by Samuel Levin
Flatland is narrated by a square who lives in a two-dimensional world. The first part of the book explains how a two-dimensional world works. But then the square is visited by a sphere (which appears as a circle that is growing larger as the sphere passes through the plane). The sphere eventually convinces our narrator of a third dimension and then takes him to visit other worlds—a one-dimensional one, a dimensionless one. The climax of the book occurs when the square says to the sphere, “Wow, I wonder what a four-dimensional world is like,” and the sphere says, “That’s ridiculous, there’s no such thing!”
The book is fun and clever, stretches your mind, and, I hoped, was a way to show how mathematical thinking can be used in more exciting ways than plugging numbers into a graphing calculator. Mirabelle read the book and, like everyone else, enjoyed it.
But it didn’t convince her to want to do math. “Flatland was really cool!” she said. “But it didn’t require doing any math . . .”
So I went to my last resort. “Do it for me, then,” I said, finally, desperately. “You’ve loved everything else about the IP. You did amazing things in the sciences; you told me you had fallen in love with learning. Your Individual Endeavor is fantastic. You’re loving the novels we’re reading. Do this one bit for me, for the Independent Project, just to prove to all the teachers who said we couldn’t that we can, that we can do math in the IP.”
Well, that worked, sort of. She agreed to keep trying math. But the problem was, she was just doing the same things she had done in traditional school—stuff she hated. The only difference was, rather than being told to do it by a teacher (which hadn’t worked), she now had a reason to force herself to do it. She was doing it because she was invested in the school’s success, because the school was her own, because she was responsible for it, because she cared about her peers and our rise and fall, and because I had asked her to do it. So she was doing math, yeah, but she wasn’t really learning any more about math than she had been before, and she certainly wasn’t learning to like or appreciate it.
I see, in those first few weeks, neat writing in Mirabelle’s journal about math: “Linear equations: things that continue steadily either positively or negatively. Exponential equations: things that . . . ,” and it continues. On another page is a definition for unit circles, and after that a list of symbols (radians, trigonometric functions). You can see she’s copying these from a book, or at best reading a sentence, trying to memorize it, and then writing it down. I guess it was good that she was trying. It was a start. But it wasn’t good enough.
I realized, at some point, that there was no point pushing on if she wasn’t going to be interested. Otherwise I was making the same mistake I was trying to fix. So in a last-ditch effort, I suggested Mirabelle spend the morning trawling through the Internet until she found something math related that interested her. At lunch, she came to me with a printout of a page on biomathematics and elephant movement patterns. Mirabelle loved elephants. I mean, really loved them.
“Well,” she said, “if elephants can’t get me interested in math, nothing can. So I guess I’ll give this a shot.”
The change in her journal is so drastic, it’s like a different person started writing in it. Now the pages are filled top to bottom, with scribblings all over the place. There are probability-density equations tucked into the corners, with arrows pointing away from symbols, and little notes like “how far away from the origin something will be after this amount of time,” with an arrow linking “this” to another symbol, and another arrow from “something” that says, “elephants, people, sugar grains in my coffee.” Then there are cutouts, articles, photocopies from textbooks, calculations, errors crossed out.
She must have come to me a thousand times in those next couple of weeks with questions. She really struggled with the equations, and any calculations still gave her a lot of anxiety. But the difference was, now she was willing to do whatever it took to try to work them out, because she actually wanted to know the answer, and she actually was excited to be using math to learn about elephants.
Mirabelle will never be a mathematician. She’ll never do math in her free time. She says, herself, in an entry in her journal toward the end of the semester, “Now, I suppose it would be a lie to say that I enjoy doing math, but I really have grown to appreciate it and admire those who work in the field. This was a huge leap for me and it was a slow revelation, but I am very glad for it.” She goes on to explain what she thinks was different in the Independent Project that finally made her enjoy math. “Naturally we are all curious and interested in things. If we reverse the process and started with those things that we are naturally drawn to, then apply and learn the methods of math that are needed to further explore those subjects [we’d be much better off].” And finally, she writes, “Even if the actual work I did with numbers was not exhilarating, what I was able to discover about predicting randomness in life was very impressive and really convinced me that without math all the things I care about in this world would either not exist, not be able to be explored as thoroughly, or just remain undiscovered.”
Sam’s last-ditch strategy for luring Mirabelle into math—“Do it for me”—made me cringe a little. Was Sam going to get this pretty girl interested in mathematics by suggesting she do it for him? But then I realized that for as long as I had been in schools, teachers had been telling kids to “do it for them” in one way or another. “Do it because I said so.” “Do it because I’ll be mad if you don’t.” “Do it because I’ll smile at you if you do.” The only difference with Sam and Mirabelle was that, if it worked, she’d be doing it out of friendship. And, I told myself, weren’t there times when I had done something not out of fear or the promise of a reward, but because I wanted to connect to someone I admired or liked? Relationships, after all, are among the greatest motivators.
