From the moment he won control of the schools, less than six months after taking office, Mr. [Michael R. Bloomberg] has urged voters to judge him on education. ''I want to be held accountable for the results, and I will be,'' he said in June 2002 when Gov. George E. Pataki signed the law putting Mr. Bloomberg in charge. ''I do promise you that you will see in the very near future that we are going in the right direction.''

Regardless of which analysis is used, Mr. [Joel I. Klein]'s top testing official, Lori Mei, said that the overall trend in reading scores has been flat since Mr. Bloomberg took office, while math scores continued to rise as they have since 2000 when Mr. [Harold O. Levy] became chancellor and made improving math skills a top priority.

Kindergarten at Public School 30 in District 5, Harlem. More than 65 percent of students tested at the school don't meet state standards in math.; Parts of P.S. 30 that the Bloomberg administration says it has improved include breakfast, physical education and school safety. (Photographs by Michael Nagle for The New York Times); The mayor at Thomas Jefferson High School to mark the opening of four small academies in September. Three of their principals are gone. (Photo by Ozier Muhammad/The New York Times); Gov. George E. Pataki, right, signed a bill in 2002 giving the mayor control over the schools. The mayor has asked to be judged by the results. (Photo by Associated Press/Matt Moyer); Mayor Bloomberg giving his plan to fix the schools in January 2003. (Photo by Librado Romero/The New York Times)(pg. B4)

Rhonda Johnson paused outside Public School 92 in District 5 in Harlem, where her daughter, Raanesha, was spending the midwinter vacation in a special class for fifth graders at risk of being held back. The class is a new tutoring effort central to Mayor Michael R. Bloomberg's pledge to end social promotion.

Raanesha is precisely the sort of child -- a poor black student at one of New York's worst-performing elementary schools -- who Mr. Bloomberg hoped would benefit most from his overhaul of the school system.

But when asked the question that will hang over the mayor's head in the coming months -- are New York's public schools any better? -- Ms. Johnson expressed a view widely echoed in numerous interviews with parents and in opinion polls across the city. ''I haven't seen a difference,'' she said. ''Really, no, I haven't.''

From the moment he won control of the schools, less than six months after taking office, Mr. Bloomberg has urged voters to judge him on education. ''I want to be held accountable for the results, and I will be,'' he said in June 2002 when Gov. George E. Pataki signed the law putting Mr. Bloomberg in charge. ''I do promise you that you will see in the very near future that we are going in the right direction.''

With the mayor's re-election campaign now under way, retracing some of his steps through the school system over the last three years -- to districts where he made major announcements, schools that he singled out as models or in critical need of repair -- shows that the academic results so far have been mixed at best. On much of the school system, the main impact of the changes has been shock and tumult: start-up difficulties, dizzying and at times conflicting policy changes, high staff turnover.

For Mr. Bloomberg and his schools chancellor, Joel I. Klein, some of the shock and tumult has been precisely the point. In their view, a badly broken, maddeningly bureaucratic system that overwhelmingly failed the city's students -- especially poor black and Hispanic children -- has been permanently dismantled.

But across the city, reading scores in the third through eighth grades have been flat since the mayor's overhaul began in 2003. Math scores have risen steadily since 2000, though officials inside and outside the system trace the progress to changes made by the former chancellor, Harold O. Levy, and say recent gains are a continuation of that trend.

Since Mr. Bloomberg took charge, overall attendance rates have not changed -- with improved numbers at new small high schools offset by declines in old larger schools. School safety remains a murky picture. The four-year graduation rate has risen, yet the number of students receiving a Regents diploma, meaning they met a set of state requirements including Regents exams, has declined.

At some of the signature stops that Mr. Bloomberg made to highlight his ambitious plans for the schools, a picture emerges of two years of extraordinary upheaval. To some degree, the mayor's travels reflect his willingness to tackle the most difficult problems in the most troubled schools. And, of course, there are schools across the city where principals expressed optimism. At the same time, the mayor's inability so far to achieve clear-cut success, even where he had personally shone a spotlight, helps explain why many New Yorkers fail to see any change and why some say things are actually worse.

