The CLeaR Project at the University of Maryland, College Park (UMCP)

CLeaR is an acronym for the developers (Drs. K. Coombes, R. Lipsman and J. Rosenberg) of a computer supplement for the sophomore Multivariable Calculus course. The goals of this project are:

Since 1995, CLeaR has been experimenting with the mathematical software system Mathematica in Math 241, the sophomore multivariable calculus course at UMCP. These experiments have resulted in the writing of a book Multivariable Calculus and Mathematica, with Applications to Geometry and Physics that was published in May 1998 by Springer/TELOS. The book is used at UMCP in conjunction with the text Calculus with Analytic Geometry, 5th Edition, by Ellis & Gulick, although it is completely suitable for use with most sophomore level multivariable calculus texts--traditional or reform.

The bulk of the book consists of eight chapters on multi-variable calculus and its applications. Some chapters cover standard material from a non-standard point of view; others discuss topics that are hard to address without using a computer. Each chapter is accompanied by a Mathematica problem set. The problem sets constitute an integral part of the book. Solving the problems exposes students to geometric, symbolic, and numerical features of multi-variable calculus. Many of the problems are quite challenging. Each problem set concludes with a glossary of Mathematica commands, accompanied by a brief description, which are likely to be useful in solving the problems in that set. A more complete Glossary, with examples of how to use the commands, is included at the back of the book. In addition, the book contains Mathematica Tips, Sample Notebook Solutions, and an Index. Finally, there is an accompanying disk containing

Here is a brief description of the books' contents.

Chapter Descriptions
How it works!
In Chapters 2-9, students are introduced to those aspects of elementary differential geometry, optimization and physics that, while vitally important and most relevant to the needs of practising scientists and engineers, are usually omitted, or only treated briefly, in a traditional text: namely, numerical, geometric, symbolic, and qualitative methods. The software systems render these topics, almost untreatable in an old format, easily and stimulatingly accessible to undergraduate students.
In solving the problem sets the student brings to bear newly acquired skills in the computer system to solve non-traditional problems in multivariable calculus, elementary differential geometry, optimization and physics. The emphasis is on the geometric, symbolic, numeric, and qualitative aspects of the subject. The problems, each of which is a small project, are designed to force the student to engage in critical, analytic, and interpretive thinking beyond rote manipulation of algebra and calculus formulas.

Students do all their work in campus computer Labs. All platforms are available, and students select those they feel most comfortable with. Because of the remarkable interface, faculty barely notice any difference in the output generated by students working on different platforms.

Very little formal instruction on the platforms or software system is presented in class. Students learn about them from primer material, and from on-line help, Graduate Assistant tutors (acting as first-aiders), each other, and faculty assistance in office hours.

The effects of the project, aside from achieving the goals indicated above, include: creating a mathematical computational culture among students (they use the tools they take away from this course in other courses, in lab reports, and later on in their jobs); fostering cooperative learning (students are encouraged to work in teams, and they quickly become acclimated to cooperative problem-solving in a team setting); enhanced visual and communication skills (the interface allows the student to integrate textual, symbolic, and graphical material in an informative and effective way). Most importantly, the intellectual level of the course has been raised---without a drop in student performance.

July, 1998