The waiver program was announced by Mr. Obama in September. Many states had already adopted academic standards and other measures that conflicted with the requirements of the No Child law, which was passed with broad bipartisan support in 2002 and has been up for renewal since 2007.

State education officials had long criticized the 2014 deadline for math and reading proficiency, saying that it was an impossibly high bar and did not take into account the needs of some of the most disadvantaged children. Mr. Obama, surrounded by educators and lawmakers at the White House, said Thursday that while the goals of the No Child law were the right ones, “we’ve got to do it in a way that doesn’t force teachers to teach to the test, or encourage schools to lower their standards to avoid being labeled as failures.”

“So when it comes to fixing what’s wrong with No Child Left Behind, we’ve offered every state the same deal,” he said. “We’ve said, ‘If you’re willing to set higher, more honest standards than the ones that were set by No Child Left Behind, then we’re going to give you the flexibility to meet those standards.’ ”

Some Congressional Republican leaders criticized Mr. Obama on Thursday for what they saw as an effort to politicize education policy rather than work with Congress. The waivers came on the same day that Representative John Kline, chairman of the House Committee on Education and the Workforce and a Minnesota Republican, introduced two bills addressing some of the same issues.

Image President Obama said Wednesday that the states would be given “flexibility” in finding ways to improve their schools. Credit... Stephen Crowley/The New York Times

“Our children’s education should not be a political poker chip,” said Senator Mike Enzi, a Wyoming Republican.