The back-to-school bustle in Berlin this week is anything but banal: As thousands of young refugees take new seats in classrooms, Germany is facing the ultimate test of integration.

If the refugee crisis at the start of the last academic year played out at borders and shelters, this year the drama has shifted to the school building. While the number of arrivals has slowed considerably, it continues to grow. And many initially separated into “welcome” training to learn basic German are now moving into regular classes.

The challenges are enormous, with teacher and space shortages and soaring costs. And while Chancellor Angela Merkel’s oft-repeated mantra that Germany “can manage” will only be proven in the long-term – and is dogging her politically in the meantime – in schools, the theory is being put to the test already. Many say that if schools fail, so too will integration.

Matthias Nowak, of the Catholic nonprofit Maltese International that runs one of the largest refugee centers in Berlin, says that if they succeed, however, so too will Germany. On a recent day as young boys and girls mill about the halls after school, a lone boy hunkers down to do his homework. “They are the most important people here,” Mr. Nowak says. “They are the integration motor in this town.”

Their experience at school, he says, becomes a pathway to integration for the entire family.

The cost of education

Of the nearly 1 million asylum seekers who arrived in Germany last year, about 300,000 are school-aged. That number has grown to about 400,000 this year, due to continued arrivals. Much of it has been experienced as a surge. In Berlin, the number of refugees and asylum seekers in the system doubled between December 2015 and the start of this year, from 6,000 to 12,200.

Dieter Dohmen, president of the Research Institute for the Economics of Education and Social Affairs in Berlin, says their academic integration, from kindergarten through higher education, could cost Germany 67 billion euros ($75 billion), based on the expected number of 3.6 million asylum seekers through 2020. He adds that the system would need 15 billion to 16 billion euros ($17 billion to $18 billion) to educate them properly this year alone, or 10 percent of the annual education budget for all 16 states.

One of the biggest challenges is a lack of space. At the Berlin refugee center, Mr. Nowak says that of the 140 school-aged children here, 120 are in school while the center scrambles to find places for the others.

In Berlin, the teacher’s union says 80 more schools will need to be built by 2030, according to Juliane Zacher, a high school teacher who chairs the school politics committee at the trade union GEW. “We really need more resources to successfully cope with this,” she says.

Experts estimate Germany faces a shortage of 25,000 teachers, with 1,100 more needed in Berlin.

Josef Kraus, head of the teacher’s union GTA, says there is also need for more interpreters, social workers, and psychologists “in order to attend to publics traumatized by war and their flight,” he says.

Endless challenges

Mohammed Alsahn, a teenager from Syria, has been at the private Freie Waldorfschule Kreuzberg in Berlin since February, after the school agreed to accept refugee students. He says his biggest fear was not lack of German but the stutter he developed during the war. "I prepare all the sentences in my head, but then they can't come out. Still I've learned to be cool, quiet, and relaxed,” he says.

He deems his integration a success so far. "At first it was strange to be in a class here and not understand anything. That's not a nice feeling,” he says. Now he has an upper intermediate level of German and aims to be a doctor.

But challenges for staff are endless. Wassima Shulz, who heads the “Welcome class” at his school, says back-to-school is a state of flux. Two new kids arrived in her class this week alone. “You never know when someone is coming. They’ll say, here is your new one. This is the moment when you say to yourself, ‘calm down,’ and turn to the student and say, ‘we welcome you,’” she says. Trained in psychology, she says the trauma many students face is daunting. “Some are asking, ‘why am I here? Are my friends still alive?’”

“Some of them have a gap in their education," says Raquel Alvarez, a Spanish teacher who also is now teaching German as a foreign language. And teachers have to deal with varying levels of capability.

“You have to be so flexible,” says Ms. Shulz.

Armin Himmelrath, who co-authored the book “The Refugees Are Here – How Refugee Children Change and Improve our School System,” says Germany has shown flexibility and stands to gain. Schools were overwhelmed last year, and unprepared, but they have quickly spurred into action. “The students were there and the teachers were sitting in front of them. Schools started to change without a plan from the top,” he says.

He also says refugees have accelerated conversations already taking place, such as whether Germany's educational routing of students, locking them into vocational training or a higher ed path, is working and fair.

It’s also motivated some of the nearly half-million students in Berlin. Recent high school graduate Joshua Kriesmann, in the middle of his busy senior year last year, said while his school had asylum seekers in “welcome classes,” they remained apart. So he and classmates organized meetups – over cooking or German lessons – that he has now turned into the NGO named Students Meet Refugees. With the support of the US Embassy in Berlin, he is training 30 students to multiply projects on integration.

“Students have gotten a new perspective on civic engagement and voluntary work,” he says in perfect English with a slight twang after spending an exchange year in Oklahoma.

'This is a marathon'

But challenges remain.

“I’m optimistic Germany will be able to cope with this in principle, and I’m optimistic that the efforts are high and will remain high,” Mr. Dohmen says. “But my impression is that no one really has the full picture… We have 16 states doing quite different things. All of them have their initiatives and are doing their best. But funding in many cases is rather small, and my concern is about the quality.”

Parents and politicians have said refugees are a resource drain that could lower educational standards for all. Many teachers have been hired with little or no education experience. Those fears are at the heart of flagging polling numbers for Chancellor Merkel, who went on the defensive this week after her conservative party lost local elections in her own constituency to the right-wing, anti-migrant Alternative For Germany.

“The situation today is many times better than it was a year ago, but there is still a lot to do,” Merkel told national lawmakers this week.

Indeed, pressures have eased as the migration flow has. But Mr. Nowak says now the complicated task of forming new members of society begins, and schools are on the front lines.

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“I’m convinced we can do it. But this isn’t a sprint,” he says. “This is a marathon.” He puts Germany around Mile Three.

• Rachel Stern contributed to this report.