IN THE CLASSROOMS of Washington's public schools, another semester is drawing to a close -- while in the far-distant world where the city's school board member dwell, the work is only beginning in what will be a critical 10 months of politicking that should command the closest scrutiny of every voter.

That the board has managed to elect a president in something fewer than the 13 rounds of balloting it took last year is of small comfort; as constituted, this collection of self-centered and contentious individuals shows no prospect whatever of conducting its business responsibly under any particular member picked to wield the gavel for 12 months. Last week's choice, Eugene Kinlow, was in the forefront of those members who drove Vincent Reed to quit -- and who has consistently opposed the effort by Dr. Reed to establish some kind of model academic high school open to students from every part of the city.

All of this has stirred a certain amount of voter sentiment in favor of abolishing the elected board, as District council member Polly Shackleton advocates in "For the Record" excerpts appearing elsewhere on this page today. Though we agree that changes should be made in the District's charter to eliminate the strange financial and policy-making relationships between city hall and the school board, we do not agree with Mrs. Shackleton's inclination to abolish the elected board. Just as neighboring jurisdictions elect school boards that function responsibly, so can the District -- in less than 10 months.

That process has to begin immediately, though -- and should not be sidetracked just because things may settle down to a dull roar while the school board searches for a permanent superintendent, or because Mayor Barry and Mr. Kinlow say, as they did Friday, that they will try to get along with each other. The existing board needs to be replaced. A clean slate of candidates for the five seats up for election this year is a must.

The formation of a slate of candidates who believe in what Vincent Reed was doing and trying to do -- and, who knows, perhaps this would include Dr. Reed himself -- should not be left to Mayor Barry, either. His interest is understandable; but by inserting himself into school board politics, Mr. Barry could turn the school board election into a referendum on the mayor. There are parent-teacher organizations, citizens' groups and others in the community that can and should coalesce to agree on a platform and a slate and to get out the vote and clean up the board in November. But to be effective, the big decisions should be made before the school year ends.