When I was eleven years old, I fell in love with three things all at once: gymnastics, theater, and my teacher. He was my coach and he directed the plays at our school. He had crinkly eyes, crowded, chipped teeth, and a southern accent.
I had never heard anyone speak to students the way that he did—intense, colorful, and seductive. He told us that people secretly liked the smell of sweat, he quoted Faulkner, and he didn’t hesitate to scorch us with an angry glare when we weren’t 100 percent focused. All of our other classes and projects had the appeal of mold compared to the exultation we felt working with him. I gravitated toward his intensity as if mesmerized, and he knew that.
One day I was in the gym with him, trying to perfect my back handspring. He kept making me go back to the end of the mat, come running toward him, and just before I got to the end, do a roundoff, the move just before you hurdled yourself backward. I had to do it again and again, because, he insisted, I was on the verge of a breakthrough. As he knelt there at the end of the mat, ready to spot me, he kept shouting, “Again. Again.” I must have tried the same move thirty times. I was sick to my stomach with fatigue. I was ecstatic.
Some time ago, Leon Botstein, the president of Bard College, came to the college where I teach to give a talk about how academic institutions should handle sexual relationships between faculty members and students. Many of my colleagues in the audience bristled at what he said. As I recall, he argued that though it was wrong for a professor to have a sexual relationship with a student, it was misguided to try to use the law to prevent such relationships. The thing I remember best was his willingness to say that romance is present when one person is learning from another. He recalled the stories his mother had told him about her girlhood in Switzerland, where she studied piano very seriously with a famous, and famously intense, pianist. She told her son, young Leon, about her piano teacher, “The better I played, the closer he sat.”
As psychologists have come to realize, our emotions are deeply bound up with our thoughts, never more so than when it comes to learning. After all, motivation is the key to learning. We’ve never hesitated to motivate kids with fear of punishment or d
esire for a prize. Why not motivate them with friendship? Once I overcame my first gut reaction and thought it over, “Do it for me” started to seem like a really good idea.
But meanwhile, I had another nagging doubt about Mirabelle. Watching her over the years, I got the feeling that she liked the idea of being one of the thoughtful literary kids more than she liked actually pursuing a thought or mastering a new body of information. She liked thinking about big ideas from a distance, without having to get into the nitty-gritty of working on an idea herself, or conducting a painstaking piece of research. If she loved something, she dove in willingly. But if a topic seemed difficult to her, she circled around it, giving it a wide berth.
Like most kids in our culture, Mirabelle was brought up to believe that there were things she was naturally good at (reading and art) and things that she wasn’t good at (science and math). Watching her thread her way through school, I saw her use a familiar strategy for topics that scared or bored her: she kept them at arm’s length, doing what she needed to in order to get reasonable grades, but avoiding any real contact with the ideas, intellectual procedures, or experiences she might need to get a feel for the discipline. This way of taking a class is sort of like eating something you know is good for you but tastes bad. You try to swallow it without smelling it, and you certainly don’t let it linger on your tongue. And just as the person who eats vegetables as if they are medicine will never become someone who loves healthy food, the student who treats complex or challenging topics as if they were porcupines will never really get anything out of what they are studying. And yet, it’s a common strategy, especially among students like Mirabelle who want to think of themselves as “good” students. The sad part is, the curriculum supports such a tentative and disengaged approach.
It’s not reasonable to expect teenagers to love every subject. After all, we don’t require adults to. Yet students have few options for dealing with a course in which they have no interest. The first, very popular option is to turn their backs on a topic that seems hard or boring. They tell themselves they aren’t good at it, they don’t care about it, it has no meaning in their lives. Kids often accept the fact that they’ll fail or just skate by.
The second option is to figure out how to game the system. Many kids are savvy enough to do just enough of the homework to get an okay grade. They learn whatever procedures allow them to answer the questions (skim the paragraph in order to summarize it, memorize the science they need for the lab report, or learn the steps for conducting a statistical test). Alternatively, they may simply figure out some good test-taking strategies. Adam Gopnik, writing about taking the test for his learner’s permit as an adult, shares the advice his son Luke gave him: Skip the two answers that are obviously wrong and choose between the two plausible ones. Gopnik then zeroes in on the meaning of such a strategy: “The American social truth—that what we spend years teaching our children is essentially to spot the two obviously wrong answers.”
Lots of kids want to do okay in school without having to learn material they don’t like. These kids simply follow some version of Luke’s advice to his dad.
Finally, there are the kids who work hard in a class they don’t really like, just to get a good grade. But just because a student does well in a class does not mean he has learned anything significant or useful.
It seemed to me that a kid like Mirabelle could avoid math, with greater or lesser adroitness, as students have done for decades. Or she could give it a real go. She might not become a math convert. Why should she? But at least she could tangle with it in a deeper, more authentic way than was her custom. Rather than skimming the surface of mathematics to attain the veneer of competence, what Mirabelle needed was to tackle a smaller piece of the discipline and get into the guts of it.
