
Re: Proofs  My Thoughts
Posted:
Feb 17, 1993 2:16 PM


In article <1gkqsbINNmom@forum.swarthmore.edu> Bernie Ivens <ivens@forum.swarthmore.edu> writes: > PROOFS My Thoughts > > 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? For some students, keeping in mind a goal of the NCTM standards that we should try to reach ALL students with SOME valid mathematics, the "Proofs without words" of the MAA Mathematics Magazine might be a sufficient consolidation of explorations and theorizing. For others, the goal of rigor is worth striving for. See "Rigor and Proof in Mathematics: A Historical Report" vol 64 no 5 Mathematics Magazine. > > 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.) Yes, axiomatic mathematics is beautiful. The Bourbaki group which arose partly in response to the tragic decimation of young French thinkers during WWI, had a great impact on world mathematics. The Cambridge Report of 1962, reflects Bourbakist thinking. The Cambridge Report was an influence on the curricular movements of the middle 1960's and 1970's. See the recent book {Pi in the Sky...}.
You can talk about fractals to an engineer, and she'll be very interested, but not when you start to discuss proofs. The litmus test for mathematicians may be: Who walks out of the room when you start to talk about proofs. The ones left are sieved out, like primes, and they're the mathematicians.
But there is also some beautiful mathematics, and the variant genius Ramanujan may have provided some good examples, which just is there. Yes, it can be proved, but it's the result that's there and amazing, not a gorgeous axiomatic structure like you get with Bourbaki Topologie Generale, vol I > > 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. If you want to teach varieties of logic, and that's been a longtime goal of education, or at least what is [very approximately] called WESTERN education, you could use, Rhetoric, or Formal Logic, or...Geometry. Geometry is a good domain for studying principles of logic. However, I'm not sure that logic is a good context in which to study geometry. > > 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") > Consider this. Should we teach long division, and then have children check their answers with calculators? Should we spend hours graphing parablolas and then check the answers with graphing calculators. Should we do lots and lots of algebraic factoring and expansions and then check the results with Mathematica? Should we spend weeks and weeks doing proofs, when we can check the results with automated theorem proving? Now ATP isn't quite here yet, but it's close.
"Many areas of high school and undergraduate mathematics, by now, are "trivialized" in the sense that their problems can be solved algorithmically by existing mathematical software like Mathematica. Well, if an area of mathematics is trivialized, why should students bother to study the area? Rather, shouldn't we just teach the students how to solve the main problems in the area by applying, in a reasonable way, the appropriate algorithms in, say, Mathematica? There are two dogmatic answers to this question. The puristic answer: Ban math software systems from math education! The pragmatic answer: Don't spend time in class on any trivialized area of mathematics!" [Bruno Buchberger, RISCLinz, Johannes Kepler University. Teaching Math Using Math Software: Some Examples for the BlackBox/White Box Principle. Rotterdam Mathematica Conference, September, 1992]
> 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? (CONTENT + FORM) That body of information called geometry is a microworld of a subject marjorie senechal and others are calling "shape." Look at the SIGGRAPH notices from ACM. People are using simplicial topology and quaternions and other mathematics to do very beautiful renderings. Watch the video "Not Knot." This video was inspired by a proof, but it suggests a future in which you might prove a result by flying through a virtual space and checking off the features as you go. > > b). Do we think students are incapable of learning how to do "proofs"?
"Imagine then the kind of person coming into our graduate schools, if not today, then certainly tomorrow. Brought up from early childhood on a diet involving MTV, Nintendo, graphical calculators packed with algorithms, Macintoshstyle computers, and, in the nottoodistant future, hypermedia educational tools as well. Such a person is going to enter mathematics with an outlook and a range of mental abilities quite different from their instructors..." [Keith Devlin, Notices of the American Mathematical Society, March 91, 38, 3 ].> > c). Do we need to learn how to teach "proofs" better than we have in > the past? differently... > > d). A high school in Pennsylvania has a 5 month geometry course no > proofs at all. Is that the way we should go? Back to The Standards again. [The Standards, by the way, are now part of the "Conservative Reform Movement."] I think that we need to conceive of Shape, or Spatial Cognition, as a vertical theme running through the K12> curriculum. In such a context, I can imagine that there could be a 5 month stretch when you aren't doing proofs. But taking a radical point of view, I wouldn't call that 5 months geometry. Because, I believe Husserl was right. Geometry is no longer alive. It's dead, like Latin. You can teach Latin beautifully, and make a mess of teaching Japanese. You can teach geometry beautifully, and make a mess of teaching fractals. But Latin is dead. So is Euclid. Fractals are alive. So, and let's all wish them the best of health, are Mandelbrot, Peitgen, Devaney, Pickover, and all those other interesting people. That ultimately is going to make a difference in most classrooms. > > > 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. I'm sure you have been successful, and I've entered my opinions [as you askedopinions] with consideration for your work. The fact that you are asking such interesting questions now is [proof (?)] that you must have been asking your students interesting questions, for 30 years.
David Fowler, Teachers College, University of NebraskaLincoln. > > > ANY OPINIONS OUT THERE? > Bernie Ivens > The Geometry Forum > School Liaison > Swarthmore College, PA > ivens@forum.swarthmore.edu

