The gifted program was not a panacea. The researchers found that the district’s specialized classes had little effect on the academic achievement of students who had been specifically identified as gifted, through I.Q. tests. They are not sure why. In Broward County, as in many other places, classes for the gifted use the same curriculum and textbook as other classes. Teachers in the classes for the gifted were required to have a special certification and were encouraged to supplement the curriculum.

But the separate classes did produce enormous, positive effects for children who were high achievers but did not qualify based on the I.Q. test. A quirk in the rules helped these children: Broward requires that schools with even one child who tests above the I.Q. cutoff devote an entire classroom to gifted and high-achieving children.

Since a school in Broward rarely had enough gifted children to fill a class, these classrooms were topped off with children from the same school who scored high on the district’s standardized test. These high achievers, especially black and Hispanics, showed large increases in math and reading when placed in a class for the gifted, and these effects persisted.

What is more, while many children in the gifted program gained enormously, Mr. Card and Ms. Giuliano found no negative effects for those who remained in regular classes. Yet all of these gains came at little financial cost. The enhanced classes were no more expensive than the standard ones. They were the same size as regular classes, and teachers in the classes for the gifted were paid no more than others.

This story has twists, though.

Despite these positive results, Broward County suspended its universal screening program in 2010 in a spate of budget cutting after the Great Recession. Racial and ethnic disparities re-emerged, as large as they were before the policy change. In 2012 the district reinstated a modified version of universal screening, but it has not achieved the same results. Using data from the Florida Department of Education, I calculate that 8 percent of white students in Broward County are classified as gifted. That is twice the rate for Hispanics and four times the rate for blacks, much higher ratios than under universal screening.

One problem with the new screening program is that the previous nonverbal test, which psychologists say they believe to be culturally neutral, has been replaced with one that relies more on verbal ability. Another is that Broward parents and teachers can still influence whether children are selected. While school psychologists test students at no cost, parents can hire a private psychologist to test a child, at a cost of $1,000, and are allowed to pay for multiple tests, should a child not meet the I.Q. requirement on the first try. Mr. Card and Ms. Giuliano found evidence suggesting that private testing gives an advantage to upper-income families, who tend to be white.

Many researchers worry that I.Q. tests are biased against low-income and nonwhite children, and some recommend a more holistic approach that includes teacher referrals. But referrals produce biases, too. Matthew McBee, a psychologist who edits The Journal of Advanced Academics, which focuses on gifted education, recently called referrals “the elephant in the room,” a largely unexamined source of racial and ethnic bias in the identification of gifted children.