# Learning to rock-climb is changing how I’ll teach math.

Five years out of college—and taking a year off from teaching—I find myself in a precarious new position: dangling by ropes, 35 feet up a wall, a beginner again.

Learning to rock-climb is as exhausting and fun as I’d hoped. I’ve spent hours rising and falling, hauling my body from Point A to Point B, returning home too drained for anything but Facebook and Orange is the New Black. I’d half-forgotten how exhilarating and vulnerable it feels to begin something.

Because my mind exhales analogies (in much the same way that my body exhales CO2), I’m constantly drawing connections between rock-climbing and teaching math. In my own grunting and straining, I hear the graceless echo of my students’ efforts, but from the other side. Back in the classroom, I was the one holding the ropes, with two feet planted on flat, sturdy ground. I assured them not to worry, that I’d catch them if they fell. Now, I’m the one clinging to the wall, hoping like heck that the ropes don’t break.

Here are the lessons I’ve learned up on the wall. Not the lessons about rock climbing, but the lessons about learning itself.

1. Just because you can do it doesn’t mean you can explain it.

As a teacher, I always pressed for explanations. The approach baffled some students, who knew math demanded right answers, but had never bothered to issue detailed accounts of where those answers came from.

Now, coming down off the wall, I land right where my students once stood. “How did you make it over that ledge?” I don’t know. “Did you use that foothold, or the one further out?” I couldn’t tell you. “Did you balance with your right hand and reach with your left, or vice versa?” No idea. I’m lost in the gray mist that my students know so well: I can do, but I can’t say why. The knowledge lives in my muscles and instincts, not in my verbal, reasoning mind.

Rock-climbing has reminded me that, before you can explain your understanding, there’s an intermediate phase: knowing how without knowing why. Continue reading

# The Mental Machinery of the Chess Master

As a Psych major in college, I learned about some cool experiments. Fatal shocks. Coldhearted preachers. The staggering forces of peer pressure. I saw slobbering dogs, wailing babies, and semi-literate pigeons.

But one of my favorite experiments requires nothing more than a chess set. It begins to answer the question: What, exactly, is going on in the minds of great chess players?

# Black Boxes (or: Just Say No to Voodoo Formulas)

We’ve all got black boxes in our lives.

A black box is a little mystery that you take for granted. It’s something you use without thinking, without skepticism, without once opening the lid to peek at the workings inside. For all you know, it might be powered by wind, water, cold fusion, hamster wheels—or even some fantastic combination thereof (i.e., nuclear hamster windstorms).

I’ve got a lot of black boxes. Too many. Cars, computers, non-stick skillets—every day is a decathlon of technologies and tasks I don’t actually understand. That’s why I love math, which (ideally) has no black boxes. In math, you only use tools once you’ve studied every gear and lever so closely that you could assemble them in your sleep. Math is the realm where all boxes are transparent.

Ideally.

# Is Memorization Necessary, Evil, or Both?

At The Atlantic today, I have an essay weighing in on the decades-long debate over memorization, trying to cut a middle path between two extremes:

1. “Memorization is the enemy. It’s the antithesis of critical thinking and conceptual learning. Memorization’s defenders are wilfully blind soldiers marching for an outdated tradition.”

2. “Memorization is an essential tool for students. It’s the surest path to retaining important facts. People who denounce it are letting liberal orthodoxy get in the way of our children’s achievement.”

I’d summarize my view along these lines:

3. “Memorization is a generally-not-great shortcut. It’s better than not knowing at all, but it’s not nearly as enduring, effective, and powerful as meaningful learning.”

# History’s Greatest Chess Matches

I’ve been poking at chess lately, the way a chimpanzee might poke at a car engine. Does he understand it? Not really. Is he having fun? Sure!

Chess is much like math. Instead of problem sets, you play games, and in lieu of lectures, I’ve been YouTubing famous games from chess history. Here are 3 of my favorites, each no longer than an episode of Arrested Development – and just as intricate and clever.

The Immortal Game

In 1851, Adolf Anderssen (a math teacher) and Lionel Kieseritzky (the editor of a chess magazine) played a match that has been talked about ever since.

