Hypothesis How they teach fractions

- Present explanations of rules.
- But explanations are false. You can’t derive rules from them.
- Students are not told that.
- Students fail to understand how to derive the rules from explanations.
- Students think they can’t understand math.
- Students drop out.

The following links came up in my searches and are not meant to be criticized this way, but are simply search results that the reader can test whether these statements or hypothesis is valid.

A derivation of a fraction rule using letters. This has nothing to do with pizza or lines. It starts with algebra rules in letters and ends with algebra rules in letters. No mention of lines or pizza is attempted.

http://mathforum.org/library/drmath/view/63841.html

In contrast the following webpage discusses ways to teach students to remember the rules. It does not actually claim the rules are derived this way.

http://www.homeschoolmath.net/teaching/teaching-fractions.php

The teaching that the rules can be derived from the explanations may not be explicitly made, but it can be the message students take from the clutter of explanations and applications. This is the downside of Fractional Democracy, throwing every explanation at students as if equally valid.

Hung-Hsi Wu advocates teaching the rules of fractions starting in fourth grade as simply abstract rules not derived from anything. He advocates just presenting the definition of fraction as a data object with two numbers and rules for manipulating the data object. These are not his words, but in effect this is what is done.

One advantage of the Peano Axioms is to see what assumptions and derivations look like. From that one can see that fraction rules are not derived from geometry but are applied.

A constructive model of whole numbers is something like the von Neumann construction of ordinal sets that contain each other. Cardinality as 1 to 1 correspondences builds on some way to represent or at least set up the head segments of natural numbers. It can also be viewed as a constructive model of the Peano Axioms.

A model of fractions like these would be complicated. One question is whether such a model exists. In prior posts, candidates for such models are discussed.

These include counting by bins using the Peano Axioms where we have a parallel set of bins of different size. So we count by bins of size 1, of bins of size 2, etc. using the Peano Axioms for each bin size as if we did not know the bins contained members.

We then set up correspondences between these Peano Bin Sets and use those to represent fractions. This is what an actual definition and derivation of fractions might look like. It is rather complicated.

Another approach is to define ordinal sets that contain fractions and define fractions as ordinal sets. This is expanding on the von Neumann construction of whole numbers.

Another approach is to use a pair of 1 to 1 correspondences. So we relate 2/3 to two one to one correspondences. We then define fractions as such a pair of correspondences and derive the rules from that.

All 3 of these approaches are still in development. But these are examples of how one might try to explain fractions in more fundamental logical and set theory terms. The resulting explanations are more complicated than the rules of fractions. However, they do offer meaning.

This is like the Peano Axioms and von Neumann constructions and cardinality with head sets of natural numbers. These give meaning but are work to learn.

In this webpage, a calculator is used to “prove the rule”

http://www.sosmath.com/algebra/fraction/frac3/frac33/frac33.html

“Now prove to yourself with your calculator that both fractions are equivalent.”