In District 5, for instance, where Mr. Bloomberg began his efforts two years ago, it is not surprising that Ms. Johnson and other parents do not see all the changes as helpful. The parents of more than 42 percent of fifth graders like Ms. Johnson's daughter, Raanesha, got letters this year saying their children were in danger of being left back. Among third graders, the number was more than 52 percent.

Two years ago, Mr. Bloomberg announced his initiative to improve special education, at Public School 87 in Middle Village, Queens, pointing to the school as a model program. But for much of this year, the school's parents fought to preserve a special reading program for children with learning disabilities, as a new principal tried to dismantle it. The parents also struggled to fend off steep cuts in gym and art classes.

And at Thomas Jefferson High School in East New York, Brooklyn, which the mayor visited on the first day of this school year to cheer the opening of four new small schools in the Jefferson building, little has gone right. Three of four principals have quit or been removed. At one school, three of six teachers quit; at another, attendance is worse than it was at the old big school before Mr. Bloomberg took over.

But given the long legacy of failure at schools like Thomas Jefferson, of pervasive violence and demoralizing dropout rates, Mr. Klein said he hoped New Yorkers would recognize the size of the mayor's task.

''What I would hope people evaluate us on are the clear leadership and commitment,'' Mr. Klein said in an interview, ''and the boldness to change a school system that they know, everyone knows, was not working for the last 30 years.''

In the interview, Mr. Klein and Deputy Mayor Dennis M. Walcott, the top education official at City Hall, pointed to a series of changes that they say will pay long-term dividends: 16 million books delivered in the last two years, uniform reading and math programs, reading and math coaches to support teachers, dozens of new small high schools, and a parent coordinator hired for every school. They also said that more children are eating breakfast and lunch prepared at school and that those meals are more nutritious.

But most of these points have a counterpoint. Every school has one parent coordinator, whether it has 500 students or 5,000. For that reason alone, the impact of the new position has varied. Some schools have been more effective than others at adopting the new reading and math programs. The new small schools have aggravated overcrowding in the existing big schools and prompted debates about the equity of the money and resources being spent.

Some accomplishments that Mr. Bloomberg and Mr. Klein are proudest of -- a new training academy for principals and an aggressive effort at professional development for teachers -- are largely out of public view. And again, results have been mixed.

The administration boasts of having driven down the cost of school construction, but critics say that new schools and the state money to build them have not materialized fast enough to reduce class size.

While it is hard to compare the New York system, with nearly 1.1 million students, to any other, experts on urban education point to Paul G. Vallas, the chief executive officer of Philadelphia's schools, as proof that clear increases in test scores can be achieved in just one year.

''They managed to produce results faster than anybody would have expected,'' said Michael Casserly, executive director of the Council of the Great City Schools, a lobbying group for large urban districts.

Mr. Casserly said that Philadelphia's scores had started lower than New York's, making it easier to post gains, but he also credited Mr. Vallas, who led the Chicago schools from 1995 to 2001, for aggressive and sure-footed leadership. But even without a quick jump in scores, Mr. Casserly said that he would not fault New York.

''I think the mayor and the chancellor get enormous credit'' for streamlining the management of the district, he said. ''But the jury is still out on the instructional side.''

Mr. Vallas said it was not too soon for parents to expect tangible results. ''There is absolutely no reason why school districts can't begin to show immediate improvement if they establish an effective instructional management system,'' he said.

A New Bureaucracy

Test scores aside, many parents say the new bureaucracy is harder to navigate than the old one, that the parent councils that replaced the local school boards are powerless, that community involvement has been quashed, and that the Panel for Educational Policy, which replaced the Board of Education, is now merely a rubber stamp for the mayor.

