Sitting on a wooden chair in the green-carpeted reading area of Ms. Perez’s first-grade class, Mayor Bloomberg welcomed 18 students yesterday morning to the first day of school at P.S. 376 in Brooklyn.

After reading from “Chrysanthemum,” he fielded questions from the gifted-and-talented class about toothfairy prices and bullies. One first grader said he had hoped to see a Democratic mayoral candidate, Rep. Anthony Weiner, instead.

By most accounts, however, yesterday was one of the smoother opening days in recent years.

“Today we welcome 1.1 million children to their first day back to school, and they are returning to a system changing for the better,” Mr. Bloomberg later told reporters in the library at Intermediate School 131 in the Bronx.

“With test scores improving, graduation rates increasing, schools getting safer, and more classroom seats being added, our schools are definitely moving in the right direction,” he said.

This year will bring 73 new public schools, including 53 high schools, 15 charter schools, and five elementary schools, for a total of more than 1,400.

The Department of Education also opened 69 new gifted-and-talented classes in neighborhoods including the South Bronx, Long Island City, and Washington Heights, where students previously had little access to programs for the gifted.

The president of the teachers union, Randi Weingarten, spent yesterday welcoming students at the new United Federation of Teachers Elementary Charter School in Brooklyn, which boasts of being the first union-run school in the country.

The UFT was granted a five-year charter on July 21 by the state Board of Regents and spent the past two months transforming a rundown floor of I.S. 292 in East New York.

“This school is an oasis,” Ms. Weingarten said, sitting in one of the newly painted classrooms that are covered with colorful bulletin boards.

The school opened with 150 kindergartners and first-graders dressed in green-and-blue UFT Elementary Charter School shirts. The school will eventually grow to include classes up to fifth grade.

At the helm, the school operates with a “school leader” instead of a principal. Students are called “scholars.” A business manager functions in place of a vice principal.

More than 600 teachers applied for 14 slots, including Lucy Ogando, who said that teaching in the Bronx until last year she was shelling out hundreds of her own dollars to buy books, paper, and other supplies.

“Here we have a large budget to use for our classroom,” Ms. Ogando said. She was given $800 to buy classroom supplies, compared to just $200 last year.

Despite the optimistic outlook at the UFT’s own school, teachers across the city returned to school for the third year with no contract.

“The talk all over is why won’t the chancellor negotiate a contract? Why won’t the mayor negotiate a contract? Especially when the mayor is running on the record of teachers from last year. That’s the talk that has eclipsed everything else,” Ms. Weingarten said.

Both the mayor and Chancellor Joel Klein said yesterday that they were moving toward negotiations with the union.

“I can’t tell you that there is a day that goes by that there isn’t a conversation within the administration and probably also with the UFT. It would be great if we could come to an agreement,” Mr. Bloomberg said.

Even as school opened yesterday, more than 700 of the 6,300 new teachers were still awaiting classroom assignments. They spent the day at one of the school system’s 10 regional offices. Another 700 “excessed” teachers reported to their old schools yesterday even though their posts had been cut.

“Teachers who are waiting for a position, it’s a hardship for them, it’s a logistical nightmare for the school system to be placing that many people,” City Council Member Eva Moskowitz, chairwoman of the council’s Committee on Education, said.

Ms. Weingarten put the number of unassigned teachers at closer to 2,000.

A Department of Education spokesman, Keith Kalb, said school officials were confident the teachers would all have posts in the upcoming weeks.

During Mr. Klein’s five-borough tour of city schools, he stopped for lunch with Mr. Bloomberg at the Queens Vocational and Technical High School, which until this year had no cafeteria. Facing a row of cameras, the pair selected sandwiches, salad, and pints of milk while their teenage companions all opted for pizza and french fries.

But Mr. Klein seemed to really hit his stride at the College of Staten Island High School for International Studies, a new small high school that has a partnership with the Asia Society. The school offers Japanese, Mandarin, and Spanish language instruction.

When asked where they would have enrolled if the new school had not opened, many ninth-graders ticked off names of parochial and private schools, which caused Mr. Klein to grin.

“We’re taking everybody from the private schools into the public schools,” the chancellor said.

Before leaving he offered up his email address, jklein@nycboe.net, and encouraged students to write him.