While the Saturday sessions were voluntary, O'Brien said they have been well-attended. Students said they are useful because they get lessons from college professors and other faculty who aren't their teachers but have a deep knowledge of the subject.

Students said hearing a concept they are struggling with explained in a different way by another teacher helped them better understand the material.

In addition, their teachers were available to give them help for four hours each week before or after school. Teachers were paid more money for the extra hours they worked, and they received in-depth AP training for a week each summer.

Amber Milnes, who taught her first AP biology class at Aberdeen last year, said the training taught her how to set up labs in her classroom. "It was extremely helpful to me," she said.

Milnes received $2,600 for the exams her students passed. She said she appreciated the money but stressed that wasn't the chief reason she put extra hours into teaching the high-level classes.

"We are probably getting 10 cents an hour in pay for extra work," she said, laughing.

Aberdeen physics teacher Nathan Sloan got an additional $7,000 for helping students to pass the test, and he said the money was something of a motivator. The grading of AP homework takes far longer and is more involved, he said.

Havre de Grace also showed an increase in passing rates. In 2013, 73 students passed an AP test out of 156 test-takers, or 47 percent. That was an increase over the previous year, when 41 percent passed.

"All of our teachers are increasing the rigor in our classrooms," said Havre de Grace Principal James Reynolds. "It is opening a lot of doors for students who have the potential to take it."

In addition to paying for passing scores and holding Saturday sessions, the Havre de Grace program paid for a portion of the AP exams and allowed students to take a practice exam to prepare them just weeks before the test.

In Maryland, the percentage of graduates who have passed at least one AP exam has increased each year for more than a decade. The percentage was 29.6 percent for the Class of 2013, up from 28.1 percent the year before.

Maryland's No. 1 ranking comes in part because so many students take the courses and exams. Nearly half of all Maryland graduates are taking the exams. And far more than half of graduates will have taken a class, though about 20 percent take the courses and never sit for the exams.

Some critics have said pushing AP courses for high school students has had mixed results.

According to a Baltimore Sun analysis last year, in some schools around the state, the majority of students who get As and Bs from their teachers in AP classes are then failing the exams. And there is no research that suggests that students who do better on the exams will perform better in their freshman year of college.

Moreover, there is a pronounced achievement gap. African-Americans represented 36 percent of Maryland's graduates in 2013, but only 12 percent of the total who had passed at least one AP exam. And low-income students represented only 14 percent of the passers.

Kristin Klopfenstein, executive director of the Education Innovation Institute at the University of Northern Colorado, who has studied Advanced Placement, said at least one rigorous analysis has shown the NMSI program to have a positive effect on student achievement.

But, she said, the program is complex, and it is difficult to say which portion of it works to improve scores.

"My understanding is that the student and teacher incentive pay constitute a relatively small fraction of the total program costs, and I doubt they are an important piece of the program among students traditionally underrepresented in higher ed," she said.

liz.bowie@baltsun.com