The superintendent of the New Rochelle schools, just north of New York City, was clear: She was not closing schools just because there were cases of covid-19 in the community. That was her plan right up until Gov. Andrew M. Cuomo (D) ordered a different one.

The clash between the superintendent and the governor is a stark example of the tensions playing out in school districts throughout the country. Public health officials and school administrators are struggling to balance the ramifications of closing a school on children and families against the imperative of stemming a cascading public health crisis.

In Washington state, a center of the outbreak, a suburban district was closed, but Seattle Public Schools said it would remain open — until Wednesday, when Seattle set aside concerns about inconvenience and access to learning and opted to close for two weeks in the face of a heightened crisis. Washington Gov. Jay Inslee (D) advised all schools to prepare contingency plans in case they, too, need to close.

In New Rochelle, three schools within a one-mile radius have closed under orders from the state, but the rest of the district’s buildings are open. Many schools across the country are closing for a day or two for “deep cleaning,” although some experts wonder whether that is necessary or effective.

In other systems, including D.C. Public Schools, classes are being canceled for a day to give teachers time to prepare for distance learning, should schools be forced to close. In New York City, schools remain open, but parent-teacher conferences scheduled for Thursday and Friday will be conducted online or by phone.

As of Wednesday, 1,251 public and private schools have been closed or are scheduled to close, affecting more than 856,000 students, according to a tally by the journal EdWeek.

Globally, 22 countries — including China, Italy and Iran — closed all of their schools, leaving some 372 million children out of the classroom, according to the United Nations. Seventeen other countries closed some schools.

Covid-19, the illness caused by the novel coronavirus, has confounded virologists because it appears to be sparing children, who develop far milder symptoms than older adults. In China, where the virus was first detected, there have been no reported deaths of young children, according to the World Health Organization. But experts are unsure how easily children might be able to transmit the virus to others.

Public health experts, school administrators and parents are divided about the risks and benefits of school closures, sparking debate in communities where the virus is rampant and in places with only fleeting exposure.

Even the nation’s top disease experts seem unable to provide clear guidance. Anthony Fauci, the director of the National Institute of Allergy and Infectious Diseases, said Tuesday that a decision about closing schools depends on how far the disease has spread in a community. If the virus has spread little, there’s no point in closing, he said. But wait too long, and closings will not help.

“Everything is on the table for consideration,” he said. Closing all schools in the country would not be appropriate, he said, but it might make sense for a community “when you start to see, ‘we’re getting a little bit of danger here.’ ” The goal, he said, is to “do real mitigation sometime before you think you really need it.”

The reactions by districts to similar facts have varied widely.

The Northshore School District in suburban Seattle announced a week ago that it was closing for two weeks. That decision was made easier because the district is home to many affluent families employed by high-tech companies. The district offered laptops and WiFi hotspots to students who requested them to accommodate remote learning.

“We clearly have more resources than other districts might have,” Schools Superintendent Michelle Reid said.

But in Seattle, the state’s largest system had remained open. Officials cited factors including their inability to provide a laptop to every student who might need one.

On Wednesday, the district reversed course and said schools would close Thursday for at least 14 days.

“Together, we are facing an unprecedented health crisis in our community,” the Seattle Public Schools said in a notice to the community. “The decision to close the district was extremely difficult.”

The notice explained that though no serious covid-19 illness has been reported in children, the district must also protect adults who work in schools.

The decision to stay open had upset parents including Jan Munger, who was going to keep her children out of school even though schools were open.

“It is absolutely irresponsible to keep the schools open without stepping up sanitation in the spaces students spend most time in,” she said before the reversal was announced. After Wednesday’s announcement, she said, “That is a huge relief.”

Decisions of all kinds have drawn fire, from parents who think closed schools should be open and from parents who think open schools should be closed.

In the Elk Grove Unified School District, south of Sacramento, some parents were furious after officials announced plans to move spring break up to this week. That change came after two adults in a family with children in the school system tested positive for covid-19. Others with connections to the school system also may have been exposed to the novel coronavirus.

Schools Superintendent Christopher R. Hoffman said Saturday that the district was canceling all school activities — including a high school prom, set to take place hours later, and a playoff game for the Sheldon High School boy’s basketball team, which was defending its state title.

“I can’t tell you for sure that it’s the best decision,” Hoffman told a news conference. “This is something that has been weighing on us for weeks now.”

Hoffman said moving spring break would give him and other school officials “time to take a breath, and get a little more information.” Later, an elementary school-aged child, who was living with two adults with the infection, tested positive for the virus.

The move to cancel the basketball game infuriated parents and the team’s players, who were looking forward to playing in front of college recruiters with the hope of securing scholarships.

“You have kids who are depending on this basketball tournament to go to college,” Marvin Bagley, whose son plays on the team, told elected officials at a news conference Sunday.

In the end, the league reinstated the game, which Sheldon won, 59-58.

The questions were also intense in New Rochelle, the New York City suburb where more than 100 people have tested positive. The community became a center of the coronavirus outbreak after a 50-year-old lawyer contracted the virus and spread it to others at his synagogue and in his neighborhood.

Nonetheless, the superintendent of the New Rochelle schools said she would not close the district. Her plan was to close a school for cleaning if a teacher or student in the building was diagnosed with the virus. Closure, she said, would be particularly difficult for families without alternative child care and for those who depend on school for meals. She said students would be likely to congregate anyway outside of school, meaning the virus could still spread.

“From a strategic long-term perspective, closing schools has no ‘end game.’ Once schools reopen, a student or staff member could test positive and so the cycle of school closures would continue,” Schools Superintendent Laura Feijóo wrote in an email to parents Monday.

She was forced to change course the next day by the governor, who announced a one-mile-radius containment zone in New Rochelle. Now, three of the district’s schools are closed for two weeks, affecting just over half of its 10,500 students, while the other campuses remain open.

Cuomo acknowledged the difficulty in setting the one-mile-radius containment zone.

“You have people who say it shouldn’t be one mile, it should be two miles, it should be three miles,” he said. “You have other people. We call them deniers. ‘This is not a big deal, it’s all going to be fine, you don’t have to do anything, don’t close anything, don’t disrupt anything.’ So you have opinions all across the board.”

Michael LePore, 45, who has three children attending closed schools in New Rochelle, asked why any district school buildings were allowed to stay open. “Why are we sacrificing health and safety for some and not all?” he asked.

The teachers union also argued that all schools should close. Billy Coleman, the executive vice president of the New Rochelle Federation of United School Employees, said the union was asking the governor to expand his order. He said many families have students in multiple schools, so closing all schools is necessary to halt the spread of the virus.

“Closing three — we don’t see any purpose in that,” he said.

But some parents perceived the closure of any schools as an overreaction.

“I feel just because it’s unknown, everyone is overreacting,” said Jamie Abel, 28, whose 6-year-old daughter attends a school that is now closed. “They should be in school learning.”