M. B. Monagan
Centre for Experimental and Constructive Mathematics, Simon Fraser University, Canada
and
J. F. Ogilvie
Centre for Experimental and Constructive Mathematics, Simon Fraser University, Canada
Escuela de Quimica, Universidad de Costa Rica, Costa Rica

Although for one or two decades mathematical software for symbolic computation has been employed extensively in the teaching of university mathematics, each respective course typically emulates a traditional course without computers in treating a particular topic such as differential calculus or linear algebra. We advocate an holistic approach to the teaching and learning of mathematics involving intensive use of contemporary software not only for pedagogical purposes but especially for implementation by the users of mathematics. By expecting a learner to employ computer software for almost all mathematical operations, an instructor can emphasize the mathematical concepts and principles, involving formal definitions, algebraic derivations, numerical examples and especially graphical illustrations and constructions, and then develop the implementation with selected software. In this way not only does a student acquire a profound understanding of those principles – and limitations of the software, but he or she becomes proficient in applying the software for real problems of a scale that would be impracticable in manual work. By eliminating practically all repetitive drill and practice that is irrelevant when the computer undertakes the calculations, an instructor becomes able to cover a much enhanced range of topics within a given duration; for instance, all material – from differential, integral and multivariate calculus, linear algebra and differential equations to probability, statistics and data analysis that typically occupies six or more semester courses – might be covered within three semesters at a typical pace. That content is all that a student is likely to need for a technical career after completing an undergraduate programme in science and engineering. We argue that contemporary students of engineering and science who are not so equipped with a working knowledge of symbolic mathematical software are not being prepared properly for a technical career.

"The human mind is never performing its highest function when it is doing the work of a calculating machine." – Lord Kelvin

In this lecture we present examples of topics of which we demonstrate some benefits of teaching with mathematical software.