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Children are returning to classrooms across the nation during one of the most tumultuous periods in American education, in which many thousands of teachers and other school workers — no one yet knows how many — were laid off in dozens of states because of plummeting state and local revenue. Many were hired back, thanks in part to $100 billion in federal stimulus money.

How much the federal money has succeeded in stabilizing schools depends on the state. In those where budget deficits have been manageable, stimulus money largely replaced plunging taxpayer revenues for schools. But in Arizona, California, Georgia and a dozen other states with overwhelming deficits, the federal money has failed to prevent the most extensive school layoffs in several decades, experts said.

When Lori Smallwood welcomed her third-grade students back to school here, it was a new beginning after a searing summer in which she lost her job, agonized over bills, got rehired and, along with all school employees here, saw her salary cut.

“I’m just glad to be teaching,” Smallwood said. “After the misery of losing your job, a pay cut is a piece of cake.”

In the hard-hit states, the shuffling of teachers out of their previous classrooms and into new ones, often in new districts or at unfamiliar grade levels — or onto unemployment — continues to disrupt instruction at thousands of schools. Experts said that seniority and dysfunctional teacher evaluation systems were forcing many districts to trim strong teachers rather than the least effective.

And in some places, teacher layoffs have pushed up class sizes. In Arizona, which is suffering one of the nation’s worst fiscal crises, some classrooms were jammed with nearly 50 students when schools reopened last month, and the norm for Los Angeles high schools this fall is 42.5 students per teacher.

“I’ve been in public education north of three decades, and these are the most sweeping cutbacks I’ve seen,” said Michael Casserly, executive director of the Council of the Great City Schools. “But it would have been worse without the stimulus.”Driving the layoffs was a precipitous decline in tax revenues that left states with a cumulative budget shortfall of $165 billion for this fiscal year, according to the Center on Budget and Policy Priorities, a research institute.