Book Review: Masters of Theory

Masters of Theory: Cambridge and the Rise of Mathematical Physics by Andrew Warwick


What does it mean to excel at mathematics? And how should you teach towards this kind of excellence?

A common way of telling it is that for as long as people have been learning math, there has been a “traditional” way of teaching it. The tradition of traditional math teaching involves, but is not limited to: textbooks, nightly homework, in class problem solving performances, corrections and grades.

When did it start? Where? One of the things that really stunned me about Masters of Theory is that there is an entirely concise and known answer to this question: it started in Cambridge University, and coalesced in the 1840s and 1850s. And as to the question of “why?,” the answer is largely: because of competition and testing.

More on that development in a moment. First, some trivia.

  • Did you know that the word “coach,” as in “mathematics coach” or “basketball coach,” came into practice in the 1830s in England to describe private mathematics tutors? The word was borrowed from “coach” as a mode of transport — the most efficient, fastest, classiest way to travel. When Cambridge undergrads were reaching for a word to describe the work of someone you could hire (if you had money) to help you smoothly move through the mathematical terrain of the mathematics curriculum, they turned to analogy: coaches.
  • For a top student, James Clerk Maxwell had a reputation as a sloppy student of math at Cambridge, and his attempt at capturing his thoughts on electromagnetism in a text was basically incomprehensible.
  • Before the 19th century, paper had very little to do with learning mathematics.

These pieces of trivia fit together nicely in Warwick’s detailed narrative. There are clear stages marking the shift. Here is a summary.

Step 0: What was learning university math like before the shift?

The goal in the medieval university wasn’t technical facility with a problem solving apparatus. You weren’t supposed to be able to solve problems. The goal was knowledge of works of mathematics.

A central text was Euclid:

“The main form of undergraduate teaching was the thrice weekly college lectures in which students were taught the basics required for a pass or mediocre honors degree…The main job of the mathematics lecturer was to go through required sections of such important texts as Euclid’s Elements, ensuring that the majority of students had at least a minimal grasp of geometry, arithmetic, algebra, mechanics, hydrostatics, optics and astronomy. Lectures were run at the pace of the average student and appear…to have consisted mainly in the lecturer asking students in turn to state proofs and theorems or to solve simple problems orally.”

If you were really great at mathematics, that meant that you were the absolute best at oral discourse in mathematics. You could recite proofs, split hairs, debate and give convincing rhetorical presentations of various proofs or arguments, etc.

Step 1: Newton, Leibniz and the emergence of an extremely powerful technical apparatus for solving problems in math and physics

Then, Calculus happened:

“In the early seventeenth century, most of what was known at the time as ‘mixed mathematics’ was not overly demanding from a technical perspective. However, it was considered by the majority of scholars to be of only secondary importance to the study of ‘physics,’ the search for the true causes of natural phenomena…It was with the gradual translation of Newton’s mechanics into this new [algebraic] language in the early years of the eighteenth century that the fundamental techniques of mathematical physics, indeed the discipline itself, began slowly to come into existence.”

And this was completely different than the mathematics that was previously seen as valuable for undergraduates to learn:

“The aspect of these developments of most immediate relevance to our present concerns is that the increasingly technical nature of physico-mathematics from the mid-seventeenth century made it ever more alien to undergraduates students.”

The new technical apparatus was useful in solving a variety of problems. Yet university mathematics education at Cambridge didn’t significantly change. It continued teaching more or less as it always had:

“Even at Trinity the primary function of a college lecture was still to inform students what they should read and to test their recall and understanding of that reading by catechetical inquisition…In order to find out whether students had learned the definitions, proofs, and theorems they were required to know, the lecturer would go round the class asking them in turn to enunciate propositions and even to solve simple problem orally. the paper and pens provided in the lecture room were not therefore central to the teaching process, but enabled students to take notes as they saw fit during the oral exchanges between the lecturer and individual members of the class…The only visual aid employed by the mathematical lecturer was ‘a cardboard, on which diagrams were drawn relating to the mathematical subject before us’ (Prichard, 1897, 36). This cardboard was handed from student to student as the lecturer went round the class and seems to have functioned as a kind of primitive blackboard.”

This is 100 years after Euler, 150 after calculus, and university math class still didn’t involve learning how to solve problems.

(Going off script here for a moment to note that you were still supposed to be able to solve arithmetic problems. The point is that if you wanted to assess this, you’d do so in an oral recitation. Written mathematics wasn’t expected, and so anything beyond basic mental math wasn’t oriented towards solving problems.)

Step 2: The new mathematics is perfect for ranking students and assigning scholarships

As this is going on, Cambridge University began using the Senate House examination (a.k.a. the Mathematical Tripos) to award valuable college fellowships. If you were one of the top 2 or 3 “wranglers” (as they are apparently called in Cambridge) you were guaranteed “one of the few recognized routes by which a young man from a relatively poor background, but with academic ability, could make his way up the social scale in Georgian Britain.”

At first, it was an oral test and not focused on mathematics. But the new technical mathematics was perfect for clearly ranking students (you either solve the problem accurately, or you don’t) and the exams changed in three ways that Warwick identifies:

  1. the exam became focused more on math and natural science
  2. if you wanted to rank in the top few, you had to do a written exam, not just an oral one
  3. the exam became increasingly competitive

So, the exam, its content, and the incentives students encountered were changing rapidly. But the teaching in the university classes didn’t change at all. Which meant that before long…

Step 3: An extensive system of private tutors (“coaches”) emerged to prepare students for the examination

“The main point to take from this brief survey of Cambridge pedagogy is that professorial and college lectures, tutorial sessions, public disputations and private study were all forms of learning based in the first instance on reading or oral debate. With the gradual introduction of written examinations the preferred form of teaching began to change to suit the new form of assessment. Success in the Senate House examination depended increasingly on the ability to write out proofs and theorems and to solve difficult problems on paper. Ambitious students accordingly turned to private tutors.”

Step 4: These tutors, competing with each other for top students, soon land on a paper-based pedagogy that is distinct from what the university offers, and closely resembles what we think of today as “traditional” math teaching

There was a whole industry of tutors surrounding Cambridge University, but some tutors stood out and gained reputations for reliably producing top scorers on the examinations. The first of these elite tutors was William Hopkins:

“Hopkins’ success derived from his teaching methods, his own ability and enthusiasm for mathematics, and his reputation as a tutor. Unlike the majority of private tutors, he taught students in small classes–between ten and fifteen pupils–composed of men of roughly equal ability. This meant that the class could move ahead at the fast possible pace, the students learning from and competing against each other. Hopkins considered it an ‘immense advantage in Class Teaching when there is a sufficient equality in the ability and acquirements of each member of the class’ that ever student would: ‘hear the explanations which the difficulties of others might require, and thus be led to view every part of the subjects of his studies, through the medium of other minds, and under a far greater variety of aspects than those under which they would probably present themselves to his own mind, or would be presented by any Tutor teaching a single pupil (1854, 19-20).”

“Hopkins’s teaching methods were thus designed to optimize the benefits of intensive, progressive, and competitive learning. He also developed and exploited an avuncular intimacy with his pupils which would have been quite alien to most college lecturers and university professors…In the friendly atmosphere of his teaching room, Hopkins combined the admiration of his students with his own infectious enthusiasm for mathematics to promote the competitive ethos and a dedication to hard work.”

An entire chapter is dedicated to a later tutor, Edward Routh, whose students dominated the exams over many years. Here’s what Warwick has to say about Routh’s teaching:

“The primary method of teaching, around which Routh’s whole system was built, was the one-hour lecture to a class of not more than ten pupils using blackboard and chalk. The blackboard was a fairly recent pedagogical innovation in Cambridge at this time, private tutors having previously worked on paper with their pupils sitting next to them…Routh would begin with a ‘swift examination of exercise work’ set for the class at the end of the previous meeting. These exercises generally required students to reproduce proofs and theorems, and to solve related problems, as they would have to in the examination, and Routh would quickly discuss any errors common to several members of the class and those of an individual from which he felt the class might learn.”

Going over homework on a blackboard — this was state of the art!


“Despite the public nature of these corrections there was ‘no jesting, no frivolous word over a blunder,’ and Routh would neither give any ‘tips’ on which exercises he thought likely to appear in forthcoming examinations nor express an opinion on the relative abilities of members of the class. Having corrected the work of the previous lecture he would at once launch into a ‘continuous exposition’ of the material to be covered that day,each member of the class taking notes as fast as he was able. Routh generally led students through which he considered the best textbook accounts of each subject…”

“Revision sessions apart, Routh devoted little time to solving difficult examination problems in his lectures. At the end of each hour, rather, he would hand out about six problems ‘cognate to the subject’ of the lecture’ which were to be solved and brought to the next class.”

“Once a week Routh gave a common problem sheet to all his students, regardless of year or group. In one week the students were allowed as much time as they required to puzzle out the solutions, but, every other week, the problems had to be solved ina timed three hours under examination conditions. Each students was required to leave the problems in the pupil-room on Friday or Saturday in order that Routh could mark them over the weekend. The following Monday the marked scripts would reappear in the pupil room together with Routh’s model solutions (to save him having to waste precious minutes in the lecture) and a publicly displayed mark sheet ranking all students according to the marks they had scores. These biweekly ‘fights’ gradually accustomed students to working at the pace required in the Senate House, incited and preserved an atmosphere of fierce competition, and provided that objective measure of relative merit upon which Routh himself took care never to comment.”

He gave feedback to everyone, and everyone had room to improve:

“Routh was extraordinarily scrupulous in marking student scripts to the extent that ‘it was one of his peculiarities that he was never wholly satisfied with any work shown up to him’ (Moulton 1923). On one memorable occasion when a brilliant student, Fletcher Moulton, mischievously prepared a problem paper ‘on which no criticism could be offered,’ Routh still found room for improvement by urging Moulton to ‘Fold neatly.'”

In contrast, less successful tutors tended to group students in two and threes, didn’t recreate exam conditions, and individualized instruction more (rather than following a systematic and carefully constructed syllabus). Make of that what you will.

Step Five: All this time, university lecturers still weren’t teaching any of the new technical math that calculus and mathematical physics produced! The only way to learn any of this stuff was through the tutors. This went on for way longer than you might expect.

Step Six: Eventually, both the methods and curriculum of this system of private tutors were mainstreamed and brought into the colleges. From there, it spread to top prep schools and eventually to public schools. 


Honestly, none of this is really the main point of the book. The book’s major argument is far more ambitious than just telling this story. It aims to detail how mathematical training at Cambridge involved an extensive local, distinct, technical manner of doing math. If you wanted to learn this way of doing math, you had to be socialized into it at Cambridge. And this manner of doing math, Warwick argues, can help explain the distinctive research that into mathematical physics that happened at Cambridge.

