The Calculus Initiative Sequence

Over the past three years, the author has led an effort to develop interactive technology-based laboratory experiences for undergraduates in the first two years of calculus. This paper will focus on the first year of calculus, because these are the labs that take advantage of the ability of World-Wide Web browsers to communicate with software and thus to perform computations and generate new Web pages in real-time. All labs in the Calculus Initiative help the students to discover the geometric ideas behind calculus and to explore some of the applications of calculus to physics, biology, and engineering, but it is the first few labs that expose the student to a Web-based interface to numerical and symbolic software. The labs in the second year help the students to become actual users of the symbolic, numerical, and graphical methods that they will need in order to understand applications. Thus, students not only leave the course with knowledge of mathematics, but also they gain an arsenal of technological tools ( e.g. , Maple and Matlab) and some insight into when each tool is appropriate to use.

The author once read a posting on an internet calculus reform newsgroup that claimed ``there are no real applications of the first year of calculus.'' We disagree. We believe that the applications in our course effectively combine theory, experiment, and ideas from the client disciplines so that the applications are interesting, thought-provoking, and as challenging for instructors and teaching assistants as they are for students. The labs provide a vehicle through which students investigate large-scale applications of the mathematics they have learned. These are not pseudo-applications about the maximum area that can be enclosed by a farmer with 100 feet of fencing and whose field borders a river shaped like a parabola. Instead, these are applications developed in collaboration with faculty outside of mathematics that touch on topics that many engineering students are likely to meet again in upper-division courses outside of mathematics.

In fact, so far the main drawback of the labs has been that instructors and TAs have sometimes been stumped by lab questions whose answers rely on physical or engineering concepts that are not part of the education of a typical mathematician. There is nothing wrong with an instructor not knowing the answer to a student's question, but it is important that the instructor is comfortable saying ``I don't know, but I will find out'' rather than confuse the students by giving a vague or misleading answer. Some people have suggested that a ``teacher's guide'' to the labs would be beneficial, but there is a danger in that approach, since it is inappropriate for an instructor to require students to work on a lab that the instructor has not completed. A more satisfactory (and more expensive!) solution would be an afternoon workshop at which instructors can learn the related engineering concepts.

A central focus of many calculus reform efforts is the introduction of interdisciplinary laboratory modules. At Minnesota, students work on these modules in teams to collect and analyze data, formulate and test conjectures, and communicate their ideas clearly and effectively in writing. As a group, the students submit a professionally-written lab report that summarizes their investigation. The four applications used for the 1995-96 sequence are described below:

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Frederick J. Wicklin <>
Last modified: Fri Nov 29 12:28:19 1996