Real-World Computing Aided by Math Package
One of the most effective teaching methodologies in engineering is based on a 'What if?' paradigm," says Delores M. Etter, professor of electrical/computer engineering at the University of Colorado. In this paradigm, fundamental concepts and examples are followed by "What if?" questions to determine if students are able to synthesize related material. The first part of an answer to a "What if?" question requires a student to predict as much about the answer as possible without actually computing it. The second part involves verifying that predicted answers match correct answers, which are usually determined using a computer. If predicted answers are incorrect, then a student knows that he or she has not completely understood some aspect and a review of concepts or examples is probably needed. If predicted answers are correct, a student's self-confidence is reinforced relative to the material, and self-confidence develops intuition. Supporting the Paradigm Central to the paradigm is a discussion of the types of software most useful in supporting "What if?" questions. Since students must be able to test their assumptions and evaluate intermediate steps in order to test their intuition, the computing environment must allow access to information at all stages in the problem-solving process. The computing environment must also provide an interactive interface through which students can pose questions in terms that are as closely related to the problem as possible. If students have to spend considerable time rephrasing questions, the effectiveness of this learning process is greatly reduced. Capabilities of MATLAB MATLAB, a popular package for interactive numeric computation from The MathWorks, Inc. in Natick, Mass., data analysis and graphics, is a technical computing environment for implementing a "What if?" methodology. Its commands allow students to easily and quickly develop solutions to problems using true vector and matrix operations. In addition to performing element-by-element operations, the software also lets students use operations such as dot products, matrix multiplication and matrix inversions. Further, the package allows students to access the values of all variables used in a problem solution by simply entering their names -- very useful in tracking an error or understanding the results of computations. Plotting capabilities are a good reason to introduce students to MATLAB early in their engineering program. Visualizing relationships between parameters in an engineering problem is key to developing a solution. The software allows students to generate a plot (x-y, histogram, contour or 3D) with a single command. MATLAB also contains very powerful numerical techniques. While freshmen may not use many of these offerings initially, they will quickly find uses for techniques such as fitting a curve to data, interpolating between data points and solving a system of linear equations. Supporting real-world engineering, which includes many iterative processes, the language of MATLAB allows those processes to be easily defined and simulated. The visualization capabilities allow students and engineers to graphically follow the reaction of a system to changing parameters in an iterative process. Student Experiences Professor Etter uses MATLAB in courses ranging from introductory engineering at a freshman level, to linear systems theory at a junior level, to digital signal processing at a graduate level. The mathematical power and visualization capabilities are accessible to all levels of engineering students and expand to fit their growing needs. General comments from student course evaluations in the fall 1992 and spring 1993 semesters reinforce Etter's belief that the MATLAB environment is ideal for building intuition through a "What if?" learning methodology. Student comments include: "The MATLAB assignments provided really useful insights." "MATLAB material was helpful and usually more interesting than straight theory homework." "Use of MATLAB was helpful in visualizing..." "MATLAB is easy to explain to other students," comments Mark Coffey, a student in electrical and computer engineering. "I especially like it because it puts just about any numerical technique at my fingertips. It is also a great check for homework; even with theoretical homework, I can plug in a few numbers to see if my answers are correct." Bridge to Engineering World Since MATLAB is commonly used by engineers in research and industrial settings, students have an easier time bridging the gap between school and real-world engineering when they are familiar with the package. This is also an advantage to employers who then do not have to train new engineers. MATLAB's technical computing environment provides students with a tool that not only prepares them to be better students, but also better engineers.