At the high-school level, the city has been pushing for greater academic diversity, almost doubling the number of what are known as Educational Option programs, which aim to replicate an academic bell curve, with 16 percent of the offers going to applicants who score the highest on state English tests, 16 percent to those who score the lowest, and the remaining 68 percent to those in between. Next year, of the city’s more than 400 high schools, 264 will use the Educational Option model, up from 142 schools this year. Still, many of these so-called Ed-Opt schools have struggled to attract high-performing students, who often prefer the selective schools, and ended up with mostly lower-performing students.

In District 3, scars remain from a battle over redrawing school zones two years ago to increase diversity and reduce crowding at a group of elementary schools. At the time, district leaders also considered a plan to integrate middle schools by requiring them to give priority to students who qualify for free and reduced-price lunches for 30 percent of their sixth-grade seats. The most sought-after middle schools, which have a screening process to admit students, have long been criticized for being mostly white and Asian, and more affluent, than the district as a whole.

But that plan foundered after principals voiced concerns that it would not do anything for low-performing schools, many of which already had far higher percentages of poor students. Many parents also saw it as increasing competition for the remaining seats at the top schools.

This spring, the district tried again with the new plan, which would require every middle school to give priority to students who average a 1 on state tests for 10 percent of their sixth-grade seats, and those who average a 2 for another 15 percent of their seats. “As a district, we value diversity, equity and access for all our students and across all our schools,” said Ilene Altschul, the district superintendent, who proposed the plan, adding that educators and parents “have been working together this school year on changes that can help us support these goals.”

But that plan raised questions about whether it was legal to rely solely on scores on state standardized tests, which many educators and parents say do not accurately reflect students’ abilities. State education law prohibits officials from placing students “based solely or primarily” on their performance on state English and math tests; some city schools can and do use other tests to admit students. State education officials declined to comment on the issue.