Interactive geometry is about manipulating geometric object with a computer. DrGeoII is such an implementation within Squeak. The DrGeoII user interface takes care to provide good user feedback and intuitive manipulation. In the following paragraph I present you a simple activity example with right-angled triangle. You can click on every picture to visualize videos exposing the different parts and scenarii of the activity.
The annotation from the teacher in the document:
"Reproduce this figure using the geometric tools. You can use all the tools you want but pay attention to respect the properties of the original drawing."
The teacher drawing is prepared to graphically express the
following fact:
It is supposed the student has the knowledge related to the properties of right-angled triangle and its circle going through its summits. The objective for the student is to reproduce with the DrGeoII interactive geometric commands this drawing. The student must use his knowledge about geometry to reproduce a mathematical equivalent figure but he will be tempted to just do a visual reproduction, this is where DrGeoII and the interactive geometry is very interesting. Indeed, by direct manipulation the teacher or/and the student himself can realize there is something wrong in his construction.
In the following sections, I present several imaginative student solutions to this simple problem. Two naive solutions and two good solutions. In fact DrGeoII lets the student free to solve the problem in different way. The apparition of the interactive geometry within Squeak open new way to conceive interactive geometry activities. Indeed, while designing DrGeoII, I took great care of a smooth integration with other traditional object. Therefore it is very easy to mix Squeak objects (drawing, picture, widget, book, video,...) and DrGeoII to produce new and exciting learning experiences.
Student draws over the original drawing an identical figure without taking care of the properties of the figure. Basically it is just drawing segment and circle as with any traditional drawing software.
Then, it is very easy for the teacher to demonstrate this solution is wrong. Just moving a segment, a point, the circle shows there are problems in the construction.
In this solution the three summits of the triangle are located in the circle, nevertheless the triangle fails to be a right-angled one.
Again the interactive nature of the drawing helps to figure out it is wrong.
To do a correct figure, the student first constructs a
right-angled triangle (using the perpendicular tool). Then he uses the
property: in a right-angled triangle, the middle of the hypotenuse is
the center of the circle going through the triangle summits.
When moving the interactive figure, the student graphically
verifies it is correct.
As there is not only one solution, it is interesting to look
at other solutions student can propose. Here he uses another related
property: in a circle, a triangle defined with one segment as a
diameter is a right-angled triangle. This solution is even shorter as
it does not need the use of the perpendicular tool.
Then, check with direct manipulation of the figure, it is true
in any situation: move a point, a segment or the circle, the
construction is alway identical to the one provided by the teacher.
Here the example is about reinvesting student knowledge. In other situation it could be about discovering new properties or to express hypothesis then check for validity.
DrGeoII, as an interactive geometry software, simulates a geometry micro-world where the student can manipulate interactively geometric objects. It allows users to create geometric figures to be manipulated with respect to geometric rules. The program is usable with students from the primary through secondary levels.
In this simple example, I show you the idea of interactive geometry, I also illustrated the deep level of integration we have between Squeak and DrGeoII. Squeak encourages a constructivist approach with direct manipulation, modelisation of concept, experimentation. DrGeoII brings to Squeak the interactive geometry paradigm. When designing DrGeoII, I took care to do so that E-toys and other Squeak interactive objects can be integrated within a DrGeoII drawable. I think it is a key element for deeper hand-on activities.
If you want to know more about DrGeoII and related research work, you can contact me at <hilaire AT ofset.org>