Beyond Proving and Explaining: Proofs That Justify the Use of Definitions and Axiomatic Structures and Proofs That Illustrate Technique.

Abstract: Students' difficulties with proofs ate well documented: in particular, students typically cannot comprehend proofs, fail to 1 ealize their purpose and often consider the process ofproof to be pointless (e.g. Hare!, 1998). Tb address these issues, researchers have attempted to identify and categorize the pmposes of proof in mathematics and the mathematics classroom (e g. Hanna, 1990: Hersh, 1993) This article is my attempt to fmthet this categorization Hanna and Hersh distinguish between proofs that convince, i e proofs that an individual can use to verify that a theorem is true, and proofS that explain, i e proofs that one can use to gain an intuitive understanding of why the theorem is tme. Although proofs that convince dominate the mathematics journals, Hanna and Hersh both argue that these proofs are often not appropriate for the mathematics classroom Instead, they advocate using proofs that explain in the classroom, even at the expense of mathematical rigm Proofs that convince and proofs that explain both provide knowledge about mathematical truths, albeit in different ways They are proofs of theorems whose validity is not obvious to the reader By studying the proof, the reader gains insight into the theorem's veracity While Hanna and Hersh's distinction has been valuable, many of the proofs that are presented to students do not fall into the categoties of their taxonomy: consider proofs in axiom-based logic courses that I + I = 2 or that the sum of two even numbers is even As students never doubted these facts and have probably used them without penalty on their assignments in other courses, one cannot say that the purpose of these proofs is either to convince the student that or explain why these facts are true According to Hate! (1998), such rigorous demonstrations of seemingly obvious results frequently lead students to view the process of proof as pedantic and meaningless; some have even advocated removing these proofs from the classroom altogether (e g. Kline, 1973). I believe that students' and instructors' difficulties with these proofs stem from the fact that they do not understand the proofs' putpose The proof that I + 1 = 2 is not a proof that provides knowledge about this mathematical truth; this proof provides infmmation about the axiomatic system in which one is working and about how one can generate proofs within that system. In this article, I will describe four types of proof. I will review two types of proof that provide knowledge about mathematical truth: proofs that convince and proofs that explain Then I will describe two other types of proof ptoofs that justify the use of a definition or axiomatic structure and proofs that illustrate technique I will conclude with some pedagogical suggestions as a result of this expanded classification 1 . Proofs that provide knowledge about mathematical truth Over the next two sections, I propose fOur purposes fOr presenting proofs in the mathematics classroom I should str·ess that a decision about which category a proof belongs to is contextual and depends on the instructot 's intentions and the audience observing the proof. Proofs that provide knowledge of mathematical truths fall into two overlapping categories: proofs that convince and proofs that explain. As these types of proofs have been discussed previously in the mathematics education literature (as mentioned above), my description of them here will be brief A proof that convinces begins with an accepted set of definitions and axioms and concludes with a proposition whose validity is unknown An overview of such a proof purpose is given in Figure I The intent of this type of proof is to convince one's audience that the proposition in question is valid. By inspecting the logical progression of the proof, the individual should be convinced that the proposition being proved is indeed true