How Do You Know that You Know? Making Believe in Mathematics

Abstract

Knowing in a discipline means not only knowing its products, and their uses, but also about the genesis of its knowledge, how that knowledge is warranted, and how contrary views are reconciled. This shapes the entailments of teaching and learning of that discipline whether at age 8, 18, or 80. This lecture will focus on the discipline of mathematics, illustrating what mathematicians do when they seek to “prove” a claim, taking note of the challenges presented by the advent of computer-based proofs. Proving is a fundamental mathematical practice, learned by few students, and then only late in their education. I will consider how proving could shape the early learning of mathematics, including the use of “generic” proofs. Examples from elementary classrooms will illustrate why the warrants for shared knowledge in a field matters for the curriculum at any point.