Well, we certainly have a lively, energetic, and committed crew out on the Geometry Forum! Possibly a little trigger happy though ... I think folks have assumed the worst in response to queries from me and Bill Berlinghoff about defining "angle". We are NOT taking an axiomatic approach to geometry nor are we contemplating most of the other crimes suggested. In trying to defend ourselves, I think we've come out looking much more conservative than we are. We just wanted to know how people are approaching the idea of angle.
Maybe a non-geometric example would be less incendiary. In our ninth grade materials we define "mean absolute deviation" as the mean of the absolute value of the deviations from the mean. (Our treatment there is not so curt.) Now I have seen people argue that the thing to look at is the deviations from the MEDIAN, or that we should look at the median of the deviations. Now whatever the pros and cons of these positions might be, and however we might present the material to students, I think we need to be clear in our own mind what the different definitions are and be consistent in our own usage so that we do not have MAD meaning different things on different pages. Also, if we have not considered various alternative definitions, or even conceived of their existence, we can't initiate a very intelligent dialog with students on these alternatives. Finally, if one of the definitions is overwhelmingly accepted as "correct', and the others are considered misunderstandings or errors, then we should know that too.
Please join us in the Mid-Plane Peace Conference. --
_ | | Robert W. Hayden | | Department of Mathematics / | Plymouth State College | | Plymouth, New Hampshire 03264 USA | * | / | (603) 968-9914 | ) (603) 535-2489 L_____/ firstname.lastname@example.org