The tendency is to justify rational numbers by a resort to geometry. This is what Euler did in 1765.
If we start from the Peano Axioms, we define natural numbers and the successor function through the axioms. We then build up order, addition and multiplication through definitions, theorems and proofs.
When we get to rationals, we would like to justify the series (1,n) as n is a non-zero natural number. We then want to justify (m,n) for any pairs m and n as long as n is not zero.
We could try to axiomatize (1,n) as a successor sequence similar to the natural number sequence n. This would be a different successor function.
In this approach, we would define or axiomatize at some stage that (1,n)/p = (1,np).
For fixed n, we still have to justify the fractions (1,n), (2,n), etc. These could be justified as another axiomatic successor sequence.
So as an axiom (m,n)’ exists.
One way or another (m,n)’ = (m’,n) is what we want to get to in this approach. But we can’t define (m,n)’ as (m’,n) if we have not defined (p,n) for general p.
So we are back to defining an ordered pair (m,n) as part of our definition of natural number.
Once we get enough structure of rational numbers going, we can build the rest using recursive definitions.
So if (m,n)’ = (m’,n) is valid at some stage of our construction, then we can define addition of the same denominator recursively.
(m,n) + (0,n) = (m,n)
(m,n) + (p’,n) = [(m,n) + (p,n)]’
We can define multiplication of a fraction by a natural number recursively. We can combine multiplication of a fraction by a natural number and division of a fraction by a natural number to define multiplication of two fractions.
But we are still left with the need to get the algebraic definition of rational numbers started using ordered pairs and some basic definitions on manipulating ordered pairs.
Rather than accept this, education tries to present this as somehow implied from area or geometry. This leads to the muddling of area and rational numbers.
The weakest students are not helped by muddling concepts. The result of this muddling is a major block to understanding the logical structure of the progression from natural numbers to rational numbers.
Modern attempts to do what Euler did don’t work. Euler gave a hand waving justification to line segments to justify fractions. Then he ignores that completely and defines the procedural rules for adding, subtracting, multiplying and dividing subtractions. These rules ignore the geometric justification completely. This is because they are not based on the geometric justification. The result is not good.
Some students may think they don’t understand fractions because they don’t understand how to go from the geometry pseudo definition to the algebraic or arithmetic rules for fractions. No one does. They are simply being made to feel they don’t understand a connection that does not exist.
Other students will try to invent a connection that is false. They will think they understand how to go from the geometric pseudo definition of fraction to the arithmetic and algebraic rules for fractions. In actuality, they are just deceiving themselves. No such link was given in Euler or since Euler. There is no such link. But they think there is and they think they know what it is. And this becomes their self view of how they understand fractions. Now they are really tangled up in fallacies and self deception. This was done to them not for them.
The students should be told that number concepts are defined as objects with procedures. Full stop. We do not get numbers from geometry. The procedures for each number class are not derived from geometry in any way.
Area of sets of rational points is a separate subject. This subject has its own start and its own development. It uses rational numbers as a number concept to define the area of some sets of rational points. Eventually, this approach runs into point sets of rational points that it can not assign a rational number to as the area consistently.
That requires a new number class. This does not define the new number class. To define the new number class, we go back to algebra or analysis and define a new number class.
Then we go back to area and we modify our approach to area to define a function from point sets to the new number class.
This progression is part of 19th century foundations of math. That you can’t just point to geometric objects and get the different number classes and their procedures from them is part of understanding these concepts as distinct. If you think you get number classes such as rationals or reals from geometry including procedures for addition and multiplication o the new number classes, then you are mistaken. Standard education creates that mistaken concept as its standard result.
Properly guided projects and self discovery can lead students to recognize the impracticality of using geometry in a simple way to justify the arithmetic and algebraic rules of fractions.
The symbols of arithmetic and algebra that were developed intensively in the 15th to 17th centuries lead us to define fraction procedures using algebraic rules using these symbols. This is true whether using letters or numerals. The symbols have taken over and the math caught up to that in the 19th century with the proper logical structure and progression from natural numbers to rational numbers.
Teaching anything else is deliberately creating misconceptions. This results in either students believing they can’t understand how to go from geometry to the rules for fractions, or thinking they do. In either case, they are mistaken.
