Chapter 11 of To Kill a Mockingbird focuses primarily on the Finch children's ongoing feud with Mrs. Dubose, the virulently racist elderly woman whose verbal abuse of Jem and Scout on account of their father's decision to defend an African American man accused of rape is about to reach its boiling point. It is within the context of Mrs. Dubose's insults that Lee's young narrator, Scout, relates the story of her baton and the misuse to which it would be put. Jem is given money for his twelfth birthday and is eager to spend it, specifically, on a miniature steam engine. With the surplus money he anticipated, he would buy Scout a baton that his sister had had her eye on—a baton "bedecked with sequins and tinsel" that she hopes to use when she is older and can join the high school marching band. Scout appreciates the gesture, noting that "I thought it generous of Jem to buy one for me."
Unfortunately for Scout, and for Mrs. Dubose and Jem, the latter grabs the baton and, in a fit of fury, uses it to destroy the old woman's prized shrubs. This prompts a lecture from Atticus about compassion, and a punishment that involves sitting with Mrs. Dubose in her home keeping her company and reading to her. Adding insult to injury, Jem's temporary insanity results in the baton's destruction, as described by Scout:
He did not begin to calm down until he had cut the tops off every camellia bush Mrs. Dubose owned, until the ground was littered with green buds and leaves. He bent my baton against his knee, snapped it in two and threw it down.
So, the item that Jem bought for Scout was a baton the young girl hoped to use when she grew up and joined the high school marching band. She does not, however, have the baton for long because of her brother's assault on Mrs. Dubose's garden.