The SCHOL Project in Ordinary Differential Equations
Goals
The goals of the project were threefold: (1) to enhance students' understanding of the fundamental concepts of differential equations; (2) to provide students with a tool, in the form of a modern mathematical software system, for exploring, understanding, and using mathematics; and (3) to improve students' ability to present, in a polished and organized manner, their understanding of mathematics to faculty, peers, and others.
The effects of the project, aside from achieving the goals indicated above, include: creating a mathematical computational culture among students (they use their knowledge of a mathematical software system in other courses, in preparing 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); and enhanced visual and communication skills (the software 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.
Course Curriculum
The course follows a traditional Ordinary Differential Equations syllabus, but the emphasis has been shifted away from the rote application of closed form symbolic solution methods, and toward a numerical, qualitative, and graphical understanding. Numerical methods are covered early in the semester, and the built-in numerical differential equation solver in MATLAB, Mathematica, or Maple is used throughout the course. The use of a computer also has allowed the formulation of many open-ended and challenging problems.
The SCHOL supplement has three components: computer platform instruction; non-traditional ordinary differential equations (ODE) supplements; and original computer problem sets. Each of these is unusual in its own way.
- Computer Platforms and Mathematical Software Systems. These chapters instruct students in the rudiments of MATLAB, Mathematica, or Maple, and in the innovative interfaces through which one interacts with these systems: M-files, M-books or Cells/Publish in MATLAB, Notebooks in Mathematica, and Worksheets in Maple. They also train students to use the special features of the software that have a direct bearing on differential equations. The chapters are written with explicit and simple instructions for the most common platforms: Windows or LINUX (on PC's), Macintosh, and the X Window System (on Unix machines).
- ODE Chapters. In these chapters students are introduced to those aspects of differential equations that, while vitally important and most relevant to the needs of practicing scientists and engineers, are usually omitted, or only treated briefly, in a traditional text--namely, numerical, geometric, and qualitative methods. The mathematical software makes these topics accessible to undergraduate students.
- Problem Sets. In these sets, students bring to bear newly acquired software skills to solve non-traditional problems in differential equations. The emphasis is on the symbolic, numeric, geometric, and qualitative aspects of the subject. The problems are designed to force the student to engage in critical, analytic, and interpretive thinking beyond rote manipulation of algebra and calculus formulas.