In December of 2013, students at New York University created the only union of graduate teaching and research assistants that is recognized by a private university in the United States. Six miles uptown, Lindsey Dayton, a doctoral student in Columbia University’s history department, heard the news and got excited. Dayton had spent time working as a labor organizer with hotel and restaurant workers before enrolling at Columbia. If N.Y.U. students could win unionization, she thought, maybe student workers at Columbia, who were worried about issues like job security and benefits, could do the same. Quietly, Dayton began talking to likeminded Columbia students, and they eventually decided to try to follow N.Y.U.’s lead. Their campaign culminates on Friday, when dozens of students plan to enter the university’s administrative offices and deliver a letter to the university president, Lee Bollinger, seeking support for a union of student workers that would be affiliated with the United Automobile Workers. If they are successful in their larger campaign, Columbia could become the second private university in the U.S. whose student workers belong to a recognized union.

The campaign reflects an evolution, which has taken place in recent decades, in how universities operate. They have increasingly relied on graduate students as workers—to aid full professors in teaching lecture-hall classes, help them with research in university labs, and so on—partly to help keep costs down. The graduate students develop work experience and relationships with professors, along with receiving wages and some benefits. Unionizing could allow these students to collectively bargain and get formal help addressing grievances. At many public universities, including the University of California system, graduate students have been unionized for years, with the support of the state labor-relations boards that govern them. But that hasn’t been the case at most private universities. Employees at private companies, including private universities, have to follow a different path to unionization: they must petition the National Labor Relations Board or persuade their employer to voluntarily recognize them. In several N.L.R.B. decisions handed down in the nineteen-seventies, the board made clear that it believed students shouldn’t be able to unionize because their main relationship with their institution was as students, not employees. For a long time, this understanding went unchallenged, with students at private universities simply accepting that they couldn’t unionize.

In 2000, however, a Democratic-controlled N.L.R.B. concluded that students at N.Y.U. could, in fact, unionize as employees. Students at Columbia and other institutions, including Brown University, were emboldened and tried to gain recognition for their own unions. But in 2004, an N.L.R.B. controlled by Republicans who had been appointed by George W. Bush decided, in a case involving Brown, that students shouldn’t be able to unionize, after all; they reasoned, as boards had done until the 2000 decision, that a student’s relationship with a university was primarily educational. That essentially killed the efforts of the students at Columbia and other universities, too. A year later, N.Y.U. stopped recognizing its students’ union, a story that Rachel Aviv documented in her 2013 Profile of John Sexton, the school’s president.

But lately, the atmosphere seems to have changed. In 2013, instead of going through the N.L.R.B., the N.Y.U. students won voluntary support from their university. Following this decision, Dayton and other Columbia students thought that their own administration might be more inclined to support unionization. They also had a feeling that the current N.L.R.B., controlled by Democrats appointed by Barack Obama, might look more favorably on them if they were to petition the board along with, or instead of, being voluntarily recognized by Columbia. (A spokeswoman for the N.L.R.B. declined to comment, as did a spokesman for Columbia.) Plus, their concerns as employees weren’t getting addressed on their own. “It was kind of like, ‘Enough is enough,’” Dayton told me.

In January of this year, the Columbia students contacted Maida Rosenstein, the president of U.A.W. Local 2110, which represents the N.Y.U. students, and asked for help unionizing at Columbia. (A union for auto workers might seem like an odd choice for grad students, but it represents thousands of higher-education workers at institutions including the University of California system and Columbia itself.) The students also started getting in touch with friends and acquaintances on campus to gauge their interest. When they exhausted those contacts, they began going door to door in Columbia’s offices and research labs, introducing themselves and collecting e-mail addresses and phone numbers. There seemed to be momentum outside of Columbia, too; around the same time, student workers at Yale University were seeking union recognition from their administration.

By the fall, the Columbia students were ready to start a card drive, gathering signatures from student workers who wanted to unionize. They have since collected about seventeen hundred signed cards, which they believe represents about sixty per cent of undergraduate and graduate student workers. Seth Prins, a doctoral student in epidemiology who is one of the campaign’s organizers, said that students generally seem receptive to unionizing and have a long list of desires that they hope to fulfill: lower health-insurance costs, especially for students’ partners or children (Prins hasn’t been to a dentist in four years, he said, because the cost of dental insurance has been prohibitively high, though his health insurance does include some basic dental services), along with greater job security, more regular and predictable paychecks, and more generous family-leave policies.*

It is unclear how useful unionization will be in achieving these goals. There is little research into the status of unionized student workers versus non-unionized ones. In 2004, when the N.L.R.B. ruled that Brown students couldn’t unionize, the board suggested that doing so might complicate relationships between professors and students and even impinge upon the academic freedom of faculty members and administrators by limiting their decision-making power. Last year, however, professors at Rutgers University and the University of New Mexico published a paper that looked at how unionization had affected graduate-student employees at public universities. They found that unionized students reported better pay and higher levels of personal and professional support from the faculty advisers supervising them, as compared to their non-unionized peers; students noticed similar levels of academic freedom whether they were unionized or not.

“That’s what we were really trying to say: ‘O.K., is there any evidence for these awful outcomes that the board has predicted if graduate students unionize?’” Adrienne Eaton, the chair of the department of labor studies and employment relations at Rutgers, and a co-author of the paper, told me.

“No, there aren’t awful outcomes.” She acknowledged that differences exist between public universities, which she and her co-authors studied, and private ones, but she described them as minimal.

Of course, administrators at private universities have another good reason to resist unionization. If graduate-student workers get together and demand better pay and benefits, that could put pressure on university budgets. Bollinger hasn’t often commented publicly on the unionization question, but his administration opposed such efforts in the early aughts, and in 2004, he told a University Senate gathering, “The graduate students and the administration will be better if we do not have a union acting as an intermediary between graduate students and the faculty,” according to the Columbia Daily Spectator, the school’s student newspaper.

On Thursday, I spoke on the phone with Olga Brudastova, a doctoral student in engineering at Columbia. Brudastova, who is Russian, said that she enjoys her work and her colleagues but believes a union could persuade the university to improve job security for graduate assistants. Since international students are often barred by law from working off campus, they have few opportunities to make money if they lose their teaching or research positions. She also recalled that she had started a research position in January but didn’t receive a paycheck until April; in the meantime, she had to borrow money from friends and relatives to cover her expenses. (The Columbia spokesman, asked about Brudastova’s experience, said that he couldn’t “discuss matters involving a particular student.”) Although she said that she has been paid in a more timely fashion since then, she told me that she feels that collective bargaining could lead to more reliable paychecks. “I felt like it’s just a little bit of a mess in the system,” she told me. “There was no clear way for us to correct it. Then I heard about the unionizing effort, and I thought it was a good idea, and hopefully it works.”

*Clarification: An earlier version of this post didn't note that Seth Prins, a graduate student at Columbia, has health insurance that includes some basic dental services.