[IAEP] Turtles All The Way Down

mokurai at earthtreasury.org mokurai at earthtreasury.org
Sun May 22 11:36:16 EDT 2011

On Sun, May 22, 2011 11:09 am, Hilaire Fernandes wrote:
> Le 21/05/2011 03:17,
> mokurai at earthtreasury.org a écrit :
>> Seymour Papert also proposed creating an environment in which learning
>> math would be as easy as learning ordinary language. Smalltalk has a
>> number of kinds of number and shape objects, but I have not seen much
>> else
>> in the way of mathematical objects. I am trying to go through various
>> subjects to extract the ideas that preschoolers can absorb, and create
>> materials to encourage them to explore those ideas.

We also need symmetries (group theory and algebra more generally), Venn
diagrams (Boolean algebra and set theory), "rubber-sheet" geometry
(topology), probability, knot theory (topology), infinities and
infinitesimals, graphing (analytic geometry), and conic sections, among
other topics easy to visualize and make tactile. (It is trivial to
generate the conic sections using a flashlight in a darkened room.)

> DrGeo provides those extensions to Smalltalk for the Euclidean geometry
> field. This opens large use case in teaching programming related to
> history of math, largely based on Euclidean geometry.

It works also for non-Euclidean and projective geometry using well-known

History of math needs to be mined for its moments of adventure, discovery,
and controversy. It is widely assumed that math is perfect and
unchangeable in its nature. For example, that a theorem once proved stays
proved forever. This turns out not to be the case. Lambert thinking he had
disproved non-Euclidean geometry, and Peano thinking he had proved that
all models of the natural numbers are isomorphic are historically the two
most important instances that I know of. Gauss, Lobachevsky, Bolyai, and
Riemann realized that Lambert was wrong, and Beltrami finished off the
case by demonstrating a surface in Euclidean space with Lobachkevskian
geometry. Abraham Robinson ran with non-standard arithmetic, creating
non-standard analysis as an easier way to do calculus, and disposing of
Bishop Berkeley's ghosts of departed quantities.

> http://www.reunion.iufm.fr/recherche/irem/spip.php?article493
> http://fr.wikibooks.org/wiki/Programmation_objet_et_g%C3%A9om%C3%A9trie
> http://revue.sesamath.net/spip.php?article330
> Sorry those references are only in French, a lot of teachers exploring
> programming for math seems to come from that place.

Pas de difficulté pour moi. Merci.

> Hilaire
> --
> Education 0.2 -- http://blog.ofset.org/hilaire
> _______________________________________________
> IAEP -- It's An Education Project (not a laptop project!)
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Edward Mokurai
ج) Cherlin
Silent Thunder is my name, and Children are my nation.
The Cosmos is my dwelling place, the Truth my destination.

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