It may seem like a slight achievement: Mirabelle came to appreciate people who do math. But what she gained is more potent than you might think. Her mind won’t shut down the minute people begin to talk about math. Her eyes won’t glaze over the minute she encounters a statistic or graph in a newspaper. She’s likely to approach the world of numbers in a different way. She finished the IP with something all her previous required math courses had not given her: a feel for what it’s like to view the world through a mathematical lens, and a sense of what it takes to think in mathematical terms. She went from cautious bystander to comfortable novice.
No small feat, and much more powerful than a bunch of procedures she’s eager to forget as soon as she can. And all of that in only eight weeks. Imagine if her work on random walks had been only the first step. Imagine if she had had a full year, or four full years, learning math by getting inside of it. Who knows how far she would have gone?
One last story. I mentioned Dominic earlier, and that he was severely dyslexic and failed many of his classes. He came to the Independent Project never having read a novel, at least not in school. So the fact that we were going to be reading a novel every week, unsurprisingly, totally freaked him out.
“I won’t be able to keep up with you guys,” he said grumpily that first day of the languages. Fine, we all told him. We didn’t care, as long as he just read what he could and talked about whatever he had read in our discussions on Fridays.
I had no idea what books we would be reading, because a different person chose each week. Tim went first and chose The House on Mango Street, by Sandra Cisneros, a coming-of-age novel about a Mexican American girl. It’s a book that’s read by adults but also targeted to young adults, so it’s an easy read; perhaps Tim chose it because he was an Asian American son of a first-generation immigrant.
“Yo, I can’t do this,” said Dominic angrily on Tuesday, throwing the book down.
“Come on, Dominic,” said Tim. “I chose this book because I wanted all of you to read it. Don’t bail on me, man.”
“I feel stupid trying to read it. I feel like I did in regular school again.” This was halfway through the Independent Project, and during the sciences—like Mirabelle—Dominic had really come into his own, saying that he liked school for the first time in his life.
“None of us care if you finish it, Dominic,” said Mirabelle. “It doesn’t matter. We all have strengths and weaknesses. Look at how useless I am at math.”
But on Friday, Dominic refused to read aloud the piece of writing he did. He agreed to give it to me and Mr. Huron, as long as we didn’t show it to anyone else. It was only about a paragraph long, and simply described what had happened in the first couple of chapters of the book.
The next week it was John’s turn to choose. I expected something really difficult from him. He was the most literary of all of us and at the time was reading War and Peace for pleasure. But he surprised me by choosing Charlotte’s Web. He later explained to us that he thought E.B. White was America’s greatest author, and he gave a few of us some of his essays to read as well. But at the time, I was just happy that we had another easy book for Dominic to read.
“Fuck this, I’m watching the movie,” Dominic said on Wednesday. “I’m never gonna finish this by Friday.”
Week by week, it seemed, Dominic made it a little further before tossing the book aside angrily. Every Friday he refused to read aloud from his writing. And every Friday he seemed a little surprised that we weren’t mad that he hadn’t finished the book. When I was younger, I used to annoy the hell out of my older brother. It was essentially my purpose in life. So he’d rough me around—throw me off the couch, put me in a headlock, give me a dead arm—and it usually worked, at least in the short term, for getting me to stop. Sometimes, though, I’d do something by accident that I thought might annoy him, like knock over his glass of water. He’d reach out to pick up the water glass and, out of instinct, I’d duck and cower.
Dominic was a bit like that on Fridays. “I didn’t finish the book,” he’d say defiantly, and then brace himself as though expecting us to lash out. “Okay,” we’d say, “fine. Just talk about what you read.” Or, “Whatever, your loss, we’re abo
ut to spoil the end for you.”
Eventually we came to Dominic’s week for choosing a book. “You guys are gonna think it’s stupid,” he said.
“Dominic,” said John, “you gave all of our books a shot. We’ll give your book a shot too, even if it is stupid.”
“It’s got pictures in it,” said Dominic, still sounding like he was waiting for us to turn on him.
The book was Tales of the Weirrd by Ralph Steadman, a British cartoonist. It certainly was strange, and it wasn’t a novel—it was a collection of stories, with illustrations, about extremely eccentric people from the nineteenth century. But we all read it, and we all found it hugely entertaining. “How ’bout the guy who started to grow the beard at age four?” “What about the blind caricaturist? That was so cool!”
Dominic seemed transfixed by all of us talking and debating excitedly over the book he had chosen. On Friday he was laughing, listening, explaining, adding more backstory to the book. He had spent sixteen years unable to keep up with others as they zoomed along through their books, unable to join in or contribute, marginalized and made to feel like an outsider to reading. Now everyone was reading his book, everyone was asking him questions about it, and everyone was grateful to him for having chosen such a cool thing for us to read.
The following week was Dakota’s turn to choose a book. She was definitely the brainiest of the group. So I wasn’t surprised when she chose As I Lay Dying, by William Faulkner. Well, I thought, if Dominic hadn’t finished Charlotte’s Web, he wouldn’t make it three pages into Faulkner.
