Designing a Dynamic Geometry Course
Dr.
James M. Parks
SUNY-Potsdam
A talk given in the MAA
Special Session on Geometry in the Classroom in the Next Millenium, at the
AMS/MAA Jt. Mtgs., San Antonio, Jan. 13-16, 1999.
Abstract: Dynamic
geometry software has changed forever the way that we work in geometry and the
way that we teach geometry. At Potsdam State College all students in our
7-12 Math/Educ Program must take a geometry course. Most students take
our junior-level Elements of Geometry course (in addition many other students
take this course as an elective in our major). Over the last 3-4 years I
have made major changes in this course so that it has evolved from the
traditional lecture/text/ problem-solving course into a project-based self-discovery
course using the dynamic geometry software Geometer’s Sketchpad (GSP).
The course topics are still
somewhat traditional: Finite Geometries, Euclidean Geometry, Non-Euclidean
Geometries, and Transformational Geometry. However, all of these topics
are now addressed in the computer/GSP setting. The use of dynamic
geometry software allows the student (ala van Hiele) to visualize the problem
or concept being presented, helps them describe and analyze what they are
observing, and allows them to make abstract connections, informal proofs and
conjectures based on their observations. They are then convinced of the
result and ready to proceed to the next step of constructing a proof to
substantiate their claims and observations.
The recent development of an
interactive internet version of GSP, JavaSketchpad, will bring a new aspect to
the course. Students will now be able to ‘publish’ their work on the
internet and to share ideas in an interactive environment. This new
aspect of the course (and the material) will of course have to be
addressed. New improved pedagogical methods have been developed, and new
techniques for covering some material have been discovered* as a
consequence of the change to a Dynamic Geometry course. Examples of these
methods for covering material and of the computer projects will be given as
time allows.
*see my article
in GEOMETRY TURNED
ON: Dynamic Software In Learning, Teaching, & Research, ed. by D.
Schattschneider & J. King, MAA Notes #41.
The Topics & GSP
1. FINITE GEOMETRY
Use GSP to build models of finite
systems. These models can then be used to study the consistency of the
axioms, the independence of the axioms, to make conjectures in the system, and
as a guide to building proofs of conjectures in the system.
2. EUCLIDEAN GEOMETRY
All of the traditional Euclidean
constructions can be done on GSP, and the results can be recorded for future
use. This gives the students ‘ownership’ of their results, and they can
save them & retrieve them at any time for future use.
3. NON-EUCLIDEAN GEOMETRIES
There are dynamic models available from GSP
for the Lobachevsky/hyperbolic geometry (see "Poincare's disc"), and
for the Riemann/elliptic geometry (see "Elliptic Geom.").
4. TRANSFORMATIONAL GEOMETRY
All of the basic transformations are
available on GSP, and the user can build their own transformations from
compositions of these. Many applications which use transformations are
also available from GSP.
References
TEXTS:
1. Lockwood, J. & G. Runion, "Deductive Systems:
Finite and Non-Euclidean Geometries", NCTM, 1978.
2. Heath, Sir T., "EUCLID, The Thirteen Books of THE
ELEMENTS, vol.1, 2nd ed.", Dover Publ., 1956.
3. Hilbert, D., "Foundations of Geometry, 2nd
ed.", Open Court Publ., 1994.
4. Parks, J., "Transformational Geometry: A
Workbook", rev., Math. Dept. Publ., SUNY-Potsdam, 1997.
SOFTWARE & WEBSITES:
1. "GSP: The Geometer's
Sketchpad", v.3, & "Java Sketchpad Center", Key
Curriculum Press: <www.keypress.com>.
2. The Geometer's Sketchpad @
The Math Forum: <forum.swarthmore.edu/sketchpad/sketchpad.html>.
3. Dynamic
Geometry Home Page: <www.edc.org/LTT/DG/index.html>.