How to cultivate your own mathematical genius

It's obvious, but let's say it anyway: American schools don't teach math with the brain in mind. 

The way we teach math doesn't match much of what we know about engagement, creativity, understanding, or memory.

In our last few posts, we've described two radically different methods of teaching math: the JUMP Math approach, and the "Japanese method".

Both are really quite different from each other — JUMP is super-guided, while the Japanese approach is quite unguided.

But both methods put each student in the driver's seat, forcing them to make sense of mathematical ideas themselves, rather than blindly following a textbook's method.

The trick: use them both.

But even when combined, these two methods are still (we think) not enough. Neither method helps students truly master problems: digesting them fully, ruminating on them until the mathematical ideas contained in each problem become encoded in a student's long-term memory.

For that, we have a third piece of our math curriculum: Deep Practice Books.

Deep Practice Books are a curricular invention that we've been pioneering over the last eight years, using ourselves as guinea pigs, and refining with the help of hundreds of students. 

Like the Japanese teaching method, Deep Practice Books involve parachuting students into math problems they don't know how to solve, and helping them develop, on their own, the tools to solve them.

But unlike the Japanese method, a Deep Practice Book is highly personalized. It's a tool for students to develop their own mathematical brilliance.

I (Brandon) created the Deep Practice Book out of my own struggle to study for the GRE, a story I've never told in print. 

So, here goes. I believe a suitably grand title is in order:

The Deep Practice Book:

A deceptively simple method anyone can follow to impressively raise a math test score and ho boy cultivate actual mathematical genius

Mostly, I avoided math in school.

I was always pretty good at math — enough that I didn’t need to particularly worry about it. But never great — and I never particularly loved it.

In fact, when I found myself bored in high school, and decided to spend a year homeschooling myself, I fell behind in math. (I did, however, learn a bit of ancient Greek, which probably has proven more useful as an adult!)

And in college, I got my one required math class out of the way as quickly as possible. I didn’t even do that well in it, earning a C+, which the instructor was merciful enough to raise to a B–.

I even avoided classes that smelled like math: physics, of course, and chemistry.

(By the way: huge mistake! Since graduating college, I’ve fallen deeply, desperately in love with science — but because I never took the time to systematically understand the periodic table, it’s difficult for me to pass beyond the scientific comprehension of someone living in the 18th century.)

So when I decided to apply to graduate schools, and needed to tackle the GRE, I knew had a challenge in store for me.

The GRE is the test to get into academic graduate school — where you can get a master’s or Ph.D. The GRE is made by the same people who make the SAT, but they make the GRE on the days when they’re feeling mad.

The math problems on the GRE deal with simple math — there’s almost nothing on it beyond basic geometry — but the questions can be devilishly complex. Take, for example, this basic-looking problem:



And here was me, who had been running away from mathematical thinking for more than five years.

My one advantage was that I was already a test-prep coach for the SAT and ACT. I loved helping other people through their math pains — so maybe I could find some fun in working through my own.

I had started off working at a tutoring center, and had gotten good enough to start working privately. I had seen some initial success — my first student had improved his SAT score 290 points and gotten into Harvard. But I had also seen some darker episodes. I had lately worked with two young women for more than half a year when something troubling happened.

We worked our way through the entire SAT book, doing more than 400 math questions.

They studied diligently!
I tutored competently!

And then, with the real test less than two months away, I bought them new copies of the same book. They re-took the first test…

and got nearly all the same questions wrong.

We were aghast. We were forlorn.

I want to call attention, at this point in the story, to how weird this is. We seemed to be doing everything right — they were studying hard enough, and I was teaching clearly enough. And yet there was almost no change, even on precisely the same problems.

It was right around then that I decided to study for my GRE.

I took my first test, and got a 670 out of 800 in the math. Now, for the SAT, that’d be a fantastic score — somewhere around the 87th percentile. But on the GRE, it was the 48th percentile.

That means, if you randomly grabbed a hundred GRE-takers and put them in a line, with low-scorers on the left and high-scorers on the right, I’d be the 48th guy. Basically, average.

Ooch. I was a test-prep coach — this was my professional image on the line. I decided to use the blow to my pride as a motivator to study hard. I wrote up a study schedule for myself: I decided to take a half-test every Monday morning for the three months before the real deal.

