I am all for helping students make sense of these properties that at first seem arbitrary. And I do it in ways similar to yours, except, I am more clear regarding the distinction between definitions and axioms versus the rest that "follows from". You don't "prove" axioms (as you often seem to be doing) but at the same time you certainly must show the student that they are more than just arbitrary decisions. I generally show how these choices (i.e. the definition of exponents) make sense because other choices would prove to be inconsistent down the road and lead to contradictions. The instinctive sense of the logical structure and layering in mathematics is reachable by students in algebra, at least by algebra 2.
On Nov 27, 2012, at 5:20 PM, Peter Duveen <firstname.lastname@example.org> wrote:
> Bob, I usually give similar demonstrations for exponents. That is, I begin with the self-evident properties of exponents, and then use them and extend them to interpret negative exponents, fractional exponents, raising to the zero power, and raising to the first power, all of which are, in my opinion, not self evident. I'm not so interested in conventions as I am the demonstration of the properties of exponents extended to all reals, and ways to interpret unfamiliar concepts in terms of the familiar.