The Rule of Teaching Fractions
Every attempt to explain fractions fails and you always end up with just the rules.
An alternative version is
Any explanation of fractions in terms of meaning or fundamentals that can actually derive or prove the rules of fractions including the division rule is or will be extremely difficult to use or understand.
So if you have a constructive definition of fractions in terms of functions and sets in some way, then when the constructive approach gets to proving the rule for division of fractions, the constructive approach will be difficult to understand at best even for its inventors.
A constructive approach to fractions has to create the following
- Define or give a meaning to unit fractions.
- Postulate or derive Peano Axiom type rules for the numerators of a fixed denominator.
- Derive the rule to add fractions.
- Derive the rule for multiplication of fractions.
- Derive the rule for dividing fractions.
So if you take 1/2 * 1/3 = 1/6, you will have to find a constructive way to go from unit fractions of 1/2 and 1/3 and map to a unit fraction of 1/6.
This is possible with something like the infinite system of parallel bin sizes being counted, but it is complicated. Basically, you have to constructively define multiplication so that the bin size of 2 unit and the bin size of 3 unit are mapped by multiplication to the bin size of 6 unit. This involves a function relationship so it involves functions from two separate bin size Peano systems to a third bin size Peano system.
Hung-Hsi Wu’s various dicta are discussed at the above and are found to be apparently inconsistent.
What this blog has called Fractional Democracy, the pushing of every idea everyone ever had about fractions at kids is overwhelming and confusing is one of his dicta.
Another Wu dicta is that at grade 4, the rules of fractions have to be taught as the single definition of fractions. This amounts to just teaching that fractions as an ordered pair are a type of number. Whether this is called a new type of number is not clear.
But basically, we have Peano Axiom numbers and constructive versions of them such as ordinal number sets of the von Neumann construction type and cardinals as 1 to 1 correspondences with head segments of natural numbers. When we try such approaches with fractions we get overwhelming complexity. So we just go to the definition of fraction as an ordered pair with rules.
Progressive education methods are suited to fractional democracy since this leads to group projects and discussion. One could also say that all constructive methods for fractions will be somewhat suited to group projects, although the grade level of the group might be questioned for more complicated schemes which are sufficient to actually derive all the rules of fractions.
==The following comment from one of above links is particularly effective in expressing the pro progressive view.
The pro Wu just teach the rule position is that these other methods if they actually construct and prove the rules of fractions for addition, multiplication and division will have to be very complicated. If the justifications are not complete systems that derive the rules, then obviously they will fall short when we get to teaching the rules.
The most notorious failure in teaching the rules is to say they are points on a line full stop. This teaches no rule. You can’t derive rules for fractions from the statement that fractions are points on a line. That has no mathematical content. This is what Euler does in 1765 and then ignores it and states the rules in algebra without any attempt to derive them from some geometric basis.
At this stage, the method for deriving the rules of fractions completely amounts to the abstract ordered pair definition with some of the rules taken as part of the definition and the rest defined or derived as a type of function. This is really a pure rule based method.
Deriving the full rules of fractions from a constructive meaningful definition of fractions in terms of other more “basic” logical and mathematical entities does not really exist as far as this blog is aware. Pointing out other approaches exist. The formal method exists, but this is not the type of constructive definition that the von Neumann construction is of the natural numbers for example.
Thus as a matter of practice, at some point, one simply states the rules for fractions in abstract form, perhaps deriving some of them from others, but it is an abandonment of a constructive definition of fractions that leads to deriving the full set of rules.
==Restatement of the Rule of Teaching Fractions
Any system to explain fractions will either fail to derive the rules or be so complicated no one can understand it or remember how it derives the rules.
Of course, the next project of New Math Done Right is to publish a series of books giving constructive approaches to fractions and attempting to make the constructions and derivations understandable. This is like squaring the circle for math ed. The approach will be based on applying the Peano Axioms to bins of separate size and constructing maps between these parallel Peano Axiom systems to represent fractions and derive the rules of fractions from them.
This project will give a mathematical basis to the existing attempts to justify or explain fractions and their rules. This is in parallel to the Peano Axioms and the von Neumann construction and cardinality as a system of one to one correspondences between sets and head segments of the natural numbers. Even that is not fully explained or documented and that will be the next publishing task of New Math Done Right.