Barbara Schierenbeck, a 59-year-old nurse in New York City, is swept up in the excitement of potentially electing Hillary Clinton the first female president. She cannot understand why her 19-year-old daughter, Anna, does not feel the same way.

“Fifteen or 20 years ago, no one would even think about a woman being president,” Barbara Schierenbeck said. “Certainly, when I was 20 years old in the 1970s, I don’t think I would even have thought about it.”

But for her daughter, electing a woman, while a nice idea, is not a motivating factor. “I want to see someone who, like, has the fervor to fight for me,” Anna Schierenbeck said. A woman will be elected president “pretty soon” anyway, she said, regardless of what happens in 2016. Why does that woman have to be Clinton?

The mother-daughter debate unfolding in the Schierenbeck household reflects a debate taking place across the country, as women of varying ages and backgrounds confront the potential milestone implicit in Clinton’s bid very differently. As her chances of becoming the first woman to be nominated by a major political party improve, many women are considering how much gender should play into their decisions to embrace Clinton’s candidacy — or not.

The generational divide in how they answer that question has added urgency to Clinton’s efforts to focus on how she appeals to younger women, who overwhelmingly vote Democratic but who might sit out an election if they are not excited by a candidate. After beginning her candidacy in April with hopes of inspiring women that this was their moment, she is now more intent on trying to forge common cause on specific issues, and less on merely shattering the glass ceiling.

Unlike in her 2008 campaign, Clinton has this year leaned heavily on her gender, often ending speeches by invoking an America “where a father can tell his daughter, ‘Yes, you can be anything you want to be, even president of the United States.’”

It is a powerful line for Clinton’s most avid supporters: college-educated women in their 50s and 60s. “For baby boomer women, in particular, it’s ‘I fought this whole war, and now we’re running out of time, and if not Hillary, then who would it be?’” said Celinda Lake, a Democratic pollster who is herself a baby boomer.

But younger women are less impressed.

Meghan Speed, a 20-year-old college junior from Concord, North Carolina, said she expected a woman to be elected president in the next 20 years, but planned to vote for Sen. Bernie Sanders in the Democratic primary because of his record on issues like income inequality.

“For me it was very difficult to wrap my mind around not fully supporting Hillary, because she is a woman,” she said. “But I came to the realization that if I am supporting her because she is a woman, that’s equally as bad as not supporting her because of her gender.”

Indeed, Clinton campaign officials — who say younger voters are generally harder to engage than older voters — have tried to reach women ages 18 to 35 with Snapchat videos and personalized Bitmoji emoticons, but also with serious attention to issues that motivate them. Putting a woman in the Oval Office is presented as an added bonus.

“Millennial women are getting more and more involved in this campaign because they’re hearing about Hillary Clinton’s proven track record fighting for women and the issues she’s talking about that matter to them like equal pay, college affordability and protecting women’s health care rights,” said Mini Timmaraju, the campaign’s director of women’s outreach.

The Democratic primary provides Clinton with an opportunity to lay the groundwork among a constituency she would rely on heavily in a general election. Fifty-two percent of women lean Democratic, compared with 44 percent of men, but young women and black and Latino women, in particular, will not participate in an election unless they are inspired by a candidate, according to polling by the Pew Research Center. In 2012, 55 percent of women who voted went for President Barack Obama, compared with 44 percent for Mitt Romney, giving the president a particular boost in critical swing states like Ohio, exit polls showed.

That Clinton’s candidacy has not yet sparked among young women the kind of excitement about making history that Obama generated among black voters in 2008 speaks to the progress women have made, said Erin Gloria Ryan, 32, formerly managing editor of the feminist website Jezebel.

“The younger generation” — of which Ryan counts herself a member — “just thinks the pipeline will magically fill up with women who are qualified enough to run for president,” she said.

The generational gap haunted Clinton in the 2008 primary: In Iowa, Obama took 51 percent, John Edwards 19 percent and Clinton just 11 percent of the caucus vote among women younger than 24. The only demographic cohort that Clinton won, exit polls showed, was women older than 65.

But Clinton largely ceded the youth vote to Obama in that campaign. She did not play up her chance to make history.

This time, Clinton, 68, anticipating that the election will hinge on her support among women, frequently talks about being a mother and grandmother. She uses gender as a way to overcome the perception that she is the establishment candidate and to counter questions about her age. “I cannot imagine anyone being more of an outsider than the first woman president,” she told CBS News. “I mean, really, let’s think about that.” And she often says, “I’ll be the youngest woman president in the history of the United States.”

That message galvanizes an army of devoted supporters, including women like Pat Higby, a 62-year-old educator in Cedar Falls, Iowa, who sounded a familiar refrain when asked why she planned to vote for Clinton:

“It’s time,” she said.

For women near her in age, Clinton represents “the apex of that generation’s aspirations for itself,” said Ryan, the writer and editor. Ambivalence about her among their daughters and granddaughters can be maddening.

“They haven’t experienced the kind of barriers that their mothers and grandmothers did — the kind of exclusions from areas of accomplishment,” Mary L. Shanley, a political-science professor at Vassar who specializes in gender studies, said of women born after 1980.

Jessica O’Connell, executive director of Emily’s List, which raises money for female candidates who support abortion rights, said younger women would become more enthusiastic about Clinton’s candidacy when they “engage more in the election and pay attention more to what the GOP is doing and how out of touch they are.”

Some of Clinton’s younger supporters are already there.

Taylor Casey, a 21-year-old college senior, said she was an ardent supporter of Clinton’s. When asked why, she mentioned issues like college affordability, criminal justice reform and conflict in the Middle East.

Asked about the excitement of electing a woman president, Casey shrugged.

“If you put her in a man’s body, she would still be exactly what I want,” she said.

© 2015 The New York Times Company