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We know that the Carole originally came from Crete up the Mediterranean and largely ended up in Normandy where dance was already an important part of the culture. We also know that the Normans are responsible for introducing dance technology, such as thick leather soles, wooden floors and better acoustics, as well as wooden instruments to Britain.
May I ask how we know all this? I'm definitely not a dance historian, but I do consider myself fairly well-read on Anglo-Norman history and this is pretty much all news to me. I have never seen a primary source reference to the particulars of Norman dance. Some of what you've said disagrees with what I do understand about the Normans. For instance, medieval people generally wore soft-soled shoes, and the Normans were no exception to this. I don't think hard-soled shoes even became common in Europe until around 1500, long after the end of any significant Norman influence on England.
I'm not saying you're wrong; I'm just interested in where you've found all this detail on such an arcane subject as Norman dance. If possible, I'd like to look into their sources and see what on earth they're drawing from.