Hundreds of thousands of elementary school students headed back to classrooms Tuesday as New York City enters a high-stakes stage of resuming in-person learning during the coronavirus pandemic, which is keeping students at home in many other big U.S. school systems.

Twice delayed, the elementary school reopening comes over objections from school principals who said the city's complicated, changing plans put them in a staffing bind. Mayor Bill de Blasio said ahead of the reopening there would be no more delays, including for middle and high schoolers set to return on Thursday.

"Let's be honest, the first days will take some adjustment, there will be a transitional reality. The first days are not going to be everything we want them to be. It's going to take several weeks -- with the in-person, with the blended online, with the fully remote -- it's going to take time to get them all to the level we want to get them to, because it's an unprecedented and massive endeavor," De Blasio said at his daily news conference, assuring parents the city would get there eventually.

That said, he also acknowledged concerning clusters in Brooklyn and Queens that are now beginning to affect the city's overall positivity rate. If that positivity rate hits 3 percent citywide over a seven-day rolling average, schools will be closed -- not just in those neighborhoods, but for the whole city. (It hit 1.38 percent Tuesday.)

For now, as many as half a million students could be in New York City public schools this week for the first time since March. De Blasio and Schools Chancellor Richard Carranza welcomed some of them in person in Manhattan early Tuesday.

With over 1 million public school students, New York City initially had a more ambitious timeline than many other big U.S. school systems for bringing children back to schoolhouses this fall. Families have the option of choosing all-remote learning, and a growing number are doing so — 48% as of Friday, up from 30% six weeks earlier, according to city Education Department statistics.

Other students are already back in the city's virus-altered version of in-person school, learning sometimes in classrooms and sometimes at home.

Pre-kindergarteners and some special education students began showing up Sept. 21 as online instruction began for the rest of the student body.

Students were originally due back Sept. 10. But the start date was pushed back, repeatedly, after the city teachers' union said it wasn't safe to open schools because of outdated ventilation systems, an insufficient number of school nurses and other issues. At one point, the United Federation of Teachers threatened to strike.

The union was still pressing for changes as recently as Friday, when the city agreed to let more teachers work from home when instructing students remotely, rather than having to come in to school to conduct online classes.

The principals' union said the late-breaking change was too much. Principals had already complained that the city was creating a staffing crunch by planning to have three different groups of teachers — one for all-remote students, another for in-classroom pupils and a third for blended-program students when they're at home.

Saying that de Blasio and Schools Chancellor Richard Carranza “have entered into grossly irresponsible staffing agreements," the Council of School Supervisors and Administrators called Sunday for the state to take control of the school system for the duration of the pandemic.

Gov. Andrew Cuomo said Monday that he understood the concern of the principals' union and that the state would monitor virus testing data to determine whether any steps need to be taken concerning New York City schools.

He also rolled out a stabilization plan for New York City's recovery Tuesday that starts with schools, even before dealing with crime or the economy.

Many other big school systems around the country began the fall term online, though some are reopening physical schools. In Florida, for instance, students opting for in-person learning returned to schools Sept. 21 in Palm Beach County, where the nation’s 10th largest school system has over 197,000 students.