(Historians of science had already argued that the technical apparatus of science — the tools, machines, etc. — involved a local, distinctive area of knowledge that grounds whatever theorems or ideas they shared. Warwick is making a similar argument about paper, examinations, the classes that coaches offered. All of these things are like the microscopes of Cambridge research into mathematical physics. It sounds better when Warwick says it.)

All of this is pretty heady; extracting just the pedagogical story from it is sort of cheap.

(Full disclosure: I only read the first half of the book! That’s where the pedagogical history mostly is.)

At the same time, all of this info deserves to be more widely known in math education. I think it’s particularly relevant to a number of perennial discussions:

  • Has there only ever been one mathematics, or has mathematics changed drastically depending on culture and time?
  • What is “traditional” mathematics teaching? When did it arise, and why did it become widespread?
  • What’s the relationship between the current way we teach mathematics and past ways of teaching it?

I’m eager to read more books like Masters of Theory. Reading recommendations are seriously appreciated.


What should a school do with an advanced 1st Grader?


Here’s the situation:

You’re the principal of a large elementary school

Ooh, spooky. Read on — it gets worse!

You’re the principal of a large elementary school, and one of the decisions your school faces is what to do with young students who aren’t challenged by their class’ mathematical work.

A certain segment of the population of educators is rolling their eyes right now. Read on, read on.

For example, in your school there is currently a 1st Grader who does math above grade level, e.g. he performs long division in his head.

His parents initiated contact with the teacher after hearing their child complain that math at school was boring. They’re concerned that he isn’t being challenged.

The classroom teacher knows that he is above grade-level in math, and is trying to meet his needs in class. The parents, however, do not think the current situation is working. The teacher reports that the student is difficult to engage during math class, and that sometimes he misbehaves during math. 

That’s the situation I posed in a little survey I recently shared on twitter. I then asked two questions:

  1. As principal, what policy would you want your school to do for someone like this 1st Grader?
  2. Do you encounter this situation, or something like this situation, in your current role in education? If so, how is it typically handled, in your experience?

Twenty-one people responded (responses here). This post is about what they said, and what I think about what they said.


Here was my process for writing this scenario. First, I thought about what I’ve seen happen in the schools I’ve worked at. Then I emailed someone with more experience with this stuff than me and asked her how this usually goes down at her schools. Then I basically ripped off her email for this scenario.

Which is a long way of saying that I was pretty sure that this was a realistic situation. Still, it was good to hear people say that it sounded right to them:

“Yes, there is a large range of abilities even in kindergarten.”

“I had 7 [6th Grade] students who tested at the 11th grade level last year.”

“My son IS this kid (and also in 1st grade).”

“Yes!  Currently have similar situations in Grades 3 and 4; it’s a very rare year where this issue doesn’t come up.”

And let’s cut to the chase: according to people who are not me, how well do schools tend to handle this scenario?

A bunch of people thought their schools were handling this pretty well. A lot of these people, but definitely not all, were talking about high schools:

“I have high schoolers and would be willing to place a student in a higher level class if they could study and demonstrate mastery on a department final exam. I have tried assigning Khan Academy on an individualized basis.”

“Rarely [is this a problem] since math classes can easily be leveled, especially on a HS block schedule. And I think it’s easier for HS math teachers to connect to more advanced courses if they’ve taught them before. (In my experience HS math teachers change courses more often than elementary teachers change grade levels).”

“Yes, there is a large range of abilities even in kindergarten… It just takes patience.”

Then a bunch of people said that this was just not working at all in their schools:

“It’s pretty terrible at our school. We don’t have the resources to do anything when kids transfer having already knowing trig.”

“Tough one in High School, especially since we have no honors programs.”

“My son IS this kid (and also in 1st grade). I completely respect the teacher for trying to engage my kid, but he needs something else. Either allow him to work with some 2nd graders (this would be some massive coordination between grades), or give him new puzzles (logic stuff or solving mysteries or…), or help him with particular weaknesses during this time (perhaps social skills or OT), or become a librarian’s helper twice a week while others work with on-level stuff.”

“Typically slight pressure (or guidance) is applied to the teacher to differentiate and provide math learning for the student at the level and depth that they need. Of course, in reality this usually means that the teacher is given a few resources, tries hard, but is unable to challenge the student in the way that they need. Strategies such as open ended questions and 3-act problem solving help, but teaching is hard with so many diverse brains in a class and many teachers do not have experience or knowledge to extend math for such a student.”

“It varies wildly by building in our district but typically the response will be a lot of hand waving about how differentiation in the class is already occurring.”

“Student is given independent work while teacher works with others. Student is given consequences for behavior when not enriched.”

Some schools do seem to have figured out ways of making sure this problem isn’t just dumped on the classroom teachers:

“In grades K-2, these students are often pulled-out for additional math enrichment opportunities. Say the math block is 60 minutes. These students are pulled-out for around 15-20 minutes daily to work on enrichment tasks. This tends to happen at the discretion of the teacher and the groups change for every unit. There are 12 units.

In grades 3-5, there’s actually an advanced math class. The class is grade-level accelerated (3rd grade learns 4th grade concepts) and the criteria for that class depends on standardized test scores. These groups are not flexible and rarely change.”

But, overall, it seemed to me that this is a problem that people have experienced, and many (most?) find their school’s response frustrating.


If the status quo is often bad, what would educators prefer? The responses hardly coalesced around one idea, but I was able to group them:

  1. Find a pedagogical solution (differentiate, use groups, better tasks, etc.) [4]
  2. Assign challenges for the kid to work independently on [4]
  3. Create something outside of the school day (e.g. a club, a course) for the kid [2]
  4. Connect the kid(s) with an enrichment specialist [3]
  5. Let the kid go to a different math class during math [4]

And, finally, there were people [4] who recommended all of these things, essentially treating #1-5 as a ladder of intervention.

Which raises the possibility that everybody on my survey would agree with this, but they were all imagining slightly different situations.

Meaning, I didn’t read these as really disagreeing much. Of course, if you could provide a pedagogical solution — including extension work — that would typically be easier in any situation. And everyone would probably agree that it would be amazing to have another teacher who could come in and handle some of the kids. AMAZING.

And I bet that 95% of teachers would agree that moving kids to another class is the most complicated solution, the break-in-case of emergency solution.

But why? Why should moving to another grade’s math class be the last resort? After all, to people outside of schools it seems like it would be the simplest approach — it requires no extra work on the part of the teacher, no extensions, it’s compatible with whatever pedagogy, and you don’t need extra personnel to make it work. Why is this unpopular?

The easiest answer would be an ideological one — teachers don’t like inequitable solutions — but on the other hand do we really mind creating inequitable solutions?

Every single thing that educators recommend besides for this kind of acceleration also exacerbates inequities. If you have an interventionist providing special challenges to a few kids — that’s inequitable. If you give extension work to some kids — that’s inequitable. If only some kids go to a math club after school — that’s inequitable, even if it’s open to all.

Some may balk at my use of “inequitable,” and I get that, because we only tend to use “inequitable” to describe certain things in education. Like, we’re used to a world where some kids learn more and some kids learn less, and that’s not “inequitable” because everybody got a fair (“fair”) chance to learn the same material in class. But just looking at the situation — one person gets one outcome, one kids gets another — that is an inequity. And if you give only some kids a chance to learn e.g. cryptography in an extension worksheet that’s inequitable too.

That’s a long way of saying that I don’t think teachers are ideologically opposed to acceleration, in the sense that teachers don’t like it because it creates inequities.


So why is moving a 1st Grader to a 2nd Grade math class behind a “break in case of emergency” label? I don’t know. Here are two possibilities:

  1. Schools think it’s educationally risky for students.
  2. It creates a huge pain in the neck for administrators.

I think both are true, but if you know your way around a school you might also know that a lot of things happen because they solve administrative problems. Which is another way of saying that they help administrators deal with parents.

Especially for younger students, moving to another room might backfire. This is both experientially true (some of my 3rd Graders do much better with consistency and a familiar environment) and also something that seems like it might be starting to trickle through in research. See: “Two studies point to the power of teacher-student relationships to boost learning”.

But I think the more likely explanation is that some parents really, really, really want their kids to be accelerated, even if it’s not appropriate. This is especially true as parents become wealthier, and this is especially true of white parents.

So when a school starts opening up the option for moving up a grade for math, all of the sudden some other parents start calling up the school.

Seriously, so many people in teaching have experienced this situation. Parents talk to a young kid, then (though they’ve seemed totally happy in class up until this point) the kid says, “Hey Mister, I was just wondering if you had any, umm, like extra math? or more challenging stuff? Because sometimes class is too easy for me.”

And a lot of the time it’s…well, it’s a lovely kid, but a kid who wouldn’t be at the top of your “I’M WORRIED THAT THEY’RE NOT CHALLENGED” list.

So what do we do? Anything, as long as we keep it in the classroom. Because as long as it stays in the classroom, the teacher is in charge.

Like Cuban & Tyack say in Tinkering Toward Utopiaclassrooms are places that are more safe from outsider touch. At the end of the day, as long as it’s happening in a classroom, it’s sort of invisible from an outside perspective. And this can be bad (see: all the sad stories above) but in a way it can be good — it’s one of the only ways schools have to protect kids from parental demands.

I’m not saying that anyone is doing this on purpose, but there’s a structure to how schools respond to unchallenged kids, and it likely exists for a reason. That reason could be to protect schools from parents pushing their kids ahead against their kids’ needs.


It seems to me that the status quo almost works. If more schools had interventionists who could come in and focus on the needs of the unchallenged, that would be amazing. (Those specialists along with teachers and parents could then decide if a kid would be better off in a different math situation.)

This raises an interesting philosophical question, which is whether schools should spend their money on the needs of kids who aren’t challenged by their grade-level material.

Of course, any “should money get spent” question in education is complicated, since money for something means money away from something else. And a lot of people think that any money for students who are doing well is money that is effectively being taken away from students who are waaaay over-challenged by the curriculum.

But I also think it’s fair to say that, as a matter of funding, as a matter of research, as a matter of journalism, which is to say “in general,” in education we do mostly focus on under-achieving students. I’m not saying that this is wrong — it makes sense for us to focus on kids that are losing the game we made for them — but I think it is true.

There is a group of kids, though, who are unchallenged and as a result school is not working for them. In a lot of situations, these kids don’t have much to do. And if they’re in your classroom, and they present classroom management issues when they’re bored? Good luck with that, teacher.

(The exception is probably high-SES districts where parental demand forces schools to come up with a policies, plus they’re more likely to have resources for handling unchallenged kids who are working above grade level in math or other subjects. The rich get richer, etc.)

So here’s my conclusion:

  • If your school has a pile of money sitting around, it might be good to spend it on something like a coach or interventionist who can focus on enrichment.
  • If your school is strapped for cash, you might still consider whether it would improve the overall situation by hiring someone who can focus on enrichment. It might improve the classroom situation enough that everyone benefits.
  • If you’re a researcher, this might be something interesting to study.

