Most potential Republican presidential contenders are renouncing the national educational standards known as Common Core. Former Florida Gov. Jeb Bush, who has championed measuring academic achievement for two decades, is doubling down.

Resistance to Common Core is growing among the party's activists, who see it as a federal incursion into local schools. Republican governors of South Carolina and Oklahoma last month joined Indiana in opting out. Louisiana Gov. Bobby Jindal, who two years ago said the initiative "will raise expectations for every child," recently compared it to Russian centralized planning and on Friday vetoed a pro-Common Core bill.

Mr. Bush, by contrast, is urging states to stick with it.

"Pressing pause means stopping forward momentum," said a letter released Monday by Mr. Bush's nonprofit Foundation for Excellence in Education. "And when that happens, things can go backwards."

Mr. Bush's embrace of Common Core, as well as his support for legalizing undocumented immigrants, is coming to define his national profile ahead of a possible 2016 presidential bid. These stances isolate him from the conservatives who dominate Republican nominating contests. Last week, they demonstrated their power by ousting House Majority Leader Eric Cantor in a Virginia GOP primary.