Geoffrey Brown, a developmental psychologist in the department of human development and family science at the University of Georgia, who has studied fathers of young children, described the phenomenon of “maternal gatekeeping — mothers play a large role in determining fathers’ roles.” Mothers can encourage and they can discourage, he said, and sometimes both at once, with mothers asking fathers to do something and then not liking the way it gets done.

In one of his research studies, looking at fathers and 3-year-olds, the effects of the father’s involvement on the child’s attachment varied depending on whether the involvement was play or caregiving, and whether it happened on workdays or non-workdays.

Research has shown that fathers’ involvement has lots of benefits, Dr. Garfield said. “We know that fathers use different words than mothers, and that helps develop the child’s expressive vocabulary, they use different language when out and about in the world.” Fathers are more likely to engage in “rough and tumble” play, he said, and they often keep changing the rules, which can be very exciting for children and helps them learn.

In the poll, 32 percent of the fathers had been criticized for being too rough, and 32 percent for not paying attention to their children. “Some things are unique to dads,” Ms. Clark said. “Being too rough and not paying attention play into some of the gender stereotypes still present in our society.”

[Read about a new ban in Britain on ads depicting gender stereotypes such as men being unable to change diapers.]

Fathers tend to engage with their children in more physically active ways, Dr. Brown said, and tend to take more risks and encourage exploration. “They might be engaging with their kids in a way, not just not harmful but actually helpful, but different from mothers.”

Mothers sometimes note with irritation that fathers may get a great deal of praise just for showing up or for getting a child dressed. But it’s insulting when fathers “face the assumption that we’re babysitting rather than parenting,” Dr. Hill said. “You wouldn’t praise a woman for getting the barrettes in straight.” A father might hear something like, “Wow, her hair is combed, congratulations!”