THE END. Your ideas/reactions/questions/challenges/readings/links in the comments, please.

“Taking a Knee in Math Education”: Danny Martin’s NCTM talk, partially transcribed

Annual Meeting and Exposition – National Council of Teachers of Mathematics

At the most recent NCTM Annual convention, Prof. Danny Martin gave a talk titled “Taking A Knee in Math Education.” (See Annie and Wendy’s posts for a summary of the talk.) It’s pretty dense at times, and he hasn’t yet published (as far as I can tell) on the second half of the talk, which focuses on what he calls a “black liberatory fantasy” of math education. I wanted a transcript of the talk to refer back to and take a closer look at, and I figured that maybe others would find that useful too. 

I started transcribing from around 35 minutes into the talk.


Equity for black learners in math education is a delusion — a compromise consistent with other historical compromises; undergirded by antiblackness; rooted in the fictions and fantasies of white imaginaries and white benevolence; held hostage by white sensibilities and sensitivities; and characterized, at best, by incremental changes that do little to threaten the maintenance of white supremacy and racial hierarchies inside or outside of mathematics education.

Given this position what do I propose as a different framing and vision of math education for black children — one committed to black self-determination, black liberation and black joy?

In the last part of this presentation I draw from recent work with two colleagues where we engage in what we are calling liberatory fantasy. Specifically, black liberatory fantasy, in order to imagine what we’re calling a black liberatory mathematics education.

We define black liberatory mathematics education as the framing and practice of math education that allows black learners to flourish in their humanity and brilliance, unfettered by whiteness, white supremacy and anti-blackness. We view liberation as a means to a radical end rather than an end in itself. We imagine a world in which our relationality is not to whiteness, anti-blackness, systemic violence, a world in which we are not defined by survival, resistance and a fight for freedom. We imagine a world in which we define ourselves, our joys and our desires in infinite multiplicities and in which we are committed to individual and collective black fulfillment.

In planting the seeds — the initial seeds — of BLM education, we draw inspiration from black liberatory struggles of the past such as the Black Panther 10 Point program as well as recent programs like Black Lives Matter where radical demands have been made within and against racial capitalism and state violence against black people.

We note that some contemporary efforts have been made to address the needs of black students in mathematics education including work that has framed math in terms of civil rights and citizenship and work that focused on culturally specific pedagogy. However in our view these approaches share a focus on liberal notions of reform and inclusion into the system of existing math education.

We are not suggesting here that there be a singular black liberatory mathematics education and we recognize that framings under this umbrella could be appropriated in many ways — even in ways that support the existing system of math education

Our goal here is to offer one perspective in the spirit of liberatory fantasy moving beyond efforts that focus on incremental change that have historically framed math education for black learners in ways that are deferential to white logics, white imaginaries, white sensitivities and white benevolence.

In terms of framing, a black liberatory math education prioritizes liberation over integration and freedom. This form of math education is skeptical of liberal notions of inclusion and equity, of appeals to democracy and citizenship, neoliberal multiculturalism and refuses all forms of systemic violence against black learners.

Moreover we recognize that freedom is never free. The freedom to be included in and to participate in anti-black spaces characterized by systemic violence is not freedom. We are cognizant that in calling for and framing a black liberatory math education we risk valorizing mathematics in a way that maintains its status and power. However our position is aligned with S.E. Anderson who in 1970 expressed that black people should learn mathematics not because of American capitalism’s advanced forms of technology require this background but because black liberation struggle against the capitalistic system requires this knowledge.

A second critical component of black liberatory education is the ability to exercise the right of refusal of the dominant education system of math education institutions and organizations that maintain their status as white institutional spaces and schooling practices and policies that instantiate anti-black violence and white supremacist orientations. Reformists often use this as a cover for the ongoing brutality of education against black learners. In this sense, reforms in math education can be viewed as sustaining the dehumanization process because these reforms are beholden to the overall anti-black system in which math education is embedded. The goal of reform is slight modification of teaching, curriculum and assessment processes, not radical dismantling. Recognizing that many children are seemingly trapped and imprisoned by the dominant system of math education we suggest resistance in the form of refusal in and refusal of.

Principle refusal in the existing system of math education will ultimately take black people to the precipice of refusal of the system. Examples of refusal in the existing system include — and its coincident that we see this nowadays — sit-ins, walkouts, and boycotts locally and nationally should be employed by black children and parents as a way to disrupt anti-black violence and dehumanization in math education. Inclusion into anti-black spaces should not be the goal of these walkouts and boycotts. Black parents and caregivers should also protest under-assessments and nullifications of their children’s abilities and refuse the tracking of black children to lower level and remedial tracks. Black parents  should refuse their children’s participation in remediation and research programs that are premised on deficit orientations and are designed to diagnose, repair and remediate — fix — black children.

These calls for refusal should not be construed with a call for segregation or racial isolation. Just as we are calling for humanizing treatment of black people we expect black people will recognize and value the humanity of others.

With respect to everyday practice: we suggest that a black liberatory math education is designed and directed first and foremost by liberation-seeking black people including parents, caregivers, community members, black teachers, and black students.

Within this new system we believe that every black child should learn mathematics for the purpose of understanding the history of black people of the United States. A k-12 curriculum — or several — devoted to the numbers of black life and history would at a minimum: help black children to understand black peoples’ incorporation into US society, quantify the ways in which the US racial state and economy were built on the labor of black people, and understand the disparate impact of systems and structures like education and prisons on the lives of black people.

We do not propose the implementation of curricula in top-down fashion nor suggest that it’s the only way to proceed. However we do believe that knowledge of self is key in meaningful math education, and the spirit of self-determination in a curriculum in support of a black liberatory math education should be co-developed first and foremost by liberation-seeking black people including parents, caregivers, community members and black students. We propose annual community reviews of all math curricula and assessments used with black children.

We propose that black children be taught by knowledgeable, liberation-seeking black teachers and independent community-controlled schools that stress collectivity and black humanity. All teachers, black and non-black, should be vetted by black parents, community members and children. Teachers should be required to live in or near the neighborhoods where they teach, and required to take training in restorative justice practices.

Drawing again on Anderson, we propose that free tutoring and math classes in community settings that are open to adults and children outside of school context, black college students and knowledgeable community members would teach these classes. Black college students with strong knowledge of math for example would serve weekend and summer internships in black communities, paid for with work study and summer research funds.

We propose the development of easy to understand and up-to-date resources such as a black parent’s guide to math education that allows black parents and caregivers to understand how school mathematics functions from many different perspectives including: curricular, assessment, teaching, and how practices like tracking and teacher recommendation for gifted programs are used against black children. Relatedly, we propose distributing comprehensive easy to read pamphlets which explain the pitfalls of financial shortcuts found within the world of the black consumer.

We propose that by the time they finish elementary, middle and high schools, all black children engage in at least one capstone project where they apply mathematics to propose solutions to challenges faced by black children — by black people. We propose that black parents, community members and children be consulted on community development projects so they can suggest ways to embed culturally relevant mathematics installations and activities in community spaces such as parks, playgrounds, barber shops, beauty salons, bus stops, community centers, community gathering spaces, health services, waiting rooms, neighborhood museums and other such contexts.

In stating these minimal components of a black liberatory math education we also assert that these are necessary but not sufficient. Revolutionary change and the building of a new and different system requires a commitment to such components.

Some people might say that my perspective is too pessimistic. [laughs] They will say that it ignores the good will of allies in the ongoing struggle for black liberation. They will point to the gains made in civil rights over the past few decades. In response, I offer this analysis of civil rights by Carol Anderson, professor of African American Studies at Emory:

She says in her book “Eyes Off the Prize”:

How could all of the blood, all of the courage, and all of the martyrs of the Civil Rights Movement still leave in its wake a nation whose schools are more segregated than ever, where more than half of all black children live in poverty, and where the life expectancy of African Americans has actually declined? And how could a movement with so much promise still leave more than six million African Americans trapped and dying in the “underclass”? The answer lies, I believe, not so much in the well-documented struggle for civil rights, but in the little known, but infinitely more important, struggle for human rights.”

For too long civil rights has been heralded as the prize for black equality. Yet those rights no matter how bitterly fought for could only speak to the overt political and legal discrimination that African Americans faced. Human rights, on the other hand, especially as articulated by the United Nations…had the language and philosophical power to address not only the legal inequity that African Americans endured, but also the education, health care, housing and employment need that haunted the Black community.”

It turns out, the civil rights, the prize, was historical compromise, a historical compromise from a demand for human rights. Equity is a similar compromise in math education.

In conclusion, I say that mainstream math education has traditionally invited black people to participate on its terms. Expecting this system to reform itself from its foundational purposes and fundamental character to a new state of validating and valuing the humanity of of black people is unrealistic in the face of evidence otherwise. Traditional discourses of equity and inclusion have been self-serving within liberal white imaginaries, white supremacy and anti-blackness. They have been inadequate for black liberation. The kinds of changes advocated for within mainstream math education discourses are welcomed and accommodated within the self-correcting systems of white supremacy and anti-blackness because they represent no real threat to these systems. Much in the same way that one can not fix capitalism, only replace it with a different system, because capitalism will seek to exploit, under any conditions or threats, liberation-seeking black people will recognize that mainstream math education cannot be fixed in service to black liberation.

Citing W.E.B. Du Bois, historian Ibrahim X. Kendi expressed the following: Instead of using our energy to break down the walls of white institutions, why not use our energy to refurbish our own? I will add that we must continue to find ways to take a knee on behalf of black children and in service to black liberations. Thank you.


Q: Thank you Dr Martin for your talk and for all of your wisdom all the time. Could you maybe…in your framework could you maybe say like a little bit about where you think people should start like this room is full of people who might be interested in doing this with different intersections like are there things that you think people could do to start doing this work?

A: I always have this response to what I’ve been called solution-on-demand — I don’t think you’re doing that right now.

Well, but I would say, Step Number One, Step Zero even, would be to hear me, first of all, to just hear me open ears, open heart, let it soak in, it may not be today, maybe tomorrow, maybe a week from now, maybe a month from then. But just hear me first of all. What am I saying? Why am I saying it? I think that’s Step One: what sense are you making of it?

Follow up. If you have questions about something I’ve said, ask for a clarification so we can begin a conversation.

In terms of the pragmatics of what do I do when I go back tomorrow: Obviously I can’t tell any particular person what they should do because I don’t know the context the children and I don’t want to essentialize whiteness, white people, black people, blackness…

But maybe Step One after Step Zero is sort of the internal work. The self-reflective work. One simple question is when you go back to a classroom with black children you have to ask yourself, why am I here? Why am i here? We have pre-service students at UIC Chicago who say that I’m here because I love all children, and it’s not true. It’s absolutely not true. And as hard as that may be to hear and to think about and to process, there are some of us in this room, despite all of our best intentions, who don’t love all children. And particularly we don’t love all black children for whatever reasons. And those reasons can be found internally. I think one has to come to grips with that. Why am I here? Because certainly the children are asking. Why are you here? What purpose are you going to serve in my life?

