There are almost no students in the city’s gifted programs who are learning English, have special needs, or are in temporary housing. Put together, they make up less than 10 percent.

“What we have right now is something we should be ashamed of,” said James Borland, who directs gifted-education programs at Teachers College at Columbia University.

While most gifted programs are housed within traditional schools, BSI is one of five citywide schools that enroll gifted children exclusively. The citywide schools are some of the hardest to get into, essentially requiring 4-year-olds to land a near-perfect score on the standardized test used to determine who is “gifted.”

Districts used to be able to set their own admissions criteria for gifted programs. That changed in 2007, when the city standardized entry based on test scores, in part to increase diversity. A non-verbal test, also intended to address inequities, was added in 2012. Yet today’s gifted programs remain segregated.

That isn’t surprising since test scores are closely linked to socioeconomic status, said Allison Roda, who spent years studying New York City’s system and wrote a book titled Inequality in Gifted and Talented Programs.

“You’re never going to integrate gifted-and-talented classrooms that way,” she said.

The current administration has tried its own methods to diversify gifted education. Perhaps the most sweeping effort was the launch of new gifted programs in four districts in Brooklyn and the Bronx that had gone years without. Those programs started this year, admitting third-grade students rather than kindergartners and using measures other than the traditional gifted tests to decide admission, including grades and teacher recommendations. Both moves could level the playing field by making it less likely that students test into gifted programs based on the advantages they bring from home.

In those new programs, 70 percent of students are low-income, 49 percent are black and 39 percent are Hispanic, according to the Department of Education.

Officials recently announced the more expansive third-grade admissions criteria would apply to another school: P.S. 191 on the Upper West Side, which has been embroiled in a long-standing rezoning debate.

Students at P.S. 191 are largely black, Hispanic and poor—and gifted programs are often seen as a way to help integrate schools. Roda criticized that approach, saying gifted programs just lead to segregation within school buildings.

“It is a way to attract white, higher-income families to a school. But once you do that, it’s like gentrifying a school,” she said. “You walk down the hallway, and you can tell which classroom is gifted and talented and which classroom is general education.”

The Department of Education did not make anyone available for comment on gifted-education issues, despite repeated requests. In an email, a spokesman wrote: “We’re committed to increasing diversity and expanding high-quality elementary education for students and families—including through Gifted & Talented programming.”