As you look at the concept of characterization in Jem Finch, I think you could actually use the idea that his bravery proves he has become more mature.
Look at the change in Jem from the beginning of the book to chapter 28. At the beginning of the story, Jem is the propagator of Boo Radley as a "haint," or "ghost." He is the one who teaches Scout and Dill about "hot steams." One of the clearest indicators at the beginning of the story, however, of Jem's immaturity, comes in chapter 6 when he goes to the Radley place to get a glimpse of Boo at night. When Scout is apprehensive he is exasperated with her:
I declare to the Lord you're gettin' more like a girl every day!
By the end of the story, however, Jem's sense of maturity displays itself in the form of bravery and protection of his sister. As the two walk home alone from the school pageant at night, Jem hears something rustling behind them. Scout, positive Jem is just trying to scare her, attempts to call his bluff. She knows he is serious, however, with this:
"Be quiet," he said, and I knew he was not joking.
Within one year, Jem has gone from an immature boy who tells ghost stories to trick and scare the little sister who annoys him. By the end of the story, however, he is brave enough in the face of real danger to protect her. The rest of this climactic scene, as you know, includes Jem fighting with the unknown attacker and telling Scout to run. His bravery here directly proves his maturity.