What does Atticus do in court that the children never saw him do even at home?
In court, Atticus loosens his tie and collar before giving his closing argument. "He unbuttoned his vest, unbuttoned his collar, loosened his tie, and took off his coat." Usually when someone does this, it indicates fatigue or exhaustion. However, Atticus may be doing this instead to give himself a more casual look, which may appeal more to the jury and the audience.
During Atticus's cross-examination of Mayella, he calls her "Miss Mayella" or "ma'am." Mayella, who is lower class, believes that he is mocking her when he does this. She lashes out at him angrily during her testimony because she thinks that he is trying to make fun of her: Your fancy airs don't come to nothin'—your ma'amin' and Miss Mayellerin' don't come to nothin', Mr. Finch!"
Atticus is characterized by his stoicism and professionalism. However, at the end of the trial, he needs to show the jury that he is also a simple man, one of them. His formal language and attitude distances himself from the jury, audience, and witnesses, and so in order to get their attention, he has to "loosen up."