It’s commencement season, and if you haven’t been to a ceremony this year, you’ve seen pictures of one in the newspaper, on TV or on your sister-in-law’s iPhone.

If you are like me, at some point you have asked yourself a variation of this question: What is the deal with those hats?

A close-fitting skullcap and a stiff flat square bolted together with a button suitable for hanging a tassel is like some kind of psychological test: Look quickly and nothing could be more familiar. Look again and it’s outrageously strange.

The hat’s roots are in the medieval church, from about the time the first European universities were founded, when the only kind of education was religious and students and professors were all members of some kind of priesthood. The “mortarboard” hat survives in our secular times as a sign of continuity in an institution that hates change.

On July 1, University of Southern Maine will welcome a new president, its fourth since 2011. Glenn Cummings will join the army of college presidents around the country who are charged with making a stalled public higher education system work again.

They are all facing many of the same challenges. As in Maine, America’s public universities are struggling to maintain too much infrastructure and competing for students in a marketplace where choice has exploded. As in Maine, state governments have been withdrawing public support for decades. Costs have skyrocketed, creating debt burden and high expectations from a generation of students and families.

Schools are rated by how many applicants they turn down, not how much their students learn. As the prices go up, the impact of a degree on future earnings has never been higher. Once an engine of upward mobility, the university degree is becoming one more bulwark of a rigid class system.

Presidents like Cummings are charged with balancing cost and quality in new ways that increase opportunity without sacrificing value. The biggest challenge will be recognizing what parts of the tradition to keep, what new work to take on and what, like the mortarboards at graduation, are leftovers from another time that could be let go without much sacrifice.

IMPORTED AND DOMESTIC

Don’t let the hats fool you. The American university as we currently understand it is not as old as it looks. It was assembled from parts – imported and domestic – in the 20th century, and it was fertilized with huge doses of federal money.

It started in the 17th century with colleges on England’s Oxford-Cambridge model, where communities of students led by faculty acquired a habit of inquiry and a broad understanding of the world’s knowledge, preparing them for any position that they may be called to fill. In Maine, Bowdoin, Bates and Colby colleges were all founded on this idea.

The next piece came from Germany in the mid-19th century: the research institution. Here, programs were built around scientists doing specific work. Graduate students learned by working with a master, and together they pushed the boundaries of world knowledge. Johns Hopkins was the first American university built on these principles and has been a leading source of scientific discovery, especially in medicine, ever since.

The third piece was home-grown. In 1862, Abraham Lincoln signed the Morrill Act, which created land-grant universities. States were given federal land to sell with the proceeds used to endow universities. The idea was to give a fast-growing nation what it needed to industrialize, and the new institutions were supposed to educate the masses as well as share expertise in agriculture, engineering and science. The mission was public service, and the method was the practical application of knowledge.

These parts came together in the early 20th century, but the final piece was not added until the end of World War II, when the federal government – especially the Defense Department – chose to spend billions of dollars funding research at universities instead of building national labs. As the Cold War heated up, the money kept rolling in.

Fueled by cash and a steady flow of students, public universities expanded at a record pace. Enrollment tripled between 1955 and 1970. Public higher ed was available to almost anyone who wanted it, and for the student it was nearly free.

What’s now USM was created in 1970 with the merger of the University of Maine at Portland and the Gorham Normal School. It arrived at the crest of America’s higher-ed spending boom.

THE NEW AMERICAN UNIVERSITY

That pace has not kept up. Federal research funding plateaued, and states began to withdraw their support. In Maine, we were also hit with an aging population that is producing fewer college-age students every year. High school graduates have more choices than their parents or grandparents: public, private and for-profit. In many places we now have a system that has too many buildings, too few students and revenue problems as far as the eye can see.

That’s the situation that Cummings will inherit July 1. While the future is uncertain, the parts he has to work with are not. We still need schools to educate young adults (the English model), support research (the German model) and serve the greater community with practical application of knowledge (the land-grant model).

The question is how these elements will come together in a package that is relevant to our times. There are plenty of critics who predict that it won’t come together at all.

Kevin Carey, author of “The End of College: Creating the Future of Learning and the University of Everywhere” predicts that universities will be unbundled by technology. Researchers don’t need the university infrastructure, he reasons, when they can network with each other and funding sources. Students don’t need to pay outrageous tuitions if they can assemble the skills and knowledge they need online and prove their competencies with a virtual portfolio that would tell employers far more than a college diploma ever did.

But not everyone is ready to see the bricks-and-mortar campuses disappear.

Michael M. Crow, the president of Arizona State University, has presided over explosive enrollment growth that was not slowed by massive cuts in state funding. As the university got bigger, it also got better at producing students who can compete with any of their peers. For instance, ASU trailed only Harvard and the University of Michigan in the number of students who received Fulbright awards to study and teach abroad in 2013.

Crow’s approach, which he has branded as the “new American university,” breaks with some traditions while keeping others, including the link between research and teaching.

Instead of measuring success by how many students he rejects, Crow has aggressively worked to bring in as many students as possible. Enrollment grew from 58,000 to 76,000 in a decade between 2004 and 2013, and it is on pace to reach 100,000 by 2020. He has recruited in unusual places, working with the elders of the Mormon church to attract students and with the president of Starbucks to make it easy for baristas to go back to school.

Costs are controlled with aggressive use of online resources, including testing. Students with educational weaknesses have access to coaching and guidance.

And instead of dividing departments into self-contained units, he encourages professors to form multi-disciplinary teams that work on real-world problems. This is the approach used in cutting-edge scientific research, where it has become common for chemists, biologists and engineers to work together. At ASU, organizing work this way allows enrollments to increase without students feeling that it has grown “too big.”

ASU has been called a “credential factory” by critics who say it values administrators over faculty members, but it’s hard to argue with its record of open access and excellence.

HINGE POINT

In 1963, University of California President Clark Kerr talked about being at a “hinge point” in the history of higher education, and no one could tell which way the gate would swing.

It didn’t take long for Kerr to find out. Students rose up in the anti-Vietnam War protests of the 1960s and targeted the marriage between the university system and the defense establishment as the axis of evil. Politicians made campus unrest an issue. Ronald Reagan ran his first campaign for governor of California on a promise to restore order, and famously said in 1969, “If there has to be a bloodbath, let’s get it over with.”

The protests, culminating with the murder of students at Kent State in 1970, had the effect of driving a wedge between academia and the rest of the culture. The universities became seen as distant outposts of left-wing thought in a country that was getting more conservative politically. This has not been good for the university or society.

The constellation of factors that created the post-war college spending boom are not coming together now and probably never will again.

Today’s public universities are like a graduation hat – something that looks familiar even if we don’t know exactly how it’s supposed to work. Now it’s up to people like Cummings to figure out which parts of it still make sense.