SALT LAKE CITY -- Robert Kovach was frustrated. There he sat in the first row of the classroom trying to write an essay about "Things I Like to Do," but every time the 16-year-old put pencil to paper, he came up blank.

The distractions didn't help. On one side of the room, a 7-year-old girl from Montana was excited about her drawing of an imaginary house she hoped her family would one day own. On the other, a boy from Seattle was giggling over a science experiment.

"Why don't you go out into the hallway where it's quiet?" his teacher, Marilyn Treshow, suggested. Fifteen minutes later, Kovach returned with a composition that caught her by surprise.

"I like to go off where no one can find me and just ponder my thoughts about different things I've been through, such as living in a shelter, being looked upon as someone who'll never have anything in life," he wrote. "It makes you feel like you just want to explode . . . .

"I also like to think of how I am going to better myself in my future life, like getting my own business and owning my own house, because I am not going to end up like this."

This one-room schoolhouse, wedged under a highway viaduct on the southwest side of Salt Lake City -- the "Crossroads of the West" -- goes by many names: "The Shelter School," "The Viaduct School," "The School on 6th Street." But to the 14 homeless pupils who range in age from 6 to 16 and claim an equal share in rootlessness, it is known simply as "The School With No Name."

"They're just like other children except they know more about life and learn it sooner than other kids," said Treshow, an employe of the Salt Lake City school system who has been teaching at the three-year-old school for two years. "I'm consistently amazed by how many places they have seen and the troubles they've had, but what really comes through is the strength. These are very strong kids."

There is no place in this country quite like The School With No Name, which doubles at night as a shelter. Students get their hot lunches not in a cafeteria but in a soup kitchen across the railroad tracks at St. Vincent DePaul. The playground is not a grass field but a wooden hut piled high with donated clothes in which the kids burrow tunnels to play hide-and-seek and war.

And the centerpiece of the classroom is not a multiplication table or a list of the alphabet but a large map of the United States that boldly proclaims, "Places I Have Been." The map is crisscrossed by scores of black threads showing where their parents have searched for work and a sense of place.

"Colorado, Washington, California, Oregon, Wyoming," Robert Kovach said, listing a few states where his family has lived or traveled through in the last three years. His brother and two sisters sit with him in the classroom. "What I'd really like is for Dad to get a job and for us to get a decent house. It's no fun living like this."

This is the only school in the nation known to serve only homeless children in a public shelter. It was started in 1984 by the city's public school system with Traveler's Aid International, which runs the family shelter here, to educate children who otherwise would not -- or could ot -- attend school.

Most American cities try to put homeless children in regular public schools, but few are able to attend full time.

"Proof of residency requirements often prevent them {homeless children} from going to local schools, and more often they just fall through the cracks," said Maria Foscarinis, Washington counsel for the National Coalition for the Homeless. "The fact that a place like Salt Lake City has such a school points up how widespread the problem has become."

According to national advocates for the homeless, families with children constitute the fastest-growing segment of the nation's homeless population, most of whom are single men. The Coalition for the Homeless estimates on the basis of a 1986 national survey that families now make up about 30 percent of 250,000 to 2.2 million homeless people in the United States.

Coalition officials point to cutbacks in federal antipoverty programs and the increasing scarcity of housing for low-income people to explain this statistic. Since 1981, money for low-income housing programs, including rent subsidies and public housing, has been cut nearly 70 percent: from $32 billion to $7 billion, according to the U.S. Department of Housing and Urban Development.

Without such support, many poor families have sought a temporary haven in public shelters or moved to other cities when jobs and unemployment benefits are lost. And in the West, Salt Lake City -- The Crossroads -- happens to be a place where many such families end up, latterday "Okies" in the land of the Latter-day Saints.'You Have to Go Where There's Work'

Bob and Iris Kovach, Robert's parents, came here in March from Longmont, Colo. They are fairly typical of the approximately 2,200 homeless families who spend time in the shelter here each year -- white, young to middle-aged and originally from other states.

Bob Kovach, 43, a thoughtful, soft-spoken man, lost his truck-driving job late last November. The family had lived in Longmont for about a year, far from rich but making ends meet. Bob brought home $800 every month, $350 of which paid the rent for their three-bedroom apartment.

