City Connects costs less than $800 per student annually — about 6 percent on top of the typical cost to educate one. An analysis of the program carried out by the Center for Benefit-Cost Studies in Education at Columbia found that it generates a return of at least $3 for every dollar spent. “Providing the program to 100 students over six years would cost society $457,000 but yield $1,385,000 in social benefits” — higher incomes, lower incarceration rates, better health and less reliance on welfare, according to the analysis. If City Connects were a company, Warren Buffett would snatch it up.

Morris Academy opens early — breakfast is provided, along with before-class tutoring. It’s open until 6:30, as well as on some Saturdays and during the summer. Students can choose among clubs for chess players, step-team dancers and bloggers. The robotics team competes with high schools nationwide. During lunchtime and after school, tutors offer one-on-one help to struggling students. An in-house clinic provides medical, dental and psychological services.

Community school funds enabled Mr. Mazzaroppi to deliver the emotional support that battle-scarred children badly need — recruiting a squadron of social workers, training teachers to counsel students and teaching older students how to mentor their younger classmates. “Our problem wasn’t lack of an academic strategy but our inability to answer students’ pleas for help,” he says. Now, remarkably, Morris Academy students are more likely than their peers citywide to say they feel safe in school and believe that their teachers care about them.

After-school and summer programs not only keep poor kids off the streets, but they also give them the academic leg up and the array of opportunities that better-off families can afford to buy. When he was the chief executive of Chicago’s public school system, Arne Duncan, the former United States secretary of education, opened 150 community schools. “Making every school a community school — that’s got to be our collective vision,” he asserted.

Results-hungry policy makers expect test scores to rise overnight, but getting students engaged in their own education must come first. A recent evaluation of Baltimore’s community schools concluded that the schools whose students did best academically were those in the program longest.

“The key is perseverance,” says Mr. Gaither. “When you hold the course, you get more than what you pay for.” His experience bears him out. Since adopting the community schools strategy a decade ago, Wolfe Street Academy has moved from being the city’s second-worst-performing elementary school to its second-highest.

New York rarely does things by halves, and community schools are no exception. In the span of just two years, 51,616 students started attending schools like Morris Academy — more students than in the entire District of Columbia school system. Most of them go to one of the 94 “renewal schools,” the city’s lowest-performing schools. Patience is in short supply in New York, however, and these troubled schools have just three years to show substantial progress.