As the city’s 1.1 million public schoolchildren returned to the classroom, the United Federation of Teachers yesterday ratcheted up its pressure on the Bloomberg administration to add more teachers to the union’s ranks.

Joined by a bevy of Democratic politicians and parent leaders, the president of the teachers union, Randi Weingarten, stood in front of a middle school in Lower Manhattan and demanded that Mayor Bloomberg take measures to reduce class sizes in the city.

Attending the event was the running mate of gubernatorial candidate Eliot Spitzer, state Senator David Paterson, who said that while the city should “prioritize” efforts to shrink class sizes he didn’t think the city should be required to spend part of its education budget on reductions.

The battle over class sizes is emerging as a front-burner issue, with the mayor and the union split over how potential extra operating money from a settlement in the Campaign for Fiscal Equity schools financing case should be spent. Ms. Weingarten said yesterday that she wants the education department to hire enough teachers to get class sizes in line with the rest of the state. The UFT says city class sizes are more than 50% higher than the statewide average of 20 students.

Bloomberg officials are resistant to the idea of a wide-scale hiring effort and say they would rather focus resources on attracting more competitive teachers through signing bonuses and differential salary scales for math and science specialists.

Meanwhile, across the Brooklyn Bridge, Mr. Bloomberg took attendance in a first-grade class during a visit to P.S. 8 in Brooklyn Heights, where he spent the morning speaking with elementary school students on their first day back after summer vacation.

Department of Education officials tout community involvement as a catalyst for improved standardized test scores — and morale — at P.S. 8, and the mayor said being there inspired “real confidence that we’re on the right track.”

City principals and associate principals, represented by the Council of School Supervisors & Administrators, yesterday entered their fourth year without a contract. Tension between the city and union leaders reached a fevered pitch last week, when Chancellor Joel Klein sent a letter to principals saying that he would spend more than $5 million to place 44 “excessed” assistant principals — those whose jobs may have been cut as a result of budget cuts or school restructuring — in unneeded office jobs rather than force them on schools that didn’t want them.

The number of assistant principals reassigned to office jobs is now at 52, a Department of Education spokesman, Keith Kalb, said.

Yesterday, Mr. Klein took a whirlwind, five-borough tour, making stops in classrooms at P.S. 31 in northeast Staten Island, the Girls Preparatory

Charter School in Manhattan’s East Village, three small high schools that share a South Bronx campus, and M.S. 137 in Ozone Park, Queens, in addition to P.S. 8.

The South Bronx high schools he visited are among the city’s 331 “empowerment schools,” which are given more control over in-house decision-making in exchange for increased accountability for student performance.

During a press conference in front of the Girls Preparatory Charter School — a three-year-old institution that serves about 150 girls in kindergarten and grades one and two — the chancellor said “downsizing bureaucracy,” heightening accountability, and setting up more charter and small secondary schools were among his priorities.

One of a dozen new city charter schools to open yesterday was the United Federation of Teachers Secondary Charter School in the East New York section of Brooklyn. The school had received nearly 700 applications for 125 sixth-grade slots in the charter school, which will eventually serve students in grades 6 to 12.

The opening of this school — the UFT debuted its elementary charter school, also in East New York, last year — should help quash the notion that the teacher’s union opposes charter schools, the UFT’s Ms. Weingarten, who visited the new school yesterday, said. She referred to the two schools as proof that the charter model is possible “within the confines of a labor agreement.”