When U.S. schools reopened for in-person learning last year, many districts had clear metrics for thresholds that would trigger school closures due to Covid-19. This year, more are taking a wait-and-see approach.

As districts around the country reopen amid rising cases and hospitalizations, many haven’t developed formal contingency plans that lay out what it will take for them to close a school or quarantine students. Officials say they are wary of setting benchmarks that may repeatedly change as they did last year. The contingency plans, or lack thereof, have some parents clamoring for more transparency while others advocate for schools to remain open as much as possible.

School and health officials in a range of cities including San Francisco, Dallas and Miami said they would make decisions to quarantine or close schools due to the coronavirus based on each situation.

“There’s no magic number,” said Jennifer Finley, the health services director at Dallas Independent School District. Ms. Finley said a host of issues, including staffing flexibility across the district’s 230 schools, make it difficult to settle on a predetermined scenario that would trigger a closure.

Nationwide, officials said they would be weighing factors such as infection and vaccination rates, the guidelines from the Centers for Disease Control and Prevention and their local health department’s guidance.

Students were welcomed to a school in Dallas on Aug. 9. Officials in the city will look at myriad metrics to determine whether a closure is warranted.

Photo: Brandon Wade/Associated Press

Families, politicians and educators are determined for in-person classes to return after last year’s constant disruptions, but the spread of the Delta variant has complicated reopenings. The variant accounted for 98% of new cases by Aug. 14, and the number of new cases on a seven-day, rolling average in the U.S. rose more than three fold from 39,066 cases on July 19 to 130,926 cases on Aug. 19, according to CDC data. The number of new hospitalizations of patients with Covid-19 more than doubled in the same period.

The CDC in July changed its guidance for masks in schools, recommending that everyone in K-12 schools wears one indoors regardless of vaccination status.

Meanwhile, districts around the country are already seeing quarantines and closures due to the virus. More than 10,700 students and school employees in Hillsborough County, Fla., last week were either in isolation because of a positive Covid-19 test or in quarantine due to potential exposure. And in Maricopa County, Arizona’s most populous county that includes Phoenix, as of Aug. 20 there had been at least 126 outbreaks at K-12 schools since July. Students accounted for 83% of the 353 cases.

CDC data shows that child-related Covid-19 cases are rising. The agency reported the seven-day average of child hospitalizations between Aug. 7 and Aug. 13 was 272, up from the previous seven-day average of 223.

As the Delta variant sweeps the globe, scientists are learning more about why new versions of the coronavirus spread faster, and what this could mean for vaccine efforts. The spike protein, which gives the virus its unmistakable shape, may hold the key. Illustration: Nick Collingwood/WSJ The Wall Street Journal Interactive Edition

In New York City, the largest public school system in the country, officials said they are following summer school closure and quarantine policies and will announce any changes or additions for the fall soon.

Chicago public school policy doesn’t identify a threshold for closing schools and says a “brief pause in an entire school’s operations is a serious step, but may be necessary in some circumstances.”

In the event of more outbreaks, districts in states including Florida, Texas and Mississippi said they would close schools on a case-by-case basis as needed. Some parents said they wished they heard more about what would be done.

Elizabeth Radi, of Collier County’s Naples, Fla., said she was concerned crowding on her son’s bus and in his school could lead to infection. “Our schools did not plan for this,” said Ms. Radi, who has a 16-year-old son in a public school and a 7-year-old granddaughter in a private school.

Elizabeth Radi, of Naples, Fla., said she was concerned about crowding on her son Jeremiah’s bus.

Photo: Elizabeth Radi

Chad Oliver, a spokesman for the Collier County school district, said bus drivers space students apart, when possible, and that educating children in a pandemic is “a fluid process.”

Natalya Murakhver, a New York City parent, said she is eager for her two children, ages 7 and 11, to get back into in-person learning. She acknowledged potential risks of the virus but said children under age 12 have a very low risk of serious illness.

Ms. Murakhver said she believes stress from disruption and isolation is threatening the cognitive, social and emotional development of children. “We need to make every effort possible to restore normalcy for kids,” she said.

Districts that have already reopened for the fall are adding more preventive measures. Some, in Texas, Arizona and Florida, are mandating indoor masking in defiance of state laws and executive orders.

Culver City Unified District, outside of Los Angeles, said on Tuesday it will require students who are 12 and older to receive the Covid-19 vaccine by mid-November to help avoid closure decisions, likely making it one of the first public school districts to require students to be inoculated.

At Dallas Independent School District, school officials will be looking at myriad metrics to determine whether a school campus closure is warranted, such as how many linked cases are sprouting within a campus and absenteeism among quarantining staff that might make it difficult to provide in-person learning, said Ms. Finley, the district’s health services director.

Three linked cases discovered on one campus would prompt a closer analysis of that situation, Ms. Finley said. The district’s ability to trace known positive cases is another pivotal factor, she said.

An elementary school in the Bronx in New York City last week. City officials said they were following summer school closure and quarantine policies.

Photo: Brittainy Newman/Associated Press

The district, which started reopening campuses Aug. 5, had 506 positive cases between Aug. 1 and Aug. 20, including 213 students, according to the district dashboard. But there haven’t been any situations that have resulted in school closures yet, according to a district spokeswoman.

Some districts that have already seen closures made the decision based on how the virus spread. Babcock Neighborhood Schools, which has a K-8 school and a high school in Florida’s Charlotte County, announced last week that it would be closing from Aug. 16 through Aug. 25.

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A spokeswoman said the charter school doesn’t have a set threshold for when people should quarantine and buildings must close. But when 25% or more teachers and staff were absent last week, the school seemed to have reached the tipping point, she said.

Amid a shortage of substitute teachers in Southwest Florida, Babcock was forced to close, the school’s principal Shannon Treece said.

“We no longer had enough people to stretch and cover all the positions,” she said.

Write to Ben Chapman at Ben.Chapman@wsj.com, Lee Hawkins at lee.hawkins@wsj.com and Yoree Koh at yoree.koh@wsj.com