Richard Shaw remembers finding some of his parents' old college textbooks when he was in fifth grade and using them to teach himself chemistry.

In the summer after eighth grade, David Reiter took a special mathematics course at Johns Hopkins University, covering three years of high school math in eight weeks. He studied first-year calculus in ninth grade, then more advanced math for two years at the University of Maryland. But last year, as a senior at Wootton High School in Rockville, Reiter said he didn't take any math courses at all.

"I concentrated more on extracurricular activities," Reiter said. "The math was getting pretty esoteric and I just wasn't that interested in it."

This week Reiter and Shaw, a senior at Bethesda-Chevy Chase High School, are among 140 students from around the country being honored here as Presidential Scholars -- 121 chosen for academic achievement and 19 for achievement in the arts. They were picked by a presidential commission as the top students in the country in the 23rd year of the competition, which carries a $1,000 prize.

They are scheduled to receive medallions this afternoon from President Reagan at a ceremony on the White House lawn. Yesterday they heard Supreme Court Justice Sandra Day O'Connor and several members of Congress. On Monday night they were honored -- and more than a dozen of them performed -- in a special show at the Kennedy Center.

The academic winners were picked in a nationwide search that began with test scores for about 2 million students. Two winners -- one female, one male -- were chosen for each state, the District, Puerto Rico and Americans living abroad. An additional 15 top students were chosen at large, and 19 were selected in talent competitions in the arts.

The nine winners from the Washington area include one from a D.C. public school, Peter M. Ferren of Wilson Senior High in Northwest; two from private schools in the District -- Marc S. Abramson of Sidwell Friends and Lisa Weintraub of Georgetown Day School; three from Montgomery public schools, Shaw, Reiter and Leslie E. Holt of Walt Whitman High School in Bethesda; one from Howard County's Mount Hebron High, Loretta Y. Lowman; and one from Arlington's Yorktown High, Anne Y. Matsuura.

"It's a nice honor for me and for the D.C. public schools," said Ferren. "My parents could have sent me to private school, but I would never have gone elsewhere . . . . In public school you're given more freedom to make it or not make it. It's up to you. I did not feel I needed the push that private education gives."

But Weintraub, who lives four blocks from him in the Chevy Chase section, switched to Georgetown Day after attending Lafayette Elementary School and Deal Junior High. In her application for Presidential Scholar, Weintraub said she enrolled in private school in 10th grade because of "the lack of enough skilled teachers in the public school system . . . . When I graduate from college, I want to come back to D.C. to make teacher qualification tests much more stringent and teachers' salaries much higher, so that we attract better educated teachers to the public schools."