Scout matures by the end of part 1 because she has learned to walk away from fights and control her temper.
Scout has a serious temper and a tendency to let fists fly instead of talk things out. Atticus tries to explain to her that fighting is not the answer, but she is too immature.
Atticus’s defense of Tom Robinson causes difficulties for Scout, because she has become a target of both kids and adults.
Atticus had promised me he would wear me out if he ever heard of me fighting any more; I was far too old and too big for such childish things, and the sooner I learned to hold in, the better off everybody would be. I soon forgot. (ch 9)
Despite her irritation and what Cecil Jacobs says about her father, Scout does not run his nose in the ground. Instead, she realizes she would disappoint her father if she fights, and she decides to walk away.
I drew a bead on him, remembered what Atticus had said, then dropped my fists and walked away, "Scout's a cow- ward!" ringing in my ears. It was the first time I ever walked away from a fight. (ch 9)
For Scout to walk away shows enormous personal growth. Atticus explained to her that people would say things she would not like, but she had to be strong. She took in the lesson and began to develop self-control by the end of part one. This would prove useful, since her family was only at the beginning of their troubles.
To Kill a Mockingbird is a coming of age story, and one of the biggest ways Scout grows up is in her ability to walk away from a fight.