In fractional democracy, every application of fractions or idea of what fractions means are pushed at students as all being equal in structure and logic and worth. Applying fractions is the same as a concept of what they mean in fractional democracy.

In fractional aristocracy, the meaning of fractions as objects comes before applications in worth, logic and structure.

Confusion load is related to cognitive load. Confusion load will prevent the person being able to sort out what is what easily. Confusion load can lead to people giving up. If the confusion load is high enough, this is what they will do.

Not having a textbook which is logical and structured contributes to confusion load. In fact, it creates enough confusion load to prevent understanding the logical structure of math. No one in recent centuries has ever understood the logical structure of math without using a book to help them.

Current approaches to fraction instruction illustrate fractional democracy and confusion load.

We might also add structural confusion load. This is the cumulative dissonance of not understanding the logical structure of math up to the point you are trying to learn it. Structural confusion load is fought by having a textbook with the logical structure of math from the Peano Axioms and basic set theory up to fractions, if one is at the point of studying fractions.

Do constructivists spontaneously reconstruct the Peano Axioms? The Dedekind Recursion Theorem? Dedekind came up with the idea that you had to prove the existence of functions recursively defined before you could define addition as a recursive function of repeated successor.

x + 0 = x

x + y’ = (x+y)’

where ‘ is successor, e.g. 0’=1, 2’=3, etc.

Spontaneously reinvented the Peano Axioms, recognizing the need for the recursion theorem before defining addition, defining addition recursively and then proving the associative and commutative laws using mathematical induction does not happen. Moreover, most of math education research is done by people who never read material related to this, especially Dedekind’s own book.

Recognizing the need to prove that recursive definitions for functions are valid, and that the function exists seems to have seldom occurred. We have had the Landau Foundations of Analysis book which tries to avoid the recursion theorem of Dedekind, but few to none react to that book by reinventing the Dedekind Recursion Theorem.

Math foundations is the meaning of elementary math. When we skip the meaning of elementary math, we have to expect students fill the gap with something wrong. There is a missing link in their thinking. These structural errors have to retard progress and eventually produce their own type of load. The load of not understanding the previous part of the course or previous courses deserves a name itself.

The load of not understanding the logical structure of the math up to the point one is learning deserves a name. Structural confusion load is the name suggested above. This is a major load.

Missing Math Foundations Load is another load we should recognize. Not understanding the meaning of elementary math is a load. It retards progress and makes it harder to learn math. One is reduced to procedural memorization by the load of not understanding the math foundations of elementary math.

Math foundations lightens the load by removing what is not the explanation or meaning. Fractional democracy pushes in every idea that comes to the mind about fractions. Math foundations prunes those away to the correct structural definition of fractions as an ordered pair of natural numbers with procedures to manipulate that data for addition, subtraction, multiplication and division by fractions.

Math structure is about cutting down the load of what you need to know to understand the math point you are at. The shortest path from the Peano Axioms to fractions is the math foundations path. Learning this path gives you the lowest load path to understanding what fractions mean. All other paths are higher load because they are longer and higher confusion load since they are the wrong path.