Q. We recently moved to the Mt. Vernon section of Fairfax County, and our 15-year-old daughter, like the 13-year-old you wrote about from Arlington, needs things to do.

Our daughter is finishing her sophomoe year at her old high school, so she won't know anyone in our new neighborhood this summer.

She is very musical -- plays the piano, viola, violin, oboe and flute, and loves singing, dancing and some sports, like swimming gymnastics and softball. Like all people her age, she's very interested in getting her driver's license, but she's not old enough yet. In the meantime, transportation is a problem.

A. The Mt. Vernon Nursing Home not only could use your daughter's musical talents, but needs teen-agers to help the nurses; to write letter and read to patients, and to participate in the Adopt-a-Grandparent Program.

The DeWitt Army Hospital, Ft. Belvoir, is looking for Boy and Girl Fridays to work with patients, and to be messengers and receptionists. The Red Cross enlists teen-agers as young as 13 and places them in these and other slots in your neighborhood, says Adelaide Krizek, who runs the Mt. Vernon chapter at Groveton High (765-0775).

She also places "Volunteens" to work with the special Olympics and with the Fairfax County recreation program for the mentally and/or physically handicapped, starting June 26. The program needs help on their field trips, in arts and crafts and in dancing and games.

Also, 15-year-old volunteers are needed on the Bloodmobile and to help man the United Community Ministry's food and clothing store in your area.

Credit for any Red Cross work can be transferred to any chapter anywhere. And the Volunteens have parties. A picnic and dance are planned this summer for the 250-300 young workers from the county.

The Alexandria Archeological Research Center wants helpers, 14 and older, to survey sites, go on digs, sift for artifacts and sort and mend them: an exciting project since the city is full of pieces of history. For more information, call Steve Shephard, 750-6200.

Your daughter also may be interested in the August band camp at Mt. Vernon High School, or in working at the Summer Drama Workshop at Mt. Vernon Unitarian Church, June 30-July 25. Director Staci Plutchok is especially looking for a volunteer in music to back up the improvisation experience offered children under 13. Her number: 765-8939.

Even though transportation is a problem, we hope you'll figure some way for your daughter to get to one of the enrichment programs in the city.

One of the best in the Summer Theater Conservatory at the National Cathedral School, 3109 Wisconsin Ave. NW. There will be classes in acting, movement, mime, voice production and diction for grades 9-12 every day from 9-4, June 23-July 18. Cost: $400. Call 537-6348.

Six crackerjack math teachers from public and private schools have their second "Adventures in Mathematics" this summer, also on the Cathedral campus. The two sessions, each lasting three weeks, explain "what math teachers would like to cover during the school year, if only they had time," says Joe Wolfson, one of its founders.

Somehow they make us old socks think that we, too, could have liked math, if only we had had a chance to learn about computer math, geometry and art and symbolic logic with its truth trees and truth tables (whatever they are).

The 13, two-hour minicourses cost $100 each, and are taught daily from June 30-July 18, or July 21-Aug. 8. They're open to any math lover between grades 7-12, although a younger whiz might be accepted. For more information, call 703-521-2982, or 301-229-4817, or write AIM, 17 Duvall Lane, Gaithersburg, Md. 20760.

Parents of younger children might be interested in two computer language courses to be taught this summer in the gifted and talented programs at American U. and George Washington U. The AU program (686-2186) has four two-week sessions scheduled, starting June 23 for 5 1/2 hours a day. Cost: $200. Other subjects are also taught, some for children as young as the first grade.

The GWU course, produced by the Reading Center, will have 5th-9th graders learning use of a computer by analyzing and solving problems in an urban environment.

The Center also has a reading program for the gifted at Sidwell Friends, 3960 37th St. NW, where children between 2nd and 9th grades will read -- and dramatize -- excerpts from the world's great stories. Both GWU courses cost $350, last 3 hours and run from June 23-July 11.