“It will basically change 200 years of practice,” Ms. Weingarten said.

The case challenges a decision to shut down a program that provided tax breaks to donors who funded scholarships that families could use at private schools, including religious ones. Montana discontinued the program after a ruling by its high court found that it had violated a provision in its Constitution that prohibited the use of government funding, either directly or indirectly, for religious purposes. Such provisions, known as no-aid provisions or Blaine Amendments, exist in 37 states and were driven initially by anti-Catholic bias. They are now seen as the last line of defense against widespread acceptance of school voucher programs.

The conservative-leaning high court could finally settle whether vouchers blur the constitutional line between church and state.

Lawyers at the Institute for Justice, the libertarian law firm that is representing three Montana families, said the case could produce one of the most significant education rulings in the last half-century. The Supreme Court has ruled that states may include religious schools in publicly funded school choice programs, but a ruling that would essentially require it would be groundbreaking.

“If we win this case, it will be the U.S. Supreme Court once again saying that school choice is fully constitutional and it’s a good thing,” said Erica Smith, a senior lawyer at the Institute and co-counsel in the Espinoza case. “And that will provide momentum to the entire country.”

Education Secretary Betsy DeVos has proposed a $5 billion federal tax credit scholarship program that would allow states to adopt initiatives much like the one Montana struck down. Ms. DeVos, who sent her own children to private religious schools, tried to defeat the Blaine Amendment in her home state, Michigan.