When state test scores came out earlier this week, leaders of the Achievement School District breathed a sigh of relief: After two bumpy years in which test scores fell or stayed flat, the state-run turnaround district’s original schools posted significant gains, especially in math.

State and ASD officials said the third year of scores showed that overhauling persistently low-performing schools without displacing their students, the district’s unusual model, takes time to pay off.