From Harper Lee's To Kill a Mockingbird, one thing we learn about Jem Finch is that he's more levelheaded than his younger sister Scout. Unlike Scout, Jem does not engage in fist fights every chance he gets. We witness Jem's levelheadedness early in Chapter 3, just as Scout is attacking Walter Cunningham, blaming him for her having gotten into trouble in school. But instead of letting Scout attack Walter, Jem puts a stop to it, as we see in Scout's narration:
Catching Walter Cunningham in the schoolyard gave me some pleasure, but when I was rubbing his nose in the dirt Jem came by and told me to stop. "You're bigger 'n he is," he said ... "Let him go, Scout." (Ch. 3)
Jem's understanding that Scout shouldn't attack someone who is smaller than she is, or attack anyone at all, certainly shows that Jem is much calmer and more levelheaded than Scout.
However, boys will be boys, and, as a result of his youth, Jem gets involved in a game with Dill and Scout during the summer in which they reenact the rumors of Boo Radley's life. Later, though, he shows a youthful attempt at compassion by trying to pass Boo Radley a letter in which, as Jem phrases it, they ask Boo "real politely to come out sometimes, and tell us what he does in there--we said we wouldn't hurt him and we'd buy him an ice cream" (Ch. 5). Though the letter is not the ideal neighborly invitation, it's clear Jem's heart is in the right place for, as Jem explains, he thought Boo would feel better if he came out and sat with the kids for a while. Jem later tells his father that he wasn't intentionally making fun of Boo, which is believable. Jem was beginning to understand that Boo was trying to interact with the children through the gifts he left in the knothole in the tree in his property, and Jem honestly wanted to induce Boo to come out into the world for a genuine neighborly exchange. However, due to his age, Jem handles his desires poorly and performs activities that come off as making fun of Boo.
Hence, all in all, two things we learn about Jem is that he is fairly levelheaded for his age and feels a genuine sense of compassion, like his father, but, due to his age, Jem demonstrates his compassion poorly at times.