Arthur A. Fletcher, the Relublican nominee for mayor, criticized Democrat Marion Barry's presidency of the D.C. Board of Education yesterday, but cited specifics that Barry said both were wrong and offered further indications that Fletcher was a "stranger" to the District of Columbia.

As part of his criticism of Barry's tenure as presidnet of the board, Fletcher accused Barry of failing to improve discipline in city school classrooms as he said Barry had promised to do when he became board president in 1972.

Barry responded that he had never promised to improve classroom discripline, but rather had campaigned on a platform of restoring discipline to the board itself, when at [WORD ILLEGIBLE] conducted many of its [WORD ILLEGIBLE] in chaos.

He must have [LINE ILLEGIBLE] he was getting on [WORD ILLEGIBLE] and going somewhere," [WORD ILLEGIBLE] said. "He didn't know which discipline I was talking about."

Fletcher also criticized Barry as "irresponsible" for allegedly taking "children out of the classroom to picket down at the White House when they should have been learning (in school)."

Fletcher apparently was referring to a 1972 demostration protesting the welafare policies of the Nixon administration, in which Fletcher at one time was assistant secretary of labor. Barry was one of the sponsors of the deomstration, whch the board had voted to support and allowed elementary school teachers to passout leaflets about the protest in school.

"If Mr. Fletcher were to check his facts," Barry said, "he's find out that it happened on a Saturday. School's not in on Saturday."

Fletcher's initial criticism of Barry occured during the taping of a 12-minute debate between the two that will be aired Sunday at 10:05 p.m. on radio station WMAL-AM. Reporters were allowed to listen to the debate.

In response to a question about education, Fletcher said, Barry had left the school board in 1974 to run for City Council without fulfilling any of the promises he made when he become board president two years earlier.

"The fact is that there is no way you can measure his presence on that board to indicate that it improved at all," Fletcher said. "I have all kinds of questions as to what he's going to do in the mayor's office that he couldn't do as president of an elected Board of Education."

The joint broadcast appearance by the two, the second in as many days, came at a time when each has begun to intensify personal attacks on the other, with the Nov. 7 general election only two weeks away.