So I think it’s hearing me, reflecting on the self, and working with those parents, valuing and respecting and seeing their humanity through all the things that I mentioned in the talk. If you can’t get over your ability to see black children as children, and allow them to be children, to make mistakes, to not have to rip up their paper in their presence, to not have to be upset — and I’m not romanticizing — to be upset because someone didn’t put away their cell phone that it escalates to the police and arrest. There’s a lot of work in between that thought and that action. So a lot of it comes back to this.

And we don’t have the utopia. We don’t have the system that I’m talking about. So it’s the work in the current thing. So you have to find ways in your own context to refuse inside the system. There are risks involved, clearly. There is risk involved. People have to decide what level of risk they want to do.

So I could keep going on and on and move out to the bigger and bigger and bigger, but a lot of folks want those take home answers and I can’t give take home answers, but I can say, think about these issues and how they might apply. Twist them and turn them in whatever way you think is oriented in the direction of black liberation, liberation of black children and people.

Black children should not have to go to school and experience the kinds of violence that we know that they experience. There is no road to justice. Anybody that talks about a road to justice is not really interested in justice. Justice is right here, right now, in this moment. I don’t have to travel a road to get to equity. It’s here! It’s here!

So I could go on, but you get the idea.

Q: Dr. Martin again, thank you for your presentation. Do you have any examples of programs either in or out of school that you could point to that might exemplify what you’ve been talking about?

A: There are some traditions, independent black schools obviously, and you (to the questioner) know a lot about those.

And there are some efforts that are sort of on the cusp because again if we had the answer we would be doing it.

The Algebra Project is an interesting — really interesting — context in the sense of its underpinnings, its commitment to black life, civil rights, black humanity. There are things that could be done differently, clearly.

Some of the work that’s been done with the production of scholars in HBCUs in mathematics for example. Not the fact that we’re trying to fit into the system but the work that those people have undertaken. Dr. Shabazz. We learned about some folks here at Howard. Those efforts that people create those counter-spaces and it might not be well-publicized in the news, EdWeek, etc. but there are probably many examples that we’re not aware of where people are working locally and doing these things, working with groups of black children. I’ve done Saturday programs, “math for moms,” working with black mothers and other mothers to help them help their children in ways that they think are effective.

I think there is scholarship that needs to take place. There are folks like myself, like my colleague here and other colleagues — we need to be doing work where we are. We’re in the academy. We need to be countering the knowledge production that says black children come to school with no pre-mathematical knowledge for example. The teacher in the classroom, the parents — they can’t counter that up close. We can.

So our writing, our voice, is another thing that can be doing some work. But I think it can be multiple venues, big and small, Algebra Project is the essence, the idea of that, liberatory Bob Moses own historical beginnings, Saturday community-based programs, taking over community spaces and trying to figure out how to make those culturally relevant as I said, being involved in the redevelopment of your community, gentrification not to be involved in that, but to say stop, we refuse this, we want other options and other kinds of things, and then it gets very personal too.

So this notion of refusal, I’ve said this many many times. I have a five year old on the verge of going to Kindergarten. Very nervous, very very nervous. Five year old male child. And it’s not giving up on public education because the public education that we have now is the public education that many children don’t deserve. We need new and different public education. We may have to at some point homeschool him. Because my level of trust in the system — not you as individuals — but in the system is getting lower and lower every day. I don’t want to come home one day and find that my precocious brilliant son is sitting in the principal’s office in handcuffs because of some silliness, some anti-black silliness.

So I think that individual black families to the degree that they can (if they don’t have the resources other people should be supporting them) have to make tough decisions to go against many of the things that we’ve been led to believe we should be following to be, you know, to be “American.” Those lures are very powerful. Democracy. Citizenship. You can problematize every one of those. There’s black citizenship, and there’s American citizenship, and those are not the same and those have never been the same.

Democracy. Democracy for whom? Look at the history of democracy for us. It’s not something we should be swallowing in a non-critical way. We have to make individual decisions, we have to look at efforts like the big things, the community things, it’s multi-faceted.

And if we had the answer we’d be doing it yesterday, but obviously we have one system and that might be part of the problem. I’m not suggesting chargers or anything of that nature, but if the response is — what are you going to do with all those black kids if you take them out of the system. Think about that response for a second. It says, black people have no option, no alternatives, but to go back to the system that is dehumanizing. That makes no sense to me! That makes absolutely no sense! When you have cornered a people where they have nothing else except to go back to the fire, that people might be in trouble. They might be in trouble.

Q: My name is Lindsey Black and I’m an elementary teacher in the DC area. I really appreciate your insights in the topic at hand. I specifically appreciated your identification of a liberation seeking black teacher. I was wondering if you could highlight what the vetting process would be like to find a liberation-seeking black teacher.

A: So one thing we know is that we lose a lot of potential black teachers. They come to the universities, struggle for various reasons, all sorts of institutional structural reasons, and they’re committed to black people, black children, black joy, but the things we’ve set up to vet them at that level, take them in a different direction.

We know that there are people there who are interested in black people’s humanity — humanity. We need to find ways to keep those folks engaged. Not rescue them, not save them, but keep them engaged. Somehow, some way. And it’s difficult work.

The vetting process beyond that? As I said, we need to find ways and avenues for parents to be at that front, first, second, third, level to say no way — or, yes way, we feel comfortable with this teacher teaching our children. I’m turning my child over to you 8, 9 hours a day, expecting them to be safe, expecting them to learn and grow, etc. I need to have a say in that.

Not necessarily just the selected few, but come out to a community meeting, fill up a room like this. Why do you want to teach in this school? I know it’s not in our existing system, but this is the liberatory fantasy beast. Be vetted by the black children! Why can’t black children be allowed to have conversations with the people who might be teaching them at some point — of all ages. Who are you? Where do you come from? Why do you want to teach me? What do you know about me? What do you know about my people? What do you know about my neighborhood? What do you think about me? Do you think I’m brilliant? Tell me! Do you think I’m brilliant?

“Umm well…” No, no no, no, it’s yes or no. It’s yes, or no.

We can fantasize in many different ways and there are some practicalities to it as well. I think parents should be right there. One, first line of defense. It’s not the parent, it’s the caregiver. Somebody’s gotta do the vetting. Alright. I know we have these systems and structures where other people do the vetting for us, but I have to raise the question: is this good enough? I don’t think it is. Well-intentioned, you do the hard work everyday, it’s not about individuals, it’s about structures and systems. And the system that we have, in my view, is not the one that is committed to black liberation, and to black people and to black communities. And that may be trouble for other children and groups as well, and people have to fight those fights.

Thank you.

My Talk on Recent Growth Mindset Research

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Last night I gave a talk about growth mindset research at the Global Math Department. They just posted the recording, here, and I previously shared links to some of the research we discussed last night here.

My main point from the talk was that the world of teaching for a growth mindset is talking about something quite different than what mindset researchers talk about. Dweck’s Mindset and Boaler’s Mathematical Mindsets are not talking about same thing.

For instance, YouCubed and Jo Boaler’s Mathematical Mindsets lays out a comprehensive vision of what “growth mindset” classrooms look like. This includes things like ending timed work, using more visual representations, ending tracking, plus many more things besides.

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This has very little direct impact on mindset, though. This becomes especially clear after a careful look at what psych researchers themselves are doing with mindset research.

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The original work comes from Dweck in a series of papers in the 70s and ‘80s. She noticed that, when presented with a challenge, some students would behave helplessly (e.g. giving up, “I’m bored,” getting upset, etc.) while others seem motivated by the challenge. What could explain these different patterns of behavior?

Dweck retreated to goals to explain behavior. Different people react to challenges differently because they have different goals. Some people have learning goals, while others have performance goals — basically a desire to look good.

OK, but then why do people have different goals?

That’s when Dweck lands on implicit theories of intelligence, i.e. mindset. People who have fixed mindsets believe that intelligence is essentially something you’re stuck with, hence they don’t have learning goals and are only left with looking good. (Presumably everyone likes looking good, it’s just that if you believe that intelligence is malleable you believe in learning.)

(As an aside, what does it mean to believe that intelligence is malleable other than believing in learning? In other words, is Dweck’s theory tautological?)

(Another question: why stop at mindset? Why doesn’t Dweck explain where mindsets come from? I guess, from the interventions they’ve designed, we can get a sense for one place that mindsets can come from.)

This leads mindset researchers to essentially make two predictions:

  • Mindset matters. Having a growth mindset leads to good things (e.g. academic achievement, good relationships, professional success, etc.)
  • We can do something about it — we can turn some of those bad outcomes into good outcomes with our interventions.

Those interventions can be very brief, as short as 45 minutes, and they can be done using completely standardized online materials. Pretty much every intervention has the same basic structure: science tells us that the brain can grow, that challenge is good, and hard work can lead to good things happening. Then there are some reflective prompts for discussion or writing.

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(Some really good intervention materials from mindset researchers devoted to making this stuff free can be found on PERTS.)

One last question to round off the researcher version of growth mindset: how could such a brief intervention have any significant impact at all? It seems like magic. Not so, say Yeager and Walton in “Social-Psychological Interventions in Education: They’re Not Magic.” The key is recursive effects, positive feedback loops where the intervention marginally increases mindset, which then marginally promotes good learning, which then further effects mindset, which then &c., &c.

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So the claim is that growth mindset interventions can have major effects. But just how major? That’s the question of the first of the recent papers we talked about, the 2018 meta-analysis of existing research.

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How important is growth mindset for academic achievements? When the researchers pooled a comprehensive selection of results together, they found a very modest average effect. (Check out the paper and you’ll see some discussion of a fairly high gap between some studies that find fairly strong effects and others that don’t, more heterogeneity than you might expect. They aren’t able to easily find an explanation.)

And what about mindset interventions? The main point is that they find extraordinarily weak average effects for mindset interventions, as shown in this forest plot below. (The diamond represents the mean effect. Note just how many interventions are touching or less than 0 in their effect sizes.)

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And these results seem roughly in line with what Dweck and her recent collaborators at PERTS are finding in their top-notch mindset experiments. They’ve been trying to address criticisms of mindset research while doing work that would prepare mindset research to scale and reach many students.

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This move of the focus of mindset experiments to scale has quietly involved a number of other shifts as well.

  • Away from testing in just one context
  • Less researcher involvement in the mindset treatment
  • Standardized materials
  • Big studies

Correspondingly, there have been two changes in the sorts of results that the PERTS team have been publishing with Dweck.

  • Moving away from claims that mindset interventions have an impact on everybody, now a focus on at-risk students.
  • Much more modest effects being claimed.

