try to include variety and a chance to practice. Here is a simpler activity, but I still call for students to do a bit of proof-completion in the second prompt:

DM8ANArXkAItunW.jpgSometimes when I talk to other teachers about examples, they tell me they’re worried that kids will just try to unthinkingly copy the model. I do know what they mean, but it’s not what I see with my kids. I think that part of the reason is that I reserve example-analysis for when I worry that the math is going to be difficult, even overwhelming for many students. There is certainly a way to misuse these activities, and perhaps if I used these sorts of tasks on less complex material I would see unthinking imitation.

One of my jobs is to help students see things that they can’t yet see — things like the logical structure of a good mathematical argument, or the way just a tiny bit of information about a shape can guarantee a whole lot more. When things don’t come naturally to my students, what I’m learning to do is to design an activity that opens up a little window into the mathematics so they can learn to see new things.

Addendum (1/6/18): I just came across this lovely line from Paul Halmos:

A good stack of examples, as large as possible, is indispensable for a thorough understanding of any concept, and when I want to learn something new, I make it my first job to build one.

Something I Wrote on NCTM’s Membership Forum about Equity and NCTM Policy

I also think there should be an opportunity to review NCTM’s policy commitments and their relation to equity at the national conference. It’s not at all clear to me that the policies that NCTM gets behind lobbying for in DC help to reduce inequality. This should be a matter for member discussion.

My understanding — just from things that I’ve read — is that much of NCTM’s lobbying happens as part of the STEM Ed Coalition []. Here are aspects of the STEM Ed Coalition platform that trouble me, from the perspective of equity:

  • They call for expanding accountability measures and testing to include science, but testing regimes are frequently used to support inequity in practice.
  • They call for private money to flow into education to support STEM education, though in practice private money has been used to support inequity.
  • The foundation for their STEM platform is the belief that STEM jobs are necessary for national security and economic reasons. This kind of nationalistic justification is often used to support inequities in education — after all, it’s in the national interest to have our very best students, and only our very best, in STEM.

If NCTM wants to put equity at the center, then NCTM policy needs to be revisited at the conferences.

Teaching, in General


If you give a quiz covering lots of different topics, you’re going to get a lot of different mistakes. Which leaves you with a dilemma: how do you address those mistakes?

Yesterday’s quiz in geometry was a review quiz, so the topics were from all over the place:

  • angles in isosceles triangles
  • inscribed angles in a circle
  • area of triangles, parallelograms and trapezoids
  • congruence proofs

As expected, kids distributed their not-quite-there work fairly evenly across these topics. (OK so that’s not true, there were a lot of issues with the congruence proofs. There always are and always will be. Sigh.)

Here were two bad options for returning the quiz:

  • Try to address all the issues with individual comments. First, it’s a game of whack-a-mole that is guaranteed to drive me insane. Second, what should I do? Try to leave perfect hints? Say nothing, and let kids figure out on their own what they did wrong? Show them the correct way to answer the question, and thereby eliminate anything for the kids to actually think about when I return the quizzes?
  • Pick just one thing to focus on. Reteach that one thing in a careful way, then return the quizzes and ask kids to revise.

The second of the two options is great when there the mistakes are in the same galaxy. (I wrote about this in a post, Feedbackless Feedback.) But, I’m realizing now, this isn’t a terrific move when the mistakes are distributed across many topics. Because on what basis should I pick something to focus on reteaching? Any choice would be equally bad.


While reviewing the class’ quizzes, I found myself falling into written comments, at least until I figured out what else to do with the quizzes.

I used to write long, wordy comments that were essentially hints on the margins of the page. (“Great start! Have you tried multiplying both sides of the equation by 3?”) I came to dislike those sort of comments, as they just focus focus focus attention all on THIS problem. But I don’t particularly care about whether a students gets this problem correct; I care about the generalization.

