“The frustration of the court matches the policy movements across the states and across the federal government over the same time,” said Eric A. Hanushek, a fellow in education at the Hoover Institute at Stanford University, who testified for the State of Connecticut in defending what it spends on schools. “It’s not that the states or the federal government has said ‘We’re not going to do anything about schools.’ In fact, there’s been a concerted effort to try to improve things, but it is rather faddish and we do what people currently think is in vogue, and it just hasn’t worked.”

He added, “Just doing more of the same is unlikely to lead to a different result.”

Rather than tell the General Assembly to figure out a way to even out the gap in resources — which is what most courts in similar cases have done — the judge called for a radical reimagining that starts with the question of what schools should do: What are the goals for elementary students, or high school graduates? Then, he said, the state should decide how much money schools require so that all students, rich and poor, reach those goals. In the kind of rational system the judge proposes, you determine what you are trying to do before you decide what you are going to spend.

Over the past half-century, nearly every state has had a similar case about how to finance its schools, some more tortured than others. In New Jersey, for example, the state’s highest court has issued more than 20 rulings in a case that has stretched over four decades. David G. Sciarra, the executive director of the Education Law Center in Newark, which has helped litigate many school funding cases across the country, said he was watching pending litigation in Kansas, New Mexico, Pennsylvania, South Carolina and Washington.