Unemployment was not a new experience for the Kovachs who, like many working poor, are shaken by the slightest tremors in the U.S. economy. Before Longmont, the family spent two years in Vancouver, Wash., and nine years in Denver, where Bob worked as an emergency medical technician for a private ambulance company. His working hours in Denver -- 48 on, 24 off -- were "awful for family life," he said, so when he was laid off from that job in 1984, he and Iris decided to make a fresh start in another part of the West.

Friends told them that Vancouver was a beautiful place, with good schools and work available in the timber industry. There Bob landed a 9-to-5 job for $1,000 a month in the warehouse of a wood pulp manufacturer.

Iris, 35, stout and dark-haired with piercing blue eyes, recalls their time there as one of their best. Robert and his siblings -- Joe, 14, Christy, 8, and Michelle, 5 -- attended school regularly, the family lived in an apartment near the Columbia River, and they were able to save a little money. After six months, Iris was overjoyed that they had $500 in their savings account.

But Bob was a casualty of a company layoff, and a coworker steered him to the trucking job in Longmont. Robert, then 12, hated saying goodbye to friends "I'd just gotten to know," but he understood: "Times are hard, and you have to go where there's work."

When Bob lost that job, too, he tried to find another in Longmont because he and Iris were "tired of moving around." But weeks passed and his unemployment insurance ran out. They pawned their television, furniture, even their wedding bands, to pay the rent, but eventually they had to move out. Life seemed so hard there, they began to think it couldn't be worse somewhere else.

Bob heard that Utah was nice -- "We're a born-again family, and I knew there were a lot of religious people here" -- so he telephoned the state's employment services office and was told there was work in construction and maintenance.

The Kovachs piled into their 1971 Ford Pinto and, with Robert, Joe and Christy in the back and Michelle in Iris' lap, took their time making the 600-mile trip. To save money, they spent three nights in the car on the side of the road, each night reserving their customary hour for Bible study. Iris, a devout believer in cleanliness, twice took time to wash the family's clothes, she said, once on the bank of a Colorado stream.

They arrived the night of March 5 with $5.10. After Iris changed $2 into quarters, Bob made several calls from a downtown phone booth to find shelter. "The police said they didn't have funds to help us," Bob said. "The guy on the phone got us in touch with Traveler's Aid. That's how we got into the shelter."Salt Lake City Disappoints Semi-Skilled

This city and its namesake, the Great Salt Lake, lie at the nexus of three interstate highways -- 15, 80 and 84 -- that lead to San Francisco and Los Angeles to the west and southwest, Seattle to the northwest and various points, including Omaha and Chicago, directly east. It is the biggest city from Phoenix to Canada and from Denver to the Pacific Ocean.

John Naisbitt, author of "Megatrends," listed Salt Lake City among 10 American cities with the greatest growth potential in the next decade, primarily because of its location, quality of life and the growing number of high-tech and computer-related industries.

But for many semi-skilled workers and their families who arrive here hoping for a change in luck, Salt Lake appears to be something less than a "megacity." Copper, steel and oil, which historically accounted for the city's growth, are depressed industries here as elsewhere.

"About half our clients come to Salt Lake City from other states. They travel by car mostly, with all the worldly possessions they can pack, hoping to find work and a better life," said Priscilla Solarz, director of the Traveler's Aid office here where homeless people are helped with job referrals and directed to public shelters. Traveler's Aid operates three shelters here with a combined capacity of 400 and all have been fully occupied for more than a year. Last year, 7,791 people found emergency shelter here.

"Some come to Salt Lake specifically. Others find themselves with broken cars or empty pockets while passing through from the oil fields in the Southwest, the timber forests in the Northwest, or California. But all of them are pretty desperate," Solarz said.

Like every major American city, Salt Lake City has not been immune to the problem of homelessness. But in this strict and sparkling place, where an X-rated movie theater cannot be found, where the search for a beer can be futile and where 40 percent of the 218,106 citizens adhere to the conservative moral standards of the Church of Jesus Christ of Latter-day Saints, the contrast between civic ideal and human reality can seem particularly stark.

"This is a new problem for Salt Lake City. We didn't really start seeing large numbers of homeless people here, including entire families, until two or three years ago," said Steven Holbrook, Mormon church member, a former state legislator and longtime community organizer who has actively brought together church and city leaders to address the issue.

"The challenge is pretty unique here for two reasons: Salt Lake is the capital of Mormonism, and traditionally Mormons have taken good care of their own. But what do you do when you're confronted by growing numbers of people who, for one reason or another, can't help themselves?

"Secondly," he said, "there is a civic willingness here to be out front in setting an example for other communities to follow. People want to keep Salt Lake the clean city it's always been, but we want to address the problem in a positive way."

