What vital life lessons do Scout and Jem learn in chapters 12 and 13 of Harper Lee's To Kill a Mockingbird?

Humility and its antithesis, arrogance, are two lessons Scout and Jem learn in chapters 12 and 13 of Harper Lee's To Kill a Mockingbird.

Scout and Jem learn a lesson in humility from Calpurnia the day she brings them to church with her when their father is out of town. Walking home from church, Jem notes that Calpurnia does not speak the way the rest of the African Americans speak when she is with them at the Finch household; instead, she speaks with proper grammar. However, the children also noticed that while she was at the church speaking with her fellow African Americans, she used the same slang and dialect they use. When Scout and Jem want to know why she speaks "nigger-talk to the--to your folks" when she knows how to speak grammatically, Calpurnia's response gives them a lesson in humility. First, she explains that she would seem out of place if she "talked white-folks' talk at church, and with [her] neighbors." Second, and more importantly, she explains that it is "not necessary to tell all you know." Just because she is literate and knows how to speak grammatically does not give her the right to show off her knowledge at every chance she gets. Most importantly, she knows that showing off her knowledge won't change things; it will only infuriate people. As she further explains to Jem and Scout,

"You're not gonna change any of them by talkin' right, they've got to want to learn themselves, and when they don't want to learn there's nothing you can do but keep your mouth shut or talk their language."

In other words, Calpurnia's answer shows that she is willing to accept the folks around her as they are. Her willingness to accept those around her shows that she is an extremely humble person, and Scout and Jem learn a lesson about humility by having this conversation with her.

Scout and Jem learn the exact opposite of humility from their Aunt Alexandra when she comes to stay with them. Contrary to Calpurnia, Aunt Alexandra insists in believing she is better than those around her and in acting in a way that shows her superiority. Since the Finch family has been well-educated and wealthy for multiple generations, Aunt Alexandra feels that the Finch family is a family of high quality. She is very surprised to learn that Atticus hasn't taught his children to value their family background and tries to get him to do so. Atticus walks into Jem's bedroom to give both children a lecture about how they are the "product of several generations' gentle breeding" and of a family that has been very influential throughout all of Maycomb County for multiple generations. Finally, at Aunt Alexandra's encouraging, he tries to impress upon the children that they "must try to behave like the little lady and gentleman that [they] are."

However, Scout breaks into tears and Jem is stunned because they know these are not their father's own thoughts, who has only ever said that "Fine Folks" are people who could use their intelligent minds well. The children's reactions bring Atticus to his senses, and he humbly retracts the lecture. But what is most interesting is that Aunt Alexandra tries to teach the children the exact opposite of what Calpurnia teaches them; Aunt Alexandra tries to teach them how to think of themselves and act as if they are better than others, which is the exact opposite of humility. It's arrogance, and this is a lesson that the children reject.