And it didn’t want to repeat the tragedy that had befallen my two hard-working students.

It was at this point that I did something rather random, without understanding why I was doing it: I re-copied all the math questions I had gotten wrong on that diagnostic test into a binder. And on the cover, I wrote (in big, cocky letters) “HOW WE BEAT THE MATH.”



And I obsessed over the problems. Since they were in a special binder, it seemed natural to do so — this was my binder of Impossible Problems, my binder of pain. 

Gradually, it became my binder of math love.

I didn’t just learn to solve them, I learned to explain them to myself. I made sure I didn’t write down my work or the answers in the binder (because then I wouldn’t really have been able to re-solve the problems), but  whenever I had a question, I made sure to write it down:

Wait, how do you add fractions, again?
Why does the area for a trapezoid use the average of the top & bottom?
How the heck does that ugly permutation formula work?

By filling the binder with questions, and by obsessing about the answers, I learned the math so deeply I think I could have explained it to a fourth-grader.

And then, as Monday approached, I prepared to take a new half-test.

The night before the new test, I did my second oddball, I-didn’t-really-grasp-the-profundity-of-what-I-was-doing thing: I re-solved all the problems in the binder.

And was horrified when I got half of them wrong.

Remember: I had been obsessing over these problems the whole week. I had these problems down: I thought I understood them perfectly clearly.

And I got half of them wrong.

This was my first hint that human brains didn’t evolve to do GRE math. Nor did they evolve to do SAT math, or ACT math.

If I wanted to do really, really well on this test, I realized I needed to study in a fundamentally different way than twelve-plus years of schooling had prepared me to study. I needed to identify every mathematical idea I found confusing, and put it into a foolproof system that would allow me to understand it — and engrave it into my long-term memory.

Over the course of the next three months, I added problems to my binder almost religiously. And I did whatever it took to understand them — read answer explanations, pose questions, ask friends.

But all of this wouldn’t have amounted to much had I not re-solved all of them from scratch at least once each week — each and every problem I had previously entered in.

As I re-solved those problems on fresh paper, something delightful happened: I began to get them right, every time. And quickly, too! Initially I struggled with the problems, weaving back and forth inside my brain to figure out what the next step might be. But now the next steps came easily.

Before, I could only see a single step at a time — now, after re-solving the problem three or four times, I could see the whole thing at once. I could chop the problem up into tiny moves, and deal with each of those moves quickly.

And that wasn’t even the best part! About once or twice a week I would be re-solving a problem for maybe the fifth or sixth time when I’d realize that I had been an idiot. I had been solving a problem by doing a long series of steps — but if I just reconceived the problem, looked at it from a different perspective, the entire thing would be easy, could be solved in one or two moves.

Math, I realized, was simple. It was elegant. These insights were glorious — when I had them it felt like the sky was opening, and a beam of light was shining down directly on me. I could almost imagine I could hear angels singing.

And I recognized that this was why mathematicians did it — modern mathematicians, and the great mathematicians of history who had originally discovered the methods I was now uncovering myself. They were chasing the sublime high of mathematical insight.

How often, I asked myself, did I experience this in all of high school?

Maybe once or twice.

But now, studying for a standardized test — engaging in perhaps the least glamorous math learning task Western civilization has devised! — I was experiencing these epiphanies once or twice a week.

I had stumbled upon, I realized, a way of dependably building math expertise. And I was seeing it pay off: almost each week, my GRE practice test score rose. In fact, it rose quite predictably — going up about as many points as problems I had mastered in the previous week.

When I entered in 10 problems, my score went up 10–20 points.
When I entered in 20 problems, my score went up 20–40 points.

The week before I took my real test, I counted the problems I had copied into my binder — 104. And on my practice tests that week (full ones) I scored an 800 and a 790.

When I took the real GRE, I scored an 800 — a perfect score. Not bad for someone who avoided math in school.

But much better than the score was my newfound sense of myself as a mathematician. I realized that no mathematical concept was beyond me — I could understand anything, given enough time and effort. And I could enter it into a foolproof system, and, by repeatedly re-solving it, comprehend the ideas even more fully as time went by.

And I could even like it. Because to really understand something — to make sense of it inside and out, forwards and backwards — is sweet, and worth the struggle to achieve it.

Over the years since then, I’ve helped hundreds of students build their own collections of impossible problems — “magical math binders”, as one of my students has dubbed them, or “deep practice books”, as I call them.

