>So rather than talk about the properties of addition, you would rather >define and understand addition in terms of the counting operation? >This can be done but it requires a fairly sophisticated understanding >of counting - a level of sophistication which most six-year-olds do >not have.
I wouldn't bother to define anything, nor would I bother to tell them that something like a "property" even exists. That's for much later.
>Indeed, I'd like to see you first write down the axioms or properties >of counting, and then use those to explain associativity of addition. >Or do an example: starting with your written down axioms of counting, >explain why 6=5+1=1+2+3. (Make sure you do this in a way that a >six-year-old will be able to follow.)
I would do nothing of the sort. I would postpone the concept of associativity to senior high or so, when they first need to face it in its formal embodiment. The fact that 6=5+1=1+2+3 doesn't need to be explained to a kid, it evolves naturally as the kid's number intuition develops: count to 6, you get to 6; count to five then one more, you get to 6; count one, then two more, then three more, you get to 6. This is so intuitively obvious that there's no need to spend any time on it ! I first faced modern math when I was in senior high, and I had trouble taking it seriously, because everything that my teacher passed down to me was basically, as I saw it then, reiterations of the obvious, but done in a rather complicated way - and it took me a while to see the point of it, and even now, years after I got over that hump, I hardly see the point in complicating with formalism something that's much easier acquired with intuition. There will be time later for the formalism !
>An abacus involves a "digits" representation of a number, and this is >a much more sophisticated concept than the "length of a rod" >representation. Indeed, to understand digits, you first must >understand both addition and multiplication.
We have fingers - our hands are our abacuses, and couting with fingers is intuitive. An abacus is just an extension of our hands. Counting is, in fact, well upstream from continuous realizations such as the length of a rod. And as I pointed out before, the length of a rod is a false representation of a number: I can call the same length "1", "2" or "3.141592...", depending on how and where I use it. But a digit is a digit, and a finger is a finger.