The education of children, for all its complexity, still comes down to one element: the teacher. For all the debates over back-to-basics and open classrooms, standardized tests and creativity, it is still the teacher who makes it work, or not work. The children will remember, not so much the new math, but the teacher.

In a two-story brick elementary school in Prince George's County, there is a teacher who by the account of her superiors, her peers and her students, makes it work. Her name is Elsie Neely: Mrs. Neely, the sixth-grade teacher in Room 20, on the second floor at Ridgecrest Elementary in Hyattsville. She has taught elementary school for 17 years, since she was 20. For the last 11 years, she has taught sixth grade at Ridgecrest.

Year after year she teaches the same things: long division and percentages and decimal points and photosynthesis and punctuation and adverbs and prepositions and where Russia is and what such words as oxbow, onyx, endive, accost, plantain, regiment and disconsolate mean. But each year is a new class, and each year she starts fresh, as if it were the first.

Every summer, when classrooms are assigned, there are new sixth graders who pray they get Mrs. Neely. New teachers want to discover her secret. A Montgomery County elementary schoolteacher named Gene Carpenter, who was her student teacher three years ago, says the secret of Mrs. Neely is this: "She walks that thin line between student and teacher. I used to describe her to friends of mine as a grown-up kid. She knows what kids think. She knows how they respond. But she is always the teacher."

When her students talk about Mrs. Neely, they say they like her because: She is nice, pretty, cool, funny, kind, a good dresser. She makes you want to learn, she never makes fun of you, she never talks down to you, she calls you at home on the telephone and you can call her. She makes you feel like somebody special, even when you feel dumb because you're reading at a third-grade level and you're 11 years old, even when you can't make friends.

She tells you stories, like the one about how a nun beat her when she was in Catholic school in Baltimore. She helps you understand your work. She listens when you tell her what you did at recess and what you watched on TV last night and what you dreamed and how your parents fight and how it felt when your grandfather died. She comes to your birthday parties. She teaches you to be a lady, to be a gentleman, to be aware of other people's feelings. She knows things nobody else knows: how to make a Christmas tree out of chocolate milk cartons and how to tell an oak tree from a maple. But maybe the best thing about her, the thing that makes her different from almost every other teacher you ever knew, is this: "Mrs. Neely never yells."

There is nothing magical about a good class, but Mrs. Neely always has one. This year, her class is 17 boys and 16 girls: Shawn, Darren, Timothy, Kenneth, Douglas, Earl, Alphonso, Melvin, Thomas, Joseph, David, Michael, Reginald, Tony, Troy, Charles, Devin, Marcie, Veola, Andrea, Christie, Hosana, Tammie, Kumelym, Cindy, Lori, Michelle, Cynthia, Diane, Pennie, Peggy Lee, Tracie, Phillia. They are the sons and daughters of blue-collar workers, vice principals, teachers and businessmen and at least one welfare mother. Most live in Hyattsville, but some are bused from Lewisdale and Takoma Park. They are black and white children of the close-in suburbs, and they think of Washington as a faraway place where danger lurks.

Monday through Friday, from 8 a.m. to 2 p.m., with 30 minutes for lunch and 30 minutes for recess, Mrs. Neely presides over Room 20, an elegant woman in high-heeled leather boots and fine wool suits with silky bow-tied blouses, in a world of pale green cinderblock walls, blackboards and wooden desks with metal chairs that scrape noisily against the scuffed tile floors. The principal talks to the class through the white box placed high on one wall, next to the big round clock with the black numbers. Room 20, like many elementary school classrooms, is rendered beautiful by a teacher with a talent for decorating with colored construction paper. This winter a big paper snowman covers the door, and paper snowflakes dangle from the ceiling.

No one is humiliated in the class of Mrs. Neely, who remembers her own childhood humiliations in Catholic school. She grew up poor in the heart of Baltimore, one of nine children born to a postman and his wife, and she remembers how she felt when she was suspended for pretending with her friends, on the city bus to school, that she was rich and had a maid. She has a special feeling for the troublemakers and talkers and students who just can't sit still--the "demons," she calls them--because, she admits with a grin, she was one herself, back at St. Francis Academy in Baltimore. "Reformed demons make the best teachers," says Mrs. Neely. "I know all the tricks."

It is not supposed to happen, but it happens anyway in elementary schools: Children are pegged at the start as good kids or bad kids, smart kids or dumb kids. Each student has a record, and each year the new teacher can't help finding out what the last teacher thought about each child in the class. This fall a fifth-grade teacher warned Mrs. Neely that one boy in particular would ruin her class.

