(as I get my education in math, I'm going to be more inclined to believe that the electrical engineers are doing it all wrong, even though the driving reason for that belief is that I didn't want to be an electrical engineer when I was 18.)

I'm reminded of the infamous "delta function" - it's something that makes no sense from the perspective of standard undergraduate calculus, but engineers used it all the time... and it always gave the right answer. That last part is the part that really drove the mathematicians up the wall. ;) (Someone eventually did come up with a formalization in which it made sense, though.)