I can't resist replying to the 1/22/94 posting by Joe Malkevitch about the geometry curriculum, in which he claims that "the teaching of axiomatics makes little sense."
Without meaning to offend, I suggest that -- whatever the merits of his conclusion -- his argument does not stand up well to an (informal) axiomatic analysis.
He says: "We must remember that very few students who study geometry in high school go on in science, no less in mathematics. The goal of high school mathematics should not be to locate future mathematicians per se." I agree, and accept that as Axiom 1.
Then he says: "In any case, the teaching of axiomatics makes little sense. Research geometers have relatively little interest in such systems and they make even less sense as I stated for non-math. types." I have two problems with this. (1) IF the goal of high school mathematics is NOT to locate future mathematicians per se (Axiom 1), then why should we be guided by the interests of research geometers? (2) The claim that they "make even less sense...for non-math types" has indeed been stated, but without a shred of supporting evidence. I suggest that it is an implicit Axiom 2, which is by no means self-evident. I do not accept it.
He goes on to say: "Modern geometry has some wonderful applications. Centering a curriculum around these applications at the very least shows how important math is to our current world...." I agree with this, too. That is, I agree with the self-evident statement (Axiom 3) that centering a curriculum around applications shows how important math is to our current world.
However, implicit in the subsequent discussion of examples is an unstated (and undemonstrated) theorem which, I infer, must be something like this: 'Because mathematics is useful in our current world, therefore it must be interesting (or at least more interesting than axiomatic geometry) to high school students who will not go on to major in science or mathematics.'
I suggest that this statement is yet another axiom which has been masquerading as a theorem in many discussions about the geometry curriculum. Has there been a systematic research study of student attitudes that has demonstrated this? I don't know of one. I think, rather, that this is a case of many mathematicians projecting their likes and dislikes, "observing" what they have already decided they will find.
Prof. Malkevitch presumably finds the applications of graph theory and the like fascinating and impressive, and hence may tend to notice the students who share his enthusiasm about these things. I, on the other hand, regard the fact that mathematics has real-world applications only mildly interesting and not at all amusing. (If that had been the "pitch" for mathematics when I was in school, I would have become an English major. When I was a college freshman, it was not calculus, but a seminar on the axiomatic basis of the number systems, that turned me on to mathematics.) In my 30+ years of teaching first-year college students (most of whom have been non-science majors), I have tended to notice those who share my predilections about the subject. I suspect that neither one of us has the whole answer; even in mathematics, it may well be "diff'rent strokes for diff'rent folks."
I am grateful to Prof. Malkevitch for the COMAP paper reference; I've already requested a copy of it. I note, however, that all the people he lists as contributors are mathematicians, and -- given Axiom 1 -- I'm not completely comfortable with using their opinions as our only guide.
Having said all this, let me concede that my position within the writing team for MATH Connections is that of the dissenting minority viewpoint. Within the next day or so, I'll post for comment our team's tentative outline of 10th grade mathematics, to the extent that we have it now. You will notice that it is largely non-axiomatic and still very fluid. Please comment freely on this proposal; your timely comments can still have a significant impact on how we handle the material.