Numerical algorithms - enhancing presentation while maintaining rigour in introductory courses: A minimalist approach to course modernization
John Alan Belward · 2002
Recent academic developments which include a changing student profile, the focus of contemporary research, and social trends have combined to pose significant challenges in first year undergraduate mathematics courses. Issues which arise include the content level, the need to maintain rigour, addressing the needs of specialist and non-specialist students, the need to equip students with useful, applicable, techniques, and our desire to present a picture of the important problems and directions in modern mathematics and its beauty and excitement. Calculus reform has made a significant impact but more needs to be done. There are large areas of research in which the computer has the role of an experimental tool. The use of software packages is widespread. This has produced the need for something akin to an instinct which can identify the correct or incorrect functioning of a package or black box. To acquire this instinct some knowledge and experience of the behaviour of numerical algorithms is needed. As a consequence the way in which calculus is taught needs to be changed. It also needs to change because the computer has caused major changes in the theoretical directions of mathematics. These influences can be used to enhance courses whose content contains the essential foundations of the subject. The foundations will not change, but investigations of numerical algorithms, for example, can pose the same fundamental questions that are to be found in texts dating back a century at least. Well founded approximation methods provide exact rigorous statements. Numerical experimentation can provide insight. There is no need to present a grab bag of computations whose output is of doubtful validity. This presentation will briefly review the knowledge levels of entering students; it will describe some important applications and it will attempt to show how some of the challenges can be met.