And in a few posts to follow, I’d like to help you build and maintain your own — if you've the hankering to fall in love with math, too.

Group math lessons AND personal math puzzles


A problem:

There's wonderful value in learning math as a group: students can help one another, and every day be reminded that everyone can understand math.

But there's wonderful value in learning math as an individual: each student can spend time struggling with whatever puzzles bamboozle him or her.

Each of these has its advantages and disadvantages — we need a synthesis.

The synthesis that many American schools choose often doesn't convince students that everyone can learn all of K-12 math, and also doesn't give every student a collection of math puzzles that bamboozle them.

Our basic plan:

We spend about an hour a day on corporate math lessons — in K-8, using the JUMP Math curriculum. (The JUMP approach excels at rapidly breaking down complex big ideas into understandable tiny ideas, and then helping students arrange these tiny ideas together.)

We also, though, have students individually play with off-curriculum math puzzles (starting with the puzzles of James Tanton), with the goal of finding puzzles that still stump them after (say) 10 minutes of focused struggle. We have them collect those problems in their personal Deep Practice Book, and help them make small-breakthrough after small-breakthrough. (We don't just tell them the answer.)

Because we understand that all memories deteriorate after time, we have students regularly re-solve (and re-approach) all the problems in their Deep Practice Books — perhaps once a week. As time goes on, and they re-solve a puzzle three, four, five, or more times, something wonderful can happen: problems that had once been unsolvable become easy, and even obvious. Puzzles that had been vexing and hateful become delightful and friendly.

And every once in a while, the student will realize that there's a much more beautiful way of unravelling the puzzle. This is a momentous discovery: it's as if the clouds roll back and a trumpet reveille sounds from the heavens! How often did moments like that happen in your K-12 math experience?

As students engrave these puzzles into their brains, they'll grow to love math more, and understand it much more deeply than is now possible in the conveyor-belt math approach that most schools use.

Our goals:

We think it's possible for all students to perfectly understand the K-12 math curriculum.

We think it's possible for all students to grow to love (at least in small part) the process of doing and understanding math.

We think math can become the place in the curriculum where students most develop their growth mindset — that they see math is something difficult that they can do. We can shatter the myth that only some people can understand math.

If you walk into our classrooms, you might see:

Walking into our classrooms, you might stumble upon one of our whole-group math lessons: expect to see the teacher posing a score of tiny questions to a focused group of students. During independent work time, you might see a student frowning intently as she tries yet another way to solve an especially diabolical puzzle. When she fails (again!), she goes through a problem-solving methodology, to better tease out any clues that could help her crack the riddle.

Some specific questions:

  • Are James Tanton's puzzles good to start in grade school?
  • What other good options are available for mathematical puzzles?
  • I mention here developing a single, standard problem-solving methodology. That would be great — if we could train the kids in just one, then performing it would become a habit, one that could extend what our kids are able to do throughout the rest of their lives. But: what problem-solving methodology should we try out? (I do have the beginnings of this, and will be working on it over the summer. Presumably, too, the kids in the classroom can slowly evolve an even better one!)
  • Lee, are the kids in your class too diverse in ages/math abilities to do any kind of group lessons with? Or are there enough little kids to jump into JUMP at the lowest levels? An alternative (that still uses JUMP) is just to have kids work through the workbooks themselves, with the teacher giving assists when needed. Corbett Charter School did something like this (though not with JUMP), and an amazing math teacher that both you and I know suggested that this "kids working by themselves" method might suffice to help kids learn the whole K-12 math curriculum (though she didn't think it was ideal).
  • Math circles haven't come into this description at all. They're magical. Where should they fit?

Can a (new kind of) school change the world?


I'm obsessed with societal collapse. Economic inequality? Cultural dissolution? Systemic poverty? Environmental degradation? Substance abuse? The depression epidemic? Racial unrest? Ideological polarization? These are the topics that keep me up at night.

Though: I'm not despondent about these. Not only is there hope — I think our society is even making important progress on some of these fronts, progress that goes largely unrecognized in the media.

But a good outcome isn't a foregone conclusion. We live in the middle of a story whose ending is still up for grabs. From my vantage point, it's reasonable to expect that we'll screw the whole thing up (and take half the biosphere with us) and, at the same time, reasonable to expect that we'll get society right (and create a world truly worthy of Homo sapiens).

