How do Scout and Jem's way they think of their father change from the beginning of Chapter 10 to the end of Chapter 10 of "To Kill a Mockingbird"?Why?

Well, look at the very first sentence of the chapter to ascertain some of their feelings regarding Atticus before the gun incident.  It states, "Atticus was feeble."  So, they thought he was old and lame (literally and figuratively).  Scout goes on to clarify that she and Jem thought of his older age that it "reflected upon his abilities and manliness."  They felt he was not much of a man, and had abilities that weren't as cool as their friends' fathers.  Scout says that "there was nothing Jem or I could say" about Atticus when their peers bragged about all of the cool things their fathers did.  Atticus worked, read, wore glasses, and "didn't do anything" exciting or dangerous.

The main event that changes their attitudes towards Atticus is him killing old Tim Johnson, the rabid dog, in one clean and effortless shot with Heck Tate's gun.  That is completely awesome in their eyes.  First of all, they didn't even know Atticus could shoot; he refused to teach them to shoot their air-rifles, and didn't have a gun that they knew of.  So, to see their dad walk out into the road, toss his glasses aside, and swiftly and efficiently shoot a dangerous dog, it was quite a shock.  It was a pleasant one though.  Jem and Scout realize that their dad really IS cool, and CAN do something awesome.  Scout is so excited by this fact that she tells Jem that they'd "really have something to talk about at school" now!  But Jem, respecting his dad's wishes to keep his talent on the hush, tells her not to tell anyone, because if he'd wanted people knowing, he would have said something.  Jem then declares, with pride, "Atticus is a gentleman, just like me!"  In this chapter we see their attitudes turning from shame and embarrassment to pride and respect.