“We know that the prefrontal cortex is not fully developed until the 20s, and some people will ask, ‘Why are you trying to improve prefrontal abilities when the biological substrate is not there yet?’ ” said Adele Diamond, a professor of developmental cognitive science at the University of British Columbia in Vancouver. “I tell them that 2-year-olds have legs, too, which will not reach full length for 10 years or more  but they can still walk and run and benefit from exercise.”

Executive function involves three important skills. The first is the ability to resist distractions or delay gratification to finish a job: to finish the book report before turning on the television. The second is working memory, the capacity to hold and manipulate multiple numbers or ideas in the mind,  for example, to do simple addition or subtraction without pencil and paper. The third is cognitive flexibility, the ability to appreciate another person's point of view and to adapt when demands change  when recess is canceled, say, and there’s a pop quiz in math.

Image SELF-CONTROL Young boys take turns listening and speaking without interrupting each other. Such exercises can help children become more self-possessed at an earlier age. Credit... Penny Farster Narlesky

Researchers can rate these abilities with some precision by giving young children several straightforward mental tests. In one, youngsters sit in front of a computer and when a red heart appears on the left side of the screen, they strike a key on the left, and when it appears on the right screen they strike a key on the right. Most of them do well on this.

But when scientists change the rules, and have the children strike a key on the right when the symbol appears on the left, and vice versa, the test gets harder. The number of errors they commit, and the time it takes the children to answer, are considered measures of their ability to regulate themselves. Other similar kinds of tests can track improvements in working memory and intellectual flexibility. Researchers have designed school-based curriculums intended to improve each of these abilities. In a study published in 2007, Dr. Diamond led a team that compared one of these programs  called Tools of the Mind  to a standard literacy curriculum, in several preschools in the Northeast. The Tools program features a variety of exercises, including a counting activity in which children pair off. One child counts a given number of objects from a pile and separates them, and then the other child checks the sum. The “checker” has a sheet of paper with a list of numbers, each beside a corresponding number of dots: for example, four dots line up beside the No. 4. By placing the objects on the dots, the child can see whether the count was accurate. This double-checking is intended to force the “counter” to be more careful and to stall the other child’s impulse to grab an object.