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The CLeaR Project:
Multivariable Calculus at the
University of Maryland, College Park

Using mathematical software to enhance undergraduate education.

Goals

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

Materials

Since 1995, CLeaR has been using the mathematical software system Mathematica in Math 241, the sophomore multivariable calculus course at UMCP. These experiments have resulted in the book Book Cover

Multivariable Calculus and Mathematica,
with Applications to Geometry and Physics,

published by the TELOS imprint of Springer-Verlag, which can be used as a supplementary textbook along with any multivariable calculus text. (At Maryland, we are using it along with the primary text Calculus with Analytic Geometry, 5th Edition, by Ellis and Gulick.) The CLeaR materials have three components: nontraditional multivariable calculus supplements; original computer problem sets; and help on how to make most effective use of Mathematica. Each of these is unusual in its own way.
Multivariable Calculus Supplements
In these chapters, 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. The software systems render these topics, almost untreatable in an old format, easily and stimulatingly accessible to undergraduates.
Problem Sets
In 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.
Mathematica Help
One of the goals of the CleaR materials is to train students to a level of proficiency in the mathematical software system Mathematica at which they can use it as an effective tool for further significant applications to problems in geometry, physics, and engineering. Help in achieving this end is provided through glossaries of Mathematica commands at the end of every chapter, a comprehensive glossary at the end of the book, a Mathematica Tips chapter, and a diskette with several sample Mathematica notebooks packaged with the book.

Results

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.
Comments may be directed to the authors at the following addresses.
Kevin Coombes (krc@math.umd.edu)
Ron Lipsman (rll@math.umd.edu)
Jonathan Rosenberg (jmr@math.umd.edu)