More than a decade after No Child Left Behind established a stronger federal role in public education, the Senate on Thursday approved a rewrite of the much-criticized education law that would return much of that power to the states.

The 81-17 vote comes a week after the House passed its own rewrite and sets the stage for what could be contentious negotiations over the federal government's influence in education policy.

The Senate version would leave in place the law's annual testing schedule. But, in a major shift, it would give states and districts more control over whether and how to use those tests to assess the performance of schools, teachers and students.

The legislation, sponsored by Republican Lamar Alexander of Tennessee and Democrat Patty Murray of Washington, would prohibit the federal government from requiring or encouraging specific sets of academic standards, such as Common Core.

Alexander called the bipartisan vote "remarkable" and said it sends a message to the many superintendents, teachers and parents who have told him that No Child needs to be overhauled.

"We hear you, we know you want to end the confusion, end the anxiety and the feeling that you're not in charge of your own children."

Sen. Murray said the legislation represents a "strong bill that all sides can be proud of." But she suggested a tough battle ahead to merge the differing House-Senate bills. "Their bill really represents an unacceptable partisan approach," she said, referring to the House bill.

Education Secretary Arne Duncan offered cautious praise after Senate passage.

The bill, he said, makes progress toward replacing No Child Left Behind, but "still falls short of truly giving every child a fair shot at success by failing to ensure that parents and children can count on local leaders to take action when students are struggling to learn."

Duncan and other Democrats wanted stronger accountability measures worked into the Senate bill, but an effort to do that failed Wednesday when lawmakers voted down an amendment that would have required states to identify the nation's lowest-performing schools, and make sure the schools have turn-around plans for improvement.

Much like the Alexander-Murray bill, the House legislation, sponsored by Rep. John Kline, R-Minn., also lessens federal involvement in schools. It turns over power to the states to assess school performance and prevents the Department of Education from pushing Common Core or other standards on schools.

But unlike the Senate bill, the House version allows federal money to follow low-income children to public schools of their choice, an issue known as portability. Democrats do not support it, and the Senate voted down the idea last week.

Senators earlier Thursday rejected a series of amendments, including one from Sen. Ted Cruz of Texas, a GOP presidential candidate, that would have eliminated federal testing mandates and given states more say in how students are assessed.

Also defeated was a proposal championed by Democrats that would have expanded pre-K programs for children of low-and-moderate income families.

No Child Left Behind had bipartisan support and was signed into law by President George W. Bush in 2002. It mandated annual testing in reading and math for students in grades three through eight and again in high school. Schools had to show student growth or face consequences such as cuts in funding.

Critics complain there is too much testing and the law is too punitive on schools deemed to be failing.

Congress has tried to update the law since it expired in 2007, though its mandates remained in place. The Obama administration began issuing waivers to dozens of states to get around some of the law's strictest requirements when it became clear they would not be met.