Computer algebra has been an issue in mathematics education in the Netherlands for some years already. This does not imply that the discussion on this phenomenon has resulted in an agreement; no consensus on the role of computer algebra in the mathematics classroom has emerged so far.
Below, I describe some recent developments in my country from a personal perspective. I confine myself to mathematics education at upper secondary, pre-university level.
Firstly, I describe the Dutch situation concerning curriculum and assessment. Secondly, I briefly consider the first educational experiments with computer algebra. Thirdly, the rise of the graphics calculator is discussed. In the end, computer algebra comes into the picture again, but now in a hand-held format. I conclude with an imaginary jump into future.
Understanding the developments in my country requires some knowledge of the organisation of the curriculum and assessment.
As far as the curriculum is concerned, it is important to notice that there is no detailed curriculum that prescribes which topic should be taught when. The curriculum is defined by a description of skills and concepts that will be assessed by the end of secondary school; the schools are free to choose how they get there. They can also decide on the textbooks they want their students to use. It is because of this relative freedom that the final assessment is so important.
The final assessment at upper secondary level consists of two parts: a school set assessment and a final national examination that is externally set and internally graded. This national examination is very important for the implementation of technology; if a certain technology device is not allowed at the final examination, it will not easily become popular in the classroom. The current regulation is that the graphics calculator is required at the final examination, whereas computer algebra is excluded. Two arguments guided this legislation. Firstly, it would be hard to organize a national examination throughout the country with computer access for all the candidates; hand-held computer algebra was not yet available. Secondly, the financial aspect was important. If computer access was required at the final examination, schools would need money to buy them, whereas hand-held technology devices are supplied by the students themselves.
For the school assessment, the authorities recommend the use or partial use of a computer, but again, the schools are free to decide. I have the impression that the number of schools that use a computer in their examination is increasing. The computer is often used in combination with investigation tasks where a written or oral report forms the assessment.
Obviously, the Dutch policy on technology is a careful one. Information about the different strategies concerning technology use and assessment in other countries can be found in [4].
The first project on computer algebra at upper secondary level started in 1990. The idea of this two-year project was to develop short instructional units that were tested in pilot schools. Although the production of these materials was useful, the project as a whole was not very successful. This was caused by the lack of computer facilities at schools and by the difficulties students had with the user friendliness: they had little `computer literacy' and a windows interface was not available. Obviously, the time was not yet ripe for the implementation of computer algebra at this level.
By the end of this project, a group of volunteering teachers decided to continue the work. This group, called CAVO, existed until 1998 and was a lively and important platform for further development and discussion (see [6]). In the mean time, however, the graphics calculator came on the market, and attracted much attention.
The development of the graphics calculator elicited discussion on which technology platform should be used in secondary education (see [1]). The Dutch authorities decided that the implementation of the graphics calculator would be the first step to take. Therefore, they supported a research project on this issue in 1992. This project was carried out by the Freudenthal Institute, a research group on mathematics education. Later it became an integrated part of a larger curriculum development project called Profi. Results of the Profi-project included student textbooks that integrated the use of the graphics calculator, and experimental examinations that required the availability of a hand-held graphing device. The role of the graphics calculator in this project is summarized in [5].
Some educators and teachers were, however, opposed to the implementation of the graphics calculator. Their arguments were that computer algebra is a much more sophisticated mathematical tool, and that a graphics calculator is only a temporary step backwards compared to the possibilities that PC's offer. In 1996, however, a questionnaire revealed that PC's were hardly ever used during mathematics lessons, although they were available in schools. This supports the idea that real implementation of technology requires that the student has direct access to the device. The limited mathematical power of the graphics calculator is not an important disadvantage: it allows teachers, textbook authors and examination boards to have sufficient time to carefully integrate a graphical and numerical tool without having to cope with computer algebra in the mean time.
The choice of the graphics calculator may be a temporary preference indeed: the symbolic calculator raises the issue again.
Nowadays, computer algebra is also available in a hand-held format. A first pilot experiment using the TI-92 revealed that the students appreciated this machine as an `algebraic calculator', but not so much as a dynamic geometry tool [3].
When the Dutch Association of Mathematics Teachers became aware of the possible impact of symbolic calculators on secondary mathematics education, an Advisory Board on Computer Algebra and Symbolic Calculator was formed. In May, 1998, this Board concluded that:
In the fall of 1998, the Freudenthal Institute conducted an explorative case study using the symbolic calculator. This machine turned out to be quite useful in investigation tasks. The sophisticated use of variables and parameters, however, was not always clear to the students. Furthermore, some students were reluctant to use computer algebra for the application of techniques that they had not yet mastered manually.
At present, many research questions concerning the role of computer algebra in secondary education are still unanswered (see [2]). No decisions on its implementation in the Netherlands have been made so far. In the next few years, I expect three developments to take place.
Firstly, teachers, examination boards and school book authors will get used to the graphics calculator and will take advantage of the pedagogical possibilities that these devices offer.
Secondly, research will be carried out concerning the role of computer algebra in the learning of mathematics and, more specifically, in the learning of algebraic concepts. Such a study was started recently by the Freudenthal Institute.
Thirdly, research will be carried out on the possibilities of computer algebra as a wide-range technology tool. A project that focuses on the use of a computer algebra environment in combination with a text editor (to write mathematical reports) and an Internet browser has been started at the Algemeen Pedagogisch Studiecentrum, an institute for improvement of (mathematics) education.
It is my hope that these developments will lead to a carefully considered implementation of computer algebra in secondary education.
Paul Drijvers (Freudenthal Institute, Utrecht, Niederlande, P.Drijvers@fi.uu.nl)
Der Artikel von Paul Drijvers ist ein Vorabdruck aus
J. Grabmeier, E. Kaltofen, V. Weispfenning (Eds.) Handbook of Computer Algebra Springer, 2000 http://www.springer.de/math/prep/compalg.htmlWir bedanken uns bei den Herausgebern für die freundliche Erlaubnis.
Heiko Knechtel (Bückeburg, mailto:HKnechtel@aol.com)
Wolfram Koepf (Leipzig, mailto:koepf@imn.htwk-leipzig.de)