Peano Axioms are about the starting of the natural numbers, counting, order of natural numbers, and the definitions of addition and multiplication of natural numbers.
On these, we can define place value notation and its algorithms and prove its properties.
This gives an understanding of the structure of elementary arithmetic. This is quite valuable to understanding pedagogy as well.
Current instruction has logical gaps and is really a form of behavior and of drill. It has to be since it doesn’t teach the above structure. So its conceptual teachings must be limited, although still not nothing. However, what is taught as the concepts in actual classrooms may be incorrect since the teachers are not taught nor do they use the above structure.
Lance Rips et al have done studies on learning based on their understanding the Peano Axioms. This type of research is accessible if one has some knowledge of the Peano Axioms and proofs by induction.
This type of self development is a way to plan the school year for a grade and deal with learning problems of students caught in the gaps of what is explained to them. Each of these gaps is a potential stumbling block. Currently, they get over them by an act of will, i.e. of memorization and drill.
The mental energy for such may become limited in some individuals. So that the lack of instruction of the structure uses up their resources for forcing themselves forward when they are not being taught the actual structure.
This then shows up in lack of forward progress and even regression.
Fraction concepts and even techniques can depend on the understanding of whole numbers and their logical structure and even meaning. When this is not taught, or is limited, as it is now, then fractions may simply be a symbolic language devoid of meaning to the students.
In that case, they will simply repeat the same errors over and over again with little progress.