What are some quotes showing Jem maturing?

At the beginning of the novel, Jem is an immature, naive child, who fears their reclusive neighbor Boo Radley. In chapter 1, Jem gives his animated description of their "malevolent" neighbor. Scout says,

"Jem gave a reasonable description of Boo: Boo was about six-and-a-half feet tall,  judging from his tracks; he dined on raw squirrels and any cats he could catch,  that’s why his hands were bloodstained—if you ate an animal raw, you could never wash the blood off. There was a long jagged scar that ran across his face; what teeth he had were yellow and rotten; his eyes popped, and he drooled most of the time" (Lee, 13).

In chapter 2, Jem again displays his childhood innocence by attempting to explain Miss Caroline's new way of teaching to Scout. However, Jem's explanation is completely wrong; he tells Scout:

"I’m just trying to tell you the new way they’re teachin‘ the first grade, stubborn. It’s the Dewey Decimal System" (Lee, 15).

As the novel progresses, Jem begins to mature and develop into an understanding, sympathetic boy. In chapter 14, he attempts to ease the tension between Scout and Aunt Alexandra by saying,

"Scout, try not to antagonize Aunty, hear" (Lee, 139). He elaborates on his reasoning by telling Scout, "Naw, it’s—he’s [Atticus] got a lot on his mind now, without us worrying him" (Lee, 139).

In the next chapter, Jem once again displays his maturity by loyally refusing to leave Atticus when the Old Sarum bunch surrounds him. When Atticus tells Jem to go home, Jem refuses to leave the scene. Scout says,

"We were accustomed to prompt, if not always cheerful acquiescence to Atticus’s instructions, but from the way he stood Jem was not thinking of budging" (Lee, 154).

Following the Tom Robinson trial, Jem loses his childhood innocence and becomes jaded with Maycomb's racism. However, he gains sympathy for innocent, defenseless creatures. In chapter 25, when Scout is about to squash a harmless bug, Jem stops his sister. When Scout asks why she is not allowed to squash him, Jem says,

"Because they don’t bother you" (Lee, 242).