President Trump signed an executive order on Thursday that directs the federal Department of Education to come up with a plan for its own demise.

Only Congress can abolish a Cabinet-level agency, and it is not clear whether Mr. Trump has the votes in Congress to do so. But he has already begun to dismantle the department, firing about half of its staff, gutting its respected education-research arm, and vastly narrowing the focus of its civil rights division, which works to protect students from discrimination.

Mr. Trump’s long history of attacking the Department of Education represents a revival of a Reagan-era Republican talking point. It has unified Democrats in fiery opposition. But is shuttering the department possible? And if not, how has Mr. Trump begun to use the agency to achieve his policy goals?

What does the department do?

The Education Department was founded in 1979. Its main job is distributing money to college students through grants and loans. It also sends federal money to K-12 schools, targeted toward low-income and disabled students, and enforces anti-discrimination laws.

The money for schools has been set aside by Congress and is unlikely to be affected by Mr. Trump’s executive order. But oversight of the funds could be reduced and moved to other federal agencies.

Those federal dollars account for only about 10 percent of K-12 school funding nationwide. While Mr. Trump has said he wants to return power over education to the states, states and school districts already control K-12 education, which is mostly paid for with state and local tax dollars. The federal department does not control local learning standards or reading lists.

The agency does play a big role in funding and disseminating research on education, but those efforts have been significantly scaled back by the Trump administration.

It also administers tests that track whether American students are learning and how they compare with their peers in other states and countries. It is unclear whether those tests will continue to be delivered, given drastic reductions in the staff and funding necessary to manage them.

Still, closing the department would not likely have much of an immediate effect on how schools and colleges operate. The Trump administration has discussed tapping the Treasury Department to disburse student loans and grants, for instance, and Health and Human Services to administer funding for students with disabilities.

Can the Department of Education actually be closed?

Any effort to fully eliminate the department would have to go through Congress. Republican members would most likely hear opposition from superintendents, college presidents and other education leaders in their districts; schools in Republican regions rely on federal aid from the agency, just as schools in Democratic regions do.

“They are going to run into opposition,” said Jon Valant, an education expert at the Brookings Institution. “They have a laser-thin majority and a filibuster to confront in the Senate.”

Even if Congressional Republicans stuck together to support closing the agency, Dr. Valant predicted their constituents would protest, given the department’s role in distributing money from popular programs like Pell grants, which pay for college tuition, and I.D.E.A., which provides support to students with disabilities.

“It’s a very hard sell,” he said. “And I am very skeptical that is where this administration wants to spend its political capital.”

It’s worth noting that the attempt to abolish the agency is part of a larger conservative agenda to roll back the federal role in education and direct more money toward private-school vouchers and home-schooling. Trump allies have ambitions to cut the primary federal funding stream to K-12 schools, known as Title I — although doing that, too, would most likely require action from Congress.

How much power does the department have?

Even as Mr. Trump has vowed to close the department, he has begun to use the agency’s powers.

In January, the agency announced an investigation into Denver Public Schools for converting a girls’ bathroom into an all-gender facility. It is also investigating a series of conferences for students of color in the Ithaca, N.Y., public schools, and has created an “End D.E.I.” web page, encouraging individuals to report instances of “divisive ideologies and indoctrination” in schools.

In addition, Mr. Trump’s executive order on “ending radical indoctrination in K-12 schooling” directs the department to develop and disseminate “patriotic” learning materials.

It is unclear how much those orders will change practices in classrooms, given its small role in K-12 education.

The agency does issue regulations on how civil rights laws apply to various groups of students, including disabled students, L.G.B.T.Q. students, racial minorities and girls. One of the administration’s favored strategies is to argue that when schools allow transgender students to use the bathrooms or play on the sports teams of their choice, it is a violation of girls’ rights under Title IX, a law that protects students from sex discrimination.

Mr. Trump has shown less interest in other elements of civil rights law. He has fired government lawyers who investigate schools that fail to provide equal access and services to children with disabilities, for example.

Currently, more than 70 percent of the department’s $224 billion annual budget goes to the federal student aid program, which has also become a frequent Republican target. Mr. Trump is seeking to restrict public-sector loan forgiveness, and has said fewer students should attend four-year colleges.

The agency provides more than $90 billion in new loans to students annually, which are distributed by colleges and serviced by the federal government through private contractors. It also offers $39 billion in Pell Grants annually to low-income students, which generally do not need to be paid back. It administers the federal work-study program and gives grants to students who promise to work as teachers in hard-to-staff subjects or schools.

It has had opponents since the beginning.

Opposition to the Department of Education is today associated with Republicans. But the agency began its life with fierce opponents on both sides of the aisle.

President Jimmy Carter established the department, often known simply as Ed, in 1979, fulfilling a campaign promise to the nation’s largest teachers’ union, the National Education Association. He did so over the objections of his own presidential transition team and many in Congress — including fellow Democrats.

Some staunch liberals believed all of the issues affecting children — health care, cash welfare and education — should be handled by a single federal agency, then known as the Department of Health, Education and Welfare.

Still, over the next four decades, Ed became a part of the beltway firmament, popular with Democrats and many Republicans, too. Many of the programs Ed oversees are sources of bipartisan comity, such as funding for vocational education.

Gareth Davies, a historian who has written about the founding of the Department of Education, said the revival of conservative opposition to the agency shows “just how far the G.O.P. has moved in the past two decades, from compassionate conservatism to culture wars.”

Mike Petrilli, president of the Thomas B. Fordham Institute think tank, and a former Department of Education staffer under President George W. Bush, called the gesture toward shuttering the agency a distraction from problems like the record-low reading scores of American students, released in January.

He suggested that Mr. Trump should host a governor’s summit in Washington to focus on the problem, particularly on the question of whether screen time is harming children’s academic abilities.

“If you wanted to solve this problem and show leadership,” he said, “you would talk about the real crisis.”