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Re: Proofs  My Thoughts
Posted:
Feb 18, 1993 12:43 PM


I think it is a skill that anyone can acquire. It's just that often the students do not have the necessary background skills to think and break something down in a logical manner. They are plopped right down into the infamous 'Two Column Proof' and either they get it or they don't. There is hardly any work done to develop the deductive skills necessary to really grasp proof.
Part of the problem comes with the reputation, as it were, that geometry proofs have. They are supposedly something that students have never done before. True, in a way, but what about in algebra when they were asked to 'show' that two equations are equal to each other? There they did a 'proof' of sorts, but didn't have to write down the 'whys' of each step. How about trig identities? Why don't we have them write those out, giving explanations for each step?
There are a lot of places in the existing curriculum that proofs  logical, deductive processes which show that something is true  could be done. And I don't necessarily mean abstract algebra and number theory. Why is it that geometry has become the showpiece of this supposedly onetime experience? Is it necessary to terrorize kids (and you must admit that some of them are) with something that they have done before, but has never been labelled as such, and consequently they think they don't, or won't understand it. They use rules and theorems in algebra and other subjects, but never has so much attention been paid to it as has in geometry.
Someone (Pamela, I think) mentioned that they use Discovering Geometry by Michael Serra. Some people have expressed their dissatisfaction with this book because it doesn't do 'formal' proofs until Chapter 14, and often a class will never get that far. Yet Chapter 1 (preceeded by Chapter 0) is on Inductive Reasoning. Exercises throughout the book encourage and require the kinds of thinking skills that are really necessary to understand the process of proof. Chapter 13 is about Logic. If and when students get to Chapter 14, I suspect they are a bit more capable of actually comprehending proofs than if they had been done in Chapter 3.
I don't teach geometry. In fact, I don't teach. I have my secondary certification, student taught, and have tutored many kids in the last five years in all high school subjects. I currently administrate The Geometry Forum, a bulletin board about all kinds and levels of geometry. Previous to that I was involved in writing three sets of workbooks and videotapes dealing with threedimsional geometry and teaching it in a high school setting. I have never used Mike's book. I do own three copies, though, and might accept a teaching position if I get to teach geometry with his book. No, it's not perfect, but it's a lot closer to how I feel geometry ought to be taught than most other books available.
(Disclaimer: the publisher of the workbooks I helped write is the same one who publishes Mike's book. While this is how I know about Discovering Geometry, it's not why I like it. It may be one reason why we like the publisher, though.)
Someone else mentioned the software currently available which allows manipulation of geometric objects while maintaining established relationships. While this is not a replacement for formal proof by any means, it does help the students to understand what they may be asked to prove. It's one thing to ask a student to prove that the medians of a triangle intersect. It's quite another to ask them to investigate, using Sketchpad or one of the other programs, the medians of a triangle. What do they notice about the medians? Odds are pretty good that they will discover that they all intersect. _Now_ ask them why. Isn't that a much more interesting and intriguing way to go about things, than to ask them to prove things they aren't sure are even true?
Yes, I think proofs are important in mathematics. I was a math major in college. I also thought proofs were fun in high school, but that's due not in small part to the fact that I understood them completely the first time. I think it's more important to develop those skills that are necessary to comprehend proofs. Sometimes this may need to be done at the expense of actual rigor, but isn't it the point of education to teach people to think and reason instead of regurgitate? Any student taking geometry in ninth or tenth grade is capable of doing proofs _if they are taught the necessary thinking skills before hand_.
Annie
Annie Fetter  The Geometry Forum  Voice: 215 3288225 Project Coordinator  Swarthmore College  800 7567823 annie@forum.swarthmore.edu  Swarthmore PA 19081  Fax: 215 3287824



