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:

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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.

What if schools can help most people become good at almost everything?

I'm launching an advanced academic reading course at the University of Washington Bothell campus this morning — but in lieu of a post, a quote! This comes again from Andrew Ng, whom the MIT Technology Review dubbed one of the top innovators in the world under age 35 —

But often, you first become good at something,  and then you become passionate about it.   And I think most people can become good at almost anything.

What if schools could help most people become good at almost everything? What if schools could help most people become passionate about almost everything?

Such is our quest.

Why value-free education is impossible

In How to Raise a Wild Child, Scott D. Sampson writes:

beauty, truth, and goodness are all essential aspects of learning and education. Value-free education is impossible.

It's occurred to me recently that I haven't done a good job explaining that what we're trying to do with our network of schools isn't just to teach kids more things. It's not just to make them smarter and more skilled, better prepared for the needs of the 21st century.

Our goal, rather, is to cultivate a certain kind of person. 

Though he comes from a very different tradition, the Protestant theologian James K. A. Smith (in his jaw-dropping book Desiring the Kingdom) writes something intriguingly similar:

I’ve been suggesting that education is not primarily a heady project concerned with providing information;

rather, education is most fundamentally a matter of formation, a task of shaping and creating a certain kind of people.

How do we "shape a certain kind of people"? By helping them think more wisely about the good life — and helping them experience pieces of the good life while they're at our schools. James K. A. Smith again:

What makes them a distinctive kind of people is what they love or desire — what they envision as “the good life” or the ideal picture of human flourishing.

So, to bring together this insight with our core values:

Our schools aren't merely trying to teach kids better. We're striving to cultivate a certain sort of people — Renaissance men and women, who find all aspects of the world fascinating, relish developing mastery in all manner of fields, and work to construct lives of purpose and meaning.

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.

Love before mastery

I've recently realized — or maybe re-realized — how useful it can be to put love before mastery.

You'll remember that the three über-values of our schools are love, mastery, and meaning. The order of those three is important: love (i.e. interest, passion, desire) comes before mastery, and supports it.

Want your child to become really, really good at something? Help them fall in love with it first. At least a little.

Kristin and I had forgotten this, I think, a little while ago, when we signed our five-year-old up for a swimming lesson. It didn't take: he was terrified to put his face in the water, and didn't trust the instructor.

Now, a half-year later, our son is clamoring for lessons. The difference? He's spent more fun time in the water. He's come to love the water, and wants to learn how to do more in it. 

Goodness: now, in baths, he borrows my goggles, and sticks his head in the water.

Love comes before mastery.

Now, it's more complicated than that: mastery builds love, too. As educational psychologist Jerome Bruner wrote:

We become interested in what we become good at.

So we shouldn't become simpletons with this! But a helpful, general rule seems obvious:

When we want kids to become great at something, we need to first help them fall in love with it.

Continuous feedback

A problem:

Feedback is necessary, but terrifying. 

Feedback is crucial for building skill and understanding. A school that takes "mastery" seriously has to not just endure feedback, but embrace it (teachers and administrators as well as students!).

Yet receiving feedback is famously difficult. Every piece of feedback threatens our sense that I'm just fine, thanksverymuch. And if we build lots of feedback into a school and it doesn't work out, school could become Hell.

We need to proceed cautiously, but forcefully.

Our basic plan:

1. We make a ritual in the school: every assignment and production gets feedback. 
2. The feedback should be specific. Getting a "B" is unspecific feedback; getting a suggestion for precisely what to consider doing differently next time is specific feedback.
3. The feedback should be succinct. Being told five things to do differently next time is too much for most people to focus on; being told just one thing is more helpful.
4. The feedback should be a suggestion. Because no one is innerrant, feedback shouldn't pretend to be authoritative; feedback should (usually) be given as advice — "why don't you try this next time?"
5. The feedback should prompt a response. Students shouldn't be expected to take feedback passively; they should respond to it, perhaps commiting to trying it out next time, or explaining why they'll go in a different direction.
6. The feedback should come from the community. Teachers should be wiser than their students (otherwise, why are we teaching?), but they aren't the only sources of wisdom. Suggestions should come from teachers, from other students, and from the student him or herself.
7. The feedback should accumulate. Advice should be collected, and students should be prompted to see if they're incorporating it into their recent projects. (This can serve as an ego-builder: "I may still be struggling, but not with the same old things!")

