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 Subject: RE: Fractions, concept and calculations Author: Mathman Date: Oct 1 2004
On Sep 30 2004, lanius wrote:
> Greetings,

How do you teach conceptual understanding of
> fractions? What tools have you found to be effective, and likewise
> how do you teach computational fluency with fractions and what tools
> are effective?

A high school teacher, I found similar problems.  The difficulty is possibly to
do more with time than anything else, since teachers must now present so many
concepts and include them in the scheme of using the calculator and computer...
concepts need time.  Neither they nor we see everything at a glance, and we need
to see it several times in several different ways in order to be able to
abstract and form a sort of whole picture in our minds.  Books have been written
on this by psychologists from the early days of general public education. there
is the natural concept of a number of things, and the abstract concept of
"number".  Some arithmetic manipulations are strictly "techniques".  Seeing
meaning in the end result is a different concept.

A self-taught programmer, I challenged myself to produce an algorithm very
early on that would print out a division to anynumber of decimals.  It was only
during that process that I realised the number of detailed steps we go through
[generally quickly andwithout thought of those processes] in order to do a
"simple" problem in long division.  So it is with fractions.  There seemed to be
a world of difference to some students betweeen calculating a routine algorithm
[do this this way] sum of fractions, and really seeing the significance of the
result.  It is indeed a major problem in teaching, and the difficulty needs to
be understood and respected. As you suggest, these students need not be entirely
dull.  One of the great thrills in teaching is to see the enlightenment as the
meaning dawns.

A lot might depend on how you present the problems.  For students
havingdificulty, to see differences in size, assuming an exercise done in which
some of the results would include values like 6/7 and 7/6, the exercise should
be completed by some simple drawing to indicate these values for visual
comparison.  While talking about weights and measures, I always had the class
pass around several known weights to get a "feel" for what they were doing when
calculating and converting.

Also, I kept a strict analogy throughout all topics, such as "Adding is
counting, and you can add only *like* things." This applied to all topics
including algebraic quantities, and vectors or whatever.  They kept getting the
same message over and over.  So, fractions can be added if they are alike, and
that is *why* denominators must be made the same, in order to make them "alike".
...and so on.

Fractions are extremely important for more advanced studies.  This is not only
in terms of their many practical applications, but in order to understand their
extension into rational algebra, ratio of many terms, proportions, and variation
which has many further applications.  Let's not forget also trigonometry.  That
is, the really important message for them to understand is the concept of what a
fraction really is.  There is also the inevitable fact that not all are born
with the ability to abstract [generalise] as readily as others.  It's all a
matter of time as the brain coalesces random thoughts into a whole.  It's a most
difficult task for teacher and student with perhaps no single answer.  Students
in more advanced studies meet the same difficulty trying to learn to see the
graph of an algebraic function before they actually draw it, or to get a feel
for what might be happening physically as described by function or graph.  It
amounts to the old expression:  "Familiarity breeds contempt."  So, when all
else fails, ...practice, practice, practice.

If it might help, I found that a study of early ideas in algebra helped to see
what was happening in some manipulations.  Numbers change during claculations.
Letters just shift around.  Study of (a/b) + (c/d) to (ad+bc)/(ad) or
(a/b)/(c/d) to (ad/bc) and so on an lend itself to helping them see what is
happening with numbers.  Also impression of fundamental ideas like adding like
things, or equivalence [perhaps the most fundamental of all], and so on might
help wit hunderstanding of structure.  But only by comparing numbers with some
form of visual drawing or real image can such ideas as relative size have
meaning.

I apologise for the length of this reply, and hope it helps a little.

David.