A Proposal for Filling the Gap Between the Knowledge of a CAS and its
Application in the Classroom

AUTHORS: Justo Cabezas (1), Maria de la Vega Vara (2),
         Eugenio Roanes-Lozano (3,*), Eugenio Roanes-Macias (3),
         Pedro Ortega (5), Concepcion Romo (3)

(1) High school Professor (Ret.), Badajoz, Spain
(2) High school ``Sierra de San Pedro'', La Roca de la Sierra, Badajoz, Spain
(3) Universidad Complutense de Madrid, Algebra Dept., Madrid, Spain
(4) High school ``Celestino Mutis'', Madrid, Spain &
    Universidad Autonoma de Madrid, Madrid, Spain

(*) Corresponding author. e-mail: eroanes@mat.ucm.es

EXTENDED ABSTRACT:

We strongly support the use of new technologies in the mathematics classroom
because they improve concept acquisition through an increased experimentation
and the possibility of avoiding tedious computations.

Nowadays personal computers have an affordable price and there are excellent
software pieces for mathematics, like computer algebra systems (CAS), dynamic
geometry systems (DGS)...  We believe that the mathematical education of the
XXIst century requires computers with mathematical software in the classroom
plus well trained teachers.

Spain is organized in 17 regions with their own parliaments and the possibility
to decide on many topics of educational policy.  Extremadura is a region in SW
Spain, sharing a long border with Portugal, and consists of two provinces:
Badajoz and Caceres.

An innovative project supported by the regional government has transformed all
classrooms of all high schools (12-18 year old students) in Extremadura into
computer labs, with a personal computer for every two students running a local
version of Linux (LinEx).  Only free software is allowed to be used in these
classrooms.

The local government also organizes free courses in ``teachers centers''
located at the big villages in order to introduce the different pieces of
software to the teachers.

Although the situation in Extremadura can be considered ``ideal'', the
percentage of teachers that really use computer tools in their classes is low.
Moreover, among those who use them, the number of lessons treated with the
computer is also low.

If the teachers have the hardware, the software, the training and a positive
attitude, why are most of them reluctant to apply these media in their classes?
We believe the main reason is that there is a gap between their training and
the activity they have to develop.  Their training and the books available are
usually focused on mastering the tool, not to be able to develop applications
to be used by a high school student.

Consequently, regarding the use of CAS, we are developing a collection of
activities that try to fill this gap.  They are completely finished
applications (to be spread both printed and in electronic format) that can be
directly used in such an ideal environment.  They have been chosen according to
their suitability or the determinant roll that the CAS has in that particular
application.  For instance, we have developed in detail, at the student's
level, among others:

o permutations (bijections), using the advantages of exact arithmetic for
  dealing with big numbers

o formal continuity of functions, using the possibility of calculating
  (symbolically) limits

o probability calculus in continuous random variable using probability density
  functions and the possibility to compute definite integrals instead of using
  statistical tables or statistical computer packages

o ...

The characteristics of the environment suggests to choose easy-to-use
cross-platform compatible software, so that the students can also work at home,
despite the operating system they have at their home computers.  Non-commercial
pieces of software that fulfill this requirements are the CAS Maxima/wxMaxima,
the DGS GeoGebra and the spreadsheet OpenCalc of OpenOffice. We have chosen
wxMaxima to develop the applications mentioned above.