Our goals:

We hope to...

• Create a culture of mastery.
• Create a culture of embracing criticism. When we hear a kid say, "Sure, I know that this is good — but tell me how I can improve!", we'll know we've succeeded.
• Create a culture of mutual help. When we hear one kid tell another, "I was really impressed by how you've changed in x," we'll know we've succeeded.
• Build resilience. Kids are not these fragile things; they're strong. They don't need to be fazed by criticism. (But the only way to learn that is by dealing with criticism.)

If you walk into our classrooms, you might see:

• Students scribbing down brief feedback after another student completes a speech, or a piece of art, or a meal.
• A student and a teacher conversing — and maybe arguing! — over the quality of a recent assignment. (When we get a frank but respectful exchange of views, we'll know we've succeeded.)
• Students spending a few minutes each week reviewing previous feedback about (e.g.) essay writing so they can incorporate it into their new essay.

Some specific questions:

• Should we aim for a specific ratio of positive to negative feedback? There's a danger in getting too much negative feedback. There's a danger, too, in getting too much positive feedback. Perhaps it would be best to allow the student to say how what ratio they'd like to receive. (That might make students feel more in control, and thus make them more likely to embrace the feedback they receive.)
• How should we store this feedback? Should a regular assignment be that students copy in their feedback (or a segment of it) into a year-long Google document?
• Should students give teachers feedback? Doing so could help teachers (and thus everyone) improve more quickly. It could demonstrate our respect of our students.

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?

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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).

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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.

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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.

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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.)

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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.

Faith-based math

Our school shall have no faith-based math. Before I set off an Internet flame war (or is it too late already?!): I'm not talking about religion right now. Except maybe I am?

The Calvin & Hobbes strip above really nails the experience of many students in math class. Doing well in math amounts to taking things (formulas, for instance) on the authority of the textbook. Students who do well in math class are those who can best memorize these bits of dogma.

Obviously, this has nothing to do with actual mathematical understanding.

I know that this idea sounds incontestable — and, well, it is. Of course students should understand what they're doing in math!

Yet this principle is broken in nearly every textbook, in nearly every class.

I'm reminded of this today as I prepare my economics lesson for the afternoon. We're reading a popular book on economics — I won't mention the title — and are trying to understand how supply and demand curves shift when products are taxed.

The students are struggling to understand it. They're model students: reading carefully, testing their comprehension. But they're frustrated. I should be able to help them, because I should have a full understanding of the topic at hand.

The thing is: I don't. And the book is no help.

The book — at least this portion of the book — is, in effect, faith-based. It doesn't explain taxation the way it claims to. It doesn't matter how hard the reader works: they're stuck in faith-based math (or, in this case, faith-based economics). They're forced to kowtow to the author, and simply assume the theory makes sense.

Ack. Uck.

I'll see what I can do for the class — I may need to bring in an outside economist to help us make sense of this. I'll certainly own up to my own non-understanding, and help the students explicate the gaps in their understanding.

That is, I'll help them see what they don't see.

And that's useful, in the short-term. But here's a long-term promise we can make for our school:

When studying any analytical, reasoning-based subject, students will never be expected to take anything on faith. We'll inculcate them in the truth that, if some idea (a math formula, an economic concept, a chemistry… chemically-thing!) has been understood by someone else's mind, it can be understood by their mind.

And we'll rear them in the conviction that achieving this understanding — capturing its complexity in their own head — is one of the most beautiful experiences available to us humans.

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?

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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 clever — that 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.

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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.

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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.

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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)

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?

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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.

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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.

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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.

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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.

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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.

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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!

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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.

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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

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.

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(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.)

Our Trinity, #2: Mastery

Ours can be a school of mastery.

Let's assume that we succeed in our crazy goal of helping students fall in love with many (or most) of the subjects they study — that by middle school our students are entering adolescence convinced that history, biology, math, astronomy, and so are are desperately interesting. What next?

I suggest: we can help them learn precisely how to excel at any task they set themselves to.

Here, we can get help from science. Psychologists have hacked talent, and the world is only beginning to wake up to it. Most classrooms, driven by a century of inertia, still work off the assumption that kids who lack native skill in a subject (math, for example, or writing) probably won't be able to get more than passable in it. (I've found this idea almost universally held, though only rarely expressed.)