And I'm obsessed about figuring out how we can move away from the bad ending, and toward the good one.

I say this because lately I've realized that almost no one knows this about me. (Not my friends; not even my wife! That was an intriguing conversation.)

And I say it because, at some level, my goals for this school — this new kind of school — are bound up with these questions.

Can a school — a new kind of school — help mend the world?

Not save the world, mind you. Save is all-or-nothing. Mend is a more realistic goal. Mend allows us to count half-steps, allows us to take pride in making improvements at any scale, allows us to work with others.

So: can it?

Three possible routes

Obviously, this question of "can a school mend the world?" is an old one. It's what launched the common school movement in the mid 1800s, what launched Dewey's Progressive movement in the early 1900s, what launched Maria Montessori's and Rudolf Steiner's schools in the mid-1900s.

I can count (at least) three routes that people have pursued as to how a type of schooling can do this. The first — ideological indoctrination — I think misguided (and entirely inappropriate for our school). The second two — developing skills and cultivating understanding — I think promising (and entirely fitting).

Route #1: Ideological take-over of society? Nah.

There's a famous essay — well, famous among historians of American education! — that advocates that schools be ideologically-charged: that they communicate the true view of the world and radicalize the students, who will then go on to launch the revolution that will change society.

(It's funny: the author I'm thinking of was a Communist, but what I just wrote could equally well describe any number of Republican or Democratic writers currently writing about education.)

The author was George Counts, a previous partner of John Dewey who, in the midst of the Great Depression penned the pamphlet "Dare the School Build a New Social Order?"

I love the chutzpah of the pamphlet. Heck, I love the chutzpah of just the title! (I bet George Counts' wife knew where he stood on mending the world!)

It's a short piece. If you haven't read it before, and have yet to fulfill your doctor's daily recommended dosage of fiery midcentury call-to-revolution rhetoric, can I suggest you take a skim through it?

Counts argues that schools should help bring about the socialist revolution:

If Progressive Education is to be genuinely progressive, it must... face squarely and courageously every social issue, come to grips with life in all its stark reality, establish an organic relation with the community, develop a realistic and comprehensive theory of welfare, fashion a compelling and challenging vision of human destiny, and become less frightened than it is today at the bogies of imposition and indoctrination.

This is the moment I probably should make something clear: George Counts was a Communist, and I'm not. (Though, oddly, I'm wearing this Communist Party t-shirt right now! In my defense, it was still dark when I picked my clothes this morning.)

George Counts, of course, failed in his attempt to make the teaching profession an extension of the Communist Party. And in retrospect, it's almost impossible to imagine he could have succeeded. Politics follows Newton's Third Law of Motion:

For any action, there is an equal and opposite reaction.

If well-meaning people on the Left try to bend schools to their will, then well-meaning people on the Right will step in to thwart them. And if well-meaning people on the Right try to do the same, then well-meaning people on the Left will step in.

George Counts' mistake was thinking that the schools could stand outside the rest of American society — that they could influence without being influenced (except by him!).

Mending the world by ideologically charging the schools: a losing game.

Route #2: Building skills? Yes.

But there are other routes to mending the world: one is by building crazy-mad skill.

I'm teaching a high school course in moral economics this year, and this week we've talked about human capital. "Human capital" is a term from economics, invented when economists started taking seriously that the resources that lead to economic well-being aren't just oil and machines and large stacks of bills: they include the grand sum of skill, natural talent, knowledge, experience, intelligence, judgement, and wisdom that reside inside people and contribute to their ability to make a living.

Human capital, to be clear, is a very expansive idea. Sci-fi author Robert Heinlein once wrote:

A human being should be able to change a diaper, plan an invasion, butcher a hog, conn a ship, design a building, write a sonnet, balance accounts, build a wall, set a bone, comfort the dying, take orders, give orders, cooperate, act alone, solve equations, analyze a new problem, pitch manure, program a computer, cook a tasty meal, fight efficiently, die gallantly. Specialization is for insects.

All of these, even, fit cheerfully within "human capital." (In fact, one of the primary criticisms of the concept is that it's too inclusive, but that's a different topic.)

Why do we care about this? Because human capital is one of answers to the question "why are some people more successful than others?"

