AMERICANS like criticizing their government. On any given day in any large U.S. city, elected officials field a bewildering array of complaints, ranging from relatively trivial matters of garbage pickup and snow removal to deeper issues of police brutality and development.

And most officials believe they have a responsibility to respond to these problems, whether great or small.

"You the Mayor?" The Education of a City Politican

By Barabara Ackermann

Dover, Mass: Auburn House Publishing Co.

$24.95

But solving them can be difficult. City bureaucracy often has a life of its own, with each city agency performing its own particular function in its own particular way. An outsider trying to get anything done in a city government must clear a path through an administrative jungle simply to understand what is going on.

"You the Mayor?" is written from this outsider's perspective. While 10 years on the Cambridge City Council--including two as mayor--should qualify Barbara Ackermann as an expert in the quirks of city government, one gets the sense that she never really feels confident about her abilities. Rules of politics and government that are second nature to a native like Councillor Alfred E. Vellucci, now the mayor, are an unfolding mystery to a relative newcomer like Ackermann.

And Ackermann's perspective is exactly what makes "You the Mayor?" an informative book--not only about Cambridge but city politics in general.

Ackermann does not lecture. She describes the the things she learned while trying to solve some of her constituents' problems.

SOME lessons come easily. In the early chapter "Nine Ayes," Ackermann describes her discovery that much of the council's business is simply a matter of form--her fellow councillors approve virtually any order she brings them, as long as it does not involve money. Attempts to deal with the semiautonomous Redevelopment Authority teach her the three I's that often dominate city administration--inertia, indolence and incompetence.

But most of the problems with which Ackermann tries to deal are too complex to be neat little dictums. One of the book's major themes is the escalation of violence in the city during the 1970s, beginning with the takeover of University Hall in April, 1969.

Ackermann sees the so-called "Peace Wars" as a tragic example of violence begetting violence. Police who were not trained to handle peaceful demonstrations let events get out of control and did not know how to respond.

While Ackermann was mayor in 1972, a 16-year-old high-school dropout named Larry Largey was found dead in a police cell. A police autopsy claimed he died of a drug overdose. City activists claimed he was beaten to death by the police.

News of Largey's death touched off a wave of riots in Roosevelt Towers, the East Cambridge housing project where he lived, as angry youths protested what they believed to be a murder of one of their own.

In one of the book's most moving scenes, Ackermann enters the Towers and confronts Largey's 16-year old peers, sharing their anger and desperation. "Other children were weeping," she says of the moment. "This was Larry's wake we were at. I wept too."

Despite Ackermann's efforts to discipline the police force--eventually, the whole force was sent back to the police academy for a week-long refresher course--she writes with a good deal of sympathy for the officers. She suggests that the police are often ill-equipped to deal with the rage that society creates in incidents like the Roosevelt Towers riots. Police did not make the problem, she writes, they were simply a part of it:

I told a reporter we were all at fault: The Police Department that didn't teach its officers the job for which they were hired. The Housing Authority that ran the Towers as a slum. The School Department that made Towers and other project kids feel unwanted. And the city manager, Council, School Committee, and mayor who had suspected all that and had not managed to change it.

Ackermann writes beautifully. "You the Mayor?" reads more like a novel than a political memoir. But the book begins to fall apart after the discussion of Largey's death.

Ackermann's treatment of her next four years on the council seem tacked on to the end of the book. Although the content remains fascinating, the chapters drift away from the emotion at Roosevelt Towers.

An epilogue containing Ackermann's advice to council newcomers seems similarly out-of-place. "You the Mayor?" is most intriguing when it describes Ackermann's development as a politician, instead of the lessons she learns. By adopting the tactics of fiction, Ackermann prepares her readers for more than they get.

THERE are also notable omissions. Ackermann makes scant mention of the problems she faced as Cambridge's first woman mayor--perhaps because woman councillors were commonplace by the time she took office.

More puzzling is her treatment of the conflict within the Cambridge Civic Association (CCA)--the good-government organization that endorsed her--over the issue of rent-control.

Although she discusses the growing rift during the years 1972-73 between the CCA and one of its original founders, Councillor Edward A. Crane '38, Ackermann makes no mention of the final day of the 1973 council session, when Crane and another CCA councillor broke ranks and tried to repeal the city's rent control ordinance.

But on the whole, "You the Mayor?" works well. It provides splendid portraits of city officials--particularly the giant Ed Crane and the often preposterous Al Vellucci, the independent Independent. It explains clearly and eloquently the problems of being an elected official in a city like Cambridge, where councillors have "accountability without authority."

But the most important accomplishment of "You the Mayor?" is the way Ackermann documents the day-to-day frustration of city government, how she responded to the anger of a city unable to solve its problems.