Algebra , Computer Algebra , and Mathematical Thinking

Mathematical symbolism in general—and symbolic algebra in particular—is among mathematics’ most powerful intellectual and practical tools. Knowing mathematics well enough to use it effectively requires a degree of comfort and ease with basic symbolics. Helping students acquire symbolic fluency and intuition has traditionally been an important, but often daunting, goal of mathematics education. Cheap, convenient, and widely available technologies can now handle a good share of the standard symbolic operations of undergraduate mathematics: differentiation, integration, solution of certain DEs, factoring and expansion in many forms, and so on. Does it follow that teaching these topics, and even some of the techniques, is now a waste of time? The short answer is “no.” On the contrary, as machines do more and more lower-level symbolic operations, higher-level thinking and deeper understanding of what is really happening become more, not less, important. Numerical computing has not made numerical viewpoints obsolete; neither will computer algebra render symbolic mathematics obsolete. The key question is how to help students develop that bred-in-the-bone “symbol sense” that all mathematicians seem to have. What really matters is that students use mathematical symbolism effectively to pose worthwhile problems in tractable forms. Once properly posed, such problems are well on the way to solution, often with the help of technology. The longer answer, explored in the paper, concerns choosing mathematical content and pedagogical strategies wisely in light of today’s technology.