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 problems.
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 <firstname.lastname@example.org>
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.