Not a math or physics test. A coronavirus test — one she would administer herself.

Ms. Hammermeister, a 17-year-old high school junior, entered the tent erected in the schoolyard along with some classmates — all standing six feet apart — and picked up a test kit. She inserted the swab deep into her throat, gagging slightly as instructed, then closed and labeled the sample before returning to class.

It took less than three minutes. The results landed in her inbox overnight. A positive test would require staying home for two weeks. Ms. Hammermeister tested negative. She now wears a green sticker that allows her to move around the school without a mask — until the next test four days later.

“I was very relieved,” she said happily. In addition to feeling safe around her classmates and teachers, who all tested negative, she feels like less of a risk to her grandmother, who eats with the family every day.
The self-administered test at the high school in Neustrelitz, a small town in northern Germany, is one of the more intriguing efforts in Europe as countries embark on a giant experiment in how to reopen schools, which have been shuttered for weeks and which are now being radically transformed by strict hygiene and distancing rules.

Restarting schools is at the core of any plan to restart economies globally. If schools do not reopen, parents cannot go back to work. So how Germany and other countries that have led the way on many fronts handle this stage in the pandemic will provide an essential lesson for the rest of the world.

“Schools are the spine of our societies and economies,” said Henry Tesch, headmaster of the school in northern Germany that is piloting the student tests. “Without schools, parents can’t work and children are being robbed of precious learning time and, ultimately, a piece of their future.”

Countries across Asia have already been making the leap, experimenting with a variety of approaches. In China, students face temperature checks before they can enter schools, and cafeteria tables are outfitted with plastic dividers.

In Sydney, Australia, schools are opening in staggered stages, holding classes one day a week for a quarter of the students from each grade. Hong Kong and Japan are trying similar phased reopenings. In Taiwan, classes have been in session since late February, but assemblies have been canceled and students are ordered to wear masks.

For now, Europe is a patchwork of approaches and timetables — a vast laboratory for how to safely operate an institution that is central to any meaningful resumption of public life.

In Germany, which last week announced that it would reopen most aspects of its economy and allow all students back in coming weeks, class sizes have been cut in half. Hallways have become one-way systems. Breaks are staggered. Teachers wear masks and students are told to dress warmly because windows and doors are kept open for air circulation.

Germany is keeping a wary eye on the rate of virus spread as it moves to reopen.

Germany has been a leader in methodically slowing the spread of the virus and keeping the number of deaths relatively low. But that success is fragile, Chancellor Angela Merkel has warned.

On Saturday, the reproduction factor — the average number of people who get infected by every newly infected person — which the government wants to stay below 1, crept back up to 1.13.

With still so little known about the virus, many experts say mass testing is the only way to avoid the reopening of schools becoming a gamble.

The school in Neustrelitz is still an exception. But by offering everyone from teachers to students free tests twice a week, it is zeroing in on a central question haunting all countries at this stage in the pandemic: Just how infectious are children?

Evidence suggests that children are less likely to become seriously ill from Covid-19 than adults. But small numbers of children have become very sick and some have died, either from the respiratory failure that causes most adult deaths or from a newly recognized syndrome that causes acute inflammation in the heart.

An even greater blind spot is transmission. Children often do not have symptoms, making it less likely that they are tested and harder to see whether or how they spread the virus.

The prospect that schoolchildren, well-documented spreaders of the common flu, might also become super spreaders of the coronavirus, is the central dilemma for countries looking to reopen while avoiding a second wave of deadly infections. It means that school openings could pose real dangers.

“That’s my biggest fear,” said Prof. Michael Hoelscher, head of infectious diseases and tropical medicine at Munich University Hospital, who oversees a household study in Munich that hopes to shed light on transmission inside families.

Manfred Prenzel, a prominent educationalist and member of a panel advising the German government on its reopening, said children represent the most intractable aspect of this pandemic: asymptomatic transmission.

A study published in Germany last week by the country’s best-known virologist and coronavirus expert, found that infected children carried the same amount of the virus as adults, suggesting they might be as infectious as adults.

