Hundreds of thousands of children in the US are enrolled in schools that are without behaviour problems, assemblies, playgrounds or even classrooms. The rise of the internet and of new teaching software means that “virtual schools” now offer a cheaper alternative to traditional bricks and mortar educational provision.

But as the movement takes hold across the US - virtual schools can be found in at least 34 of the country’s 50 states - so does the controversy surrounding it.

This week, charter school commissioners in Maine, one of the states yet to open cyber schools, rejected one application but approved a second. However, its opening could yet be thwarted, with state legislators calling for the money to instead be used for existing schools.

The objections raised against the virtual schools - that they have a mixed record of success and siphon off public funding - go to the heart of national concerns about the project. Critics have described the schools as a “slash and burn campaign” that would compromise standards and lead to vast profits for private business, funded by taxpayers’ money.

But the schools’ operators say they are providing an alternative for children who do not thrive in traditional classrooms. The students might be “accelerated learners”, bullied or bored with the pace of learning in regular schools, they say.

US education secretary Arne Duncan has said that the schools could play a “huge role in increasing educational productivity.reducing wasted time, energy and money”.

But teaching unions, whose members’ jobs are threatened by the concept, remain unconvinced. Bob Tate, a policy analyst at the National Education Association, the largest US teachers’ union, told TES that virtual schools suffered from “very high student-to-teacher ratios”, delivered “dismal” results and produced “high profit margins”.

“We believe the face-to-face relationship between student and educator is critical to improving student learning,” he added. “And students’ interactions with each other are an important part of their socialisation into society.”

In Europe, unions fear that the idea could cross the Atlantic to countries still suffering in the euro crisis.

John Bangs of Education International, a global federation of teaching unions, said that virtual schools were seen as a cheap option, but they did not focus on quality. “Various companies are pressing this as an inexpensive solution for developing countries,” he said. “I think it is a deceptive and siren voice that could also see the virtual schools advocated as a solution for some of the southern European countries like Greece [that are] trying to cope with the economic crisis.”

According to the Brookings Institution, a respected US think-tank, a survey of 20 virtual charter schools in 14 states shows that the cost of online learning is “roughly half that of traditional public schools”. But The New York Times has reported that despite lower costs, operating companies still collect nearly as much taxpayer money in some states as normal schools.

Questions over the quality of education persist, too. A Stanford University study looks at test results in eight virtual charter schools in Pennsylvania between 2007 and 2010. Taking student background and prior attainment into account, it finds that “all eight cyber schools perform significantly worse than their traditional public school counterparts” in reading and maths.

There is also unease about for-profit companies effectively running the virtual schools. In June 2012, applications for two virtual charter schools in Maine - with online curricula provided by the US’s two biggest virtual school companies, Connections Academy, owned by Pearson, and K12 - were withdrawn after concerns were raised about the influence that the for-profit firms might have over the schools.

The applications were resubmitted, but only the Connections school was approved on Monday. The K12 application was rejected, with some charter school commissioners reported to be concerned about the company’s academic record in other states.

K12 and Connections Academy have both defended their track records. A K12 spokesman said the company respected the independence of the schools that used its services. Data showed that the longer they used the K12 programme, “the better they perform”, he said.