Dill challenges Jem's bravery by daring him to get Boo Radley to come out of the house in chapter one. After three days of strategizing, Dill modifies the challenge for Jem by telling him he just has to touch the Radleys' front door. Relieved that Jem can now demonstrate his courage more easily, he does it quickly and successfully. For the most part, Jem's understanding of courage is that it should be a demonstration of manliness. For example, Jem tells Scout the following after she leaves a tire in the Radleys' front yard and he must fetch and return with it in chapter 4:
"Nothin' to it. I swear, Scout, sometimes you act so much like a girl it's mortifyin'" (38).
Therefore, according to Jem, acting like a girl is a sign of cowardice, not courage. Additionally, both Scout and Jem compare Atticus with other fathers in town. They think that their dad is boring and old, not courageous. Scout explains her father as follows:
"Atticus was feeble: he was nearly fifty. When Jem and I asked him why he was so old, he said he got started late, which we felt reflected upon his abilities and manliness" (89).
The word "manliness" seems to be tied to the children's perception of courage. However, when Atticus takes down a mad dog with one rifle shot in chapter 10, their attitudes about him change. This event prepares them for their encounters with Mrs. Dubose in chapter 11. This is when Jem learns that losing his temper does not demonstrate courage. Also, chopping off all the tops of an old lady's camellia bushes isn't manly, either. Doing that only shows a lack of self-control. Jem does learn the following from Atticus about Mrs. Dubose's courage, though:
"I wanted you to see something about her--I wanted you to see what real courage is, instead of getting the idea that courage is a man with a gun in his hand. It's when you know you're licked before you begin but you begin anyway and you see it through no matter what . . . Mrs. Dubose won, all ninety-eight pounds of her. According to her views, she died beholden to nothing and nobody. She was the bravest person I ever knew" (112).
Atticus teaches Jem that patience when dealing with other people's differences can be courageous just like Mrs. Dubose was patient and committed to overcoming her addiction to pain pills. Therefore, Jem learns that not allowing others to challenge your resolve is another way to show courage. Courage is not necessarily a demonstration of manliness or bravery as much as it is living and dying "beholden to nothing and nobody."
Jem has changed by finally realizing that a man holding a gun does not constitute courageousness, rather a feeble 98lb woman on her death bed, who wants to leave this world "beholden to nothing and nobody."
He questions Atticus about why he defends those who seem to be guilty, and those who everyone believes to be guilty, only to be told that courage is, "knowing you're licked before you begin but you begin anyway and you see it through no matter what."
He starts off the narrow minded child, questioning everything but develops into this more mature character by the end.. review the bets made between Dill, Scout and Jem about slapping the Radley house, going back for his pants and cutting off Mrs. Dubose's azalea bushes. Those are all points that discuss courage.