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It was a stormy graduation day in Cambridge on Saturday. Drenched with rain, the students in their gowns looked like penguins emerging from the sea. Parents wielded golf umbrellas, the sword and shield of the English middle classes. The downpour could not extinguish their justifiable pride: a degree from a university often ranked number one in the world is still a glittering prize.

GREENWELL SPRINGS, La Rebekah and Kevin Nelams moved to their modest brick home in this suburb of Baton Rouge seven years ago because it has one of the top-performing public school districts in the state. But starting this fall, Ms. Nelams plans to home-school the couple’s four elementary-age children.The main reason: the methods that are being used for teaching math under the Common Core, a set of academic standards adopted by more than 40 states.