MILWAUKEE — Weekday mornings, three of Tracie Weldie’s children eat breakfast, make beds and trudge off to public school — in their case, downstairs to their basement in a suburb here, where their mother leads them through math and other lessons outlined by an Internet-based charter school.

Half a million American children take classes online, with a significant group, like the Weldies, getting all their schooling from virtual public schools. The rapid growth of these schools has provoked debates in courtrooms and legislatures over money, as the schools compete with local districts for millions in public dollars, and over issues like whether online learning is appropriate for young children.

One of the sharpest debates has concerned the Weldies’ school in Wisconsin, where last week the backers of online education persuaded state lawmakers to keep it and 11 other virtual schools open despite a court ruling against them and the opposition of the teachers union. John Watson, a consultant in Colorado who does an annual survey of education that is based on the Internet, said events in Wisconsin followed the pattern in other states where online schools have proliferated fast.

“Somebody says, ‘What’s going on, does this make sense?’ ” Mr. Watson said. “And after some inquiry most states have said, ‘Yes, we like online learning, but these are such new ways of teaching children that we’ll need to change some regulations and get some more oversight.’ ”