The largest study ever of school cell phone bans finds that they offer decidedly mixed results, with teachers reporting fewer distractions when students lock their phones away during the school day, but little evidence the bans quickly bring improved academic achievement or better behavior, as many advocates have hoped.

The study, by scholars at Stanford University, Duke University, The University of Michigan and the University of Pennsylvania, compiled data from Yondr, a California startup that makes lockable pouches for schools, businesses and entertainment venues. Published Monday by the National Bureau of Economic Research, it looks at data from about 4,600 schools and is the first nationally representative look at cell phone bans.

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