The words — the theme song of a children’s cartoon — were being bellowed by six grown men huddled on a makeshift stage in a hotel banquet room.

The song leader, an education specialist, held up a baby rattle.

“Give them alone time,” one man offered.

“Follow their lead,” another said.

“Have stuff around that they can interact with,” a third suggested.

All were correct. And why wouldn’t they be? They were stay-at-home fathers observing a presentation on children and play.

The men are part of a group called the National At-Home Dad Network, which on an early fall weekend had gathered here for an annual retreat (and a rare night without the kids). The men — 100 in total — had traveled from all over: the Midwest, Canada, Washington State. Over two days, they would attend a workshop on seatbelt safety and bro out at a Colorado Rockies game. They traded recipes — Tex-Mex spaghetti squash, lentil soup, piled into a box in the lobby — and asked questions of a panel of working women. (“Is it weird when your husband gets you a gift with your own money?”; “Who handles your finances?”) The men exchanged email addresses and made plans to meet up in playgrounds across the country.

By Sunday, they left, as the convention organizer put it, “better men, better husbands, better fathers.” It was the largest gathering of stay-at-home fathers ever, according to the organizers.

Some may wonder why fathers need a convention at all. But these men said the answer was simple: They wanted other dads to talk to.

At-home mothers have every support resource in the book, as well as a changing vernacular for how to refer to them (they too are “working moms”). Yet when it comes to dads who are the primary caretakers of their children — a group that is growing swiftly, both in size and visibility — the resources remain dismal. Few books. Fewer community groups.

“You’ll hear many guys describe it: I’m alone on an island in a vast sea,” said Jim O’Dowd, the conference organizer, who is a former mechanical engineer and a father of four. “There’s no history, no social structure, no guidebook. A guy jumps into this blind.”

And yet, he is also more visible than ever. According to a June study by the Pew Research Center, stay-at-home dads now account for more than 16 percent of at-home caretakers, a number that has more than doubled over the past decade (and still does not factor in dads who work part time).

By no means are single-earner households the norm in this country. And yet along with women’s economic rise — 23 percent of wives now outearn their husbands — has emerged a new kind of male caretaker: the out-and-proud involved dad.

Sure, he raises his children differently than a woman would. But he’s also there by choice. He isn’t a product of the recession, necessarily. And, according to a Boston College survey, a majority of his full-time working brothers wish they could join him — if their wives’ incomes only allowed.

“There’s been a feeling for a long time that dads are not capable, that if dads are in the home, moms are still directing, that dads are not interested in that caretaker role,” said Matt Schneider, a 39-year-old former teacher who had traveled to the convention from New York, where he is a founder of a dads group with 1,100 members. “That doesn’t jibe with what we see every day.”

The roots of the National At-Home Dads Network actually began two decades ago, with a dissertation project at Oakton Community College in Des Plaines, Ill. Robert Frank, now the chairman of the college’s behavioral and social science department, was working toward his Ph.D. in psychology while caring for his two children. When asked one day by a professor what he did for a living, he replied without thinking twice, “I’m a stay-at-home dad.”

She stared back at him, wide-eyed. “I’ve never heard that phrase before,” she told him. “You’ve just found your dissertation topic.”

Over the next two years, Mr. Frank surveyed 371 men who said they were the primary caregivers to their children, and determined that 63 percent said they felt isolated versus 37 percent of mothers in the same position. He began a series of at-home dad meetings — before the days of Google, their existence spread by word of mouth — and his findings culminated in a book, “Parenting Partners,” which publishers told him could not have just “dad” in the title.

His group fizzled somewhere around the 10-year mark, in part because his children were grown by then. He piled his old research notes in boxes in the garage, and moved on to other issues. He said he rarely gives interviews on the subject.

And yet Mr. Frank had started a movement that endures. In recent months, the engaged father has become a subject in men’s magazines, as well as a bevy of advertisements (among them: Cheerios and Tide). At Boston College, the study of the “New Dad,” as it’s called, is now the major focus of the university’s Center for Work and Family. At the dads’ convention, researchers from Notre Dame and Arizona State were looking at stay-at-home dads in the context of social class and identity.

And while there is new research showing that the daughters of active fathers are healthier and have higher self-esteem, the research has not focused just on the positive. A study at the University of Toronto recently determined that the so-called “motherhood penalty” — the idea that moms are penalized at work — applies to men, too, only worse. While women who talked about their children at work were deemed worse employees but better women (read: taking on their feminine role), men who talked about being a parent at work were viewed as both lesser workers and lesser men.

“Has there been an attitudinal shift? Yes,” said Brad Harrington, the executive director of the Center for Work and Family. “But we’re still in a period of transition.”

For its part, this dads’ network is working to change the stigma. Today there are subsets of the group all across the country, including men who gathered in five cities last month to celebrate “International Babywearing Week” (that is, a week to honor parents with small children strapped to their chests). One member hosts a conference for dad bloggers, called the Dad 2.0 Summit, and the network’s president, Al Watts, recently published an essay collection with Hogan Hilling called “Dads Behaving Dadly.”

A couple of years ago, when Huggies created an ad that many of the men found offensive — the idea was to put the diapers up to a bumbling “Dad Test” — the men got together and wrote the company with their objections. Huggies pulled the ad, and was even a sponsor at this year’s conference.

That camaraderie was in full display in Denver. Mr. Watts played master of ceremonies, while Greg Washington, a part-time football coach in Madison, Wis., encouraged men to share photos of their children. A dad who works part time as a graphic artist designed the program, and the opening video montage — a mash-up of news clips about modern fatherhood — was created by a dad, too. (At the end: a female newscaster asking, “Is dad the new mom?” To which a chorus of male voices shouted at the projector, “No!”)

But these men are used to that question, or at least what it represents. They see it at the playground as they scan the grass for other dads to talk to, or from male friends who, as Mr. Washington put it, “don’t always get it.” They hear it in the innocent question from a neighbor — “Are you babysitting today?” — or the pediatrician who asks, “Should I speak with your wife?”

Mr. Harrington, of Boston College, recalled a story from a few years back in which a man he knew — carrying his baby snuggled under his jacket on a cold day — emerged from a park trail to a circle of police officers because he looked suspicious.

“One of our first dads meetups was at the Whitney Museum,” said Mr. Schneider, a father of two. “There were five of us with our babies, having lunch. It was like we were one of the exhibits. We had people coming up to us saying, ‘What is this?’ ”

The good news is that the culture has started to catch up. In April, Daniel Murphy, the Mets second baseman, ignited a fury of radio chatter after missing two games to be at home with his wife and newborn son. When the chief executive of MongoDB Inc., a software firm, announced that he would step down this year to spend more time with his family, he noted that, “As a male C.E.O., I have been asked what kind of car I drive and what type of music I like, but never how I balance the demands of being both a dad and a C.E.O.”

Companies like Facebook and Change.org are among a group that have begun to offer generous paternity leave policies, and as a 2014 survey of dads revealed, 89 percent said it would be an important criterion in looking for a new job.

And then there are the fathers who are wearing their dad badges each day, proudly.

“Find somebody you don’t know and thank them for coming here today,” Mr. Watts prompted the conference attendees, each of whom got a gift bag (a water bottle and beef jerky) underneath his seat. Good-natured heckling was allowed, but only because each of these men shared a kind of understanding.

“You know, anybody who cares about their kid can learn to change diapers and do laundry and cook — and all of those things that are part of being an at-home parent,” Mr. O’Dowd said. “But you can’t overcome social stigma on your own. So while we want guys to feel like they can come here and bond with each other, we also want to enlighten the world a little bit. To say, you know, dads can do this gig, too.”