What I’ve fallen into is, whenever possible, writing a quick example that’s related (but not identical) to the trouble-problem (the problem-problem) on the page. I do this below on the second question:


Then, I ask kids to revise the original on the basis of the example (or anything else they realized).

After writing a few of these example-comments, I realized I was taking a lot of time doing this, and repeating myself somewhat. I also realized that I don’t know if I could repeat this on every page for the congruence proofs, as the problem itself was reasonably complex:

Hard to read. I use the highlighter to flag errors, but this student highlighted the triangle himself.

I wasn’t sure what to do. Then, I remembered something I had read from Dylan Wiliam — I think it’s in Embedded Formative Assessment. His idea there was that you can give all the class’ comments to everyone, and then kids have to decide which comments apply to them.

I thought, OK, I can work with this. So I quickly (quickly!) made a page of examples, one for every mistake I saw on the quiz:

ums-copier@saintannsny.org_20171115_095927 (1)-page-001.jpg

My routine in class went like this:

  • Hand out the examples for revision.
  • Hand back the quizzes with comments.
  • Search for an example that’s relevant to your mistake.
  • Call for revision on the basis of the examples. Work with friends, neighbors. Of course, I’m available to help.
  • Then, try the extension task.

This was my first time trying this, but I thought it went well. Solid engagement, really good questions, no unproductively stuck students.

When you do something good in teaching, you never really know if it’ll work again, but I’ve got a good feeling about this one. It feels like a lot of what has already worked for me, but in a better order.


Harry Fletcher-Wood is very nice and has a lot of interesting thoughts about feedback. As such, Harry and I very nicely disagree about a pretty interesting question about feedback: how can you teach people how to give better feeedback?

The usual caveats apply: I am not a teacher teacher, but Harry is involved in teacher education, and I have no idea if I’m right on this.

In any event, Harry recently published a really cool post where he tried to synthesize a lot of the research on feedback into a decision tree:


Now, this is awesome as a synthesis. But just because something is a good description of feedback doesn’t mean that it’s useful prescriptive advice. My favorite example of this comes from Pólya’s strategies for mathematical problem solving. Alan Schoenfeld has a nice way of putting it in Learning to Think Mathematically — the strategies have descriptive, but not prescriptive validity:

In short, the critique of the strategies listed in How to Solve It and its successors is that the characterizations of them were descriptive rather than prescriptive. That is, the characterizations allowed one to recognize the strategies when they were being used. However, Pólya’s characterizations did not provide the amount of detail that would enable people who were not already familiar with the strategies to be able to implement them.

In other words, just because a heuristic is a good description of practice doesn’t mean that it is an effective pedagogical tool. And that’s precisely my concern with Harry’s decision tree.

Feedback is a high-level concept that describes a TON of what happens in teaching. And any guidelines for how to give feedback effectively are also going to be high-level in a way that reminds me of Pólya’s moves like “find a simpler problem” or “draw a picture.”

And just as Pólya’s moves struggle because they aim to guide problem solving in geometry, algebra, topology, etc., all areas of math, Harry’s decision tree seems to me an attempt to guide feedback in all areas of teaching — math, history, medical school, etc.

Of course, Harry doesn’t intend for this to be the only thing guiding students, but neither did Pólya. My question is whether these generalizations themselves are helpful, beyond whatever ways that teacher educators can make them concrete and specific for teachers.

But what’s the alternative?

I don’t know yet. I can say a few things now that I couldn’t a few years ago:

  • I think domain-specific — math-specific, history-specific — generalizations will be more useful than domain-general ones.
  • I think that the generalizations can productively come in the form of instructional routines.

And, with this post and the other one, I now have two generalizations I can make about giving feedback in math class.

First: if there’s a problem that a lot of students have trouble with, consider a reteaching/revising cycle like the one in this image:

Screenshot 2017-11-15 at 1.24.03 PM

Second: if mistakes are sprinkled across too many topics, consider something like the revision routine I described in this post.


