Satire's mission is to expose vice and folly in both individuals and institutions (religious, political, social, etc.). Satire uses irony, hyperbole, and incongruity as tactics to expose weaknesses. How does The Prime of Miss Jean Brodie excoriate using these strategies?
First, look at Miss Brodie herself. She's a maverick (writes her own curriculum and defies the standard) and fearless (she has affairs with various men, flying in the face of morals for nineteen-thirties Scotland). She also is two-faced: she believes that not having an affair with Lloyd is honorable but adjusts her rules later. Is Spark saying that Miss Brodie is full of folly? Is she making huge mistakes? You can examine what Miss Brodie says about herself, what other characters say about her, and how consequences fall out by the very end of the play. What ironic twists occur? How does the unexpected drive action in this play?
One consequence of Miss Brodie defying the standard curriculum is her conflict with the headmistress. These are two opposite, incongruous women. So who wins? Is the standard curriculum wrong? Did Miss Brodie do right to challenge it? If you believe so, and find enough evidence to prove, you could argue that the novel satirizes the knowledge students were required to master at that time. Or you may wish to argue that it is Miss Brodie Spark is exposing as a fraud and wrong.
Critic Kelly praises Miss Brodie in some respects for taking on a stody, limited historical era, saying that Miss Brodie's romantic idealism is admirable. Read this essay to see how Miss Brodie's approach is complex and how you could fault both her and the educational system she teaches in. There are some good lines to quote. Also read the historical context notes to see that in some ways, you can view Miss Brodie as emblematic of the political movements of her time. Her fascination with fascism illustrates that she is ignoring the evil in Mussolini and Hitler's encroachments and instead celebrating the romantic ideals (Aryan and ancient Greek mythology -- the idea that certain Europeans descended from the "best of" cultures, while the rest were expendable). How does Miss Brodie leave a trail of bodies behind her -- not as many as dictators, but dangerous nonetheless? Miss Brodie is totalitarian in her running of the classroom. Might she be a mini-Mussolini in her own tiny domain? For a character to be a subject of satire, she must represent qualities that are bad in several people -- a general human trend. Is Miss Brodie an exaggerated, hyperbolic example -- narcissism run amok?
Let's say the curriculum is bad; is Miss Brodie right to challenge it the way she did? Are her means a subject of satire? You can easily prove Miss Brodie made some major mistakes. She tends to use students and lovers as pawns in her personal game (Mr. Lowther as a romantic partner, and asking her students to engage in lies or keep secrets.)Likewise, the lothario Mr. Lloyd and his abuses are a likely subject of condemnation.
One element of satire not present is a distanced narrative tone. Each character is complex, and no character is so flat you can't understand some of his or her perspective. The play's tone is not as distant, mocking, and condemning as, say, a Swift or Voltaire piece might be.
Why call it The Prime of Miss Jean Brodie? Any irony there? Any hyperbole and incongruity? The title says much!