Teaching the Proof Process

A large gap usually exists between the type of thinking expected of students in the upper level undergraduate mathematics courses and that expected in the calculus sequence. Sometimes this gap is bridged with an Introduction to Proof course using a text such as Smith, Eggen, and St. Andre (1990) or Barnier and Feldman (1990). However, when no bridging course is available, we teachers of upper level courses must teach not only content but also how to do proofs. Over the years, one of my teaching responsibilities has been to introduce proof techniques to junior/senior mathe matics majors (including students seek ing secondary certification in mathemat ics) as they learn the content of the courses. Although students have had a casual acquaintance with proving theo rems, most have had nothing as all encompassing as what happens when they take a junior/senior level number theory, abstract algebra, geometry, or real analysis course. It is extremely difficult for undergradu ate students to acquire the ability of prov ing something to be true. Too often they examine the textbooks, read the proofs, and try to emulate them. But they are looking at the finished product and trying to create from scratch something that is