Promoting Metalinguistic and Metamathematical Reasoning in Proof-Oriented Mathematics Courses: a Method and a Framework

Abstract

Students’ apprenticeship into proof-oriented mathematical practice requires that they become aware of a range of conventions and assumptions about mathematical language, reference, and inference that rarely appear as explicit components of the content of undergraduate courses. Because these matters of interpretation are often pre-conscious for students and taken for granted by instructors, students may fail to apprehend the problems at hand and thus misunderstand the mathematical community’s solutions. So, we argue that proof-oriented instruction must simultaneously help students understand problems of ambiguity and reference – which requires them to engage in what we call metalinguistic and metamathematical reasoning – as well as provide some informal language for discussing possible solutions to these problems. We provide three illustrative episodes from real analysis classrooms to highlight our common way of framing this class of learning challenge and the pedagogical difficulties they entail. In each case, we highlight the use of analogies as a method of helping students apprehend a problem of mathematical language, reference, or inference and discuss possible solutions. This paper’s contributions are thus both practical and theoretical: the instructional method is intended to serve as one possible approach to training students in mathematical conventions and assumptions, and it helps demonstrate our theoretical perspective on this class of teaching and learning phenomena.