My bet is that a lot of knowledge about teaching looks like this. It’s not that there isn’t knowledge about teaching that accrues, but that we look for ways to scale things out of their contexts. Then we call those things myths and talk about how we have to kill ’em.

In general, generalizations about teaching are hard to come by. But nobody teaches in general. All teaching is intensely particular. These kids. These schools. This idea.

Some people are skeptical of the possibility of making generalizations about teaching, and the vast majority of people are cheery about making sky-high generalizations that cross every context. There’s a middle position that I want to find. There’s a sweet spot for knowledge about teaching, though I don’t know if we’ve all found it yet.

Addendum: On Discovery and Inquiry


I appreciated some of the disagreement that got aired as a response to my last piece, on discovery. In particular, some told me that guided-inquiry or discovery really is more memorable than other forms of instruction.

Either because the stuff you learn from discovery is more meaningful (and hence more memorable):

Screenshot 2017-11-07 at 9.31.30 AM

…or because discovery involves surprise, and surprises are more memorable and lead to stronger learning:

Screenshot 2017-11-07 at 9.29.10 AM

I’m not satisfied with either of these arguments.

The thing about discovery activities is that the new idea — by definition of discovery — comes at the end of the activity. That means that kids are spending most of the activity thinking about stuff besides the new, often difficult, idea. It takes time to understand new ideas — to make them meaningful, to “own” them — and most of the time in a discovery activity is spent thinking about other stuff.

That’s certainly the case for the triangle angle activity that I critiqued in my post. While working on the activity, a student’s attention is drawn to many mathematical things — the angles, protractors, adding angles — and only very little of the time is spent thinking about what exactly a triangle’s angles sum to. (This is especially true if the idea is truly new to a student — they’ll only be thinking about the sum once they discover it, towards the end of the activity.)

It’s also true in the trapezoid/triangle area task that I shared. There was a ton of excitement precisely because my class hadn’t discovered the relationship between bases and area yet. That was where the joy was coming from — that also means that they were thinking about the discovered relationship for comparatively little of the time spent on the activity.

As I argued in the original post, that’s OK for me. It was fun and beautiful, and kids should have a chance to articulate slippery patterns and feel the pleasure of discovery. That’s part of math that I enjoy sharing with kids.

Anyway, that’s my response to the idea that discovery is more memorable because it’s more meaningful. Ideas are meaningful when you have time to get used to them, and that’s precisely what gets lost in a discovery activity.

As far as the idea that guided inquiry is surprising, and surprising stuff is more effective: why can’t you structure an explanation to elicit prior knowledge and surprise students? Aren’t explanations sometimes surprising? I think they can be.

Of course, how to craft effective explanations — that surprise and really engage students — is not easy, but it doesn’t get any easier if we don’t talk and write about it. That was part of my argument in Beyond “Beyond Explaining.”


This is all theory, though. What happened in class today, after the weekend, after the memorable discussion on Friday?

I ask them to find the area of a trapezoid and…it’s like Friday never happened.

Wait what?

Hold on what do you mean the same as a triangle?

Could we go over this again?

The only kid who remembered how to find the area of the trapezoid — and I promise this is true, and not just me making up details to annoy advocates of discovery — was the kid who had connected Friday’s lesson to a formula that she once knew.

No guys, it’s the sum of the bases times half the height.

This is sort of surprising and disappointing. Friday’s class was so good! And nearly everybody was involved in the inquiry/discovery/discussion. It felt wonderful and it was fun.

That class, for me, was discovery that’s about as good as it usually gets. And yet it failed to stick over the weekend.

And yet this isn’t that surprising. The kids didn’t get a chance to practice the idea on Friday because we spend the class time uncovering some super-cool math. Kids need practice to remember ideas, and discovery takes a long time. This is just how it goes.

But if it’s not surprising, it’s also not disappointing. It was a lot of fun, and everybody was involved. It’s not what my class is like every day, and it would probably frustrate kids if it were.

