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.