Notes from David and Goliath

As human beings we are hardwired to seek the approval of those around us. Yet a radical and transformative thought goes nowhere without the willingness to challenge convention. “If you have a new idea, and it’s disruptive and you’re agreeable, then what are you going to do with that?” says Peterson. “If you worry about hurting people’s feelings and disturbing the social structure, you’re not going to put your ideas forward.” As the playwright George Bernard Shaw once put it: “The reasonable man adapts himself to the world: the unreasonable one persists in trying to adapt the world to himself. Therefore all progress depends on the unreasonable man.”

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Courage is not something that you already have that makes you brave when the tough times start. Courage is what you earn when you’ve been through the tough times and you discover they aren’t so tough after all. Do you see the catastrophic error that the Germans made? They bombed London because they thought that the trauma associated with the Blitz would destroy the courage of the British people. In fact, it did the opposite. It created a city of remote misses, who were more courageous than they had ever been before. The Germans would have been better off not bombing London at all.

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When people in authority want the rest of us to behave, it matters—first and foremost—how they behave.

This is called the principle of legitimacy, and legitimacy is based on three things. First of all, the people who are asked to obey authority have to feel like they have a voice—that if they speak up, they will be heard. Second, the law has to be predictable. There has to be a reasonable expectation that the rules tomorrow are going to be roughly the same as the rules today. And third, the authority has to be fair. It can’t treat one group differently from another.

Big Fish-Little Pond Effect

We compare ourselves to those in the same situation as ourselves, which means that students in an elite school—except, perhaps, those at the very top of the class—are going to face a burden that they would not face in a less competitive atmosphere. Citizens of happy countries have higher suicide rates than citizens of unhappy countries, because they look at the smiling faces around them and the contrast is too great. Students at “great” schools look at the brilliant students around them, and how do you think they feel?

The phenomenon of relative deprivation applied to education is called—appropriately enough—the “Big Fish–Little Pond Effect.” The more elite an educational institution is, the worse students feel about their own academic abilities. Students who would be at the top of their class at a good school can easily fall to the bottom of a really good school. Students who would feel that they have mastered a subject at a good school can have the feeling that they are falling farther and farther behind in a really good school. And that feeling—as subjective and ridiculous and irrational as it may be—matters. How you feel about your abilities—your academic “self-concept”—in the context of your classroom shapes your willingness to tackle challenges and finish difficult tasks. It’s a crucial element in your motivation and confidence.

The Big Fish–Little Pond theory was pioneered by the psychologist Herbert Marsh, and to Marsh, most parents and students make their school choices for the wrong reasons. “A lot of people think that going to an academically selective school is going to be good,” he said. “That’s just not true. The reality is that it is going to be mixed.

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So what happened to Stephen Randolph at Harvard? I think you can guess the answer. In his third year, he took quantum mechanics. “I didn’t do well,” he admitted. “I think I might have gotten a B-minus.” It was the lowest grade he’d ever received. “My perception was that either I wasn’t good at it or I wasn’t good enough at it. Maybe I felt that I had to be the best at it or be a genius at it for it to make sense for me to continue. Some people seemed to get it more quickly than I did—and you tend to focus on those people and not the ones who are just as lost as you are.

“I was excited by the material,” he continued. “But I was humbled by the experience—humbled as in, you sit in the class and you don’t understand and you feel like, ‘I will never be able to understand this!’ And you do problem sets and you understand a little bit of this and a little bit of that, but you always think that the other people in your class understand it a lot better. I think one of the things about Harvard is that there’s just so many smart people there that it’s hard to feel smart there.” He decided he couldn’t go on.

Inverted-U Curves and Class Size

That’s what is called an inverted-U curve. Inverted-U curves are hard to understand. They almost never fail to take us by surprise, and one of the reasons we are so often confused about advantages and disadvantages is that we forget when we are operating in a U-shaped world.

Which brings us back to the puzzle of class size: What if the relationship between the number of children in a classroom and academic performance is not this:

or even this:

What if it’s this?

The principal of Shepaug Valley Middle School is a woman named Teresa DeBrito. In her five-year tenure at the school, she has watched the incoming class dwindle year by year. To a parent, that might seem like good news. But when she thought about it, she had that last curve in mind. “In a few years we’re going to have fewer than fifty kids for the whole grade coming up from elementary school,” she said. She was dreading it: “We’re going to struggle.”

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Inverted-U curves have three parts, and each part follows a different logic. There’s the left side, where doing more or having more makes things better. There’s the flat middle, where doing more doesn’t make much of a difference. And there’s the right side, where doing more or having more makes things worse.

