Suppose we have defined fractions algebraically and worked out their rules including for adding fractions with different denominators. Now we are ready to apply this algebraic construct to length. We consider a function from point sets on the fraction number line to the set of fractions.
The fraction number line has all the fractions on it. We can define a length for some sets of fractions. At this stage, we can only define a length if it is a fraction of natural numbers, i.e. a rational number. So if the length would be an irrational like square root of 2, we are not ready for it yet. So at this stage, lengths of square root of 2 do not exist for us. So if we took a right triangle with legs of length one, we can’t give the hypotenuse a length at this stage.
For two fractions, we can define the length of the line segment between them as equaling the larger minus the smaller. At this stage, we work with unsigned fractions.
If two line segments overlap, we can work out that the sum of the two lengths minus the length of the overlap equals the length of the whole line segment. This assumes the line segments all stop and end at rational numbers, so we have lengths assigned for them from our length function.
This rule extends to non-overlapping line segments, where we can add the lengths and the overlap is zero, so we subtract zero. At this stage, we limit this to a finite number of line segments.
We have applied rational numbers to give a simple theory of the length of rational number points sets. This theory works in the way we expect. It satisfies intuitive rules.
Length of rational point sets is an application of rational numbers, it is not the definition of rational numbers. The rules for adding rational numbers do not come from rules for combining the length of line segments. Instead, we have defined rational numbers first. Then we define a function from rational number point sets to the rational numbers. That function results in rules for combining lengths of point set that satisfy intuitive requirements.
In no way do we get the algebraic rules for combining or manipulating fractions from any geometric theory of length or measurement. That would be putting the cart before the horse.
However, the false impression that fractions and implicitly the rules for fractions come from length or area is made all the time. This is not helping students and especially not weaker students. Confusing them on the logical structure and deceiving them to think the algebraic rules for fractions somehow come from geometry is a disservice. Moreover, since it is wrong, it is not teaching concepts to make them think it is right.
Nor can we try to resort to Euclid. Euclid did not have the symbols of arithmetic and algebra developed from the 15th to 17th centuries. So Euclid did not derive the rules for using these symbols in algebra and arithmetic when dealing with fractions. Euclid and the Greeks may have had a concept of rational number, but they did not use it to derive the rules of algebra worked out in the 15th to 17th century using the symbols developed in that time.
The modern era does fractions first using the symbols of arithmetic and algebra developed in the modern era. These rules for fractions are not derived from geometry. They are algebraic rules that are from inside algebra. They are then applied to geometry by having a function from rational point sets to the rational numbers. Later we deal with point sets that contain irrational numbers or where the mapping is to an irrational number like the square root of 2. Here again the name of Dedekind appears prominently.
Muddling fractions and length and area is not teaching concepts. What happens is a false impression is made that the rules for fractions somehow come from geometry. But since this is never actually explained, and the rules of fractions never derived from geometry, it is a deception. The students then learn the rules for fractions by using them over and over. If that doesn’t happen, they don’t learn them.
There is a false attempt to pretend to teach concepts of fractions while actually only teaching procedures. Since the claim to teach the concept of fraction starting from geometry is false, there can not be conceptual teaching or learning. That means the learning is procedural only.
If the lesson plan is to teach fractions by concepts from geometry and this lesson plan is fundamentally flawed, then it simply takes up time from procedural practice. If that time is not allocated, then the students don’t learn fractions either conceptually or procedurally. For many students, they continue not to understand fractions or be able to apply the rules of fractions accurately at higher grade levels.
One fundamental source of the confusion students have over fractions is they are led to the false belief that they should understand the algebraic rules for fractions from geometry. This false impression never helps them. Neither at early grades nor at later grades.
Fractions are an algebraic concept. The algebraic rules of fractions come from algebra not geometry. The system of symbols invented in the 15th to 17th centuries did not exist when Euclid was writing and Euclid never wrote down derivations of our algebraic rules using our symbols from geometry.
Students should be taught that fractions are an algebraic invention, an invented type of number. We invent rational numbers and then we have to invent a function from rational point sets to the rational numbers. This rational length function or rational area function satisfies most of our intuitive notions. However, it is incomplete because there are point sets it does not assign a length or area to.
That requires inventing a new number type or types. Those new number types are then used as point sets and as lengths and areas and other geometric measures. These are the output of the new functions of length and area that go from rational point sets to real numbers or from real point sets to real numbers.
We see again that math foundations comes first and pedagogical content knowledge comes second. Perhaps they may spiral as well. We can’t understand the pedagogy of teaching fractions, length and area until we disentangle the concepts logically. As it is now, the concepts are taught backwards and in a muddle. This is a defeat for PCK without math foundations since it failed to recognize this and correct it.
PCK and math foundations have to go together. Math foundations fills the gaps and removes errors in our concepts of elementary math. As long as we have such beams in our eye, we are disabled to remove them from the eyes of others.
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