Teachers do not get that addition is defined from counting by one. They may grasp that intuitively at times. But they don’t understand this is the complete definition with no more needed and that all properties of addition follow from this.

This is why teachers need to read the Dedekind book.

http://newmathdoneright.com/2012/05/03/richard-dedekind-essays-on-numbers-on-line/

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## About New Math Done Right

Author of Pre-Algebra New Math Done Right Peano Axioms. A below college level self study book on the Peano Axioms and proofs of the associative and commutative laws of addition.
President of Mathematical Finance Company. Provides economic scenario generators to financial institutions.

Is there a concise explanation of why “definition” is such a significant issue in elementary school? If definition wasn’t defined until the 19th century, does that mean all of the math before that made no sense? What about the billions of students who have a rich number sense and intuitive arithmetic fluency, as well as conceptual ability and creativity, but who never learned through formalism?

I admit you can light New York City without the Peano Axioms.

But let us go back to the Euler 1756 Algebra.

http://newmathdoneright.com/2012/08/01/eulers-elements-of-algebra-gaps-in-logic/

”

In the same manner, therefore, as positive numbers are incontestably greater than nothing, negative numbers are less than nothing. Now, we obtain positive numbers by adding 1 to 0, that is to say, 1 to nothing; and by continuing always to increase thus from unity. This is the origin of the sequence of numbers called natural numbers; the following being the leading terms of this sequence:

0,+1,+2,+3,+4,+5,+6,+7,+8,+9,+10,”

This is from Euler, see above link for link to Euler text in English. You can quickly skim the first few pages and get an idea of Euler’s thinking on natural numbers, addition, etc.

Euler does not really know how to set it up very well. He is partly right in the quote here. But Grassmann and Dedekind were familiar with Euler’s book and found it wanting.

Grassmann invented the recursion equations but using plus still.

x+0 = x

x+(y+1) = (x+y)+1

Then Dedekind added the primes, so

x+y’ = (x+y)’

I discuss using a pair of number lines instead of a single number line to teach addition.

http://newmathdoneright.com/2012/05/13/pair-of-number-lines-and-successor-identities/

As far as I know, I am the first to suggest that. I would appreciate any reference prior to me.

The pair of number lines teaches addition much better than a single line. It corresponds to the x and y in the above identities.

The pair of number lines gives you a better explanation. You can also do a fake proof from the pair of number lines back to the additive identities.

With the additive identity equations and pair of number lines you are set up as teacher with a powerful tool kit to explain. The students have a tool kit to work with.

You can then work on teaching how to go from the right additive identities to the left ones. This can be done using the pair of number lines as well as the equations.

This gives you structure, which is one definition of meaning in math. In this way of thinking, the meaning of something in math is the structure from where you start to how you get there. This is similar to your idea of teaching them how to think.

The pair of number lines also agrees with your idea of how to teach people to work out of being confused. If a person is confused on addition, the pair of number lines gives them a better explanation than the single line. This is in part because you have two numbers being added, each gets its own number line.

The pair of number lines also makes “counting on” more distinctly identified as addition from the point on the lower number line based on the number on the upper number line, see the link, although I did not use the phrase “counting on” at that point.

If you compare what I am offering here based on Grassmann, Dedekind and Peano compared to Euler it is a real change. To use your approach, Euler is confused. GDP is the way to unconfuse yourself starting from Euler. The pair of number lines is a tool for this that students can use. It can also help in projects like proving the commutative law or going from right to left identities.

The following link is on a study of Grade K commutativity of addition. It studies how Kers progress according to the authors.

https://scholar.vt.edu/access/content/user/wilkins/Public/JECP_2001.pdf

A pair of number lines would help in teaching Kers commutativity of addition.

Commutativity of addition is linked to the right additive identities going to the left ones. That is an easier lesson than commutativity.

Commutativity seems so obvious, it is hard for the teacher to grasp the students don’t get it. However, the right and left additive identities are obviously different to teacher and student. So going from right to left is obviously doing something. This makes it easier to understand. Once you learn to go from right to left additive identities, commutativity is much easier to understand.

Another point is that with right additive identities and pair of number lines, the K teacher doesn’t have to understand it as well herself.

Consider this example

4+2′ = (4+2)’ [Right successor identity] [Counting on from right]

2’+4 = (2+4)’ [Left successor identity’] [Counting on from left]

If we assume 4+2 = 2+4, then we can substitute and use transitivity to get that

2’+4 = 4+2′

Then we can do it on number lines.

So if we have commutativity 4+2 = 2+4, then we have 2’+4 = 4+2′.

This example shows that commutativity can be broken down and demonstrated.

To avoid using the word successor, we can say count on by one or next.

We can restate the whole thing as

Assume that counting on from 4 by 2 equals counting on from 2 by 4.

Then it follows that counting on from 4 by 2′ equals counting on from 2′ by 4.

Or counting on one more than 2 starting from 4 is the same as counting on by 4 starting from one more than 2.

This can be illustrated with the pair of number lines.

If the student is having trouble grasping commutativity, this gives them some half way points to work with. Right now, those half way points lack names and lack articulated structure and the teacher has to sort of make them up herself on the fly more than with this approach.

Because this approach is based on the correct theory, it is more robust as it is deployed into thousands of classrooms with different people in different circumstances. However, it has notations and methods that make the theory concrete and give identifiable points so that it doesn’t get lost in the interpretation by each teacher in each class.

Without this, it becomes hit or miss what is taught to students struggling with commutativity and what half way points they try to understand.

With this machinery, the K teacher will understand it better herself better than she does now. Using the machinery also results in the teacher being able to progress herself in the depth of understanding. The machinery gives you points to ask questions about which deepens understanding. That applies to the teacher explaining to the K student, or self studying or talking to other teachers.