Sharon Bulova, chairwoman of the Fairfax County board of supervisors. (Jeffrey MacMillan)

The meeting of the Fairfax County Board of Supervisors had ended. But several members were still arguing on the dais, even as board Chairman Sharon Bulova (D) urged them to leave.

Moments earlier, Supervisor Catherine M. Hudgins (D-Hunter Mill) had accused her two newest colleagues of political grandstanding.

“This is childish play, that once you don’t get your way, you want to come back and change it,” Hudgins told Kathy L. Smith (D-Sully) and Daniel G. Storck (D-Mount Vernon). “You can’t have it always only your way.”

Such public vitriol is almost unheard of in polite Fairfax County. But it happened Tuesday night, spurred by growing tension over the budget and the still-new presence of Smith and Storck, who in January joined a board whose other eight members have served together for at least seven years.

At issue was the ceiling the board should set for next year’s property tax rate. County Executive Edward L. Long had recommended a maximum increase of four cents per $100 of home value, which fell about $45 million short of what schools Superintendent Karen Garza says she needs to meet rising education costs.

Smith, herself a former school board member, joined Storck to call for raising the ceiling for property taxes by six cents, which would fully fund Garza’s budget request.

“My whole goal in this was to make sure we had a full and open dialogue with the community,” Smith said Wednesday, arguing that a higher tax-rate ceiling would allow the board to consider more of what residents ask for during community meetings that lead up to the final tax rate being set in April. “When I did the math, six cents would have let us have that conversation.”

Supervisor Jeff C. McKay (D-Lee), who chairs the board’s budget committee, proposed a compromise increase of five cents. He lobbied his colleagues to support that measure by pointing out that Fairfax expects an increase in state funding next year of as much as $17 million.

“I thought the motion that I put on the table could have fully funded the schools, was respectful to the taxpayers and was the only responsible motion that could have achieved enough support to pass,” McKay said.

The six-cent ceiling failed 7 to 3. Only Smith, Storck and John W. Foust (D-Dranesville) voted in favor.

But five board veterans rallied around McKay’s compromise proposal — enough for it to pass, except that Smith and Storck opposed it.

Instead, the board passed the four-cent ceiling by a vote of 7 to 3. Only Smith, Storck and Pat Herrity (R-Springfield) — who is skeptical of the school system’s budget needs — voted against that measure.

Several board members expressed frustration with Smith and Storck for refusing to compromise, arguing they should have agreed to the five-cent increase once their six-cent proposal failed. In effect, those supervisors said, the two new lawmakers hindered county efforts to fund schools instead of helping them.

“It was a case of the perfect getting in the way of the possible,” said Bulova, who backed the five-cent measure. “There was not support for six, and they were not willing to compromise on the five.”

McKay was more blunt, calling the vote “a rookie mistake” and “political posturing.”

Storck said he had expected the four-cent proposal to fail and was planning on proposing another, higher rate.

“My way of thinking was somewhere around 5 1 / 2 cents,” he said.

The real anger erupted later in the meeting, after school advocates lobbied several supervisors to employ a parliamentary maneuver that would let the board reconsider the failed motions.

Supervisor Linda Q. Smyth (D-Providence) argued that revisiting a vote that already passed could damage the board’s credibility.

“We should present the idea of being a well-managed county, not playing games with setting a tax rate,” Smyth said Wednesday.

The vote to reconsider ended in a 5-to-5 tie, meaning that the motion failed. Both Smith and Storck voted for it.

After everyone finally left the meeting room, the bickering spilled onto social media.

“I’m not going to debate a colleague on Facebook, people know what happened,” McKay wrote after criticizing Smith’s online characterization of the meeting’s outcome.

Smith replied: “Jeff McKay I agree, no reason to debate on Facebook, but if you don’t want to debate you also shouldn’t attack your colleagues on Facebook.”