JOURNAL OF MATHEMATICAL BEHAVIOR 15, 221-237 (1996) The Role of Euclidean Geometry in High School HUNG-HSI Wu University of California When I was a high school student long ago, the need to study Euclidean geome- try was taken for granted. In fact, the novelty of learning to prove something was so overwhelming to me and some of my friends that, years later, we would look back at Euclidean geometry as the high point of our mathematics education. In recent years, the value of Euclid has depreciated considerably. A vigorous debate is now going on about how much, if any, of his work is still relevant. In this article, I would like to give my perspective on this subject, and in so doing, I will be taking a philosophical tour, so to speak, of a small portion of mathematics. This may not be so surprising because after all, in life as in mathematics, one does need a little philosophy for guidance from time to time. The bone of contention in the geometry curriculum is of course how many of the traditional “two-column proofs” should be retained. Let me first briefly indicate the range of available options in this regard: 1. A traditional text in use in a local high school in Berkeley not too long ago begins on page 1 with the undefined terms of the axioms, and formal proofs start on page 22. There is no motivation or explanation of the whys and hows of an axiomatic system. 2. A recently published text does experimental geometry (i.e., “hands-on ge- ometry” with no proofs) all the way until the last 130 pages of its 700 pages of exposition. Moreover, the presentation in these 130 pages fails to give a clear picture of why undefined terms and axioms are needed, what precisely the undefined terms are in that context, and what role axioms play in mathe- matics. 3. The text of a new curriculum discusses only experimental geometry. There are no proofs, two-column or otherwise. This work was partially supported by the National Science Foundation. This article is a slightly expanded version of a lecture presented to the Bay Area Mathematics Project on July 27, 1992. I wish to thank Alfred Manaster for his constructive criticisms. Correspondence and requests for reprints should bc sent to Hung-Hsi Wu, Mathematics Dept., University of California, Berkeley, CA 94720-3840. 221 222 WU As a reference point, I should recall that the NCTA4 Standards (1989) recom- mends that two-column proofs should “receive less emphasis” (p. 159). It cer- tainly does not recommend their elimination. To begin our discussion, first allow me to quote something written by a nonmathematician some time ago: He studied and nearly mastered the six books of Euclid since he was a member of Congress. He began a course of rigid mental discipline with the intent to improve his faculties, especially his powers of logic and language. Hence his fondness for Euclid, which he carried with him on the circuit till he could demonstrate with ease all the propositions in the six books; often studying far into the night, with a candle near his pillow, while his fellow-lawyers, half a dozen in a room, filled with air with interminable snoring. Can you guess who the author is? Of course, it is President Lincoln writing about himself in his Short Autobiography (Faber, 1983, p. 45). In some sense, all I am going to do is no more than elaborate on what President Lincoln wrote a century and a half ago. So let me begin by burning my bridges and declaring at the outset that there must be proofs in a geometry course. The rest of this article is spent justifying this claim and outlining what I consider to be a reasonable way to implement such a recommendation. So why proofs? Proofs are the guts of mathematics. Producing a proof of a statement is the basic methodology whereby we can ascertain that the statement is true. Anyone who wants to know what mathematics is about must therefore learn how to write down a proof or at least understand what a proof is. The sciences also use the same methodology to deduce complex phenomena from first principles. Thus all who want to study science would benefit from learning about proofs as well. For the others who are outside of science, it comes down to what Lincoln wrote, that learning how to prove theorems is an excellent way to sharpen one’s mind. In a larger context, if anyone has any wish at all to find out how human beings can distinguish right from wrong or true from false, he or she would find in mathematical proofs the purest form of how this is done. Now I would like to make some comments about geometric texts that put most or all of their weight on experimental geometry and intentionally slight the theorem- proof aspect of geometry. Perhaps the argument is that because hands-on experi- ments are as efficient at arriving at the truth as abstract arguments, why not bypass this arduous task of writing down proofs altogether? In case such a statement does not immediately strike the reader as being silly, let me try to convince you with a simple example. A standard problem in number theory is to find integer solutions to equations of the following type (the Fei-mat-Pell equa- tion): x2 - 1141y2 = 1. HIGH SCHOOL GEOMETRY 223 This is of course the same as looking for positive integers y so that 1 + 1,14 ly2 is a perfect square. (We exclude the obvious solutions: x = 1, y = 0.) This is a problem tailor made for experimentation on the calculator. Starting with y = 1, 2, 3, . we can work our way up. The case y = 1 is no good because 1,142 is not a perfect square, for the simple reason that the square of any number must end in 1, 4, 5, 6, or 9. In fact, nothing works up to 100. For example, with y = 99, we get I + 1,141 (99)2 = 11,182,942 so that for the same reason it is not a perfect square. Similarly, nothing works up to y = 100,000. For example, with y = 23,456, we get 1 + 1,141 (23,456)* = 627,759,870,977, and because it ends in 7, it is not a square. If you try y = 45,678, then 1 + 1,141 (45,678)2 = 2,380,673,319,445, and 1,542,9432 = 2,380,673,101,249 < 2,380,673,319,445 < 2,380,676,187,136 = 1 ,542,9442. In fact, for all integers y all the way up to 1025, 1 + 1141~2 is never a perfect square. In terms of experimentation, one would have given up long before this and concluded that this particular Fermat-Pell equation has no integer solutions in x and y. But in fact, we can prove that there are an infinite numbers of integer pairs x and y that satisfy this equation, the smallest being: y = 30,693,385,322,765,657,197,397,208 and x = 1,036,782,394,157,223,963,237,125,215. So much for the power of experimentation. r Now this is not to belittle the importance of experimentation, because experimentation is essential in mathe- matics, What I am trying to do is to point out the folly of educating students to rely solely on experimentation as a way of doing mathematics. Mathematics is concerned with statements that are true, forever and without exceptions, and ‘I have taken this example from Stark (1978) 224 wu there is no other way of arriving at such statements except through the construc- tion of proofs. One school of thought claims that secondary school is not the place for students to learn to write rigorous, formal mathematical proofs, and that the place to do this is in upper division courses in college. Similar claims have also been put forth by other curriculum reformers.* With the preceding discussion at hand, let us approach these claims from a rational perspective. To call a spade a spade, a so-called “rigorous, formal mathematical proof” is just correct mathe- matical reasoning, no more and no less. As we are in an age when mathematical knowledge is at a premium, it does not seem proper that correct mathematical reasoning should be suddenly declared too profound and too difficult for all high school students and must be reserved for a few mathematics majors in college. If this country still has any intention of producing high school students that are first in the world in mathematics and science by the end of the century,3 the correct move should be to put more emphasis on the substantive part of mathematics (i.e., proofs) and not less. In a broader context, mathematics courses are where the students get their rigorous training in logical reasoning; this is where they learn how to cut through deceptive trappings to get at the kernel of truth, where they learn how to distinguish between what is true and what only seems to be true but is not. They would need all these skills in order to listen to the national debate and make up their minds about such knotty issues as the national deficit and the environment, for example. Learning how to write correct proofs is a very impor- tant component in the acquisition of such skills. (see again the previous quote of Lincoln). Because the high school students should all be voting to help mold the destiny of this nation as soon as they reach 18, any proposal that would deprive them of this training, so vital to their task at hand, must be regarded as ill advised at best.