District of Columbia budget director Gladys W. Mack sent a letter recently to city school board member R. Calvin Lockridge saying the board's funding request for the 1983 school year is "totally unrealistic," given the city's financial crisis.

Lockridge's response to Mack was simple: He crumpled up the letter and threw it into the wastepaper basket.

Lockridge, as chairman of the board's budget and finance committee, is the school board's chief spokesman for the schools' fiscal needs. His reaction to Mack's letter is indicative of the animosity, some of it personal, that underscores the school board's relationship with the mayor's office, especially over the school budget.

Each side says the other is to blame for the lack of communication and cooperation. Both sides acknowlege that the situation has prevented the school system for the last three years from organizing its budget well before school begins in September and avoiding the last-minute layoffs, shifts of teachers and principals and threats of furloughs of the recent past.

At stake this time is the $289 million budget that school officials say they need for the 1982-83 school year. That amount is $47 million more than the school system received this year. But school officials say the $289 million will allow them to lower the pupil-teacher ratio in elementary and junior high schools, add more guidance counselors and restore certain programs that have been cut out, like drivers' education.

Without that $289 million, as many as 200 teachers could be laid off, according to Arthur G. Hawkins, the schools' chief budget officer. About 400 teachers lost their jobs last year.

Lockridge said the budget proposal merely reflects the public's desire, expressed at a series community meetings, for a lower pupil-teacher ratios.

But in her letter to Lockridge, Mack urged the board to keep to the $249 million budget mark the Mayor Marion Barry has set for the schools. That is $7 million less than the school system received this year. Mack argued that the schools can still provide "quality" services with the lower amount of funding since a decline of about 4,400 students is expected in the fall of 1982.

She also criticized the board for failing to close or consolidate several underused school buildings. She said the city would have to take from other agencies or raise either the property or sales tax to give the schools the full amount requested.

Both Mack and school officials say they would be willing to sit down and discuss their differences. But with the mayor's office scheduled to present its entire budget to the City Council five weeks from now, no meeting has been set. And school officials insist on sticking to the $289 million figure, no matter what Mack or the mayor say.

"The community is confused, they don't know who to believe," Mack said. "I think this just helps to foster the negative image the schools already have in the community."

School board president Eugene Kinlow in turn has accused Barry of "political posturing" over the school budget to score points with the voters for next year's mayoral election.

"It's a situation where the school system is everybody's favorite whipping boy. But I think Marion is misjudging what the public reaction will be" to his stand on the school budget, Kinlow said.

In her letter, Mack said that every city agency kept within the mayor's suggested budget guidelines, except the schools.

But part of the problem is the unusually entangled political process surrounding the school board budget. It is the only budget in the city government in which all the District's 25 elected city officials -- the 11-member school board, the 13-member council and the mayor -- are involved, and it is thus a "highly politicized" matter, in Barry's words.

Barry, as mayor, has the authority to propose a general budget total for the schools but has no control over the exact way the money is spent. That is the function of the school board. The board, however, cannot set the total amount of funds because it has no revenue-collecting authority.

The council in turn must approve the final budget for the schools, but it too cannot control specific expenditures or the initial budget mark proposed by the mayor.

Barry says he "can't win" when it comes to the school budget. If he proposes more money for the schools, taxpayers sometimes complain because they don't see enough improvements in the schools. If he proposes holding down funding, the school board and some public school parent groups complain that he's antieducation.

For the last two years, the mayor has proposed drastic reductions to the school budget. But each time the council restored some funds. Last year, school officials got $28 million restored to the 1982 budget with the help of heavy lobbying from a highly vocal group of parents called Parents United For Full Funding.

Council member John A. Wilson (D-Ward 1), chairman of the council's finance committee, said he voted to add funds to the school budget last year, but this time will urge that other council members reject the board's request for $289 million.

"Every year this whole situation just absolutely gets on my nerves," Wilson said. " . . . Every year they school board members propose a budget that makes no sense. And if we on the council are opposed to it, they say we're opposed to educating children," Wilson said.

"It's very simple why the council and the board can't get it together," Wilson said. "They want more money than we've got, and they don't show any marked improvement in the schools with what they do get."

Wilson also complained that no board members have asked to sit down with him to iron out differences over the 1983 budget. He said he would be willing to support a proposal by board member Frank Smith Ward 1 that calls for a 1983 budget of $270 million and a systematic closing of under-enrolled schools. The proposal has received scant attention from the rest of the board.

Council member and mayoral hopeful Betty Ann Kane, who is also a former school board member, said the annual hassling over the budget will not end until city and school officials agree on a specific formula for deciding how much money the schools need. She has suggested forming a "school finance commission" consisting of school and city officials and members of the public to devise such a formula.

In the middle of the budget fight is School Superintendent Floretta D. McKenzie. McKenzie has been successful so far in her short tenure at getting the mayor to release additional money at critical times for the schools.

But the new superintendent, the fourth in the last six years, is careful not to step on the toes of her employer, the school board. She said she believes it would be inappropriate for her to deal with the mayor directly on the 1983 budget since that is a function of the school board.

McKenzie said that even though enrollment is declining, the schools still need more teachers now that new stiffer academic standards are in place in both elementary and senior high schools, and slower students will need additional help to catch up.

She said she believes it is unfair for the mayor and Mack to criticize the board for not closing more schools this year since school closings require a long process, including public hearings. Besides, she said, the system realizes a savings of only $100,000 to $150,000 a year for each school closed.

Of the school board's relationship with the council, McKenzie said, "Maybe things do escalate to a point where it's harder to compromise." But, she added, "I do believe all of us want to give as much to the schools and to the kids as possible."