Being older than Scout, Jem tries to explain many things to Scout including the Dewey Decimal System and entailments. He doesn't report them correctly which reveals his immaturity:
When I asked Jem what entailment was, and Jem described it as a condition of having your tail in a crack, I asked Atticus if Mr. Cunningham would ever pay us.
Jem is turning into a sort of parent to Scout as most older siblings do. He is learning to help dictate appropriate behavior which he does in the case of Walter Cunningham's altercation with Scout:
I stomped at him to chase him away, but Jem put out his hand and stopped me. He examined Walter with an air of speculation. “Your daddy Mr. Walter Cunningham from Old Sarum?”
Jem is growing increasingly boastful and seemingly brave:
Jem seemed to have little fear of Boo Radley now that Walter and I walked beside him. Indeed, Jem grew boastful: “I went all the way up to the house once,” he said to Walter.
Jem is a young man bent on learning by experience, he is also an inquisitive mind that will spend time on his own just thinking:
Atticus kept us in fits that evening, gravely reading columns of print about a man who sat on a flagpole for no discernible reason, which was reason enough for Jem to spend the following Saturday aloft in the treehouse. Jem sat from after breakfast until sunset and would have remained overnight had not Atticus severed his supply lines. I had spent most of the day climbing up and down, running errands for him, providing him with literature, nourishment and water, and was carrying him blankets for the night when Atticus said if I paid no attention to him, Jem would come down. Atticus was right.
Jem is a growing young man with lots of room to go. But he certainly approaches his world and relationship with Scout from the perspective that he's working on doing right and thinking about how things should be done.