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Re: Two languages
Posted:
Oct 29, 1995 9:55 PM


I feel somewhat responsible since I was the one who brought up the idea that anaytical reasoning can be thought as a "thought experiment".
I believe there is just misunderstadning between Tony and Mark, but both of you have made some very good points.
I think all of us in doing our research constantly doing thought experiment in conjunction with our analytical procedure, the weight of the experimenting depends on the property of the work. But, not only that the experimental observation of any sort need to be followed by the analytical prove or disprove, a good thought experiment always rely on require solid knowledge and good analytical reasoning capability to start with and to connect. We should all trying to stand on the shoulder of the giant in order to reach high. Now a day, most advanced experiment all require many nontrivial "proven theories" to interpret the observed data because "direct measurement" are not available.
My personal opinion toward the math Ed is we could allow students to do it during the early learning phase of the subject, then we should expect them to understand how it was wrong and how it was right and the real learning come when they really see or construct the right ways ( I can expect if this is posted in nctml, there will be many educators will jump on me saying that : see, Hsu is insisting on his right way which may not make sense to students, I hope people here will not misinterprete me as it) and excercise with in doing various types of problems to obtain sharpened skills and deepened uhnderstanding. I also believe that the emphasis on the experimenting should be brief, students ( and all of us) gonna do it anyway, and because the nature of mathematics which requires much less experiment of any sort than science and other discipline. (IMHO). I sympthize with Tony, if I understand him correctly, that he concerns the without precise and more thorough description, such an idea of "doing experiment in math can be good for students" may lead to severe problems due to the limitted understanding of the elementary teachers in the idea of "thought experiment" so that they encourage students to pursuit it too heavily and too freely. Two bad consequences could result (1) they arrive at conclusion which is totally incorrect but not be redirected by teachers and carry that misconception through out their life; (2) they stagnate at the stage that they always experiment things from their primitive mathematic knowledge and understanding, since there are so many technologies they can use to experiment with and not be encouraged to develop their own analytic capability and understanding. From the talk about "constructivism" and "real world problem" with many math Ed specialists, I arrive at the similar concern..
Cheer! > > Is it possible that Tony and I speak two different languages? > When I say that one can discover patterns experimentally and > then prove or disprove them (by counterexample), I use the > word "prove" and "counterexample" in the only sense they > have: "Prove" means "give a mathematical proof" and, of > course, "counterexample" means a welldefined example for > which the assertion is false. I'm not sure I know WHAT > Tony is talking about. > > On a simple level, a student draws a few isosceles triangles > and sees that the base angles seem to be equal. The next > step is to try to prove it must be the case. > > A student looks at odd numbers and conjectures that they are > all prime. Hmmm....9 doesn't work. > > (These are oversimplified.) > > Start with a postive integer N. If N is even, divide it by 2, > and if odd, triple and add 1. Continue this process until 1 is > reached. Are there any patterns relating N to the number of steps > before 1 is reached? In every known example, sufficiently many > iterations will eventually terminate with 1, but it has never > been proved. Nevertheless, interesting things are known. > > Similarly for Fibonacci numbers. You don't have to start with > 1 and 1, but can begin with any two numbers. The ratio of > consecutive terms always approaches the Golden Ratio. This is > nicely investigated on a calculator, as is divide and average > for computing square roots. Many simple conjectures can be > proved rigorously by highschool students. > > This is what I meant. > >  Mark >