So, at least this time, anecdote matches argument. And since we started practicing finding the area of trapezoids today,  it’s getting a lot more meaningful for my kids.

A typically wishy-washy take on discovery in math class

I think of myself as a non-discovery math sort of teacher, but every once in a while I find myself asking kids to discover stuff. I recently did this in my geometry classes, with a dot-paper area activity.

I really like dot paper. On dot paper you can make precise statements about area that typically generalize nicely to non-dot scenarios. The same basic relationships apply, it’s just easier to see them in a dot world.

My idea was to share this picture with students, and ask them to find the area of each shape. (This is the version that I marked up under the camera in class. None of the purple or blue ink was in the original.)

Picture 1.JPG

All of these shapes have the same height — 4 — but they otherwise differ. The first is a rectangle that has area 8. Next was a parallelogram: also area 8. Then a trapezoid. Many students came up with the idea of drawing a line to split it into a triangle and a rectangle — together they have area 8. This didn’t occur to every kid, though, so there was a good deal of neighbor-asking and chat to get the hang of those trapezoids.

Spoilers: every shape here has area 8! There’s something else they share too: the sum of their top and bottom bases is always 4. (The weirdest case is the triangle, that has a bottom of base of 4 and [arguably] a top base of length 0!)

So that was what I wanted my kids to come to notice and articulate. And I wanted it to be fun, and feel like they were discovering something new.

Not because I think that learning is more effective when kids discover something new, or that they’re working on their discovering skills or anything. Just because I think it’s fun for them to uncover patterns. It’s a cool part of math, and I’m trying hard to share more of the cool parts of math with my students, along with working on their skills and knowledge etc.

Here then is my take on discovery in math class:

  • It can be fun to discover cool stuff about math.
  • It takes longer for kids to understand something via discovering, and doesn’t really confer a learning advantage.
  • But if the activity is relatively brief and I can spare the time, why not? I want my students to think math is fun and cool.

This is my unprincipled take. I like discovery for fun and color in class, and I don’t feel the need to aim for 100% efficiency in every second of my teaching.*

Whether you feel such a need — or need to feel such a need — probably varies a lot depending on your school, administration, students, etc. My guess, though, is that the vast majority of teachers don’t feel this need, and probably are correct in this feeling.

Lots of discovery activities are uninteresting to me. Though I absolutely love the practice exercises in Discovering Geometry, the discovery activities largely leave me cold. Here’s an example of the sort of thing I’ve fallen out of love with:

dg discovery.png

My issue with this activity isn’t that it’s discovery. It’s that it’s not such a cool pattern (most kids have seen it before taking a geometry class), so discovering it isn’t as surprising or fun. The activity takes a while — do it once, check, do it twice — and all that is probably to protect against the risk of incorrect measurement, which is another tedious aspect of this discovery activity. And, at the end of all this, what cool math have you uncovered? Relatively little — just a sum. You don’t see any new relationships or geometric structure that guarantees that the triangle will have such a sum.

Why bother with all this? I’ve decided that this sort of discovery activity isn’t much use for me. But that’s not a principled objection against discovery — it’s just that I don’t think this type of activity is worth it.

Speaking of “types of activity,” I think it’s fair to categorize this angle sum activity as discovering something easy to articulate. Check an example, check an example, check an example, woah it’s always the same. You don’t uncover the geometric relationship in this activity. I think that’s part of what makes it not much fun and sort of tedious.

I think discovering something hard to articulate (I’m not quite sure what to name this) tends to be more fun, more cool.

To illustrate this, here’s the conversation that went along with my “area 8” activity in class.

I began the conversation with a prompt to my students: what do all these shapes share? how do these shapes differ?

I called on James first.

James: They all have heights and bases of 4.

I said that I didn’t entirely get that, and asked if anyone also saw that. Robin came up to the board to point. She also subtly refined Jame’s claim.

Robin: A lot of them have a common theme that either the height is 4 or the base of 4.

