Saturday, February 28, 2015

Fractions.... ugh !

I have read many complaints online from math teachers.
Specifically on Linkedin [sci Math primary/secondary education]:

I would like much to some expert tell me if I'm wrong!.
There (at the SciMath group of the Linkedin) I wrote this:

I'm not an expert, and surely I am wrong in thinking that I will develop.
Anyway I'll try, and I'm sure I will not say anything new, but I'd do at least a
little humor:

I think we are talking about the understanding of the fundamentals of elementary
If you have not settled well the fundamentals of elementary mathematics, when it
comes to fractions can not understand anything. (And I'm not oblivious to this
The difficulty of the fractions is that it requires an understanding of the
And the division, requires an understanding of subtraction, and ultimately of
the sum.

Electronic calculators that are available to all people, both children and
adults do not differentiate between operations: all are on the same level
represented by a key with a painted sign. But conceptually, the knowledge
necessary to understand each of the elementary operations should be
superimposed, and with a particular order. (Although sometimes, as a first
track, in some calculators the sum key is larger)

First we must become familiar with numbers, identified and recognized.
And it seems that we are forced to choose the base 10 by a consensus not really
know where it came from.

From this point, propose a specific order to teach children basic
As is evident, we must begin by the sum.
But the next step, not subtraction.
The next step is multiplication.
That is, from knowledge of the positive integers, to play with them.
And games and experiments can be the sum, or multiplication.
The amount is usually depicted as a line, that is a dimension, although not
Multiplication games, involve working in two dimensions.
Squares and rectangles: Here, as a game that takes place in a plane or a sheet
of paper should work both multiplication as operation.
I think it's important vision of multiplication as an array of big dots in a
Here you should begin to explain about numbers that can be obtained from a
multiplication: it should be clear that there are some numbers that can not be
the result of a multiplication.
At this point, children know and can recognize all the numbers perfectly, and
can clearly see that there are numbers that can not fully populate a square or
I think at this time of learning the knowledge of the concept of prime number is
as important as knowledge of multiplication tables. (The prime numbers are the
numbers that you never will find as a result in any multiplication table, and at
the same time any number can be obtained by multiplying prime numbers between
them). The prime numbers are the basic bricks of the complete construction of
the numbers building.

And then I think you have to start again:
We must expand the knowledge of numbers.
In our world, we are used to seeing positives and negatives:
Are protons and electrons, positive and negative charges, matter and antimatter,
attraction and repulsion, money in the bank and debt (currently precisely, more
debt than money), in the old photographic culture negative film, and copies
positive, and finally even the worst example: positive and negative

After familiar with the expansion of natural numbers to integers, you can begin
to explain a new operation: subtraction.
And the rest can use all the analogies offered by nature to experience:
But there is one that is best suited: Matter and antimatter.

Everybody knows that matter and antimatter destroy each other.
And simultaneously, you can create equal amounts matter and antimatter at the
same time from nothing (they can not humans, but somewhere in the universe
happens: To be clear I'm doing a simplification that would make hair tip to a
physics expert)
Imagine games where positive numbers are eliminated with the negative, or the
creation of equal amounts of positive and negative elements from nowhere, with
no any change in the effective amount of all elements above board (the sum of
all them).
Even I have examples in software made by me some years ago.

And I think this is the time to speak of zero. Zero is both "nothing" and also
an undetermined but equally positive and negative elements or matter and
antimatter, pennies in the pocket and debt.

I think the point where we are, the films they have seen children, and games
that will surely develop or imagine, no child will miss in a reflection that use
matter and antimatter as imaginary tool for learning.

At the same time, we could start with the mechanics of operations as it has
always has been done: Subtraction with borrowing and the usual mechanics to
solve these operations.
But with the confidence that the result will always be within the set of the
known numbers. Because we are familiar with a result of both matter and
antimatter, money or money we owe.

We know to add and multiply, and we know subtract.
All the way I'm trying to go, we must never forget that mathematics is a fun
game, and a place to experiment:
Although not essential not think it be reflected by any curriculum, we must not
forget that we have not tried the multiplication of positive and negative
numbers, separated, or mixed. As representable integer multiplication above a
plane, the results are evident. And I guess as a fun game, yet I believe I can
assure you that with correct results guaranteed, although I can understand the
discomfort that may feel a teacher, helpless in front such unusual

And when we got here, we are ready to face the division.
The result of a division is a sharing out, it is also a proportion.
We're not prepared to afford the mechanism of decimal division.
We would have to broaden our knowledge of numbers:
At this point, we only know the positive and negative integers: natural
Before starting to split, one must know the integer division: the sharing

At this point, the division is the inverse operation of multiplication.
When we have the result of a multiplication in one plane (a rectangular matrix)
we can see directly the result of two divisions simultaneously. A square
displays the result of two splits. A square, the same sharing out twice.

Experimenting with splits that to make it easier, at first by simplicity should
be positive numbers, you can reach to see a problem with the distributions of
numbers that are not the result of a multiplication already met the first heard
of multiplication: the prime numbers.
Therefore, we must expand for the third time knowledge of numbers. Must
integrate rational numbers, the fractions.
Each fraction is the representation of a new kind of number: A number that needs
two numbers, one above and one below (or where you want to put). And are both as
a ratio and the description of a sharing out. But in the educational process
should be clear that they are a new type of number: which allows to get the
result of the inverse operation of multiplication for numbers that we could not
arrive until now.

I'll stop my thoughts here.
The explanation about fractions extremely lengthen this already too long
And I will not do. From now just do a sketch, but I think I managed to
substantiate the basis necessary to prepare the fractions with a little more
comfort than detected in this dialogue so interesting:

At this point, with the basis described, we would have the ability to understand
a little better the fractions, equivalent fractions, the role of prime numbers
in simplifying fractions, and the ease with which we have faced the positive
numbers and negative, integrate them into reflections on fractions, if only to
play but to see also that mathematics is a solid and coherent construction that
which is far from being threatened by the antics that can devise a kid.
And then face the complex world of operations with fractions.
And I think we should stick with the same order we have followed so far:
The sum of rational numbers, multiplication, subtraction and division.
And once experienced this, or at the same time addressing the division with
The decimal number as a result of a division is somewhat flawed alternative way
of expressing a rational number also called fraction.

Greetings, and thanks for the opportunity to participate in this dialogue.
This dialogue has been very useful, because it gave me the necessity of
writing, and writing it, I think I have clarified my thoughts a bit. Thank You.
And thanks for the patience if you arrived here.