The handsome young woman's home is on an inner-city street that rapidly is becoming a chic address. The bang of a heavy door shuts out the sound of noisy children traipsing past. The smell of wood from a glowing fire permeates the air, and a gray cat darts fitfully up and down the oak staircase.

The woman's dark eyes are smiling. She is 33 and a single parent. By most objective reckoning, she has attained much of "the dream." She is well educated and her only daughter, age 8, is in a private school. Traditionally, this not only bespeaks position for her, and a better-than-average education for her child, but also assures her daughter a better chance at college and a better job after that.

Few touches within the woman's home reflect her civil rights activism in the 1960s, but that experience is etched upon her consciousness. "If anybody would have told me my child would have been in private school 10 years ago I would have told them they were crazy," she laughs, with a graceful wave of slender fingers.

But she feels a bit of alarm. The trouble is not primarily academic. There are daily pluses for her child but there is the matter of cultural identity. The woman has a sense of emotional unease about the predominantly white environment in which she has placed her daughter, and she feels the jury is still out on what its effect will be.

"It's an entirely different world from the one in which I grew up and I must admit that I'm not as comfortable in it as she seems to be," said the woman, setting down a ceramic mug of creamy coffee as we talked.

"You know the classic stereotype of the black person who is raised so much around whites and is part of neither the white nor the black world and is an object of ridicule -- I wouldn't want that for her," she says.

There's the realization that she's the modern black woman, more "American" than her mother. But riding just under the surface are the pent-up furies, the residue of generations of discrimination and brutality. She must find a way to help her daughter develop without dishonoring her roots or jeopardizing her appreciation of the wider world and of universal human values.

The women is one of thousands of black parents with children in the estimated 80 private schools in the Washington area. For most -- as with many white parents of average means -- it's an extreme economic sacrifice. But as one put it, "I can't leave an inheritance, so this is it."

Yet many share this concern of instilling a sense of cultural identification in children who spend their school days in predominantly white environments, who come home all too often to "Fonzie" and "The White Shadow" for a baby sitter, and who are often greeted at night by a single parent or two-paycheck parents too tried to fill their difficult and important parenting function.

Part of this woman's unease is that the institutions that gave her her sense of moorings are but fragile factors in the life of her child. Her growth was nurtured by a strong black church, a cohesive community, an extended family, and a school in which the black experience was stressed.

"The harm done by the weakening of these institutions in so many of our lives is so insidious," she says.

The job of cultural identification is primarily a family function, and it is a problem that black families share with Jewish parents of Hispanic ones. But this problem is complicated by a degree of racism in private schools -- often unconscious -- that is reflected in a sometimes unbalanced presentation of issues and a scarcity of black teachers.

"In any school, you're going to get some things; some things you are not," she says. "I had to make a determination of what would be good; what would be important that she wasn't going to be given. I knew she would be missing a sense of history. By this for a black standing of the world from a black perspective . . . of being able to look at a situation and understand if there is an impact due to race . . . to understand how different races of people look at things."

The woman has talked with other parents and they sometimes trade stories. Here are some examples:

An 8-year-old girl comes home and tells her mother she's to be in a skit the next day. What part is she playing? "The maid."

A school dispute erupts over whether a controversial black public figure should be invited to speak, and some black kids didn't know which side to take. s

Third-graders have a relay race in which they scamper with an eraser atop their heads. A child with corn rows is asked to keep score because she has an unfair advantage -- her hair style will help the eraser adhere. "The one time it is an advantage to be black and they change the rules," her mother laughs, only half in jest.

A sixth-grade history class is studying Reconstruction. A teacher says that, after the Civil War, black people could do anything . . . implying that this included legally questionable acts -- a complete misreading of that turbulent era -- and the entire class turns around and looks daggers at a black boy. He abruptly leaves.

Rising to fill our coffee cups, this mother told me how she goes about instilling history and identification. "I try to listen closely and point out different ways of looking at things. For instance, when my daughter studies colonial history, which is fraught with different perceptions of the same event, I think it is important for her to know there were people here before the Mayflower. I don't think her teachers consider this as important to point out as I do. But there is a commonness of the experience of ethnic minorities that I think she should know about . . . ."

I know parents who are history buffs, who use resources at the Smithsonian, who go to plays and the Archives, who fill under their home with books, who talk casually over the dinner table with their children -- using as starting points incidents the children encounter or news events.

Private schools in general provide a sense of individual caring and concern that the larger, more impersonal public schools find it difficult to provide. But part of their crucial obligation to provide importance and dignity won't be met until the scarcity of black teachers is tackled. A survey of 34 private schools found 19 black teachers out of a total of 895 teachers. Some private schools have sought black teachers vainly; others have unreal qualifications. "I want a black English teacher -- from Harvard," one administrator told the Black Student Fund, which is working on this problem. a

Black teachers would give an extra dimension to white students and teachers as well. At too many schools, the most conspicuous black staff is the service staff.

"There are such painful aspects of teaching cultural identity -- sometimes the kids resist your always talking history and identity," the woman continues. tIt points out to them differences they wish didn't exist. When they're young, differences are painful. They want to be ONE with their situation. I try to explain I'm not talking about right or wrong, simply that people are different." She instinctively knows that a people who believe that its history is irrelevant is on its way to cultural extinction.

Her fitful gray cat finally settles down, curling up near the fireplace that by now shows only embers. But she is not serene. "I am not sure," she says, passing the back of her hand against her forehead, "that the trade-in at the end will be worth it."