Then Liam chimed in to make it precise and accurate:

Liam: The height is always 4. The bases are different though.

Luiz: Yeah, the bases are either 2 or 4…or no they vary. Sometimes it’s 1, 2, sometimes it’s none.

Then Sara chimes in. She started articulating a generalization — she was WORKING HARD to try to articulate some sort of generalization. Her first one wasn’t entirely accurate though. I loved how she put it as a question.

Sara: Wait, does that mean that any shape that has a height of 4 has an area of 8?

The class and I (and Sara) agreed: this just could not be true. (Luiz says: well you could have a base of 1000.) Jess tried to get clearer about what was special about all these shapes.

Jess: No this is just because they are all parallelograms and…wait the third one is sort of confusing. What is that?

[Insert a minute of discussion about which of these are trapezoids and which are parallelograms.]

After this, I decided that we wouldn’t be able to restart the conversation unless I summarized and took some notes. So I wrote some notes on the board.

This time, though, I decided to take a heavier hand to draw attention to something really cool. The kids hadn’t noticed it yet, and I wanted to make sure that they did.

I wrote the numbers that you saw in purple ink in my image. I wrote the top and bottom bases.

Sara: So couldn’t you say that it’s base 1 plus base 2, times the height and that’s the area. Like for shapes in general.

Very close! There are some gasps and agreements.

Samantha: So wait does that work for everything?

Sara: Yeah that’s what I’m asking.

Me: That’s a good question. I’m trying to find some dot paper.

Joe: Wait so does it?

That question just hangs there for a second. And here’s a choice I could make. I could act coy, refuse to answer, and insist that the thinking here come from the students.

But then you get this weird dynamic in class where kids never know if they’re getting a straight answer from the teacher or not. I don’t like that dynamic. I like it when kids ask questions about math, and I like that they can get a straight answer out of me. And would they spend more time thinking about this cool relationship if I answered that question, or if I refused to?

“The answer is yes, sort of.”


Sara: Does it have to have parallel lines? Does it have to be a trapezoid…wait does a parallelogram count as a trapezoid.

Good thinking, Sara!

While there’s thinking kind of just toppling out of Sara’s head, I’m searching for a blank piece of dot paper, because there’s something that I realize might help. I don’t want them to get too used to the area 8 case — that might lead to a false generalization, since Sara keeps on saying that it’s double the sum of the bases. (That’s true when the height is 4, but not when the height is something else.)

So I draw this:

Picture 2.JPG

We clocked in at about 10 minutes there. No question that this was not as effective as a worked example or something else more carefully designed for learning. But I wasn’t aiming for efficacy. I was aiming for those half-articulations, those gasps, that enthusiasm. And as long as I don’t come to worship those gasps and chase them exclusively, class will be a bit closer to being fun, cool.

Addendum: this follow-up post.



YouCubed is Sloppy About Research


There’s a lot of sloppy talk of science that gets tossed around education. Every teacher knows this — or least, I hope they do — and I try not to get too worked up about it. I also try not to get worked up by people who wear backpacks on crowded subways. Not that it’s OK, but lots of people do it (the backpack thing) and picking any one person to bark at hardly seems like it would help.

The other thing is that people can be really passionate about sloppy science. This happens all the time, it’s nothing special about education. But passion makes it hard to talk critically about the research without it seeming like you’re attacking everything else that the person is passionate about.

I think a lot of the time it’s because we assume that the attack on the research isn’t really about the research, it’s about everything else it supports. It’s like, Why are you bothering to poke holes in [research that supports X]? You’d only do that if you were really against [X]. 

Which leads to an interesting question. Is it possible, at all, to avoid this trap? Is it possible to critique sloppy use of research without being heard as if you’re trashing a person, their organization, and everything they stand for?


What if you ask a lot of rhetorical questions — does that help?

