These pieces originally appeared as a weekly column entitled “Lessons” in *The New York Times* between 1999 and 2003.

[THIS ARTICLE FIRST APPEARED IN *THE NEW YORK TIMES* ON APRIL 12, 2000]

## A Teacher in the Trenches of the Nation’s Math Wars

In 1982, the nation’s 13-year-olds were tested in math. One question was: If a bus holds 36 soldiers, how many buses do 1,128 soldiers need? Seventy percent did the long division correctly, yet few got the right answer: 32 buses. More common was “31, remainder 12.”

Problems like this have been at the core of our “math wars,” an ongoing debate about how to teach. “Back-to-basics” advocates say we don’t drill students enough. In effect, they focus on the 30 percent who could not divide 1,128 by 36. But their opponents worry more about children who can’t picture that the “remainder 12″ requires a whole bus with 24 empty seats.

Math curriculum issues are hard to settle because so many of us fear the subject. Well-educated adults joke about never having been good at math when estimating a restaurant tip but would never brag that they couldn’t read. Many who demand that schools drill students more are themselves examples of how that method hasn’t worked.

In 1989, the National Council of Teachers of Mathematics proposed a curriculum that not only played down rote practice but also left an impression that basic skills could be ignored. This guaranteed a back-to-basics reaction. A new set of standards being released today at the council’s meeting in Chicago is more balanced, but unlikely to end the math wars.

The council members insist that there is no single way to solve math problems, and that students should experiment with alternatives. This infuriates traditionalists; they want the “right way” memorized, and children tested on it.

In Massachusetts, Nancy Buell, a fourth-grade teacher, recently circulated a petition among the state’s math teachers protesting renewed emphasis on drills. She claims that such drills won’t work well for any of her suburban Brookline students — mostly middle class, but about a third low-income, bused from Boston or living in public housing.

I visited Ms. Buell’s classroom recently. She spent several days exploring a single division problem with her fourth-graders: if juice cans come in cases of 24, how many cases are needed to give a can to each of 250 children?

She told her students first to write the question as a “number sentence” (250 ÷ 24 =) and then discuss it in groups. Traditionalists don’t see why this should be a subject of discussion — it’s a straightforward long division problem. But in classes like Ms. Buell’s, some students solve it by addition, adding 24’s until they get to 250. Others subtract, taking 24’s from 250 to see how many they need to reach zero. A few see it as multiplication: 24 times what equals 250?

In one group, a girl reasoned that there were four 25’s in 100, so 250 students would include two of these hundreds, and then another half a hundred students. If a case had 25 cans, she said, exactly 4 plus 4 plus 2 cases (a total of 10) would be needed. But with only 24 in a case, each would be short a can, and so 10 cans must come from an 11th case.

For another group, Ms. Buell drew a rectangle and divided it into 24 sections using parallel vertical lines. How many horizontal lines would be needed to make at least 250 squares? Children could count squares for the solution.

Another group said the answer was 11 cases, 14 remaining. Ms. Buell asked whether the “remainder” was kids or cans. While the word-problem answer could be either 11 cases with 14 cans left over, or 10 cases with 10 cans still needed, only “10, remainder 10″ correctly solves the number sentence, she showed.

These students may come to understand math better than most adults, but when they reach adulthood, will they be more comfortable than their parents calculating restaurant tips? Few states have retained reform curriculums long enough to judge their effectiveness, but Nancy Buell insists that if students are taught properly, math proficiency will flow easily from their experiences. “Trust me,” she said, “kindergartners can split up a plate full of cookies fairly!”

Some new evidence comes from Professor James Stigler of the University of California and Professor James Hiebert of the University of Delaware, who videotaped American and Japanese classes. From their book, “The Teaching Gap” (Free Press, 1999), it seems that math instruction in Japan, which ranks high on international exams, is similar to Ms. Buell’s methods.

But Ms. Buell is unusual in this country, and odds are against American schools copying her techniques. It takes more training than most of our teachers have, and more time for planning than our schools allow. In the hands of poorly trained teachers, nontraditional methods can cause children to get neither basics nor understanding. Japanese teachers spend less time teaching and more time meeting to compare strategies.

There are certainly faster ways than Nancy Buell’s to teach the answer to 250 ÷ 24. But in learning math, a straight line may not always be the best path between two points.