A Virginia company leading a national movement to replace classrooms with computers – in which children as young as 5 can learn at home at taxpayer expense – is facing a backlash from critics who question its funding, quality and oversight.

K12 Inc. of Herndon has become the country’s largest provider of full-time public virtual schools, upending the traditional American notion that learning occurs in a schoolhouse where students share the experience. In K12’s virtual schools, learning is largely solitary, with lessons delivered online to a child who progresses at his or her own pace.

Conceived as a way to teach a small segment of the home-schooled and others who need flexible schooling, virtual education has evolved into an option to traditional public schools for an increasingly wide range of students – high achievers, strugglers, teenage parents and victims of bullying among them.

“For many kids, the local school doesn’t work,” said Ronald Packard, chief executive and founder of K12. “And now, technology allows us to give that child a choice. It’s about educational liberty.”

Packard and other education entrepreneurs say they are harnessing technology to deliver quality education to any child, regardless of ZIP code.

It’s a proposition, and one that has attracted support in state legislatures. But in one of the most hard-fought quarters of public policy, a rising chorus of critics argues that full-time virtual learning doesn’t effectively educate children.

“Kindergarten kids learning in front of a monitor – that’s just wrong,” said Maryelen Calderwood, a school board member in Greenfield, Mass., who unsuccessfully tried to stop K12 from contracting with her city to create New England’s first virtual public school in 2010.

People on both sides agree that the structure of public education is not designed for virtual schools. How, for example, do you pay for a school that floats in cyberspace when education funding formulas are rooted in the geography of property taxes? How do you oversee the quality of a virtual education?

“There’s a total mismatch,” said Chester Finn, president of the Thomas B. Fordham Institute, a conservative think tank, who served on K12’s board of directors until 2007. “We’ve got a 19th-century edifice trying to house a 21st-century system.”

Despite questions, full-time virtual schools are proliferating.

In the past two years, more than a dozen states have passed laws and removed obstacles to encourage virtual schools. And providers of virtual education have been making their case in statehouses around the country.

K12 has hired lobbyists and backed political candidates who support school choice in general and virtual education in particular. From 2004 to 2010, K12 gave about $500,000 in direct contributions to state politicians across the country, with three-quarters going to Republicans, according to the National Institute on Money in State Politics.

“We understand the politics of education pretty well,” Packard told investors recently.

K12’s push into New England illustrates its skill. In 2009, K12 began exploring the potential for opening a virtual school in Massachusetts in partnership with the rural Greenfield school district.

But Massachusetts education officials halted the plan, saying Greenfield had no legal authority to create a statewide school. So Greenfield and K12 turned to legislators, with the company spending about $200,000 on Beacon Hill lobbyists.

State Rep. Martha “Marty” Walz, D-Boston, wrote legislation that allowed Greenfield to open the Massachusetts Virtual Academy in 2010. She acknowledged that the language was imperfect and didn’t address issues of funding or oversight but said she couldn’t wait to craft a comprehensive plan.

“You do what you need to do sometimes to get the ball rolling,” said Walz, who accepted at least $2,600 in campaign contributions from K12, its executives or its lobbyists since 2008, said the National Institute on Money in State Politics.

That scenario is repeating nationwide as K12 and its allies seek to expand virtual education.

About 250,000 students are enrolled in full-time public virtual schools in 30 states, according to Susan Patrick of the International Association for K-12 Online Learning, a trade association. Although that’s just a fraction of the country’s 50 million schoolchildren, the numbers are growing fast, Patrick said.

K12 teaches about two out of every five students in full-time online schools. Its next largest competitor is Baltimore-based Connections Education. The rest of the industry consists of smaller operators and some nonprofit virtual schools.

The company, which began in two states in 2001, now teaches about 95,000 students in virtual schools in 29 states and Washington, D.C.

And it plans to grow. “We are now that much closer to our manifest destiny of making a K12 Inc. education available to every child,” Packard said in a call with Wall Street analysts this month.

It’s a promising business. In the past fiscal year, K12 had revenue of $522 million – a 36 percent increase from the prior year, according to securities filings. Its net income after a series of acquisitions was $12.8 million. Packard earned $2.6 million in total compensation.

A former Goldman Sachs banker, Packard, 48, was working as a consultant with McKinsey and Co. when he got a call from Michael Milken, the financier who pleaded guilty to securities fraud in 1990 and later became a philanthropist partly focused on education.

Packard joined Milken’s education investment holding firm and ran one of his companies, a chain of preschools. About the same time, Packard was trying to find an online math course for his 6-year-old daughter. Frustrated by the dearth of options, he saw a business opportunity.

He founded K12 in 2000 with $10 million from Milken and Larry Ellison, CEO of Oracle Corp., maker of software and hardware systems. William Bennett, education secretary under President Reagan, became K12’s chairman, bringing his conservative bona fides and political connections to a company that originally aimed for the home-schooling market. Bennett resigned from K12 in 2005.

K12 sells a variety of ways to learn online. But its core business – and the one proving most controversial – is full-time virtual public schools.

For Tyler Hirata of Dumfries, Va., going to school used to mean waking up at 6 a.m. and clambering aboard a bus. Now he snoozes until midmorning and pads downstairs to the computer in his home.

“This is fantastic!” said Tyler, 8, who had attended a Prince William County school but was enrolled this fall in the Virginia Virtual Academy – a public institution run by K12 and open to any student in the commonwealth.

Tyler said the best thing about taking third grade online is that it requires less than three hours a day. His mother is more excited that, for the first time, Tyler is reading fluently on his own.

“The K12 program is phenomenal,” said Michele Hirata, adding that Tyler blossomed with her daily one-on-one attention. Virtual school has been equally positive for her fifth-grade daughter, Gennifer, 10, a fast learner who spends five hours a day practicing gymnastics, she said.