In the shadow of a Manhattan housing project, Public School 33 is coming back to life. A new principal has brought a wave of optimism, test scores are way up, and middle-class families who used to avoid the school are enrolling their children. In Brooklyn, teenagers who might have dropped out of school are getting diplomas through a special evening program. In the Bronx, a small high school is graduating its low-income students at a faster clip than are New York City's larger, older high schools. Promising signs like these are some of what Mayor Michael R. Bloomberg is banking on as he seeks re-election November 8. Not long after taking office in 2001, the Republican businessman persuaded the New York state legislature to give him nearly unprecedented control of the biggest school district in the country. He invited voters to retain or replace him based on his work improving the schools. Of all the Bloomberg administration's changes, the new approach to instruction has drawn by far the sharpest and most prolonged criticism. Though the mayor suggested he would take a back-to-basics approach, his top deputies rejected the highly scripted model of "managed instruction" that's popular in cities with large numbers of children behind on their skills. Eliminating the dozens of curricula in use around the city, they chose a language arts program, Month by Month Phonics, that they say gives children both reading skills and plenty of practice with good books. Critics charge it is woefully short on systematic instruction in basic skills such as decoding. They also picked a strongly conceptual mathematics program, Everyday Math, that some see as shortchanging computational skills. The Bloomberg administration contends that this approach, spread citywide, is starting to pay off. On the 2005 state language arts tests, 59 percent of 4th graders scored at grade level, up 10 percentage points from the previous year. On the math tests, 77 percent of 4th graders scored at grade level, up 9 points in one year. Top aides acknowledge privately that poor test scores could have been disastrous for Mayor Bloomberg. And his critics are frustrated that good ones seem to have immunized him from criticism. Many still rue the administration's instructional choices, contending its literacy methods are about how to teach, not what to teach.