Is there anybody out there who teaches geometry and has good experiences with teaching more-or-less average students how to do proofs on various topics - say, proving that if you have a parallelogram and its diagonals are perpendicular, then it is in fact a rhombus?
Or are there thoughts on whether proving that sort of thing is in fact a worthwhile endeavor for students to engage in, for at least part of the time in a year-long course in geometry?
And, if you do teach proofs in your geometry class, do you differentiate between things like 'AB=CD' and 'segment AB is congruent to segment CD'? Should one differentiate between the symmetric property of equality, which says that for all x, if x = y then y = x, on the one hand, and on the other the theorem that congruence of segments and angles is symmetric as well (i.e. if seg AB is congruent to seg CD, then seg CD is congruent to seg AB), or is that too nit-picky for students to be bothered with? And, if you don't bother with that, then what level of precision should one strive to reach? What level of mastery with either flow-chart type proofs, 2-column type proofs, or paragraph-type proofs do other teachers attempt to instil in their charges?
Oh, to foreshadow some responses, I do have the book on Rethinking Proof written by de Villers or whatever his name is (the fellow from South Africa) published by Key Curriculum Press. And, yes, I do try to point out that not all relations are symmetric - if John is older that Juan, then it is not true that Juan is older than John; and not everything is transitive - If Bill likes Maria, and Maria likes Jakov, then it does not necessarily follow that Bill likes Jakov.... And, yes, I understand that proof - while it is the key that underpins all of mathematics today, making it all verifiable - nonetheless has limitations, as Goedel showed about 60 or 70 years ago, since any relatively complex formal mathematical systems will contain statements that cannot be proved either true or false. And, yes, I understand that the NCTM in 1989 suggested de-emphasizing (but not, as some interpreted at the time, eliminating) two-column proofs in geometry classes. And, yes, I understand that inductive thinking is key to making new conjectures in mathematics and science that can later be either proved or disproved. But it does seem to me that it is a good idea for students to at least get a taste of what mathematicians actually do, so that they can see that there is a base of over 2200 years of mathematical proof for all of the mathematics that they use.... That it's not all just guesswork and suppositions....
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