Math Forum Internet News

Volume 3, Number 52A  -  December 1998 Discussions

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30 December 1998                                 Vol. 3, No. 52A

  THE MATH FORUM INTERNET NEWS - DECEMBER 1998 DISCUSSIONS

This special issue of the Math Forum's weekly newsletter  
highlights interesting conversations taking place during
December of 1998 on Internet math discussion groups. 

For a full list of these groups, with links to topics covered
and information on how to subscribe, see:

            http://mathforum.org/discussions/

Replies to individual discussions should be addressed to
the appropriate group rather than to the newsletter editor.

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                     DECEMBER SUGGESTIONS:                    
                   
CALC-REFORM - a mailing list hosted by by e-MATH of the 
American Mathematical Society (AMS) and archived at
http://mathforum.org/epigone/calc-reform/
  
- Pedagogy of "Big Theorems" (5 Dec. 1998)
  http://mathforum.org/epigone/calc-reform/zheldstehty/

  "...proving 'big theorems' is not pedagogically effective. 
   Proofs are simply too long and too overwhelming for students. 
   It is hard to learn 5 or more steps at once, the brain gets 
   overloaded...." - Kazimierz Wiesak

  "Proofs are not intended to build intuitive understanding.
   But saying 'accept this!' is probably worse pedagogically,
   depending, of course, on goals...." - Walter Spunde

  "My piano teacher always amazed me by playing complicated 
   Bach pieces at sight. She explained the method: she
   recognized groups of notes (chords and chord sequences for 
   example), and hardly had to think when playing them; her 
   fingers knew what to do, and she had to think only for the 
   broad sweep of the piece. Whereas when I played at sight, 
   each note involved finger juggling and thinking. No wonder 
   I found it difficult (and still do)! Polya wrote about the 
   value of teaching mathematical ideas using repetition but 
   with variation, and pointed to the composers for examples.  
   Perhaps sight-reading is another example where we could
   learn from the composers." - Sanjoy Mahajan

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GEOMETRY-PRE-COLLEGE 
http://mathforum.org/epigone/geometry-pre-college/

- "Ellipses inscribed in triangles" (30 Nov. 1998)
   http://mathforum.org/epigone/geometry-college/chaxbrayman/

  "In playing with ellipses inscribed in triangles, I have come 
   to these conclusions... an infinite number of ellipses can 
   be inscribed in any triangle..." - Steve Sigur
  
  "...it's wisest to make these assertions for 'conics', rather 
   than 'ellipses', since that stops you from having to worry 
   which type of conic a given one is... the relation between 
   a conic and its perspector is a projectively invariant one, 
   while that between a conic and its center is an affinely 
   invariant one. The projective invariance implies that the 
   formulae are the same in all systems of trilinear coordinates, 
   so in particular they are the same in barycentrics or 
   orthogonal trilinears. One of the many advantages of 
   barycentrics over orthogonal trilinears is the fact that 
   they also detect affine invariance by the property that the 
   formulae involve no constants (such as the edge-lengths). 
   In particular, the formulae that give Q in terms of P have
   this property." - John Conway

The discussion continues at:

- "Trilinears vs. Barycentrics" (4 Dec. 1998)
   http://mathforum.org/epigone/geometry-college/zahclermjing/
   http://mathforum.org/epigone/geometry-college/khunyimpha/

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MATHEDU - a mailing list set up to discuss issues in 
Mathematics/Education at the post-calculus level, archived at
http://mathforum.org/epigone/mathedu/
  
- "continuous functions" (9 Dec. 1998)
   http://mathforum.org/epigone/mathedu/quulwheldtwand/
  
  "What reasons, if any, should be given to students, for why 
   continuous functions should be studied, as opposed to all 
   functions? Does the importance of continuous functions really 
   depend on the laws of physics? Quantum phenomena aren't 
   continuous: nor are the results of throwing dice; and 
   computers don't act in a continuous way, at least in the 
   Turing machine abstraction." - David Epstein

  "What about ordering in the mathematics undergraduate 
   curriculum? Based on people's experiences, which should come 
   first: discussing real analysis (at the introductory level, 
   i.e. so-called "Advanced calculus" in many departments); or 
   instead discussing complex analysis? Are notions of
   continuity and limits and convergence perhaps easier to 
   comprehend, for the first time, in the complex setting, 
   rather than in the real setting?" - Richard M. Kreminski
   
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  We hope you will find these selections useful, and that you
  will browse and participate in the discussion group(s) of
  your choice. 

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