Combinatorial proof serves both as an important topic in combinatorics and as a type of proof with certain properties and constraints. We report on a teaching experiment in which undergraduate students (who were novice provers) engaged in combinatorial reasoning as they proved binomial identities. We highlight ways of understanding that were important for their success with establishing combinatorial arguments; in particular, the students demonstrated referential symbolic reasoning within an enumerative representation system, and as the students engaged in successful combinatorial proof, they had to coordinate reasoning within algebraic and enumerative representation systems. We illuminate features of the students’ work that potentially contributed to their successes and highlight potential issues that students may face when working with binomial identities.
This article is based in part on work supported by the National Science Foundation under Grant No. DRL-1419973. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. The authors also wish to thank Keith Weber and the anonymous reviewers for helpful comments and insights on previous versions of the article.
This article was accepted under the editorship of Jinfa Cai.
Elise Lockwood, Department of Mathematics, Oregon State University, 064 Kidder Hall, Corvallis, OR 97330; Elise.Lockwood@oregonstate.edu