With the new school year that began yesterday, public schools in the city are flush with the most money in their history, but exactly how the new funds will be spent has yet to be negotiated, Governor Spitzer said yesterday.

The extra $1 billion windfall is the outcome of a years-long lawsuit that contended the city’s public schools were underfunded. A large portion of the new money, about $700 million, comes from the state and has strings attached: Before spending the full grant, state education officials must approve a plan proving some of it will go toward goals specified in the funding lawsuit, such as lowering class sizes.

Visiting a Bronx elementary school, P.S. 53, to wish students well on their first day of school, Mr. Spitzer said the spending plan Mayor Bloomberg and his schools chancellor, Joel Klein, sent during the summer has not yet been approved.

Asked whether city and state officials were fighting over the terms, Mr. Spitzer said, “I wouldn’t say that we are. … The contracts are being negotiated.”

The unfinished business was a departure in a start-of-the-year send-off whose theme was cooperation. “There is harmony and there is a partnership among all levels of government,” Mr. Spitzer said. He was standing at a lectern with, among others, Messrs. Bloomberg and Klein; the president of the principals union, Ernest Logan; the speaker of the City Council, Christine Quinn, and the president of the teachers union, Randi Weingarten, whom several city leaders singled out for praise. “Thank you, Randi, and thanks to taxpayers for paying,” Mr. Bloomberg said, announcing a batch of 6,000 new teachers hired this summer using $50,000 bonuses that teachers can use to cover housing expenses.

The tribute was a marked turnaround; just five months ago, in April, the mayor compared Ms. Weingarten’s union to the National Rifle Association.

When a parent in the audience, John Moncrief, asked Mr. Bloomberg about one stakeholder the mayor may have failed to mention — parents — Mr. Bloomberg said he has included families in his changes to the schools.

Mr. Moncrief, the treasurer of a Bronx Community Education Council appointed by the city to advise on education policy, privately called the response “a lot of smoke and mirrors.”

“They’re still not involving parents,” he said. “It should be parents up there instead of a bunch of politicians.”

Another advocate for city parents, Leonie Haimson, said families she spoke to after the school day ended yesterday were not pleased, reporting to her that sizes of their children’s classes had gone up, not down.

Ms. Haimson, whose group, Class Size Matters, has criticized the city’s Contract for Excellence for not laying out a specific plan to lower class sizes, applauded the state for not yet accepting the city’s plan.

Additional accountability will be another theme for the new school year, as the city sends all schools letter grades, A through F, based on a variety of measures, including scores on state tests. Schools graded “D” or “F” will face consequences including the possibility of being shut down and having principals fired.

Yesterday, the principals union leader, Mr. Logan, said he embraced the new accountability. “Give us the resources,” he said, “and we will make our city shine once again by having the best schools in the world.”

Just more than half of schools receive more funding under the city’s new formulas; the rest, deemed sufficiently funded in years past, are getting no extra money. The principal at P.S. 53, Collin Wolfe, received almost $440,000, Mr. Bloomberg said.

Mr. Wolfe said he is using the money to help “buy” three new employees: a data coach; a band teacher, and an enrichment teacher.

“I have bought in entirely to the chancellor’s belief in putting children first,” the principal said.

Meanwhile, a group of protestors marked the first day of school with a silent vigil at a Brooklyn middle school followed by a protest outside City Hall. The protesters were calling on the Department of Education to close an Arabic-language school in Boerum Hill, Brooklyn, the Khalil Gibran International Academy.

Some parents at Khalil Gibran had said they were concerned about the first day of school, but a spokeswoman for the Department of Education, Melody Meyer, said the day “went smoothly and without incident” — despite a swarm of television and press reporters greeting students as they arrived.