In 1975, the Grand Rapids, Mich., school board began sending teachers to Catholic and Lutheran schools to teach gym.

It soon added art and music, then languages and social studies and advanced and remedial reading and math.

By 1982 about 11,000 children in religious schools -- one-third of the city's school-age population -- were being taught part-time by public school teachers at public expense.

In that year a federal judge struck down the aid program as unconstitutional, an impermissible breach of the wall between church and state. That case will be argued on appeal Wednesday before the Supreme Court, the latest in a long string of "parochiaid" cases to reach the court and one of seven major church-state issues to be decided this year.

The U.S. District Court judge in the case outlawed the program even though school officials had taken down all religious insignia in the classrooms and put up signs saying "Grand Rapids Public Schools classroom" during the public-school part of the day. The school board also had paid rent for the classrooms to show public control over the space.

To the judge it still seemed as if the city was relabeling part of the private school curriculum as public, then paying for it. In some cases the city accomplished the same thing the opposite way, redesignating the teachers instead of the courses, shifting religious-school teachers to the public payroll when they taught "public" courses.

The Grand Rapids program was one of several in the state made possible by a state court ruling that the new aid program was constitutional.

The case comes to the Supreme Court under the First Amendment, which bars laws that promote or restrict religion. The court has been struggling for 37 years to figure out what forms of public aid to religious schools are allowable.

Opponents say court approval of the Grand Rapids program will open the way to all kinds of public aid to private schools previously believed unconstitutional.

Financially strapped religious groups would increase pressure on "notoriously vulnerable" legislatures, the 6th U.S. Circuit Court of Appeals warned last year in ruling against the program. States inevitably would escalate aid to church schools, the majority said, eventually "end ing public education as a major aspect of the American goal of equality of opportunity."

Proponents say such a result is hardly likely, in part because church groups are sensitive to state interference and would resist assistance that went much beyond the Grand Rapids program.

To public and religious school officials in Grand Rapids, the federal courts are bogged down in theoretical and philosophical wars over matters that have little to do with local reality.

What the courts see as unconstitutional "entanglement" of church and state, local officials see as "a spirit of cooperation" and a "good working relationship."

"There has always been cooperation," Don Lennon, the Grand Rapids public schools' director of the program, said recently. "Grand Rapids is a church city" where one-third of the students are in church schools, he said. The heads of the various school systems had for years met regularly to discuss common needs. The sports leagues play in public school facilities, and there long has been a sharing of audio-visual and other equipment.

Lennon said the program in 1982 cost the city $3 million. But the city received $6 million in state funds to run it, netting a $3 million profit. "We ran the public schools for six years without a millage increase because of this," Lennon said.

Local taxpayers who filed suit against the program argue that it was devised only as a deal between the local public and church schools to ensure passage of local school mill levies. Local officals deny the accusation.

"There was no deal struck," said William Gritter, head of the Christian Reformed Church school system. "It just did not happen." He and others say there was no need for a deal. The public school system "has always made every effort within the law to provide services we have coming to us," Gritter said. "It's just an accepted thing here."

Program supporters say it is constitutional in part because the classes taught by public school teachers were only supplementary. They did did not relieve the religious schools of the need to teach state-required courses with their teachers.

"But what would be the legal basis for not teaching other courses as well?" asked Albert Dilly, an attorney for local taxpayers opposing the program. "On what principle can you draw the line?"

Grand Rapids officials emphasize that the program ran smoothly for six years. There were no disputes, no flare-ups, no public school teachers taught religion. There was none of the political divisiveness that courts have feared from such programs.

The Supreme Court, in past rulings in similar cases, has ended up making some intricate distinctions. State grants of maps and other materials to religious school students are barred by the Constitution, the court has ruled, but loans of textbooks, presumably with or without maps, are permissible. Aid to bus parochial students to the front door of the schoolhouse is fine, but buses for field trips to a zoo or museum are forbidden.

States may now provide textbooks, buses, lunches and some testing and health services for parochial students. Such aid went directly to students, the court majority said in a 1975 Pennsylvania case, and the "indirect and incidental benefits" to the schools did not "offend the constitutional prohibition against establishment of religion."

But large-scale aid to church schools such as was proposed in Pennsylvania in a plan not unlike Grand Rapids' was neither "indirect nor incidental," the court said, and it would "ignore reality" to say a major aid program was not providing direct aid to religion.

Since then, and especially since 1980, the court has shown more willingness to favor state "accommodation" of religious beliefs.

While the court in 1973 struck down a New York scheme to give tuition reimbursements to parents of children attending religious schools, it ruled in 1983 that a Minnesota law allowing state income tax deductions for education expenses was permissible.

Most court observers predict a 5-to-4 vote in the case. They count three votes virtually certain to favor the Grand Rapids program -- Chief Justice Warren E. Burger and Justices Byron R. White and William H. Rehnquist. Justice Sandra Day O'Connor has written little in this area, but hers is a likely fourth vote on that side.

Lined up on the other side are another four virtually certain votes: Justices William J. Brennan Jr., Thurgood Marshall, Harry A. Blackmun and John Paul Stevens.

The deciding vote likely belongs to Justice Lewis F. Powell Jr., president of the Richmond school board from 1952 to 1961 and president of the Virginia state board of education in 1968.

Powell voted in 1975 to strike down the similar Pennsylvania program. And it was Powell who said a year earlier that he "would have serious misgivings about the constitutionality of a statute requiring public school teachers in sectarian schools."

But Powell in 1977 observed that "we are quite far removed from the dangers that prompted the Framers to include the Establishment Clause in the Bill of Rights . . . . The risk of significant religious . . . control over our democratic processes -- or even of deep political division along religious lines -- is remote."

Two terms ago, Powell provided the crucial fifth vote to uphold the Minnesota tuition tax deduction plan.

The lawyers on both sides Wednesday will be addressing all nine justices, but they will be watching Powell.