Recent Developments in Computer Algebra Technology and Their Impact on Mathematical Research and Teaching

Abstract

Computer algebra systems have come a long way from its infancy almost forty years ago with the MACSYMA project at MIT. This first generation of computer algebra systems, characterized by one dimensional input and output and ASCII plots, emphasized the implementation of symbolic algorithms. Graphical user interfaces did not appear for twenty years – Mathematica 1.0 in 1988. The user interfaces that characterize the second generation brought significant improvements in the presentation of mathematical output and graphical presentation. Earlier this year, nearly another twenty years later, Maple 10 is likely to be the viewed as the beginning of a new generation of computer algebra. The most revolutionary develops of this generation are

(i) the two-dimensional entry of mathematics that eliminates the need for the typing of commands and the associated syntax problems and
(ii) the ability to break free from the traditional linear structure imposed by scripts, worksheets, and notebooks.

While it is too early to assess the full impact of the new features, it is safe to say that computer algebra has entered a new paradigm.

In this talk I will briefly summarize some of the significant steps in the early development of computer algebra systems. The majority of the talk will be devoted to explicit examples that illustrate the new paradigm. In addition to new uses of the traditional “worksheet”, the examples will show how computer algebra systems can be used to develop enhanced standalone web-based applets for the presentation of research results and to provide web-based supplemental support for student learning (drill and assessment). The mathematical topics will involve calculus, linear algebra, differential equations, and number theory.