By Jennifer Medina

LOS ANGELES — On Tuesday, voters in Los Angeles will go to the polls for a mayoral primary. But much of the attention will also be on the three races for the school board, a battle that involves the mayor, the teachers’ union and a host of advocates from across the country — including New York City’s billionaire mayor — who have poured millions of dollars into the races.

The outcome of the political fight for the school board seats will have a profound impact on the direction of the nation’s second-largest school district. But the clash has also become a sort of test case for those who want to overhaul public education, weakening the power of the teachers’ union, pushing for more charter schools and changing the way teachers are hired and fired.

After years of pressing to take power away from local school boards, some advocates have directed their money and attention directly to school boards in the hope that they will support their causes, as unions have done in the past.

Last month, Mayor Michael R. Bloomberg of New York City donated $1 million to a coalition formed by Mayor Antonio Villaraigosa of Los Angeles to help elect candidates who will support the current superintendent and the policy changes he has promoted. Students First, a national advocacy organization created by Michelle A. Rhee, the former schools chancellor in Washington, donated $250,000 to the same cause.

So far, the total spending from outside groups, including the teachers’ union, has reached $4.4 million as of Friday, according to the city’s ethics commission. And Mr. Villaraigosa has said he expects to raise even more in the final days for his group, the Coalition for School Reform.

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