The development of dynamic geometry in teaching is
arousing the creation of many pedagogical Web sites that cannot
take benefit from this technology throughout their pages at the
present time. Classical techniques in figures animation on the Web
are either too limited, or too difficult to implement. The project
Cabri-Java plans in a first time to develop a Java applet allowing
to bundle all the advantages of dynamic geometry in active
figures. The realization of a complete Java application for
creating and manipulating figures is being studied.
Table of Contents
Dynamic Geometry and Teaching
The developement of Dynamic Geometry has been closely related
to the development of Direct Manipulation.
Actually dynamic geometry roots probably long time ago when
some mathematicians including Clairault (XVIII) or some more
recent authors have considered the idea of moving elements of
figures in order to illustrate (geometrical) phenomenons and to
prove theorems .
Dynamic Geometry with Direct Manipulation is often embodied in
computer environments offering to the user - teacher or student -
an opportunity to develop intellectual activities based on
geometrical knowledge. In other words the user is placed in a
microworld, in the sense of what has been called by Seymour Papert
when writing his most famous book Mindstorms,Children,
Computers, and Powerful Ideas [1]
(Cf the Logo orientation). In contrast to Logo in a microworld
centered on geometry, as implemented in Cabri or Geometer's
Sketchpad, the mathematical content to be taught plays a key role.
Through Direct Manipulation the student can construct his own
sophisticated construction, mobilizing for that more or less
complex geometrical knowlege, and he can interact directly with
the representations of the theoretical geometrical objects. Key
features of such Dynamic Geometry environments are
- a mathematically solid base of the implemented
geometry
- a user's friendly Man-Machine interface to allow for
immediate familiarity with a substantial part of the
environment
- a fast feedback of the system to allow the user to have
control of the behaviour of the objects (s)he has
constructed.
For many years, works [2,3,
4, 5] have shown the
importance of direct engagement of the learner using construction
software for manipulating of geometrical objects. . Research made
around the software Cabri-géomètre
[6] underlines the preponderant role
of student's personal investment.This frees his mind from the
difficulties he is faced with when constructing geometrical
objects that can be often difficult to draw, and he can freely
analyze problems that are posed for him in an active and dynamic
way. Cabri, as a multilingual software available on Macintosh or
PC platforms, but also on pocket calculators [7]
has today exceeded the million of copies sold around the world and
has aroused the creation of many collaborative groups such as
"Cabri clubs". Its use in classroom, but also in auto-learning out
of school time has been so much greatly developed so that hundreds
of users meet together during Summer Institutes organized about
this software.
This tool for dynamic geometry is also used in a fruitful
manner in other areas where geometry serves as mathematical
support for scientific modeling in optics, electronics, mechanics,
astronomy...
That kind of tools turns out to be also well suited in
contexts involving disabled students being in difficulties, either
at the school level (personalized support), or at the medical
level (TéléCabri
project [8] , a distance learning
project for hospitalized children).
Geometry on WWW
From its beginning, the Web appeared as an indispensable tool
to allow in the first place for a better collaboration between
teachers. They were able to exchange their courses, discuss on
concrete documents about best tools for teaching. Mailing lists
have been created, many Web sites have extended through the world;
some of them include particurlarly dynamic
geometry.
Other teaching experiences where the main centre of interest is
the creation of Web sites by pupils themselves, have spread out
recently. In this framework, pupils become to be actors of their
learning, to communicate with others classes throughout the world,
and to share their growing knowledge. Faced with this new
situation, some students who were exposed to a traditional
teaching feel comfortable in a context where personal initiative
is more requested. Generous ideas of youth to help other pupils
who perform badly at school or are experiencing momentary
difficulties (for example hospitalized) often motivate children to
give their best to put exercises and comments on the Web, or
edutainment activities or corrected versions of homework (see for
example this site of a college from Grenoble[6].
How to manage then to gather benefits of dynamic geometry and
the Web?
The first solution consists in describing in phrasing of the
problem and by screen copies, the exercises that have to be done
with the help of a software as
Cabri-géomètre. The major disadvantage of
this method is a loss of time caused by the installation of
geometrical elements of the figure before being able to really
tackle the problem posed. The play side of the learning process
through the Web is then totally erased.
A second possibility is to allow the direct transfer of files
coming from the dynamic geometry software from the Web server.
This solution needs the definition of a MIME type for the transfer
of files, the configuration often manual of the browsers as well
as the preliminary existence of the software
Cabri-géomètre on the client's computer. The client
will have to be able thus to launch the software at the same time
that the browser, as an application helper. If all these
conditions are gathered, the user will be able thus to take full
advantage of the capacities of the software, but his work will not
be directly integrated in the page Web where the figure will have
been unloaded. It will have then to make back and forths between
the software and the browser to follow for example the
instructions given in the Web page.
