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Topic: Re: Geometry/Alternatives/Gifted
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Ronald A Ward

Posts: 298
Registered: 12/4/04
Re: Geometry/Alternatives/Gifted
Posted: Feb 28, 1998 12:24 PM
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I believe I have detected in some of the posts by Wayne Bishop and his
California colleagues a particular concern about challenging highly
capable learners and an interest in whether there really are alternatives
to traditional Euclidean geometry as a means of teaching proof. I will
try to "kill two birds with one stone" by suggesting that they take a look
at the "Elements of Mathematics" curriculum, a grades 7-12 program for
highly precocious students.

When I was Associate Director for Math and Science at CEMREL, this is
one of two curricula developed at that Lab which we tried to disseminate.
Briefly, the EM program was an experimental curriculum designed to
explore the upper content limits of mathematics that students with
excellent reading and reasoning ability could understand and appreciate
[e.g., students completing the program studied groups, rings, fields,
finite probability spaces, and measure theory, among other topics].
The grades 7-9 materials were in two parts: Book 0--an intuitive
approach to mathematics--contained in 16 chapters, themselves each of
book length and separately published as such; and Books 1-3, a formal
approach to mathematics. The rest of the materials, Books 4-12, were
built upon that foundation.

Contained within Book 0 were two chapters devoted to geometry. Chapter
12 was devoted to Incidence and Isometries, and Chapter 13 included
Similitudes, Coordinates, and Trigonometry. [If I remember correctly,
these were written by Vincent Haag of Franklin and Marshall and Edward
Martin of the High School of Glasgow, Scotland.] Then, included in some
of the later books in the series were "The Elements of Geometry" by Lowell
Carmony [on staff at that time] and "Linear Algebra and Geometry With
Trigonometry" by Carmony and Robert Troyer of Lake Forrest College.

Now proof in these later volumes was based on the formal logic that
students learned in Books 1 and 2 and was quite different from what one
might ordinarily see in a typical Euclidean Geometry course. [Book 1
develops a symbolic language suitable for expressing mathematical
statements and names as well as essential notions of a proof theory; Book
2 develops the first order predicate calculus]. *

But back in the Book 0 material, Euclidean Geometry was developed through a
series of experiments, conjectures and arguments based on a dynamic approach in
terms of mappings rather than synthetic methods. The material began with
intuitive descriptions of incidence, the real numbers, and distance, with
simple, informal discussions of interior, exterior, and boundary points
of figures, as well as open sets, closed sets, and convex sets. Line
reflections of a plane were then introduced with mirrors and paper
folding. Experiments led to conjectures about composites of reflections
[translations, rotations, and glide reflections]. The resulting group of
isometries was used to describe congruence and symmetry as well as area
and volume of figures. Polygons were classified in terms of their
symmetry groups. Later, the ideas and skills usually associated with
similar figures arose from experiments with magnifications of the plane
and space. This then led to the group of similitudes, including
properties of right triangles, and the trigonometric ratios. The plane
was then coordinatized and the same study of mappings was reviewed in a
coordinate setting, leading to the idea of a vector space and providing
techniques for such procedures as finding equations of lines. Each type
of mapping was first developed in the plane and then quite naturally
extended to space as a regular feature in the course.

Thus, Book 0 provided students with a variety of intuitive experiences
and the concrete examples from which the rest of the EM books drew
motivation, illustrations, and interpretations. The mathematics was
sound and relevant; it differed from the rest of the EM books only in
style and degree of rigor. The language was correct, the concepts
significant, the experiences meaningful, but no attempt was made to give
formal proofs in the context of axiomatic or logical systems. One freely
used local deductive argumentation and implicit logic, but all concepts
were presented in an empirical setting of real situations and practical

Although I directed the dissemination effort, at no time did I ever
recommend this program for students other than the top 5% of the
population. Now, since then, the materials have gone thru considerable
revision under the direction of Burt Kaufman [the original project
founder and director], and I have not seen the most recent version. But
I believe the EM series might be an alternative approach to high school
mathematics [including geometry, specifically] which those concerned
about highly capable learners might find interesting. Materials are
currently published thru McREL. Contact is thru <>

Ron Ward/Western Washington U/Bellingham, WA 98225

* I will be happy to post detailed descriptions of the formal logic
books [1 and 2] as well as the formal geometry books [8 and 9] for
interested readers. I did not do so here to save space in the initial post.

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