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"Congratulations on your kids," said Rice, who is the director of the Hoover Institute at Stanford University. "Get 'em ready for Stanford."
It was a brief — but sweet — moment between Rice, a former secretary of state who served Republican President George W. Bush, and Buttigieg, the current secretary of transportation in President Joe Biden's administration, who welcomed a daughter, Penelope Rose, and a son, Joseph August, with husband Chasten in August.
Buttigieg, 39, didn't share how his family intends to prepare the twins for college. But earlier in the broadcast, Rice shared her own thoughts on the hot topics of education, history and race.
"Parents ought to be involved in their children's education," Rice, 66, said. "Parents ought to have a say. We used to have parent-teacher conferences. We used to have PTAs. There are lots of ways for parents to be involved and they should be."
Rice also reminded co-hosts Whoopi Goldberg, Joy Behar, Sunny Hostin and Sara Haines that she grew up during segregation in Birmingham, Alabama.
"I couldn't go to a movie theater or a restaurant with my parents. I went to segregated schools until we moved to Denver," she said. "My parents never thought I was going to grow up in a world without prejudice but they also told me, 'That's somebody else's problem, not yours. You're going to overcome it. You are going to be anything you want to be.' That's the message we ought to be sending to kids."
Rice was critical of the notions that "white people have to feel guilty about everything that's happened in the past" and that "Black people have to feel disempowered by race."
"I would like Black kids to be completely empowered, to know that they are beautiful in their blackness," she said. "But in order to do that, I don't have to make white kids feel bad for being white."
In her view, she said, the conversation about race in schools has "gone in the wrong direction."
"I have no problem with letting people know what happened, but let's remember that history is complex," she said. "Human beings aren't angels now and they weren't angels in the past and so how we teach about our history is also important."
Goldberg, 65, made a point about why teaching the dark elements of the nation's past is important. "The whole idea of teaching history is so we don't repeat it," she said, adding, "There's no way to hide the fact that white people owned black people."
How race and racism is taught in schools has become a cultural flashpoint among conservatives in recent months as Republican lawmakers increasingly criticize "critical race theory," a niche higher-education movement dating back to the '70s that they say could become more common.
Advocating for a thorough and complete approach to teaching history — "We teach the good and we teach the bad," she said — Rice said on The View she took issue with curriculum that would make "7- and 10-year-olds feel that they are somehow bad people because of the color of their skin."
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"We've been through that and we don't need to go through that again," she said.
Behar, 79, objected, saying, "That doesn't seem to be part of the plan."
"Oh, it is," Rice responded. "I'm sorry."
"In Germany they teach the Holocaust to every student," Behar responded. "They learn about their history and there are not two sides to the story."
As Goldberg was going to commercial break, Rice got the last word during the segment, saying as the outro music played, "We all have to learn about our history but we also have to recognize that we have to live together and we're going to do better living together if we don't make each other feel guilty."