Several things happen during the novel that show that Jem is growing up. He takes less interest in his sister, childhood games and trying to make Boo Radley come out. He spends more time on his own, reading and Scout notices he's become moody and is eating more--all signs that Jem is, physically, growing up and going through puberty.
When Nathan Radley fills the knot hole of the tree with cement, it is Jem who realizes how cruel Nathan is being to Boo, and it is Jem who cries about. Jem also stands up to his father in front of the jail house and tries to protect his father. Jem is broken-hearted about the verdict of Tom Robinson's trial, but is able to have a mature, logical discussion with Atticus about what happened and what the outcome might be. Jem is the one who make the decision to tell Atticus that Dill has run away, rather than turning it into a game or hiding him from Atticus. Jem also tries to fend off Bob Ewell and protect Scout at the end of the novel.
These actions on Jem's part, rather than being more concerned about not getting in trouble with Atticus or making Boo Radley come out of his house, show that Jem is growing up and maturing.
As a sign of southern respect, Calpurnia begins to call Jem "Mister Jem." It is a rite of passage for Jem because he is growing up and becoming more of an adult. It is also a sign of respect Calpurnia, a black woman, shows to her young “employer.” Jem is now 12 years old in the story and beginning to truly understand more and more of the adult world. He has a newfound respect for his father and wants to emulate his honesty and character. Unfortunately, as Jem grows in the story, he begins to leave Scout behind and even nags her to act more like a girl and not to speak to him when he is with his pals at school. Scout even notices that Jem prefers to be by himself instead of playing games with her. She also says he is moody. When Atticus loses the Tom Robinson case, Jem’s childhood idealism is shattered by the realities of an adult, racist society. However, it is the same grown-up Jem who protects Scout and risks his own life when Bob Ewell attacks them.
“Mister” Jem shows how he is coming of age in the adult world and, therefore, is leaving his childhood behind.