Fractions can be introduced as an operator. A fraction m/n can be applied to either 1) a multiple of n, or 2) to a number p such that p*m is a multiple of n.
So formally, a fraction is defined as an operation on natural numbers. A fraction as an operator has a domain of definition which is the set of natural numbers that it can be applied to.
Equality of fractions is defined when both test natural numbers are in the domain of definition of each fraction.
The sum of two fractions is defined in an operator sense. If we can apply each fraction separately to a test natural number, and then add the resulting naturals, then this is defined as the effect of the sum of the two fractions on that test number. The domain of the sum is the set of naturals in the domain of each fraction separately.
Equality of the sum of two fractions to another fraction is defined as equality on the common domain of the two fractions being tested for equality.
The product of two fractions is defined in an operator sense in the same way. One can then prove the properties of the product such as commutative.
Subtraction, negative fractions and division by fractions can all be introduced in an operator sense.
Fractions so defined as operators have the properties of numbers. They have addition and multiplication operations. They have commutativity, etc.
We can write operator equations, which are defined by the two operators being equal on their common domain of definition.
Given operator equations, we can introduce negative operators and division by operators to get solvability. The algebraic properties of these fractions as operators can be worked out.
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Howard Fehr published an article on fractions as operators in NCTM.
This article seems to have limited impact.
Every concept of fraction conceivable is thrown at students except ones that are logically structured and they explain explicitly the logic and structure of what they are doing. The current system produces confusion which reduces to just procedural learning which results in forgetting and blocking later abstractions that build on fractions. This includes real numbers, convergence of sequences, and functions on real numbers.