PERSISTENCE AND SELF-EFFICACY IN PROOF CONSTRUCTION Annie Selden and John Selden New Mexico State University, USA We first discuss our perspective and three useful actions in proof construction that depend on persistence. Persistence is important for successful proving because it allows one to “explore”, including making arguments in directions of unknown value, until one ultimately makes progress. Persistence can be supported by a self- efficacy belief, which is “a person’s belief in his or her ability t o succeed in a particular situation” (Bandura, 1995). We discuss a study of U.K. undergraduates’ perceived sense of self-efficacy with regard to proving (Iannone & Inglis, 2010). We then examine actions needed for a successful proof construction of a theorem given to mid-level U.S. undergraduates in a transition-to-proof course. We contrast those actions with the actual actions of a mathematician proving the same theorem. Key words: proof construction, persistence, self-efficacy, undergraduates, mathematicians. INTRODUCTION In this paper we first discuss our perspective on proof construction. Then, drawing on observations from a multi-year teaching experiment, point out three aspects of proof construction that appear to be especially difficult to teach. We suggest that the teaching difficulty arises from a need for students to have a kind of persistence, which in turn may depend on students’ sense of self -efficacy. After discussing self- efficacy, we consider a prior study of U.K. undergraduates’ performance and perceived sense of self-efficacy with regard to proving(Iannone & Inglis, 2010). We next illustrate the way that self-efficacy and persistence are valuable by discussing the proof construction of a specific theorem that students in the teaching experiment are asked to construct and how one mathematician approached proving it. Then, after a brief discussion, we end with some teaching implications. OUR PERSPECTIVE We view constructing a proof as a sequence of actions (Selden, McKee, & Selden, 2010). Some of these are physical, such as writing a line of the proof, and some are mental, such as focusing on the conclusion or “unpacking” its meaning. Such actions are taken in response to certain kinds of situations in a partly constructed proof. With practice, the links between some repeatedly occurring proving situations and the resultant actions will become automated, thus reducing the burden on working memory in future proof construction. Many such actions taken during the construction of a proof are not recorded in the final written proof. Thus, it may be difficult to mimic a given proof, or even parts of it, when constructing another proof.