These two methods have therefore the disadvantages to require
several preliminaries for the use of dynamic geometry through the
Web :
- to have the software Cabri-géomètre,
which does not run outside the compatible systems DOS, Windows
or MacOS.
- to have its browser configured in order to launch Cabri
directly from the downloaded figures or macros .
- to have an enough powerful computer to allow the launching
of the two softwares concurrently : the browser and the dynamic
geometry software.
Until a more adapted solution is found, some sites use
nevertheless this method as for example the site abraCAdaBRI
[10] available on the server of the
Cabri project. But all the webmasters of these sites request to
have the possibility to place animated geometry figures directly
in their Web pages.
Classical Supports for Animations
Let's study the ways to put animations done with an dynamic
geometry software directly in pages of a Web site.
QuickTime or AVI which are animations people use
mostly have formats that can be useful to code animated sequences
captured from a direct use of the software. But despite these
formats are taking up a growing part on various platforms, some
incompatibilities with some systems are remaining. Moreover the
size of corresponding produced animations remains often too
important for slow connections.
To overcome the impact of this difficulty a solution can
consist in producing graphic animations in the format of animated
Gifs, directly supported by all recent browsers. Today, many tools
allow to create such animations by a sequence of files taken from
"screen snapshots". By limiting the number of colors (a coding on
4 or 5 bits is often enough), the size of the animated files is
often smaller than the size of corresponding QuickTime or AVI
files. The official Cabri site shows some simple
examples of animations that can be produced.
These animations have nevertheless a major defect : they are
only " pictures " of an animated sequence and in front of them
pupils remain passive. Any direct commitment disappears and with
it one of the essential contributions to the Sciences of Education
of these last years.
Then, how to allow to create a real interactivity between pupil
and animated picture on the Web ?
The first solution is to use one of the most well-known
softwares in animation like Macromedia Shockwave to create
animated sequences based on a real interaction scenario ; the free
use of the Shockwave plugin allows then the integration of
animation into Web pages. It is this way which has been used in
some pages of the site AbraCadaBRI [10],
whose a copy figures hereafter.
![[Shockwave animation]](fig2.gif)
This method involves several major disadvantages :
- the cost of the software Director making
Shockwave animations often too expensive for school
budgets.
- the necessary time to create an animation (the author of
the animation shown in the above example, claims he has needed
a complete day to create it).
- the interactivity too heavely controlled by the system
reducing the freedom to explore and discover new
solutions.
- the additional memory size asked to the browser when
loading the plug-in which often causes a lockout of this
plug-in on limited configurations.
An other solution would be to realize a specific plugin
allowingto recreate the Cabri environment for figures integrated
into the Web pages. Writing this plugin with a native code
presents pros and cons. On the one hand, it is the best way to put
Cabri on the Web, but on the other hand, it is necessary to
rewrite the plugin for each existent or future system. This last
point added to the problem of the preliminary installation
inherent in plugins led us to turn down this possibility.
Cabri-java
![[Pythagoras]](fig3.gif)
- The solution finally chosen to put active figures on the
Web was to use the language object Java. This choice has been
taken for several reasons:
- the Java multiplatform support through virtual machines for
each architecture,
- the increasingly close integration of Web and Java in
recent browsers,
- the object-programming well adapted for realizing animation
of geometrical structured objects.
The project Cabri-Java in its current phase devotes in a first
time to writing an applet allowing to animate a dynamic
geometry figure by resuming as best as possible ergonomics of the
software Cabri. This is sometimes difficult to do because of gaps
contained in first tools of Java development (JDK 1.0.2). By
example, the only standard classes known by browsers do not
propose direct possibility to change the mouse cursor's form in an
applet, neither to display pop-up menu in the applet zone itself.
At the graph level, the aspect for dotted or bold lines is not
planned, unless to program it oneself, what would not be efficient
compared with the selection of native methods on each
architecture. All these gaps are nevertheless on the way to being
filled by JDK 1.2 and the new standard Java 2D API.
Once Java has been chosen, it remains to determine the way to
transmit to the applet the numerous parameters allowing to define
a geometrical figure even not very complex. A solution could be to
describe all objects and their properties with the help of PARAM
tags of the applet. This is the way chosen to communicate
geometrical data to applet in two other dynamic geometry java
projects [11, 12]
But, on the one hand it is soon very difficult to create manually
all these data and on the other hand HTML files would have become
then very heavy. The undertaken choice has been to preserve all
the files generated by the software Cabri by putting them on the
server in order the applet could read them, only one tag PARAM was
necessary to indicate the file name and its position on the Web
server.
The advantage of this solution is an easy publication, but the
disadvantage is that it requires to have the software Cabri, what
is not a real restriction for webmasters who are developing sites
devoted to the use of this software in classroom...
