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It is math time for the fifth-graders at Fernangeles Elementary School in Sun Valley, Calif., and everyone is stumped. The students have spent close to an hour puzzling over the question at hand: "What if everybody here had to shake hands with everyone else? How many handshakes would that take?" While the children, seated in small groups, debate and frown and scribble notes--and devise alternatives to the dread act of actually touching classmates of the opposite sex--their teacher, Kathy Pullman, roams the room. When the hour ends, no group has an answer. "This happens to be a particularly difficult lesson," Pullman says, unfazed. The class will dive in again after lunch.

To peer into Pullman's classroom is to glimpse why math has once again become a battleground in America's education wars. This school year, nearly half of all American elementary students are expected to learn math the way children do at Fernangeles Elementary: not in neat rows of desks, repeating times tables and memorizing theorems, but through trial-and-error problem solving, often in groups with little direct instruction and almost always with a calculator nearby. Advocates call it "interactive" or "inventive" math and insist that it sets American schoolchildren on the way to becoming "mathematically powerful."

When opponents of these currents are being polite, they call that kind of talk nonsense. And they have labels of their own. One is "whole math," a pejorative reference to "whole language," the controversial method of reading that emphasizes learning entire words and phrases over mastering phonics. Another is "new-new math," recalling the ill-fated New Math fad of the 1960s and '70s, which introduced millions of students to math arcana like set theory and congruences. (Remember those? Didn't think so.)

Resistance has resulted in pitched battles at school-board hearings and within academia over the future of U.S. math education. Educational scholar E.D. Hirsch Jr. says opposition to the newest math is "suddenly making people wake up and say, It doesn't work, it doesn't comport with reliable theories, and we're making a mistake."

It all started in 1989, when the National Council of Teachers of Mathematics, in response to the consistently poor math scores of U.S. children, issued new standards overhauling math education. Out went the stalwarts of traditional math: the rote memorization drills, the droning chalkboard lectures. In came the cool stuff: calculators and geoboards, hands-on, open-ended problems, exercises that encourage kids to discover their own route to the right answer. "The standards emphasized that you had to pay attention to how kids think," says Gail Burrill, president of the council.

Eight years later, 40 states have instituted some "standards-based" math programs in their schools, and the National Science Foundation spends $10 million a year on development of comprehensive instructional materials. California has become a leader: one cutting-edge curriculum called MathLand is used, its publishers claim, in 60% of the state's K-6 classrooms--including those in Fernangeles, where MathLand is taught in both English and Spanish. It even has its own Website: "These kids will be better problem solvers," says Pullman. "They will think more."

But talk to opponents--including newly aroused parents--and you'll hear horror stories of reformers dumping the most basic algorithms, or first-graders turning to a calculator to subtract 4 from 6, or "math" textbooks featuring lessons on endangered species and the Dogon people of West Africa. "Whole math," says molecular biologist Michael McKeown, "means less material covered in less depth with less rigor." Early last year, McKeown co-founded Mathematically Correct, a Website based in San Diego that has become the nerve center for the counterinsurgency. Here parents like Marianne Jennings of Mesa, Ariz., share dispatches from the front. Not long ago, Jennings watched her daughter Sarah, a straight-A algebra student, reach for a calculator to find 10% of 470. "It made my blood boil," says Jennings. In response, she and other parents pressured the school district to offer traditional math as a choice.

Similar small-scale revolts have erupted from San Diego to Sonoma Valley, and the state board of education is expected soon to release a new, more traditional math framework for California schools. Meanwhile, McKeown and whole-math detractors like Lynne Cheney, a senior fellow at the American Enterprise Institute, darkly warn that President Clinton's voluntary national test for eighth-graders, set for 1999, is being used to promote a whole-math agenda. Says Cheney: "Every single member appointed to the math [exam's] panel is a whole-math advocate." Department of Education officials bristle at the charge, saying the exam will test both fundamental math skills and high-order problem solving.

Is whole math really that bad? Few question the reformers' motives: to inject some much needed juice into American math education. But though the approach can certainly stimulate kids, it can just as easily leave them adrift. While the fifth-graders at Fernangeles were mulling over their handshake problem, a nearby fourth-grade class was fiddling with green and red tiles, trying to figure out how many green gates fitted into how many sections of red fence. "If there were 48 green doors," the teacher implored, "how many red fences do we need?" After a few minutes, one student incorrectly ventured, "Eighty." A few others tapped their calculators, with little success, and after 20 minutes, the solution seemed no nearer.

Of course, the most convincing defense of whole math would be evidence that it works. In a few states that have emphasized new-new math, such as Connecticut, there are early indicators of improved student performance. Critics in California, on the other hand, point to test scores in cities like Santa Barbara and Palo Alto that show at least temporary drop-offs after whole math has been introduced. One thing's certain: lukewarm results won't cut it. In the most recent worldwide comparison, the Third International Mathematics and Science Study, U.S. eighth-graders fell below the international average and miles behind Singapore, Korea and Japan.

Be patient, advocates argue. Says Thomas Romberg, a University of Wisconsin professor who helped write the revolutionary 1989 math standards: "We knew there needed to be a fair amount of research and teacher training. We knew it would take 20 or 25 years to pull this off." Parents whose 12-year-olds still can't count on their fingers may not want to wait that long.