University Faculty Find Many Uses for Mathematics Software
Beginning in 1989, a number of Vanderbilt University faculty began using Mathematica software in mathematics as well as six other disciplines. Courses were actively modified to take advantage of the software. Mathematica, a product of Wolfram Research, Inc., located in Champaign, Ill., manipulates mathematical expressions symbolically. For example, it solves systems of equations and allows expressions to be plotted in three-dimensional color with animation. The program also enables faculty to develop "notebooks" of tests, mathematical expressions and graphics to create an electronic reference book on a given topic. Several Vanderbilt professors have developed notebooks covering the entirety of their respective courses; in effect, they have written full-length electronic textbooks. Students may interact with the notebooks -- read explanations, manipulate values, plot functions and explore the various mathematical relationships -- by using just a few commands. Productivity Gains The productivity gains from using Mathematica for mathematics are thought to be substantially greater than gains achieved from using word processors for writing. For this reason the school's administrators expect an increasing number of faculty and students to use the new tools for mathematics. However, having software that can take an expression and return its derivative d'esn't necessarily help a novice understand what a derivative is. In the hands of a thoughtful teacher, though, Mathematica illustrates the interaction between numbers, mathematical expressions and graphics; with it students gain deeper insight than they might from conventional instruction. Moreover, students undertake projects that engage their creativity from the beginning of their collegiate mathematics careers. Instruction, then, places less emphasis on drill-and-practice in manipulating expressions and more emphasis on problem-design and solution strategies. Implementations Professor Philip Crooke has pioneered instructional efforts with Mathematica since 1989, and he has more classroom experience with the software than anyone else at Vanderbilt. He candidly observes that the most difficult hurdle for ordinary (rather than honors) mathematics students is introductory calculus. Combinations of math and computer phobia, and even poor keyboard skills, can cause students to be overwhelmed. Therefore, Crooke begins his course with a week or so of instruction on the computer and the core syntax of Mathematica. Vanderbilt's Computer Center offers a short workshop at the beginning of each semester that presents the basics of Mathematica, and Crooke encourages his students to attend. He says that students who take the workshop learn more quickly once they have mastered the computer-based tools. Crooke assigns one project in which students must find an algorithm for landing an airplane; it results in a polynomial that is easily plotted in Mathematica. Crooke also assigns projects involving the volumes of objects of rotation; these produce results that are both mathematically and visually interesting. As another example, Robley Williams, professor of Molecular Biology, teaches Biomolecular Interactions, which instructs seniors and beginning graduate students on the energies and speeds with which biological molecules bind to each other. Williams used Mathematica to write 35 class notebooks that integrate lecture material and exercises. Students learn a defined part of Mathematica and then make use of it to derive and graph mathematical relationships as well as to illustrate biological phenomena like cooperativity in the binding of repressor molecules to DNA. Spanning the Curriculum In sum, Vanderbilt faculty are making significant and creative pedagogical uses of Mathematica. The university's efforts are part of a national "technology in the teaching of mathematics" movement and are unique because they span a wide spectrum of the curriculum, well beyond the "calculus reform" movement. Many faculty find the software useful in research as well. In the classroom, Mathematica demonstrations enable instructors to show more interesting phenomena. But most importantly, students use mathematics to understand more realistic problems. Many of the faculty say they would not want to teach their courses without these tools. Knowledge of Mathematica gives students quantitative power that extends outside the classroom and lasts after the course is over.