Chapter 10 is an interesting chapter because it shows a growth of maturity in the way the children view their father. The chapter starts with them noting Atticus is older then the other fathers and does nothing that could be admired by others. There is even a hint that they are ashamed of this.
This changes when Atticus shoots the mad dog. They find out that he is a sharp shooter and was infamous in the county as one-shot Finch. They were now proud of their father.
Jem is confused that his father has kept this a secret and at first wants to know why Atticus would not be proud and brag about this ability. Scout immediately wants to run around and tell everyone but Jem realizes it is best to respect Atticus wishes and not do this.
There are two levels of maturity here. Both realize there is more to their father then meets the eye and realize he is a person with a past and see him in a new light. Jem, has even a stronger realization that there is no need to brag to be proud and being a gentleman and doing the right thing is more important.
It is representative of children maturing and seeing their parents as human. Jem even wants to be just like his father which is a change from the start of the chapter.
Jem and Scout's view of their father changes during the events of chapter ten. When they learn that their "feeble" older father was once known as "One-Shot Finch," they gain a new sense of respect for him. They realize that their father is not inept at the kinds of activities that children admire, he just does not value them. He would rather distinguish himself with gentleness, kindness, and by defending the innocent rather than by being good with a gun.
When Scout and Jem see Atticus kill Tim Johnson (a rabid dog) with one shot, Scout can't wait to brag to their school friends about it. Jem, on the other hand, realizes that they must keep quiet--this is not a part of Atticus's life that he is proud of. At the end of the chapter, Jem admits he wants to be like his father when he shouts, "Atticus is a gentleman, just like me!"