In keeping with the current trend of using technology to improve student learning, the Mathematics Department has recently been incorporating a computer component in its honors calculus sequence using materials developed by Professor Allen Weitsman and graduate student Gregory Rhoads. Intended for students who have tested out of the first semester of calculus and who show a special interest in mathematics, these courses have smaller enrollments, typically between 15 and 20 students from both Science and Engineering. While the traditional lecture part of the honors courses is conducted by a professor, for the last two years Greg, a 1995 Ph.D., has had the responsibility for teaching the students how to use the computer.
The computer projects use the mathematical software MATLAB, which is a high level programming language. Among the reasons MATLAB was chosen is that it contains the basic control structures that are found in other programming languages such as C and Pascal, and therefore gives students an introduction to computer programming. It also includes many built-in features for use in linear algebra, numerical solutions of differential equations, and signal processing. The School of Engineering requires students in Electrical Engineering to have access to MATLAB and has introduced it into many of its more advanced undergraduate courses.
At the beginning of each semester, Greg gives an introduction to MATLAB and then assigns projects in the recitation sections. Each assignment is accompanied by a comment sheet which introduces the necessary computer statements and gives further information on the topic of the particular assignment. Greg gives a brief discussion of each project, after which the students complete the work on their own outside of class, consulting with Greg as necessary. The projects may be done in one of the University computer labs or, in many cases, on the students' own computers.
The goal of these projects is to reinforce the students' understanding of the concepts in parallel with the lecture. Often the students write programs to implement the theory they are currently studying. These programs are then applied to problems similar to those typically assigned with paper and pencil; however, the calculations are often more complex and cannot be performed without the computer. In other projects students discover underlying theories by computer experiments, state conjectures based on their observations, and then attempt to prove their conjectures.
The smaller class size and the additional interest of the students have allowed professors to discuss more advanced material and applications, and Greg has tailored some projects to reinforce this new material. Topics of past projects have included: investigating convergence of Riemann sums in one and two variables, approximating functions of one or more variables by series and studying their convergence properties, using graphics to discover properties of functions, modeling the motion of a satellite, using Markov Chains to study long term genetic behavior, and investigating various numerical methods to solve differential equations and systems of differential equations.
Professor James McClure, who taught the honors course this spring, found that the students were quite enthusiastic about them; they enjoyed having a change from the usual routine of homework assigned from the textbook.
With the increasing availability of powerful computers, new computer-initiated scientific discoveries, and the centrality of computers in the workplace, Greg feels that this kind of training is essential for today's students.
Greg has recently accepted a position as Assistant Professor of Mathematics at Appalachian State University, where he plans to continue his work using computers in mathematics education.