You can see this in the figures from the 2015 pilot study. There was really not very much of an effect at all from the growth-mindset intervention for the general population. This is in line with the meta-analytic results.

There is an effect — again, a modest one — from the growth mindset intervention for at risk students on their GPA.

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But these results came from what was supposed to be a trial run before a much larger experiment. And those results were briefly released earlier this year in draft version. The official version hasn’t been published, but the study was widely praised for its rigor and again found modest results, mostly for at-risk students.

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Here’s where I want to start putting the pieces together. What I want to ask is what these results should entail for advocates of teaching for a growth mindset. My answer is: not much, because the relationship of all this research to what goes on in teaching was never clear.

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Here’s what I think happened. Dweck’s research begins with descriptions of helpless behavior from students, and she points to motivational problems. All of this resonates deeply with teachers, so we associate motivational concerns with growth mindset. When Boaler talks about teaching for a growth mindset, usually she means motivational teaching, e.g. if you teach with visual representations more students will experience success and they’ll be more motivated to continue engaging.

But mindset researchers were only ever trying to impact the very bottom of this causal chain, the mindset stuff. And maybe, it turns out, that wasn’t as important as they thought it was, and all the stuff in the middle (i.e. teaching) is really important.

When we dialogue with research, there are always a couple different games we can play. One game we can play is “how can this be wrong?” There’s a time for that. Another is “how can this be right?” and there’s a place for that too, but my favorite is “how can both of these things be right?”

I think it’s entirely possible that some mindset interventions had huge successes, but I also think that when we try to get clear enough about why these worked, the generalizations don’t hold particularly well. And that’s because we’re working with motivation, fickle motivation, and it depends on so many different things — what the kids happen to need, who’s running the intervention, how the kids happen to hear the message, which of their other adults were involved, etc. It seems to be very hard to standardize motivation.

But teachers who advocate for growth mindset can be right — mostly right — also, because they were never talking about just changing a student’s theory of intelligence. They were talking about the entire causal chain of motivation, including social safety, helping kids see success, making the curriculum more accessible, &c., a million things that have to do with motivation but that are waaaay beyond the purview of mindset research.

I don’t mean to be too wishy washy. Mindset is not a revolution — that’s not a hypothesis that finds a ton of support in 2018. And teaching for a growth mindset doesn’t have all that much to do with mindset, and I’d prefer that we talk about motivation rather than mindset — I think that’s what we really mean to say.

I’ll end with two questions:

What other research would we be curious to see in 2019 or any other year, relating to mindset?

What are the narrower, more specific, less potentially revoluationary but more potentially connected to the classroom, questions that we can ask about motivation? Because revolutions like Dweck’s are consistently overrated at the outset, and in education we tend to place teaching on these boats that end up being much more rickety than they at first seem.

Some recent research on growth mindset

Tomorrow night [June 5, 9 PM ET] I’m giving a webinar on growth mindset at the Global Math Department. It’s title is What does the latest research say about growth mindset?Technology permitting, it’ll be recorded for later viewing/listening. 

Here’s the session description:

Growth mindset is an active area of research, and new results and tests of the theory are constantly coming in. Come to this webinar to learn some highlights of this research, including some new controversies and subtleties. Here are some of the questions we’ll discuss: Do adults grow new neurons? How much of an impact do growth mindset interventions have? And, most importantly, what is the relationship between growth mindset interventions and teaching math for a growth mindset? You’ll leave this webinar with new questions about how mindset impacts teaching.

(Not sure if we’ll get to the adult neurogenesis stuff. If you’re interested, check this piece out.)

It occurred to me that some people might appreciate a list of readings in advance of the talk. (Reading anything in advance is totally optional. One of my jobs at this webinar is to help you get a sense for what these papers say.)

Here are some of the readings that I’m looking at while preparing for this session. Enjoy and discuss!

This is from 1988, but it’s a blockbuster paper by Dweck that’s a great place to start for thinking about what mindset means and represents in Dweck’s original research:

A social-cognitive approach to motivation and personality.

This one from 2018 and is the first major, new paper that we’ll talk about in the webinar. The link is to a gated version, and (ignoring illegal avenues like sci-hub) I haven’t been able to find a pre-print or draft to share. You can see some figures and discussion of the paper on twitter.

To What Extent and Under Which Circumstances Are Growth Mind-Sets Important to Academic Achievement? Two Meta-Analyses

How exactly is mindset supposed to work for Dweck and the mindset researchers she works with? This paper by two of her research collaborators does a great job making explicit the assumptions that are necessary for mindset theory to have significant impacts on students from short interventions.

Social-Psychological Interventions in Education: They’re Not Magic (draft version here)

And what have Dweck and her collaborators been up to lately? They’ve been studying online versions of their mindset interventions that allow for more robust testing [e.g. they don’t depend on the researchers, they test the impact of the intervention in many different locations, they can more easily run true experiments instead of going dumpster diving in data, they can have a larger number of participants, and they can pre-register their studies since they aren’t relying on extensive collaboration between researchers and teachers]. It also tests the viability of moving growth mindset up to scale. Some of these studies aren’t yet published (e.g. this), but some of them are. (See this this and this.)

Here’s a paper published in 2015 that gives a taste of this research program and relates to the 2018 preprint that made headlines, but is currently unavailable as it undergoes revision.

Mind-Set Interventions Are a Scalable Treatment for Academic Underachievement

Anything else you think we should add? What thoughts or questions do you have about all this mindset stuff?

Questions and Answers about Tracking and Ability Grouping

[I am not an expert. Maybe you are, in which case I would greatly appreciate a comment pointing out things I’m missing. Much thanks to my anonymous partner in crime, the unlinkable TracingWoodgrains. We’ve been reading this literature together, and while we don’t agree entirely about tracking this piece is a result of our work together. Any good parts of this wouldn’t have been possible without his collaboration.]

What does “tracking” typically mean in American schools?

European-style differentiation into vocational/academic tracks is rare in the US, though it used to be very common earlier in the 20th century.

Now, most elementary classrooms have tables or little within-class groups for reading and math. As kids get older, it’s more and more common for schools to create high/middle/low classes for various subjects, but especially for math. By 8th Grade, most kids are assigned a class based on past performance, and sometimes those classes are “accelerated,” meaning they take Algebra 1 in 8th Grade. By high school, high/middle/low tracking is near universal in math.

(Some of this picture I get from Loveless. Loveless also notes that there’s a lot of terminological confusion between tracking and ability grouping. I’ll use the terms interchangeably here.)

In addition, though it’s not called “tracking,” a lot of school resources are dedicated towards students who aren’t performing highly. This amounts to a kind of ability grouping and is super-common throughout k-12. It’s federally mandated ability grouping, in fact.

Who benefits from conventional tracking? Who loses?

If anyone benefits, it’s almost certainly students in the higher tracks who gain and students in the lower tracks who lose. But the effects aren’t clear, and the impact of tracking isn’t particularly well-understood.

In 1987 Robert Slavin reviewed the existing literature for elementary and secondary students and found practically no benefits for anyone from conventional tracking — but also no real harm done to any group. On this basis he argued against conventional ability grouping, seeing as it helped no one and was morally noxious.

But the studies he reviewed had limits (small size, not nationally representative). Using better data, a number of researchers (Hoffer, Gamoran and Mare, Argys, Rees and Brewer) came to the conclusion that conventional tracking benefits students in the high tracks and hurts those in the low tracks. But it’s really hard to control for the right factors in these definitely non-experimental studies, and Betts and Shkolnik raise questions about the results of these papers (summarized, as is this whole story, in this review by Betts).

(Just to mess with everybody, Figlio and Page argue that by attracting stronger students to the school (because parents seek tracking) students in low-tracks benefit, indirectly, from tracking.)

Recently, there was an experiment in Kenya — one of the very few true experiments! — where they randomly instituted tracking into some schools and measured the impact. It was positive for everybody, but there are a million differences between this context and the one in the US, starting with the number of ability groups (just 2) and class size — over 45 kids are in each classroom! It’s hard to know what to take from this study for the situation in the US.

And there was also a recent big meta-meta-analysis that found no benefits and no harm for between-class grouping, echoing Slavin.

Loveless says the evidence is inconclusive, and that’s echoed by Betts, and the fact that it’s not a clear effect tells you something about how tangly this whole issue is. But if it helps anybody, it’s probably top-track students, and low-track students would be the ones hurt by tracking.

Wait, you said “conventional tracking.” What’s unconventional tracking?

Slavin, Destroyer of Tracking, has a school-turnaround program with good results called Success for All that depends heavily on grouping students by ability. How? By adopting something like the Joplin Plan, which assesses frequently and regroups students based on those assessments. Students across ability groups show benefits in these programs. (Though, not without controversy.)

Another form of grouping that isn’t widely used is acceleration, e.g. placing 1st Graders in 2nd Grade math if they’re ready for it, and continuing down the line. There is research supporting the notion that acceleration benefits the accelerated student in completely straightforward ways — they learn things that they wouldn’t otherwise have access to. See that meta-meta-analysis for instance.

Does race impact where you’re tracked?

Using one of those large, nationally reprsentative samples mentioned above, Lucas and Gamoran (fierce opponents of tracking) found that race wasn’t a factor in track placement. Meaning, controlling for academic performance, race isn’t a further factor in deciding where a student gets placed.

Because of the gap in academic success that Black and whites collectively experience, this still means that Black students disproportionately occupy the lower tracks.

In contrast to Lucas and Gamoran, Dauber et al, found that race was a factor in track placement in Baltimore schools. It’s hard to know for sure how to fit Dauber with Gamoran’s bigger picture results.

What about other non-academic factors? Do they impact track placement?

Gamoran found that, unlike race, socio-economic status does statistically impact track placement (modestly) suggesting that, somehow, high-SES students tend to get tracked above their academic performance. Why? We don’t know for sure. Maybe it’s parents? Maybe it’s the intangibles, like being a good student with homework done and things organized because they have parents at home who can help manage the academic lives since they aren’t coming home at 10 PM from their second job?

But we can’t really know yet how precisely SES helps determine tracking.

What do we know about the quality of these low-track classes, as compared to higher-tracks?

Even defenders of tracking agree that low-track classes are often very poorly taught and that this is a major problem. Here’s Loveless: “Even under the best of conditions, low tracks are difficult classrooms. The low tracks that focus on academics often try to remediate through dull, repetitious seatwork.” Much of Oakes’ contribution is documenting the lousiness of a lot of low track classes.

How does this square with researchers who find no negative impact of tracking on low-track students? All this would mean is that instead of failing these students in low-track classes that schools typically fail these students at similar rates in untracked schools.

Speaking personally, it seems to me that the strongest argument against tracking is the state of low-track classes. Forgetting academic performance, these students need to be placed in safe, respectful, happy classrooms staffed by competent teachers who believe in and care for their students. I think we have plenty of reports showing that this is often not the case in low-track classes, and this is what I saw at the first school I taught at.