Several months ago, the Mormon church donated $90,000 worth of kitchen equipment to the St. Vincent DePaul society to refurbish a soup kitchen. Several weeks ago, the church appropriated $400,000 to help renovate a building to house two of the city's three shelters. They represent the largest church donations ever for the non-Mormon homeless.

But, as Holbrook puts it, "the feeling is that more needs to be done than just offering shelter and food. We don't want London poorhouses here, but the emergency shelter is just a stopgap solution . . . . All the other problems -- health, employment, education -- have to be dealt with as well."

Families with children seemed a good place to start.

In March, the Child Welfare League of America and Traveler's Aid International completed a study of 331 children of 163 homeless families in eight American cities, including Salt Lake City. They found that 10 percent of the children were in immediate need of health care, 10 percent had been victims of physical abuse -- nearly three times the rate of the general population -- and 43 percent of those old enough to attend school were not attending.

The 13-month-old son of one couple had lived in a car for so long that he became confused and agitated when a social worker placed him on the floor of a playroom: He apparently had forgotten how to play in an open space.

The report's findings did not surprise Solarz, 47, a former child actress who performed bit parts in more than 20 Hollywood films produced in the 1940s, including "It's a Wonderful Life" and "Miracle on 34th Street." As the director of an organization that represents a front door for the city's homeless, Solarz had taken special interest three years ago in the plight of the children.

"Nothing was being done. We were seeing more and more kids and few of them could get into school because of paperwork problems. When you don't have a home and you travel from place to place, you tend to lose track of things like birth certificates, shot records and other stuff that the school system requires," she said. "Fortunately, the school board and the city were receptive to the idea of placing a teacher in the family shelter."

School is in session from 8:45 a.m. until 1 p.m. Reflecting the transience of the children's parents, the pupils' average stay is six to eight weeks. Since September, 120 children have enrolled in and left The School With No Name.

Still, whatever brief structure the school can offer the children is "better than nothing," said Solarz, who spends most of her time ministering to the parents.

In their second-floor offices at the Greyhound Bus Station across from the Salt Palace downtown, Solarz and her staff contend with a daily tide of travail. Social workers' files reveal various reasons their clients ended up in Salt Lake. "Came from El Paso where he lost a job in an oil field. He was headed to Washington state to look for work in the fisheries. No money for gas. Needs shelter for wife and a child and temporary work," one file said.

Another client's "car broke down on I-15. Arrived from Provo. Divorced, two small children. She is without funds, very depressed and needs a place to rest," stated another.

And in the office of social worker Maxine Greer one April day, a middle-aged woman named Elaine was asking for an extension on the month her family would be allowed in the family shelter. She had arrived from Seattle in March, hoping to find work here, and had enrolled her children in The School With No Name.

"I was married 21 years. I got the clothes, the car and the kids. He got an 18-year-old girlfriend," the woman, draped in a large green coat, says as Greer makes notes.

"That's hard to take . . . .

"Well, if there's a God in heaven, he'll get his."

"You don't want to wish harm to anyone. It's bad for your health."

"Him, I do," the woman says. "I'm 40 years old. Over the hill. An old hag with teen-age kids . . . . "

"Your life goes on."

"Maybe," she says, unpersuaded. Suddenly, she brightened. "I got a job today cleaning rooms at the Orleans Hotel. I went in for the interview with this coat on because I didn't want them to see the old housedress I have on underneath. It's only $3.35 an hour, but it's a start . . . . "

"Good for you," Greer says. She waves goodbye and wishes Elaine well.

"How does anyone do it?" the social worker asks a little later, sighing as she files her notes with a pile of others. "How can you expect to raise a family on minimum wage?"'Strangers Drive By Just to Stare'

For Bob and Iris Kovach, it isn't easy. A reddish sun is setting as they sit on a cot in a cramped trailer at the family shelter. The shelter is a collection of five trailers, the schoolhouse and a couple of other buildings on a plot of gravel barely 20 yards from I-15.

The traffic noise is thunderous. Near their trailer, a huge green road sign announces Ogden, Pocatello, Reno and Airport, arrows pointing in several directions. Highway dust and grit wafts over the shelter grounds, which smell of diesel fumes and old concrete.

The orange and white trailers are divided into private compartments in which 31 adults and 33 children sleep. The Kovachs have just returned from The Broiler Cafe where, for $16, everyone feasted on veal cutlets and fries. It was the first hot dinner the family had had in more than a week.

