It seems to me that it is easier to get people to realize just that they can't use their regular language to understand what is going on than to actually explain it. People seem to have issues with understanding this primarily because of Dunning-Kruger and because of the large number of popularizations of difficult science that just uses vague analogies.

I'd ask "ok. This is going to take some math. Did you ever take linear algebra?" If yes, then I just explain things. When they answer no (vast majority of the time)I then say "ok do you remember how matrix multiplication works?" They will generally not or have only a vague memory. At that point I then tell them that I could spending a few hours or so developing the necessary tools but that they really don't have the background without a lot of work. This generally results in annoyance and blustering on their part. At this point one tells them the story of Oresme and how he came up with the idea of gravity in the 1300s but since he didn't have a mathematical framework it was absolutely useless. This gets the point across sometimes.

Edit: Your idea of using polarization as an example is an interesting one and I may try that in the future.