So is the tracking status quo bad for racial inequality?

Put it like this: if we immediately removed all US tracking and replaced them with heterogeneous classes, the result would possibly be narrowing of the black-white score gap somewhat — a bit from improving the performance of low-track students, but mostly by limiting the advancement of high-track students.

Those high-tracks don’t just contain white students (schools are also highly segregated remember), and another national priority is increasing the representation of Black and Latino students in the highest ranks of achievement. Some of the tools we have for increasing representation are universal screening for tracked gifted programs, and removing tracking would also remove these programs. Without public access to gifted programs, would wealthier, whiter students just pursue these out of school, exacerbating inequities at the highest levels of achievement?

Still, the net effect would probably be a narrowing of the black-white gap.

OK, so let’s get rid of tracking entirely.

Only if you’re willing to really restrict the amount of learning that some students are capable of — either through deliberate acceleration or by maintaining track differences — for the sake of equity. After all, the flipside of the evidence that tracking exacerbates inequalities is that it really does help some students, usually those in the top tracks. (And, if you doubt that evidence, there are still unconventional tracking methods that we could be using to further accelerate more students, deliberately, from younger years.)

The tough question here is what happens to the learning of students who are ready for more than their heterogeneous class is offering them.

But can’t you teach in a way so that everybody maximizes what they could potentially learn?

This is the golden snitch of teaching, right? You win the game if you can grab it, but it always manages to slip away.

Maybe there are schools that have pulled this off (Boaler, Burris Heubert & Levin), but they seem to be relatively rare. In general, schools that tended towards untracking amidst the heights of the anti-tracking movement inched back towards tracking (Loveless).

Another note: a lot of untracked elementary schools just use ability grouping within classes. Maybe there’s increased mobility between those groups, but teachers need to find ways to deal with heterogeneity. Pedagogies that benefit everybody with no costs are highly vaunted within education, but I’m skeptical, and there isn’t evidence that these schools provide widely replicable models.

But if you don’t remove tracking, is there any way to improve the status quo?

One approach might be to pursue some of the unconventional tracking options, though that would involve pushing against what Larry Cuban calls the “grammar of schooling.”

But there are also many examples of tracked schools that offer a good education to their lowest-tracks. In fact, Rochelle Gutierrez studied eight high school math departmenets, some tracked and some untracked. She came to the conclusion that “tracking is not the pivotal policy on which student advancement in mathematics depends.” What is crucial for her are a whole host of other factors, including strong pedagogy, school culture, and solid, shared curricular resources.

More examples of effective tracking programs that promote mobility come from Catholic schools. (Wait, Catholic schools are closing left and right as they lose students to those charter schools that politicians made such a big deal about? Whoops.) See Camarena and Valli.

Likewise, Adam Gamoran identified examples of schools with successful low-track classes, and identified features of these programs. It’s what you’d expect — high expectation, good pedagogy, making sure good teachers work with the low tracks too.

Which is more promising — expanding hetereogenous instruction or improving low-track classes?

Let a thousand flowers bloom, etc., but I think if you put me in charge of a district or a school I’d focus on improving low-track experiences. It seems to me as if there are more cases of working low-track classes than examples of successful heterogeneous programs. And, as a matter of experience, I am not sure I believe in cooperative learning as a pedagogy that mitigates the risks to high-achieving kids.

Tracking or untracking: what do you say?

Well, it really depends on the school. I think if you put me in charge of a school I’d want to follow Gutierrez in focusing on things like curriculum, high expectations for every kid, safe classrooms with comptetent teachers for every kid. Tracking or not wouldn’t necessarily be my most important decision.

I don’t think I could stomach a school that tracked strongly along racial lines. That’s not good for school culture or the experience of students in the lower-track, and so I’d probably want to untrack that school as much as possible. That said, I’d still want to see programs for students at either extreme of the achievement spectrum. (And I’d be federally mandated to provide a lot of such resources at the lower end of that spectrum.)

Otherwise, I’d be fine with tracking probably, as I’m fine with the tracking that my current school uses. And I’d be really interested in seeing if I could employ some of the unconventional tracking plans like the Joplin plan or reasonable acceleration, like letting 4th Graders take math with 5th Graders.

Does that include grade-skipping? That’s rarely used in schools, but it’s a form of acceleration.

Grade-skipping seems to generally benefit those who skip (Park Lubinski & Benbow). I’d want to be able to use it, mostly when kids aren’t happy and we think it’s because they’re unchallenged by their current grade.

Why don’t more schools use unconventional tracking?

There are two ways of putting this, I think. The first is to state, as Cuban and Tyack do, that there is a grammar of schooling that resists reform. The typical age-graded classroom is a strong feature of schools, and both of these unconventional tracking methods push against age-grading.

But why should age-grading be such a persistent element of schooling? I find David Labaree helpful in explaining this, because what educational consumers seek are either markers of distinction for their kids, or equitable access to those markers.

Learning is only of secondary value to most parents — they don’t seek learning without distinction — and so something like acceleration is very hard for schools to offer more widely without leading to chaos as parents demand ever-increasing acceleration for their kids.

Age-grading as a strong default is a compromise that helps schools manage the demands of the educational consumer.

But the Joplin Plan, and other plans that assess kids frequently to better determine and match their curricular needs, seem like they deserve more attention than they get.

So does anybody like the status quo?

Maybe not, but that’s by design. Schools aren’t designed optimally for learning or for equity. School as it exists is a sort of uneasy compromise between contradictory principles — fair access and award of distinction — and the competing demands of different groups. Some claim to have revolutionary solutions, but these probably don’t exist. You can reduce inequity, but only if you’re willing to curtail the learning of some. You can improve learning for all, but risk exacerbating inequity. This is an optimization problem with more than one possible solution. Or, as Rochelle Gutierrez says in a different context, the answer to the questions of tracking are usually “neither and both” sort of answers.

That said, in math education circles, tracking is unfairly maligned, in particular by NCTM. In Catalyzing Change they say that the research is “unequivocal” that tracking harms low-track students in permanent and irrevocable ways. Looking more broadly at the literature, it’s hard for me to agree with that take. 

Starting to think about tracking


I think my school does a nice job handling tracking for math classes.

When kids are in K-2, they stay in their classrooms and get taught math by their classroom teachers. But starting in 3rd Grade, kids split up out of their homerooms for math and get taught by a math specialist.

From 3rd through 8th, we teach six sections of math class for each grade. Three of these classes are “regular” pace class; the other three are “accelerated.” At the end of each year each grade’s math teachers and administrators get together with a huge stack of notecards and make classes for the upcoming year. We think about a lot of things — which kids would do well together, whether a class has a nice mix of personalities, and whether a kid would do better in a regular or accelerated section. And, by high school, students get to pick their pace, and kids absolutely do move between the “tracks.”

For the last few years I’ve been teaching an accelerated 4th Grade class, and it’s definitely not an easy class for me to teach (though consensus here is that accelerated tends to be easier teaching). The spread of interest and abilities is still high. (As you’d expect it to be at a school that has ~50% of students in accelerated sections.) There are two things that I find challenging. First, I don’t have nearly as many curricular resources for the accelerated 4th Grade as I do for the regular pace classes. Maybe you think it’s a social problem, maybe you don’t, but I have way more curricular tools for a struggling class than one that’s ready for more.

And the other thing is that I feel a real responsibility in this class for the kids who come in seeing math as their “thing.” There’s a special responsibility to make sure that these kids are challenged and engaged in my class since there’s nowhere else for them to go.

It’s sad but true: there’s more than one way to fall out of love with math.

Overall, I think it’s very good that my school has half the sections accelerated and half not. This gives us flexibility to make classes that we want, and it avoids some of the ways that tracking can make more problems than it solves.


If we’re looking for a good example of bad tracking, look no further than the first school that I taught at.

Here’s what happened each of the three years I spent there. The school would put students in 9th Grade sections based on what they’d heard from the middle schools. The top two sections were “honors,” and they’d study Geometry. The bottom two sections were taking Algebra 1. Once that placement was made, the rest of their high school enrollments in math were more-or-less locked in.

I used to teach the bottom two 9th Grade algebra classes, 9C and 9D (as they were so lovingly called). At the start of the year the classes would be about the same size, maybe 18 and 18 kids. Slowly, though, the 9D kids would figure out where they’d been placed. They’d tell their parents, the parents would call the school, immediately the kids would be moved to the 9C class, which would typically blossom into a lovely group of 26-28 students, leaving the remaining 8-10 stragglers behind.

I really did love the kids in 9D, but WOW that was a hard class to teach. Thanks to this artificial selection process, all the kids in that class were there because either (a) they didn’t care what section they were in or (b) their parents didn’t but mostly (c) they had diagnosed learning needs that weren’t being met, because it was an under-resourced private school.

I sometimes fantasize about going back to that school and teaching that class again. It wouldn’t be fun, but it really nags at me. Could I do better now, if I tried again? I do know so much more about teaching now, but it’s not a class that sets up a teacher for success. If I’m honest with myself…I don’t know if I’d be any better.

The class was very hard to teach. I don’t want to say unteachable, because there were good kids in that room that needed a good teacher. There were also numerous behavior problems, really all the time, just sometimes punctuated with learning. This says something about me in my first few years teaching; it might say something about me now. I’ll never really know.

But it certainly says something about the school.

I taught other low-track classes at this school — 10D, 11E. The Regents exam was far out of their reach, for the most part. We’d have a couple of passes out of each group, but it wasn’t a realistic goal for most students. The environment in the classroom was often out of control, and the school overall had this reputation for barely contained chaos.

I think I did alright there, but this is just the reality. There are hundreds, thousands of schools like it. And while tracking was clearly not the major problem at this school, there was no question who the losers and winners of this arrangement were.


How common are my experiences? How do they fit into the bigger picture?

I’ve been looking into the research on ability-grouping (within a class) and tracking (making classes by ability), trying to make sense of the state of things.

The story of this field is pretty interesting. It’s a field with a million meta-analyses — even a meta-analysis of the existing meta-analyses! All these reviews exist because there hasn’t been much first-order research since the early 1970s or so. So everyone is bootstrapping their analyses on top of the same old studies. If this is making you think that the evidence base isn’t particularly strong here, you’re getting the picture.

While there isn’t an incredibly strong research base here, there is evidence and even a sort of consensus. Tom Loveless does a nice job reporting on this for the Fordham Institute in a report titled Making Sense of the Tracking and Ability Grouping Debate. Loveless, as others do, frames the research around a debate between two researchers, Robert Slavin and James Kulik.

