Something I've had to do is tell my son that lots of other people don't care about being rational and so he shouldn't always expect to be able to convince his classmates that he's right because he's provided a good argument. I did this after hearing my son getting in an argument with another child over the value of a pawn, with the other kid insisting that the pawn had zero value. My son was upset that the other kid wasn't responding to his arguments.

You don't want to teach your kids rationality skills, have them apply these skills when interacting with classmates and then realize that their classmates don't respond well to them. It's best, I think, to let your kids know that they should always be rational with you, that they are more likely to convince you to do something if they make rational arguments, that being rational gives them special powers in understanding the world, and that later in life there will be huge payoffs from them being rational.