As Mr. Bloomberg reorganized the system and poured money into it, satisfaction slowly grew. It appeared to peak in 2009, when nearly one-third of registered voters said they were satisfied with the schools, and 57 percent with Mr. Bloomberg’s handling of them, according to Quinnipiac University polls.

But in 2010, a recalibrating of state test scores sent the city’s soaring passing rates nearly back to 2003 levels. Then Mr. Bloomberg made what the public may have judged as his biggest gaffe, naming Cathleen P. Black, a publishing executive with no education experience, as the schools chancellor. Even though Ms. Black was dismissed in April after three months, the effects on public opinion linger, analysts said. Fewer people today approve of Mr. Bloomberg’s handling of education now than in his first years as mayor.

And Dennis M. Walcott, the longtime deputy mayor who replaced Ms. Black as chancellor, has not yet made a strong impression, with 53 percent of those polled having no opinion on his performance.

A year of political attacks on teachers and their unions nationally appeared not to have registered with New Yorkers. A quarter said they had a favorable opinion of the local teachers’ union, compared with 16 percent who had an unfavorable opinion. A majority did not express an opinion.

Among parents, 34 percent volunteered that teachers were the best thing about their child’s school, more than in 2004, when 22 percent said so. A plurality of parents said teacher quality was the most important thing to look at when choosing a school.

Crowding and class sizes, which are growing, ranked highest when parents were asked to name the worst thing about their schools, at 13 percent. It is an issue frequently mentioned by the teachers’ union and parents, but not City Hall.

New Yorkers are pleased with some of Mr. Bloomberg’s initiatives: 56 percent, for example, said charter schools, which have expanded under the mayor, were a good idea. Fewer than half, however, favor having charters share buildings with traditional public schools, a situation that is increasingly common and contentious.