## Fraction confusion from mishmash of methods and applications

Fraction confusion comes from a mishmash of methods and applications.  Methods of representing or calculating with fractions and applications to length or area are conflated together in k-12 without having distinct names or references.  The result is that it is just one confused block in students’ minds.

Each distinct idea in fractions needs to have a distinct name and be distinguished from other methods or applications.  Representations of a fraction, the equivalence class, methods and applications all must be distinguished precisely.  If not, the students will be confused and will invent to fill gaps or to reduce the confusion.  The result will be they don’t understand the concept.

The education system throws at students applications, representations, mathematical concepts, pictures and wordplay to teach fractions.  The one thing they don’t tell is the logical structure of fractions as an ordered pair.  The latter would take less time and be less confusing.

Students are expected to take what they are deluged with and self discover the 19th century abstraction of an ordered pair.  This doesn’t really happen.  So the students are confused.

All the different applications, methods, and representations led over centuries with the contribution of mathematicians over time to the concept of an ordered pair of numbers with methods.  This has to be taught clearly and explicitly.

The other stuff mostly existed before the 19th century and didn’t result in discovering the 19th century of fraction until mathematicians hit on the notion in the 19th century.   This won’t happen in the classroom unless the 19th century notion is taught explicitly at some stage.

The rest has to be distinguished as precursor, application, requirement, or inspiration.  They are not our final logical concept of rational number. This has to be taught explicitly.

It is the weakest students hurt the most by the confusion.  Logical confusion does not help the weakest students. The weakest students are the least able to replicate the work of 19th century mathematicians on their own.  Anything less is a confused notion.  That is what the 19th century mathematicians proved.

Difficulties understand fractions start with difficulties understand natural numbers.  If natural numbers are not taught as Grassmann, Dedekind and Peano taught us, then everything made of natural numbers or using natural numbers in its procedures will be confused as well.

Students and pre-service teachers must be taught the Peano Axioms, order of naturals, recursion theorem, addition and multiplication and their proofs.  Moreover, this should be in the Dedekind order of topics.  Order should be taught before addition and before the recursion theorem.  This is the best way to learn order of natural numbers.

To really intuit the logic of order of natural numbers it is in some ways the only way to learn order of natural numbers.  Dedekind had a direct connection when he wrote his book.  Yet it is largely ignored because it contains many small steps as distinct numbered paragraphs and lemmas.  But without understanding those, the logic of order of natural numbers is not understood.   This hampers any understanding of natural numbers.  It leads to failing to see how addition and multiplication are order based and are methods for managing indexing and keeping the place in the count.

The failure to understand natural number logic then carries over to rational numbers.  Fractions are a confused concept if one does not have a strong understanding of natural numbers in a logical structure.