Answer:
At the beginning of the book, Jem is still very much a child. He enjoys playing make-believe games with his Scout and Dill. He fears Boo Radley. He is a sensitive and intelligent boy, but at age ten, he is still a boy. Jem ages from 10 to 13 over the course of To Kill a Mockingbird, a period of great change in any child's life. Jem is no exception to this rule. Interestingly, the changes he undergoes are seen from the point-of-view of a younger sister, which gives a unique perspective on his growth.
Like Scout, Jem has to grow up during the Tom Robinson trail. He struggles to find himself and his place in the world. For example, when Jem runs away from the Radley house and loses his pants, he does not want to tell Atticus where they are. He is not afraid of being punished, worrying instead that Atticus will think badly of him. But he grows up from that.
Over the course of the novel, Jem's biggest change comes from his transition from a young boy into a young man. When we first meet Jem he is ten.... by the end of the novel, he is thirteen. From the start, Jem and Scout are best friends. When Dill arrives in Maycomb, all three hang out and play together. As the novel progresses, Jem starts to brush his sister off, noting that she's a girl, and sometimes even encouraging her to act like one.
After the trial begins, Jem grows up even more. He begins to understand injustice, disappointment, and ill intent. Jem learns that right doesn't always win, and that good people sometimes do bad things. He acquires a deep respect for his father, a respect far beyond Atticus as a father.... he respects him as a man.