“We are saying that we want to not have to test children repeatedly in those even-numbered grades, because basically we are wasting their time,” Mr. Huberty said. Texas Education Agency data indicated that students who scored above a certain threshold on tests one year had a 97 percent likelihood of doing so the following year, he said.

Kathi Thomas, a parent who followed education issues during the legislative session, said she was skeptical of how much the proposal would alleviate testing-related burdens.

“These few kids that would be able to get out of taking the test, they will still get all of the wasted time prepping,” said Ms. Thomas, whose daughter attends middle school in the Dripping Springs Independent School District. “And on the day that they don’t take the test, they are going to be warehoused in a library, and they still won’t getting any learning out of it.”

This month, the state education agency sent a letter to the United States Education Department seeking to clarify whether Texas needed a waiver to enact the law. There is no timetable for waiver request decisions. But H.B. 866 requires its provisions to take effect no later than Sept. 1, 2015.