The resignation of Fairfax County Superintendent William J. Burkholder has left a number of School Board watchers inside and outside the system with one major unanswered question:

Why did the School Board lose the competent and popular superintendent it had tried so hard to keep?

Burkholder, 55, who was eligible to retire with full benefits last year, was persuaded by county officials, including School Board members, to remain this year and then continue under a new four-year contract through 1989. But the price for keeping him, $144,5000 this year, was high.

And when county residents and politicians discovered that the School Board was on the verge of continuing the arrangement for another four years at the even higher price of $157,000 a year -- $79,450 in salary and $77,800 in yearly payments to make up for his lost retirement benefits -- they revolted.

Last Friday, Burkholder said he would resign, leaving office in July, partly because he was fearful that the public uproar over his income would harm chances of a $74.8 million school bond referendum Nov. 6, as well as the county's bid to secure more state funds from the Virginia General Assembly next January.

Some School Board members say they should have tried to defuse the mounting public anger by staging a press conference with an army of financial experts explaining the quirks of the retirement system and why Burkholder's early retirement benefits were unusually high.

The School Board might have convinced taxpayers that the overall costs for their system's top executive would hold steady no matter what: that even if Burkholder resigned, the county would have to pay his large retirement benefits plus a salary for his replacement.

That press conference was never held. Instead, Gerald A. Fill, a School Board member running for county supervisor in the Mount Vernon District, denounced the four-year arrangement for Burkholder even though he and his colleagues had approved the similiar one-year contract unanimously 10 months earlier. And School Board members Carmin C. Caputo and Robert E. Frye indicated they were uncomfortable with the amount of the new contract.

"I think the whole situation pointed up very dramatically the problem the School Board has had for years -- a lack of leadership," said a senior school executive, who asked not to be named.

School officials and School Board watchers said the board's factionalism carries over from last year's redistricting of school attendance boundaries. They say the eight board members who are appointed by the county board are under pressure from their supervisors as well as from the neighborhoods in their districts.

"The feeling is still there that people represent eight separate districts," said one former School Board member. "That's not a healthy sign and it's going to take a strong leadership to get the cohesion back again."

Some board members have privately criticized Chairman Mary E. Collier, saying she lost her credibility as a "consensus builder" during the redistricting by pushing the interests of Supervisor Nancy K. Falck (R-Dranesville), who appointed her.

Board watchers also said the School Board did not originally call attention to the sensitive issue of Burkholder's unusual pay package last winter when it approved the one-year contract with him and then again when it agreed to the four-year contract in private negotiations. Minutes of the Dec. 15, 1983, meeting show the board outlined in detail the compensation that Burkholder was to receive, but that agreement received surprisingly little publicity.

School activist Kevin Bell, who reviewed the school budget for the County Council of Parent-Teacher Associations, said Burkholder's income arrangement was such a departure from school policy and involved so much money that the board should have felt "the obligation to say to people: 'This is what we've been doing, this is a change in procedure, and we want your input.' "

Toni M. Carney, whose term as a School Board member ended last July, said the board was concerned last year that Burkholder's initial contract would not sit well with the public. She said that the board's overriding concern then was to rebuild a school system that was demoralized before Burkholder became superintendent in 1982.

In short, the board wanted Burkholder to stay -- at any price.

"There just wasn't anyone else," said Carney. "We came down on the side that the needs of the system had to prevail.

"Maybe it might have been better to spell it out and try to educate the people about the situation the School Board was up against and take the chance" that the public would understand, Carney said.