Kelly Clarkson Says Her Kids Don’t Understand ‘Fame’ and ‘Think Everybody’ Performs on Stage

Clarkson tells PEOPLE about teaching her daughter River Rose and son Remington Alexander “the gratitude and appreciation factor” of their lifestyle

Kelly Clarkson and River Rose Blackstock

Having remained humble throughout her career, Kelly Clarkson wants to make sure her kids do the same.

At SiriusXM’s Next Generation Industry & Press Preview event in New York City on Wednesday, the pop superstar, 41, opened up to PEOPLE about having her kids, daughter River Rose, 9, and son Remington Alexander, 7, join her on stage at concerts — and how they think it’s a normal activity.

“Mine come out with me all the time. They recently did in Vegas when I did these shows in the summer, but they ask to. I don’t ask them,” Clarkson tells PEOPLE, noting that “it’s hard” to teach her kids, whom she shares with ex-husband Brandon Blackstock, about the special nature of her famous lifestyle.

Kelly Clarkson, Remington Alexander and River Rose

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The “Miss Independent” singer often holds “dance parties” at home with her children, who’ve also watched their mom perform and appear on television since they were young — and they don’t necessarily understand the difference between the two.

“It’s so normalized for them. So, they don’t even see it as anything weird,” says Clarkson. “They’re like, ‘Can I bring my friend?’ And I’m like, ‘OK, well, maybe we’ll just bring you right now.'”

One of her focuses as a parent is clueing her children in on how fortunate they are to grow up around the glitz and glamour of her entertainment career.

Kelly Clarkson

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“They don’t know the fame thing yet. They just think everybody can do that, right? Which is kind of cool, but at the same time you want to teach them the gratitude and appreciation factor,” explains Clarkson, before quipping: “Privilege is definitely taught in my household. They’re annoyed with it, probably.”

The “Since U Been Gone” performer’s daughter recently got a taste of what it’s like to be a musician, as the new deluxe version of Clarkson’s Chemistry album features River’s vocals on the track “You Don’t Make Me Cry.” The clip used in the song was recorded when River was 5 but held until now, and the NBC talk show host says her daughter “was so excited” for the song to come out.

“They wanted to listen to it all the time before it came out, and now it’s out. So, now she gets excited about it, but she gets a little nervous,” says Clarkson, adding that she’s realizing River may be getting to an age where she starts to feel a little self-conscious at times.

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“I didn’t even think about it, but, you know, it’s now kind of brought to my attention of like — hey, she’s gonna get to that stage where she’s like, ‘Mom, don’t do this,’ or ‘Don’t do that,’ but she loves it right now,” explains the Grammy winner. “She actually calls it her song.”

One of the perks of releasing music that River may not yet be aware of is the fact that she’ll get paid royalties as listeners stream and purchase the song. But for now, Clarkson will oversee her daughter’s business affairs.

“She’s very well taken care of as far as the River Rose [and the Magical] Lullaby books, and my son as well. I have like a whole fund for them,” explains Clarkson, declaring with a laugh: “I don’t take the money!”