Miss Caroline is present to provide us with an outsider's view of the town of Maycomb, as opposed to the insider's view we get from Scout. She does not understand the class situation, such as Walter Cunningham's pride or the Ewells, who come to the first day of school every year and then stay home the rest of the time.
Miss Caroline's greatest mistake is misunderstanding that it she who is the oddity, for she is an "outsider," and, in a tight Southern small town being an outlander, as the Easterners call it, is a clear social disadvantage. She should have thanked Scout for her assistance, and then she would have ingratiated herself with the community since they hold Atticus Finch as one of their more prestigious citizens.
[Remember Scout's mention that Miss Caroline was from Winston County--a county of Northern Alabama that was sympathetic to the North in the Civil War. Thus, Winston County and its inhabitants were in great disfavor in Southern Alabama, whose roots are very Confederate.
I, too, must agree with bullgatortail. Miss Caroline shames Scout and belittles her father. Neither of these things would fly today. One could assume that Miss Caroline's ego simply got into the way of her dealing with Scout. Today, a reader such as Scout would be praised. The biggest problem is that Miss Caroline's attitude impacts Scout so much that Scout's love for school changes.
I agree with bullgatortail. The core of Miss Caroline's mistake is her ignorance and her ego. She comes into a community with very little understanding of their ways, and expects that she will be able to reach them and educate them. She knows so little of human nature, that she does not realize the value of a first grader who can read and what devestation she perpetrated upon Scout for condemning this.
Miss Caroline's biggest mistakes were making Scout feel ashamed and inferior because she could read so well; and declaring that Atticus--perhaps the wisest and most intelligent man in Maycomb--"does not know how to teach." Miss Caroline's swelled head over what she thought was her modern education and superior intelligence deflated considerably by the end of her first day.
One could argue that Miss Caroline should have researched the area and the students she was to teach before school started. The conflict with Walter might have been avoided if she understood the poverty level and the pride of the students she was teaching.
While I agree that criticizing Atticus was a mistake, I think the bigger mistake was telling Scout to stop reading at home. I can't imagine a teacher telling an advanced child they need to stop learning. Miss Caroline only knew one way to teach the students to read. If Scout was already reading, she didn't know what to do. It seems silly for Miss Caroline to want Scout to stop reading at home so she can be taught to read at school.
Miss Caroline really wasn't a good fit for the class she was sent to teach. She wasn't in tune with them culturally. She made a mistake because of this when she tried to get Walter to take a quarter for his lunch. That would embarass him but she doesn't realize it. She makes a differnt sort of mistake by criticizing Atticus to Scout. This seems extremely improper for a first grade teacher.