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Topic: Fractions, concept and calculations

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Subject:   RE: Fractions, concept and calculations
Author: Mathman
Date: Oct 21 2004
On Oct 21 2004, ihor wrote:
>One thing It does do is
> confront  the misconception that increasing only the denominator
> makes the fraction larger.

Misconception appears to be as much a part of the human makeup as conception.

Fractions are perhaps one of the more important tools to master as early as
possible because of what follows:  Ratio, proportion, similarity, ...studies
which have even further ramifications in studies in science or more advanced
mathematics.  However, it is an unfortunate part of human nature that we do not
all grasp everything shown to us, nor all at the same time, no matter the
efforts of the teacher.

Is there a better way? I'm wondering how we'll judge that when the large
majority already do gain some control over the study, and we've had centuries to
perfect any approach to it.  For some, even headstands won't work.  A difficult
task at the best of times [ fractions, not the headstand.]

"intuition" is a fine word.  Some have it, and other don't, at least not to the
same extent.  After as many years teaching and practicing, I have no idea how to
either measure or teach that.  All I've been able to do is to give the best
explanation possible, and let their individual intuition work and develop. I
don't think I can teach to improve that.  I can show different, perhaps better
ways to do one thing or another, but the arithmetic of fractions [not
necessarily the physical representation] does eventually boil down to memorising
method, with or without deeper understanding.  Ability to recognise the same
pattern and thus the same approach to solution in rational algebra pays off as
the skill through practice [at what point DO they understand?] puts the method
into automode as the larger problem develops into a solution.  I'm comparing
that to practicing scales on the piano.  Music is built on structure [a
wonderful article in Scientific American on the patterns in the music of Chopin
comes to mind], however intuitive it becomes, and not having to think about
where to place fingers allows for other, better things to develop more naturally
in the long run.  All can learn to play, but not all can learn to play well, no
matter the teacher.  Some things simply are not taught, but are a product of the
individual.  The teaching having been done equally as well for one as for the
other.  Teachers should stop beating themselves to death over it.

My sister in law is a consumate artist.  I draw stick-people.  She could
improve my stick-people, but will not make me into an artist; and her teacher
likely had to do a lot less for her.


> I'm interested in hearing from others about effect
> ive strategies they have tried to get students to get a better a
> intution for fractions rather than just clever way to remember the
> rules.

Thanks - Ihor

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