Harmonizing a computer algebra system with a regular mathematics course

Abstract

This paper documents a four-month experiment in which a class of undergraduate primary school teacher trainees were taught the third of their normal four mathematics content course entirely through the medium of a computer algebra system (CAS). The course content included topics in number (continued fractions, real numbers), geometry (vector geometry, i-j components, transformations), algebra (polynomials, matrices and their combinations), trigonometry (sine and cosine rules, angles of any magnitude, periodicity), statistics (sampling, correlation, subjective probability), and historical and multicultural special topics. The course content was not specially chosen for this experiment, and was, in fact, carefully constructed to fit in with the other three mathematics courses which the students take. This paper discusses not the theoretical issue of what could be done with a CAS in this setting, but the problems and benefits which arise when the CAS is actually used for a normal mathematics course.

A number of practical issues caused difficulties. This paper describes briefly the nature of these difficulties and how they were resolved or at least ameliorated. More significantly, a number of time-related issues emerged which were not so easily resolved. For example, the overhead in learning how to use the CAS meant that, in effect, the teaching term was reduced from 14 weeks to 10, suggesting that learning a CAS for a single course may be impracticable, and that learning a CAS for just one segment of a course almost certainly is. Furthermore, although it was planned to use other software packages for special purposes (CAD to obtain accurate diagrams, SPSS to handle the statistical work etc), the learning overhead of those software packages prohibited their use. Some parts of the course content could not be accommodated within the CAS. For instance, the historical and cultural special topics required the students to do research in the library, where computers for student use were not available. Other topics were difficult to present using the CAS. To some extent, these difficulties were overcome by using other (simple) software, but the overhead in learning that software also proved uneconomic, and these topics were therefore covered by a normal, but brief, lecture process.

Eclipsing all of the above, though, was the emerging confirmation that the existing course, and the traditional examining process for that course, could simply not remain intact. By far the most important aspect of this was the growing appreciation by the students that the expectations of them had changed because of the powerful new medium for "doing" their mathematics. That is to say, the very nature of the students' mathematical processes changed as a result of "doing" mathematics with a CAS. This aspect of the experiment is the main focus of the paper.

The examination at the end of the course was conducted in the same environment as the "lectures", with one exception. Course assignments, notes, handouts etc. during the course were maintained as HTML files or as live CAS documents on the instructor's computer, and students accessed these on demand over the network. This was not considered secure enough for examination purposes, and so the examination was distributed on floppy disc at the appointed time. The students' examination "scripts" were in the form of saved CAS files, which were printed out later at the instructor's leisure.

The paper concludes by comparing this experiment to other work published elsewhere.