If you think about the class-size puzzle this way, then what seems baffling starts to make a little more sense. The number of students in a class is like the amount of money a parent has. It all depends on where you are on the curve. Israel, for example, has historically had quite large elementary school classes. The country’s educational system uses the “Maimonides Rule,” named after the twelfth-century rabbi who decreed that classes should not exceed forty children. That means elementary school classes can often have as many as thirty-eight or thirty-nine students. Where there are forty students in a grade, though, the same school could suddenly have two classes of twenty. If you do a Hoxby-style analysis and compare the academic performance of one of those big classes with a class of twenty, the small class will do better. That shouldn’t be surprising. Thirty-six or thirty-seven students is a lot for any teacher to handle. Israel is on the left side of the inverted-U curve.

Now think back to Connecticut. In the schools Hoxby looked at, most of the variation was between class sizes in the mid- to low twenties and those in the high teens. When Hoxby says that her study found nothing, what she means is that she could find no real benefit to making classes smaller in that medium range. Somewhere between Israel and Connecticut, in other words, the effects of class size move along the curve to the flat middle—where adding resources to the classroom stops translating into a better experience for children.

Why isn’t there much of a difference between a class of twenty-five students and a class of eighteen students? There’s no question that the latter is easier for the teacher: fewer papers to grade, fewer children to know and follow. But a smaller classroom translates to a better outcome only if teachers change their teaching style when given a lower workload. And what the evidence suggests is that in this midrange, teachers don’t necessarily do that. They just work less. This is only human nature. Imagine that you are a doctor and you suddenly learn that you’ll see twenty patients on a Friday afternoon instead of twenty-five, while getting paid the same. Would you respond by spending more time with each patient? Or would you simply leave at six-thirty instead of seven-thirty and have dinner with your kids?

Now for the crucial question. Can a class be too small, the same way a parent can make too much money? I polled a large number of teachers in the United States and Canada and asked them that question, and teacher after teacher agreed that it can.

Here’s a typical response:

My perfect number is eighteen: that’s enough bodies in the room that no one person needs to feel vulnerable, but everyone can feel important. Eighteen divides handily into groups of two or three or six—all varying degrees of intimacy in and of themselves. With eighteen students, I can always get to each one of them when I need to. Twenty-four is my second favorite number—the extra six bodies make it even more likely that there will be a dissident among them, a rebel or two to challenge the status quo. But the trade-off with twenty-four is that it verges on having the energetic mass of an audience instead of a team. Add six more of them to hit thirty bodies and we’ve weakened the energetic connections so far that even the most charismatic of teachers can’t maintain the magic all the time.

And what about the other direction? Drop down six from the perfect eighteen bodies and we have the Last Supper. And that’s the problem. Twelve is small enough to fit around the holiday dinner table—too intimate for many high schoolers to protect their autonomy on the days they need to, and too easily dominated by the bombast or bully, either of whom could be the teacher herself. By the time we shrink to six bodies, there is no place to hide at all, and not enough diversity in thought and experience to add the richness that can come from numbers.

The small class is, in other words, potentially as difficult for a teacher to manage as the very large class. In one case, the problem is the number of potential interactions to manage. In the other case, it is the intensity of the potential interactions. As another teacher memorably put it, when a class gets too small, the students start acting “like siblings in the backseat of a car. There is simply no way for the cantankerous kids to get away from one another.”

Here’s another comment from a high school teacher. He had recently had a class of thirty-two and hated it. “When I face a class that large, the first thought that I have is ‘Damn it, every time I collect something to mark, I am going to spend hours of time here at the school when I could be with my own kids.’” But he didn’t want to teach a class of fewer than twenty either:

The life source of any class is discussion, and that tends to need a certain critical mass to get going. I teach classes right now with students who simply don’t discuss anything, and it is brutal at times. If the numbers get too low, discussion suffers. That seems counterintuitive because I would think that the quiet kids who would hesitate to speak in a class of thirty-two would do so more readily in a class of sixteen. But that hasn’t really been my experience. The quiet ones tend to be quiet regardless. And if the class is too small, among the speakers, you don’t have enough breadth of opinion perhaps to get things really going. There is also something hard to pin down about energy level. A very small group tends to lack the sort of energy that comes from the friction between people.

And a really, really small class? Beware.

I had a class of nine students in grade-twelve Academic French. Sounds like a dream, doesn’t it? It was a nightmare! You can’t get any kind of conversation or discussion going in the target language. It’s difficult to play games to reinforce vocabulary, grammar skills, et cetera. The momentum just isn’t there.

The economist Jesse Levin has done some fascinating work along these same lines, looking at Dutch schoolchildren. He counted how many peers children had in their class—that is, students at a similar level of academic ability—and found that the number of peers had a surprising correlation with academic performance, particularly for struggling students. In other words, if you are a student—particularly a poor student—what you need is to have people around you asking the same questions, wrestling with the same issues, and worrying about the same things as you are, so that you feel a little less isolated and a little more normal.

