Backstage at an Ohio rally in Columbus this summer, Michelle Obama skipped into the room to meet supporters who had earned the privilege of having dinner with her husband. President Obama followed, and when one of his guests offered an outstretched hand, Obama returned a bear hug. “We don’t do handshakes in our family,” he said. “We do hugs.”

The Obamas talked about their daughters’ piano lessons, their struggles with student loan payments, the grind of the campaign trail. Michelle Obama grasped the hands of her guests as she spoke. Later, the guests said they felt at ease, that President Obama and his wife were regular people.

The Obama campaign fervently hopes that the president will make a similar impression when he takes the debate stage Tuesday. Supporters and advisers see an opportunity in the town-hall format for him to connect directly with his questioners.

In the first debate, in Denver, Obama appeared sluggish and gave rambling responses. The performance surprised supporters and critics alike. Now, Americans are looking to see whether he can do better. Supporters who have met him are especially puzzled.

“When I turned off my television, I started crying,” said Paulette Camp, who owns a country store with her husband in western North Carolina and met Obama backstage in Columbus. “I thought, ‘Oh, my God — we just lost this election.’ I couldn’t believe it. I don’t know who that person was.”

In 2008, Barack Obama won eight states that had voted for former president George W. Bush in 2004. In six of those, including Virginia, more voters turned out in 2008 than in 2004 — and the Democratic gain closely followed the increase in the number of overall voters.

She was among the winners of one of the Obama campaign contests, which include “Dinner with Barack,” “Backstage with Barack,” “NYC with Barack, Jay-Z and Beyonce.”

Obama’s contests encourage supporters to give money and share valuable information about themselves with the campaign. They offer camera-ready portraits of a campaign in which even a volunteer has a chance to meet the president.

Perhaps most important, though, the contests can showcase Obama’s ability connect with ordinary Americans on their terms, something he did not do well in Denver.

“We heard you just graduated!” Obama said to Grace Litano of Miami when she met him at the Democratic National Convention in Charlotte. “We definitely need more public health officials, so what you’re doing is awesome.”

Litano is a single mom, a native of Peru and a hospital case manager who recently completed her master’s degree in public health. She won a contest to attend the convention and meet the president by donating $10 to the campaign. Standing in line at Obama’s hotel near Charlotte with other contest winners, Litano knew that Obama had studied her biography and that this is what politicians do, but she was impressed anyway.

“He’s so easygoing, you just forget that he’s president,” Litano said, recalling that Obama told jokes but also asked after her 6-year-old daughter, Katelyn, and talked about his support for the Dream Act.

The story is similar for other winners who have spent five, seven, 11 minutes with the president. Granted, they are all donors and volunteers who will vote for Obama, but they were downright giddy, and surprised at how down-to-earth they found him to be.

“We all had similar stories to tell,” recalled David Garcia, a music professor at the University of North Carolina at Chapel Hill who won a “Dinner with Barack” contest and dined with his wife, two other couples and the Obamas in Washington last week. The president had a cheeseburger, and the first lady shared her fries with the table, Garcia said.

“They were talking about their daughters,” Garcia recalled. “It was real and it was genuine.”

Obama has held 17 such contests since last October, giving more than 70 supporters from across the country the chance to have dinner, attend a Los Angeles fundraiser with George Clooney or watch the president shoot hoops in New York with basketball stars.

The connections made in these settings may be explained by the fact that these are people who are already supporters. And it also may be that the winners are chosen for their helpfulness to the campaign. Most are from swing states. More than a few are teachers. Others are students, small-business owners or government workers whose stories resonate with the president’s message. Inevitably, they all have a lot to talk about.

“He asked us everything about our lives — where went to school, where we lived, our neighbors, our businesses, our customers,” recalled Camp, the store owner from North Carolina. “He was so interested in our lives, like when you just meet someone where you just clicked. He’s really good with people, and this town hall will be people. So I think that part of him is going to really shine.”