AFTER years of budgetary chaos and disruptive teacher strikes, Chicago's 410,000 school children return to class today with notebooks, lunch boxes, and a rare promise of stability. ''Kids and parents can now be confident the schools are going to open on time and run for the next four years,'' says G. Alfred Hess, executive director of the Chicago Panel on School Policy. This fall's smooth opening of Chicago schools translates into high marks for Mayor Richard Daley. Last spring, Illinois enacted a law that for four years grants Mr. Daley sweeping powers over the schools and teachers' union, including full control over school board appointments and a nearly $3 billion budget. Daley's self-proclaimed school ''revolution'' advances a national trend toward mayors taking more control of big city schools. New York Mayor Rudolph Giuliani and Washington Mayor Marion Barry are lobbying for similar authority. Boston eliminated its elected school board four years ago in favor of one appointed by the mayor. In Chicago and elsewhere, the school board takeovers by City Hall are popular among businesspeople and other taxpayers who have demanded more political accountability for poor students and failing urban schools. Many students are also delighted. Hugging each other and slapping high fives, upperclassmen celebrated the reopening of Lindblom High this week after Daley's new school managers overturned a decision by the outgoing board to close the city's most prestigious inner-city school. ''It's a great academic school,'' said sophomore Tiffany Cook, an aspiring lawyer who joined hundreds of students who protested to save Lindblom from the old board's budget axe. Daley's team, by rethinking the budget, can afford the $2 million to $4 million needed to keep Lindblom and five other underutilized schools open. ''What you have seen is a revolution,'' proclaims Paul Vallas, chief executive of the seven-member school management team. ''We have taken aggressive action to downsize the central office, deal with the trade unions, and make the tough budget decisions. But there has been no fallout, no phone calls. All we're getting are overwhelming pledges of support.'' In a short two months, Mr. Vallas has led Daley's corporate-style management team in imposing financial order and market discipline on the nation's third-largest school system. The team, operating under a streamlined, five-member school board, has impressed observers with tough-minded actions that include: * Adopting a balanced budget designed to guarantee schools ''four years of financial and labor peace.'' This year's $2.6 billion budget includes $240 million in cuts. * Laying off 1,700 people, including 1,000 maintenance workers whose jobs will be contracted to private companies and about 200 bureaucrats, or 13 percent of the central-office staff. * Increasing the staff of budget officers and inspectors to help combat waste. * Concluding a four-year teachers' union contract that provides a 3 percent annual pay raise. * Announcing a $600 million program to construct about five schools a year and repair dozens of dilapidated buildings between 1996 and '99. If necessary, officials say they will seize people's homes to obtain optimal sites for new schools. ''The old board was fashioned on keeping business as usual and chopping a few twigs off the tree. This board is doing radical surgery,'' says Mr. Hess. Critics, however, contend that the push by Daley and other big city mayors to corporatize urban school systems will do little to solve the complex problems of educating students. ''This is a cheap organizational answer to a tremendously deep problem,'' says Gary Orfield, a Harvard University education expert. ''City schools are symptoms of a gigantic collapse of opportunity, the family, and income,'' he says. ''People want to suggest that these issues will be solved by having a mayor run the schools. But there's little evidence that mayors have any special insight into dealing with low-income students in collapsing communities.'' ''If we think this will produce educational breakthroughs, we are wrong,'' he says. Daley's team disagrees, saying better management is key to allowing Chicago schools maximize resources and foster innovation. ''We've set the stage so that if new money is available from Springfield, it will be targeted toward the classroom instead of going to some budget hole or union contract,'' says Vallas. Chicago's reforms will make it harder for Illinois legislators to reject raises in state education spending, which now provides less than 50 percent of school funds statewide, he says. Still, Daley's team acknowledges that the bottom line is not a balanced budget, but higher test scores, attendance, and graduation rates. Ambitiously, the team seeks to cut Chicago's 52 percent dropout rate in half, reduce truancy by two-thirds, and lower from 75 to 50 the percentage of students scoring below the national average in math and reading. Some initial steps include: * Dispatching ''intervention'' teams to the 149 schools that have failed to meet state education standards for three years. After an evaluation, the school board can fire, lay off, or transfer all employees without hearings. * Building 12 new alternative schools for violent youths and dropouts at a cost of $10 million. * Expanding the authority of principals and elected school councils set up in 1988 to promote school-based educational reforms. Still, Lynn St. James, Daley's new chief education officer, admits that boosting achievement will be especially hard in the worst pockets of Chicago's school system: the high schools. Wracked by gangs, drugs, and violence, many high schools have shown stagnant or dropping test scores in recent years, despite reforms. Even at Lindblom, which ranks among the top 10 Chicago high schools, test scores have declined over the last five years along with the enrollment of less highly qualified students. Ms. St. James, a former principal at Lindblom, says teachers need more time to develop new teaching methods better suited for ''today's video-active youngsters.'' The old models of drilling and lecture-discussion-test don't work, she says. ''Our biggest problem has been our failure to change as the students changed,'' says St. James, noting rising truancy. ''We're driving them away in droves.''