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If you’re a teacher and you want to make sure that your students get a foundation in a basic topic before moving onto a more complex, related topic, it may make sense to overlearn the first topic before tackling the second with the goal of revisiting the latter at a later date.

he asks them to choose three examples that help them tell their parents a deeper story: one that shows they have recognized both a personal strength and an area in which they are struggling. Most students, he says, have never thought about their learning in this way. Nor have most of their parents.

kids learn to advocate for themselves

“What do I do well?” and “How can I build on this?”

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?