[IAEP] Looking for Concrete "Fraction Experiences"

Maria Droujkova droujkova at gmail.com
Thu Jul 14 08:40:36 EDT 2011

On Wed, Jul 13, 2011 at 3:38 AM, <mokurai at earthtreasury.org> wrote:

>
>
> > Special bonus points for anyone who can come up with an example of
> > division with fractions (ex: 1/3 divided by 1/2)
>
> 1/2 goes into 1 twice.

When the divisor OR the quotient are whole, people do use fraction division
in life. Many people will conceptualize "How many halves are there in three
pizzas?" or even "How many quarters are there in three halves?" (the last
one is a stretch) as division.

However, after messing with fraction division for about a year (see
http://naturalmath.wikispaces.com/Divide+a+fraction+by+a+fraction ) I
believe people who don't have PURE MATH purposes avoid conceptualizing
division of a fraction by a fraction, when it's not immediately clear the
result is a whole number. Instead, they conceptualize it as TWO operations
(multiplication and division) where at least one number is whole.

The pure math purposes have to do with extensions of operations. In
mathematics, figuring out how operations work for all types of numbers and
even non-number entities is a very strong value. As such, we want to
subtract greater numbers from smaller ones, take square roots of negatives,
and multiply anything whatsoever (zeros, ordered arrays, transformations,
etc.) This extension value definitely tramps any muggle values such as
cognitive accessibility or ease of calculation. There are strong
mathematical reasons for holding the extension value dear. We just have to
realize these reasons don't necessarily apply to eating pizzas, or even to
math-rich professional practices such as nursing (let me know if you want
"Proportional Reasoning in Nursing Practice" study).

> In fact it goes into any whole number N by dividing
> N objects into 2 pieces each, giving 2N pieces. Similarly, it goes into
> 1/3 twice 1/3

There - you conceptualize it through whole-number steps. These steps are
entirely sufficient for dividing pizzas.

You only need to re-conceptualize these steps (at a significant cognitive
cost, as my teaching experiments indicate, if you go beyond the example of
1/2) as division by a fraction if you are going for the mathematical value
of figuring how fraction division works.

There are no utilitarian or artistic purposes, that I could find in more
than a year of looking for them, in conceptualizing the separate steps as
division by a fraction. In practice, nurses, pizza cooks, carpenters and so
on don't "really" divide by fractions - they work with numerators and
denominators separately.

I would suggest exploring reasons behind the math value of stretching
operations, for example, talking about how inefficient it would be to
program operations separately for different types of variables.

Cheers,
Maria Droujkova
919-388-1721

> , or 2/3. If you divide a circle into sixths, you can easily
> see that a third of the circle (two pieces) is two-thirds of half the
> circle (three pieces), in just the same way that, for example, two beads
> is 1/4 of eight beads.
>
> It has been done in detail, and is available on various OER sites, some of
> which are given at
>
> http://wiki.sugarlabs.org/go/Open_Education_Resources
>
>
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