"She said, 'That boy is going to be a terror. He threw a chair at me. He won't make two weeks.' " Each day, Mrs. Neely says, the teacher would ask for a report. "She'd say, 'Has he acted up yet?' I told her, 'He's getting A's in my class.' " And he was.

Mrs. Neely says she never checks her students' records to see what other teachers have said about them. "I don't want to prejudge anyone," she said. "They may have been a demon for someone else, but they may not be for me."

Some of Mrs. Neely's students believe she has some sort of supernatural instinct that tells her when someone is doing things he shouldn't. "She's got eyes in the back of her head!" said an eighth-grader named William Rogers, known as Roger, the third of four children in his family to have Mrs. Neely for the sixth grade. "One day Shawn and I were playing around, throwing crayons. Mrs. Neely's back was turned. She was talking to someone else. I threw a crayon. She turned around and said, 'Roger, go pick that up.' I couldn't believe it. She knew exactly where it was. I said, 'God.' I didn't throw any more crayons after that."

"She taught us how to behave," the reformed 13-year-old crayon thrower said. "You don't have to be a regular lady or gentleman. You have to be aware of other students. She taught us they have feelings, too."

Sixteen-year-old Luz Kaplan from the Dominican Republic, who speaks English with only a trace of an accent, says she will never forget Mrs. Neely. Luz was in her class three years ago, when she had been in the United States only one year and was assigned to the sixth grade because she could not speak English. "I was scared. I thought, 'What if this lady doesn't treat me well?' But the first day I said, 'I'm going to have a good time in this class.' It was just her face, the way she acted.

"I used to say, 'I can't do it, I can't do it. It's so hard.' She'd say, 'Luz, you can do it. Try, just try hard.' She really taught me. She made me feel good. I wanted to come to school every day. I wanted to learn. It made me happy to see Mrs. Neely. By the end I could understand almost everything."

Luz remembered how hard it was, being 13 years old in the sixth grade, a girl from a foreign country who looked different from all the others. "Some kids used to bother me. They'd say, 'Look at that girl, she's from another country. She's 13 in the sixth grade.' Mrs. Neely would hug me and tell me not to worry about it. She told me I was special. She can make people better."

One day shortly before Christmas vacation, Mrs. Neely began a discussion about Poland. She was very upset when martial law was declared there, and each day she would tell the class the news she had heard on her car radio on the way to school. Someone in the class said that President Reagan had given a speech about Poland, and then one girl volunteered, with feeling: "I don't like Reagan."

Mrs. Neely asked why, as she often does when her students give opinions.

"I don't want to tell you."

"Is it racial?"

"No." The girl looked as if she might cry. "I don't like him because he's putting my mother out of work."

Mrs. Neely looked sad and asked the student where her mother worked. She said her mother worked at the Veterans Administration. Several other students, obviously sensing the distress of the girl whose mother was going to lose her job, raised their hands.

"My mother used to work for the government till Reagan came. Then she had to leave."

"My mom works for the government, and she's probably going to get laid off, and she doesn't know how to do anything else."

After class that day, Mrs. Neely said she would never have pressed the girl about her dislike of Reagan had she known the reason.

Mrs. Neely rarely talks about what she does. She just does it. But if asked, she can talk for hours about teaching and the sixth grade. "When you tell people you're a teacher, they say, 'God bless you,' like you're a martyr because you're enduring this horrible life of a teacher. They think kids are crazy," she says. "But they were the same way, too. Kids aren't different. School isn't that different."

Sixth grade, she says, has always been the last year before adolescence, before dating, before changing classes and being taught by seven teachers instead of just one. It is also the last year that most students fall in love with their teachers.

Mrs. Neely can speak of teaching as a cherished calling. She can also say, as she walks down the hallway on the day before Christmas vacation, as excited as the students: "Where else can you get two weeks at Christmas and three months in the summer off?"

The students do not know, and probably would not believe, that Mrs. Neely dislikes some things about teaching--for example, grading hundreds of papers, getting up early so she can get to school by 7:30 every morning, never having enough time to go out for lunch, having to sit through boring meetings. And because she has her own restless moments, she understands those of her students. There is a certain flow about her class. The children move around, within reason, and so does Mrs. Neely, a teacher who can be seen hanging from the jungle gym bars and running around the playground with the children during recess.

