She asked them to rate, on a scale of one to five, how much they would value potential new programs, like medical services, tutoring or summer activities. One of the questions asked how they would value “an opportunity for parents to sit at the decision-making table,” which left one parent perplexed.

“Meaning?” Maria Pena, a 31-year-old mother of three, asked.

Ms. Carter explained that she could “sit at the table” with P.S. 112’s “community school team,” a group of parents and staff members who would decide what programs the school needed.

“I would definitely sit,” Ms. Pena said. “The problem is if they would listen.”

A number of experts argue that high levels of parent engagement are critical to turning around troubled schools. The theory is that when parents are more involved, students are less likely to be absent or to have discipline problems, and that parents will give their children academic support at home and will lobby politicians to properly fund schools.

Ms. Fariña has embraced this view, making parent engagement, as measured by surveys, a key component of schools’ annual ratings. The teachers’ union contract signed last year required teachers to spend 40 minutes each week on activities like meeting with, writing to or calling parents, replacing a requirement that they spend that time tutoring after school.

Karen L. Mapp, a senior lecturer at the Harvard Graduate School of Education and a consultant on family engagement to the federal Department of Education, said that besides training parents to be more active in schools, it was equally important to train school staff members how to communicate with parents, especially across racial and socioeconomic divides.

Dr. Mapp said that often the biggest problem was that staff members looked down on the families they served, seeing them as hindrances rather than as potential partners.