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These success-against-all-odds stories are captivating. It’s hard to overstate how much “Stand and Deliver” — the 1988 movie about an A.P. calculus teacher who overcame the odds when all his low-income Latino students passed the exam — has influenced many advocates’ perceptions about what an A.P. class can do. And things like this do happen; “Stand and Deliver” is based on real events. But they’re anomalous. Yom credits his success to a number of things: a math department that lays out clear expectations from ninth grade on about what students need to know to get to A.P. calculus, a mentor who has taught A.P. calculus at Lincoln High for 16 years and his own ability to devote countless hours to his students. But once Yom is married and has children, he told me, it simply won’t be sustainable to continue spending so much time with his classes.

Even if students don’t pass the test, there is reason to believe that simply taking A.P. courses is valuable. After all, many students receive passing grades in their courses while still failing the A.P. exam. But because so much focus is on the test — the College Board tracks only participation and outcomes from the tests, not the classes — and because numbers are so much easier to measure than the far more intangible benefits of teaching and learning, the real value of A.P.s can be hard to assess. It seems logical to assume that taking a more rigorous course can have benefits in and of itself: by opening horizons, by sending a message to students that they are capable. And many teachers and students feel that way. Calid Shorter, 17, who was in Fuchs’s A.P. government class this past year, says she was one of his best teachers. “They really care,” he says. “Pushing me into classes has been a benefit — it’s given me more of a go-getter mind-set.”

Is it effective to be investing the time and resources in a program whose benefits seem so difficult to pin down?

Daniel Isquith, who has taught math at Calhoun for eight years, said he was initially “worried the kids would burn out” during the long classes. But he reorganized his lessons into 15-minute chunks, with a little breathing room in case things ran over: a 15-minute lecture, 15 minutes of problem solving, then 15 minutes of group work, capped by a final 15 minutes in which the students have to summarize what they did in class — a gem, he said, that the old schedule did not permit. During two-hour classes he changes things up just as often, to keep the students engaged. “Once you live in this and get a sense of pacing,” Mr. Isquith said, “it’s incredible what you can accomplish in terms of real actual understanding versus proficiency.”

The judge held forcefully that the use of a work for educational purposes weighs "strongly" in favor of a defendant's claim of fair use

the scope for "fair use" of factual, informative copyrighted works is larger than that of fictional, "creative" works.

In works of ten chapters or more, the use of one chapter is fair. In works of fewer than ten chapters, it is not fair to use more than 10 percent of the work, counting front and back matter (notes, bibliography, index, etc.) as well as the main body of the work. When instructors stayed within these bounds, the judge found that the third factor weighed in favor of fair use; when they went beyond these bounds, she found the opposite.