This is the problem with really small classes, Levin argues. When there are too few students in a room, the chances that children are surrounded by a critical mass of other people like them start to get really low. Taken too far, Levin says, class-size reduction “steals away the peers that struggling students learn from.”

Can you see why Teresa DeBrito was so worried about Shepaug Valley? She is the principal of a middle school, teaching children at precisely the age when they begin to make the difficult transition to adolescence. They are awkward and self-conscious and anxious about  seeming too smart. Getting them to engage, to move beyond simple question-and-answer sessions with their teacher, she said, can be “like pulling teeth.” She wanted lots of interesting and diverse voices in her classrooms, and the kind of excitement that comes from a critical mass of students grappling with the same problem. How do you do that in a half-empty room? “The more students you have,” she continued, “the more variety you can have in those discussions. If it’s too small with kids this age, it’s like they have a muzzle on.” She didn’t say it, but you could imagine her thinking that if someone went and built a massive subdivision on the gently rolling meadow next to the school, she wouldn’t be that unhappy.

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A half-hour drive up the road from Shepaug Valley, in the town of Lakeville, Connecticut, is a school called Hotchkiss. It is considered one of the premier private boarding schools in the United States. Tuition is almost $50,000 a year. The school has two lakes, two hockey rinks, four telescopes, a golf course, and twelve pianos. And not just any pianos, but, as the school takes pains to point out, Steinway pianos, the most prestigious piano money can buy.6 Hotchkiss is the kind of place that spares no expense in the education of its students. The school’s average class size? Twelve students. The same condition that Teresa DeBrito dreads, Hotchkiss—just up the road—advertises as its greatest asset. “[Our] learning environment,” the school proudly declares, “is intimate, interactive, and inclusive.”

Why does a school like Hotchkiss do something that so plainly makes its students worse off? One answer is that the school isn’t thinking of its students. It is thinking of the parents of its students, who see things like golf courses and Steinway pianos and small classes as evidence that their $50,000 is well spent. But the better answer is that Hotchkiss has simply fallen into the trap that wealthy people and wealthy institutions and wealthy countries—all Goliaths—too often fall into: the school assumes that the kinds of things that wealth can buy always translate into real-world advantages. They don’t, of course. That’s the lesson of the inverted-U curve. It is good to be bigger and stronger than your opponent. It is not so good to be so big and strong that you are a sitting duck for a rock fired at 150 miles per hour. Goliath didn’t get what he wanted, because he was too big. The man from Hollywood was not the parent he wanted to be, because he was too rich. Hotchkiss is not the school it wants to be, because its classes are too small. We all assume that being bigger and stronger and richer is always in our best interest.

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The definitive analysis of the many hundreds of class-size studies was done by the educational economist Eric Hanushek, The Evidence on Class Size. Hanushek says, “Probably no aspect of schools has been studied as much as class size. This work has been going on for years, and there is no reason to believe that there is any consistent relationship with achievement.”

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The logic of the inverted-U curve is that the same strategies that work really well at first stop working past a certain point, and that’s exactly what many criminologists argue happens with punishment.

What is an Advantage?

Suppose you were to total up all the wars over the past two hundred years that occurred between very large and very small countries. Let’s say that one side has to be at least ten times larger in population and armed might than the other. How often do you think the bigger side wins? Most of us, I think, would put that number at close to 100 percent. A tenfold difference is a lot. But the actual answer may surprise you. When the political scientist Ivan Arreguín-Toft did the calculation a few years ago, what he came up with was 71.5 percent. Just under a third of the time, the weaker country wins.

Arreguín-Toft then asked the question slightly differently. What happens in wars between the strong and the weak when the weak side does as David did and refuses to fight the way the bigger side wants to fight, using unconventional or guerrilla tactics? The answer: in those cases, the weaker party’s winning percentage climbs from 28.5 percent to 63.6 percent. To put that in perspective, the United States’ population is ten times the size of Canada’s. If the two countries went to war and Canada chose to fight unconventionally, history would suggest that you ought to put your money on Canada.

We think of underdog victories as improbable events: that’s why the story of David and Goliath has resonated so strongly all these years. But Arreguín-Toft’s point is that they aren’t at all. Underdogs win all the time. Why, then, are we so shocked every time a David beats a Goliath? Why do we automatically assume that someone who is smaller or poorer or less skilled is necessarily at a disadvantage?

One of the winning underdogs on Arreguín-Toft’s list, for example, was T. E. Lawrence (or, as he is better known, Lawrence of Arabia),  who led the Arab revolt against the Turkish army occupying Arabia near the end of the First World War.