Charles Wheelan, a professor of public policy at Dartmouth, writes in Naked Economics:

True, people are poor in America because they cannot find good jobs. But that is the symptom, not the illness. The underlying problem is a lack of skills, or human capital. The poverty rate for high school dropouts in America is 12 times the poverty rate for college graduates. Why is India one of the poorest countries in the world? Primarily because 35 percent of the population is illiterate.

Now: this isn't the whole story. Poverty is a complex beast, and it has more causes than a dearth of human capital: systematic racism, classism, sexism, and so on. But human capital explains a crucial part of what holds some people back (and allows others to leap ahead).

The wonderful thing, of course, is that schools do provide human capital: reading, writing, math, and so on. The terrible thing is that they seem to not do it particularly well.

Take reading. Diane McGuinness unpacks a research finding, in Why Our Children Can't Read (And What We Can Do about It)

about 17 percent of working adults, thirty-three million people, are both well educated and sufficiently literate to work effectively in a complex technological world. We are dooming the vast majority of Americans to be second-class citizens. 

And E.D. Hirsch writes, in The Knowledge Deficit:

Reading proficiency… is rightly called "the new civil rights frontier."

There's a defensiveness that can pop up when people criticize schools. To be clear, I'm not criticizing public schools in particular: it's been demonstrated that private schools don't do a much better job.

There's also a defensiveness that can pop up when people suggest that people in poverty lack skills — the idea can appear to people as "blaming the victim." But does anyone really want to argue that children born into intergenerational poverty wouldn't benefit from reading much better, from excelling at math and science and computer programming and everything else?

A new kind of schooling can deliver human capital. Heck, we can develop superpowers — recall that this is Big Goal Number Two of our school! And we can do so without stirring up the ire of the political Left and Right, the way ideologically-charge interventions do.

We can empower people — especially marginalized populations. We can help people read well, write well, and think well. And by doing so, we can help mend the world. 

Charles Wheelan again, citing Marvin Zonis:

Complexity will be the hallmark of our age. The demand everywhere will be for ever higher levels of human capital. The countries that get that right, the companies that understand how to mobilize and apply that human capital, and the schools that produce it… will be the big winners of our age.

I'm not concerned with our schools being "winners" of our age. I'm obsessed with cultivating children and adolescents who have the capacity to win for themselves, and for others.

And we can do this.

Route #3: Expanding understanding? Oh yes.

There's one more route, I think, that a new kind of school can take to helping mend the world: expanding comprehension about how the world really works.

On this blog, I've been concentrating on describing our vision for elementary school, because that's what we'll be opening with in 2016. Our high school program is a decade out — we'll be growing the school organically with our opening classes of kids.

But boy, am I excited to be starting a high school.

I'm a high school teacher, and I love my job precisely because I get to spend my days peeking into how the world hangs together. A stranger, looking over a list of the social science courses I teach, might be confused —

  • Moral Economics
  • Evil
  • Happiness
  • Philosophical Worldviews
  • World Religions
  • Political Ideologies
  • The Next 50 Years
  • Ancient History
  • Moral Controversies in American History

The thing that connects them is my obsession with how society works. Why can we explore space but still have poverty? Why do some people behave horrifically to others? What is the good life? How do ideas drive society? Where is technology taking us? Where do we come from? And so on.

Many students don't get the opportunity to deliberate on these compelling questions in school. Most schools aren't designed to reflect on issues like these every single day. Most schools aren't designed to help students ask probing questions, identify and overcome their biases, and develop hard-won wisdom.

Ours can be! (In fact, this is our school's Big Idea Number Three.)

The thing to keep in mind is that mending the world is possible. We know that, because we've seen it.

Steven Pinker's recent book on how some things (especially rates of violence) really have been getting better — The Better Angels of Our Nature — helped convince me of this. From that he wrote a short essay, "A Two-Minute Case for Optimism," that appeared on (and I love this) Chipolte bags. The essay concludes:

“Better” does not mean “perfect.” Too many people still live in misery and die prematurely, and new challenges, such as climate change, confront us. But measuring the progress we’ve made in the past emboldens us to strive for more in the future. Problems that look hopeless may not be; human ingenuity can chip away at them. We will never have a perfect world, but it’s not romantic or naïve to work toward a better one.

We can have a better world. To some degree, every school everywhere — every teacher who teaches — is already creating this world.

