After the natural numbers, 0, 1,2, etc. are introduced, the fractions are introduced. What are they? Did they already exist? Or are they an operator on natural numbers?
We can think of 1/2 as a function. Its domain is the even natural numbers. For each number 2n, it matches as output n. So the graph of the function are the ordered pairs (2n,n) for n any natural number.
Suppose that 1/2 is introduced in dividing something in half to share it. The thing divided is implicitly treated as already a multiple of 2. So if it is a donut, we cut it in half, but the two halves essentially already existed in this way of thinking of 1/2.
If we have a pizza and it has 6 slices, then 1/2 is 3 slices. The pizza is either already cut, or is first cut. The physical cutting is not what fraction means. The fraction comes into play after the cutting happens.
Something is broken in half first. Then the two halves are shared. So the mathematics part comes in after the natural number 2 already exists in the problem.
The same applies to marking 1/2 points on a ruler and so “dividing” it. The marks are drawn physically and then we take them as given and say that we have divided the ruler into 1/2 units. Marking and cutting are physical acts not mathematical ones. The math comes after the physical act. The physical act either creates a new natural number, such as two halves by cutting, or it reveals that there was a pre-existing granularity that corresponded to an inherent natural number in the problem that was capable of being divided by 2.
The point is that all these methods of teaching fractions are really fractions as operators, just covertly. In each case, there is considered to be a natural number that exists and this natural number is what is divided by 2.
So a donut is considered to have already been 2 halves. The 2 is divided by 2 to get 1, i.e. one of the halves.
A pizza of 6 slices is divided into 2 by taking 3 slices. The 6 slices pre-existed before dividing them by 2 to get 3. The dividing by 2 does not happen until we change our view to associate the number 6 with the pizza.
In the same way, we alter our perception of the donut and say it corresponds to the number 2 before we divide by 2 to get 1.
In each of these cases, the fraction as function point of view applies as soon as we change our view of the original natural of the thing as having associated with it a natural number divisible without remainder by the denominator. So the fraction as function view is being applied as soon as we change the number we associate with the object or situation. The new associated number is always one in the domain of the fraction as function. So we are always applying the fraction as function concept, even when we don’t explicitly say so.
In fraction democracy, every application or idea about fractions is presented as equally important. The student is expected to sort it out logically. They don’t. When we analyze fraction democracy, we find that all the cases presented are really a covert application of fraction as function. The application comes after a new number is associated with the problem that can be divided by the denominator.