VAN WERT, Ohio — Ken Amstutz’s phone did not stop buzzing long enough for him to think about what could happen in 24 hours. He was fielding questions from public officials and the national news media, keeping tabs on planned protests and coordinating a meeting with United States Marshals.

“Nothing like this happens in Van Wert!” said Mr. Amstutz, a small-town school superintendent.

Then on Thursday, Betsy DeVos, the education secretary, and Randi Weingarten, her antagonist and president of the American Federation of Teachers, descended on this small, rural school district for a highly anticipated meeting of two polarizing education leaders. In a town that voted overwhelmingly for President Trump, and that takes pride in its public school system, educators hoped the two leaders could find common ground.

The area remains avowedly Republican but is leery of the budget cuts that Mr. Trump has proposed for federal aid to poor school districts. Parents also have little use for the Trump administration’s push to expand school choice and access to private-school vouchers. They want their neighborhood schools to work, because that is what they have.

“We’ve always voted Republican,” Mr. Amstutz said. “Public education is one area where Republicans have done nothing for us, really.”

Ms. Weingarten and Ms. DeVos made for strange traveling companions. Ms. Weingarten called Ms. DeVos’s confirmation a “sad day for children” and has accused the billionaire secretary of undermining public education by bankrolling school choice initiatives in Michigan.

Ms. DeVos is a staunch supporter of parochial and charter schools and has challenged teachers’ unions. While she says she supports public schools, she has been sharply critical of their performance and believes in expanding options for parents.

The two leaders have also clashed over Mr. Trump’s proposed budget, which would cut the Education Department by $9 billion, targeting programs for poor students, even as it funds a $1.4 billion school choice initiative.

Ms. Weingarten challenged Ms. DeVos to visit a public school system, and the result was the visit to Van Wert, a small district with a robust early childhood program, a nationally recognized robotics team and a community school that helps at-risk students graduate.

“It was clear that this community has invested heart and soul into the students here,” Ms. DeVos said, praising the district.

Ms. Weingarten said the visit “proves that support for public schools transcends politics.”

For teachers and students here, fights over funding and policy are not abstract. Jen Arend, a literacy specialist at Van Wert’s early childhood center, is paid with federal Title I money, which provides additional resources to districts with high concentrations of poverty. Mr. Trump’s budget proposes increasing Title I funding but using it for school choice initiatives.

“I hope she sees we’re more than a line item in the budget,” Ms. Arend said of Ms. DeVos.

At Van Wert High School, Spencer Teman, a senior, showcased his robotics team’s 18-by-18-inch robot, Suzanne, which won the state robotics championship this year. “In our school, opportunities are everywhere, and they should stay here,” said Mr….