Our school can be part of that effort.

A School for Difficult, Exhilarating Math

The problem:

Math is more than following someone else's recipe. Math is about prolonged puzzled, creative daring, and brilliant insights.

In my last two posts, I argued that we need to make math as simple as possible. If we're going to be risky, let it be in making mastery too easy.

That sounds snarky, but I mean it seriously. It's our duty to students to all but guarantee that they'll succeed in coming to a full understanding of math.

But that's not enough. It's not enough that all our students excel at math. Our job is also to lead them to love math.

How can we do that? How can we lead kids to mathematical infatuation?


Well, there are a number of ways we'll be pursuing this, but one major route:

Bring in creative puzzles. Puzzles that are challenging. Puzzles that can't be unraveled right away — that need to be put away and considered hours or days later. Puzzles that offer multiple solution methods. Puzzles that require creative daring — trying something that, on the face of it, might seem strange or stupid. Puzzles that, week by week and month by month, grow creativity.

Some of these puzzles will incorporate old ideas — the concepts that the kids have learned in previous years, only shown in an unfamiliar form. Others of these puzzles will preview new ideas — the concepts that the kids will be learning in the following months and years.

What matters is that the puzzles not just be technically difficult, but conceptually cleverthat they be about ideas.

As a class, the goal isn't merely to get the right answer, though that's (of course) very important. The goal is also to explore diverse routes for finding that answer.

Good, complex math puzzles typically can be solved using multiple methods. For example, consider a classic math puzzle: What's the sum of all the whole numbers from 1 to 100?

There's a straightforward way to solve this: just add away! 1 + 2 + 3 + 4… + 99 + 100 = 5,050.

There's a clever way to solve this: Look for similar pairs. 1 + 100 = 101 2 + 99 = 101 3 + 98 = 101 … 50 + 51 = 101.

There are 50 pairs. Each pair adds to 101. 50 * 101 = 5,050.

There's a weird way to solve this: Add sets of 10s, and spot the pattern. Sum of 1–10: 55. Sum of 11–20: 155 Sum of 21–30: 255 Sum of 31–40: 355 … Sum of 91–10: 955 And then add all those together: 5,050.

On their own, students will come up with all these methods — and more besides! Our teachers need only give them the encouragement to do so. (According to the excellent book The Teaching Gap, this is actually pretty close to how math is taught in Japan, and to a lesser extent Germany.)

One of these super-challenging problems can be given each week. At the week's end, our students can present their methods. And teachers can lead the class in exploring how, ultimately, each method is exactly the same thing.

This is deep mathematical understanding.

Diverse routes lead to fuller understanding.


But the results could be even better than that. Devising (and valuing) diverse routes to solving puzzles changes the nature of math. No longer is math something out there to be obeyed — it's something in you to be explored.

Diverse routes make math personal.

As a class, we might award a prize each week to the method that is the most clever, and to the method that is easiest to perform in your head, and to the method that is the weirdest!

It's not that there's a single right method, and many wrong methods. It's that there are many methods, each beautiful or ugly or useful or pointless in its own way. 

Math is an expression of humanity.  It's a human thing, not a robot thing.

To best appreciate these puzzles, we might collect them (and our favorite methods) in binders, and encourage students to re-visit them from time to time.


The focus of this post is to talk about love, not mastery — but mastery is exactly what will result as students slowly internalize these puzzles and their methods. As students write these ideas in their long-term memory, they will become more and more brilliant at mathematical problem-solving.

The SAT and ACT are made up (nearly exclusively!) of these sorts of puzzles. Bizarrely, a curriculum of creative math, of loving math, will end up being the best standardized-test-prep curriculum imaginable.

Not that we're putting much weight on that.


In brief:

Alongside a micro-scaffolded curriculum of tiny mathematical discoveries (based on JUMP Math), our school will also have a curriculum of unguided math puzzles. We might have 1 super-challenging problem per week. Students can work on the puzzles by themselves, or in teams. At the week's end, students will present their methods, and the teacher will help the class explore why each method works.

Our hope is that this won't just help raise kids who are adept at math — but kids who truly enjoy it.

A School for Complete Mathematical Understanding (2 of 2)

(Also entirely true!)

Complete Math Understanding and Social Justice

In my last post, I identified a huge problem with traditional schools: they don't reliably bring all students up to a complete understanding of math.