Delightfully, this is wrong. False. Mis-conceived.

The higher realms of performance, the psychologists tell us, are open to us all.

That is: Anyone can excel at math. Anyone can draw realistically, and beautifully. Anyone can write lucidly. And so on, and so on.

What's needed isn't just practice — flashbacks to the "10,000-hours rule" — but a certain type of practice, done (yes) repeatedly over a long period of time. And psychologists have been uncovering what that type of practice (dubbed "deliberate practice") looks like.

To engage in deliberate practice is to target a specific goal, and to measure one's progress toward it. It's to constantly adjust the difficulty level of a challenge, so that one is always working at the full extent of one's abilities. It's to break down complex routines into simple tasks, perfect those simple tasks, and then re-assemble them into their (now perfected) complex routines.

(For a very helpful distillation of deliberate practice that expands on what I've just written, check out this blog post.)

Deliberate practice is painful. But it works wonders. And anyone — with certain commonsense limits — can use it to become impressively better in any domain.

No one has devised a school built on deliberate practice. No one (so far as I know) has done a from-the-roots-up rethinking of what schooling could look like if talent can be built by anyone.

We can do that.

And we can go further than deliberate practice — we can cultivate a culture that values excellence and self-overcoming. And we can do this in a number of different ways. In our history curricula, we can highlight brilliant inventors, crafty leaders, and ingenious artists. In our assemblies, we can laud students who have struggled the most. Perhaps we'll find it good to group students into different houses (I'm imagining Hogwarts here) based on how they best approach talent acquisition — those who benefit from competition in one house, and those who benefit from a non-competitive environment in another.

I've been speaking of the sub-set of cognitive psychology called "expertise studies," but we can also adopt some of the most helpful discoveries of cognitive psychology more generally. Cognitive psychologists, for example, have decisively answered the question of how we can remember what we learn forever. They've worked out useful insights into how creativity functions — how people generate new ideas and solutions. All of these, too, can be brought into our curriculum.

I've focused here on how we can do this all in middle school, but certain aspects of it should start in grade school — for example, we should craft our math curriculum with a full knowledge of deliberate practice, so kids at the very least aren't wasting their time on unchallenging problems, or forgetting what they've worked hard to learn. Before we teach them how to acquire expertise, we should build some aspects of it into everything they do.

A few provisos to what I've just written:

1. None of this means, by the way, that we should take an intensive, Tiger-Mom / Korean-prep-school-esque approach to any aspect of our school. I'm allergic to these. I think they're (typically) bad places to raise humans. We should aim for a school culture that exalts excellence, and encourages students (and teachers!) to pursue it. But coerced practice is not (typically) useful practice. And such forcefulness can threaten to poison everything else.
2. I've been speaking too blithely here — I understand (everyone understands, I think) that not literally everyone is able to, for example, excel at drawing, or writing, or math. People with significant neurological damage, for example — or people with a deep, learned aversion to a certain subject. (I'm reminded of the Jack Handy quip: “To me, clowns aren't funny. In fact, they're kind of scary. I've wondered where this started and I think it goes back to the time I went to the circus, and a clown killed my dad.” Such a person would have a very difficult time excelling at clown school.)
3. I want to be sure that this "virtually anyone can develop mad skills" isn't confused with the "blank slate" hypothesis — the idea that everyone is born with a perfectly equal predisposition to develop talent. I once believed that, but psychologists tell us it's wrong. Some kids really are born with a higher or lower propensity to learning math (or writing, or art, or whatever). But the beautiful thing is that this isn't determinative: a student who doesn't have a predisposition toward learning to do math really can excel at it, with the right sort of practice.

Finally, I think there are some real-world implications to all of this. In my caffeine-fueled dreams, I nurture hopes that this could be a school to change the world. Well, if we really can hack talent in practice (the way that psychologists have hacked talent in theory), that is something the world needs.

Marvin Zonis, the University of Chicago professor of global economics, wrote:

The demand everywhere will be for ever higher levels of human capital [skills and talents]. 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.

Again, this is something the world needs. And it's something that we can provide.

A final question: I'm not sure about the word "mastery" to carry all this meaning. I've also been kicking around some other choices: excellence, expertise, genius, and talent. Any thoughts, y'all?

(Next up: wisdom.)