Expert Cognition During Proof Construction Using the Principle of Mathematical Induction

The purpose of this study was to identify and analyze the observable cognitive processes of experts in mathematics while they work on proof-construction activities using the Principle of Mathematical Induction (PMI). Graduate student participants in the study worked on ``nonstandard mathematical induction problems that did not involve algebraic identities or finite sums. This study identified some of the problem solving-strategies used by the participants during a Cognitive Task Analysis (Feldon, 2007) as well as epistemological obstacles they encountered while working with PMI. After the Cognitive Task Analysis, the graduate students participated in two semi-structured interviews. These interviews explored graduate students\u27 beliefs about proofs and proof techniques and situates their use of PMI within the contexts of these beliefs. Two primary theoretical frameworks were used to analyze participant cognition and the qualitative data collected. First, the study used Action, Process, Object, Schema (APOS) Theory (Asiala et al., 1996) to to study and analyze the participants\u27 conceptual understanding of the technique of mathematical induction and to test a preliminary genetic decomposition adapted from previous studies on PMI (Dubinsky \& Lewin 1996, 1999; Garcia-Martinez & Parraguez, 2017). Second, an Expert Knowledge Framework (Bransford, Brown, & Cocking, 1999; Shepherd & Sande, 2014) was used to classify the participants\u27 responses to the semi-structured interview questions according to several characteristics of expertise. The study identified several results which (1) give insight to the mental constructions used by mathematical experts when solving problem involving PMI; (2) offer some implications for improving the instruction of PMI in introductory proofs classrooms; and (3) provide results that allow for future comparison between expert and novice mathematical learners