To a good teacher every classroom appearance is a performance -- but rarely so much of a performance as the one William J. Bennett is putting on in a limited-engagement teaching sweep through eight American schools this fall.

Bennett, the frequently embattled secretary of education, brought his show to Washington's Banneker Academic High School Tuesday to honor the first day of school, delighting 27 students, 61 journalists and assorted TV cameras and Education Department functionaries with a fist-punching lecture on "The Federalist Number Ten." He mugged for the cameras before getting down to work. He shot questions at the class and cracked jokes. He urged the kids to read the daily newspapers. And he drove home with a philosopher's zeal what he considers the principles that underlie American government.

The traveling education act is part of Bennett's attempt to dramatize the importance of teaching -- which, as his press release says, "is of course at the heart of the educational process." It is an ingenious use of the kind of symbolic politics this administration has found so crucial -- and which, Bennett has made clear, are uppermost in the "bully pulpit" conception he has of his role.

But unlike Mickey Mouse initiatives such as the teacher in space, this teaching swing actually benefits its participants. Eight schools (all chosen for some educational success story) get a jolt of prominence and a festive feel on the first day of class. About 200 students get an impassioned history lesson; perhaps three times that number of journalists get a refresher course. Afterwards the kids trots out meticulously photogenic and articulate reactions for the TV cameras. "The way I see it," Joseph Razza, age 16, told a cluster of D.C. journalists, "Secretary Bennett is a gladiator in the public arena. The best weapon in that arena is public relations, and this was very good public relations." When it turned out a tape had stuck and would have to be rerolled, Razza cheerfully said the whole thing over again, changing "best weapon" to "sharpest sword."

The secretary, meanwhile, got to do one or two things that went beyond PR. Though he has never been an elementary school or high school teacher, he has emphatic views on how to teach -- specifically, how to incorporate American "values" in the curriculum. One of the many furors he touched off last winter stemmed from his comment that more people would "recognize the urgency in Nicaragua" if they'd been taught their American history properly in school. Since Bennett is also involved in a word-fight over religion in public life and voluntary school prayer, and has sharply criticized numerous aspects of so-called progressive education, the question of what he would want to demonstrate as ideal teaching has a slightly more than academic interest.

Bennett, as usual, was not fazed by such delicate matters. He did not skirt references to either the religion-in-public-life controversy (listing "religious zeal" among the causes of faction that can bring about civil discord) or the tricky matter of comparative government, which has proved prime ground lately for hysterical battles over "moral equivalence." Reaching what he called "the heart of the heart of the heart," Madison's theory of factions as inevitable given human nature, he was, for him, unusually measured: "When you come across other systems of government based on other theories, use this one -- whatever you think of it, use this one as a baseline, because it's the basis of the system that's operating here."

As noted frequently, Bennett paces. At Banneker he jiggled chalk and perspired and punched fist into hand to make each point. He elicited almost every point through rapid-fire questioning rather than assertion -- sure sign of a confident teacher. He clearly had a ball.

In fact, he had so much fun with the confident, articulate 11th-graders at elite Banneker that the exercise at times seemed to lose its point. Banneker is, after all, a bright and airy and well-painted place; there are "Banneker Achiever" sweatshirts for sale in the lobby and NEA stickers on the wastebaskets. Not many of the teachers affected by the workings of Bennett's department -- or the students either -- would find it familiar. And even at Banneker it was only the first day of the year. Robert Steptoe, the 28-year veteran teacher who got his class back from Bennett for the second and third and fourth day and thereafter, praised the secretary's performance -- but in response to further questions allowed politely that he didn't think he had had anything to learn from it in terms of pedagogy.

And then there was the rest of Bennett's audience -- an audience which, however young, is already wise in the ways of PR. Asked whether any students at Banneker wanted to be teachers, 11th- grader Wendy Woods, who said she had enjoyed the class, answered carefully. "I guess there may be a good number," she said. "But most Banneker students are interested in being something above that, like a doctor or a lawyer."