Many great principals sidestep school system regulations. If a rule gets in the way of good teaching and learning, they go around it. Usually, however, they try to keep such digressions from becoming widely known.

Is that a good idea? Wouldn’t more candor from good principals help systems fix bad regulations? Or would that just get them fired?

We may soon find out. In a new book by two education experts, Richard G. Trogisch, principal of the District’s highest-performing public high school, reveals that if a rule makes no sense to him, he pretends it is not there.

Chester E. Finn Jr. and Jessica A. Hockett, in their book “Exam Schools: Inside America’s Most Selective Public High Schools,” say this about the School Without Walls, where Trogisch has been the principal for six years (exam schools require all students to meet certain academic requirements for admission):

The school “seems simply to disregard those rules that it finds dysfunctional. The principal firmly adheres to the view that, in a bureaucracy like DCPS [D.C. Public Schools], it’s far ‘better to seek forgiveness than to ask permission.’ And so, for example, [George Washington University] faculty members sometimes teach SWW classes even though they’re not ‘certified’ to do so.”

Finn and Hockett don’t actually name Trogisch, but the principal cheerfully confirmed to me that the book describes his views accurately. It is one of the themes of “Exam Schools” that selective public high schools such as the School Without Walls (usually known as Walls) often find that system rules don’t work for them. Some suffer in silence. Some protest. Some, like Trogisch, do what they want anyway and apologize if they get caught.

Do other exam schools behave that way? Finn and Hockett, in their eye-opening book, identify 165 such schools in the United States. Their analysis suggests few of them are run by rebels such as Trogisch. Many try to work out compromises.

Only five of the 165 exam schools Finn and Hockett identify are in the Washington area. Besides Walls, they are Benjamin Banneker Academic High School, McKinley Technology High School and Phelps Architecture, Construction and Engineering High School, all in the District; plus the Thomas Jefferson High School for Science and Technology in Fairfax County. There are also selective programs at high schools such as Richard Montgomery and Montgomery Blair in Montgomery County, Eleanor Roosevelt in Prince George’s County, and Dominion in Loudoun County, but they do not involve the whole school, as the 165 on the book’s list do.

Most, but not all, of these schools are high performing. Jefferson has the highest test scores in the country. Walls, Banneker and McKinley have the city’s highest proficiency rates for secondary schools — all above 86 percent. But Phelps, a relatively new program not yet sought after by the city’s best students, has proficiency rates of only 47 percent in math and 58 percent in reading, compared with the D.C. secondary school averages of 46 and 42 percent, respectively.

Finn, a longtime education pundit who heads the Thomas B. Fordham Institute, and Hockett, an education consultant with a doctorate from the University of Virginia, suggest that more exam schools in some places would be a good idea because “we downplay excellence at great cost, not only to our economic competitiveness but also perhaps to reform of the education system itself.”

They also acknowledge that there is little research on whether exam schools do a better job of educating top students than regular schools. What research there is suggests they don’t.

Trogisch wishes his school got more support. Former D.C. schools chancellor Michelle Rhee gave him no love. But he says her successor, Kaya Henderson, has promised to help him get more funds for his overcrowded building and overworked staff. How will that turn out? Henderson hasn’t said what she thinks of principals who act first and seek approval later.