Right before Mathfest in LA, a presentation on "Alternative Approaches to Geometry" was given as part of the (by invitation only) Project NExT Program. (Similar presentations were made at last year's Project NExT activities right before Mathfest in Providence.)
A paper entitled "Geometry is Alive -- Long Live Geometry!" kicked off a contributed paper session (with 37 speakers) on Geometry In The Classroom In The Next Millennium at the Joint Meeting of the AMS/MAA in San Antonio, 14-16th January.
On Friday, 13th February 1998, a whole-day geometry symposium entitled
Exploring New Frontiers in Geometry: in the World Around Us and in Our
Classrooms (co-chaired with
David Henderson of Cornell
University) was held, in Philadelphia, as part of
the AAAS Annual
Meeting and Science Innovation Exposition.
For detailed information, click here.
All of the pictures here were taken at A Hands-on Discovery-based Approach to Geometry, a three hour short course given at the College of Charleston, Charleston, South Carolina, on 13th March 1998, during the 77th Annual MAA Southeastern Section Meeting.
The geometry minicources/short courses are be based to a significant degree
on the philosophy and methods of
David Henderson of Cornell University, whose book
Experiencing
Geometry on Plane and Sphere (
Prentice Hall), whose second edition is entitled Experiencing
Geometry in Euclidean, Spherical and Hyperbolic Space,
shows that geometry can be approached in a hands-on, discovery-based way,
so as to take advantage of natural curiousity and lead people into sometimes
unfamilar waters where they learn for themselves and emerge excited.
Too often, geometry is presented to students in an axiomatic way, divorced
from reality and their own experiences. Many students--even good ones who
may major in mathematics--have learned to hate geometry before they ever
show up in college-level classes. Yet, recent successes at colleges and
universities in the USA and overseas show that for both liberal arts students
and mathematics majors, curiousity about practical, real-world considerations
(e.g., symmetry, locations, distances, areas and navigation) can be turned
into a genuine desire to learn mathematics when the material is approached
in the right way.
We will offer a hands-on, discovery-based workshop which shows by doing how one can take advantage of such natural curiousity. For instance, spherical geometry has vanished from the curriculum over the last fifty years, leaving students and teachers alike quite ignorant about the geometry of the surface we live on. Yet it is not hard to motivate students to explore spheres, where the geometry is quite different from the familiar planar case.
Three provocative questions--with non-obvious answers--which can be used to generate thought and discussion, are:
Hands-on experiments with paper, rubber-bands, string, wool and spheres lead to natural considerations such as: what is the intrinsic geometry of a surface? (what does it look like to a bug?) Is our world (approximately) spherical? How can we tell? What shape is the universe?
Apart from the practical applications of geometry to navigation, optics, astronomy, surveying, drafting, art, computer vision, robotics, graphics and molecular modeling, we have found that students appreciate geometry for its own sake when they have discovered a lot of it for themselves, with instructor encouragement and careful guidance.
We will show how to make geometry alive, hands-on, interactive, and fun. The emphasis is on on discovery, understanding, cooperative and group learning and applications rather than formal axiomatic proofs. Instructors and students can discover and work together as a team in this context, and if the instructor so desires, the class can also get away from the standard rigid lecture/exam system.
This approach was used successfully in Spelman College's 300 level geometry
course in the springs of 1997 and 1999, whose participants included mathematics majors
and future teachers, as well as English and Psychology majors.
Personal observation:
I am one of a generation of mathematicians whose university education included no geometry at all, either the undergraduate (University College Dublin, Ireland) or graduate (Cornell) levels. I saw plenty of good old Euclidean geometry at school, up to the age of about 15, but that was it. Looking back on my generally excellent notes from my days at UCD, I see hardly any pictures of any type. A few in complex analysis, metric spaces and topology, but very few. Certainly none in algebra, which is what I chose to specialize in at Cornell. (I later found out that then, as now, there was a wealth of geometry courses on offer at Cornell - presently 8 or 9 are offered at the undergraduate level alone - but surprisingly I was essentially oblivious to that during my five years there.)
I entered university unaware that one could pursue a degree or a career in mathematics, mistakingly believing that despite the subject's undeniable beauty, it was all as old as the hills, static, and at best a servant of other disciplines. I left graduate school labouring under similar illusions concerning geometry. Two of life's greatest pleasures have been discovering that I was wrong on both counts.
With any luck, we are getting closer to the day when books with the word
"geometry" in their titles can all be expected to be generously illustrated.
Perhaps we are coming out of the dark ages during which formalism and
abstraction have been emphasized to the exclusion of any pictorial intuition.
(This is http://www.spelman.edu/~colm/geom.html,
click here to return to main page.)