It took me awhile to grok why thinking about multiplication as scaling made sense, but given how much it comes up in discussion I thought it was pretty important to understand. I think having been taught that multiplication was repeated addition as a kid made it difficult for me to think about multiplication as scaling. Once the "light came on", however, it was very intuitive. It's definitely better to think about repeated addition as being a "trick" that just happens to work like multiplication on natural numbers, versus thinking that multiplication IS repeated addition.

Edit: I've heard a similar statement made re: exponentiation-- that it's not "just repeated multiplication". I'd love to have the similar flash of understanding re: exponentiation that I had when I learned to think about multiplication as scaling. Is the "exponentiation is not repeated multiplication" statement true, and if so, does anybody have a recommendation on something I can read about it?

I would dearly love to be more in touch with math than I am. I stopped at intro college algebra and never felt overly comfortable with trig. From talking with friends who have deeper math backgrounds I can tell that calculus would add a richness to my experience that I lack, if only because I can see a richness that my basic understanding of algebra brings when I talk to people for whom algebra isn't intuitive. I hope I can get it together enough to spend some time learning more about math before I end up old(er) and grey(er).