Jem's interaction with Miss Dubose serves as a starting point for his maturing. At first, he approaches the situation as a child, simply correlating an unpleasant task with punishment. As he begins to understand her morphine addiction and suffering, he realizes that things are not always as they appear. He goes through a huge emotional upheaval upon her death.
In the second part of the novel, Jem begins to show huge strides in his progressing maturity, often much to the dismay of Scout. For instance, Jem understands Atticus' stress leading up to Tom Robinson's trial, despite Atticus remaining calm and collected. Jem attempts to quell Scout's childish behavior, such as her antagonistic behavior towards Aunt Alexandra and general unruliness, for Atticus's sake.
Jem shows a huge moment of maturity when he tells Atticus that Dill has been hiding under Scout's bed. He breaks a code of his childhood but does so because he is genuinely concerned for the people that care about Dill and realizes that he needs to tell an adult. Scout, still being a child, feels immensely betrayed by this.