TUCSON, Ariz.—BASIS charter school in Tucson, Ariz., may be the most unassuming high school ever to receive positive treatment from the national media. Nothing about its location (downtown next to a Bank of America and across from Target) nor its building (a rather dated former daycare center) indicates that it could compete with the monolith, state-of-the-art high schools that surround it. Yet not only does it compete with local schools that have vastly more money and resources, according to analyses by Newsweek and U.S. News and World Report, it bests them and schools all across the country. In 2008, Newsweek...