All students in K-6 education are led to believe one or the other of these mistaken beliefs. This is a 100 percent of all students effect using the standard approach. This is a major block to understanding.
The structure of the symbols developed in the 15th to 17th centuries and the progression of number concepts from naturals to rationals has been covered up and a confusion left in student minds. This does not serve them well.
The opposite approach is to explain the history and the structure of arithmetic and algebra as a progression of object oriented number classes. Each number class has its own procedures that use its own data. These are not derived from geometry or area in any way. Students should know that.
If teaching concepts has any meaning, it means students know they don’t get fractions as a type of number or the rules of manipulating fractions as a type of number from any geometry at all. That is what they should know if they are taught concepts.
If students don’t know this, and they don’t, then they were not taught concepts. They were taught only procedures. This is exactly what happens in Euler’s 1765 Elements of Algebra. The geometry misdirection is given, and then the procedures ie rules for adding fractions are given. These rules are not derived from geometry in Euler 1765 Elements of Algebra nor are they in arithmetic classes nor in algebra classes in schools. Yet students are led to believe they are.
The students are then led to believe either 1) they don’t understand how to go from geometry to the rules and therefore they can’t understand rational numbers or 2) to falsely believe they do know how to go from geometry to the rules of fractions. In either case, they are mistaken and therefore they have not learned the concept of rational numbers. 100 percent of students are thus deceived as a result of this type of education.
Zero percent of students understand the concept of rational numbers. When zero percent of students understand the concept and 100 percent of students are mistaken, then they were not taught the concept. This instruction has failed. This instruction then blocks further steps, i.e. going from arithmetic to algebra.
Euler uses the same rules for fractions with letters standing for any number. So if the students are deceived at this point on where these rules come from for numerals, they are likewise deceived on where they come from for letters standing for any class of number.
Thus students go into algebra and are quickly lost. They are deceived on the logical structure for the algebraic rules for fractions. This means their understanding of slope is also not conceptual.
Thus algebra can only teach procedures, since it intentionally creates false concepts that are fallacies. It can’t be teaching concepts since it teaches fallacies on the basics. So it must be teaching procedures. Thus all the education on algebra that is based on fractions is procedural and not conceptual. Applications can’t change that.
Moreover, if students or anyone else does not understand natural numbers, then they will not understand fractions of natural numbers. The concepts of natural numbers are the Peano Axioms.
We see that progression as well from Euler 1765 to Grassmann 1861, Dedekind 1888 and Peano 1889. In Euler, he tells us there is a progression of natural numbers by adding one. But then he does not derive the rules of adding these natural numbers or of place notation. He simply jumps to those rules without deriving them from the series. It is the same as with the line segment justification of fraction to the rules of fractions.
What Grassmann, Dedekind and Peano did was put in the logical steps that go from the progression of natural numbers, 0,1,2, etc to the definitions of addition and multiplication and the proofs of their properties. From there one can construct the rational numbers logically and prove their properties.
Euler does not do that for naturals or rationals. Instead, he gives pseudo definitions without attached methods to define or do proofs and jumps to rules for addition and multiplication of naturals or fractions without filling in the links. This leaves procedural learning only since the concepts are not taught. The Euler approach is what is repeated in modern education today.
This leaves logical gaps for natural numbers and rational numbers. The gaps from what they are defined as to the procedures attached to them. The students fill this gap by either a) thinking they don’t understand math and are not good at it or b) imagining that somehow they know the links when hey don’.t In either case, conceptual learning is not happening. So all they can have learned is procedures.
This carries over to slope and everything built on slope including graphs including graphical calculators. Graphing, applications, etc. can not make up for the gaps and the false instruction that either they can’t understand math or they do when they don’t.
The impact of this tower of confusions and fallacies is to bring their learning to a halt. They then fall back to memorization. You can’t teach the concepts if you don’t teach the concepts. The concepts are not in Euler, they are in 19th century math that filled the gaps in Euler. If you teach Euler 1765 you do not teach the concepts, you teach the procedures. If you claim you teach the concepts then you teach deception. This eventually brings progress in learning to a halt as algebra progresses. This then leaves memorization of procedures to pass the course.
Only teaching the foundations of math can teach the concepts of elementary math. Falsely saying that area leads to the rules for fractions can not help anyone. It is false in Euler 1765 and it is false in 2012. No one is helped by saying this lie.