“In the current situation, we have to warn against an unlimited reopening of schools and nurseries,” concluded the study supervised by Christian Drosten at the Berlin-based Charite hospital.

The Robert Koch Institute of public health, Germany’s equivalent of the C.D.C. in the United States, found that children get infected in roughly equal proportions to adults.

Other studies, including two from China, suggest that children may be less contagious than adults, possibly because they often do not have the symptoms that help spread it, like a cough. Researchers in Iceland and the Netherlands did not identify a single case in which children brought the virus into their homes.

“The evidence is not yet conclusive,” said Richard Pebody, team leader for high threat pathogens at the World Health Organization. His advice on school openings: “Do it very gradually and monitor the ongoing epidemiology very closely.”

That is easier said than done.

For now, Europe’s school openings are as varied as its countries. Denmark opened primary schools and nurseries first, reasoning that young children are the least at risk and the most dependent on parents, who need to return to work. Germany allowed older children back to school first because they are better able to comply with rules on masks and distancing.

France is opening preschools on Monday before phasing in primary and middle school children later in the month. High school students will keep learning remotely for now.

Belgium, Greece and Austria are all resuming lessons for select grades in coming weeks. Sweden never closed its schools but has put in place distancing and hygiene rules. Some hard-hit countries like Spain and Italy are not confident enough to open schools until the fall.

One precondition for any country to open schools, epidemiologists say, is that community transmission rates be at manageable levels.

Early evidence from countries that have led the way in lowering community transmission and opening schools looks hopeful, said Flemming Konradsen, director of the School of Global Health at the University of Copenhagen.

Denmark, after letting younger children back more than three weeks ago, announced last week that the reproduction factor of the virus remained below 1. Older students will be allowed to return to school on Monday.

Germany, Europe’s biggest country, announced last week that all children would see the inside of a classroom again before the summer break after a two-week trial run in high schools had not stopped overall transmission numbers from falling. Officials hope the rise that was reported over the weekend was a blip instead of a sign that the loosening is already reviving the spread of the disease.

Many argue the benefits of opening schools — to economies, parents and the children themselves — far outweigh the costs so long as hygiene rules are put in place. Disadvantaged children in particular suffer from being out, said Sophie Luthe, a social worker at a Berlin high school.

“We have been losing children; they just drop off the radar,” Ms. Luthe said. “School is a control mechanism for everything from learning difficulties to child abuse.”

But teaching in the time of a pandemic comes with a host of challenges: In the high school in Neustrelitz, roughly a third of the teachers are out because they are older or at risk.

There are not enough classrooms to allow all 1,000 students to come to class and still keep six feet apart, which means at most a third can be in school at any one time. Teachers often shuttle between classrooms, teaching two groups at once.

At the same time, the virus is spurring innovation.

Teachers in Denmark have moved a lot of their teaching outdoors. German schools, long behind on digital learning, have seen their technology budgets increase overnight.

“Corona is exposing all our problems,” Mr. Tesch, the headmaster in Neustrelitz, said. “It’s an opportunity to rethink our schools and experiment.”

That’s why he did not hesitate when an old friend, who co-founded a local biotechnology company, offered the school free tests for a pilot. Mr. Tesch said he hoped the testing would allow him to increase class sizes safely and restart activities like sports and the orchestra.

Many experts advocate more testing in schools but so far it remains the exception. Luxembourg, tiny and wealthy, tested all 8,500 of its high school seniors before opening schools to them last Monday.

Some students and teachers in Neustrelitz were skeptical when they first heard that the school would offer voluntary biweekly tests.

“I didn’t want to do it at first,” recalled Kimberly Arndt. “I thought, ‘What if I test positive? I’d be pegged as the girl with corona.’”

The incentive to test is high: A negative result allows students to wash and disinfect hands in bathrooms where lines are much shorter. Corona-negative students do not have to wear masks, either.

Mr. Tesch, the headmaster, acknowledges that his school is able to test only because he was offered free kits. Normally they would cost around 40 euros, or $44, a piece. But the government, he said, should consider paying for similar testing at all schools.

“It’s a lot of money,” he said, “but it’s cheaper than shutting down your economy.”