Scout matures in the first part of the novel because she learns empathy, or looking at things from another person’s point of view.
During Part One, Scout’s journey is a long one. Before the serious part of the book even starts, Scout is preparing for the tumultuous times ahead. Atticus knows that Scout will be tested. He tries to teach her what she will need to know. The main aspect of this is empathy. Scout has difficulty putting herself in someone else’s place.
When Scout starts school, she symbolically enters the real world. As she passes this threshold, she faces her first test in her conflict with her teacher Miss Caroline.
"[If] you can learn a simple trick, Scout, you'll get along a lot better with all kinds of folks. You never really understand a person until you consider things from his point of view-until you climb into his skin and walk around in it." (ch 3)
Atticus is trying to tell Scout that to understand someone you have to see things from their perspective. Scout has trouble doing this at first. She struggles to understand Jem as he gets older. Part of the sign of her progress is that she tries to understand her brother, and realizes that she does not.
Sometimes I did not understand him, but my periods of bewilderment were short-lived. This was beyond me. (ch 6)
Scout does not fully understand other people at a deep level until the end of the book, but by the end of part one she has made some real progress. She has demonstrated one of the key themes of the book: you can never really understand a person until you walk a mile in their shoes.