Sen. Lamar Alexander, R-Tenn., is chairman of the Senate education committee and a former U.S. Secretary of Education. (AP Photo/Manuel Balce Ceneta)

This story has been updated.

Sen. Lamar Alexander (R-Tenn.), who is leading the Senate’s effort to overhaul the nation’s main federal education law, committed something of an education-reform faux pas Wednesday morning when he pushed back against the suggestion that all charter schools are public schools.

“There are some private charter schools, are there not?” Alexander said, speaking at the Brookings Institution during an event about school choice.

Charter schools are publicly funded but run independently by nonprofits or for-profit companies. They enjoy strong bipartisan support in Washington, and the federal education department treats them as public schools, but whether they should be called public or private is still a matter of debate. In calling them private, Alexander — a champion of charters and school choice — crossed a semantic battle line.

Charter school critics argue that charters amount to a privatization of public schools because they are run by organizations that don’t answer to the public and in some states aren’t subject to key rules that apply to government agencies, such as open meetings and public records laws. Advocates for charter schools, meanwhile, say that charter schools are unequivocally public schools because they are open to all children and don’t charge tuition.

The debate is over more than language. Though federal and state education regulators refer to public charter schools, courts and other regulators have wrestled with the question of whether charter schools should be treated as public or private institutions. In some cases, charter schools have themselves argued that they are private institutions.

For example, in 2013 the National Labor Relations Board ruled in favor of a Chicago charter school and deemed it a private institution. Therefore, teachers at the school must organize under laws governing private-sector rather than public-sector employees.

Despite the legal gray areas, it was unusual to hear a proponent of charter schools like Alexander — who also is a former U.S. Secretary of Education — refer to charter schools as private and insinuate that some aren’t subject to federally mandated testing.

He made the statement during Brookings’ release of its annual Education Choice and Competition Index, which rates school districts according to the degree of latitude they give parents in deciding which schools their children should attend.

Grover J. “Russ” Whitehurst, director of the Brown Center on Education Policy, corrected a member of the audience who, in asking a question, had referred to “private, parochial and charter schools who are not subject” to annual federally mandated tests.

“Charter schools are subject to the same tests as regular public schools,” Whitehurst said.

Public charter schools,” Alexander interrupted.

“Well they’re all — charter schools are public schools,” Whitehurst said. “Charter schools, I guess as we define it, are public schools that operate under charters from the state rather than private, so they’re subject to the same tests.”

A moment later, Alexander clarified: “I guess I’m thinking of public schools operated by nonprofits or a private company, but you define those as a public charter school? I agree with that.”

After the event, Alexander clarified further: “Obviously I misspoke this morning. Charter schools are public schools and are subject to the same tests as all other public schools in their state.”