For the past 31 years, since I stumbled across amazing things happening at Garfield High School in East Los Angeles, my main topic as an education writer has been schools whose low-income students have been raised to unexpected heights of academic achievement. There are many schools in the Washington area that have done that. What about those that haven’t?

By my count, in about 20 predominantly low-income high schools in the District and Prince George’s County, the passing rates on Advanced Placement exams have been stuck below 10 percent. Yet other high schools full of impoverished kids in those same two school districts have done better on the challenging college-level tests.

Why are some succeeding and others not?

The Eli and Edythe Broad Foundation has released a remarkable study answering these questions from a national perspective. It found six districts among contenders for its annual Broad Prize for Urban Education where black students in AP “were improving passing rates quickly enough to gain on their white peers while increasing or keeping participation levels steady.” They were: Cobb County and Fulton County in Georgia, the Garland Independent district in Texas, Jefferson County in Kentucky, Orange County in Florida and the San Diego Unified district in California. The researchers identified four reasons for their success:

1. Searching for more academic talent: In many cases, this meant enlarging gifted programs for younger students far beyond the designations based on high IQ scores that most districts use. In Fulton County, the settlement of a court-ordered desegregation plan in the early 2000s included a big expansion. The district went from two to 58 elementary schools with gifted-education teachers, and from 300 to almost 2,000 elementary school students getting gifted services.

2. Giving more high school students access to challenging courses: A surprising finding, at least to me, was that the move to smaller high schools in some urban districts reduced the variety of course offerings, including AP. I know of some small charter high schools that have plenty of AP courses, but I can see how smaller schools in big districts might be shortchanged. The researchers said: “San Diego has opened several small schools and is now moving back to large ones in part because of the lack of opportunities for specialized courses.” The districts doing well with AP tend to give PSAT tests to all students to identify the many who are overlooked for AP but whose test scores show they are ready. The researchers said that approach is based on this fact: “A College Board study showed only a 0.28 correlation between AP exam passage and grade point average, while the correlation with PSAT scores was 0.5 to 0.7.”

3. Giving AP students more support: These six districts provided AP preparation sessions in the summer, on weekends and after school. Some of them started Advancement Via Individual Determination (AVID) classes. The AVID program is designed to get average students, usually black and Hispanic, ready for AP by teaching them how to manage their time and by providing regular tutoring on how to think through their lessons, not just memorize. Garland’s program is particularly robust, with AVID classes starting in elementary schools.

4. Keeping the programs going even if gains are small: Even in the districts that are narrowing the gap between black and white students, “it will take years at the current rate to close the gaps for AP participation and passing rates,” the report said.

Some schools across the country with many impoverished students have moved faster than that, but they have employed the same principles — involving more kids, creating more challenging courses, supporting those students and sticking with the program even if progress is slow. We need more schools doing that.