BECAUSE BAD RAPS and stereotypes tend to linger long beyond any basis they may have had in fact, the latest news from the District of Columbia's public schools should be shouted from the rooftops: the city's schools are showing signs of scholastic improvement, with reading and mathematics test scores on the increase for the second year in a row. Yes, we all know that two years isn't enough to retire the cup.Still, the progress is significant and encouraging, and tremendous credit goes to Superintendent Vincent E. Reed -- who has been the driving force behind this progress.

Dr. Reed is among the first to note that the schools still have a long way to go, since average test scores remain below national norms. But after a decade of declining or stagnant achievement, a two-year upswing has put the city's third-graders just one month behind the national norm in math and five months behind in reading; in 1978, they were four months behind in math and eight months behind in reading. Sixth-graders scores gained an average of five months in both subjects from 1978 to 1979 and another two months by May of this year. This leaves these test scores three months behind national norms in math and one year back in readings.

News from the ninth grade is worse: after slight gains in both reading and math in 1979, there was a three-month improvement in reading scores this year, but no change in math -- and these scores are all distressingly low. It means that the District's ninth grade is still 3 years behind the national average in math and nearly 2 1/2 years behind in reading. Greater emphasis on these higher grade levels clearly is in order; it is not enough to celebrate improvements of younger children while continuing to let the teen-agers who are next to go into the working world do so at terrible disadvantages they don't deserve.

That there have been similar patterns of test-score improvements in other big-city systems does not diminish the story here. Some of this general progress may be due to nationwide educational efforts -- Sesame Street, Head Start and so forth; but more interesting is the fact that many of these other school systems have done what Dr. Reed has done here; put the emphasis on basic reading and math -- the good old phonics and numbers -- in a "competency-based curriculum" that serves as monitor and guide for teachers. The teachers, by the way, have not been undermining this program; not only did their union endorse it three years ago, but teachers were involved directly in its planning, as were principals.

This renewed dedication to the teaching of fundamental skills, coupled with more efficient operations and a more disciplined atmosphere, has made a difference. But there are plenty of gray clouds: sudden and severe budget cuts, however justified, are causing disruption and low morale among principals, teachers and their dwindling constituency -- and the resurgence of bush-league behavior on the school board is infuriating even the most die-hard defenders of suffrage at all costs. So far, Superintendent Reed has weathered it all with amazing patience and dedication. For now, hope of any more improvements depends on keeping in public education here the "fourth R" -- which stands for Reed.