How much do partial explanations help in fractions?

If an explanation is not something that anyone has developed in writing to derive a rule or all the rules of fractions from, how much does it help in explaining fractions?  Aren’t ideas like a fraction is a point on a line that have never been developed into a way to derive the rules of fractions proven to be bad explanations?

How many explanations have a starting point that is well defined and then prove even one rule of fractions even in limited circumstances? Circumstances stated and used in a proof?

A pseudo explanation is they state the rule and then they give some auxiliary statements and don’t try to derive the rule using letters and simply assert that the rule in letters follows from the claimed explanation.   Do such pseudo explanations help?  Do they contain misdirection?  A seeming explanation that can’t derive the rule is actually clutter?

Occam’s razor of math ed is remove all explanations that don’t derive any rule.  If you can’t put the explanation into letters and derive the rule in letters, then the explanation is probably not working.

How many explanations are written down in words and symbols and used to go from a well defined starting point to a rule of fractions?  None?

What is the best example? An actual piece of writing that does it?  Not something that has been around decades or centuries but doesn’t work.

Is fractions cluttered with these non-explanations?  If we used Occam’s razor to eliminate any explanation that can’t be stated in letters and used to derive a rule of fractions in letters, then we would make fractions easy?

Is this what Hung-Hsi Wu is saying?  Or is close to saying?  Or is the underlying motivation?  If an explanation has never evolved to the point of stating its own starting point in letters as rules and definitions and then derived a rule of fractions, then throw it out.

What is left are the rules of fractions themselves.  Everything else doesn’t pass the start and end in letters test.

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About New Math Done Right

Author of Pre-Algebra New Math Done Right Peano Axioms. A below college level self study book on the Peano Axioms and proofs of the associative and commutative laws of addition. President of Mathematical Finance Company. Provides economic scenario generators to financial institutions.
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