![[Cabri-java figure]](fig4.gif)
How to do in practice to publish an active figure ?
- It is necessary to put on the Web server binary files Java
(.class) of the applet or a non compressed zip archive of these
binaries. (One can also reference an other server that have the
applet binary through the parameter CODEBASE).
- In Web pages, the whole tag describing the applet can be
for example:
<APPLET CODE="CabriJava.class" WIDTH=600 HEIGHT=400
ARCHIVE="CabriJava.zip">
<PARAM name = "lang" value = "en">
<PARAM name = "file" value = "figures/College/Star">
</APPLET>
The "lang" parameter can for the moment take " fr" values
for the French messages and "en" for the English ones.
The "file" parameter indicates the path (here the relative one)
of the figure on the server.
- Other decoration parameters have been added like tags for
Web page's background :
<PARAM name = "background" value =
"images/bg.gif">
<PARAM name = "bgcolor" value = "#F0F0A0">
<PARAM name = "border" value = "0">
In its current preliminary version, the CabriJava applet
already allows to put active figures in Web pages, in which the
user can drag geometrical objects while preserving the geometrical
properties defined in their creation. A demonstration page is
available on the Cabri project's server [13]
and allows to better test the possibilities.
Even if all the possibilities of the software Cabri are not yet
transcribe in Java (loci, conic...), a first pedagogical use has
already been done on the Mathematical Server of La Réunion
[14]
To the current stage, Cabri-Java suffers from slowness of
virtual machines proposed with browsers. But a new generation of
VM (virtual machines) is coming, implementing JIT (Just-In-Time)
compilation technology, allowing to obtain a feedback really
sufficient during objects draging .
Future Work
![[bicycle]](fig5.gif)
In parallel with the work consisting in completing the
CabriJava applet to integrate in it most of the
possibilities offered by the software itself, the project is
developing into two ways:
- implementing "a priori" or on action animation
aspects : Cabri already allows to animate figures on user's
action by using the "animation spring" tool as in the opposite
figure. But, it seems also useful to be able to animate figures
on the Web immediately from their display in case of no
intervention from the user. In order to specify the drag that
should be undertaken, a supplementary parameter of the applet
will be able for example to transmit messages to named objects
of the figure through a script language already defined in the
framework of Cabri-script project for communication between
Cabri figures.
- conceiving and realizing a Java application for
constructing and manipulating Cabri figures: the equivalent of
a multi-platform Cabri. Even if this program will never have
the fluidity of Cabri native applications, it will be very
helpful to create figures that could be then used by
CabriJava applet. An other interest of such application
will be to enable a best knowledge of dynamic geometry by
allowing a larger distribution of its approach and its use in
many active figures throughout Web pages. To develop such an
application presenting a same user- interface on all platforms,
it is envisaged to use Java Foundations Classes defined
by SUN [14]
References
- Papert S "Mindstorms,Children,
Computers, and Powerful Ideas" 1980 New-York
- Schneiderman B "Direct
Manipulation: a Step Beyond Programming Languages", IEEE
Computer (16)8 57-69 1983
- Laborde JM "Des connaissances
abstraites aux réalités artificielles, le concept
de micromonde Cabri" dans Environnements Interactifs
d'Apprentissage avec Ordinateurs, pp 29-41, Eyrolles Paris
1996
- Laborde JM & Strasser R,
"Cabri-Géomètre : A microworld of geometry for
guided discovery learning", Zentrablatt für Didactik
der Mathematik 5 p. 171-177, 1990
- Schumann H "The design of microworlds
in geometry based on a two-dimensoinal graphics system devised
for second education", INT. J. MATH. EDUC. SCI. TECHNOL.,
1993, VOL. 24, N°2, 231-250
- Official Cabri project site
http://www-cabri.imag.fr/index-e.html
- Texas Instruments Cabri site
http://www.ti.com/calc/docs/cabri.htm
- TeleCabri project site
http://www-cabri.imag.fr/TeleCabri/
- Desigaux M. Collège Jules
Flandrin
http://www-cabri.imag.fr/TeleCabri/PassionRecherche/
- Martin Y. abraCAdaBRI site
http://www-cabri.imag.fr/abracadabri/
- Jackiw N. JavaSketchpad site
http://forum.swarthmore.edu/dynamic/java_gsp/
- Joyce D. Geometry applet
site
http://aleph0.clarku.edu/~djoyce/java/Geometry/Geometry.html
- Kuntz G. Cabri-java project
1997
http://www-cabri.imag.fr/cabrijava/
- Hakenholz E. First real experiment of
Cabri-java 1998
http://www.ac-reunion.fr/pedagogie/covincep/icosaweb/GeomJava/activite/Quatre/Docs/Pytha/Pytha.htm
- Sun Microsystems Java Foundations
Classes
http://java.sun.com/products/jfc/