“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.”
†
Marita’s Bargain
“ALL MY FRIENDS NOW ARE FROM KIPP.”
1.
In the mid-1990s, an experimental public school called the KIPP Academy opened on the fourth floor of Lou Gehrig Junior High School in New York City.
*
Lou Gehrig is in the seventh school district, otherwise known as the South Bronx, one of the poorest neighborhoods in New York City. It is a squat, gray 1960-s-era building across the street from a bleak-looking group of high-rises. A few blocks over is Grand Concourse, the borough’s main thoroughfare. These are not streets that you’d happily walk down, alone, after dark.
KIPP is a middle school. Classes are large: the fifth grade has two sections of thirty-five students each. There are no entrance exams or admissions requirements. Students are chosen by lottery, with any fourth grader living in the Bronx eligible to apply. Roughly half of the students are African American; the rest are Hispanic. Three-quarters of the children come from single-parent homes. Ninety percent qualify for “free or reduced lunch,” which is to say that their families earn so little that the federal government chips in so the children can eat properly at lunchtime.
KIPP Academy seems like the kind of school in the kind of neighborhood with the kind of student that would make educators despair—except that the minute you enter the building, it’s clear that something is different. The students walk quietly down the hallways in single file. In the classroom, they are taught to turn and address anyone talking to them in a protocol known as “SSLANT”: smile, sit up, listen, ask questions, nod when being spoken to, and track with your eyes. On the walls of the school’s corridors are hundreds of pennants from the colleges that KIPP graduates have gone on to attend. Last year, hundreds of families from across the Bronx entered the lottery for KIPP’s two fifth-grade classes. It is no exaggeration to say that just over ten years into its existence, KIPP has become one of the most desirable public schools in New York City.
What KIPP is most famous for is mathematics. In the South Bronx, only about 16 percent of all middle school students are performing at or above their grade level in math. But at KIPP, by the end of fifth grade, many of the students call math their favorite subject. In seventh grade, KIPP students start
high school
algebra. By the end of eighth grade, 84 percent of the students are performing at or above their grade level, which is to say that this motley group of randomly chosen lower-income kids from dingy apartments in one of the country’s worst neighborhoods—whose parents, in an overwhelming number of cases, never set foot in a college—do as well in mathematics as the privileged eighth graders of American’s wealthy suburbs. “Our kids’ reading is on point,” said David Levin, who founded KIPP with a fellow teacher, Michael Feinberg, in 1994. “They struggle a little bit with writing skills. But when they leave here, they rock in math.”
There are now more than fifty KIPP schools across the United States, with more on the way. The KIPP program represents one of the most promising new educational philosophies in the United States. But its success is best understood not in terms of its curriculum, its teachers, its resources, or some kind of institutional innovation. KIPP is, rather, an organization that has succeeded by taking the idea of cultural legacies seriously.
2.
In the early nineteenth century, a group of reformers set out to establish a system of public education in the United States. What passed for public school at the time was a haphazard assortment of locally run one-room schoolhouses and overcrowded urban classrooms scattered around the country. In rural areas, schools closed in the spring and fall and ran all summer long, so that children could help out in the busy planting and harvesting seasons. In the city, many schools mirrored the long and chaotic schedules of the children’s working-class parents. The reformers wanted to make sure that all children went to school and that public school was comprehensive, meaning that all children got enough schooling to learn how to read and write and do basic arithmetic and function as productive citizens.
But as the historian Kenneth Gold has pointed out, the early educational reformers were also tremendously concerned that children not get
too much
schooling. In 1871, for example, the US commissioner of education published a report by Edward Jarvis on the “Relation of Education to Insanity.” Jarvis had studied 1,741 cases of insanity and concluded that “over-study” was responsible for 205 of them. “Education lays the foundation of a large portion of the causes of mental disorder,” Jarvis wrote. Similarly, the pioneer of public education in Massachusetts, Horace Mann, believed that working students too hard would create a “most pernicious influence upon character and habits....Not infrequently is health itself destroyed by over-stimulating the mind.” In the education journals of the day, there were constant worries about overtaxing students or blunting their natural abilities through too much schoolwork.
The reformers, Gold writes:
strove for ways to reduce time spent studying, because long periods of respite could save the mind from injury. Hence the elimination of Saturday classes, the shortening of the school day, and the lengthening of vacation—all of which occurred over the course of the nineteenth century. Teachers were cautioned that “when [students] are required to study, their bodies should not be exhausted by long confinement, nor their minds bewildered by prolonged application.” Rest also presented particular opportunities for strengthening cognitive and analytical skills. As one contributor to the
Massachusetts Teacher
suggested, “it is when thus relieved from the state of tension belonging to actual study that boys and girls, as well as men and women, acquire the habit of thought and reflection, and of forming their own conclusions, independently of what they are taught and the authority of others.”
This idea—that effort must be balanced by rest— could not be more different from Asian notions about study and work, of course. But then again, the Asian worldview was shaped by the rice paddy. In the Pearl River Delta, the rice farmer planted two and sometimes three crops a year. The land was fallow only briefly. In fact, one of the singular features of rice cultivation is that because of the nutrients carried by the water used in irrigation, the more a plot of land is cultivated, the more fertile it gets.
But in Western agriculture, the opposite is true. Unless a wheat- or cornfield is left fallow every few years, the soil becomes exhausted. Every winter, fields are empty. The hard labor of spring planting and fall harvesting is followed, like clockwork, by the slower pace of summer and winter. This is the logic the reformers applied to the cultivation of young minds. We formulate new ideas by analogy, working from what we know toward what we don’t know, and what the reformers knew were the rhythms of the agricultural seasons. A mind must be cultivated. But not too much, lest it be exhausted. And what was the remedy for the dangers of exhaustion? The long summer vacation—a peculiar and distinctive American legacy that has had profound consequences for the learning patterns of the students of the present day.