Yes, but surely commutativity and distributivity etc. was properly taught before? I can't speak for the situation in the US - and it's been a while for me -, b

Yes, but surely commutativity and distributivity etc. was properly taught before? I can't speak for the situation in the US - and it's been a while for me -, but I remember that we were learning this stuff in that way, and these tricks, as you call them, as well as basic algebra, always seemed as a natural part of the process. I don't know, but I have a hard time believing that fundamental axioms of math were not part of the teaching. Oh, and by the way, you should check the result of your example again :)