> I find this emphasis on properties of operations misplaced - do we > really care, at that level, to formalize it that way ? Addition is > associative because to count to six it doesn't matter if I count to > five first and then one, or if I count to [three] first and then three > more.
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.
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 belong to the slide rule generation, and I find rods unnecessary. > If I had to choose any prop, I'd choose an abacus.
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.