Math Forum Internet News

Volume 2, Number 46

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17 November 1997                                Vol. 2, No. 46


 Modular Origami | Eye Opener Series | Implication & Denial



Modular origami is a branch of the art of paperfolding.
Unlike regular origami, where a figure or sculpture such as
a crane, frog, or airplane is made from a single sheet of
paper without glue or cutting, modular origami figures are
built up from multiple individually folded pieces, usually
identical. Modular origami is often very geometric. 

Valerie Vann offers her own designs and developments,
examples of other significant modular origami works, and
favorite models by other modular origami creators.


Pictures of many origami polyhedra, with instructions for
making the modules and the polyhedra, and putting the 
modules together:

    - polyhedrons with the penultimate module
    - the compound of 5 tetrahedrons
    - greater/lesser stellated dodecahedrons
    - the compound of 12 dodecahedrons


Origami and form, architecture and design: buildings, 
housewares, and furnishings; with a select list of
related sites.


For other ORIGAMI POLYHEDRA sites, see David Eppstein's 

or use the MATH FORUM's Quick Search:



A collection of Java applets, all of which depend on a piece 
of Java code that controls two eyes, that illustrate and 
thereby help to solve or prove math problems such as 
99 = 100 and:

    - Euclid's Proof of the Pythagorean Theorem 
    - Construct an n-gon 
    - The Disappearing Lines puzzle 
    - The Sam Loyd's fifteen 
    - The Sliders puzzle 
    - The Lucky 7 puzzle 
    - The Happy 8 puzzle. 
    - The Blithe 12 puzzle. 
    - The Binary Color Device 
    - Analog gadgets 
    - Wythoff's Nim 
    - Changing Colors 
    - Breaking Chocolate Bars 
    - Calendar Magic 
    - Squares and Circles 
    - Diagonal Count 
    - Flipping pancakes 
    - Four Knights 
    - Latin Squares 
    - Marriage Problem 
      From Alexander Bogomolny's Interactive 
      Mathematics Miscellany and Puzzles: 


                    IMPLICATION AND DENIAL

         A conversation from MATHEDU, an unmoderated 
         distribution list for discussing post-calculus 
         teaching and learning of mathematics.

    Questions about student understandings of implication, 
    ranging among causality, spreadsheets, Boolean functions, 
    ambiguity, proof, how to communicate reasons why a result 
    is true, how to elicit connection, mathematical language, 
    and native modes of thinking - with sample problems,
    suggestions, and references for helping clarify the
    logic of x<y => x <=y.

"One of the things that has surprised me in teaching beginning 
 analysis... is the difficulty experienced by a number of 
 students in accepting the statement: x<y => x <=y 
 (x less than y implies x less than or equal to y). Working 
 through this caused fierce arguments between some students. 
 Some of them wanted the implication to be reversed... Has 
 anyone had similar experiences? Is there any theoretical 
 framework for understanding the real difficulty for students 
 in this situation?" - David Epstein

"If students in their work up to calculus have had no trouble
 with inequalities, it may be that they were operating
 symbolically and the difference was not apparent to them.
 ... Only when you start using explicit language does the 
 difficulty crop up." - George Tintera

"It is an annual event to find the class thinking an 
 implication works the opposite way to the way I mean. One
 of the examples I habitually use is: P and Q are congruent 
 triangles. P and Q are similar triangles. Which implies 
 which?" - Robert P. Burn

"...part of learning mathematics (at least for those who 
 plan to become mathematicians) is learning to think in ways 
 that are closer to mathematical language." - Brian M. Scott

"...two things might help... The first is Johnson-Laird's 
 notion of mental models (which are not necessarily images). 
 What mental models do students have for implication? The 
 second is that if students interpret implication as in a 
 spreadsheet, then it's more like an injunction to act, 
 rather than a logical statement..." - Gary E. Davis


The Math Forum's Epigone software archives, threads, and
searches the MATHEDU discussions:


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