Paul Tanner II says: >> Its not circular since it's not a synonym. Repeated addition and scaling are properties of the operations respectively on some or all subsets of the reals, and that these properties are not what these operations *are*. They are models of properties of some or all of the operations, not the operations themselves.
Joe N says: >Sorry -- I'm not taking your explanations at face value -- you are far too inventive to be credible in these matters. Nobody talks like this but you, "models of properties" and whatnot.
I've repented - I'll try to interpret you anyway. When you talk about "scaling" as a property I'll guess you mean something like "ab = c <--> a/1 = c/b".
 That's not "scaling" - you are just a mapping an algebraic property to an everyday notion, that works out well for some fields and applications. The algebraic property also holds in systems where thinking of it as scaling doesn't work.
 Going from Q down to N we lose the nice equivalence and have to replace it with an 1-way arrow. So - things *do* change as we go up and down through the number systems. My point all along. There's no escaping it. Something else (not the above) changes going from Q to R - otherwise there would be no need. I'll guess that most high school students don't need any particularly deep knowledge of "why R?" -- but I'm just guessing.
 Thinking of "repeated addition" as if it was a competitor to the above property, seems a mistake to me. "Repeated addition" refers to something better thought of as a computational procedure. Pitting one against the other the way Devlin and you do is just a mistake, and wrong. Kids need to learn both, and not confuse them as being in competition. Computational math (and up through computability theory too!) has always been and remains important in math.
 Math that is computable is all of the math that the vast majority of students will ever need. I'm NOT saying "learn only computation, don't learn algebra (or whatever)!" Rather, I'm say that they never ever need be concerned with non-computable functions and numbers and whether algebraic properties hold for such, unless they have a special interest.