RETHINKING GEOMETRY

  Dynamic

Elementary ^ Geometry

Dr. James M. Parks

SUNY - Potsdam

parksjm@potsdam.edu

 THE CASE FOR DYNAMIC GEOMETRY

   With the introduction of computers and creative software, the pedagogy in the field of mathematics is undergoing many changes.  However, with programs like Dynamic Geometry, the aim is not to abandon the methods of deduction and proof, but rather to enlarge the audience, so that more students can share in the sense of the nature of what it is ‘to do mathematics’.

   The van Hiele_s levels of geometric thinking:
     1. Visualization
     2. Descriptive/Analytic
     3. Abstract/Relational (Informal Deduction)
     4. Formal Deduction
     5. Rigor/Metamathematical

   A main goal of Dynamic Geometry (like GSP:Geometer_s Sketchpad, Cabri Geometry II, & The Geometry Inventor) is:
   _...to bring students through the first three 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 geometric constructions, manipulate figures, observe patterns (visualize), develop conjectures & informal proofs, and build/discover counter-examples.  This allows the user to gain understanding and conviction BEFORE they attempt proof!

 HOW TO USE DYNAMIC GEOMETRY

  GSP lists three ways which it can be used to teach geometry*:
     _ Investigations:    Build & manipulate specific
                                   constructions/figures to study some
                                   particular property (makes use of the
                                   dynamic aspect of GSP).
     _ Explorations:   Open-ended experimentation (uses all
aspects of GSP).
     _ Demonstrations/Presentations:  Can vary from classroom ‘lectures’
                                                         to ‘visua’ proofs.

  Specific GSP applications/tools:
     _ Drawing
     _ Constructions
     _ Transformations
     _ Measurements & calculations
     _ Tracing
     _ Animation
     _ Recording (Scripts)

   Method of use of GSP in the course:
Geometry Topics GSP Application(s)
     _Finite Geometries                       _Drawing (intro. to GSP)
     _Euclidean Geometry                   _Constructions
     _Non-Euclidean                            _Poincaré & sphere models
      Geometries
     _Transformation                          _Transformations
      Geometry

 RESULTS

   As is usually the case, the results of a change in pedagogy such as this do not immediately appear in the exams and other means of assessment.  I have noticed a slight increase in the gpa, and I have noticed an increase in the amount of time available to cover some topics, that is the use of GPS seems to speed up the presentation of certain parts of the course.
  The students do appear to have more interest and enthusiasm for geometry, and they seem to have a better grasp of certain parts of the course (like Non-Euclidean geometry).

THE FACILITY

   The campus facility at Potsdam where I teach the course is a combination computer lab and presentation room.  It is split into two parts: one half has 12 PowerPC Macs on individual tables arranged in a ‘U’ with seating for 30, and a projection system (which includes computer, video, laser disc, and electronic overhead projection); the other half is a free-form area with tables and chairs for 30, a white board, and the same projection system.  The entire room is networked and has a laser printer on line.  We also have a site license for GSP, so the program is available on all computers on campus.


* ‘Teaching Geometry with The Geometer’s Sketchpad’, Teaching Notes, Key Curr. Press, 1995.

REFERENCES

1. GSP: Key Curr. Press, POBox 2304, Berkeley CA 94702,
  800-338-7638, <http://www.keypress.com/>.
2. Cabri Geometry II: Texas Instruments, Dallas TX,
  <http//www.ti.com/calc/docs/cabri.htm>.
3. Dynamic Geometry Home Page:
  <http://www.edc.org/LTT/DG/index.html>.
4. Geometry Turned On Home Page:
 <http://forum.swarthmore.edu/dynamic/geometry_turned_on/>.
5. The Geometer_s Sketchpad Home Page:
  <http://forum.swarthmore.edu/sketchpad/sketchpad.html>.