I am not disagreeing with either Lou or Gary, but am asking where is the role of logic in all this. Is logic important or not? If it is (say for the average student), then how much? And if, some reasonable degree is useful, does one pick it up on their own or is it somehow firmly part of the instruction. Finally, if it is part of the instruction, when and how is it part of the instruction (say for the average student)?
On Wed, 12 Mar 1997 08:43:13 -0500 firstname.lastname@example.org (W Gary Martin) wrote:
>Lou, > This is, in my opinion, a strong analysis of the role of proof in the >mathematics curriculum, especially the final sentence. I hope you are >writing up some version of this for the MT Focus Issue on Proof. > A personal experience on the misguided zeal of "EIT"'s. One of my >biggest >surprises as a curriculum developer for high school geometry came when >my >advisory board discussed a number of issues related to proof. I wanted >to >take a softer, more common-sense (I thought) approach to proof -- for >example, avoiding tiresome issues like congruence of segments vs. >equality >of their lengths. My reasons were purely pedagogical: These things trip >up >students who are perfectly capable of deductive reasoning but don't see >the >importance of all the picky detail. I was more concerned with their >being >able to see the broad picture of proof, and why proof is important. >(Note >that the Van Hiele levels are also applicable to this argument.) > My strongest allies on the advisory board ended up being the research >mathematicians. My sharpest critics were classroom teachers. I found >this >fascinating. The "real experts" cared about whether a plausible >argument >was made. The teachers were more concerned about the particular details >of >a proof. > The danger that I see is that many teachers read the Standards as >saying >rigorous proof is not important and that it should not be the sole >focus of >geometry, missing what I think is the real message -- proof should >always >be done at an appropriate level of rigor, and it should be a part of >all >areas (including geometry). And should always be aimed at deepening >students' levels of thinking about mathematics, "teasing [them] deeper >into >more arcane issues". > Gary > > >At 1:29 -0500 3/12/97, Lou Talman wrote: >... >>The crux of the matter is this: A good teacher can be precise without >being >>pedantic. Issues that an expert-in-training (EIT) will think >important will >>seem pedantic to both the novice and the real expert. Precision >demands those >>issues be dealt with; the EIT is learning this and must find them and >deal >>with them publicly--not only to convince herself that she can but to >convince >>her teachers as well. The expert knows about the issues and deals >with them >>privately. (Dealing with issues privately: One of my favorite >teachers >>taught me that it is impolite to compute in public.) The novice >doesn't yet >>know about the issues--let alone understand them or know how to deal >with >>them. >> >>In point of fact, almost all "proofs" are really plausibility >arguments. Good >>teaching is the art of casting our plausibility arguments at a level >that >>convinces the EIT but keeps him stretching for that conviction, >teasing him >>deeper into more arcane issues, forcing him gently but inexorably >toward the >>expert level. >> >>--Lou Talman