These views were conveyed vividly in a recent New York Times Poll. One-third of registered city voters said that the quality of public education had gotten worse since Mr. Bloomberg took office. Slightly more -- 34 percent -- said that it had not changed, while 23 percent said that the schools had gotten better.

The outlook was grimmer among public school parents: 46 percent said the quality of public education had gotten worse while 21 percent said the schools were better. Among the places where parents and community advocates say the schools are as dysfunctional as ever is District 5, in Harlem. The survey was conducted between Feb. 4 and Feb. 13 and had a margin of sampling error of plus or minus 4 percentage points among registered voters and plus or minus 5 percentage points among school parents.

In January 2003, Mr. Bloomberg chose the Schomburg Center for Research in Black Culture, in the heart of District 5, as the setting for an address laying out his plan to fix the schools, including everything from new curriculum to the reorganization of the city's 32 districts into 10 regions.

But the changes have had little impact on student achievement. In District 5, nearly one-third of third graders scored so low on tests last year that they had to attend summer school. And nearly half of those students failed again in summer school and were forced to repeat third grade. ''Student performance in terms of statewide and citywide standardized tests has been in the toilet for a long time, and this mayoral takeover of the schools has not changed that,'' said Rose Marie Seabrook, the president of the District 5 parent council.

The council is another example of dysfunction, Ms. Seabrook said. It is supposed to have nine elected parent members but has just four. Some were ineligible to serve, while others never showed up. One fifth-grade teacher from District 5 praised the special tutoring program, where she worked on Saturdays and during vacations, but said that the children in her regular classes had not been well served by the mayor's changes. Like many educators interviewed, this teacher asked not to be identified, citing the bitter relations between teachers and the administration and a fear of reprisal by superiors.

''The Saturday program has been really terrific because we work in small groups,'' this teacher said. ''I have been able to do some really impactful things.''

But the teacher said many new programs came without appropriate supplies or sufficient training.

''It seems that everything is being done to say it was done,'' she said. ''We're not getting the support that we need. We're not adequately trained.''

She added, ''Things are worse for these kids since they have been in power.''

In an election-year promise to expand programs for the gifted, Mr. Bloomberg said in January that programs would be created in three District 5 schools. But only a tiny fraction of students are doing well enough to qualify, not enough for even one gifted class per grade.

Mr. Klein cited his own analysis of District 5 test scores, showing overall gains of 4 percentage points in reading and 11 points in math since Mr. Bloomberg took office, and said the mayor's reforms would eventually succeed.

''We are making real investments,'' he said. ''There is no question that in a place like District 5, which has had years and years of dysfunction, of political cronyism, that you can't turn it around overnight.''

In analyzing the scores, Mr. Klein compared results on city and state tests for grades three through eight from the 2001-2002 school year, two years before the mayor's instructional changes were implemented, with 2003-2004. He also adjusted the 2004 scores, removing seventh-grade results, because a scoring problem invalidated seventh-grade results in 2002. The 2002 results from low-performing schools that were in a special ''chancellor's district'' then were added back into District 5's scores in order to compare those results with last year's.

But a comparison of District 5 scores from 2002-2003, the year before the mayor's changes, to 2003-2004 shows that reading scores dropped 2 percentage points and math scores increased 4 points, which is how the scores are reported on the Education Department's own Web site.

Regardless of which analysis is used, Mr. Klein's top testing official, Lori Mei, said that the overall trend in reading scores has been flat since Mr. Bloomberg took office, while math scores continued to rise as they have since 2000 when Mr. Levy became chancellor and made improving math skills a top priority.

Private Sector Help

One promising development in District 5 was the arrival of two privately run charter schools: KIPP Star College Prep Academy and the Harlem Children's Zone Promise Academy. In September 2003, the mayor attended the opening of KIPP Star in rent-free space in the old District 5 offices.

It was one of many visits Mr. Bloomberg has made to schools to highlight good news, and it is these visits that Mr. Klein said better illustrate the mayor's efforts. ''The mayor went to the KIPP school and opened it in Harlem, and there's a shining star,'' he said.