The beds are neat, and the family's clothes are hung smartly along a wall. Bob, who got a minimum-wage job with the city's street department two weeks before through an emergency work program, received his first paycheck earlier in the day -- $220. Minus the price of the night's dinner, the $10.50-a-week rent for the shelter and the $145 Bob owed a fellow who repaired the Pinto's transmission, the family was left with $38 to last the next two weeks.

The Kovachs receive $290 a month in food stamps, which ordinarily would be enough to last, according to Iris. "If I had a refrigerator and a stove and one of those family-packs of hamburger, I could make it last a week," she said. But the shelter has no cooking facilities and hot plates are forbidden, so Iris spends the food stamps on lunch meats, bread, fresh fruits and other food the family can eat in a day or two, while the cash she receives in change is spent on hot meals at inexpensive restaurants.

Iris could try cooking over an open fire under the viaduct as a few of the other mothers do, but "I'd feel too ashamed," she said. "At night, strangers drive by just to stare."

Indeed, the indignities of poverty often seem just as painful and hard-edged as poverty itself. No longer do the Kovachs go bowling or camping. Even their personal freedom is limited. Every day from 8 a.m. to 4:30 p.m. the manager of the shelter locks each trailer so that the mothers and children won't isolate themselves.

"He thinks we'll get too depressed," Iris said. So she spends her days "feeling worse" sitting with the other mothers watching hour after hour of soap operas and game shows on the day-room television.

"Last week," she said, "We went to a church in town, just to see what the services were like. The minister introduced us, and people gathered around to welcome us. One lady came over with her hand out to shake mine and she asked us where in town we lived. I told her the family shelter on 6th Street. And she pulled her hand back like I had a disease."

Bob takes his cap off and runs a hand through his thin brown hair. "That's the toughest part: Seeing what they have to put up with and feeling like there's nothing I can do about it," he said. Each morning, he rises at 6 and works from 7 a.m. to 4 p.m. filling potholes in city streets. He doesn't pack a lunch and doesn't eat during the day, mostly to save money but partly because he doesn't get very hungry, he said.

Aside from money, shelter and food, his biggest worry is more immediate. Before he falls sleep each night, he prays that a truck doesn't miss a turn and come crashing through the rusty fence that separates his family from the freeway.Field Trips Buoy Spirits

It is 12:15 p.m. at The School With No Name. Most children are working on science experiments Treshow has set up around the classroom. They exhibit an unusual comaraderie and a unique knowledge of the city. When a visitor asks a girl where the soup kitchen is, she replies: "Two viaducts over from the Blood Center," referring to the place where her mother donates blood for $12.

The children are abuzz over a field trip they took to a planetarium. Treshow believes in field trips. She has taken them to the zoo, the museum, even McDonald's to show them a brighter side of life.

Robert Kovach, a handsome fellow with bright brown eyes, stands by as Treshow goes over his essay. She appears stunned and is quiet for a moment. "That's tremendous, Robert. Just super," the teacher said finally, nodding in admiration.

At 16, he is a serious, well-mannered boy with a philosophy about life that is more mature than his years. "Last week, me and my brother went to Crossroads Mall downtown just to look around," he said. "It was about 1:30 and this policeman got suspicious. We told him our school got out at 1, but he didn't believe us. He said if he caught us there again, he'd get us for truancy.

"My mother tells me to try to forgive people. They don't know what it's like to have nothing," he said. "I look at it this way: They may be rich now, but the way things are going they could end up like us."

The week before, he had been accepted by a Job Corps program that will pay him minimum wage while he learns a skill. He was looking forward to that, for he wants to become a machinist. But more important, he said, "I'll be able to help the family out."

At about 12:45, a 13-year-old girl named Melinda enters to tell everyone that her family will leave for Wyoming today. The news doesn't surprise the children. They have seen classmates come and go. But Melinda is special. Although she has been there only three weeks, the cheerful girl has forged friendships, and everyone rises to say goodbye.

"Oh, you'll like it there," Treshow says, as Melinda accepts hugs from her friends. "Wyoming is gorgeous."

"Yeah," she gleefully answers. "Mom thinks we can live on a ranch when Dad gets a job."

"Who's driving?" Robert asks, fearing that Melinda's father likes to drink.

"Daddy. I get to hold the baby in my lap."

"Fasten your seatbelt," he says as the two embrace. "And make sure it snaps."