On one side are parents and at least half the teachers, who accuse the New York City Education Department of installing Ms. Garg in an attempt to make the school more traditional. These parents and teachers say that she is forcing teachers to adopt prepackaged curriculums, bringing in families with little stake in the school’s approach and retaliating against teachers who resist her policies by starting investigations focused on them. They held a rally outside department headquarters on Tuesday demanding that she be removed.

“In the nine months that she’s been there, she sort of has done everything she can to divide the community,” said Bonnie Massey, who has a son in the second grade. “She’s mistreated teachers. She’s mistreated children. She has said things that are incredibly offensive.”

On the other side are parents who see Ms. Garg as trying to return the school to its mission of serving the children of East Harlem, something they say Central Park East I has lost by increasingly catering to middle-class families.

“Debbie Meier said in one of her books that, without a powerful system of accountability, well-intentioned schools can easily become smug, secretive, tyrannical and even racist,” said Elaina Watkins, who has three children at the school and supports Ms. Garg. “And as a parent at C.P.E., I can see that is what has happened at the school.”

What is not in dispute is that, in recent years, the makeup of the school, which does not draw from a defined georgraphic zone, has changed significantly: In 2004-5, the student body was 92 percent black and Hispanic; 54 percent of the children qualified for free lunch. Today, it is 53 percent black and Hispanic, and 25 percent qualify for free lunch. In the early years, most students lived in District 4, where the school is. Now, 24 percent of the students come from District 4, and the largest share of students, 29 percent, come from District 3, on the Upper West Side.