[This is a thread that has been going on sci.math and misc.education on usenet. We're jumping in at a good point, though, I think. -Annie]
In article <1993Apr11.firstname.lastname@example.org> email@example.com (Tal Kubo) writes:
>For Euclidean geometry I don't know anything really good in English, but >there is a reasonable book coauthored by Garrett Birkhoff called "Basic >Geometry" which is out of print but possibly gathering dust on a shelf in a >high school near you. I think it sometimes uses the odious "2-column >proof" format, but it also has some good points (such as basing the >treatment of angles on what can be called a "naive theory of real numbers" >instead of the more cumbersome Euclidean development).
I think this "naive theory of real numbers" is a serious pedagogical mistake. It has the advantage for the author of making the book easy to write (and most US high school textbooks use this approach). But high school geometry students do not have an intuition for real numbers; in particular, they have no intuition for the topology of the real line.
So the "naive theory of real numbers" is a way of keeping from talking about things that *must be taught* if the student is to develop an adequate intuition. Without a knowledge of the geometry of the real line, tacitly assumed in this approach, the student will not understand calculus (though he may be able to perform the usual dog tricks of college calculus courses).
This approach makes a farce of any attempt at rigor. Axioms and theorems are "supplemented" with appeals to nonexistent intuition about real numbers. The student may well wonder why he can't appeal to his own intuition about triangles, which he probably understands much better than real numbers.
Indeed, I have found that none of the entering Freshmen I've talked to can give a coherent explanation of what real numbers are. Yet their intuition for real numbers is made into the foundation of high school geometry courses.
Of all math and science courses taught in high school, geometry is taught the worst. Nearly all the freshman math and science students I've taught are lacking the basic knowledge one would expect from a geometry course. They remember no theorems, except for the Pythagorean theorem and a few conditions for the congruence of triangles. They don't know any axiom scheme for geometry, and they can't prove anything. (Not surprising, some students I've taught have told me that in their high schools, their geometry teachers SKIPPED ALL THE PROOFS!) They can't tell a theorem from its converse, and they have no idea how to negate (or otherwise manipulate) statements with universal and existential quantifiers. Most crippling of all, THEY ARE WHOLLY UNABLE TO VISUALIZE ANYTHING IN THREE DIMENSIONS. Behold the result of removing solid geometry from the curriculum.
(I ran into this the other day in a physics workshop I'm teaching. I was trying to explain vector cross products, and the students were BEGGING me for 2-dimensional examples. They were mortified when I finally got it across that vector cross products are inherently 3-dimensional.)
As usual, incompetent teachers are largely to blame. But don't blame them too much. A few years ago I undertook a diligent search for a good high school geometry book in English. After much searching, and much rejecting of hopelessly inadequate candidates, I concluded that such a high school geometry book doesn't exist. There are no introductory high school geometry books in English which are pedagogically and mathematically sound, and which in addition cover the material which ought to be covered.
This last sentence was intended as a lead-in to a discussion: what constitutes pedagogical soundness in a high school geometry course, and what ought to be covered?