Smaller classes don't necessarily equal better education. Do the math!

("Why Smaller Classes..." would be a more accurate headline.)

The New York Times, "Economic Scene" , February 22, 2001

President Bush has been on the road this week, promoting the education package that is a centerpiece of his policy agenda. His plan, whose broad outlines enjoy bipartisan support, ties federal funds to minimum performance standards on regular tests. But it also gives states a lot of flexibility in how they set and achieve their educational goals.

One specific policy the package doesn't include: lowering class sizes across the board. That idea was dear to the Clinton administration and has been popular with many governors, most notably former Gov. Pete Wilson of California, a Republican. In 1996, the state used its budget surplus to reduce the size of classes in kindergarten through third grade.

Despite the intuitive appeal of smaller classes, empirical economists have had a hard time finding any evidence that cutting class sizes increases learning. This counterintuitive result is a puzzle that requires more than data analysis. It needs theory.

Enter Edward Lazear, an economist at the Stanford Graduate School of Business and the Hoover Institution. Struck by both the puzzle and the field's lack of theoretical underpinnings, he set out to develop a framework that would help education researchers understand the relation between class size and learning. While such a framework is always more stylized than reality, a useful theory provides predictions that can be tested empirically. And it can help explain seeming anomalies.

In a paper presented at the January meetings of the American Economic Association and coming in The Quarterly Journal of Economics, Professor Lazear started with a real-world insight that his empirically minded colleagues had generally overlooked. "If you talk to elementary- school teachers," he says, "they will tell you that their No. 1 issue is managing the class." Student behavior is critical.

One can think of classroom learning as a public good with "congestion effects," he suggested. The behavior of each student affects every other student's learning, and the more crowded the classroom the more spillovers there are. Professor Lazear developed a mathematical model to express the relation among class size, behavior, learning and the cost of teachers. The model predicts what might happen when various factors change.

The key variable is how likely it is that students disrupt one another's learning. (This disruption can occur through classic misbehavior, or it can simply take the form of asking questions everyone else knows the answers to.) Small differences in behavior can have large effects on learning and big implications for class size.

To find out how much of the time learning is actually taking place in a given class, you multiply the probability a student is not disrupting by the probability for each other student. In his model, Professor Lazear uses the same probability for every student, which means he can simply raise the probability to the power of whatever the class size is.

The results are striking. If each student behaves well 99 percent of the time, learning takes place 78 percent of the time in a class of 25; if good behavior drops to 98 percent, learning takes place only 60 percent of the time; at 97 percent, learning drops to a mere 47 percent of the time.

Reducing class size makes a much bigger difference for disruptive students than for nondisruptive ones. In the example above, cutting the class size to 20 means the best-behaved students are learning 82 percent of the time, the middle ones learn 67 percent of the time, and the most disruptive now learn 54 percent of the time.

What does this have to do with the class-size puzzle?

First, this model suggests that across-the-board reductions like California's are a foolish use of limited resources. Such reforms spend a lot of money to cut the size of classes where behavior is already good and smaller classes won't make that much difference. Adding teachers and classroom space takes resources that could be used more effectively elsewhere.

"I would prefer to see a much more focused effort on reducing class size for kids that have special needs, or disadvantaged areas," Professor Lazear said. For schools where disruption isn't a problem, it would be better to improve facilities or raise teachers' salaries to attract a better pool of applicants. (This assumes that such salary increases would lead to new hires, not just give a windfall to existing teachers.)

Second, the model explains the sorting that already takes place. Young children are in small classes because they have shorter attention spans than do high school students. Special education classes are small for the same reason. Catholic schools may have big classes, but they attract students who aren't disruptive. (In an extension of his model, Professor Lazear looks at how self-selection affects the makeup of private schools.)

Finally, it's possible to reduce the size of a class, increase student learning and still look bad compared with larger classes. This happens if, as the model suggests they should, schools put disruptive students in smaller classes. The students will learn more than they would in a larger class. But at the optimal class size (which depends on the cost of teachers), they'll still learn less than better students in a big class. Special education students, for instance, tend to be in very small classes. But they won't score as well on tests as above average students in large classes.

"The puzzle is reconciled immediately once you think about it this way," says Professor Lazear. Schools do adjust class size to their students' characteristics. "But," he adds, "that optimal adjustment doesn't reduce class size enough to counteract the fact that the kids who are in the smaller classes are not the better kids. And that's the key thing. You're reducing the class size, but not enough to offset that factor. That's what the model shows, and that's what seems to be true in the real world."

Different children and different school settings have different needs. When it comes to classroom learning, one size does not fit all.