Yet in the presidential debates, Mr. Romney seems to have shifted his position. In the first debate on Oct. 3, he said, “I don’t have any plan to cut education funding and grants that go to people going to college.” He went further in the debate Tuesday, promoting a state aid program he created as governor of Massachusetts, saying, “I want to make sure we keep our Pell Grant program growing.”

College costs have risen dramatically. In the last school year, tuition, fees, room and board averaged $38,589 at private colleges, up almost $15,000 from a decade earlier, according to the College Board. At public four-year colleges, the total bill came to $17,131, up more than $8,000.

But behind the headlines about soaring costs, the reality is more complex and wildly uneven, because a growing number of students receive financial aid, and only relatively high-income families pay those fast-rising sticker prices. Adjusted for inflation, the College Board calculates, the average “net price” changed little over the last decade at private schools, and rose only modestly at public ones.

Defending federal spending, Arne Duncan, the secretary of education, said that for more than 30 years, college prices had risen even when federal aid had not, leading him to believe there was “zero correlation.” And he noted that the big increases under Mr. Obama had come at a time of declining state support for public colleges.

Conservatives have criticized Mr. Obama’s other college affordability programs.

Mr. Obama removed banks as middlemen in making federal student loans, saving billions of dollars a year in lender fees, some of which was used to pay for increased student aid. Mr. Romney has said he would replace that policy with a marketplace approach. The president also strengthened programs that limit the size of student loan payments, based on a debtor’s income, and forgive unpaid loans for some borrowers after 10 or 20 years. So far, few borrowers have taken advantage of such options, and Mr. Obama has directed the Department of Education to publicize them more aggressively.

Scott Fleming, an education policy adviser to Mr. Romney, said those changes would cost the government billions of dollars by increasing unpaid loans. Shrinking monthly payments means extending the life of a loan, he noted, so some borrowers could end up paying much more.

The administration repeatedly makes the case that more people need to go to college — or back to college — and that much of that education should take place at community colleges. Mr. Obama has sought $10 billion in new aid for community colleges over several years, much of it to teach job skills; so far, Congress has approved $2 billion.