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For some reason, this is a very difficult lesson for us to learn. We have, I think, a very rigid and limited definition of what an advantage is. We think of things as helpful that actually aren’t and think of other things as unhelpful that in reality leave us stronger and wiser. Part One of David and Goliath is an attempt to explore the consequences of that error. When we see the giant, why do we automatically assume the battle is his for the winning? And what does it take to be that person who doesn’t accept the conventional order of things as a given—like David, or Lawrence of Arabia?

David and Goliath

At the heart of ancient Palestine is the region known as the Shephelah, a series of ridges and valleys connecting the Judaean Mountains to the east with the wide, flat expanse of the Mediterranean plain. It is an area of breathtaking beauty, home to vineyards and wheat fields and forests of sycamore and terebinth. It is also of great strategic importance.

Over the centuries, numerous battles have been fought for control of the region because the valleys rising from the Mediterranean plain offer those on the coast a clear path to the cities of Hebron, Bethlehem, and Jerusalem in the Judaean highlands. The most important valley is Aijalon, in the north. But the most storied is the Elah. The Elah was where Saladin faced off against the Knights of the Crusades in the twelfth century. It played a central role in the Maccabean wars with Syria more than a thousand years before that, and, most famously, during the days of the Old Testament, it was where the fledgling Kingdom of Israel squared off against the armies of the Philistines.

The Philistines were from Crete. They were a seafaring people who had moved to Palestine and settled along the coast. The Israelites were clustered in the mountains, under the leadership of King Saul. In the second half of the eleventh century BCE, the Philistines began moving east, winding their way upstream along the floor of the Elah Valley. Their goal was to capture the mountain ridge near Bethlehem and split Saul’s kingdom in two. The Philistines were battle-tested and dangerous, and the sworn enemies of the Israelites. Alarmed, Saul gathered his men and hastened down from the mountains to confront them.

The Philistines set up camp along the southern ridge of the Elah. The Israelites pitched their tents on the other side, along the northern ridge, which left the two armies looking across the ravine at each other. Neither dared to move. To attack meant descending down the hill and then making a suicidal climb up the enemy’s ridge on the other side. Finally, the Philistines had enough. They sent their greatest warrior down into the valley to resolve the deadlock one on one.

He was a giant, six foot nine at least, wearing a bronze helmet and full body armor. He carried a javelin, a spear, and a sword. An attendant preceded him, carrying a large shield. The giant faced the Israelites and shouted out: “Choose you a man and let him come down to me! If he prevail in battle against me and strike me down, we shall be slaves to you. But if I prevail and strike him down, you will be slaves to us and serve us.”

In the Israelite camp, no one moved. Who could win against such a terrifying opponent? Then, a shepherd boy who had come down from Bethlehem to bring food to his brothers stepped forward and volunteered. Saul objected: “You cannot go against this Philistine to do battle with him, for you are a lad and he is a man of war from his youth.” But the shepherd was adamant. He had faced more ferocious opponents than this, he argued. “When the lion or the bear would come and carry off a sheep from the herd,” he told Saul,“I would go after him and strike him down and rescue it from his clutches.” Saul had no other options. He relented, and the shepherd boy ran down the hill toward the giant standing in the valley. “Come to me, that I may give your flesh to the birds of the heavens and the beasts of the field,” the giant cried out when he saw his opponent approach. Thus began one of history’s most famous battles. The giant’s name was Goliath. The shepherd boy’s name was David.

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David and Goliath is a book about what happens when ordinary people confront giants. By “giants,” I mean powerful opponents of all kinds—from armies and mighty warriors to disability, misfortune, and oppression.

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Saul tries to give him his own sword and armor so at least he’ll have a fighting chance. David refuses. “I cannot walk in these,” he says, “for I am unused to it.” Instead he reaches down and picks up five smooth stones, and puts them in his shoulder bag. Then he descends into the valley, carrying his shepherd’s staff. Goliath looks at the boy coming toward him and is insulted. He was expecting to do battle with a seasoned warrior. Instead he sees a shepherd—a boy from one of the lowliest of all professions—who seems to want to use his shepherd’s staff as a cudgel against Goliath’s sword. “Am I a dog,” Goliath says, gesturing at the staff, “that you should come to me with sticks?”

What happens next is a matter of legend. David puts one of his stones into the leather pouch of a sling, and he fires at Goliath’s exposed forehead. Goliath falls, stunned. David runs toward him, seizes the giant’s sword, and cuts off his head. “The Philistines saw that their warrior was dead,” the biblical account reads, “and they fled.”

The battle is won miraculously by an underdog who, by all expectations, should not have won at all. This is the way we have told one another the story over the many centuries since. It is how the phrase “David and Goliath” has come to be embedded in our language—as a metaphor for improbable victory. And the problem with that version of the events is that almost everything about it is wrong.