Length, area, pizza, etc. are applications of fractions. Any geometric application of fractions is a type of measure theory. We don’t derive the rules of fractions from a geometric application, we use them to explain the geometry.
We don’t derive the rules for derivatives by geometry of slopes. We apply the rules of derivatives to geometry of slopes.
1/2 + 1/3 = 3/6 + 2/6 = 5/6
This is applied to points on a line, not derived from them.
What happens when a student is told that this rule follows from the statement that fractions are points on a line? The student can’t derive the rule from the statement that fractions are points on a line. So they think they don’t understand it.
The supposed “help” actually is a wall. The student is now stuck because he can not derive the rule for adding fractions from the supposed premise that fractions are points on a line. No one can, but the student is not told that.
The teacher doesn’t say, and likely doesn’t know, that there is no derivation of the rule for adding fractions based on the premise that fractions are points on a line. The premise in fact has no mathematical content and is completely without any mathematical value at all by itself.
However, the student is led to believe the opposite. Since he will find it tells him nothing about how to derive the rule for adding fractions, he will simply conclude he is not good at math.
Students often stop believing they are good at math at fractions. This is because fractions are “explained” by non-explanations that are completely without any ability to derive the rules of fractions whatsoever.
This is shown by the total absence of such derivations from the purported explanations. There is no Pizza Theory of Fractions. There is no derivation of the rules of fractions from the statement fractions are points on a line.
Let the reader show me wrong. Let them cite the literature where this is done, with explicit page numbers and the statement of what the premises are that lead to the derivation of the rules for adding fractions, multiplying them and dividing them.
Students are given these supposed explanations, they don’t work and they conclude they can’t do math. This happens in fractions because that is where they are set up to fail by the system. This means by the textbooks and the college math ed programs. They design this failure by their false explanations for fractions. So the students fail in lock step, year after year, generation after generation.