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.

How to move beyond the Math Wars? Make Math Easy; Make it Hard.

We live in the shadow of the 1990's "Math Wars", and crafting a new approach to the K-12 math curriculum is fraught with problems — and hate! 

On one side: the math traditionalists. They point out, quite sensibly, that learning one particular method for thinking about, say, division makes math much easier to do. They also point out that repetition is required to develop fluency. 

On the other side: the math reformers. They point out, equally sensibly, that learning one particular method for doing math doesn't provide actual conceptual understanding of the math at play: a student can learn how to "do" long division without having any glimpse of what's going on. They also point out that mindless repetition is typically at odds with any sort of enjoyment: doing problems 1–30 (odds!) rarely stirs our curiosity.

What follows: a sketch of how a new kind of STEM school could structure its math curriculum, especially in grades K–4. 

In brief: embrace extremes. We can recognize that both the traditionalists & the reformers recognize crucial aspects of math, and of the human mind. 

Our job isn't to "balance" them into some sort of middling practice. ("Balance" is typically doomed from the start.) Rather, our job is to hold the extremes together. 

Math math as easy as possible

Math is hard, and to help students value it we need to help them see that they can understand it. 

And no mathematical idea — at least, none in the K–12 curriculum — is beyond students. Virtually all students can perfectly understand everything in the math curriculum.

It's easy to say that, of course, harder to do it! To make math as easy as possible, a new kind of STEM school could use two tools:

  1. The JUMP Math curriculum. We've explained our love of JUMP in an earlier post, but to distill it: the K-8 JUMP curriculum breaks every math idea into an armful of tiny ideas, which all students can zip through. JUMP works for advanced students, for struggling students, and for everyone in between. 
  2. Deep Practice Books. We've written a little about this before, too, but to distill: students can keep collections of problems they find frustrating. They revisit these problems, asking questions of them, seeking deeper explanations, and re-solving them in diverse ways. After a few days or weeks, each problem becomes easy — and students tend to enjoy them!

Make math as hard as possible

Math is hard, and to help students value it we need to (wait for it) let them struggle. 

JUMP's brilliance comes from the fact that it carefully guides students to full understanding. But to cultivate real mathematical thinking, we need to also give students unguided experiences with math: we need to toss them into problems, and let them fend for themselves. 

Well, that's an exaggeration: it's not that we should offer no guidance, but that we should offer minimal guidance. Students need to learn how to problem-solve on their own. 

Yes, this is the opposite of the above! And both are important.

How should we do it? We know of two tried-and-true methods:

  1. Host math circles. What's a math circle? A sports team for math.
    Math circles can vary profoundly: some are all about preparing for math competitions, others are anti-competitive. Some look like traditional teacher-led courses, others are inquiry based. What they share in common is that they're led by real mathematicians, and lead children into the depths of mathematical ideas through conversation. 
  2. Attempt mathematical puzzles through the "Japanese Method". In The Teaching GapJames Stigler and James Hiebert lay out how Japanese schools put unstructured problem-solving before guided explanations. A teacher in Japan will post a challenging problem — one which students do not have the tools to easily solve. Working in small groups, students will tackle it, crafting their own tools to do so. And then the groups share their methods, and the teacher leads a conversation comparing them. Which method is easiest? Which is most elegant? Which is the most complex? By puzzling through how (superficially) diverse methods can lead to the same answer, students see into the heart of mathematics. (We've written about this idea before here.)

If there's one thing that everyone in American education can agree about, it's that math is currently taught abysmally. By reconceiving math teaching (and learning) with the insights of all sides of the math wars, a new kind of STEM school can forge a way forward.

The secret to boiling an egg (and mastering EVERYTHING ELSE)


A remarkable fact about the world: how difficult it is to boil an egg. Perhaps you're thinking right now, "what, in the universe of cooking, could possibly be simpler? You plop the egg in the water, you set a timer, boil the water, and take out the egg! Violà! A hard-boiled egg!"

Oh, I too was once naïve!

For a few months now my daily breakfast has consisted of four hard-boiled eggs, and so I've had ample opportunity to get this right. And I do, sometimes — I cook the yolk to the perfect consistency, in a manner that leaves the shell uncracked yet easy to peel off the albumen.

Sometimes. But not always. 

It's surprisingly hard. Though: I'm getting better.

Making precisely the same food every day has made me recognize that there are so many factors, even in this, the world's simplest dish:

  • Do I bring the water to a boil first?
  • Should it be a low boil, or a high boil? Does it matter?
  • Should I do anything to the water? (Some swear by vinegar; others by salt.)
  • After I take it out, should I let the eggs cool in the air, or plunge them into cool water? Iced water?

Over the last few months I've varied each of these factors, experimenting around until I've found the nigh-perfect recipe. (Which is, in case you're interested, to place the eggs in the pot, fill it with hot tap water, shake in some salt, and set the stove on "medium/medium-high" for 11 minutes. Afterwards, I take the eggs out and juggle them into an old pickle jar filled with ice water. C'est magnifique!)

Why am I talking about this?

Because in my breakfast-hacking, there is a lesson that pertains to everything we do:

Mastery comes from cycles.

Try something, get feedback — make a small change. Repeat it, get feedback — make another small change. And again. And again. And again.

Philosopher Daniel Dennett writes about this eloquently in his answer to the question, "What scientific concept would improve everyone's cognitive toolkit?" I first read it in the book This Will Make You Smarter; it's also online here.

Dennett suggests that these cycles of repetition are at the heart of what makes the natural world complex and wonderful: the biochemical Krebs cycle, Darwinian evolution — even the gasoline engine.

And then Dennett goes to human skill:

At a completely different scale, our ancestors discovered the efficacy of cycles in one of the great advances of human prehistory: the role of repetition in manufacture. Take a stick and rub it with a stone and almost nothing happens — few scratches are the only visible sign of change. Rub it a hundred times and there is still nothing much to see. But rub it just so, for a few thousand times, and you can turn it into an uncannily straight arrow shaft. By the accumulation of imperceptible increments, the cyclical process creates something altogether new.

Dennett concludes his essay:

A good rule of thumb, then, when confronting the apparent magic of the world of life and mind is: look for the cycles that are doing all the hard work.

This is how skill is made: repetition with feedback.

As I've laid out earlier, one of the three major values of our type of school is mastery. A new kind of schooling needs to lay out for students the route to building expertise — in math, in writing, in thinking, in art, in everything. And we need to do more than lay it out — we need to help excite students to achieve it, and work to achieve it with them.

Every student, and every teacher, can make stirring advancement in a great number of fields.

Our schools can be talent workshops. 

And to do it, we need to set students at the task of lovingly crafting their work, seeking advice, and experimenting with small changes.

This is how to boil an egg, and master everything else.

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?