I remember Jem showing immaturity by destroying Mrs. Dubose’s camellias out of his own anger when he was eleven years old (Ch. 11) ...
I remember Jem learned from Atticus and his gentlemanly qualities of patience and understanding when he was eleven (Ch. 11)...
Jem waited, per his sister’s request, until the school auditorium was near empty for embarrassed Scout to exit in her costume.
Now nearly thirteen, Jem showed growth by, in the end, learning to put others before himself (and in these examples, specifically, his sister, Scout). In the beginning, Jem would rough his sister around and make her follow after him instead of letting her go first. However, by the end of the novel on Halloween night, after the Maycomb County pageant...
...and when Atticus made Jem frustrated, so he impulsively decided not to follow in his father’s footsteps to become a lawyer like he’d always wanted to, when he was ten (Ch. 5).
...and during the trial chapters, Atticus’s determination and want of justice for all inspires twelve-year-old Jem to follow in such admirable footsteps.
Run, Scout! Run! Run!
In addition, during their encounter with Mr. Ewell, Jem made sure to protect Scout from their attacker and send her off to safety before him, so he knew she would get away safely.