Introduction

College calculus teaching has changed more during the past dozen years than during the previous century. The driving force behind this transformation has been our increasing ability to shift the burden of algorithmic processing from humans to machines [1]. An accompanying trend has been the expanding capacity of computer systems for graphically representating complex mathematical concepts and processes. Furthermore, the possibilities offered by electronic communication have revitalized the notion of mathematical discourse [2]. The result is epitomized by Mathematica, a fully integrated environment for technical computing [3]. Students who learn calculus with Mathematica are not learning "the history of calculus"--they are preparing for work as it is being done and will be done in the near future. And as we will see below, a new application for optimizing the use of Mathematica, known as Calculus WIZ, will present teachers and students with even more possibilities for teaching and learning.

Technology is not the only difference to be found in new calculus classes. Changes in the college classroom also reflect a pedagogical shift on the part of mathematics educators [4], from a point of view that "only the best students make it through the course," to a new attitude that mathematical knowledge should be available to all students. This attitude is reflected in the phrase "a pump, not a filter" [5]. Distinguishing characteristics of the new calculus pedagogy include the following.

1. Cooperative, rather than individualized student work ("Getting help" is not a sign of weakness or cheating)

2. Exploratory study by students, rather than chalkboard-driven presentation by a professor: "Formal definitions and procedures evolve from the investigation of practical problems"

3. Multiple representations of the subject: "Every topic should be presented geometrically, numerically, and algebraically"

4. Alternative assessments of student progress, such as reviews of portfolios of student work [6]

A successful realization of these themes is the Calculus&Mathematica project [7], developed at the University of Illinois at Urbana - Champaign and the Ohio State University and tested at thirty other sites for about six years. The project is described in the National Research Council report Moving Beyond Myths: Revitalizing Undergraduate Mathematics, as:

An innovative calculus course . . .[which] uses the full symbolic, numeric, graphic, and text capabilities of a powerful computer algebra system [ Mathematica ]. Significantly, there is no textbook for this course-only a sequence of electronic notebooks.

MU is one of the first universities in the country to teach all calculus courses as computer-based classes. In a recent report, the American Mathematical Society Task Force cited the department for being one of the first to introduce technology into mathematics classes and for coordinating it with engineering and with other departments in the arts and sciences. Mizzou Show Me Calculus and Mathematica is a shift from a memorization-based learning process to an understanding-based process. In this new academic experience, students become active partners in learning, as well as doing. [8]

Calculus courses have also changed in content. Many, if not most, of the basic concepts of calculus and its applications to science and engineering are at least a century-and-a-half old [9]. The "phylogeny" is fairly well understood: The broad, intutitive themes were laid out by the early masters, and the finer details of mathematical rigor came later [10]. Many mathematicians and their colleagues in science and engineering have dedicated time and effort to the continuing analysis of the content of the calculus sequence, and topics have been pruned from the curriculum in response to the call for a "leaner and livelier calculus" [11]. New mathematical knowledge, for example the rise of the study of dynamical systems, is also a factor in curriculum change. The effect, in this instance, has been to introduce students to differential equations and linear algebra at an earlier stage in their mathematical program [12].

Among the new "topics" to include in a modern calculus course are the technologies on which the course will be based. A few sessions devoted to Mathematica at the beginning of a course can be a good investment, with a pay-off later in students having the necessary command of their computer systems to conduct meaningful explorations. An impressive array of calculus material, including start-up tutorial notebooks, is available on the World Wide Web. An excellent starting point is MathSource [13].

Of course, many college students today have excellent command of their computer systems. We are seeing the realization of a prediction made a few years ago by Keith Devlin:

Imagine then the kind of person coming into our graduate schools, if not today, then certainly tomorrow. Brought up from early childhood on a diet involving MTV, Nintendo, graphical calculators packed with algorithms, Macintosh-style computers, and, in the not-too-distant future, hypermedia educational tools as well. Such a person is going to enter mathematics with an outlook and a range of mental abilities quite different from their instructors. [14].

Despite the evidence in favor of technology, there remain skeptical professors, even skeptical graduate assistants. The next section illustrates some of the typical attitudes and opinions through use of a dialogue based on actual email correspondence. Following that section is a closer look at the learning and teaching issues behind the technology debate, and an examination of the outcomes that should result from calculus instruction. The concluding section describes some of the emerging research in cognitive science that will inform the next generation of calculus instruction.