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Geometry and Axiomatics
Posted:
Jan 26, 1994 4:44 PM
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Geometry and Axiomatics:
Since it appears that Joe Malkovitch and I have been misunderstood,
let me restate some of the risks of using axiomatics to teach geometry.
(1) A course in Axomatic Geometry is not a course in geometry.
Twenty years ago, when experimental treatments for epilepsy
seperated the left and right brain, experimenters found that
understand of shape, and visual patterns lay in one hemisphere
but Euclidean Geometry lay in the other. The 'knowledge' developed
in Euclidean geometry did not connect to the kinds of geometry which
every citizen (and every mathematician) needs: knowledge of shape,
of visual patterns, of geometric reasoning.
(2) Fruitful study of axiomatics includes abstracting FROM a
substantial body of prior experiences. The student needs
substantial mastery of the geometry prior to an 'axiomatization'.
The axiomatics should not prevent this other level of
appreciation of geometry.
Most of my sophmore students are not 'ready' for proofs.
TO understand a proof (or even the statement of a 'Theorem')
someone should know what a couterexample is. I trained in logic and
teach a sophmore course in logic (to computer scientists) and
I can confirm that they are just now learning what a couterexample
to a mathematical statement is.
(If axiomatics is something everyone should experience, people
should also experience the limitations of axiomatics and proofs.
The essay of Morris Hirsch on Myths of Mathematics, circulated last
fall on the geometry forum is a nice college level essay on this.
It was a shock for my average students (B-B+) to face the limits
of 'proofs'.)
(3) Every citizen will be faced with communication
(or miscommunication) via through visual patterns. Many will
be faced with problems about shape and geometric
relationships. These must be learned. Work with geometry
is at the core of crucial areas of work - at
all levels from the factory floor, the construction site and
the auto-shop to advanced research on robotics and
Computer Aided Design.
Geometry must NOT be sacrificed to axiomatics.
(4) Geometry, as a child learns it and as a geometer uses it,
includes many 'geometries' - including topology and combinatorics
(e.g. graph theory). Piaget suggests that the LAST geometry children
understand is Euclidean Geometry. Experiments with
exploratory tasks with graph theory and geometry are sometimes
easier for kindergarten children than college students or
high school teachers. A revised 'geometry curriculum' will include
some of this variety, including substantial work (play?) in 3-D.
(5) A major problem with revising high school curriculum from
the point of view of High School Teachers is - education in the
60's, 70's and 80's did NOT include significant geometry.
(It confused a course in axiomatics or algebra with a
course in geometry). Most current University
teachers have learned little geometry. Most people
using geometry in their work have not had courses in
these gometries (and it shows).
(6) I am a geometry researcher and teacher. I am also a member
of a committee working with high school teachers to recommend
new high school curriculum in Ontario. I am debating these issues,
to get my ideas straight for a general task we share.
Walter Whiteley
York University
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