As is the case in many small colleges, the University of the South requires at least one semester of mathematics from each student -- but once that minimal requirement is complete, we often don't see that student in a math class ever again. As mathematics educators, then, we often have an extremely limited window of opportunity to engage a student in the process of careful, sophisticated mathematical thought and an even smaller window of opportunity to ignite the student's interest or imagination concerning things generally mathematical.
If we have just one semester with a student then, it becomes clear that the Calculus per se is not the sole purpose of the class, but should also include the Calculus (or pick your favorite introductory topic) as a medium for communicating the importance and beauty of mathematics and mathematical thought.
As such I have made concerted attempts to integrate into my introductory mathematics classes not just the Calculus or traditional Finite Math topics per se but also a broader appreciation of mathematics generally, through the use of outside general mathematics readings (such as Keith Devlin's The Language of Mathematics: Making the Invisible Visible) and through the increasing use mathematically-related technologies and software. These introductory-level math classes have gradually developed from the standard paper, pencil, chalkboard, and lecture format to more intensive electronic and web-based interactions. My current classes are "administered" through class web sites (see, for example, http://yucraft.sewanee.edu/math101/ ), and the classes now typically require that all quizzes and group projects be completed within Mathematica and submitted electronically, and the classes all require the student's participation in a web-based discussion "forum" (which requires postings of mathematically-related comments and observations related to our class, our text, quizzes, current events, etc., on a weekly basis; see http://forum1.sewanee.edu:835/webx and sign in as "guest").
The diversity of the requirements for the classes have been constructed with an overriding theme: the integration of mathematical concepts outside of the classroom and into the "everyday lives" of the students. The students cannot get by in these classes by sitting passively through lectures and working homework problems alone in their dorm room. The students are being rather forcefully encouraged to actually ENGAGE in the intellectual process of mathematics, by getting them to engage each other in mathematical interactions, and engage themselves in math-related technologies and software.
A number of challenges present themselves, however, when one moves from the use of a program like Mathematica, in class by the instructor for demonstrations, to the use of Mathematica (or some similar program, or perhaps web-based interactions) by the students in an interactive, exploratory mode.
First, it is often difficult to make this "technology-engagement" process productive for an introductory-level math student, and avoid information (and task) overload. The point of the process is to engage the student in mathematics and mathematics-related topics by offering them new avenues of exploration and presentation, while at the same time achieving the pragmatic result of facility with the computers and software. But it is all too easy to overload the student with the demands of simultaneously learning the topic (math) and the technology.
Second, it is also all too easy to use our technology in unimaginative ways (often seen in the form of websites filled with text-book-like excerpts that make no real use of the web-based technology except in terms of its great accessibility), or, on the other hand, to overwhelm the student with too much unstructured material, hoping they will "get the idea" through self-initiated exploration.
Third, many of the most useful, general-purpose symbolic manipulation programs, such as Mathematica, are just complex enough that they produce a maddeningly shallow learning curve.
Overview. To address the main issues raised above, I plan to begin development of two series of Mathematica electronic-notebooks (hereafter simply called "Mathematica notebooks"). One series of notebooks will consist of a highly interactive presentation, tutorial, and exploration of some of the principal concepts in introductory Calculus and Finite Math, in a general format that provides independence from any specific textbook, and (as much as possible) independence between individual notebooks. A second series of notebooks will consist of a highly interactive tutorial process for introducing the basic operations of Mathematica to intro-level students.
Principal Concepts Series. The notebooks covering some of the principal concepts in introductory level Calculus and Finite Math can be made relatively independent of any specific textbook because the topics (and the sequence of topics) are so standardized (although much less so for Finite Math).
For Calculus, I am currently planning separate notebooks covering: Functions, Limits, Derivatives (mechanics), Derivatives (applications), and Derivatives (applied to graphing). For Finite Math, I am currently planning separate notebooks covering: Linear Equations, Linear Programming, Logic, Set Theory, and Graph Theory. The challenge, of course, for any of these topics is to create within each notebook a tutorial-style presentation that escapes the typical text-bound presentation of so many textbooks, using the interactive, graphic, and dynamic-animation capabilities of Mathematica (and possibly even making use of the sound generation capabilities of Mathematica as well).
