By the American Talent Initiative’s reckoning, the number of students at participating schools with enough financial need to qualify for federal Pell Grants grew more than 7,200 from 2015 to 2017. That represented a 3 percent rise in total Pell enrollment at the schools and a step toward the 50,000-student growth target. Bloomberg hailed the results as a sign of commitment to widening access at top colleges.

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“These are solvable problems if we have the courage to do it, and if we want to actually do things rather than give speeches and go to rallies,” he said in an interview.

Skeptics say colleges are overstating how much they have accomplished.

“The reality is colleges have not made nearly as much progress as they tout,” Caroline M. Hoxby, a Stanford University economist who studies higher-education enrollment, wrote in an email. She said counting Pell recipients is a “crude” measure that fails to show what efforts schools make.

Too often, Hoxby said, selective colleges raise Pell recipient totals at the expense of potential students with financial need who aren’t eligible for the federal grants. Or they compete for the same Pell-eligible students while overlooking other strong prospects with similar low-income backgrounds. “The missing students are still out there, unhelped,” Hoxby said.

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Pell Grants are worth up to $6,095 a year. About 69 percent of recipients, according to federal data, come from households with annual income of $30,000 or less.

Participants in the initiative range from small liberal arts schools such as Allegheny College in Pennsylvania to the massive public system of the University of California. To be eligible, schools must have at least 500 students and six-year graduation rates of at least 70 percent. Nearly 300 meet that threshold nationwide, including many that are not participants.

It is difficult to evaluate the initiative independently because organizers are declining to release data they receive from each participant. The group’s Pell Grant data is more extensive than what is so far publicly available.

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Stephen Burd, an education policy analyst for the left-leaning think tank New America, said colleges should be applauded for advocating an agenda that departs from the usual pursuit of money and prestige. “Any effort to counterbalance that and try to reward institutions for enrolling more low-income students is a good one,” he said.

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Those involved with the initiative say raising Pell enrollment is not their only goal as they campaign for wider educational opportunity. “There’s no one answer to how you should measure it,” Bloomberg said.

Spelman College, one of the participants, wants to raise graduation rates. Federal data show 74 percent of students graduate from Spelman within six years. About half of the college’s students are Pell-eligible.

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Mark Schlissel, president of the University of Michigan, denied that schools are neglecting potential students who aren’t eligible for Pell Grants. “People get the wrong impression, that we don’t care about people who are earning 10 dollars more than Pell” cutoff levels, he said. “Which is ridiculous.”

Schlissel was one of five college and university leaders who joined Bloomberg on Tuesday in an interview with The Washington Post. He and others said the venture provides an important venue for swapping ideas.

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U-Mich. has long struggled to recruit in lower-income areas even though it offers generous financial aid. Federal data show 15 percent of its undergraduates in 2016 qualified for Pell Grants, lower than the share at many public universities. The university wants to raise its Pell share to 20 percent. In pursuing that goal, Schlissel said the university has discovered that marketing matters enormously.

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Last year, U-Mich. launched the Go Blue Guarantee, promising “free tuition” — the word “free” is crucial — to students from within the state whose families make $65,000 a year or less and have assets below $50,000. Researchers have found such tuition-free promises are a huge draw. Targeted students were twice as likely to apply to the state flagship in Ann Arbor, and if accepted, they were twice as likely to enroll.

“These kids have aspirations they might not have had before,” Schlissel said.

University of Illinois Chancellor Robert J. Jones said his school is taking a similar approach. “We learned from Michigan in terms of the power of the word ‘free,' ” Jones said. The undergraduate Pell share for the university in 2016 was 21 percent. Starting with next fall’s entering class, the flagship at Urbana-Champaign is offering four years of free tuition to qualified in-state students. The Illinois Commitment targets those with family income of $61,000 a year or less and assets of $50,000 or less.

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Jones said he sometimes hears from doubters who call it “some kind of scheme” or “a PR ploy.” He tells them, “Free means free.”

This is the kind of synergy Bloomberg wanted to promote when he launched the initiative. Aides said he has given it $4.7 million since 2016.

Bloomberg is also a major donor to his alma mater, Johns Hopkins University. He recently gave Hopkins $1.8 billion to support need-based financial aid. In 2016, federal data show, the share of Hopkins undergraduates with Pell Grants was 12 percent. The share has grown lately, and Hopkins — a participant in the initiative — wants it to reach 20 percent within four years.