This was a problem in the middle of the 20th century. This is a disaster at the beginning of the 21st. 

If I can interject a bit of social justice: the inequalities in contemporary American society as numerous as they are complicated — but there is a strong correlation between economic success and mathematical understanding. This holds through many inequalities.

There's a gap between the outcomes of males and females, but when you filter out differential math abilities, the gap gets smaller. There's a gap between the outcomes of white students and students of color, but when you filter out differential math abilities, the gap gets smaller.

Obviously — obviously! — these disparities are not reducible to math performance. There is sexism, and it matters. There is racism, and it matters.

But there's good evidence to say that if we provide a way for all students to excel at math, we will make a significant stride toward reducing inequality in American society. This is something worth fighting for.

All right. So: how can we accomplish this?

Step #1:  JUMP Math

We'll start by using the gold standard for curricula that achieve full comprehension: JUMP Math.

JUMP is published by a non-profit organization from Canada, the brainchild of John Mighton: an actor-turned-playwright-turned-math-tutor-turned-Math-Ph.D.-turned revolutionary-curriculum-designer. (Y'know, one of those people).

(Not that it particularly matters, but you might have seen Mighton before — he played the inspirational teacher in Good Will Hunting.)

The heart of JUMP Math is the insight that each math concept — even the very most complex ones — can be broken down into smaller and smaller chunks, until they're small enough for students to understand in mere seconds. Students come to understand (not merely perform) each chunk quickly, and then jump onto the next micro-concept.

Emphasis on quickly. In JUMP, students move from insight to insight, with only a small bit of struggle in between. There's little of the floundering that makes many (many) students feel that they're just spinning their wheels, that they'll never understand math.

People don't like floundering. People don't like struggle without hope. People love to struggle and achieve.

Video game makers understand this. In the last few decades, they've mastered the psychology of struggle and reward, and have made video games into feedback systems so well-suited for human brains that they are nearly addictive.

JUMP stokes the ego. JUMP (metaphorically) turns math into video games.

Learning anything — feeling the change from not-knowing to knowing well — feels fulfilling. Learning quickly feels especially fulfilling.

a (Crucial) Side Note

Crucial side note: small struggles are not enough. To be psychologically healthy, humans also need big struggles — we need to take on enormous projects that we're not confident we'll be able to solve.

In our school, our math curriculum will also have another component — baffling puzzles that students will need hours and weeks to unravel; puzzles that will allow for creativity and individualized solutions.

Our school's math curriculum will be both/and: students will fully learn the core K-12 math curriculum through a micro-scaffolded JUMP Math curriculum, and they will cultivate their creative brilliance through non-scaffolded puzzles.

I'll be blogging later on the second half of this.

End of side note.

What about Struggling Students?

Some students, of course, have more difficulty learning math. (Again: people are not blank slates.) That doesn't mean their mathematical understanding has a ceiling. 

JUMP Math works wonderfully for them, too. Teachers simply break the micro-concepts down into still-smaller chunks — however small the student needs to quickly and fully understand the concept.

Every student can learn one more concept. Every student can learn another concept after that. There are no ceilings in math.

This psychological insight is perhaps the most revolutionary piece of JUMP. My students who use JUMP report having new faith in their abilities to learn. JUMP teaches that anything is possible in learning.

Learning to teach the JUMP Math way is an art: one of the most joy-inducing skills I've honed as a teacher.

How is This Different?

Traditional math books have two phases: they introduce the concept (the first couple pages of each chapter, replete with 2-3 sample problems), and then they ask students to apply the concept (the next few pages, featuring about 20-30 problems).

JUMP Math doesn't do that — it teaches new concepts through the very problems it presents.

Every question enlightens. Students learn constantly. No problem is wasted.

I recall, when I was in high school, staring blankly at my math book, reading the sample problems a third, fourth, and fifth time, wondering what I wasn't getting.

(I also remember stabbing my book in frustration. Lost a good pen that way!)

JUMP, again, makes learning math easy. It makes achieving a fundamental skill of the 21st century simple, something everyone can do.

This seems, to me, a fundamental human right.

But Wait, There's More!

Any curriculum that did all the above would be excellent, but JUMP Math goes an extra step.

Instead of asking students to merely perform math, JUMP leads them into the messy guts of understanding.

