Epsilon Mathematics Courseware - Calculus and Linear AlgebraDon Taylor
School of Mathematics and Statistics, The University of Sydney
This package is essentially a set of lecture notes for first year calculus and linear algebra transcribed to CD-ROM for Windows 95/98/NT. A full installation requires about 180Mb of hard disk space; alternatively, the program can be run from the CD-ROM. The navigation engine is Asymetrix ToolBook, which is installed along with Epsilon as part of the initial setup. This allows the authors to include interactive exercises and to illustrate the material with graphics, animations and sound bytes.
A graph drawing program GraphMAX is provided and can be accessed through the Toolbox menu. This menu also gives access to the Windows calculator, tables of formulae (standard derivatives, integrals and trigonometric identities) and a glossary.
There is no printed documentation but the small help file is all that is needed to use the program.
Epsilon covers many of the topics found in first year mathematics courses but it is nowhere near as comprehensive as the CD-ROMs that accompany the calculus texts by Hughes-Hallett et al. or Stewart. On the other hand, the simplicity of the interface and the alignment of the material with a typical Australian syllabus may make it an attractive supplement for students wanting additional exercises and explanations.
Basically the CD-ROM takes the form of a book in which pages are turned by clicking the navigation bar. It contains 21 chapters grouped into 4 sections: Numbers and algebra; Calculus; Matrices and linear systems; and Vectors and vector calculus. Exercises are interspersed throughout the modules and each chapter consists of a Concepts section followed by Applications and concludes with a set of additional exercises. Most problems can be done by following the patterns in the text. The navigation bar includes a "contents" button that lists the subsections within the Concepts portion of each chapter but there is no general search engine. The interactive exercises are not bug-free. Quite often I received the error message "Not a valid OpenScript term" when attempting a problem.
The Calculus section begins with functions, limits and continuity and concludes with differential equations and functions of several variables. The linear algebra sections cover matrices, determinants, linear transformations, vector functions, eigenvalues and eigenvectors. Despite the name of the package there are no e-d definitions to be found. Limits and continuity are defined intuitively. At times this leads to rather sloppy definitions. For example, instead of "if x is close to a, then f(x) is close to f(a)" I would prefer to see something like "f(x) can be brought as close as we please to f(a) provided x is taken close enough to a". In the definition and in all examples of definite integrals, only continuous functions are considered. There are no proofs and the style and the notation is often pre-tertiary: cis(q) is used for cos(q) + i sin(q), and there are occasional references to pronumerals.
Each chapter has an audio introduction and there are sound bytes throughout each module giving the pronunciation of key mathematical terms. In all there are 53 sound clips.
The section on matrices has excellent animations illustrating the definitions of matrix multiplication, determinants and cofactors.
A major drawback is the Glossary. It is hard to know why the distributors bothered to include it. It is surely an embarrassment to the authors of the main package. It is trite, often wrong, uninformative and badly written. There are no links (in either direction) between it and the main text. It should be ignored.
In the accompanying printed notes it states that the material is intended to be a supplement to other study resources. On the other hand, the readme.txt file on the CD-ROM and the Epsilon web page at http://www.maths.monash.edu.au/Research/publications.html suggest that it could be used as the only source for the material covered. In fact it is the prescribed text at the Caulfield campus of Monash University, where it was developed. Unfortunately, the user cannot modify the package and so it is not possible to adapt it to meet local conditions. This may not be a problem for the Concepts sections but the Applications tend to draw on a much wider range of background knowledge in physics and engineering and this could limit its usefulness.
In summary, it is a compact offering of additional material for first year mathematics and pre-calculus which will be most useful in service courses that have a similar syllabus.
This software was primarily designed for first year Engineering students, some studying by distance education. It was certainly always intended as a supplement, but never as a set of lecture notes (in fact, there are separate outline notes available on the Web).
We hoped that by using neat screen design, providing many interactive exercises; using animations, graphs, buttons, hotwords, photographs and sound clips; we could cater for new learning styles through a lively presentation, quite different from what may be achieved in most lectures, tutorials, or printed study material. We aimed for clarity and brevity. To have adopted the modus operandi of "transcribing lecture notes" would have guaranteed failure. We didn't do so.
Although some publishers currently produce CD-ROMs that seem to be very little more than transcribed textbooks, we saw no point in making such a limited use of interactive multimedia tools. Students use pen, paper, calculator and the GraphMAX package while using Epsilon. (Very few use the Glossary, but it was included in version 2.0 because some student reviewers of version 1.1 requested one.) Engineering students do have a wider knowledge of physics and engineering that we feel should be drawn on, in illustrating the wide range of applications of both the language and techniques of calculus and linear algebra.
Error messages have been reported appearing in some systems which had a previous version of runtime ToolBook installed. Reinstalling the new runtime version fixes all these small problems.
One of our many editorial decisions was that we were unlikely to advance the understanding of these students by presenting e-d definitions. The title of the courseware has calculus connotations, and also signifies that this is our small contribution to mathematics education. We certainly wish to make improvements for future versions, while maintaining the spirit and design choices inherent in the current software. The very favourable reviews by our students are encouraging us to do so.
Cristina Varsavsky and Alistair Carr
UniServe Science News Volume 15 March 2000
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