Quentin Thompson talks about himself with the candor and distance of a man looking back on his childhood.

"I played a lot and thought school was a joke," said Quentin, an 11-year-old pupil at Alexandria's Charles Barrett Elementary School.

His classmate Jill Sturge displays an equal sense of sophisticated hindsight. "I used to have a problem of putting myself down," she said. "I'd say, 'They can do it but I can't.' I thought everyone was better than me.

"My teacher taught me I can do it."

Jill and Quentin have much in common. They like to talk about their futures. They are black and live in a low- to middle-income section of Alexandria.

And not too long ago they were considered underachievers, potential casualties of an educational system still struggling to find an antidote to the wide academic gap between black and white children in the same classrooms.

The closest thing in the school system to a cure, say some Alexandria parents and educators, is Barrett Elementary, which Jill and Quentin have attended since kindergarten and which has developed a variety of innovative programs designed to improve minority achievement.

Last week, Alexandria school officials announced that, for the first time, black children in its 12 elementary schools had scored above the national norm of 50 percentile points on the Science Research Associates test, a nationally accepted instrument for measuring mathematics, reading and language skills.

Two years ago, the city's black elementary pupils averaged 47 out of a possible 100 percentile points on the test; their white classmates averaged 83.

Shocked by the discrepancy, Superintendent Robert W. Peebles, former head of the racially sensitive Stamford, Conn., and Marshfield, Mass., school systems, launched a major effort to improve minority achievement. This year, $920,552, or 1.5 percent of the school budget, is earmarked for programs to help minority and disadvantaged students.

Barrett's formula for helping pupils identified as potential underachievers is not expensive, but it is time-consuming.

The principal and some of the teachers make house calls every week to talk with parents. Teachers and parents tutor pupils after school at least twice a week. And training in special test-taking skills, with an emphasis on psychological preparation, has become a regular part of the classroom curriculum.

At the school, emphasis is put on uninterrupted instruction and completing assigned tasks. Many pupils carry a time card divided into 15-minute blocks so they can keep track of how they perform in the very short run.

At the foundation of Barrett's program is raising children's self-esteem and expectations. "The teachers, they drill you and they want you to do good," said Quentin, who was elected class treasurer this year. "But you have to want to do good."

"They want us to be something important," said Jill. "I think they want us to succeed."

Many educators doubt the validity of improved test scores as indicators of learning. It is too easy, they say, to teach pupils test-taking tricks without teaching them important critical-thinking skills required to put a good education to use.

"It worries me that, if that's the only thing we're doing is teaching kids to take tests, that we'll be raising false hopes," said School Board chairman Timothy S. Elliott.

But Barrett, with 278 pupils, has gone a step further than simply teaching test-taking skills, say parents, teachers and school officials. The number of black pupils in the gifted and talented program has risen, the overall number of pupils on the honor roll has increased from 48 to 68, and the number of pupils held back a grade has declined, said school officials.

"The {performance} gap within the class" has been narrowed, said Carolann Sharp, a white parent, about her child's third grade class. "The bottom has been brought up to grade level. It has enabled the third grade teachers to do more science and enrichment so that the entire class has benefited."

Barrett is an unlikely success story in some ways. Its student population is the most economically polarized in the city. About 38 percent of the pupils are from families under the poverty level. Most of the white pupils, 34 percent of the total, come from the upper-middle-class Beverley Hills community.

Many of the minority pupils' families are being forced to move out of apartments undergoing renovation in the western Arlandria area of the city, a disturbance that has made Barrett's job of getting children to concentrate on schoolwork more difficult.

Principal Betty Hobbs, a teacher at Mount Vernon Elementary School for 15 years before she became Barrett's principal in July 1985, is a key player in the school's effort to improve minority achievement.

"Many of our black students may know more than their tests show," said Hobbs. "The key is teaching them how to transfer the knowledge onto a test. We teach them not to give up, how to relieve the frustration. It tends to teach them to believe in themselves."

Hobbs is a firm believer in the power of praise, and she insists that pupils look her in the eye and speak in full sentences. Each week, she calls at least one pupil into her office to praise him or her in front of her secretary, Debra Monday.

Also, Hobbs hands out pencils inscribed "I'm Proud of You -- From Mrs. Hobbs."

Yancy Ramos, 8, got one Thursday. "I'm going to take this home and read it to my mother," she said, clutching it tightly. "Maybe she'll be proud of me."

Every Friday, and often during the week, Hobbs makes house calls. Alone, or with a teacher, she visits parents who are not sending their children to school regularly. She also makes visits if pupils have discipline or homework problems, or if they seem unusually sad or disruptive.

Sometimes she telephones before she arrives, but often she goes unannounced. Many of the parents have no telephone.

"Sometimes I'll just drop by, sit on the stoop or stand in the yard," she said. "I think {parents} know I care. I talk with them, not down to them. You must first make the parents feel you are with them, that we are a team. It takes both of us."

Regardless of the gains schools make, most educators believe that a child's socio-economic background is a major contributor to academic success. Closing the gap, argue educators in Alexandria and elsewhere, is an uphill fight if parents do not reinforce, even indirectly, the skills and attitudes that a child may learn school.

It has been particularly difficult to get black parents involved in Barrett, say some parents and school officials. Many of them work two jobs, are sharing cramped temporary quarters with friends or family, and are not used to working with school officials. White parents dominate the PTA, although they solicit and encourage black parent participation.

Loretta Thompson, Quentin's mother, said the only way black parents may be willing to get more involved is if they are urged to do so by blacks already active in the school.

"Black parents, especially in mixed schools, we have a tendency to feel a little intimidated by the school. I felt that way," said Thompson, supervisor of administrative services at a local power company. "The first time I went to a PTA meeting, I felt like a minority. I looked around and said, 'I have to let myself be heard.'

"The PTA is very good," she said. "The feeling of intimidation was my own."

To keep parents informed, students take home weekly bulletins with snippets on their small triumphs ("Andrea Wiley is moving right along on addition and subtraction. Thomas Bell can write his first and last name . . . . One of Tyler Yowell's tadpoles is now a frog.").

Pupils in Ann Douglas' class carry home a notebook each week with her comments about their work. Parents are encouraged to write back.

"Some of them have home lives that are so unstable," said Douglas, who teaches a special class for students who are old enough, but are not academically ready, to start first grade. "I may never have seen a parent or talked to a parent, but at least there's a line of communication" through the notes.

A group of nine pupils from many racial and economic backgrounds sat in Hobbs' office one day recently. Throughout the year, she had been trying to get them to improve their attitudes about school and take more responsibility for their work.

First, they reversed roles with the principal. The pupils called her Betty, which made them giggle, and they demanded to know why she did not do her homework the night before. She addressed them by their last names and gave every excuse imaginable.

When the game ended, Hobbs reminded the pupils that school was out for the summer in another week and asked them what they thought they had learned from their talks.

"I learned to look you straight in the eye," said one.