>Joe, numbers are not quantities. Quantities are physical, they include a number, a unit, a direction, if needed, and for goodness sake, a freaking context.
My goodness. I was speaking of the good old world, as Dickens might say, not your personal universe.
Here is Isaac Newton: "Algebraic quantities are of two sorts, affirmative and negative; an affirmative quantity is greater than nothing, and is known by this sign +; a negative quantity is less than nothing, and is known by this sign -."
He follows up with a money example.
Here is Euler: "The calculation of imaginary quantities is of the greatest importance."
Lest you think taking number as "quantity" is merely archaic usage, check these:
I understand the distinction you are pointing to, though; nice as it is, it doesn't seem particularly germane in this context, that of understanding negative numbers and their rules.
Saying that a negative number "is a mathematical concept" (well, by golly, its abstract!) does nothing to explain what it is. How are they different from the whole numbers a child already knows about? What good are they? Why are the rules (esp. the infamous one) such as they are?
What's your lesson look like? What are the key points for a child?
R.H. says: >Mathematics deals only with the number part of all that.
And which part is that? Does your concept of number include separable components as well? What makes a real number real, but an imaginary number a figment of the imagination?