A colleague of Mr. Levitt’s, the Nobel laureate James J. Heckman, presents a similar argument on the science of early brain development: “Skill begets skill, and early skill makes later skill acquisition easier. Remedial programs in the adolescent and young adult years are much more costly in producing the same level of skill attainment in adulthood. Most are economically inefficient.”

But these economists are selling teenagers short. As the Temple University psychologist Laurence Steinberg points out in his path-breaking new book “Age of Opportunity: Lessons From the New Science of Adolescence,” during the past decade neuroscientists have realized that adolescence, like early childhood, is a “period of tremendous ‘neuroplasticity,’ ” during which the brain has solid potential to change through experience. During the past generation, educators have focused on the growth between birth and age 3, but the teen years may be just as important for shaping a person’s future.

As any teacher will tell you, there’s no good way to manage a ninth-grade math class in which some students are working on quadratic equations while others are still mastering multiplication and division. There’s a “mismatch between what many students need and what schools can provide,” Jonathan Guryan, an economics professor at Northwestern University and faculty co-director of the University of Chicago’s Urban Education Lab, points out.

The tutoring program tackles this problem with intensive support, providing a safety net for students who have fallen far behind. Working two-on-one, the tutors, most of them recent college graduates, can individualize instruction to suit each student’s needs. They are trained not only in how to teach math but also in how to relate to these teenagers.

“It’s friendship and pushing — they nag them to success,” Barbara Algarin, the executive director of Match Education, which runs the tutoring program, told me. “These students can make remarkable progress when they appreciate that their tutor is in their corner. The math connection leads to better study skills and a love of learning. Grades improve across the board.”

American parents spend $7 billion annually on tutoring, in some cases as much as $400 an hour, to reassure themselves that they are giving their children every advantage in the academic rat race, and research on the impact of tutoring backs them up. But the high cost of traditional tutoring has made it hard to offer this kind of help to the students who need it most. Because the Match tutors earn about $16,000 a year plus benefits, about what other public service programs like City Year pay, the Chicago program is a relative bargain, costing about $3,800 a year for each student. (To put that figure in context, New York City spends more than $20,000 per pupil, and even more in schools serving poor neighborhoods.)

“Just a few years of this type of intervention could bring almost all students up to grade level,” Jens Ludwig, a professor of social service administration at the University of Chicago and the director of its Crime Lab, told me. “By then they can benefit from what’s being taught in regular classes and have real hope for a high school diploma.”

To Mayor Rahm Emanuel, who plans to expand the program, the lesson is plain. “What this shows is, don’t ever throw the towel in on the kids,” the mayor said last March. What’s happening in Chicago shows that, without breaking the bank, the lives of adolescents can be turned around.