JUMP helps all students clearly understand somewhat-obtuse concepts that I recall merely memorizing.

Students understand why order matters in subtraction and division. Students understand why order doesn't matter in addition and multiplication. Students understand why you can't divide by zero. (Hint: it has nothing to do with blowing up the universe.) And so on.

Again, I'm pretty "good" at math: I got a perfect score on my GRE Quantitative, for example. But I regularly learn new things when I teach with JUMP. Big things. Things I never thought to ask about. Things that make me aghast I didn't know them before.

JUMP Math makes it simple for every student to develop full mathematical understanding. We'll ground our curriculum in it — and move beyond it, too.

I listed, in the last post, four things we should be able to promise students vis-à-vis math. By explaining JUMP, I think I've handled the first two of them — complete (2) understanding and (3) solving of the K-12 curriculum.

I haven't touched on (4) remembering everything that students learn, and (4) allowing students to be active learners, rather than passive receivers.

But I suspect I'm pushing the upper bounds of how long a blog post ought be already. I'll look forward to addressing those in future posts!


In Brief:

Understanding math is (and will continue to be) crucial in the 21st century. Yet our brains aren't built for it. What's needed — and what our school will set itself to delivering — is a math curriculum that takes seriously how difficult and unnatural math learning is, and then helps students master it entirely. To do this, we will start with the JUMP Math curriculum, and build from there.


For Further Reading:

John Mighton has written two books — The Myth of Ability and The End of Ignorance. Both are excellent, though start with the first. For a quicker overview of JUMP, however, take a peek at these two excellent posts in the New York Times Opinionator column — "A Better Way to Teach Math," and "A Better Way to Teach Math, Part 2."

A School for Complete Mathematical Understanding

both of these... The thing is, this is actually true — for all genders.

The problem:

The 21st century rewards those who can think mathematically, but the human brain isn't built to do that.

A new kind of school can train students — all students — to understand math, all the way from 4+5=9 to statistics and calculus. Our goal is nothing less than that. And there's good evidence that this isn't just a utopian wish.

But before we continue with the optimistic thinking, we have to make something perfectly clear:

Math is unnatural. Math is hard.

Sure, there are a few rare humans who apprehend new mathematical ideas as readily as the majority of us apprehend the plots of, say, Michael Bay movies. Those people face entirely different problems in math class — we'll ignore them for right now.

For the grand majority of us, learning math is difficult. Vexing. Boring.

We shouldn't be surprised by this — the human mind wasn't designed to do complex, abstract math. Being able to solve for x wasn't of particular use on the African savannah.

In contrast, the human mind was designed for learning a different abstract, rule-based, and knowledge-heavy system: language. Each human language has a syntax and vocabulary that is far more complex than the sum total of everything we ask students to learn in K-12 math.

Yet every virtually every 6-year-old speaks their local dialect flawlessly, while few 18-year-olds have mastered math.

We're built for language. We're not built for math.

Maybe there's some species out there who can do complex math automatically — but it ain't us.

We need to face this reality — until we do, we'll underestimate the difficulty of math learning, and sell kids short. From this dismal starting point comes the outline of a plan:

We need to make learning math easy.

Once we understand how alien math learning is, and yet how crucial it is to success in the 21st century, we're left with the resolve to re-invent math learning so every student can succeed.

This is already being done. We can piggyback on it, and make it even better.

By understanding how human cognition works, we can lead all students to learn math, and learn it well. We can help people succeed. We can have a better world.

We should be able to promise a few things about math to every student who comes into our school:

1. They will be active learners, not passive receivers. 2. They will be led to fully understand every concept in the curriculum. 3. They will be adept at solving every mathematical problem they encounter. 4. They will easily remember everything they learn, all the way up until they graduate.

In the next post, I'll sketch out how.


(A note: I'm worried about two things in this post. First, that what I've just sketched out will seem too optimistic. Is it really true that all kids — even those who aren't predisposed to math — can master calculus? There's excellent evidence that, excepting students with interesting neurological difficulties, the answer is yes. Second, I'm terrified that I've wrongly conveyed the sense that math can't be enjoyable. This would be horrible, as I love math: love teaching it and learning it for myself. The joy of fully grokking a math concept is one of the sweetest pleasures I know. Hopefully I'll be able to explain how we'll bring the love of math into our school in a future post.)