E-mail: schwarze@swhamm.de

There are many problems associated with the nature of teaching math, especially geometry: visualization of geometric properties in the range of its validity, proofs, showing the connection between geometric theorems and sometimes a lack of simple real life problems that lead to a geometrical problem.

In this paper I want to show in which way the internet can be used (beneath other media for exploration and visualization) to enrich the learning process in a stimulating way- for both the teacher and the learner. I would like to put the main emphasis on pre-college geometry.

Today the academic again reminds us of the learning process being first of all an individual, autonomous, exploratory and self-responsible process, which will then be highly efficient if it takes place in a rich environment.

A change from the focus on the teacher and the teaching process to the learner and the learning process will demand a change of the teacher's role.They will have to change from being an instructor to a 'learner enabler', somebody who accompanies and supports the individual learning process.

The learners will have to change from being a mere recipient to an active creator of their learning process. They have to be encouraged and enabled to take control of their own learning process, which means defining the aim, selecting the content and evaluating what they have learned.

In this classroom situation both teachers and learners need appropriate materials and tools to support the learning process mentioned above.

Here dynamic, interactive programs amd hypertexts, too, play an important role since the first generation of dynamic tools had been created. Since the internet and java-capable browsers have become more popular and can be used in school because of better hardware in the computer-labs teachers have more possibilities for getting information, for communication, interactivity, cooperation, for exchanging materials ready for use, etc.

I 'd like to describe how WWW-resources may help to publish the idea of new ways of learning , how to use dynamic geometry software to create more situations for investigation and exploration in the classroom:

learn:line - a (german) platform of information and cooperation (going to be translated into English)

The educational server of North Rhine-Westfalia called "learn:line" has been built up to be a workshop for teachers and students for educational topics and subjects- one of these is "computational geometry" for k12-teachers with a little "corner" for interested students, too.

In this work area teachers will find helpful materials how to teach with dynamic geometetric tools like Cabri, Sketchpad, Euklid. They will find an online-tutorial that demonstrates the special "educational power" -focussed on teachers who hardly have any knowledge and experience in dynamic geometry .They can see the possiblities and advantages when using these tools in the classroom with the help of examples and a lot of figures for themselves.This website gives information on good new media for geometry, supports teachers in experimenting with them, presents classroom situations where dynamic tools will gain ground for discovering and pursuing individual ways of learning and broaden the individual knowledge network.

Like the famous math forum in Swarthmore I would like to offer well-working classroom materials, lesson plans and ideas, that had often been discussed and modified in teacher training. German teachers start to use the corresponding discussion area only tentatively than their collegues for example in the United States- perhaps because of the lack of connection with the WWW at home and the costs for telecommunication that are still rather high in Germany.

Ka's Geometriepage und Mathe-Gallerie -more examples on my "private" homepage- (http://kunden.swhamm.de/Geometriepage/)

In general: good learning-environments based on hypermedia are advantageous for learning and understanding mathematics.The understanding of math requires the knowledge of details, conclusions and relations between single objects. If subjects are offered in a linked-up, but not linear structure it will be easier to build up a network of mathematical, geometric knowledge in the learner's mind.

Good learning-environments (not only for geometry) should offer e.g.:

• guided tours
• survey of the learning subject,table of contents, glossary
• meta level map (where am I?)
• some (recommended) provided paths
• different modi of representation and levels of activity
• hypertextual structures with internal and external relations
• some helpful biographical facts of famous mathematicians, real-life application,...
• exercises and contextual help
• possibilities to integrate other documents and visualazations- either static or dynamic, interactive tools and demonstrations

A glance at the market of new offline-media shows that only a few products match up to almost all criteria so that they could be called examplary in they way mentioned in the beginning. Learning environments in the WWW are suitable in particuar: HTML-based hypertextes are open, can easily be modified and beeing published in the internet, they can be improved, developed, discussed and can include practical advises for teachers etc.

Exemplary learning-environments for geometry (written in german) do not exist at the moment. But we will find some first attempts:

1. At the math department of the University of Bayreuth (Germany) the java-based dynamic geometry tool GEONET has been developed. Based on this tool there are some little hypertexts that offer diffent ways through the subject, for example " the order of the quadilaterals" and some (applied) exercises, too. Students of math education have dealt with further subjects (Thales, Pythagorean theorem). http://www.did.mat.uni-bayreuth.de

2. Supposing that learning takes place in an individual way and knowing that different students will choose different explanations and examples to understand essential geometric theorems and their proofs, I drafted a learning unit based on a worksheet about the Pythagorean theorem and its different proofs. I collected different proofs with a wide range of interactivity, knowing that students prefer different accesses to these proofs and related theorems. Some of the downloaded resources only visualize the theorem or are hypertexts that make understand the idea of the proof before it is written down in a more formal language. Some of the chosen websites contain java applets allowing the students to manipulate and experimentate interactively with dynamic constructions.

In addition to this students can use the PC-versions of Cabri, Sketchpad or Euklid to prepare some explanations for the other students.

So an oriented way of learning is encouraged by which thinking, finding out and thus learning and connecting different knowledge can be improved.

I used this learning unit in a ninth grade geometry class to stimulate the pupils to come up with their own way of building up their knowledge network; this meant that pupils have to decide for themselves what information is important, how much and what sort of help they require.

After a phase of individual work the students who worked in groups of two or three had to present their investigations and their results to all students of the class. It seems to be evident that discussion in a group of two or tree could intensify the real understanding of a given context. In addition to this better students had to analyse informations at a higher level. www.ham.nw.schule.de/projekte/swmathe/Uonline/

Beneath the PC-version of dynamic geometry tools like Cabri, Sketchpad or Euklid java-based constructions sometimes can complete demonstrations and geometric constructions for exploration in the classroom.

Some of the special advantages of these Java-applets are:

• complexe construction are just ready for use and can be combined with worksheets, that are appropriate for the special classroom situations
• most java applications for geometry constructions only "allow" manipulations that are provided by the constructor himself (an advantage for special problems where students otherwise can "destroy" the construction)
• the applicatons run at any platform and do not require a special programm
• the teacher can build up a library ready to use and integrate in his own concepts or in the intranet

Some examples of websites with dynamic geometry applets:

Manipula Math: http://www.ies.co.jp/math/java/geojava.html

Cabri Java

JavaScetchpad: http://www.keypress.com/sketchpad/java_gsp/

Cut-the-knot

IcosaWeb: