Faced with 70,000 students or more who are years behind in obtaining the credits needed to graduate from high school, New York City is at the forefront of a movement to recognize that for a significant number, high school might stretch into five, six, even seven years. In an effort that has expanded across Mayor Michael R. Bloomberg’s second term, the city has spent nearly $37 million to identify and cater to students who are at the biggest risk of dropping out and has already contracted for $31 million more in programs. The staggering numbers of those who are far...