Fractions are taught as a mishmash of disparate ideas. These freely mix all types of concepts and applications of fractions a though they are on an equal logical plane.
Fraction democracy is where every idea associated with fractions is taught as if they were all equal.
Fraction hierarchy or fraction aristocracy teaches that some fraction concepts are more equal than others.
Fraction democracy leads to concept confusion and chaos in the brain.
Fractions should be taught not as pre-existing in pizza but as a step by step process. We start with fractions as operators not on 1 but on multiples of the reduced denominator.
Each fraction is a distinct operator. Each operator with a different reduced denominator has a different domain than all the others with different reduced denominators. This is a good place for self-discovery. This can and should be proven as a theorem.
The rules for adding, multiplying, subtracting and dividing fractions as operators come from the operator model.
We then abstract the operator model and create fractions as their own object with their own member data and rules on their member data.
We then apply the new number concept to model geometry. We say a circle is a set of rational number points whose squared distance from the center is less than some number. This is like N. J. Wildbergers rational trig and rational geometry.
When we do fraction democracy and just push every idea about fractions at students at the start, then they become confused forever in most cases. This hurts the weakest students most.
Doing natural numbers first and then fractions as operators on natural numbers builds in a logical progression. This is the only way to save students from a mishmash concept of fractions that is meaningless.
The current approach does not teach concepts. It blocks teaching concepts. Moreover, it leaves procedures as all students can learn.
There is no clear conceptual path given to students by the mishmash of every concept and application of fractions pushed on them as though they were logically equivalent. Fraction democracy does not work. That is what is proven by schools now.
Fraction aristocracy teaches fractions as operators and then fractions as a new class of numbers.