Many students and teachers complain that math is a sequence of random drills with no connection or structure. What is the plan? How does it fit together?
The answers to these questions comes from structure. Structure is how parts of math fit together.
Abstraction is the architecture of the structure of math. The nodes and a pair of nodes as a link are abstractions.
Natural numbers are a topic in math. The structure of the natural numbers is given by the Peano Axioms.
Induction is itself an abstraction. Hermann Grassmann recognized that induction is the key abstraction to define addition and multiplication of rational numbers and prove the properties of these functions.
Dedekind then rebuilt the structure Grassmann started and filled in gaps. Peano then picked out some results in the middle of Dedekind’s book and made those a starting point as a set of axioms that would capture the essence of natural numbers.
The structure of the natural numbers as a sequence shows up over and over again. There are other ways to constructively set up such a sequence such as the von Neumann structure of natural numbers.
These ideas lead to a better understanding of fractions as ordered pairs of natural numbers. Understanding the abstractions in natural numbers is fundamental to understand a fraction as an ordered pair of natural numbers.
This takes away the randomness and substitutes meaning instead. The meaning is the abstraction. The abstraction is key to the structure.
We have removed the applications. The applications are not the meaning of natural number. They are possible as applications because the abstractions of natural number apply to that situation.
We understand an application by recognizing that it is suitable to the abstractions in the natural numbers. That includes induction.
The motivation is the insight of how it fits together. The motivation is going from seeing it as random drills to seeing it as a structure that has parts that fit together.
The applications can illustrate the abstractions and structure of natural numbers. This way they teach the natural number abstractions and structure.
If we do applications willy-nilly, it is likely that the wrong concepts are taught. Ones that are pre-Grassmann. Thus emphasizing applications can lead to wrong teaching that creates problems as abstractions like fractions as ordered pairs are taught later.
Good teaching materials based on the Peano Axioms are the only way to prevent teaching the wrong concepts of natural numbers.