BOISE, Idaho — In the struggle to fix the nation’s public schools, the old red-state, blue-state idea is looking as dated as Dick and Jane. You can hear the change in the voice of Gov. C. L. Otter, a Republican here in one of the most deeply conservative corners of the country, when he expresses a brotherhood bond with Rahm Emanuel, the Democratic mayor of Chicago and former Obama administration chief of staff.

“I could empathize with Rahm and what he was going through,” Mr. Otter, better known as “Butch,” said about the recently settled teachers’ strike in Chicago during an interview here in the State Capitol.

“It’s not the teachers,” Mr. Otter said, paraphrasing Mr. Emanuel’s tough-guy script from a news conference at the height of the standoff. “It’s the union bosses.”

Chicago’s fight may be over, but in Idaho, where a three-part proposition on performance pay, tenure and technology in the classroom is roaring toward Election Day, the debate over schools has morphed into a harsh discussion about whom the voters should trust. And as Mr. Otter’s attack line shows, the political and social battle lines are blurred — neither predictably conservative nor liberal, and often tinged with emotion about what schools can and might be.