NEW YORK » The teacher held up a card with a number on it, then looked at the 4-year-olds waving their hands eagerly in front of her. "Anderson," she said, calling on a small boy in a blue button-up shirt and a sweater vest.

"Five," Anderson said, correctly.

"Good boy, Anderson," the teacher said. Then she turned to the rest of the class. "Are you ready?" she said, and then, "Go!" At that, the children jumped up and down five times as they counted: "One! Two! Three! Four! Five!"

This exercise, which held a prekindergarten class in Brooklyn riveted one morning last week, was not an effort to introduce high-impact aerobics into preschool. It was part of an ambitious experiment involving 4,000 children, lasting more than six years and costing $25 million, and designed to answer a fundamental question: When it comes to preschool, what actually works?

As Mayor Bill de Blasio prepares to greatly expand New York City’s preschool offerings, much debate has focused on how the expansion should be paid for, and less on what actually constitutes an effective prekindergarten program – one that will, as the mayor says, shrink the achievement gap between children of different racial and economic backgrounds. The mayor often says that studies have found that prekindergarten can have long-term benefits for children. But while that is true, experts note that the studies of programs that are decades old, and were extremely small, intensive and costly do not single out what the key elements were, nor are they practical to duplicate on a large scale today.

"We’re motivated by them; we just can’t replicate them," said Dale Farran, a professor of education and psychology at Vanderbilt University.

Part of the problem is that rigorous research is expensive and time-consuming. The study involving the children in Brooklyn, who attend Public School 221 in Crown Heights, will gauge whether a certain math curriculum can create lasting improvement in students’ math and language skills, as well as their likelihood to persevere in the face of academic challenges. It will track roughly 4,000 children who enter prekindergarten in 69 schools and community-based organizations next fall, and continue following them through at least the third grade. Half of them will get the curriculum, called Building Blocks, and the other half will not. Later on, if there is sufficient funding, a subset of each group will get a supplementary math program in kindergarten, in small groups or in the form of intensive tutoring.

The study is being financed by the Robin Hood Foundation, the antipoverty organization that is a favorite philanthropy of many in the hedge fund industry, with help from two family foundations, the Heising-Simons Foundation and the Overdeck Family Foundation. While Robin Hood is hardly a natural ally of the de Blasio administration, the research it is financing could end up providing important information for officials in New York City and elsewhere trying to design high-quality prekindergarten programs.

Michael Weinstein, the chief program officer at Robin Hood (and a former economics columnist and editorial board member of The ), said that his motive was simple: He was interested in the promise of early childhood education to fight poverty, but unsatisfied by the existing research, which did not provide clear guidance as to which programs were the most cost effective.

"We pride ourselves, correctly or not, in having an evidentiary basis for making the grants we do," he said, describing Robin Hood’s approach as one of "relentless benefit-cost calculations."

Robin Hood hired MDRC, a respected social policy research organization, to carry out the study, and Robin Hood and MDRC convened a group of experts to advise them on what curriculum they should examine.

The experts recommended Building Blocks, which was developed by two education professors at the University of Denver, for two primary reasons, Weinstein said. First, short-term studies had suggested that it improved children’s math and, to a lesser extent, verbal skills. Second, other research had indicated that preschool math skills were a good predictor not only of math achievement but also of reading ability in elementary school, as well as graduation from high school and college attendance. In addition, studies suggest that prekindergarten teachers currently spend little time on math.

"The interest here is not in math per se," said Pamela Morris, the director of the Institute of Human Development and Social Change at New York University, and a principal investigator on the study. "It’s really math as a lever to intervene in a broad set of outcomes for kids."

To ensure that the teachers in the study are using the curriculum correctly, they are being trained now, with weekly visits from coaches.

Last week at P.S. 221, for example, Emily Hamlin, 31, a teaching coach, looked on as the teacher, Joanne Perrina, 44, used a moose puppet, called Mr. Mix-Up, to help the children learn to count. A child would tell Mr. Mix-Up what number to count to; then, when he would invariably lose his way or skip a number, the children would hold up their hands to tell him to stop.

Later, Perrina divided the students into small groups. She supervised two students in Dinosaur Shop, in which one child (the buyer) asks the other child (the shopkeeper) for a certain number of plastic dinosaurs, which the shopkeeper counts out. Then the buyer has to count out the right amount of play money to pay for the dinosaurs, and the shopkeeper has to count the money again. Hamlin, meanwhile, helped two children use the Feely Box a wooden box with holes on each side that helps children learn to identify shapes, like triangles and trapezoids, by feel.

The study’s investigators will look over time at children’s test scores and grades, grade retention, special education placement, as well as so-called executive function a term that encapsulates a range of skills, from focus and short-term memory to impulse control, that experts think are closely tied to children’s success in school. The importance of tracking the students through the third grade is that most studies have found that academic gains from prekindergarten tend to fade by then, either because other children catch up or because the quality of elementary education is so variable. Weinstein said he hoped that, if the study found improvements that lasted through third grade, he would be able to raise enough money to extend it.

In any case, if the results are significant, he said, "We will obviously pour resources into the Building Blocks program, and we expect people around the country to do the same."

Building Blocks is currently used in all of Boston’s prekindergarten classes. In New York City, where the system is much larger, and prekindergarten classes are based in a combination of public schools and community organizations, there is no mandatory curriculum, although all are supposed to be "evidence-based," and to advance the Common Core standards. In its funding plan for expanding prekindergarten, the de Blasio administration has set aside money for research and evaluation, but it is not clear what form that will take, or how the administration will measure the effects of a program that is so far from uniform.

Deborah Phillips, a professor of psychology at Georgetown University who has studied the prekindergarten program in Tulsa, Okla., said that the Robin Hood study was important for two reasons: It would follow the children through third grade, and it would test the effectiveness of Building Blocks when taught by the same kinds of teachers, with the same range of qualifications, who currently make up New York City’s prekindergarten teaching pool.

"It’s certainly not just a hothouse assessment of, ‘Can it work under strong conditions?’" she said. "It’s really asking the question: Will it work?"

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