How does Atticus stand by values that he tries to impart to his children?

Atticus not only tries to teach his children fairness, he sees racism as a kind of "madness" that takes over the citizens of Maycomb.  Atticus represents reason in the face of that madness and shows this reason both realistically and symbolically.  In the scene at the jail when he confronts the lynch mob, it is symbolically significant that Atticus takes with him a light buld (the light of reason) and a newspaper (words rather than weapons), but he does not take a gun.  He uses the power of reason--with the help of his daughter--to quell the angry mob, and it is Scout's singling out of a single man, Mr. Cunningham, that makes the crowd dissipate.  Similarly, when Atticus shoots the mad dog, Tim Johnson, a rabid dog, we see a foreshadowing of the courtroom fight between reason and madness.  Atticus represents reason against the madness symbolized by the mad dog.  Atticus requires Jem to read (logic, reason) to Mrs. Dubose to apologize for the frenzy--a kind of madness--that led Jem to cut off the tops of Mrs. Dubose's flower.  Early in the novel Atticus reminds Scout to "fight with [her] head" as opposed to her fists--reason against madness again.

The real contest between reason and madness, however--which all these other events lead up to--is the trial itself.  With calm demeanor and reasoning, Atticus effectively proves that Tom Robinson, whose right hand was damaged in a cotton gin and is useless, could NOT have beaten and raped Mayella Ewell.  He shows that Mayella was beaten by someone who led exclusively with his left, thus casting suspcion on her father, Bob Ewell. Jean Louise (Scout), the adult narrator, lets us know that Atticus made talking about rape sound as dry and matter-of-fact as legal briefs, thus calming the rage, gossip, and heightened emotions that seemed destined to propel this case.  Despite the guilty verdict, reason has made a significant step forward, and we see this in the length of time the jury is out: in most such cases, the jury would return immediately with a guilty verdict, but Atticus has made the jury think and has paved the way for justice in other trials.  He truly believes there is a good chance for an appeal until Tom runs and is shot, UNreasonably, 17 times.

This conflict between reason and madness does not end with the trial or even with Tom's death.  Atticus continues to model his calm, non-violent beliefs when Bob Ewell spits at him.  Rather than fighting back or even acting outraged, Atticus calmly removes his glasses and wipes them off--the epitome of "turning the other cheek" in a moment of Christlike charity.  He later says that if letting Ewell spit at him saves Mayella another beating, it is worth it.

As the novel ends, Atticus worries that he is not standing up for his values if he hides the truth about Ewell's encounter with his children.  He actually thinks Jem has killed Ewell, but Sheriff Tate convinces him that bringing the story out in the open would be hurtful to Boo Radley, who has saved the children's lives.  Scout, demonstrating that she has truly learned the values Atticus has tried to impart, states that bringing Boo's actions into the limelight would be like "killing a mockingbird."  She learns about civility, kindness, nonviolence, and reason from Atticus.