In an email to donors and surrogates obtained by CNN, campaign manager Danny Diaz highlighted Bush's answer, saying he knocked the question "out of the park" and achieved "the best response in our online metrics."
Bush has vigorously supported Common Core's national education standards in the past, which has led to some uncomfortable questions -- and at times confrontations -- from voters on the campaign trail this year as the issue has become a major subject of concern in recent years from critics who describe the standards as federal overreach.
Conservative radio host Laura Ingraham, one of Bush's fiercest critics on the right, tweeted he was "good" with his answer on education standards.
At the debate, Bush was asked if he thinks those who oppose Common Core constitute a "fringe group of critics," as stated by Education Secretary Arne Duncan.
"No, I don't," Bush replied, reiterating his position that he's maintained on the campaign trail this year that he doesn't think the federal government should be involved in creating content or curriculum.
Common Core was created by a bipartisan group of governors and educators, and states that adopted the standards received financial incentives by the federal government. Other candidates on the same stage, like New Jersey Gov. Chris Christie and Wisconsin Gov. Scott Walker, formerly supported the standards but have since dropped them in their respective states.
"I'm for higher standards, measured in an intellectually honest way, with abundant school choice, ending social promotion," Bush said to applause in Cleveland, while also highlighting voucher programs he created as governor.
"And we had rising student achievement across the board, because high standards, robust accountability, ending social promotion in third grade, real school choice across the board, challenging the teacher's union and beating them is the way to go."
The email comes after Bush got a slate of mixed reviews after his performance in the crowded debate Thursday night. With Donald Trump dominating the stage, the other nine candidates struggled to break through and generate as many headlines as the real estate titan.
Bush didn't make any gaffes or errors, but he was criticized by some for lacking enthusiasm. In an Iowa focus group conducted by CNN
, for example, voters agreed that Bush gave no wrong answers, but he didn't create a spark in the group.
"It's not like I disagree with him, it's not like I don't agree with a lot of what he says," said Julie Mueller, a small-business owner and wife of the Pella mayor. "He has a very good record in Florida, but I'm suffering from a little bit of Bush fatigue and he's just not that interesting ... I think it's the last name, and maybe that's too bad."
At a watch party hosted by Bush's campaign in Cleveland, supporters and others who were interested in Bush gave their most robust applause to his answer on Common Core but had no response when he talked about how he would deal with ISIS.
The email from Diaz also laid out a series of goals that Bush hoped to accomplish in the debate, like showing that he's a "doer" by talking about his record and maintain an "optimistic" tone, all of which Diaz felt like Bush succeeded in doing.