1. Students could and should be asked to develop, explore, theorize. After that process is it possible to prove seemingly great conclusions? Should we rely on insight, totally?
2. To me the "beauty" of geometry is the development of provable theories based on terms(undefined or defined), assumptions (axioms or postulates), and previously proven theories(theorems, propositions,etc.)
3. Any methodology - discovery, constructions, discussions, experimentation which will help students of geometry understand more completely is wonderful. Let's follow these eye opening discoveries with proving the concepts in an organized way. Let's document, keep records carefully, use previously developed ideas to help develop new ideas.
4. I wonder if that part of the mathematics educational community that wants to eliminate or greatly diminish the use of proofs is succumbing to the frustration of many instructors. (to avoid the word PROOF some texts use flowcharts, conjecture- "a rose is a rose")
5. SOME QUESTIONS a). Are we just interested in students learning a body of information or are we also interested in a logical development justifying conclusions that we make?
b). Do we think students are incapable of learning how to do "proofs"?
c). Do we need to learn how to teach "proofs" better than we have in the past?
d). A high school in Pennsylvania has a 5 month geometry course- no proofs at all. Is that the way we should go?
I have taught geometry for 30 + years. I have enjoyed combining the excitement of students exploring and discovering with the excitement of proofs- and I think, very successfully.
ANY OPINIONS OUT THERE? Bernie Ivens The Geometry Forum School Liaison Swarthmore College, PA firstname.lastname@example.org