Counting by bins obeys the same 5 Peano Axioms as counting by “true” ones. In fact, there are no “true ones”.
A sequence of bins is as good as the von Neumann construction as an implementation of the Peano Axioms and thus the concept of counting by one.
Once we realize that the 5 Peano Axioms all apply to counting by bins, we can teach both multiplication and fractions better.
Multiplication is repeated addition of bins.
Fractions are counting by bins with partial bins. In rigorous math terms, we have a Peano Axiom sequence of 2 bins, where we don’t look inside. So 3 means bin 3.
At the same time, we have a sequence of counting by one where we match each pair in the sequence to one of the bins in the bin sequence we set up first. Call this the bin splitting sequence.
Now when we count along the bin splitting sequence, we can stop at one node in a bin. This gives us a fractional bin.
We count the whole number of bins and this is matched to the whole bin sequence. The split bin sequence then gives us the idea of a fraction as a partial bin.
This is a rigorous explanation of fractions, because we actually can model using math the data, the bin size and bin node, and the whole number part.
Adding fractions from different bin sizes, requires a common denominator bin and Peano Axiom split bin sequence of that bin size.
So to add 1/2 to 1/3, we have two split bin sequences of bin size 2 and size 3 and then another bin size of size 6.
We map the bin size of 2 as matching the bin size of 6, i.e. one bin one bin vote, regardless of the elements inside.
Thus for a count of one bin of size 2, we map that to a count of one bin of size 6. So a node in the bin of size 2 equals 3 nodes in the bin of size 6.
Similarly the bin of size 3 is mapped to a bin of size 6 and we give two nodes in the bin size of size 6 to each node in the bin size of size 3.
We can now add 1/3 to 1/2 and get 5/6 by this mapping.
This set of mappings actually allows us to derive the rule for adding fractions of different denominators.
We should not be surprised how complicated the set of functions are that allow this to be derived.
Fractions have an infinite nature, i.e. 1/2 represents an infinite set of pairs of numbers. This complexity will show up in any fundamental representation of fractions in terms of logic, set theory and functions.