In it, Anderssen leads his pieces in a suicide charge. He sacrifices a bishop, both rooks, and the queen – nearly all his best material – in exchange for nothing but pawns. It’s like a parent chanting, “Go! Go! Go!” as he sends his children sprinting out into onrushing traffic. He’s a madman. Yet in spite of the seemingly crippling losses, he checkmates Kieseritzky on the 23rd move. The madman wins.

Anderssen’s play is as spooky as a zombie movie. The attackers seem to have no regard for their own safety. They’ll destroy themselves to bring you down. Continue reading

# How Fast is Exponential Growth? (Or, Yao Ming Confronts the Vastness of the Universe)

On this humble brown planet, we’re used to things growing at a steady pace. Trees add a ring every year. Families expand by one child or marriage at a time. Even in their most extreme months of food-gobbling growth spurts, teenagers will sprout at most a few inches. All of these are examples of linear growth (or something close to it). It’s modest, approachable – something the human brain has no trouble grasping.

But not all growth is like this. Take the old story of the sultan and the beggar. The beggar comes before the sultan, pleading for some rice to eat. When the sultan asks how much he needs, the beggar cleverly points to a nearby chessboard. He asks the sultan to put 1 grain on the first square, 2 on the next, 4 on the next, 8 on the next, and so on, doubling the number of grains for each successive square.

The sultan agrees, not realizing that on the 64th and final square, he’ll need to stack 600 trillion pounds of rice – enough to cover Rhode Island to a depth of 1400 feet. That’s exponential growth. It may start slow, but it quickly reaches dizzying heights.

This kind of growth occurs surprisingly often. Look at the human population, or the number of Facebook users, or even the amount of money you make on a brilliant investment. Most of these growth patterns eventually hit some kind of ceiling (the planet will hold only so many humans; Facebook has begun to max out its user base; and your investment can climb only so high), but as long as exponential growth is at work, change proceeds far faster than our linear minds are accustomed to.

To explain this idea to my students, I start with a graph of the function f(x) = 2x, using a one-inch scale. Continue reading

# Three Sentiments (or, Ode to the School Year)

Fast year, right? Summer is here. The seniors can be found draped across their desks, exploring stages of hibernation so deep that they are yet uncharted by the medical community. It’s all very festive.

And into this start-of-summer breeze, I’d like to offer three sentiments.

First, guys, let’s be honest. We didn’t always getting along. I assigned too much homework, gave too many quizzes, wrote tests that stumped you. When I relented even slightly, you celebrated like a labor union that had scored a victory against the cigar-smoking management. I pushed you too hard, expected too much, demanded the unfair or even the impossible. This brings me to my first sentiment: Continue reading

# The Quadratic Formula Must Die! (or, Long Live the Quadratic Formula!)

Algebra students are often compelled to memorize the following jumble of symbols:

Every adult I meet seems to remember this equation. Some quote it proudly. Others recall it grudgingly, fists clenched. Some people sing it (to the tune of “Pop Goes the Weasel,” though the rhythm’s a little forced). Many believe it captures the essence of mathematics: mystical formulas, taught to everyone, comprehensible to few.

I’m ambivalent. I can’t deny the quadratic formula’s usefulness, but I can’t let its tyranny go unchallenged, either. This formula has planted itself, like a virus, in the minds of otherwise healthy adults. Our society demands that its members know this equation, but makes no mention of its context, its uses, its colorful history.

I come not to praise the quadratic formula, nor to bury it, but merely to shed a few rays of light. What is it for? What is its story? And where on Earth does it come from? Continue reading

# A Ray of Light

There are moments of teaching I like to remember – episodes of cleverness, compassion, success. And then there are the other moments, the ones that my thoughts tend to flee, the ones I prefer not to think about. This is a story about both.

One Friday after school, a student came to me with questions. As a 12th-grade transfer, she found herself struggling to catch up with the students who had already spent years in the crucible of our intense charter school. To graduate that year, she needed to take my Statistics class concurrently with Algebra 2. She was failing them both. Continue reading