“I think that was just what we needed,” Taylor said.

In joining the city’s “Diversity in Admissions” program, Taylor is trying to address striking differences between her school and others. Citywide, about 77 percent of students are poor and almost 70 percent are black or Hispanic. Last year, BSI’s poverty rate was 23 percent, and less than 10 percent of students were black or Hispanic.

The disparity is not unique to BSI, or to gifted education. Citywide, about 73 percent of gifted students are white or Asian, and the poverty rate averages around 43 percent.

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.”

Recent efforts, however, appear to have fallen short. According to city data, recorded in annual school diversity reports, the percentage of black and Hispanic and low-income students enrolled in gifted education has remained about the same over the last two years.

Borland said the city could instead move back toward allowing districts more flexibility in how they decide who’s “gifted.” Instead of one test cut-off score, students would be admitted based on how they compared to their local peers.

“That makes sense because you need a different program based on how you compare to kids in your class,” and not, for example, students in another borough, Borland said.

But ultimately, he said, “I would not base admissions on tests.”

The new enrollment policy at BSI is likely to have a small effect—one that could take years to play out. Because siblings of current students get priority in enrollment, precious few kindergarten seats are open in any given year. And even if BSI meets its enrollment target of 40 percent low-income, it would still be far below the city average for student poverty.

But Taylor says the school has to start somewhere. She admits some parents have questioned whether the initiative would impact performance.

Her response: “Sometimes the parents don’t have time to advocate for their kids, but that doesn’t mean their kids can’t do as well.”

Other parents have welcomed the change, and started an information campaign to encourage more families in the community to sign their children up for testing.

“This is a big deal,” said Sara Mogulescu, the parent of two students at BSI. “It’s important to a number of us.”