Student mathematics scores on the Scholastic Aptitude Test plummeted 10 points in Alexandria and nine in Arlington, but overall scores rose dramatically in the District and Montgomery County, according to highly mixed local results released yesterday for the widely watched examination.

Verbal scores in Alexandria and Arlington and both math and verbal scores in Prince George's County stayed the same or remained very close to those of last year. Both categories of scores in the District and Montgomery County rose by more points than did national scores, although the District average remains below the national average. In Fairfax County, the increase equaled the national rate.

The results showed Prince George's County and District students lag behind their counterparts nationwide. Arlington, Montgomery and Fairfax counties scored comfortably above the national mean.

Scores were released by area school systems and the College Board, which did not have available a breakdown of scores for minority students or individual schools.

The release of scores for local school systems followed Monday's announcement that national scores on the aptitude examination went up for the fourth year in a row and made the largest single-year gain in 22 years. The national verbal average was 431 and the national mathematics average was 475, up five and four points respectively.

The SAT, which is taken by about a million college-bound seniors every year, is divided into verbal and mathematics sections, each with a possible score ranging from 200 to 800.

In Arlington, mathematics scores dropped nine points to 497, still above the national average of 475. Arlington County schools spokeswoman Margaret Heckard said one explanation is that more students -- including, presumably, the less talented ones -- took the test this year: 73 percent of seniors this year, versus 65 percent last year. Nationally, only 37 percent of students take the test.

In Montgomery and Fairfax counties, however, officials said the rate of seniors taking the test increased -- in Fairfax, to a record 80 percent -- and scores went up as well.

In Alexandria, math scores dropped 10 points to dip below the national average. "The word for it right now is puzzling," said James P. Akin, Alexandria's executive assistant for research.

Akin said the city's math score showed an unusually high increase last year, and this year's score could be a return to the expected level.

In the District, James T. Guines, associate superintendent of instruction, credited the improvements to the city's intensive efforts to improve academics, which he said already have shown some payoff in elementary and junior high school test scores. "We're making some inroads," he said.

The District results, however, included private school students -- about half of those who took the test -- and Guines could not immediately assess how much of the gain came from the public schools.

In Montgomery County, where the verbal scores were the highest since 1974 and the math scores were the highest since at least 1973, officials were happily puzzled.

"I would say the same thing when they go down," said James Myerberg, the county's coordinator of testing. "I don't really know why."

In Fairfax County, the 515 average score on the math examination was the highest since at least 1973, school officials said. The math score increased by four points compared with last year, and the verbal rose by five points.

"I think it's because we put a strong emphasis on academics," School Board Chairman Mary E. Collier said.

In Prince George's County, the math scores stayed steady and verbal scores increased only three points. The county is engaged in an intensive program to improve its academics.

"While we feel good about the fact that they didn't go down," Superintendent John A. Murphy said, "we still have hopes they will go significantly higher in the future."