Use of strategic knowledge in a transition-to-proof course

The ability to construct proofs has become one of, if not the, paramount cognitive goal of every mathematical science major. However, students continue to struggle with proof construction and, particularly, with proof by contradiction construction. This paper is situated in a larger research project on the development of an individual’s understanding of proof by contradiction in a transition-to-proof course. The purpose of this paper is to compare proof construction between two students, one graduate and one undergraduate, in the same transition-to-proof course. The analysis utilizes Keith Weber’s framework for Strategic Knowledge and shows that while both students readily used symbolic manipulation to prove statements, the graduate student utilized internal and flexible procedures to begin proofs as opposed to the external and rigid procedures utilized by the undergraduate.