John C. Rothberg folded his legs under a green chair, placed his hands neatly in front of him on the table and began politely answering questions from members of the Alexandria City Council.

Rothberg, 48, a Harvard-educated economist and former Fulbright scholar, sat straight-backed like a student being quizzed by his teachers as he tried to win a $50-a-month job.

The position Rothberg was seeking was one of three forthcoming vacanies on the city's school board, whose members are appointed by the council in a process nearly as controversial as the board itself. The council is scheduled to make its selections Tuesday.

"I don't know half of the people running," said council member Donald C. Casey (D), who has participated in the selection rites twice before. "I think the system ought to be changed, but I don't know what we can do."

Casey is not alone in his bafflement. "The system is inadequate," says Paul W. Wyman, 40, a systems analyst and another of the nine school board candidates. "I was surprised we will be appointed by so many lame ducks."

Five of the seven Alexandria council members will not be returning to the new council, whose members were elected early this month and will be sworn in July 1.

Although the school board's $34 million budget is awarded by the City Council, the board is legally a state agency. But any change in the method of choosing its nine members must be approved in Richmond. And state legislators, mindful of the power the current system gives fellow officeholders, are reluctant to change.

The question has been tinged by racial overtones in the General Assembly, where opposition to school board elections goes back 20 years, according to Democratic state Sen. Edward Holland of Arlington.

"In the 1950s Arlington had an elected school board which was opposed to the position of massive resistance to integration proclaimed by the Virginia state government," Holland said.

"The Arlington board took steps toward integration following the Supreme Court's decision in Brown vs. Board of Education. When they did, the General Assembly took away the election privilege."

Holland said repeated efforts in the legislature to make the Arlington board elective failed because of these "historic reasons." Opposition to elective boards, Holland added, has been "symoblic of state control over education."

Holland, state Sen. Wiley Mitchell (R-Alexandria) and several Alexandria School board members point out that the appointive process is extremely difficult for newcomers because the City Council tends to reappoint any incumbent who wants the job.

If their reasoning is accurate, that means seven newcomers are currently running for one seat. Two incumbents, Claudia Waller and board Vice President Shirley N. Tyler, are considered to have excellent chances of being reappointed.

The resulting scramble in Alexandria - or any Northern Virginia locality - is hardly a nonpolitical contest. Board candidates lobby council members and line up key supporters to plead their cause.

Rothberg's backers inlcude Sen. Mitchell, who said he mentioned to several council members that Rothberg deserved their "keen consideration."

Rothberg, who has been active in PTAs at two Alexandria schools, also telephoned all of the council members to explain his positions on issues and urged his supporters to do the same. His interview with the City Council lasted 15 minutes.

Another candidates, lawyer Harold G. Hernly Jr., won support from former Democratic state Sen. Armistead Boothe, while incumbent Tyler also has the blessing of Sen. Mitchell.

"It's almost a popularity contest and you can become influenced by the number of telephone calls you receive," says Republican council member Robert L. Calhoun.

"Personally, I think the board needs new blood and if other things are equal I might vote for a newcomer over an incumbent just to get some fresh viewpoints," he said.

Besides Tyler, Rothberg, Waller, Wyman and Hernly, candidates for the three-year appointments are James Harvey Harrison Jr., 52, a budget analyst; James M. Snyder, 64, a professor at George Mason University; Maxwell Moore, 57, a businessman, and Robert H. Anderson, 55, a pediatrician.

All nine are scheduled to appear at a League of Women Voters forum Monday night.