The four Democrats running for mayor complain that Mayor Bloomberg excludes parents and teachers from education decision-making, but none of them would relinquish any of the control over the school system that Mr. Bloomberg won from Albany.

In the 2001 campaign, Mr. Bloomberg pledged to convince state lawmakers to give the mayor authority over the public-school system. It was a goal mayors had sought unsuccessfully for decades, yet Mr. Bloomberg achieved it by his sixth month at City Hall. At the time, city leaders, academics, and even union leaders said winning control was a great triumph for the mayor, and he said he wanted and expected to be judged by his record on education.

Now, as the mayoral campaign accelerates, the Democratic candidates are trying to poke holes in the mayor’s education record, expressing concerns, for example, about the curricula that Mr. Bloomberg’s chancellor, Joel Klein, installed on a citywide basis. Instead of attacking mayoral control, some said they have problems with the way Mr. Bloomberg has exercised authority over the school system, and others said they oppose specific policies that Mr. Bloomberg has implemented.

The former Bronx borough president, Fernando Ferrer, who is leading in the polls, was the harshest critic of the way Mr. Bloomberg has used mayoral control.

He said he supported Mr. Bloomberg when he won control of the school system.

“I certainly didn’t support, however, eliminating a public board that made its decisions about important policy and budget matters out in the open,” he said. “I believe that a public board that does these things in a transparent way, that’s accessible to parents, is the way we should be going.”

Mr. Ferrer said he warned years ago that without a strong public board, decisions would be made in back rooms of government buildings. He said it seems that’s what has happened during the Bloomberg administration.

Under Mr. Bloomberg, he said, the Panel for Educational Policy doesn’t openly discuss such issues as the adoption of the budget, contracts, and policies.

“It’s beyond toothless,” Mr. Ferrer said. “It barely has gums.”

He said that on Day 1 as mayor, he would change the role of the panel so that it would be more involved, turning decision-making into a more inclusive and transparent process. But he insisted that he would not cede any formal power to the panel, whose members are appointed by the mayor and the borough presidents.

The Manhattan borough president, C. Virginia Fields, also said the panel isn’t playing the role it should be playing in policy-making. She said that the panel votes on decisions that have already been made, and that the newly created parent councils, too, have not been “brought into meaningful decision-making.”

“I know that the legislation was not intended to exclude others whose voices could help in shaping reform decisions, like educators, like parents, administrators, or elected officials. And we have basically been excluded,” she said. “Decisions are made. They’re handed down.”

Ms. Fields said as mayor she would be more inclusive. “The reform, and giving a mayor jurisdiction, I think, provides an enormous opportunity,” she said, “but it has to be done in a way that it’s not just top-down management of that system but inclusion, so we can get at the real reforms that help the children in our schools.”

The other contenders also supported mayoral control, but they were less critical of the structure of the system and more critical of Mr. Bloomberg’s specific policies.

“The mayor’s biggest accomplishment on education was to get control of the school system, and his worst failure was to squander that control,” the City Council speaker, Gifford Miller, said. “He’s confused getting control with an actual accomplishment on behalf of kids. Just because he’s running the school system doesn’t mean that kids are learning any better. What means that kids will learn better is if he actually runs the system better.”

Mr. Miller objects to the pedagogy Mr. Bloomberg has implemented in the city’s classrooms and the way the mayor has chosen to direct school resources.

“I would have focused on the fundamentals, on smaller class sizes, on attracting and retaining more quality teachers, on safer schools and stronger after-school programs,” Mr. Miller said, “instead of a bureaucratic reshuffling that’s leaving parents out, instead of blaming teachers and driving thousands of them out of the system, instead of blaming Albany and refusing to do anything about class sizes.”

Mr. Miller said it seems as though Mr. Bloomberg “wants control without accountability.”

Rep. Anthony Weiner said he supports mayoral control and as mayor would “make good use of it.”

