Many of us who teach mathematics have been using technology in our classrooms for over a decade. Having a computer available on a daily basis has become standard for me -- in fact, I feel bereft if I enter an assigned classroom a few days before the beginning of the semester and discover that there is no computer in the room. Our students, after learning calculus with a laboratory component, have moved on to upper-level courses with new expectations about the learning environment. We knew there would be substantive differences, compared to calculus, in the way computers would be used in theoretical courses.

Ellen Maycock is Professor of Mathematics at DePauw University.

However, many of my colleagues and I anticipated that there would be common themes in the uses of technology in upper-level classes. Indeed, two conferences -- the FIPSE-supported Conference on Technology in the Upper-Level Curriculum, held at St. Olaf College (1994), and a subsequent NSF-UFE conference, Exploring Undergraduate Algebra and Geometry with Technology, held at DePauw University (1996) -- were based on the idea that there would be a fundamental similarity in the use of technology in these upper-level, axiomatic courses.

I had, in the early and mid-1990’s, developed laboratory materials to teach abstract algebra with a computer laboratory component (Parker, 1995, 1996 ). Additionally, as part of a Mellon Foundation grant to DePauw, I was chosen to be one of a pilot group of faculty to introduce technology into a course that had been previously taught without any. I decided to experiment with our department’s introductory real analysis course in the spring of 1999, and I confidently planned my syllabus assuming that I could use my successful pedagogical techniques from abstract algebra in real analysis. Because of this underlying assumption, I was unprepared for the problems that I encountered in this experimental real analysis course. After having candid discussions with students midway into the semester, I realized that there were subtle but distinct differences in the nature of our department’s upper-level courses that meant that technology would play a significantly different role in each.

I present here my experiences in three upper-level courses: abstract algebra, real analysis, and geometry. Certainly, there are recurring themes in the use of technology in all of these courses, and I can only repeat here what many others have also experienced in their classrooms. For me, the use of a laboratory component has been very successful in all three courses, but only after I understood that I could not automatically transfer the format from one course to another. I share these observations to encourage the reader to consider thoughtfully how technology can enhance the learning environment in each classroom. I would like to pose the bulleted conclusions on page 5 as research topics to be explored more fully, both in theoretical pedagogical discussions and in controlled classroom experiments.