[Newbies] Proper fractions?

[johnps11 at bigpond.com]

> Hey, guys:
>
> 	(And who came up with that whole "improper" terminology? Some guys with
> small numerators, I'd bet....)
>
> 	===Blake===

I spent many years teaching remedial maths.  The whole
proper/improper/mixed number concept messes with many kids learning
arithmetic with fractions.

As I recall,  proper fractions used to be written as a sum of partial
fractions with numerator 1. For example:

3/8 = 1/4 + 1/8 so it was written as a proper fraction as

1  1
-  -
4  8

2/5 = 1/4 + 1/10 + 1/20

1  1  1
-  -  -
4  10 20

12/7 = 1 + 1/2 + 1/3 + 1/42

  1  1  1
1 -  -  -  (a mixed number)
  2  3  42

If it was clear from context that a fraction was being discussed it would
be written just by listing the denominators:

1 2 3 42

This caused much confusion, is 2 4  3/4 or 9/4 (i.e. a mixed number or a
proper fraction)?

2 2 4 was always 11/4 though (as the second 2 had to denote 1/2).

A fraction written any other way (excepting continued fractions) was
considered 'improper'.

Later usage allowed for larger numerators.  Improper fractions were then
fractions where the numerator was larger than the denominator, as we use
today.  Proper fractions are always < 1.

1 3/4 as meaning 7/4 is called a mixed number, and use of mixed numbers is
a cause for many errors,  as if written messily it can easily be read as
13/4 or even 1/14.

To my mind a class that displays as a mixed number should be a subclass of
Fraction, as the default of displaying a fraction as an improper fraction
has no room for ambiguity.