More than half a century after Brown v. Board of Education, the Supreme Court is still wringing its hands over whether race can determine where a child goes to school. Can a school district look at two neighboring families, one black and the other white, and send the children to different schools because of their skin color?

In a 5-4 opinion last Thursday, the court took a gratifying but sadly limited step in the right direction. Racial sorting, it said, is not permitted -- except when it is.

Chief Justice John Roberts, who was joined by Justices Antonin Scalia, Clarence Thomas and Samuel Alito, wrote the plurality opinion in Parents v. Seattle. Justice Anthony Kennedy agreed that Seattle and Jefferson County, Ky. (in the companion case of Meredith v. Jefferson County), had crossed a constitutional line by embracing racial balancing for its own sake. But in important respects he disagreed with the Chief Justice, and his separate, gravely disappointing concurrence was the most important opinion. His view -- providing the decisive fifth vote -- is now the settled law.

The schools in Seattle had never been segregated by law (unlike those in Jefferson County), but the city's minority and white families tend to cluster in different neighborhoods. Worried about racially "isolated" classrooms (which really meant classrooms with too few whites), the district adopted a high school assignment plan to promote more racial togetherness. Incoming ninth graders could request a particular school, but if there were already too many whites or minorities in that school, too bad. The children of two of the white plaintiffs were not assigned to the schools of their choice and ended up commuting -- four hours a day. The parents voted with their feet, opting for private schools.

Seattle's plan was extraordinarily crude, as the court noted, and that crudeness was central to sinking the whole scheme. The school board had divided students into only two racial groups: white and nonwhite. If schools were half-Asian, half-white, that was fine; if they were 30% white with the rest Asian, the numbers had to be adjusted. The city, which has a rich racial and ethnic mix, has not been the slightest bit concerned that blacks might have insufficient contact with Asians or Hispanics. "Racial balance" in practice has meant "enough whites." Asians have been viewed as having more in common with blacks than whites, contrary to the views of almost all social scientists.

When the case reached the Ninth Circuit, Judge Carlos Bea noted (in his dissent) that the intellectual foundation for Seattle's policy was pure racial stereotyping. The school authorities assumed "that all white children express traditional white viewpoints and exhibit traditional white mannerisms; all nonwhite children express opposite nonwhite viewpoints and exhibit nonwhite mannerisms," and therefore racially balanced classrooms are needed so that "white and nonwhite children will better understand each other."

Ending racial "isolation" and promoting "better understanding" were two sides of the same coin, in the imagination of the school district, which thought interracial contact inevitably brought "understanding." In 2003, the U.S. Supreme Court had signed on to a variation of this appreciating-diversity theme. When students have "the greatest possible variety of backgrounds," classroom discussions are "livelier, more spirited, and simply more enlightening and interesting," the court claimed in the decision upholding race-based admissions to the University of Michigan law school.

Livelier classroom debates, however, were not Seattle's primary aim. It had an ambitious cultural agenda: fostering "racial and cultural understanding" and creating a life-long desire to "socialize with people of different races." Seattle believed that racial balancing plans were the students' first and (for some unexplained reason) last chance to learn citizenship in a "multi-racial/multi-ethnic world."

What about schools as places of academic learning? Did moving kids around the city to get the racial numbers right have a positive impact on how much math kids learned? Surely that is the bottom line that truly matters. The Seattle school board simply ignored that question.

Chief Justice Roberts dismissed it as irrelevant to the case since the plan was constitutionally deficient on other grounds. It was deficient mainly because the means chosen to achieve the stated ends were "extreme." There was not only the problem of the questionable white/nonwhite dichotomy, and the commitment to racial balancing for its own sake. "Diversity" was a legitimate concern, in the view of the plurality, but it had to be defined more broadly. Moreover, he said, less overtly race-driven means could be found to create racially mixed schools.

If this is foggy, there are still moments of arresting clarity in the plurality decision, such as the final sentence: "The way to stop discrimination on the basis of race is to stop discriminating on the basis of race." Alas, it's hard to know how much weight to place on that sentence: Much of the plurality opinion seemed to suggest that a more nuanced racial balancing plan, presenting slightly different circumstances, might be permissible. All this is in sharp contrast to the principled, bracingly direct concurrence of Justice Thomas: "Racial imbalance is not segregation," he wrote, dismissing one of the more fashionable sophistries of our day, "and the mere incantation of terms like resegregation and remediation cannot make up the difference."

Justice Kennedy showed himself willing to sign on to more subtle race-based school assignments than were on display in Seattle or Jefferson County -- and so, in another case, on another day, his decision might be different. "Diversity, depending on its meaning and definition, is a compelling educational goal a school district may pursue," he wrote. Racial distinctions, apparently, are fine as long as they aren't the sole ingredient in a public policy mix -- like spice in a soup, good when they add flavor, but they should not be overwhelming.

But with no clear recipe to provide guidance, only the taster can decide what is right. And so, alas, with Justice Kennedy we end up with the court's familiar and utterly unprincipled formula: Racial sorting is sometimes OK, but sometimes not. It all depends.

Where does this leave us? What precise limits does the Constitution place on race-conscious policies? Unwilling to simply follow Justice Harlan's famous dissent in Plessy v. Ferguson and declare that "our Constitution is color-blind, and neither knows nor tolerates classes among citizens," the Supreme Court has been trying to draw bright lines between the permissible and impermissible for many a decade -- and mainly producing a lot of what can only be described as barely decipherable gibberish.

Thursday's decision was an important opportunity for the high court to settle this issue. Four justices seemed ready to do so; the signals were mixed. But in any case, they would not have had Justice Kennedy to create a majority. "As an aspiration, Justice Harlan's axiom must command our assent," Justice Kennedy wrote. "In the real world, it is regrettable to say, it cannot be a universal constitutional principle."

Why not?

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Ms. Thernstrom, vice chair of the U.S. Commission on Civil Rights and a senior fellow at the Manhattan Institute, is co-author with Stephan Thernstrom of "No Excuses: Closing the Racial Gap in Learning" (Simon & Schuster, 2003).