I have read many complaints online from math teachers.

Specifically on Linkedin [sci Math primary/secondary education]:

http://www.linkedin.com/groups/Fractions-ugh-69765.S.5941088157259800577

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

mathematics:

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

state)

The difficulty of the fractions is that it requires an understanding of the

division.

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

operations:

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

mandatory.

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

plane.

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

rectangle.

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

temperatures.

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

situations.

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

numbers.

Before starting to split, one must know the integer division: the sharing

out.

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

letter.

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

decimals.

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.