CAME: Computer Algebra in Mathematics Education

Invited Workshop at the Weizmann Institute, Israel, August 1-2, 1999

Exploring CAS as a pedagogical vehicle towards expressiveness and explicitness in mathematics

Weizmann photo
Weizmann Institute: Feinberg Graduate School with Koffler accelerator tower behind.

Themes of the workshop

Computer Algebra Systems (CAS) such as Derive, Mathematica and Maple are becoming of increasing interest in mathematics education. At present, though, we see too little interaction between the CAS research community and the main mathematics education community. This meeting was intended to serve as a bridge between these two communities. We wanted to address two issues in particular: It is a long-standing problem in mathematics education to connect research and classroom practice; as CAS technology increasingly impacts on mathematics curricula, the challenges and opportunities for mathematics education to inform curriculum change are considerable. The CAS-related research questions that we wished to highlight at the workshop were to do with what we labelled explicitness and expressiveness. In using a CAS, a particular explicit symbolism is forced: each input requires a particular forced way of viewing things and expressing relationships and the output needs to be interpreted similarly. Because a CAS is based on a programming language, it provides expressive power for its users, it is possible to express ideas (mental objects) in a concrete form (visible objects). It seems reasonable to assume that in the tensions set up between thoughts and explicit expressions there is considerable scope for researchers to understand mathematical learning better.

Pre-proceedings papers (Research fora, Keynote talks)

The complete set of papers for the four research fora, and the keynote talks (to navigate, use the bookmarks facility in Acrobat Reader):

The papers in separate parts are available below.

PDF documents can be viewed and printed using the free Adobe Acrobat Reader, which is available for all major computer systems. Download from: USA, United Kingdom, France, Germany, Australia.

Research fora

The four research fora were modelled on the format of the "research forum" used at PME (Psychology of Mathematics Education) conferences: an invited presentation and a reaction were prepared and published in advance in the Pre-Proceedings. Each forum then consisted of talks by the presenter (30 minutes) and the reactor (20 minutes) followed by questions and discussion.

Forum 1: Jean-baptiste Lagrange (presenter) and Steve Lerman (reactor),
A didactic approach to the use of CAS to learn mathematics
[CAME-Forum1.pdf (70 KB)]

Forum 2: Edith Schneider and Ted Eisenberg,
The use of CAS in teaching mathematics: Reflections on possibilities and problems of cooperation between theory and practice
[CAME-Forum2.pdf (52 KB)]

Forum 3: Paul Drijvers and Kaye Stacey,
Students encountering obstacles using CAS: A developmental-research study
[CAME-Forum3.pdf (138 KB)]

Forum 4: Nurit Zehavi and Anna Sierpinska,
Didactical use of CAS in story problems
[CAME-Forum4.pdf (67 KB)]

Reactions and comments to the fora collected during the workshop: Comments.pdf (11 KB)

Keynote talks

[CAME-Keynotes.pdf (46 KB)]

John Berry: CAS as a mentor for the apprentice mathematician

Richard Noss: Rethinking abstraction in the light of technology [Powerpoint slides from the talk (94 KB)]

Amitai Regev: Computer algebra in mathematical research

Explicitness and expressiveness, research and practice: Summary and discussion

In this session, Phillip Kent spoke briefly about the idea of expressiveness, and you can read some expanded notes on the subject here: The participants then split into groups for discussion of the following points:
  1. The links between "theoretical" work and classroom practice -- with respect to the role of CAS
  2. The place of CAS research and CAS-related activities within mathematics education as a whole
  3. The expressiveness of CAS
  4. The role of "techniques" (praxeology) -- as introduced in Jean-Baptiste Lagrange's presentation
  5. Black boxes and transparency
  6. Top-down versus bottom-up approaches to curriculum design -- with respect to CAS

Overview of national activies with computer algebra

One participant from each of the countries represented at the workshop was asked to provide a summary/overview of CAS projects (curriculum or research projects) going on in their country. The following document is the result:

Journals related to CAME activities

The editors of the following journals were present at the workshop, and expressed an interest in receiving contributions relating to computer algebra and mathematics education:

Discussion of post-workshop activities

It was agreed that the present web page would form the informal proceedings for the workshop. Some of the presented papers may be developed into full papers for appearance in the journals mentioned above.

Another CAME workshop was proposed to be held just before or after PME-25 in Utrecht, Netherlands, in July 2001. There was a suggestion that any future workshops should put a stronger emphasis on the WORKING of participants, with less time spent on presentations and discussions. Keep an eye on the CAME diary for future discussion of this.

In the meantime, CAME members will be meeting more and less formally at venues such as the annual ICTCM conference in the USA, and the bi-annual Derive/TI92/TI89 conference (July 2000 in Liverpool, UK).

Attention was also drawn to a forthcoming ICMI Study on The Future of Algebra, which begins with a conference in Melbourne, Australia, in December 2001.

Workshop organising committee

Nurit Zehavi, Weizmann Institute, Rehovot, Israel. ntzehavi@weizmann.weizmann.ac.il

John Monaghan, University of Leeds, Leeds, UK. j.d.monaghan@education.leeds.ac.uk

Phillip Kent, Imperial College, London, UK. p.kent@mail.com

Giora Mann, Levinski College of Education, Israel.

Joel Hillel, Concordia University, Montreal, Canada.

CAME reports - CAME home page

This page last modified on 24 July 2000.