But critics say online education is really driven by a desire to spend less on teachers and buildings, especially as state and local budget crises force deep cuts to education. They note that there is no sound research showing that online courses at the K-12 level are comparable to face-to-face learning.

Here in Memphis, in one of the most ambitious online programs of its kind, every student must take an online course to graduate, beginning with current sophomores. Some study online versions of courses taught in classrooms in the same building. Officials for Memphis City Schools say they want to give students skills they will need in college, where online courses are increasingly common, and in the 21st-century workplace.

But it is also true that Memphis is spending only $164 for each student in an online course. Administrators say they have never calculated an apples-to-apples comparison for the cost of online vs. in-person education, but around the country skeptics say online courses are a stealthy way to cut corners.

“It’s a cheap education, not because it benefits the students,” said Karen Aronowitz, president of the teachers’ union in Miami, where 7,000 high school students were assigned to study online in computer labs this year because there were not enough teachers to comply with state class-size caps.

“This is being proposed for even your youngest students,” Ms. Aronowitz said. “Because it’s good for the kids? No. This is all about cheap.”