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The mayor set the 3 percent threshold over the summer, when average positivity rates were hovering around 1 percent or below. He has been explicit that the number is less of a strictly scientific measure and more of a symbol intended to reassure parents, educators and the union.

In a recent interview, Michael Mulgrew, the union’s president, said he thought the 3 percent threshold was sound. He cited warnings from experts that even if there was low transmission in schools, infection could still spread from the broader community into schools, increasing the likelihood that students and staff members might carry the virus into their homes and neighborhoods.

Mr. Mulgrew said he was dismayed that schools were closing so soon, and asserted that expressions of frustration about a shutdown from some New Yorkers seemed hypocritical.

“We had a lot of criticism from people when we were opening schools,” he said. “They didn’t want them open. A lot of that came from the very same people who are yelling now that they want them open.”

He also called on New Yorkers to take the virus seriously in order to drive down numbers again. “If we want to keep our schools open, it’s up to everyone else” to take precautions, Mr. Mulgrew said.

While city parents have come to expect contradictory messages about schools from the mayor and governor, Mr. Cuomo said during a recent news conference that he did not plan to intervene in the city’s effort to close schools.

But he said “the problem is not coming from the schools,” and encouraged the city to come up with new metrics for a shutdown once schools reopen. The governor has said the state will force schools to close only if they are in a region where the seven-day positivity rate reaches 9 percent.