“I would return to the basics in curriculum,” Mr. Weiner, who represents a Brooklyn-Queens district, said. “There’s far too much focus on things like the condition of the bulletin board and how chairs are arranged and far too little focus on instruction.”

Mr. Weiner said if he were grading the mayor’s education record, he would give Mr. Bloomberg an “F” for retaining experienced teachers, an “F” for reorganizing the bureaucracy, an “F” for making the schools safer. He said the only subject Mr. Bloomberg would get an “incomplete” in is test scores, explaining, “Maybe it will take a few years for them to work out the kinks.”

He said Mr. Bloomberg has been such a “bumbling” education mayor that Albany might even take away mayoral control when the law is up for renewal in 2009.

The mayor’s press secretary, Edward Skyler, lambasted the Democrats for their criticisms of Mr. Bloomberg’s education record.

“It’s always great to hear from career politicians who have worked in public service for decades but who never had either the courage or inclination to do anything about our city’s failing schools, to now try to weigh in on an issue they ignored for their entire careers,” Mr. Skyler said. “The mayor took the reins of a system that failed generations of children and is implementing real reforms so our children get the education they need and deserve.”

An expert on New York City education issues, Sol Stern, said: “It’s true that Bloomberg scored a political victory and led the fight for mayoral control, but this was a change whose time had come. It was in the air. There was an overwhelming consensus that the old Board of Education had outlived its usefulness.”

He continued, “It’s true that Bloomberg led the fight.” But, he asked: “Four years later, how much credit should he still get merely for getting control of the system?”

Mr. Stern, an author and Manhattan Institute scholar, said the question remains unanswered, but he said it’s clear what the mayor and his political opponents will say during campaign season.

The Democrats. Mr. Stern said, “will argue that he has exercised that authority in a very negative way that has actually damaged the schools.” Mr. Bloomberg, meanwhile, will “try to trumpet his accomplishments, drawing in all his allies and supporters from the business community and the foundations.”

One former mayor who sought in vain to win control of the school system, Edward Koch, said that at election time it’s fair to criticize anything. But he said: “We’ve had a system 40, 50 years where the mayor has not had control. You can’t turn that around in one year or two years or four years. It’s hard to do it, but I think it is being done. When Bloomberg was willing to accept responsibility he did something very, very important for the city. It’s not going all honey and roses, but we’re making progress and that’s what counts.”

Mr. Koch said he doesn’t agree with everything the chancellor has done, but he said he applauds the Bloomberg administration for “trying new things.”

He said advocating a more powerful Panel for Educational Policy, as Mr. Ferrer does, seems like “reinstalling the old Board of Education that took the power away from the mayor.”

Another of Mr. Bloomberg’s supporters, the president of the Partnership for New York City, Kathryn Wylde, said it’s too soon to say if any of the mayor’s education policies have worked.

“The experts all say that the results of school-reform initiatives don’t come in for 6 to 10 years,” she said. “I guess they’ve never heard of term limits.” New York City’s mayor is now limited to two four-year terms.

But Ms. Wylde said the results of mayoral control itself are clear: “From what I see, the results of the mayor being able to take action to make the schools safer, to end social promotion, to put in place a standardized curriculum, the sort of actions the mayor and the chancellor have been able to take on so many fronts in such a short period of time, all suggest that things are going in the right direction, and probably doing so more quickly than we had any right to expect.”

A leading education historian, Diane Ravitch, who announced months ago that she planned to vote for Mr. Bloomberg but is now calling herself “undecided,” said she agrees with many of the Democratic candidates that there is a lack of transparency and a lack of accountability in the new system.

“Just to say ‘If you don’t like it, don’t vote for me’ is not enough accountability,” she said. “When the election’s over, where’s the accountability?”

She said she’s not sure if anyone, working under the existing law, would handle mayoral control better than Mr. Bloomberg has done.

“If another candidate were to say I’m in favor of mayoral control and I’d do a better job,” Ms. Ravitch said, “I don’t know, I keep coming back to the question of the lack of any oversight.”