What problems does Billy encounter upon entering the second grade in the novel, The Year of Billy Miller?

In second grade, Billy encounters his share of problems, none of which have to do with his intelligence. There is Emma Sparks, a bossy girl who behaves more like an adult than a little kid, who makes fun of Billy and says snide things to him throughout the novel. Billy holds two red markers up to his head and makes a devil face at her, but he is seen by Mrs. Silver, the teacher, who frowns. Billy worries that Mrs. Silver believes he was making fun of the red chopsticks she wears in her hair, and that he is not nice. He gives her a few silver-colored gifts, such as a paperclip and nail, which she accepts, telling Billy that she knows he is nice.