Grant Number: DUE: 9552241
Following national trends, the math faculty at Bates has in recent years introduced many changes into our single-variable calculus courses: the incorporation of technology in the form of graphing calculators, writing assignments, longer projects, collaborative work, etc.. In an attempt to carry our success with these changes into the sophomore math sequence, we sought and were awarded an NSF-ILI grant to create a computer-based laboratory/classroom designed for Linear Algebra and Vector Calculus instruction. With Matlab and Maple as the primary software packages, the lab is used for regular class meetings as well as routine homework and larger projects. Other math courses, including Differential Equations, Mathematical Modeling, Mathematical Models in Biology, Statistics, and various special topics courses have also made heavy use of the facility.
For the Linear Algebra course we have developed three major projects. Students work on these collaboratively, but present their work individually in the form of essays. In one project, students develop a version of discrete Fourier Analysis and use it to understand some simulated data that would be impossible to investigate by hand calculations. The use of computers in this project allows students to focus on the key concepts of basis and orthogonality, yet makes the applicability of these concepts to real-world problems clear. In addition to the larger projects, a number of shorter laboratory exercises are spread throughout the semester, along with some in-class computer use. Student instruction in the use of Matlab is incorporated in the exercise handouts, so that they are easily adoptable by new faculty who have not taught with Matlab before.
The computers in our new laboratory allows us to extend to Vector Calculus one of the best parts of the Calculus reform: the capability to enhance students' understanding of the theoretical material using numerical and graphical tools. For example, in one lab, students use computer programs to numerically estimate directional derivatives, gaining a solid understanding of their definition and uses. Another lab has students parameterize and animate the motion of objects along various curves and their tangent lines. Students investigate numerical approximations of double integrals. They compare second-order Taylor series approximations to functions by plotting the level curves of both, or of their difference. They use the symbolic manipulation power of Maple to take the drudgery out of finding partial derivatives for use in the second derivative test or in finding Taylor series. Numerical approximations of integrals makes important results like Green's Theorem easy to illustrate, while allowing the students to concentrate more on setting up the integrals than on actually trying to evaluate them.
At our poster session, we will illustrate some of the labs we have created, and be on hand to address relevant issues such as: Is all this effort to design labs worth it? Are students learning more and is their understanding increased by our particular implementations of technology? What plans do we have to continue to improve what we are doing?
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