That is partly because it is much smaller than other programs with ’60s roots like Medicaid ($576 billion) and food stamps ($68 billion). It is aimed at young children, who cannot be faulted for their poverty. And unlike most federal programs, Head Start bypasses state officials and directly finances local groups, including nonprofits and school systems. Many have built relationships with members of Congress, who typically view the program as a source of community services and jobs.

While Head Start was known for years as a poverty program that worked, even its friends had come to believe that it did not work as well as it could. A national study, started in the early 2000s, found modest cognitive benefits that faded out within a year. Critics noted that Head Start’s decentralized structure allowed wide variation in quality. And in 2003, a Republican attempt to cede control of the program to some state governments brought bitter opposition from Democrats, who feared it could lead to Head Start’s demise.

But four years later, Congress passed a bipartisan law that retained federal control while requiring periodic audits of classroom quality, with groups in the lowest 10 percent forced to compete to keep their grants. “The fact that both parties were behind it meant you couldn’t just end it on a whim,” said former Representative George Miller, a California Democrat who pushed the overhaul. “Programs understood they had to step up their game.”

The monitoring began in 2012 with an observational tool called CLASS, which is devised to measure teaching quality. Developed at the University of Virginia, it quantifies three aspects of a teacher’s performance: instructional support, emotional support and classroom organization. In essence, it gives the government a report card on each of its nearly 1,600 Head Start programs.

In the first four years, about 120 of those programs lost all or part of their grant.

In addition to the more stringent oversight, other factors that may explain rising scores include an increase in funding per child (18 percent in the last five years) and better teacher training. Nationwide, the share of Head Start teachers with a bachelor’s degree has risen to 73 percent, from 47 percent a decade ago.

In Jacksonville, the Urban League had run the program for 17 years, but low CLASS scores in 2012 opened it to challenge. Before the competition occurred, inspectors found so many health and safety violations that they rushed the program into interim management. Lutheran Services was selected to take the program over in 2014 and now serves 1,800 children.