This article uses a classroom episode in which a teacher and her students undertake a task of proving a proposition about angles as a context for analyzing what is involved in the teacher's work of engaging students in producing a proof. The analysis invokes theoretical notions of didactical contract and double bind to uncover and explain conflicting demands that the practice of assigning two-column proofs imposes on high school teachers. Two aspects of the work of teaching—what teachers do to create a task in which students can produce a proof and what teachers do to get students to prove a proposition—are the focus of the analysis of the episode. This analysis supports the argument that the traditional custom of engaging students in doing formal, two-column proofs places contradictory demands on the teacher regarding how the ideas for a proof will be developed. Recognizing these contradictory demands clarifies why the teacher in the analyzed episode ends up suggesting the key ideas for the proof. The analysis, coupled with current recommendations about the role of proof in school mathematics, suggests that it is advantageous for teachers to avoid treating proof only as a formal process.
Patricio G. Herbst, 4202E School of Education, University of Michigan, 610 East University Avenue, Ann Arbor, MI 48109-1259; email@example.com