"I know how they feel. I hate to sit still at meetings. I never sit at my desk," she says. It is usually quiet, but not absolutely quiet in Room 20. Sometimes the students get noisy, but this doesn't particularly bother Mrs. Neely, a woman who says she never gets headaches. "Eventually, they get themselves quiet. They don't realize it, but I'm trying to teach them self-discipline."

When she first started teaching 17 years ago, it was hard to keep order, but then she found the formula that takes so much work: "I noticed some days they were well-behaved, and that was on the days when I was really teaching. When I wasn't, they acted up. All kids want to learn. They'll be quiet if they're learning something." And so there was no need to yell.

Mrs. Neely seems to glide through the days, as serene in the afternoon when she drives out of the teachers' parking lot in her black Toyota Supra as when she arrived at 7:30 a.m. Gene Carpenter, the former student teacher whose role model is Mrs. Neely, says it can be deceptive, watching her. "It's not as easy as she makes it look. It's hard. Her personality brings out the best in kids."

Just as no one is humiliated in Mrs. Neely's class, everyone is praised, no matter how small the accomplishment.One of her students this year is a boy named Tony, who dreams of being the heavyweight Golden Gloves champion of the world. He is 13 years old, in the sixth grade, and he says, "Mrs. Neely never makes fun of me. My other teachers made fun of me. They made me feel like I was dumb . . . I'm not as dumb as I used to be." Each time Tony finishes an assignment, Mrs. Neely makes a fuss and has the class applaud. This makes Tony happy, and he flashes his biggest smile. He reads books on motorcycles and boxing in class, but Mrs. Neely doesn't mind. "At least he's reading. That's the important thing."

Her interest in her students does not end with the last day of school. Mrs. Neely's sixth-graders grow up, and she goes to their Sweet 16 parties and their basketball games and their high school graduations and their college graduations. She saves every card they send her, every note, every scrap of paper, every letter, every Valentine, every drawing.

One day last fall two of her former students, now high school seniors, knocked at the door of Room 20. They told her that Catherine Sourlis, who had been her student six years before, had been killed in a traffic accident involving a drunken driver and it would mean a lot to Catherine's family and friends if Mrs. Neely could go to the funeral.

It was 9:30 a.m. The funeral was at 11 a.m. Mrs. Neely went to the vice principal and suggested that the aide take over her class. She said it was important to her to attend the funeral of a former student. The vice principal said she couldn't go. She returned to her class, thought it over and went back to the vice principal. "I asked him if he could take over my class. He said no." She went back to her class again; this time she made up her mind. She went to the vice principal for a third time, telling him she was taking a day of personal leave. Then she went back to say good-bye to her class. The vice principal appeared moments later. "He said he would take my class."

In all Mrs. Neely's years of teaching, Catherine Sourlis was the first of her students who died, or at least the first whose death she has found out about. When she returned from the funeral, her students asked if she cried. Each year, it seems, someone's grandfather or grandmother dies, and sometimes the children cry in class about death. When Cindy's grandfather died this year, another child began crying hysterically. His father had died last year. Mrs. Neely comforted him at length. "I told him, 'I know how you feel. My husband died.' "

Mrs. Neely knows so many of her students' secrets, but they are not yet at the age when they want to know hers. That Mrs. Neely is a widow with two sons to rear--one aged 8, the other 18--rarely enters the thoughts of the children in Room 20. They want to know how many kids she has and how old she is (she is 37, but she never tells them that), and if she has a pocketbook to match each of her five pairs of boots and if she ever wears pants. When her husband died 2 1/2 years ago, she was absent for a while, and "when she came back, everyone was extra nice," recalled William Rogers, the former crayon-thrower. "No one got on her nerves. We tried to cheer her up. We told her how good we were for the substitute. We gave her cards. We had flowers for her. Everyone understood how she felt. We were getting sick of that substitute. The substitute wouldn't let you talk. She got right back into teaching. You couldn't really tell she was hurt."

Mrs. Neely's students, who dream of becoming football players and baseball players and lawyers and secretaries and American presidents (Melvin wants to be the first black president and Tracie wants to be the first woman president), don't know that their teacher has dreams, too. They do not know, for example, that Mrs. Neely, who holds a master's degree from Howard University and earns $27,000 a year, wants to travel around the world. She wants especially to visit Russia and China. She wants to be rich and she hopes that an invention of hers-- a gadget that she wants to keep secret for now because of patent problems--will help make the dream come true.

Mrs. Neely does not mind that she does all the listening in Room 20.

"To them, I'm a teacher. And that's it."