Students at Central Park East High School, one of several now receiving city credit for college-level courses its teachers developed.

At Harry S. Truman High School, juniors in an honors English class arrange their desks in concentric circles to discuss Marxist and feminist theory in the American literary canon.

At Central Park East High School, students taking the Mt. Sinai Careers course develop research projects on the health sciences while interning in hospital departments like pediatrics, orthopedics, and Mt. Sinai’s morgue.

And at East Side Community School, seniors compare ancient Greek tragedies.

The courses are as challenging as any Advanced Placement class, their teachers say: To pass, students must demonstrate not only deep knowledge but also the kind of critical thinking required for success in college. But last year, when the Department of Education moved toward giving high schools credit in their annual letter grade for exposing students to college-level work, the courses did not count.

This year, they are among 52 courses in city high schools to get the department’s “college and career preparatory” stamp of approval, meaning that students who pass them typically stay in college after many ill-prepared students drop out.

The shift happened after the department proposed a new “college preparatory course index” that gave credit in part to schools where students passed “college-level” courses such as Advanced Placement or International Baccalaureate courses or classes through the City University of New York College Now program.

Schools that struck out on their own to build college-preparatory courses were shut out of the extra points. So for this year, when information about course offerings started counting toward schools’ grades, the department offered to certify courses as “college and career preparatory courses” if their schools could prove that students who passed ultimately graduated ready for college-level work.

The certification program is part of the department’s recent efforts to scrutinize what happens inside classrooms after many years of leaving that up to principals and teachers alone. It seeks to reward schools for offering advanced classes that are homemade, said Nancy Gannon, the director of the city’s Office for Academic Quality, who is overseeing the effort.

“We have the opportunity here to give the school credit for pushing kids in a way that’s equally valuable to an Advanced Placement class,” she said. Officials are also recruiting teachers who get the department’s seal of approval to help other educators develop college-level courses of their own.

Last spring, schools proposed more than 100 courses for certification, and over the summer, department officials pored over the graded student work, syllabi, and other curriculum materials. They also looked at how students who passed the class and graduated were faring in college.

Ultimately, the department certified 52 courses at 34 schools, turning away more than half of the applicants. The stamp of approval went to classes about business law, philosophy, gastronomy, and personal finance, in addition to more typical courses in English, math, and science.

“The measure we’re using here is the Common Core learning standards,” said Gannon, referring to new standards that are geared toward college readiness. “Many that fell short were on their way there but hadn’t arrived yet. The texts weren’t rigorous enough, for example, or they were asking people to do a summary of a text rather than really do analysis.”

The schools that did make the cut include both Stuyvesant High School, which is highly selective, and John Dewey High School, which has struggled so much that the department recently sought to close it. Arts-focused schools are represented, as are schools that aim to prepare students for careers.

And both established schools and small schools opened under the Bloomberg administration had courses certified. (Schools had to have at least one graduating class to apply, and at 25 graduates had to have taken the high-level course, allowing the department to track whether they remained enrolled after their first semester of college.)

Gannon described Truman’s English class as “remarkable” for its ability to get students to talk at length about literary theories.

The class was developed five years ago when teachers realized that some of Truman’s strongest students were entering 11th grade with a basic knowledge of American history and literature already in place, according to teacher Zulay Martinez. The school wanted to push the students further by introducing them to literary theories they had never heard of, but are features of many college literature courses.

“Because they were honors students, we thought that would be a great opportunity to push them further,” she said. “We said, let’s look at it from a feminist perspective, a psychoanalytical perspective, a Marxist perspective.”

In the class, students routinely arrange their desks in two rows, to facilitate Socratic seminar-style dialogue, Martinez said.

“The idea that everything we need to know is already within us,” Martinez said. “We tell them, ‘You’ve read the story, you know the theory, all you need to do is put two and two together to answer the questions.’”

At a recent seminar on the classic short story “The Yellow Wallpaper,” students debated questions like “What’s the relationship between the narrator and women in the story?” and “Does the text support the cult of domesticity?”

Other schools shot for the same level of rigor but did not hit the mark. A Queens school’s pre-calculus course had its strengths, such as asking students to apply their mathematical knowledge to real-world problems, Gannon said, but students were rarely asked to explain their thinking in writing, a central demand of the Common Core standards.

Gannon said the department was also disappointed with the quality of feedback teachers were giving.

“It was really limited to basic cues or crossing things out,” she said. “There wasn’t much constructive and detailed feedback that would contribute to student learning from their mistakes or successes.”

Department officials said they gave feedback to each of the schools that applied and in many cases invited ones that did not meet the certification requirements to tweak their programs and reapply this year. Applications open next week.

Schools that apply must commit to sharing their course materials with the department, which will make them available to other schools in earlier stages of developing college preparatory courses.

The department is also recruiting successful applicants to guide other teachers who hope to bring higher-level courses to their schools in the future. In August, Joanna Dolgin, a teacher at East Side Community High School, led a seminar for teachers from six other schools in the New York Performance Standards Consortium. Schools in the consortium require students to demonstrate mastery through real-world applications of knowledge, rather than simply by passing state exams.

Department officials said teachers from 21 consortium schools participated in the trainings this summer. The department is not yet offering  these seminars to other schools, but there are plans to expand the effort in the future.

Dolgin teaches a semester-long advanced English class called “Tragedy” that has students read classic works such as Antigone, Othello, and Oedipus Rex, and discuss the roles that big-picture concepts like tragedy, power and identity play in them.

The course isn’t perfect for all students, Dolgin said, and replicating it is likely to be a challenge. “Designing effective courses where the students are engaged requires a tremendous amount of time and collaboration,” she said.

Adding new college-level courses is hard work but well worth the benefits — which far exceed getting a boost on the city’s progress report, school leaders say.

“The course would be the course whether they certify it or not, whether we get credit on our report card or not,” said Bennett Lieberman, Central Park East’s principal. “We have not developed these courses with the report card in mind per se, but with getting our kids college ready. We’ve been working on that longer than the [department] has been grading us.”