Anyway, let’s talk about YouCubed. I think there’s something that — if we slow down, turn off passion, turn on curiosity — we can agree is a serious mistake. Here’s a popular quote from their popular page, Mistakes Grow Your Brain:

When I have told teachers that mistakes cause your brain to spark and grow, they have said, “Surely this only happens if students correct their mistake and go on to solve the problem correctly.” But this is not the case. In fact, Moser’s study shows us that we don’t even have to be aware we have made a mistake for brain sparks to occur.

When teachers ask me how this can be possible, I tell them that the best thinking we have on this now is that the brain sparks and grows when we make a mistake, even if we are not aware of it, because it is a time of struggle; the brain is challenged and the challenge results in growth.

Indeed — this does sound really, really surprising! So, applied to math, it sounds like if you solve an equation incorrectly you learn from that experience (brain sparks?) in a way that you wouldn’t if you had solved it correctly. The mistake you made causes struggle — even if it’s not a struggle that you’re aware of. You might not feel as if you’re struggling, but your brain is on account of the mistake.

The citation of Moser is very clear, so, ok, let’s go upstream and check out Moser. Though a lot of research is behind paywalls, a lot of it isn’t and a quick Google Scholar search gives us a copy of the paper, here.

The study was an fMRI (EEG, see below. -MP) study. Here’s my understanding of the paper. The researchers wanted to know, on a neurological level, what’s different about how people with a growth mindset or a fixed mindset react to mistakes. So they gave everybody a test, to figure out if they had a fixed or a growth mindset. Then they hooked subjects up to fMRI EEG machines. While in the machines, participants worked on a task that yields a lot of mistakes. Researchers recorded their neural activity and then analyzed it, to see if there was a meaningful difference between the fixed and growth mindset groups, after making errors.

Spoilers: they found a difference.

Also: the whole paper is premised on people being aware of the errors that they made. This is not a subtle point, buried in analysis — the paper mentions this like nine times, including towards the end where they write, “one reason why a growth mind-set leads to an increased likelihood of learning from mistakes is enhanced on-line error awareness.”

This is so clearly different than what the YouCubed site says that I’m starting to doubt myself. The paper seems to be entirely about what happens when you realize you’ve made a mistake. Yet it’s cited as supporting the notion that you learn (“brain grows”) from mistakes, even if you aren’t aware of them.

Is there something that I’m missing here?


This doesn’t seem to me like an isolated issue with YouCubed.

A while ago Yana Weinstein and I made a document together trying to collect errors in YouCubed materials, explain why, and suggest improvements. I don’t have much else to say about that, except that if you’re interested you might check it out here.


What strikes me about YouCubed is that the errors just seem so unnecessary. The message is a familiar one, and I’m OK with a lot of it: don’t obsess over speed, think about mindset, don’t be afraid of mistakes. But there’s this sloppy science that gets duct taped on to the message. What purpose does that serve?

There’s also the question of why so few people in the math education community talk about this. I mean, it’s not like we lack the critical capacity. Every so often I see people in math education whip out their skeptical tools to tear apart a piece of research. Why not with YouCubed?

I don’t want to be cynical, but I want to be truthful. The first reason, I think, is because the message of YouCubed (besides the science) is widely admired. A lot of teachers love it, and nobody wants to ruin a fun time.

But I don’t think we have to worry about that. We can talk about the science of YouCubed in a way that doesn’t entangle the rest of the YouCubed message.

The second reason is, I think, that YouCubed and Jo Boaler’s popularity makes it difficult for the most visible people in math education to seem critical — no one wants to turn on one of their own. Especially since Boaler has often been subject to unreasonable attacks in the past — nobody wants to be unfair, cruel or sexist to her.

I’m actually very sympathetic to that. But this is also why it’s important for people who aren’t part of the research or PD world to have platforms to discuss ideas. We don’t have the reputation or the connections to lose, and so we can take a closer look and ask, wait, does that really make sense?


Actual neuroscientist Daniel Ansari points out that I don’t know the difference between an EEG and an fMRI, which is true.