William W. Johnston
and
Alex M. McAllister
Centre College
Abstract
At present, we are designing and developing a two-term freshman level survey course in mathematics: Introduction to Mathematics. The key innovation of this new course is the implementation of a unified presentation of the many fields of mathematics. One central component is the development of technology-based examples and exercises. In this proposal, we are applying for an ACS Teaching with Technology Fellowship for Spring 2001 to support the development of the technology component of this new course.
Our view of the current mathematics curriculum
In a standard mathematics curriculum, students typically study the single-variable calculus in their first two terms, the multivariable-calculus in a third term, and then finish this "calculus sequence" with a course in differential equations. Although the calculus sequence has many benefits, it also has some important weaknesses. Most significantly, the calculus sequence provides students with little understanding of the broad scope of mathematics. Most students never complete the calculus sequence and we often wonder if many promising students drop out of the calculus sequence before they have been exposed to "that piece of mathematics" that might have excited their interest and stirred their imagination. Even mathematics majors cannot gain an appreciation of the multi-faceted aspects of mathematical study from the calculus sequence; most students have little understanding of the many areas of mathematics unless they enroll in the corresponding upper level courses in their junior or senior year.
Our response to this view
Recent technological advances in symbolic manipulation and graphing utilities are leading the national mathematics community to reconsider the "how and why and what" of the mathematics we teach in our introductory courses. At Centre College, we have chosen to create a two-term survey course that will expose freshman students to the broad spectrum of ideas that mathematics entails, while preserving the study of the most central ideas of the calculus. Our primary objectives for this course are:
With these objectives in mind, we have established a basic syllabus for such a two-term survey course that we are calling Introduction to Mathematics. Last summer we began developing a manuscript to support the teaching of this course and we hope to introduce such a course into our curriculum in the 2002-2003 academic year. The introduction of this course is dependent on the completion of a supporting manuscript (including the technology component for which we are seeking an ACS Teaching with Technology Fellowship) and the approval of both the mathematics program and curriculum committees of Centre College.
Our Proposal for Spring 2001
The opportunity to develop Introduction to Mathematics is provided by the advent of new technologies -- particularly, computer algebra systems that are mathematically sophisticated. Nationwide, many projects have made innovative use of these new technologies to reform and revitalize the teaching of mathematics. Our project is in the spirit of this national trend and is an innovative extension of current approaches to introductory mathematics courses. Therefore, we are applying for an ACS Teaching with Technology Fellowship for Spring 2001 to support the development of the technology component of this new course.
Technology is an essential ingredient of Introduction to Mathematics; indeed, technology is the reason we are able to complement the teaching of calculus with further mathematical topics of broad interest and applicability. We intend to utilize the computer algebra system Maple for the course and we need to develop appropriate examples and exercises for a wide variety of topics. This family of topics includes:
1. graphing of polynomials,
2. understanding limits graphically and numerically,
3. computing derivatives,
4. the area problem and Riemann sums,
5. Fourier series,
6. power series and analytic functions, and
7. least squares regression.
On the ACS Technology website, we noted the project of Summer 2000 Fellow Warren D. Craft: Text-Independent, Interactive Instructional Materials for Introductory Calculus and Finite Math. Craft's project utilized Mathematica, and we believe our project will complement his by providing resources to support the use of Maple (Maple and Mathematica are the two major computer algebra systems utilized by mathematics teachers). We have also chosen the family of topics to enhance and complement Craft's previously funded project.
The final product of this project will be "class planning modules" for each of the above topics and will be available through the Centre College Mathematics Department homepage by June 2001. Any mathematics professor can use these modules to guide a class through each of the family of topics. These modules will not simply be on-line versions of textbook material and will not be dependent on any particular text. Rather, they will provide a hands-on, discovery experience for the students in the Maple environment and make essential use of the symbolic manipulation and graphing capabilities of Maple in a highly interactive setting.
Technical Requirements and Resources
We are well acquainted with the mathematics that we are considering for this "technology infusion," and we have ample experience in teaching this material to diverse groups of students. We have worked extensively with Maple and web-development software, and we possess all the technical skills required for the completion of this project. Dr. McAllister is currently the webmaster for the Mathematics Department of Centre College and examples of course webpages for both Dr. Johnston and Dr. McAllister can be found http://web.centre.edu/mat/cweb.
Furthermore, some examples of very preliminary work in using Maple in a second-semester calculus course can be found at http://web.centre.edu/alexmcal/.
We are planning to work on this project during the spring term of 2001. We both have light teaching loads during that term. In particular, we will each be teaching two courses (rather then the standard three-course teaching load), and we intend to use the "extra" available time to develop this project's class planning modules for publication on the web and for incorporation into our current manuscript.
The technical requirements for this project are straightforward and are already in place. Centre College has well-established computing resources for faculty and readily accessible technology classrooms and computer laboratories that can be utilized for teaching Introduction to Mathematics. Centre College provides us with versions of Maple, word processing software, and web development software that are needed for the project. Furthermore, Centre students have ready access to Maple, and technical support is adequate (through both the Waterloo Corporation and Dr. Samuel Fee and Shane Wilson of Centre College's Department of Information Technology Services).
Assessment and Dissemination of Materials
The overall goal of this project is to facilitate the effective use of technology in the teaching of mathematics via several class planning modules. We will evaluate the success of these modules in fulfilling this goal in three ways. As each module is implemented in an actual class during the next two academic years, the instructor will note the students' interaction with the module and suggest appropriate revisions to enhance and improve the module for future courses. Immediately after implementing the module, the instructor will complete a detailed evaluation of the module. Finally, the students will also complete a special end-of-term evaluation of the modules and of the modules' effectiveness in promoting student learning. Further evaluation of these modules will be made in the subsequent years that Introduction to Mathematics is taught at Centre College.
We plan to share the results of this project with our colleagues by posting the project modules on the Centre College Mathematics Department homepage http://web.centre.edu/mat. In addition, the first time that we implement this course, web-based course materials will be available for our colleagues through our departmental listing of course webpages at http://web.centre.edu/mat/cweb.html. We will post links to these resources on the ACS technology resources webpage as well.
Ideally, the Introduction to Mathematics courses could be used to serve all "calculus level" students, not just future mathematics or science majors. Such a change in the mathematics curriculum would be a major innovation, and the technology component of this course is crucial to its success. After we have further developed Introduction to Mathematics, we intend to share our ideas with a broad audience and we anticipate this innovation will be of significant interest to mathematics programs both at ACS institutions and at institutions across the country.
William W. Johnston
Department of Mathematics
Centre College
600 West Walnut Street
Danville, KY 40422
johnston@centre.edu
Alex M. McAllister
Department of Mathematics
Centre College
600 West Walnut Street
Danville, KY 40422
alexmcal@centre.edu