The Senate education panel began marking up a bipartisan bill to replace No Child Left Behind on Tuesday, with Democrats and Republicans going to great lengths to hold together a delicately crafted consensus around the proposal.

Members of the Senate committee on Health, Education, Labor and Pensions withdrew more amendments than they acted on, as they tried to hold back proposals that did not have bipartisan support.

“I know you’ve been working hard to make this a bipartisan bill, so I’ll withdraw (my amendment) at this time,” Sen. Elizabeth Warren (D-Mass.), told the committee chair, Sen. Lamar Alexander (R-Tenn.) and the ranking Democrat, Sen. Patty Murray (Wash.). Warren was referring to an amendment she wrote that would require states to identify high schools with low graduation rates and devise a plan to improve those rates.

It was one of about a dozen amendments that committee members proposed but then withdrew when it was clear they didn’t have support across the aisle. The panel voted unanimously on several of the most innocuous amendments, including a proposal from Sen. Al Franken (D-Minn.) to allow states to use computer adaptive tests when they measure student performance annually. The only party-line vote occurred when Warren unsuccessfully tried to amend the bill to require states that evaluate teachers to publicly explain how the method they choose is “reasonable” and “reliable.”

They left the most contentious issues — regarding private school vouchers, equitable distribution of federal dollars and the best way to hold states accountable for educating disadvantaged students — to be debated by at a later point by the full Senate.

But sprinkled throughout the polite discussion Tuesday were hints of the larger political battles to come over the update of the country’s main federal education law.

Sen. Tim Scott (R-S.C.), proposed an amendment that would let federal dollars designed to help states educate poor children “follow the child” if that student transfers from a high-poverty school to another public school or a private school.

Scott called the voucher plan a “Pell grant” for K-12, noting that the federal government gives financial aid to low-income college students to use at the college or university of their choice, whether public or private.

“If we’re really going to get serious about helping low-income children, school choice is the way to get them there,” Scott said. “I will withdraw my amendment and offer it on the floor. I do think we need a robust debate for a path to actually work for kids stuck in the wrong zip codes.”

Murray immediately said she was opposed.

“This allows federal funding to flow from the public school system to private schools,” she said. “It really dilutes the funds and stretches the dollars that are tight already a lot thinner.”

Scott said a “classic example” of a successful voucher program is the D.C. Opportunity Scholarship, which Congress created in 2004. A 2012 investigation by The Washington Post found problems with the program.

Scott drew a sharp rebuke from Sen. Sheldon Whitehouse (D-R.I.), who argued that Claiborne Pell, the late Rhode Island senator and namesake of the college grant program, would not support the idea of using federal tax dollars to pay tuition at K-12 private schools.

“Being from Rhode Island and having had a long and affectionate relationship with Senator Pell, we can have this discussion but please let’s not drag Pell grants in here,” Whitehouse said. “Nobody should believe Senator Pell would endorse such an amendment. I am certain he would not.”

Alexander said he would support Scott’s amendment during the full Senate discussion of the bill.

“The Pell grant is a voucher, and Sen. Scott is proposing a voucher,” Alexander said. “He’s proposed we take the $14 billion we spend for low-income kids and turn it into $1,500 vouchers and let it follow that child to the school that the child’s parent wants to attend. That is a very useful proposal. We’ll have other proposals about school choice. ... We’ll continue that discussion on the Senate floor.”

No Child Left Behind was due for reauthorization in 2007, but previous attempts to rewrite the law collapsed amid partisan debates on Capitol Hill about the proper role of the federal government in local schools.

“If fixing No Child Left Behind were a standardized test, Congress would have earned a failing grade for each of the last seven years,” Alexander said.