He wouldn't know what you were talking about, unless he had happened upon the Advanced Placement BC calculus class that Sue Steckel taught last school year. Even so, he would have had to decipher, through the students' laughter, the nicknames they invented for the symbols and concepts of calculus.

Giggling, covering her face as if to hide from a heresy, Ms. Steckel recalled: "I tried to tell them over and over, `If you talk like this to another mathematician, he's going to laugh at you.' They'd say: `We don't care, it's our thing. It's our in-joke.' "

Those insider names for the indefinite integral (which looks like a hook) and the factorial symbol (a bang-like exclamation point) were part of a distinctive strategy students developed to handle one of the most difficult high school courses: Advanced Placement BC Calculus, the harder of two advanced placement calculus courses. In the rare schools that offer it, it is avoided by all but the most fearless in math.

Yet this past year, at Manhattan's Friends Seminary, a private high school on East 16th Street, a group of four juniors and six seniors transformed themselves into a team whose closeness became the key to their success. They were not math nerds, their teacher insisted. There was an actor, a musician, a budding fashion designer. Not all were straight-A students.

But, displaying teamwork and camaraderie unusual even in this close-knit Quaker-run school, the class wrote calculus songs, acted out calculus game shows, exchanged nightly phone calls and e-mail messages, and cheered one another on so heartily that failure became impossible.

At year's end, not only did all of them pass the nationally administered advanced placement exam, but all earned 5's, the highest score possible on the exam. It was the first time in the school's history that an entire class of so many students had received 5's. Last year, only 38 percent of 30,000 students in the nation who took the test earned a 5. "It's really extraordinary" for an entire class to do so, said David Arnold, head of the upper classes at the school. "I've never seen anything like it."

each other and feed off one another."

"Serious kidding around" is another key to success, Mr. Sizer added, noting that "when things are fun, you pay attention."

How do the students explain working so well together, eluding both the detachment and the competitiveness that can plague high school classrooms? Caleb Rabinowitz, a student who played a goat in one of the class's calculus skits, said, "We weren't stabbing each other in the back trying to get 100's."

Trying to describe the calculus song he composed, Homer Steinweiss joked about its content. "Captain Hoo-oo-ook, Captain Hoo-oo-ook," he sang, trailing off. "It was meditative, like a mantra," he said.

Even errors were respected. "No one was going around making stupid mistakes," said Jenny Rothchild, one of two girls in the class. "When somebody made a mistake, everyone was like, `Oh, what a good mistake.' And when we finished a really hard problem, Daniel Terry would go, `That was good, we should clap for that problem.'"

While Friends has long offered Calculus AB, the easier Advanced Placement course covering one college semester, last year was the first time it offered BC, the more difficult two-semester equivalent.

At the outset, Ms. Steckel had no idea how the BC class would evolve. "I told them at the beginning of the year, `I've never taught this class before,' " she said. "They felt chosen; they felt special. I did play that up: `You guys are the hardest math class we've ever taught.' You could almost see it in their faces: `O.K., we're going to get through this.'"

Yet the students were hardly grim in their determination. "I didn't feel like it was a thing we had to go through," Mr. Rabinowitz said. "We all signed up, knowing the dangers - like a group of explorers trekking through the Arctic."

Like an expedition, the class adhered to a strict schedule. "If you didn't learn this chapter one night, you'd have to do two chapters the next night," Miss Rothchild said. "You'd blow off another class's work, but there was no question of blowing off calculus."

No one remembers whose idea it was. but early in the year, without prompting, the class split into three study groups. Students said that greatly eased the pressure, hanging successes and failures on the teams rather than on individuals.

The nicknames were also spontaneous. Looking back, Ms. Steckel said, it made sense that the students would attach affectionate terms to math concepts. "All that is a way of making light of something that's really hard," she said. "It's giving them ownership of it."

run the class. Trust the kid, and the kid is going to take more initiative."

Thirty years old, her long hair clipped loosely back, the informal, energetic Ms. Steckel looks almost like a high school student herself. In fact, when she started teaching at Friends 10 years ago, she was only a couple of years older than her students. When she applied at Friends, she displayed an enthusiasm for a subject many teenagers, especially girls, try to avoid.

"When they said, `Why should we hire you right out of school?' I said, `Because I think calculus is the neatest thing I have ever seen,' " she said. "I think it is so cool, I talk about it at parties. I think it is like magic."

That attitude played well with her students, as did Ms. Steckel's use of material from eight different textbooks and her frequent consultations with teachers at other schools.

But she gave a lot of credit to the students for the success of the course. She said they found a real pleasure in being together, and funneled it into their work. Friends from outside the class looked at them blankly when they told calculus jokes, but they didn't care, she said. "Some of it's about being willing to take a risk," she said.

Mr. Arnold said that while the results were exceptional, the class room dynamics did not surprise him in a school that emphasizes Quaker values of noncompetitiveness and openness and is regarded as among Manhattan's dozen finest private schools.

In fact, during the national exam which is administered by the Educational Testing Service, one student burst out laughing when a problem reminded him of a class joke. And meeting together three months later students and teacher still chuckled about obscure references to "Jenny's closet of numbers" (a term they invented to help visualize different kinds of infinities) or "an ant of negligible height" (which they used when describing an angle of elevation viewed from ground level).

Some of the students will go on to college math courses, and those who are seniors this year will take a class in math at New York University. But some have other plans. "It was a good period of my life, my math years, but I guess I'll probably move on," said Mr. Steinweiss, a musician who is considering a philosophy major at the State University of New York at Purchase.

Meanwhile, the course's success has been good advertising for the coming year, as 17 students have signed up and some are doing summer work to prepare, Ms. Steckel said. The size of the summer class made some of last year's kids gasp.