The answer to this question can be found in an early conversation that happens between Blanche DuBois and Stanley Kowalski. Stanley actually states that Blanche is a teacher, and then he asks for her confirmation on that statement. Blanche does confirm with a single word that she is a teacher, and then Stanley asks a follow-up question. He wants to know what subject Blanche teaches. Blanche again answers with a single word. She tells Stanley that she teaches English, and Stanley makes a self-deprecating comment about the fact that he was never that solid of an English student. It should be fairly clear to audiences that Blanche doesn't want to talk about her job. Stanley picks up on this, and he moves the conversation in a new direction. It makes sense that she doesn't want to talk about it: she was dismissed from her job for an inappropriate relationship with a student.

Blanche is one of two main characters in Tennessee Williams' play A Streetcar Named Desire, which debuted in 1949. In the play, Blanche is an English teacher, though she leaves her position when the play begins and moves to New Orleans to live with her sister, Stella, and her sister's husband, Stanley Kowalski. Blanche claims that she left her job as a teacher because of her "nerves." However, it is revealed later in the play that Blanche was actually dismissed from her teaching job after having an affair with a student.  

Blanche's profession is an important part of her characterization. Blanche uses poetic, lyrical, and over-wrought language, which places her in sharp contrast to Stanley, her antagonist, a Polish immigrant who speaks with coarse language. The combination of their dialects -- Blanche's poeticism and Stanley's grittiness -- are important components of the style of "lyrical realism" that Tennessee Williams pioneered in his plays.