Legislation rewriting the No Child Left Behind education law finally gained traction this week, and the Senate Democrat whose committee passed the bill said on Friday that progress became possible because lawmakers were irritated by the Obama administration’s offering states waivers to the law’s key provisions.

“Some of us on both sides of the aisle were upset with them coming out with the waiver package that they did, so that spurred us on,” Senator Tom Harkin of Iowa, who heads the Senate education committee, said in an interview. “It gave us a sense of urgency.”

Mr. Harkin’s committee voted 15 to 7 on Thursday to approve a bill that would greatly reduce Washington’s role in overseeing public schools. It was co-sponsored by Senator Michael B. Enzi, the Wyoming Republican who is the committee’s ranking minority member. Mr. Harkin called it “a good compromise bill” that would have bipartisan support in the full Senate.

But Secretary of Education Arne Duncan, who had long criticized Congress for failing to rewrite the law, on Friday criticized the Harkin-Enzi bill, saying it compromises too much, particularly on teacher evaluations and student-achievement goals. “There are huge — significant problems with the current draft,” he said. “Though there are some things in this that I consider positive, others are quite concerning.”