The plot revolves around Sylvia Barrett, an idealistic English language teacher at an inner-city high school who hopes to nurture her students' interest in classic literature (especially Chaucer and writing). She quickly becomes discouraged during her first year of teaching, frustrated by bureaucracy, the indifference of her students, and the incompetence of many of her colleagues. The title of the book is taken from a memo telling her why a student was being punished: he had gone "up the down staircase". She decides to leave the public school (government funded) system to work in a smaller private setting. She changes her mind, though, when she realizes that she has, indeed, touched the lives of her students.
The novel is epistolary; aside from opening and closing chapters consisting entirely of dialogue the story is told through memos from the office, fragments of notes dropped in the trash can, essays handed in to be graded, lesson plans, suggestions dropped in the class suggestion box, and most often by inter-classroom notes that are a dialogue between Sylvia and an older teacher. Sylvia also writes letters to a friend from college who chose to get married and start a family rather than pursuing a career. The letters serve as a recap and summary of key events in the book, and offer a portrait of women's roles and responsibilities in American society in the mid-1960s.
An inter-classroom note in which the older teacher is translating the jargon of the memos from the office includes the memorable epigram "'Let it be a challenge to you' means you're stuck with it." Calling a trash can a circular file comes from the same memo: "'Keep on file in numerical order' means throw in waste-basket."

Question: What sort of school does Sylvia Barrett teach at?
Answer: High school.
Alternative answer: an inner-city public high school