She starts to fiddle with the program, typing in different numbers.

"Now if I change the slope that way... minus 1... now what I mean to do is make the line go straight."

As she types in numbers, the line on the screen changes.

"Oops. That's not going to do it."

She looks puzzled.

"What are you trying to do?" Schoenfeld asks.

"What I'm trying to do is make a straight line parallel to the y axis. What do I need to do here? I think what I need to do is change this a little bit." She points at the place where the number for the y axis is. "That was something I discovered. That when you go from 1 to 2, there was a rather big change. But now if you get way up there you have to keep changing."

This is Renee's glorious misconception. She's noticed the higher she makes the y axis coordinate, the steeper the line gets. So she thinks the key to making a vertical line is just making the y axis coordinate large enough.

"I guess 12 or even 13 could do it. Maybe even as much as 15."

She frowns. She and Schoenfeld go back and forth. She asks him questions. He prods her gently in the right direction. She keeps trying and trying, one approach after another.

At one point, she types in 20. The line gets a little bit steeper.

She types in 40. The line gets steeper still.

"I see that there is a relationship there. But as to why, it doesn't seem to make sense to me.... What if I do 80? If 40 gets me halfway, then 80 should get me all the way to the y axis. So let's just see what happens."

She types in 80. The line is steeper. But it's still not totally vertical.

"Ohhh. It's infinity, isn't it? It's never going to get there." Renee is close. But then she reverts to her original misconception.

"So what do I need? 100? Every time you double the number, you get halfway to the y axis. But it never gets there..."

She types in 100.

"It's closer. But not quite there yet."

She starts to think out loud. It's obvious she's on the verge of figuring something out. "Well, I knew this, though... but... I knew that. For each one up, it goes that many over. I'm still somewhat confused as to why..."

She pauses, squinting at the screen.

"I'm getting confused. It's a tenth of the way to the one. But I don't want it to be..."

And then she sees it.

"Oh! It's any number up, and zero over. It's any number divided by zero!" Her face lights up. "A vertical line is anything divided by zero-and that's an undefined number. Ohhh. Okay. Now I see. The slope of a vertical line is undefined. Ahhhh. That means something now. I won't forget that!"

6.

Over the course of his career, Schoenfeld has videotaped countless students as they worked on math problems. But the Renee tape is one of his favorites because of how beautifully it illustrates what he considers to be the secret to learning mathematics. Twenty-two minutes pass from the moment Renee begins playing with the computer program to the moment she says, "Ahhhh. That means something now." That's a long time. "This is eighth-grade mathematics," Schoenfeld said. "If I put the average eighth grader in the same position as Renee, I'm guessing that after the first few attempts, they would have said, 'I don't get it. I need you to explain it.' " Schoenfeld once asked a group of high school students how long they would work on a homework question before they concluded it was too hard for them ever to solve. Their answers ranged from thirty seconds to five minutes, with the average answer two minutes.

But Renee persists. She experiments. She goes back over the same issues time and again. She thinks out loud. She keeps going and going. She simply won't give up. She knows on some vague level that there is something wrong with her theory about how to draw a vertical line, and she won't stop until she's absolutely sure she has it right.

Renee wasn't a math natural. Abstract concepts like "slope" and "undefined" clearly didn't come easily to her. But Schoenfeld could not have found her more impressive.

"There's a will to make sense that drives what she does," Schoenfeld says. "She wouldn't accept a superficial 'Yeah, you're right' and walk away. That's not who she is. And that's really unusual." He rewound the tape and pointed to a moment when Renee reacted with genuine surprise to something on the screen.

"Look," he said. "She does a double take. Many students would just let that fly by. Instead, she thought, 'That doesn't jibe with whatever I'm thinking. I don't get it. That's important. I want an explanation.' And when she finally gets the explanation, she says, 'Yeah, that fits.' "

At Berkeley, Schoenfeld teaches a course on problem solving, the entire point of which, he says, is to get his students to unlearn the mathematical habits they picked up on the way to university. "I pick a problem that I don't know how to solve," he says. "I tell my students, 'You're going to have a two-week take-home exam. I know your habits. You're going to do nothing for the first week and start it next week, and I want to warn you now: If you only spend one week on this, you're not going to solve it. If, on the other hand, you start working the day I give you the midterm, you'll be frustrated. You'll come to me and say, 'It's impossible.' I'll tell you, Keep working, and by week two, you'll find you'll make significant progress."

We sometimes think of being good at mathematics as an innate ability. You either have "it" or you don't. But to Schoenfeld, it's not so much ability as attitude. You master mathematics if you are willing to try. That's what Schoenfeld attempts to teach his students. Success is a function of persistence and doggedness and the willingness to work hard for twenty-two minutes to make sense of something that most people would give up on after thirty seconds. Put a bunch of Renees in a classroom, and give them the space and time to explore mathematics for themselves, and you could go a long way. Or imagine a country where Renee's doggedness is not the exception, but a cultural trait, embedded as deeply as the culture of honor in the Cumberland Plateau. Now that would be a country good at math.

7.

Every four years, an international group of educators administers a comprehensive mathematics and science test to elementary and junior high students around the world. It's the TIMSS (the same test you read about earlier, in the discussion of differences between fourth graders born near the beginning of a school cutoff date and those born near the end of the date), and the point of the TIMSS is to compare the educational achievement of one country with another's.

When students sit down to take the TIMSS exam, they also have to fill out a questionnaire. It asks them all kinds of things, such as what their parents' level of education is, and what their views about math are, and what their friends are like. It's not a trivial exercise. It's about 120 questions long. In fact, it is so tedious and demanding that many students leave as many as ten or twenty questions blank.

Now, here's the interesting part. As it turns out, the average number of items answered on that questionnaire varies from country to country. It is possible, in fact, to rank all the participating countries according to how many items their students answer on the questionnaire. Now, what do you think happens if you compare the questionnaire rankings with the math rankings on the TIMSS? They are exactly the same. In other words, countries whose students are willing to concentrate and sit still long enough and focus on answering every single question in an endless questionnaire are the same countries whose students do the best job of solving math problems.

The person who discovered this fact is an educational researcher at the University of Pennsylvania named Erling Boe, and he stumbled across it by accident. "It came out of the blue," he says. Boe hasn't even been able to publish his findings in a scientific journal, because, he says, it's just a bit too weird. Remember, he's not saying that the ability to finish the questionnaire and the ability to excel on the math test are related. He's saying that they are the same: if you compare the two rankings, they are identical.

Think about this another way. Imagine that every year, there was a Math Olympics in some fabulous city in the world. And every country in the world sent its own team of one thousand eighth graders. Boe's point is that we could predict precisely the order in which every country would finish in the Math Olympics without asking a single math question. All we would have to do is give them some task measuring how hard they were willing to work. In fact, we wouldn't even have to give them a task. We should be able to predict which countries are best at math simply by looking at which national cultures place the highest emphasis on effort and hard work.

So, which places are at the top of both lists? The answer shouldn't surprise you: Singapore, South Korea, China (Taiwan), Hong Kong, and Japan. What those five have in common, of course, is that they are all cultures shaped by the tradition of wet-rice agriculture and meaningful work.* They are the kinds of places where, for hundreds of years, penniless peasants, slaving away in the rice paddies three thousand hours a year, said things to one another like "No one who can rise before dawn three hundred sixty days a year fails to make his family rich."

CHAPTER NINE

Marita's Bargain