Mathematica Tutorial Series. The notebooks offered as a series of tutorials for Mathematica itself will each be relatively brief and are initially planned for a few basic but critically useful topics: Evaluating Input & Numerical Calculations, Manipulation of Lists and Tables, Defining Functions, Plots and ListPlots, Showing Graphics. Once a student works through this short series of interactive notebooks, that student will be just sophisticated enough to benefit from the extensive help files and examples that come built-into Mathematica.
The essential resources for this project are already available and I already personally have most of the programming skills and experience needed to eventually complete most aspects of the project.
Hardware, Facilities, and Facility Support. The University of the South has a multitude of well-established, well-networked computer laboratories and several computer-based multimedia teaching classrooms, equipped with the latest in Macintosh-based computer systems, and serviced and maintained by a separate department for Academic Computing Services.
Software and Software Support. Mathematica is available as a keyed application in almost all of the computer laboratories, and is also available as a keyed application for all University students to download to computers in the dorm rooms. Wolfram Research supplies continuing technical support for its Mathematica product and often works closely with Academic Computing Services personnel to address questions or problems that can often arise with regard to the use and performance of such sophisticated software.
Faculty. Other University faculty of particular relevance to the project include Dr. Clay Ross, recent former Chair of the Department of Mathematics and Computer Science, and Dr. Karen Yu, Assistant Professor in the Department of Psychology. Ross is the author of Differential Equations: A Mathematica-Based Approach, (1995, Springer-Verlag), uses Mathematica extensively in his upper-level courses, and is a persistent advocate for the use of technological and software innovations in Mathematics education. Yu is also a determined advocate for instructional use of technology and has persistently emphasized the need for the continuing improvement of the computer facilities, software, and instructional technology available here at the University. Both of these colleagues will undoubtedly provide valuable expertise and advice
Personal Resources/Abilities. I have used Mathematica extensively over the past decade or so, both in research and in teaching, and have served as "web-master" for my own course websites now for several years.
The introductory courses in which I plan to make use of such Mathematica materials have been fairly consistent in the regular assessment of both knowledge and performance through the use of weekly quizzes, monthly tests, and monthly small-group projects consisting of material typically too challenging for an in-class test. The grade distributions, syllabus timing, and in many cases the actual student-submitted materials are available from the past two years for comparison with future performance with these augmented materials. In addition I have recorded more subjective evaluations of the time and effort I have perceived for students to overcome the initial challenges of Mathematica and a largely paperless classroom model. Finally, I will be collecting individual student evaluations concerning their perceptions of the learning process and usefulness of the notebook series.
The Mathematica notebook series will be available for download from the project/class website, and will be fully functional for anyone having access to the Mathematica program, and fully viewable (though not editable) for anyone that will take the time to download a free version of MathReader from the www.Wolfram.com website. Students' responses to, and perceptions of, the materials and learning process will be available on the class forum/discussion website (viewable by "guests" without the need for passwords or invitation), and I will post periodic updates of project assessments/comments and Mathematica notebooks on the project website for public perusal.
I will also investigate the possibility of directly emailing ACS colleagues, particularly those in Mathematics and Computer Science departments, to alert them of the availability of such materials and the opportunity to provide comments/feedback via email or class discussion website.
Further related projects slated for the future include the following: (1) The development of a series of interactive Mathematica notebooks to accompany Keith Devlin's stimulating general mathematics text The Language of Mathematics: Making The Invisible Visible (1998, W. H. Freeman and Company) which I have found to be an excellent companion book for an introductory-level college mathematics class; (2) research into more efficient, convenient ways to use mathematical fonts and notation in web-based or browser-based materials, which would then make it much easier to produce even more accessible mathematics instructional materials; and (3) in combination with item (2), research ways to use the increasingly flexible Java programming to make more dynamic and interactive web-based mathematics instructional materials.