In all, the mayor has spurred $200 million in private donations to the schools, a record sum and a point he stressed in a visit to Public School 17 in Brooklyn, to trumpet a partnership with the Robin Hood Foundation to renovate school libraries.

Corporate donors and private philanthropists remain big supporters of the mayor, even as some conservative commentators who were most excited about mayoral control of the schools have expressed disappointment with the outcome so far.

When the mayor took office, P.S. 87 in Middle Village was a school that worked, parents and officials agreed -- especially for disabled students.

The principal, Arlyn Brody, emphasized new technology and strategies that Mr. Bloomberg said he wanted replicated. The school offered dozens of activities, including computer learning programs, music, sports and yoga.

But when Ms. Brody retired in June 2003, things began to unravel, parents said. Her successor served only one year. This fall a new acting principal started cutting programs, including art, gym and a reading program for students with learning disabilities. An occupational therapist who left for maternity leave was not replaced until parents insisted.

Parents said they complained to the local and regional superintendents, but that nothing was done. ''You held a press conference at our school and praised the success of our special education students,'' Regina Asaro, the parent of a third grader, wrote to Chancellor Klein in late November last year. ''You said we were a model for the city of New York and that other schools should follow our lead. I implore you to step in and prevent this school from failing and failing our children.''

Ms. Asaro and other parents said that things had improved since then, but that not all the cuts had been restored. A hiring committee recently chose yet another principal.

Mr. Klein acknowledged P.S. 87's troubles, but questioned whether the mayor or the chancellor in a system of more than 1,300 schools could be held responsible when, for instance, a principal fails to quickly replace a therapist on maternity leave. ''What are we supposed to do?'' he asked.

''When you have to replace, as we do, 300 principals a year,'' he added, ''there are going to be some bad ones.''

Education experts often point to stable leadership as a crucial ingredient in successful schools. But like P.S. 87, the small high schools that opened in September in the Thomas Jefferson High School building in Brooklyn have experienced turmoil.

Mr. Bloomberg went to Jefferson on the first day of this school year because it offered a powerful symbol, with four small schools opening and extra police presence as part of an aggressive safety initiative.

''We need the kind of leadership, the kind of communities, the kind of change in school culture that these four distinguished school leaders are going to bring to us,'' Mr. Klein said at a news conference that day.

The chancellor did not mention that the founding principal of one of the four schools, the F.D.N.Y. High School of Fire and Life Safety, had been replaced even before school started. By late November, the principal of another school, the World Academy for Total Community Health, was dismissed. And in January, a third principal, of the Performing Arts and Technology High School, was pressured to retire.

Mr. Klein conceded that mistakes were made but also said the administration deserved credit for acting swiftly to fix them. ''The principals who were chosen were not right,'' he said. '' I'll take the responsibility.''

In theory, small schools improve attendance, and most of the city's new small schools exceed the citywide average of about 81 percent. Three of the small schools at Jefferson have attendance between 81 and 87 percent. The fourth, the High School for Civil Rights, (the only one where the original principal is still in place) has attendance of less than 72 percent, five points lower than when Mr. Bloomberg took office.

At the Performing Arts and Technology school, better attendance has not translated into better achievement. Les Ford, the head of the Nia Theatrical Production Company, a community group helping to run the school, estimated that nearly half the 108 students were failing math and science.

Mr. Ford said that three of six teachers had quit, feeling overwhelmed. ''The disruption in leadership creates chaos,'' he said.

Police statistics show an overall decline in behavior problems at Jefferson, but Mr. Ford said the lawlessness that pervaded for years was still an issue.

He said that school officials hoped to visit every child's home in an effort to turn things around. But he added that the small schools needed more support from senior officials.

''I am afraid that we are going to be just four small schools like Thomas Jefferson the big school,'' he said. ''It's different, but I can't say yet that it's better.''