A Dynamic Geometry Course

Dr. James M. Parks

SUNY-Potsdam

parksjm@potsdam.edu

Abstract:  Over the past 3 years I have made major changes in our junior-level geometry course so that it is now a self-discovery course which uses the Geometer’s Sketchpad (GSP) software.  The course topics are still traditional: Finite Geometries, Euclidean Geometry, Non-Euclidean Geometries, and Transformational Geometry.  However, all of these topics are now addressed in a computer/GSP setting.  The recent development of an interactive internet version of GSP, JavaSketchpad, will bring a another new aspect to the course: now students will be able to ‘publish’ their work on the internet and to share ideas in this interactive environment.  New improved pedagogical methods have also been developed, and new techniques for covering some material have been discovered as a consequence of these changes.

The case for Dynamic Geometry

  With the introduction of dynamic geometry software, like "The Geometer's Sketchpad (GSP)", "Cabri Geometry II", & "The Geometry Inventor", pedagogy in the field of mathematics has been changed forever.
   The aim of these programs is not to abandon the methods of deduction and proof, but rather to enlarge the audience, so that more students can experience the nature of what it is like to "do mathematics_!
    In the van Hiele’s studies on the way students learn geometry, the following series of levels of geometric thinking were discovered:
     1. Visualization
     2. Descriptive/Analytic
     3. Abstract/Relational (Informal Deduction)
     4. Formal Deduction
     5. Rigor/Metamathematical
   Rather than ask the student to begin their study of geometry at the level of 4. Formal Deduction, which is what most textbooks do, a main goal of dynamic geometry software is:    “ ... to bring students through the first three (van Hiele) levels, encouraging a process of discovery that more closely reflects how mathematics is usually invented: A mathematician first visualizes and analyzes a problem, making conjectures before attempting a proof.”*
    In other words, empower the user with the tools to produce accurate geometric pictures & constructions, manipulate figures, observe patterns (visualize), develop conjectures & informal proofs, and build/discover counter-examples.
   This, in turn, allows the user to gain understanding and conviction BEFORE they attempt a formal proof!

* “Teaching Geometry with The Geometer’s Sketchpad”, Teaching Notes, p.1, Key Curriculum Press, 1995.

MA404 Elements of Geometry, the course

  The course covers the following four main topics:
         1.  Finite Geometries
         2.  Euclidean (& Hilbert's) Geometry
         3.  Non-Euclidean Geometries
         4.  Transformational Geometry

    In each topic there are activities using the GSP software to help visualize and analysis the material being covered.
    For example, in Finite Geometries one of the first topics is the concept of consistency of an axiomatic system.  This is an excellent time to introduce the software and to build models of the systems being studied (see Example1).  These models can also be used to study the independence of the axioms.
   In Euclid's Geometry, propositions which involve construc-tions are ideal for using GSP.  The students are not only able to follow Euclid's arguments step-by-step, but they are able to "see" what the given argument really does (or doesn't) do.  And, they can record their work, not just the end product, but their actual construction steps used to get their final result (these are called "sketches" & "scripts" in GSP).  The students thus have "ownership" over their results (see Example 2).  This approach can also be used in Hilbert's geometry.
   In the Non-Euclidean Geometries, there are extensive dynamic models of Poincare's disc and Spherical (Elliptical) Geometry available from GSP, which help tremendously to reinforce the axioms and many of the results for these geometries (see Examples 3 & 4).
   In Transformational Geometry the GSP software really excels.  Not only are all of the usual basic transformations available, but the user can build their own transformations from compositions of these transformations.  This really helps drive home the Classification Theorem for Isometries (see Example 5).
   A new aspect of the GSP this year is "JavaSketchpad", a program which allows the net user to _publish_ their work on their webpage and to interact with sketches published by others on the internet.  This program allows the user to share their work with others who do not have 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, revised & enlarged", Math. Dept. Publ., SUNY-Potsdam, 1997.
 5.  King, J. & D. Schattschneider, ed's., "Geometry Turned On: Dynamic Software in Learning, Teaching, and Research", MAA Math Notes 41, 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>.
4.  "Geometry Turned On" Home Page: <forum.swarthmore.edu/dynamic/geometry_turned_on/>.