Given a denominator n, such as 8, we can consider fractions as having a successor type relation. So
(m,n)’ = (m’,n)
where ‘ is successor, so m+1 = m’.
So (3,8) = (2′,8) = (2,8)’ = (1′,8)’ = (1,8)” = (0′,8)” = (0,8)”’
==Excerpt Grade 4
Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.
- 4.NF.3. Understand a fraction a/b with a > 1 as a sum of fractions 1/b.
- Understand addition and subtraction of fractions as joining and separating parts referring to the same whole.
- Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions, e.g., by using a visual fraction model. Examples: 3/8 = 1/8 + 1/8 + 1/8 ; 3/8 = 1/8 + 2/8 ; 2 1/8 = 1 + 1 + 1/8 = 8/8 + 8/8 + 1/8.
- Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction.
- Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to represent the problem.
The common core standard for Grade 4 is not far from a Peano Axiom based approach to rational numbers. The example given shows a fraction from repeated add one unit, i.e. repeated successor.
Recall that Keith Devlin said that addition was not repeated successor or should not be taught that way and he was not advocating teaching recursion in K-12.
We see recursion in the above example.
3/8 = 2/8 + 1/8
This is recursion.
(m’,n) = (m,n) + (1,n)
(m,n)’ = (m,n) + (1,n)
(m,n)’ = (m’,n)
(m,n) + (p’,n) = [(m,n)+(p,n)]’
“Justify decompositions, e.g., by using a visual fraction model.”
This is the weak part in the standard. We can justify the decompositions using recursion, definition of successor for fractions, and definition of addition recursively for fractions.
If we have taught successor and recursion for addition of whole numbers already, then extending these concepts to fractions is easy. The common core standard almost gets there in some ways, but still has a gap when it says “justify decompositions by using a visual fraction model.”
The object oriented approach to fractions are that they are data together with rules for manipulating that data. The common core standard gets close to that. But in falling back on visual decompositions it moves away from it. Students will fill that gap with fallacies. That is why they don’t learn fractions properly and still have problems in higher grade levels.
Fractions are an abstraction and this needs to be made clear. The proportion approach is also seeing fractions as an abstraction that is a relation. The object oriented approach is actually simpler, which is why it is embedded in the procedures, whatever explanation is wrapped around them. By avoiding saying this, the lesson will tend towards being just procedural because it won’t explicitly say what the abstract nature of a fraction is, i.e. as an object oriented data structure with member data and rules.
Going the extra effort to use the more abstract terminology has a big payoff. It is not far from what is done in the common core standard already. But the common core falls back on visual models instead of finishing the job with the abstraction. We are teaching abstractions when we teach fractions. There is no way to avoid that.
All the attempts to try to make fractions seem like they are not an abstraction fail. That is because they are. Even Euclid recognized that fractions are an abstraction and need an abstract treatment in math.
The 19th century abstraction is easier to understand than Euclid. The 19th century abstraction of an ordered pair of data with rules is almost the same as the symbols based approach developed in the 15th to 17th century with the new symbols of math. Because of this, teaching the procedures for fractions does most of the work to teach the 19th century abstraction for fractions.
The real hard work is setting up the natural numbers as an abstraction. Once that is done, then going the next step of rational numbers as an ordered pair of naturals with rules specified by definitions or proven by theorems is not difficult. Doing it this way avoids confusion. This helps the weaker students most.