In To Kill a Mockingbird, Jem does not want to interact with his younger sister Scout when they are at school. He even tells her this on the first day when they walk to school together. He tells her that she is “not to bother him” at school, which includes not approaching “him with requests to enact a chapter of Tarzan and the Ant Men” or other requests.
Even though Scout and Jem spend much of their time together, it is important to keep in mind that Jem is four years older than Scout. This means that he does not want to seem babyish in front of classmates his age by playing with her at school. In other words, like many (or probably most) older siblings, Jem does not want Scout to embarrass him in front of his peers. In fact, he tells her that she is not to “embarrass him with references to his private life…”
It is clear that Jem loves Scout and is also extremely proud of her. When the two Finch children first meet Dill, Jem brags that Scout has been able to read since an early age even before school, saying, “Scout yonder’s been readin‘ ever since she was born, and she ain’t even started to school yet.”
However, because Jem is probably concerned about appearing babyish in the eyes of his classmates, he does not want her to “tag along behind him at recess and noon.” He wants Scout to stay with other children her age when they are at school, or to “stick with the first grade,” while him stayed with children his age.
Moreover, it is not just the difference in their ages that makes Jem feel awkward about spending time with Scout in school. Scout is a tomboy who does not always understand social situations. We see this in many scenes in the book. Her lack of social awareness impacts her interaction with other children in school. For instance, when Scout attacks Walter Cunningham in the schoolyard, Jem comes to his defense. He tells Walter, “Scout here, she’s crazy—she won’t fight you any more,” and invites the boy to join them at their home for lunch.
Nevertheless, Jem loves Scout. He assures her that regardless of the new rules that govern their relationship when they are in school, they can still play together at home “but …school’s different.”