In the days following the shooting deaths of 20 children in Newtown, Conn., some of the most ardent gun rights advocates called for a new conversation on how to address gun-related violence in the United States.

But National Rifle Association officials and some leading Republicans signaled over the weekend that they would continue to resist any comprehensive change in gun laws, while calling for armed personnel to be placed in all schools and a discussion about violence in popular culture.

“I do believe better security in schools is a good place to start,” said Sen. Lindsey O. Graham (R-S.C.), who rejected the notion of government action to ban sales of the kind of assault-style weapons used in the Newtown shootings. “I don’t suggest you take my right to buy an AR-15 away from me, because I don’t think it will work,” he said.

White House officials said they were not encouraged by the NRA approach and reiterated the administration’s commitment to regulating assault-style weapons and high-capacity magazines.

“I don’t think it’s what will work,” said Wayne LaPierre, the NRA’s executive vice president, in an appearance on NBC’s “Meet the Press.” He characterized legislation to ban assault weapons authored by Sen. Dianne Feinstein (D-Calif.) as “a phony piece of legislation . . . built on lies.”

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Sen. Joseph I. Lieberman (I-Conn.) , interviewed Sunday on CNN’s “State of the Union” show, said the NRA statements were “really disheartening.”

“I had hoped they would come to the table and say ‘everything on is on the table.’ What this does mean is that the kind of new regulation of guns that President Obama and Vice President Biden and a lot of people would like to see enacted early next year is not going to happen easily. It’s going to be a battle.”

Meantime, around the country state legislators and school board members began their own debate over the NRA schools initiative, with officials from Maine to Minnesota and South Dakota and Texas discussing whether it would be good policy to have guns in the hands of teachers or others in the schools as a way to curb violence.

“The only thing that stops a bad guy with a gun is a good guy with a gun,” LaPierre said Sunday, repeating a theme from a Friday news conference in which he announced the new NRA campaign to place a police officer or an armed guard in every school.

Sen. Charles E. Schumer (D-N.Y.) said Sunday on “Meet the Press” that LaPierre was “so tone deaf he actually helps the cause of us passing sensible gun legislation . . .”

Citing polling data, Democrats have said that they think there is an opportunity to exploit a gulf between the membership of the NRA — particularly powerful in rural areas — and the group’s outspoken leadership in Washington.

Hunters and NRA members are asking for a ban on military assault-style weapons, said Rep. Mike Thompson, the Vietnam veteran and avid hunter from California who has been named point man for House Democrats on the issue. Longtime NRA backers, such as Rep. Gene Green (D-Tex.) questioned last week whether the NRA had become too closely associated with the Republican Party and was losing some of its political clout with Democrats.

Sen. Robert P. Casey (D-Pa.) , a Second Amendment booster, told the Philadelphia Inquirer last week that he was so “haunted” by the Newtown shootings that he would vote for an assault weapons ban and legislation barring magazines that hold more than 10 rounds of ammunition.

“I just believe that in light of what’s happened, in light of measures we can take to lessen the chances that will happen [again], that these are two steps we can take,” said Casey, who was just reelected.

NRA leaders disputed the idea of a gulf between them and their members. Grover Norquist, the anti-tax activist and NRA board member, said in an interview that the left consistently underestimates how committed NRA members are to the Second Amendment. The organization includes not only hunters, he said, but shooting enthusiasts and an increasing number of people acquiring firearms for self-protection. He said Friday’s announcement of a school safety initiative was just what the nation needs.

“This is exactly the right position,” Norquist said. “It is positive, it is real and it will trump the proposals for new gun laws by the liberals. This proposal will help make things better. The message from the left has been: ‘We don’t care about your kids; we care about furthering the liberal agenda.’ ”

Norquist said that board members were briefed by LaPierre and others in a Friday afternoon conference call that reported strong support from NRA members nationwide, especially from women.

LaPierre generally dismissed the idea of participating in a panel being convened by the White House, unless it concentrated on gun safety.

“If it is a panel that’s just going to be made up of a bunch of people that, for the last 20 years, have been trying to destroy the Second Amendment, I’m not interested,” he said on “Meet the Press.” He did call for increased government regulation in one area: improving a national database with information about the mentally ill.

“We have a mental-health system that has completely and totally collapsed,” he said. “We have no national database of these lunatics. Twenty-three states are still putting only a small number of records in the system and a lot of states are putting none,” making it impossible to adequately screen gun buyers.

His proposal to place armed personnel in every school provoked discussion throughout the weekend. In South Dakota, state Rep. Betty Olson (R) said that she would co-sponsor legislation authorizing teachers and other school employees to carry concealed weapons in schools, which could provide better protection than having a police officer assigned. “If you’ve got a uniformed officer in a school who’s the only one with access to a gun, he is going to be the first one” shot by an intruder, she said. Concealed weapons for staff members might be a better approach, she said.

Olson said in an interview Sunday that she had spoken with other state legislators with children in public schools who favor the approach, because it is cost-effective and more likely to protect students against an armed intruder.

“We have a hard time coming up with money just to fund the school operations even without hiring a full-time police officer,” she said. The legislation she will co-sponsor would allow “any teacher or school employee that passes a background check and has a concealed weapon” to voluntarily bring the weapon to school “to protect themselves and the students.” She said schools now are unprotected. She noted — as did officials from the NRA — that mass shootings have taken place in schools with signs that declare them to be “gun free.”

“I have a real problem with this gun-free-zone approach,” she said. “These latest school massacres all took place in gun-free zones.”

Former Arkansas congressman Asa Hutchinson, who leads the NRA “National School Shield” program, estimated it would cost more than $2 billion “if you put a federally funded program for an armed person at every school.” But, he told CNN’s Candy Crowley on Sunday that the NRA could help with training programs for volunteers that would allow the school project to proceed quickly.

He called for individual schools to make the decision whether to have armed personnel on site. “I think it is terrific that the NRA is willing to fund experts and solutions that will be provided free of cost to the schools,” he said.