Jem changes in the course of the novel. How does Harper Lee show this?
To Kill a Mockingbird, though narrated by the young Scout, is really Jem's story of growing up during the difficult events in Maycomb. He is one of the kids for the early part of the novel, playing silly games and acting out silly plays with Scout and Dill. Then several events kind of set him apart from the younger two. One is his overalls being repaired by, apparently, Boo Radley. Jem is clearly shaken by the incident, and it begins his journey of awareness of life beyond his own front porch. Next is his episode with Mrs. Dubose. He sees her as nothing but a bitter, ranting woman, until Atticus forces him to do some time at her bedside and eventually explains the true courage Jem had witnessed firsthand as Mrs. Dubose kicked her morphine addiction. Another life lesson.
When the trial comes, Jem is intent on the legal issues and can't wait to get to the trial. As the events and evidence unfold, Jem is positive the jury can do nothing short of acquitting Jim. He understands what has happened here (unlike Scout), so he is crushed, literally, when the guilty verdict is returned. The morning after the trial is over, Jem is talking with Miss Maudie, and she pays him the respect of speaking to him as if her were an adult. This shift in thinking is solidified and symbolized when Jem no longer gets his own mini cake but gets a slice of the "adult" cake instead.
Scout and Jem start out in this novel as children; by the end, though, Jem has seen a little more of life's reality probably more than anyone would have liked. Though Scout tells the story as she sees it, this is the story of Jem's journey into manhood.