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. Teman, 18, who is set to attend Miami University of Ohio next year.

Van Wert, which serves 2,000 students two hours northwest of Columbus, is an unlikely place for a national showdown on education policy, which is largely driven by big-city politics. But it illustrates how politics and public education do not always match up.

About 80 percent of Van Wert’s 13,584 votes in November went to Mr. Trump, and the town is considered among the most conservative in the state.

But “education is different,” said Linda Haycock, a newly elected state school board member who represents Van Wert and 22 other small northwestern Ohio counties. “Because in rural districts, they have the most to lose.”

Mr. Amstutz said that in Van Wert’s schools, “we struggle every day to make ends meet.”

The school district is more than 95 percent white, and half of its students qualify for free or reduced-priced meals — an increase of 10 percentage points from when Mr. Amstutz took over as superintendent 10 years ago.

The school system’s $22 million budget supports five schools: an elementary school, a middle school, a high school, an early childhood center and an alternative school.

The district receives about $2 million in federal funding. Title I pays for teachers who work with struggling readers in kindergarten through third grade. And Van Wert uses the federal Title II program, which funds professional development and is targeted for elimination, to reduce class sizes.

Spending federal money or any other taxpayer funds on vouchers for private school tuition is looked on harshly. Ms. Haycock called it “really theft.” “It’s saying we passed a levy to go to our school district, and it’s really going somewhere else,” she said.

Educators hope Ms. DeVos saw during her visit that they make do with what they have.

In the elementary school, 60 percent of students are below the poverty line, and the school had to hire many support staff members, including a social worker whose caseload swelled to 40 this year, to help students and families meet basic needs and manage hardships.

“Out here in the middle of cow country, we have this amazing opportunity to serve kids who are no different than kids anywhere else,” said Kevin Gehres, the elementary school’s principal. “Our funding is the blood, sweat and tears of our community, and we are held accountable for that.”

In the middle school, principals have overhauled scheduling to give teachers control of nearly 300 minutes of a 400-minute day.

The district has a 96 percent graduation rate, but Van Wert High School’s principal, Bob Priest, said he knew of 86 unfilled jobs at the local farm and food processing company. Too many students leave school unable to fill the jobs because they cannot pass a drug test and show up to work on time, he said. So the school has an internship program that pairs students with local businesses.

Van Wert educators said they believed their biggest threat was school choice. An expanded voucher program would be “potentially catastrophic” for the district’s finances, said Mike Ruen, the district’s treasurer.

About 400 students now take advantage of a state open-enrollment policy, which Ms. DeVos endorsed during her visit. It allows students to attend an out-of-district school and take $6,000 in state per-pupil funding with them.

Most of them attend schools in a neighboring suburb. About 20 students are enrolled in an online charter school that has a 39 percent graduation rate. And a local vocational school takes 80 percent of the funding for each student who transfers there.

Only one private school competes directly with Van Wert public schools: a small Catholic elementary school in town that the public school system provides special education services to, mostly at no charge. A Catholic high school 15 miles away is less of a draw, but could become one if parents receive vouchers. “I don’t think people are against choice,” Mr. Amstutz said. “But when you talk about expansion, taking money away from public schools, it gives people heartburn.”

Ms. DeVos made no commitments to changing course on the federal budgetand policies, but said she needed to digest the visit.

Ms. Weingarten, who has been invited to tour a school of choice with Ms. DeVos, said her union would continue to fight vigorously against budget cuts and publicly funded vouchers.

“I think today was important, and I hope that she saw things that will make her reflect on the importance of public education to our country,” Ms. Weingarten said of Ms. DeVos. “Time will tell.”