I worked with Geometer's Sketchpad, Cabri, and related programs which using geometric constructions (ruler and compass, transformations etc.), measurements and calculations to create both static diagrams, and animations (loci etc.) for geometric explorations, for stimulating conjectures, for investigating relationships and possible proofs, for illustrating concepts etc. Used in the dynamic mode, these programs offer fundamentally different experience to the user than a traditional single drawing can. In fact, they make public the kinds of variations of detail in diagram which many of us had 'in our heads' but had difficulty showing to the students. By varying the irrelevant details, the student is encouraged to focus on the 'invariant' central facts (such as the medians of varying triangles meeting in a point). On site which contains samples of drawings and scripts, as well as demonstration versions of the programs Cabri and The Geometer's Sketchpad is the Dynamic Geometry page at the Math Forum
These programs were originally developed for high school teaching. They are now used in teaching middle school and high school, for teacher training, for undergraduate geometry teaching, and for explorations and presentations in research. A good sample of how these programs can be used for an undergraduate geometry course is Jim King's recent book: Geometry Through the Circle with The Geometer's Sketchpad, from Key Curriculum Press.
As part of this project, I visited Key Curriculum Press in Berkeley (the publishers of The Geometer's Sketchpad) and discussed ways of using it for undergraduate teaching, possible modifications to make it more suitable for this level of geometry, for scripting 'micro worlds' and other related issues. [They have an interesting program for 'Scholars in Residence' if you have some expertise and some interest in developing materials for use in high school mathematics teaching.]
Within this project, Jonathan Slater an undergraduate student in our Pure Mathematics Honours program and in the Concurrent Education program also worked on scripts for several micro worlds (projective, spherical and hyperbolic geometry in the plane). These provide limited tools for geometric constructions and transformations within plane models of the other geometries. He has now presented this work on a the web on his dynamic geometry page, along with some pointers to other intersing pages.
Working with several undergraduate students and a high school student on I. Yoglom's Geometric Transformations I, MAA, we found the program to be an essential tool for understanding the problems and the solutions.
I have experimented with occasional use of The Geometer's Sketchpad in a previous geometry course. Next fall I will make regular use of the program in Math 3050: Introduction to Geometries, for work with plane congruences and isometries, as well as some exploration of general problems in plane geometry.
One interesting question to consider, in the theory of geometries, is "What geometry is dynamic geometry?" What are the objects, the transformations, etc.? When are two dynamic geometry sketches equivalent? This is now a subject of direct investigation by Jonathan Slater and myself.