Your identification of "straw men" in my previous posting is helpful; it was not at all clear to me from your earlier postings that they were, indeed, straw men. Perhaps we are closer to agreement than it first seemed. (Perhaps not.) Let me identify a few of yourstraw men; the first is named Bourbaki:
> There is a level of precision in language and thinking appropriate to > any given situation. 9th and 10th grade is one thing, Bourbaki is another.
Isn't this characterization of my position a bit extreme? Clearly, Bourbaki is not even in the picture; their formalism far surpasses anything Euclid ever imagined in even his most austere dreams. The real issue, which you identify here by contrast only, is what level of precision IS appropriate to 9th and 10th grade? That's the point of contention between us. There are actually two parts to this question: (1) What level is appropriate for the 9th and 10th graders of today who have come through an arithmetic- oriented "standard" elementary curriculum, and (2) What level is appropriate for the 9th and 10th graders of tomorrow, who (it is hoped) will have come through a much richer elementary curriculum that will already have included many of the exploratory experiences you have cited?
I agree with you and Gary Martin that
>}> Angles are a lot more problematic than we generally acknowledge! >}> Most texts dispose of them in a few lines.
This is precisely why we writers and teachers (not necessarily our students) need to have a firm, explict grasp of the way(s) in which our intuitive approaches to them EVENTUALLY crystallize into more formal mathematical concepts.
> What I am arguing for is for using MULTIPLE APPROACHES TO > GEOMETRY.... Remaining trapped within one approach puts > unnecessary blinders on the students, and reduces the numbers of > students we can reach.
I agree. Here's another of your straw men:
> you appear to believe that by "protecting" students from ever > encountering the rotational point > of view, we can teach them about > the sum of the angles in a polygon in a nice antiseptic way.
Of course not! My comments did not say (or suggest) that students should be "protected" from ANY point of view that would help them develop their geometric intuition. My concern is that we not be content to use intuition alone as sufficient justification for explicit, formal geometric statements (which then are subject to misinterpretation based on the differing intuitions of individual students). Your final comment focuses my confusion about what you espouse: