Top lawmakers from her own party are challenging Hillary Clinton’s nascent policy agenda by pushing for a more ambitious approach

Leading Democrats challenged Hillary Clinton to take a more ambitious, unabashedly progressive approach to higher education reform on Tuesday, in a sideways challenge to a candidate promoting the advantages of vocational training over traditional four-year degrees.

While the party’s leading presidential contender completed the second day of her New Hampshire campaign swing, lawmakers back in Washington were seeking to shape her largely unformed policy agenda by pushing it to go beyond a focus on subsidising community colleges that has already been adopted by Barack Obama.

As Clinton spoke at a two-year community college, Senators Brian Schatz, Chuck Schumer and Elizabeth Warren joined 30 largely progressive congressmen in the House of Representatives to introduce a resolution aiming to make sure all students at public institutions – not just those at two-year community colleges – can graduate without debt.

“When it comes to making college affordable, I’m hopeful that debt-free college is the next big idea,” Schumer said in a joint statement.

However, such an approach would requite costly federal subsidies and measures to force down university tuition fees: intervention that could alienate some voters and leave Democrats vulnerable to charges that they are seeking to make taxpayers subsidise what are often lucrative college degree courses.

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Clinton has already made student debt a big feature of her campaign, pointing it out twice this week as one of the biggest impediments to economic growth and social mobility.

“One of the big problems we have growing the economy of tomorrow is the cost of higher education,” she said at a New Hampshire Technical Institute event on Tuesday. “It is pretty daunting,” she added, pointing out that, at $33,000 per year, New Hampshire has the highest average rate of debt for graduates in the country. “You can’t expect people to take on that much debt and easily pay it off.”

For now, though, Clinton has focused on encouraging students to opt for shorter courses at cheaper community colleges, arguing that many of the institutions need to “add prestige and distinction” and that there needs to be “most respect” for vocational training.

“When you look at higher education,” she said. “The US invented the community college. We have a head-start, but it’s going to require community colleges to reinvent themselves and explain what they offer. That’s why I really support President Obama’s efforts to raise the profile of community colleges and reduce the cost.”

The day before, at a furniture factory in Keene, Clinton also said she “really approve[s] of the president’s proposal to try to make community college as free as possible,” but again urged some students to follow demand from employers for more technical courses.

“I mean, the amount of tuition is so high, both in the two-year and the four-year schools, but that still doesn’t help unless we somehow provide incentives for more people, younger and older, to go into these trades.”

Campaign aides were not immediately able to comment on the disconnect between Clinton’s comments in New Hampshire and the simultaneous pressure from Democrats in Washington, and it is almost certain that Clinton will turn to the broader question of student debt in the weeks ahead.

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Nevertheless, the issue shows the continued pressure from progressives in the party to gently nudge Clinton to the left in the absence of any serious opposition from other candidates for the Democratic nomination.

“This broken system has to change and that’s only going to happen if Democrats running for office, up and down the ticket, stand up and speak out in favour of a big, bold idea like debt-free higher education,” said Adam Green, co-founder of Progressive Change Campaign Committee.

Research by PCCC, which favours the more full-throated liberal agenda of politicians such as Senator Warren, claims that “a combination of federal aid to states, more aid to students, and innovation that reduces the underlying costs of college” could achieve the aim of debt-free graduation.

Though costly, such a goal would also attract broad support from the US public, according to polling for the Progressive Change Institute by GBA Strategies, which shows that likely 2016 voters support the federal government helping states achieve debt-free college by 71% to 19%.

A survey by Gallup released this week also shows US parents worry about having enough money to pay for their children’s college education more than any other common financial concern.