Title: Investigating Provers' Understandings of Combinatorial Proof: Preliminary Findings
Abstract. Combinatorial proof is an important topic in enumerative combinatorics with applications in statistics as well as other mathematical domains. However, only two previous studies in the mathematics education literature have investigated this topic, and much is still unknown regarding how undergraduate students think about and engage in combinatorial proof. To address these gaps in the literature, I interviewed 8 mathematicians and 5 undergraduate students in upper-division mathematics courses. Preliminary analysis of these data corroborate and provide greater nuance for findings from previous studies, and additionally I discovered that students benefited from a particular instantiation regarding scalar multiplication while engaging in combinatorial proof of binomial identities. This study is part of a broader research project aimed at examining students' and expert mathematicians' beliefs about combinatorial proof as a proof technique, as well as understanding methods that students and expert mathematicians use to find combinatorial proofs of binomial identities.