For three days, the mostly male, all-white, conservative Michigan Christians hosted the mostly female, all-white (and mostly older) New York liberal Jews in their homes. They drove them in their pickup trucks to a firing range and, later, to a prison museum. They ate ice cream. They talked about guns, immigration and, of course, President Trump.

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Going into the trip, the participants had all kinds of ideas about one another. “When I first heard of it, I thought, ‘Why would I want to do this?’ ” says New Yorker Martha Ackelsberg. In some ways, Ackelsberg fits the stereotype that Republicans have about liberals. Before retiring, she was a professor at Smith College, where she helped create the women’s studies program. She is a lesbian with short hair who wears sensible shoes and a faded, pale-blue backpack.

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As skeptical as she was, she found herself intrigued by the idea of the Michigan encounter, which she heard about through her synagogue, B’nai Jeshurun. Still, she couldn’t sleep the night before the flight. Would she have to stifle her beliefs, lie about who she was?

Caleb Follett signed up to host Ackelsberg. To liberals, he’s a prototypical conservative: a white Christian heterosexual with a shaved head who supported Trump and endorses the idea of a border wall. He has a small arsenal of weaponry in his rural Michigan home, including an AR-15.

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In his job at a prison outside Lansing, Follett almost never gets to talk about politics — one of his passions. So when someone from the Michigan Corrections Organization asked him to take part in this exchange, he said yes. “I knew that Jews are God’s chosen people, but at the same time, I knew that a majority of American Jews are liberal. And I wanted to know why. If they are God’s chosen people, how could they be liberal? It was a quandary for me.” At the same time, Follett and his wife were concerned about letting strange New Yorkers into their home.

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In their own ways, then, Follett and Ackelsberg were wary of one another, based on ideas they had about each other’s extremism. Some ideas were true; some not. This is the case for most of us.

Americans on each side imagine that almost twice as many people on the other side hold extreme views than actually do, scholars Daniel Yudkin, Stephen Hawkins and Tim Dixon explain in a new report, “The Perception Gap.” The survey, conducted by the nonpartisan organization More in Common and the polling firm YouGov, was taken just after the 2018 midterms.

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Real and consequential differences separate Americans, but the more divided we get, the more mistakes we make. For example, Democrats estimate that about half of Republicans would admit that racism is still a problem in the United States, when in reality 79 percent of Republicans say so. Republicans, meanwhile, think fully half of Democrats would say that “most police are bad people.” The actual percentage is 15 percent.

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Some amount of the time, we are fighting ghosts, not real people. And the more inaccurate our perceptions, the more likely we are to describe our opponents as “hateful” and “brainwashed,” the study found.

Meanwhile, two major institutions meant to help us become more informed — the news media and college — seem to be having the opposite effect. The more time Americans spend consuming news from a range of sources, the more inaccurate their views of the other side. Of the 13 kinds of news media considered in the study, the worst offenders, correlating with the most skewed perceptions, were Breitbart News, the Drudge Report and conservative talk radio. Nine others were also associated with inaccurate perceptions — including the New York Times, The Post, Slate, BuzzFeed and Fox News. Only one — network news — seemed to have a positive effect.

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Additionally, the more education that Democrats, in particular, acquire, the more ignorant they seem to be about Republicans. Democrats with a postgraduate degree are three times as inaccurate in their perceptions of Republicans as Democrats who dropped out of high school. Interestingly, education does not seem to have this effect on Republicans. But Republicans are, on average, just as ill-informed about Democrats for other reasons.

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Ackelsberg was a good shot, to her surprise. Follett gave her a short lesson at the firing range. At the prison museum, he told her stories about what it is like to be a corrections officer in an overcrowded, understaffed prison system; she didn’t flinch. She was tougher than she looked.

That first night, Follett took Ackelsberg and two other New Yorkers back to stay at his home. His wife is Filipino, to Ackelsberg’s surprise. She’d immigrated to the country eight years before.

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Then Follett took everyone into the basement to proudly show off his weapons. Ackelsberg was alarmed by this display. Simultaneously she felt that she was growing to really like Follett, who was so exuberant and open.

She found it hard to hold it all in her head. She could not help but want to change his mind on gun control, even though she knew how unlikely it was. Still, she discovered there was value in hearing him. She learned that he held complicated views on immigration. He would also learn that she did not want an open border, as he’d assumed.

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That summer, to complete the exchange, the Michigan conservatives flew to New York City and stayed in the apartments of the Jewish liberals. Most had never been to New York before. They visited the 9/11 Memorial, attended Shabbat services and ate kosher fried rice in Chinatown. Some corrections officers visited the Trump Tower gift shop to buy Make America Great Again trinkets, while the New Yorkers waited outside.

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They all talked about how close they felt to one another, something they struggled to explain.

“I feel like this brought out the best of me,” Ackelsberg told me. She’d been able to hold fast to her convictions while also feeling open and curious. “I wish I could appear everywhere in my life the way I felt called to appear then.”

Follett found himself feeling unusually emotional around the New Yorkers. “This had a very therapeutic aspect to it,” he said, “because we were being heard, genuinely heard, by those that we felt misunderstood us.”

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In New York, he stayed with Ackelsberg and her partner in their Washington Heights apartment. They had their first long conversation about gay rights. He was opposed to gay marriage, and he did not change his mind. Nor did she, of course. She felt saddened by that distance but also grateful to know him. “It’s still really interesting to me — and hard to describe to people — that it matters so much that I have these people who feel like friends. Who are so different from me. I’ve never been in relationships like this before.”

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It feels disorienting, she says, but also freeing — the way it feels when you look at one of those drawings that appears to be an old lady but then morphs into a young one right before your eyes. It’s a kind of awakening, one that makes life richer but also more complicated.