For those seeking a summary, here’s a condensed version of their debate:

Robert Slavin: booo tracking, you have no evidence

James Kulik: yay tracking, actually we do have evidence

Robert Slavin: no, that’s just evidence from gifted and accelerated programs that are poorly controlled, they no count

James Kulik: no they count

Robert Slavin: yo also I find tracking morally repulsive

James Kulik: really? i like it

That’s sort of it. If you like words, here’s Loveless’ summary:

Kulik finds that tailoring course content to ability level yields a consistently positive effect on the achievement of high ability students. Academic enrichment programs produce significant gains. Accelerated programs, where students are taught the curriculum of later grades, produce the largest gains of all. Accelerated gifted students dramatically outperform similar students in non-accelerated classes. Slavin omits studies of these programs from his analysis. He argues that the gains, though large, may be an artifact of the programs’ selection procedures, that schools admit the best students into these programs and reject the rest, thereby biasing the results.

Loveless is correct to point out that this debate is intractable, though, because Slavin actually finds tracking morally problematic and ugly. The burden of proof for Slavin is on schools that want to track, which explains why he can be so opposed to Kulik, even though they don’t seem to disagree very much at all about what the research shows:

Three things are striking about the Slavin-Kulik debate. First, the disagreement hinges on whether tracking is neutral or beneficial. Neither researcher claims to have evidence that tracking harms achievement, neither of students generally nor of students in any single track. Second, accepting Slavin or Kulik’s position on between-class grouping depends on whether one accepts as legitimate the studies of academically enriched and accelerated programs. Including these studies leads Kulik to the conclusion that tracking promotes achievement. Omitting them leads Slavin to the conclusion that tracking is a non-factor. Third, in terms of policy, Slavin and Kulik are more sharply opposed on the tracking issue than their other points of agreement would imply. Slavin states that he is philosophically opposed to tracking, regarding it as inegalitarian and anti-democratic. Unless schools can demonstrate that tracking helps someone, Slavin reasons, they should quit using it. Kulik’s position is that since tracking benefits high achieving students and harms no one, its abolition would be a mistake.

Loveless seems to be taking a compromise position in all this. “The research on tracking and ability grouping is frequently summarized in one word: inconclusive,” he writes. Since the research is inconclusive, he recommends a live-and-let-live strategy. Schools should have the freedom to choose their tracking structure, he says, but they need to be aware of the ways that each model can fail.

Tracking’s issues are well-known these days. Loveless calls for high-standards for the lowest tracks, and for ending what’s sometimes called the “teacher tracking” of putting the least skilled teachers with the lowest tracks. There need to be clear pathways out of the lowest tracks, a real effort to make sure that there’s room for students that start in one place to end up in another.

Untracked schools have problems of their own, though. “On the political side, anti-tracking advocates need to assuage the fears of parents that detracked schools will sacrifice rigorous academic training and intellectual development for a dubious
social agenda,” he writes, and this seems sensible to me also. The really ambitious students in my accelerated 4th Grade class do have needs, and their parents are legitimately concerned about meeting them.

If this seems wishy-washy and balanced, well, sometimes things just shake out that way.


If you think about it, isn’t it sort of weird that tracking doesn’t have clear and measurable benefits for the top groups in the research? Think about it. How often should ability grouping help strong students? Like, roughly, what percentage of the time should the top-group academically benefit from tracking?

I’d say 100% of the time. Roughly.

Teaching kids more stuff because they’re capable of learning more stuff is the single simplest idea in education I’ve ever heard. There is nothing to it. It’s just teaching more. Add to that the way they’re isolated from some of the toughest-to-teach students in the school, and this seems like it ought to be the clearest slam dunk on the educational menu.

So why should there be anything other than the clearest possible data signalling this? I’m not asking, why hasn’t the educational establishment recognized the evidence? I’m asking why the data isn’t super-clear. Why isn’t there a huge effect? Why isn’t it unambiguous in every single case that top-students benefit from ability grouping?

Two explanations:

  • Ability grouping is not necessarily acceleration. Some teachers don’t use a “top” class as a chance to do anything differently at all. You meet the standards? Great, you’ve met the standards. Let’s chill. (Or, let’s look at cool things on the side that don’t accumulate as knowledge.)
  • The skills students learn in a top group maybe aren’t measured.

Thinking about my own teaching, I think both of these things are probably happening in a lot of “top” track classrooms. I certainly do try to cover topics in my accelerated class that I don’t in my regular paced classes. But sometimes it’s just that we cover the same material but without as much stress, because there isn’t a clear vision I have of what an accelerated class ought to look like. I get little help from the available curricular resources, which are really all about fleshing out support for struggling students over kids who are ready for more.

I’m not complaining about this, mind you, but I think it’s true. There are probably a lot of teachers out there who aren’t making significant curricular changes between their tracks.

The second thing is true also. I try very hard to avoid racing ahead in the “standard” sequence of arithmetic skills with my accelerated class. The easiest way for me to handle an accelerated class would be to just march through the curriculum, teaching 5th, 6th, 7th grade standards to my 4th Graders. But this could create problems for the kids and my colleagues. If I unilaterally decide to teach e.g. fraction division, then I’m stepping on the toes of the 5th or 6th Grade teacher, who now has a handful of bored kids who are skilled at this because I decided to keep marching. 

OK, so the department should make a decision. But once you just set a class off accelerating through the curriculum, you’ve suddenly created a track that is relatively impenetrable to kids who start out of it. Somehow, they’ll have to catch up to join, and that’s going to have to happen outside of class. The only way to get ready for an accelerated class would be to be accelerated already, an unsavory Catch-22.

I think what I try to do is to flesh out the standard, grade-level topics with things that don’t have a strong showing anywhere in the standards. Right now my 4th Grade class is taking a deep dive into probability, a topic that only sort of shows up again in the 7th Grade Common Core standards. Earlier in the year I shared a bit of graph theory. We studied angles at a depth that will only really show up again when they’re studying diagrams in high school geometry.

My dream would be to have a curriculum that had a clear vision for what kids who are ready for more could dive into, beyond the grade-level expectations. There would be to keys to making this work. First, the additional skills would have to actually build and develop throughout the year — we want to equip accelerated students with something useful that builds their mathematical knowledge. But we also want a fresh start each year or so, so that kids can move in between the tracks without requiring some sort of catch-up.

I think something like this would give clarity and purpose to classes that otherwise have no choice but to plow ahead in the standard sequence.


On the margins, should US schools have more or less tracking? I think the answer is probably “better tracking.”

Even Slavin, opponent of tracking, admits that there is evidence for certain kinds of tracking in the elementary years, especially something called the “Joplin plan.” (Named after Joplin, Missouri, the district that gets credit for its invention.)

Joplin-style tracking cuts across grade-levels. A school might have an hour for reading instruction, and each student in the school would go to a classroom they’ve been assessed as ready for. So a 4th, 5th and 6th Grader might be reading similar books together in the same room, working on the same vocabulary. It’s a kind of limited breakdown of the age-grading system, really an artifact of the early 1960s.

Slavin, opponent of tracking, calls the experimental results of studies of the Joplin plan “remarkably consistent” and in support of the program.

Which makes sense, right? This is the simplest possible educational idea: teach kids more when they seem to be ready for more. And, as an extra bonus, since the kids are heterogeneously grouped for most of the day, you don’t run the risk of creating really problematic tracks that lead to wildly varying places. By the nature of the plan, there is curricular guidance for kids who are ready for more. This should work 100% of the time.

And Slavin put his money where his mouth is, co-founding Success for All, a school improvement program that has something very much like the Joplin plan as its cornerstone. (His co-founder is Nancy Madden, another Johns Hopkins education professor. Madden and Slavin are married.)

And, ironically, Slavin’s program has been critiqued for its use of tracking. (Also for its use of scripted lessons, which will never make teachers happy.)

It seems as if the situation is that Slavin’s preferred sort of tracking would be good for students and good for equity and mobility. He’s a noted critic of tracking and ability grouping, and is deeply aware of all the traps. Success for All reassesses students every two months, and students are expected to move between groups. This is the form of tracking with the strongest research pedigree. And yet it comes in for criticism.

What’s confusing about this to me is that we aren’t a country that is shy about grouping students by ability. Loveless notes this: “Ability grouping for reading instruction appears nearly universal, especially in the early grades.” In the elementary years, this is usually within-class grouping, e.g. red group sits at this table and blue is on the rug, etc. But by the time students reach high school, the near universal pattern is separately tracked classes, more like what my old school did.

Here’s the puzzle: why do some forms of grouping and tracking attract more ire than others? Is it just a matter of the devil we know vs the one we don’t? Familiarity breeds begrudging acceptance? I don’t know.

But looking at programs like Success for All and thinking about what happens in the math classrooms that I’ve seen, it seems to me that purely from the standpoint of mathematical learning, there is probably a better way of doing things. Here, as bullet points, are my takeaways from all this, with the most doable items near the top:

  • We could use curricular materials that go beyond the standards for each grade level of math, so that classes who are ready for more can dig in without dashing through the standards.
  • Something like Slavin’s reading plan could be useful in elementary grades. Keep the heterogeneous groups, in general, but assess students every couple of months or so to place them in a class that’s right for them. (If we could clone Success for All minus the scripted lessons it would probably be more popular. Though I’m sure there’s probably something lost when we do that too.)
  • Maybe, even in the upper grades, it would be helpful to split the year into two halves, with an ability grouping move in the middle. Or maybe it wouldn’t help at all. But it might be interesting for a school to try something like that and see how it goes. Maybe?

But all these are speculative recommendations. Overall, I don’t get the sense that there is a huge gap between research and practice because (as Loveless notes) there isn’t a great deal of clarity from the research.

Instead, there are promising ideas with research support (like Slavin’s). This doesn’t exclude the possibility that there are other good ideas out there, and it seems likely to me that if a school or parent body thinks that tracking or untracking is necessary for their students, they’re probably correct.

Teaching right triangle trigonometry, presented as a series of problems

How can a decimal be a ratio?

  1. Sketch a tower that has a 2:3 height to width (h:w) ratio.
  2. Sketch one with a h:w ratio of 5/4.
  3. Sketch one with a h:w ratio of 0.4, 0.45, and 1.458.
  4. My tower has a h:w ratio of 1.00001. What can you tell me about my tower?

Special (“Famous”) Right Triangles (Chapter 10, Geometry Labs)

  1. Two of my right triangle’s sides are 5 cm and 5 cm. Use the Pythagorean Theorem to find the third side.
  2. Scale the “famous” half-square to find that third side more quickly.
  3. Two of my right triangle’s sides are 1 cm and 2 cm. Why can’t I be sure how long the third side is?

Which is steeper? Part I

Screenshot 2018-04-22 at 8.50.44 PMScreenshot 2018-04-22 at 8.50.49 PM

Screenshot 2018-04-22 at 8.57.53 PMScreenshot 2018-04-22 at 8.58.11 PM

Screenshot 2018-04-22 at 9.01.40 PM

Meet Tangent

  1. Play with the tangent button on your calculator a bunch. If you find anything cool or interesting, write it down.
  2. Tangent takes an angle and gives you the slope of its ramp. Make a smart guess: what’s tan(40)? tan(80)? tan(1)?
  3. Check out this table. What do you notice? What questions do you have?


