IN THE CONTINUING dispute between the mayor and the school board, no one has a corner on the truth--neither side is completely right. But the school board's failure to cooperate with the mayor and his budget director, Gladys Mack, gives the mayor the stronger claim on public support. The other day, the board's main negotiator, the intemperate Calvin Lockridge, crumpled the latest correspondence from Mrs. Mack and threw it in the wastebasket. How's that for good government?

The big question the board must answer at once is how it would operate the schools under the $249 million school budget Mr. Barry has proposed. The board, in its submissions to the District Building, has never answered this question. The mayor has sent proposed budget marks to all city agencies and the school board, asking that they set their funding priorities under that minimal budget and let him know what services and programs would have to be eliminated to meet the budget mark.

The board's only formal response has been to tell the mayor that it needs $289 million, not $249 million, and to send in a line-item budget for that amount. Arguments are made that a board of elected officials should not even have to have its budget approved by the mayor. And some board members observe that the mayor's budget mark is at least $4 million--and possibly as much as $14 million--less than the school system's budget for the current school year.

Throwing letters in the trash, arguing over the budgeting process and pointing to declining numbers will not win the school board any battle for public opinion. The mayor can just as quickly point to the declining enrollment in the schools. One report now estimates that there are 38,000 vacant seats in the public schools. Those vacant seats are evidence of the need to close schools, a step that could save money and lessen the school system's budget needs. Yet even with the closing of more schools, the question would remain as to how much money the school system needs beyond the $249 million the mayor has proposed and what that money would be used for. How large would classes have to be, for example, on that budget? The burden is on the school board to answer those questions now.