“Remember, it’s Bloomberg,” a mother said early yesterday morning as she dropped off her son at his Bronx elementary school. “The mayor of New York City is Bloomberg.”

Wanda Torres’s son, Noah, is a newly minted fourth-grader at P.S. 62. He had learned that Mayor Bloomberg would be visiting his school yesterday, but he was having trouble remembering the important name.

On a day marked by praise for Mr. Bloomberg’s efforts at improving the schools, Noah may have been the only one to suffer such a memory lapse.

Asked how she likes P.S. 62, Ms. Torres immediately referred to Mr. Bloomberg’s move in 2002 to take control of the school system.

The law that granted the mayor control comes up for renewal in June. The deadline has made the issue of how the public schools should be governed a top talking point this school year.

If Ms. Torres is evidence, mayoral control is also an important topic among ordinary parents.

“I tell you, after the Mayor Bloomberg thing, that he stepped in — awesome,” she said.

She said the school’s staff has improved and that she has a better time navigating the school system in general.

“There’s not as much mayhem,” Ms. Torres said.

Elected officials also heaped praise on Mr. Bloomberg, making choreographed appearances at P.S. 62 and other schools.

“I think if you look around the country, at the layoffs and the problems that are happening, this school system serves as a shining beacon,” the president of the teachers union, Randi Weingarten, who has often criticized aspects of Mr. Bloomberg’s education plans, said.

“We’ve lifted all boats, and the boats are starting to sail,” the president of the principals union, Ernest Logan, another sometime critic of Mr. Bloomberg’s, said.

Assemblyman Ruben Diaz Jr., a Democrat of the Bronx, referred to talk that Mr. Bloomberg is considering seeking a third term as mayor. “Maybe, just maybe, he needs another four years to continue with education,” Mr. Diaz said.

Mr. Bloomberg deployed the strongest language of the day to defend mayoral control, saying that there would be negative consequences if authority over school decisions were devolved from a single source.

“I know of no organization that can possibly be run by committee,” Mr. Bloomberg said. “The Soviets tried it, and I don’t think that worked so well.”

Others raised problems yesterday.

The city’s public advocate, Betsy Gotbaum, said she was concerned by overcrowding after visiting a school in Washington Heights.

On a visit to the Excellence Charter School, which is housed in a sparkling new 90,000-square-foot school building in Bedford-Stuyvesant, the president of Brooklyn, Marty Markowitz, became visibly agitated.

“Listen to me,” he said to the schools chancellor, Joel Klein, as the two toured a classroom, “we have some public schools that are starving for these kinds of resources.”

Mr. Klein replied that some schools are doing as well as Excellence with more modest budgets.

Mr. Markowitz was not convinced; he said that while he supports charter schools, he is “conflicted” about the extra resources they sometimes receive from private donors.

“I really believe the jury is out on this whole thing,” Mr. Markowitz said, walking out the door.