The supposed explanations of fractions fail to derive the rules of fractions from well stated premises. Students are led to believe that somehow the rules of fractions or their very definition comes from the purported explanations. However, they don’t.
As students struggle to go from the supposed premises to the rules of fractions they fail. So they decide they are not good at fractions.
It never occurs to students that the math ed establishment is pushing a false pedagogy on them that in fact represents itself as doing more than it does. The students accept the authority of the schools and textbooks that the supposed explanations of fractions should lead to the rules for fractions that are actually taught and tested.
Since the rules that are taught and tested don’t follow from the supposed explanations, the student has to conclude, they don’t get math. So they mentally drop out of math. This happens when fractions are introduced.
If students were told that fractions were an invention that can be applied but not derived then they would not feel that they were not getting it. Pizza is an application of fractions not the source of fraction rules.
Any application of fractions to geometry is a model that consists of the following parts at a minimum.
- The geometric figure is considered to be made of points.
- These points are modeled using fractions, i.e. we have a rational point set.
- The rules of fractions already exist.
- We define the geometric figure as a type of set of rational numbers on a line or pairs of rational numbers in the plane.
- We then use the rules of fractions to work out the rules of the geometric object modeled as a set of fractions.
Students are instead taught the opposite. They are told the geometric figure exists and has properties already. They are told these then imply the rules for fractions. However, this is false and so the student is confused. They are in fact blocked. So they either memorize the rules or drop out.