The essence of the Addition operator within arithmetic How do I characterize the addition operator, without defining its properties using set theory or the peano axioms, so that someone with special needs could understand and apply it in the context of arithmetic and at most algebra? What is the definition of the addition operator in the simple context of algebra and arithmetic, and how is it understood?
 A: I will try to write what I understood from Prof. Herbert Gross's teachings and innovation.
The addition operator operates on numbers, so a good knowledge of numbers can give us some insights. If we follow this PowerPoint by Prof. Gross, we have something very interesting to look at

In a similar way, $3$ inches is a $\color{grey}{quantity}$ in which the $\color{red}{adjective}$ is $\color{red}{3}$ and the $\color{blue}{noun}$ (unit) is $\color{blue}{inches}$ As quantities, $2$ fingers is not the same as $3$ inches. However, as $\color{red}{adjectives}$, the "$\color{red}{3}$" in "$3$ fingers" means the same thing as the $\color{red}{3}$ in "$3$ inches".

And from this presentation we have

If the students are asked to put two tiles, they will probably do this


And if they are asked to put those two tiles along with three more tiles, they will probably do this
$$ {\Huge \color{blue}{\blacksquare\blacksquare \qquad \blacksquare \blacksquare \blacksquare}} $$

And they will count those tiles as "one, two, three, four ,five", and thus know that there are five $\color{blue}{tiles}$.



$$
\begin{aligned}
{\Large \mathbf{\text{How to apply it in Algebra}}}
\end{aligned}
$$
Let's say we have $3~\color{red}{apples}$ and $2~\color{blue}{oranges}$ and if I ask you how many fruits do I have, you will answer $5$. So, what you have basically done is that you did the following translation
$$
3~\color{red}{apples} \rightarrow 3 ~\mathbf{fruits} \\
2 ~\color{blue}{oranges} \rightarrow 2~\mathbf{fruits}
$$
And then you added "3 fruits" and "2 fruits" in the same way as we added the tiles above.
Now, let's say we have any noun $x$, such that $3x$ means we have $3$ of those $x$(in the same way as $3$ fingers, or $3$ inches) and $2x$ means we have $2$ of those. If we say that we are given $3x$ and $2x$ then how many $x$ do I have in total? We can surely apply that tiles example once again to see things clearly, because it is stated that $x$ is any noun. Let's visualize $x$ by some strange looking figure and draw $3$ and $2$ of them

We again find, by counting, that we $5$ of $x$'s or $5x$.

As the question strictly doesn't allow Set Theory's or Peano's Axiom (works of Gottlob Frege, Bertrand Russell) definition of addition so I wouldn't touch on that and end my answer here.
A: So after some thought, I've come to the conclusion that a definition of addition in the context of only algebra and arithmetic is impossible, as there are multiple uses of addition even in these two disciplines. Addition should therefor be introduced as a man made, hence useful calculation, "way", or "method" of calculation that is very useful in making collections bigger, or lengths longer.  What matters when introducing this operation is emphasizing its uses in most situations imaginable.
For sets and such, addition could be characterized by using toys or number blocks, where a collection of toys consists of two separate collections of toys, where the joining represents addition. This is effective in allowing students to learn that addition of toys is equivalent to combining toys 

and a number line be used to show addition as the combination of lengths

The number line approach could be introduced after students become familiar with "tactile" addition, to further the idea that addition can be generalized to the concept of length, which is more abstract than quantity.
