A proofs course that addresses student transition to advanced applied mathematics courses

Introduction. At many institutions, the standard transition for undergraduate students from the calculus sequence to upper level courses in mathematics involves a proofs course. One of its purposes is to mature undergraduate students and change their perspective from problem solving to theorem proving. In such a course, students learn about the abstract nature of mathematics while at the same time learning how to construct basic proofs, how to read mathematics, and how to write mathematics. Of course, it is impossible to teach how to prove without proving something! Proofs courses often introduce concepts and topics from a variety of mathematical fields, thereby providing a sample of advanced pure mathematics. A survey of some recent textbooks designed for proofs courses indicates the wide variety of topics used to introduce the concept of proof. For example, Schumacher [11], Eisenberg [4], and Fletcher and Patty [5] focus on number theory, axiomatic approaches to examining the real numbers, and the cardinality of sets. Rotman [10] offers less of a sampling of higher mathematics, but grounds the proofs in mathematics more familiar to students, including geometry, trigonometry, and properties of polynomials. Of course, the treatment is much more precise and rigorous than the students may have seen and does develop and use more advanced mathematics in these more familiar areas. D'Angelo and West [16] provide a more extensive sampling of advanced mathematics, including discrete mathematics (probability, combinatorics, graph theory, and recurrence relations) and continuous mathematics (sequences, series, continuity, differentiation, and Riemann integration).