One of the most prominent literary techniques used to show Jem's character development is irony. Because the story is told from Jem's younger sister, Scout, Harper Lee does an excellent job of showing how Jem is growing up and maturing faster than his sister through Scout's frustration with him, and the distance that grows in the playful friendship from the beginning of the story to the end.
Consider first, the big picture. At the beginning of the story, Jem is the impetus for keeping boredom at bay. He creates stories about Boo Radley, engages Scout and Dill in the "Boo Radley game," and tests his own bravery by daring the others and himself to do things which scare them. He and Scout fight a lot about childish things. By the end of the story, however, Jem is the one who protects Scout on the long journey home from the school pageant. He grows from playmate and rival to protective older brother.
Then, consider the details of how this change takes place, from Scout's point of view. All she knows is that her brother is acting differently (and in her mind acting like he is superior), playing less, and becoming more and more annoying:
Running after it he called back, "Atticus is a gentleman, just like me!" (Chapter 10)
Jem was twelve. He was difficult to live with, inconsistent, moody. His appetite was appalling, and he told me so many times to stop pestering him I consulted Atticus. (Chapter 12)
His maddening superiority was unbearable these days. (Chapter 14)
The irony in using a very limited first person point of view is humorous because as an audience, we see that much of Scout's confusion is due to what she cannot yet understand. In many ways, the changes she speaks about in Jem are likely also happening to her, but she is unaware of them. It is also ironic (and therefore funny) that the things Scout finds most annoying about Jem's growing up are all things that a grown up would welcome. His sense of independence, his quiet attitude and thoughtfulness, and his desire to flee childhood would only be annoying to another child.