At the beginning of part 2 of the novel, Scout sees that Jem is beginning to change. She states,
Overnight, it seemed, Jem had acquired an alien set of values and was trying to impose them on me: several times he went so far as to tell me what to do.
This reflects Jem entering adolescence or beginning to come of age. He now is beginning to think of himself as an adult and Scout as a child rather than of himself as just her older brother.
When Scout goes to Calpurnia in distress about the changes that have come over Jem, Calpurnia confirms he is maturing, saying,
I just can't help it if Mister Jem's growin' up. He's gonna want to be off to himself a lot now, doin' whatever boys do, so you just come right on in the kitchen when you feel lonesome.
Jem's entry into adolescence coincides with the Tom Robinson trial. Because he sees white adults in his community acting badly about the trial, such as convicting Tom despite all the evidence that shows he is innocent, Jem takes the outcome very hard. When he reacts harshly to Scout questioning him about racial hypocrisy and refuses to answer her, she goes to Atticus. He explains to her that while Jem is going through the process of coming of age, there are certain things he won't want to deal with, such as the town's racism, until he is a little older. In the meantime, he will be difficult:
Atticus said that Jem was trying hard to forget something, but what he was really doing was storing it away for a while, until enough time passed. Then he would be able to think about it and sort things out. When he was able to think about it, Jem would be himself again.
Both children mature over the course of the novel. Evidence of Scout's maturity comes at the end of the novel when she stands on Boo Radley's porch and is able to see the world through his eyes for the first time. She comes to understand that
Atticus was right. One time he said you never really know a man until you stand in his shoes and walk around in them. Just standing on the Radley porch was enough.