4. Suppose we graphed this thing, slope/tangent as a function of degrees. What would that graph look like?

5. What are some possible sides for a 35 degree ramp?

Which is Steeper? Part II

New steepness compare_ angle vs. slopeNew steepness compare_ angle vs. slope (1)New steepness compare_ angle vs. slope (2)

Draw two ramps whose tangents are very close, but not quite, equal. Extra points for cleverness.

Moving Between Degrees and Tangent

  1. What’s the first slope that is unskiiable?


2. Try this Desmos activity about slanty hills.

Solving Ratio Equations

  1. Solve: x/5 = 1.3
  2. Solve: 5/x = 1.3
  3. Which of these is more difficult for you to solve?

Scaling, Ratios, and Solving Triangles (Geometry Labs, Chapter 11)

  1. Figure out as many things as you can about the triangle in this diagram.

Unit Triangles - 1.pptx

2. This one too.Unit Triangles - 1.pptx (1)

3. All of these as well.Unit Triangles - 1.pptx (2)

4. Check out these 20 degree ramps.

Screenshot 2018-04-22 at 9.11.48 PM.png

5. How tall is the top ramp? How wide is the bottom?

6. Which way should you spin this to make the problem as nice as possible? What angle of incidence do you prefer? What’s your height, your width?


7. (Ala Geometry Labs, Chapter 11): Given one hypotenuse and one leg of a right triangle, what other parts can you find?

Sine is a Trig Functions Also, I Guess

  1. Sine!

And then Cosine

  1. Cosine!


Reposting: Thoughts about Future of NCTM Conferences I had at NCTM Nashville (in 2015)

[I originally wrote and circulated this as a Google document, which had the benefit of making it easy to update. Over the past few years, lots of people offered interesting comments on that doc, so definitely check those out. Now that the document has been stable for a few years, I thought I’d repost it on this site to make it slightly easier to find.]

  • Fundamentally, I want NCTM conferences to be places where long-lasting professional relationships are formed. I do not want it to be a place whose primary purpose is for people go to sessions.
  • Overall, the quality of NCTM sessions is mixed. Once at Nashville a group of us found an empty room to sit around and chat in because we didn’t see any sessions that we wanted to go to.
  • I’m not sure that I even want more higher quality sessions to attend, though. The mix-and-match nature of session attendance doesn’t really excite me as an opportunity to learn about teaching.
  • I loved the MTBoS booth. There were moments of community around that booth. People go there so that they can talk to people they’ve never talked to before, we played with toys and I met some new people.
  • The MTBoS booth was like a small island of community in a den of icky educational consumerism. I really dislike the sales-pitching of the exhibition floor.
  • On Thursday afternoon I left a session and felt exhausted. I had a weird hankering for some math (I had been working on a problem on the plane) and I realized how little math-doing there was at these conferences. Isn’t that a shame?
  • I went from there to the MTBoS booth and played with Christopher Danielson’s math toys. I saw a crowd gathered around the booth, I saw people waiting for a turn to play with his tesselating turtles or his pattern-making machine.
  • Once NCTM reorients itself towards fostering community, I think it’s going to start seeming very important to figure out how to create spaces for doing math together with other people.
  • I love books. Usually when I walk into a bookstore, I have a hard time leaving without buying something. I walked out of NCTM without buying any books.
  • I went to a bookstore in Chicago a few months ago. I pulled off four books from the shelf, settled in a corner and flipped through them. Others were doing the same. Some people were talking to each other about their selections. It was a space for loving books.
  • The NCTM bookstore is another missed opportunity to make a communal space, I think.
  • I noticed that people congregate around the outlets outside of sessions. People end up sitting there. Any space like this is a chance to help people form connections.
  • I think NCTM is going to start including more formal social events, and this is good. I think NCTM is going to start providing more online spaces for presenters, and this is good too. But the real goal needs to be making sure there are nooks and crannies throughout the conference where people can come together around some shared experience.
  • I’m sure there are things like “fire code” that I’m not considering, but is there any good reason not to have a few rooms where you let we inmates run things? A place to chat, a place to take a group of people and sketch some things out. I’m talking about making sure there are open rooms with tables where people can continue a conversation.
  • As a speaker, now: there are always people who want to talk at the end of a session. It’s sometimes tricky to know where to go. I wish I could just say, “Here’s where I’m going to be if you want to continue the conversation.” In that way I could sort of pitch a more extended experience.
  • In short: yes, formal social events; yes, improved web experiences; but also, NCTM sub guides in advance of conferences; a hangout area with “hosts” to help make connections; a “Do Some Math” area with volunteer facilitators; spaces to go after a session; spaces to go instead of a session; spaces to read and fall in love with books together; fewer speakers; more sessions that are carefully vetted for quality; more places to play with toys; “Post your favorite math problem on an index card and glue it to the wall!”; invite Zome to take over a conference room; more spontaneity, more community and more math.

Added on 2/28/16

  • On twitter [] there was an interesting conversation about whether teachers ought to be given the keynote presentation slots.
  • Keynote speakers play a role in attracting people to NCTM conferences, and so it makes sense to choose keynote speakers whose names are recognizable. I think it’s lamentable that classroom teachers aren’t recognizable names in math education. There’s a status hierarchy with teachers at the base level and consultants, academics, CEOs and journalists all hovering above us classroom folk. On one hand, this is only natural: the work that it takes to build up a personal brand, recognition and influence has very little to do with teaching children. If you’re interested in being well-known enough to influence education widely, that is a journey that will probably lead you out of your classroom.
  • This is a shame, though, because academics, consultants, CEOs, journalists are not doing the work of teaching, and so they often get it wrong. They often gravitate to issues that aren’t at the heart of the practice, or their thinking doesn’t develop in the way it might if they were forced to test their ideas over the course of years of working with children. There is no replacement for developing ideas while being a classroom teacher. Math education is worse off for not having high-status teachers who are able to speak and write with authority about math education.
  • (To be clear, there is also no replacement for visiting and seeing many different classrooms when it comes to making generalizations about teaching. And doing research well is immensely challenging but it enriches the profession. I don’t think the world should exclusively be run by k12 classroom teachers. That would be its own sort of disaster.)
  • So, what are we going to do about it? Thrusting teachers into the big lights wouldn’t fix anything, I think. True, it might raise the status of some teachers such that they could draw in people the way Jo Boaler’s name does. But could that status really be sustained while remaining in full-time classroom work? How do you develop talks and build an attractive brand without missing enormous amounts of time for conferences?
  • (The exception to this rule seems to be Jose Vilson. It seems that the laws of gravity don’t apply to Vilson, I don’t know how he does it. Truly amazing!)
  • I don’t want to advocate for some sort of requirement for keynote or featured presentations to include k12 teachers. Instead, I want NCTM to create infrastructure for gradually raising the profiles of classroom teachers. I think this could be done with the artful combination of fellowships, researcher-teacher partnerships that result in joint publications, awards, mid-level speaking profiles, and a million other things that I’m not smart enough to think of.
  • If there is a systemic critique I would make of NCTM, it’s that it’s entire leadership structure reflects a PD orientation that goes from researchers to PD providers to coaches and then to teachers, as recipients. By this I mean that board service is nearly impossible to pull off while being a classroom teacher, and that the model of the conferences seems to be of maximizing traditional PD delivery (even when it’s delivered by teachers). One thing that we’re seeing from the internet and the MTBoS is that this is just one model of how teachers like to develop professionally. Creating more opportunities at conferences for teachers to interact in ways that classroom teachers might find more natural — like teaching mathematics and talking and writing about practice — would benefit the status of teachers.
  • (To this, David Wees would add modeling and rehearsing teaching techniques. Wouldn’t that be a cool thing to do at a conference!)

An idea for teaching quadratics to 8th Graders

I haven’t seen a curriculum that develops quadratics quite in this way, but I’m having trouble giving up on this approach and going with anything else I’ve found. What do you think? Here’s how the unit would go:

FIRST TYPE OF EQUATION: y = (x + a)(x+b)

Step One: Learning to solve (x + a)(x + b) = c equations by treating them as multiplication equations. An important idea is that these equations can have 1, 2 or 0 solutions.

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Step Two: In particular, learn how to efficiently solve (x + a)(x + b) = 0 and other similar, non-quadratic equations.

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Step Three: Study these types of equations as functions. Check out what the zeroes represent in y = (x + a)(x + b).

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(I’ve already done these activities in class while experimenting during the week before spring break.)

Step Four: Make generalizations about the graphs of these equations — about where the line of symmetry is, whether is curves up or curves down.


Step Five: Check out a new type of equation x^2 + b = c or x^2 - b = c. (I mean that b is non-negative.) Learn to solve these equations by using what you already know about solving linear equations, with the new twist of taking roots of each side. And notice that sometimes these equations have 2, 1 or 0 solutions, and learn precisely what sorts of equations will have

Step Six: Graph these new equations, y = x^2 + b or $y = x^2 – b$ especially in the case when b is square. All that stuff above about lines of symmetry, zeroes, etc., study that but for these equations.

Step Seven: Big idea time. There are two equivalent ways of expressing many of these quadratic equations. No factoring, no multiplying binomials yet. Just notice: some of these y = (x + a)(x + b) equations produce the same graphs as y = x^2 + c! (Mostly when c is a square.) Let’s give arguments for why this is true, arguments about the zeroes, the lines of symmetry, and that these two equations share a vertex.


Step Eight: Learn to multiply binomials, like (x + a)(x + b), and become equipped with a new algebraic way of doing the work of recognizing equivalent quadratic functions. Here we’ll especially focus on a difference of squares, (x + a)(x - a).

Step Nine: Teach the rest of your quadratics unit at this point — including whatever other factoring you need to teach — while frequently asking the question “Will these equations produce the same graph or nah?”


This all seems to me a nice way to gradually build quadratics knowledge. If pushed on my design principles, I’d say that (a) I’m trying to be sensitive to the fact that the different types of equations that fall under ‘quadratics’ are of widely varying complexity and (b) I’m trying to make sure not to teach a connection between two mathematical objects before students have a chance to really become familiar with the different mathematical objects. (In other words, students would see lots of equations in factored/standard form before trying to connect them via multiplying or factoring one into the other.)

Is there any curriculum that structures a unit in a way that even roughly resembles this? I can’t develop too much of my own curricular stuff given my teaching load (four different courses: 3rd, 4th, 8th and Geometry) but I would love to try teaching this upcoming quadratics unit in something like this way.

Any materials or approaches you’ve seen for quadratics that resemble this? Can anyone talk me out of this approach? Where would I run into trouble, if I went against better judgement and developed my own materials for this unit?