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Topic: Mathematical maturity and lower-order knowledge & skills

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Subject:   RE: Creativity and intuition
Author: Mathman
Date: Jan 9 2005
Is not much of what is being said here true in any field?  How do we explain the
5 yr old chess whiz, or the miracle of autistic calculation?  Most do not have
such wonders to observe, but many do at some point "see the light".  I asked a
chess master once how he came to be such, and he replied that he had played
*many* games of chess.  One day, he suddenly saw the entire palying board as a
single entity.  We learn to read in a similar fashion.  We do not see, or count
the letters as we read.  We read entire words, phrases, and even sentences at a
glance, but that was learned over much time and practice.  We then see a larger
image and so a far different picture than beginning readers who struggle with
each word.  And we can not pass on what we see to those who do not yet see the
larger picture.  The y need as much time as did we, and tht varies from
individual to individual.  So it is with mathematics.  Some see x's and y's all
over the page, no pattern, until with some practice it dawns on them one day.
I've seen a good student NOT see a difference of cubes staring her in the face,
making solution simple instead of complex.  How many time have we seen a student
who can see the circle. Add a line, no problem; another no problem.  One more?
...nothing.  How many problems have occurred that have seemingly invisible
solutions until either a line is added, or in algebra some quantity is added and
then subtracted to change the structure?  Such "insight" can be learned, but it
can be a very tedious process for most of us, and all we can do is to give
opportunity for someone to put it all together.  I doubt if it can be taught,
just recognised.


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