[IAEP] Looking for Concrete "Fraction Experiences"
mokurai at earthtreasury.org
mokurai at earthtreasury.org
Thu Jul 14 14:01:01 EDT 2011
On Thu, July 14, 2011 8:40 am, Maria Droujkova wrote:
> 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
> in life. Many people will conceptualize "How many halves are there in
> 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.
and providing consistent rules for them, maintaining algebraic identities
such as commutativity, associativity and distributivity.
ab = ba
(ab)c = a(bc)
a(b + c) = ab + ac
> mathematics, figuring out how
to make (a matter of definition, not discovery)
> operations work for all types of numbers and
> even non-number entities is a very strong value.
Although non-commutative (matrices) and even non-associative (octonions)
structures are of great importance.
> As such, we want to
> subtract greater numbers from smaller ones, take square roots of
> and multiply anything whatsoever (zeros, ordered arrays, transformations,
> etc.) This extension value definitely trumps 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.
in order to arrive at the general rule, which we can then extend to
> These steps are
> entirely sufficient for dividing pizzas.
Fractions are completely equivalent to whole numbers of equal-sized
pieces. It makes no difference to the result whether we use 3/8 or talk
about 3 pieces of 1/8 each. But I would like to lead children through the
two-step process to the algebraic rules, particularly
(a/b)/(c/d) = ad/bc
without going through (a/b)/(c/d) = (ad/b)/c in between.
> 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.
Well, that's what I want to find out. Are we making the individual steps
sufficiently simple so that they can become obvious? What kind of practice
is required to make them obvious? The most complicated mathematics is made
up from steps so simple and obvious that even an utterly stupid computer
can cope with them. Children can actually understand them, and put them
together into more complicated ideas that become equally obvious over
time. It's just like learning language, which starts with memorization of
words and patterns, and soon becomes habit with sufficient practice. Only
the children can tell us how much practice is sufficient, and what kinds.
> 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.
Not in most of ordinary daily life, as opposed to work in engineering,
science, statistics, and such. But it does turn up occasionally in
cooking, sewing, carpentry and a few other areas.
* If my pancake recipe calls for 1 3/4 cups of flour, and I only have 1
1/4 cups on hand, how much egg, milk, blueberries, and so on should I add
to make a partial batch? Well, obviously I should multiply every
measurement in the recipe by (1 1/4)/(1 3/4). Multiplying top and bottom
by the denominator of the two fractions (4) gives 5/7.
* If it takes 2 3/4 yards of cloth to make this item, how many can I make
from a bolt of cloth 20 yards long?
> 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.
But we have to do that anyway. Computer processors are built with separate
integer and floating point hardware or microcode. If we want rational or
complex arithmetic in a programming language, or vector and matrix
operations, there must be a library or other segment of code for each.
You are correct about finding the maximal structure where integer
arithmetic holds, however.
> Maria Droujkova
> Make math your own, to make your own math.
>> , 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
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