The first is that, if you are interested in The Trivium, you really ought to read Dorothy Sayers’ essay, “The Lost Tools of Learning.”
Second is that, if you read Sayers’ essay, you’ll see that the Trivium was introduced in a set order: first Grammar, since you gotta start with the ABCs (hence grammar school), then Logic, because tween-agers have entered the “pert” stage wherein they enjoy argument and want to disagree with adults, and finally Rhetoric, which requires a mastery of grammar and logic as a base.
All of this was meant to prepare the student for University study, which would entail the Quadrivium. This is why 15 year-olds would be sent off to University in Elizabethan England. Also, in understanding that the Quadrivium approached each topic from a qualitative or “right brain” approach. So mathematics wasn’t adding, although I’m sure they could calculate just fine. But a big part of it was understanding the symbolic and archetypal power of the numbers. Think in terms of “sacred number” and “sacred geometry.” And if that doesn’t make much sense to you, watch thisvideo (and buy the book Decoding Design that the video is plugging):
In other words, the Quadrivium was very Magical Worlds-esque, if you will. Again, if you’d like a feel for that, I’d recommend this really cool book as a primer:
There’s that. Then there’s the larger idea of bringing structure to Wizard Academy and using the Quadrivium or Trivium as a basis for doing that. That’s a bigger topic that I’ll save for later as I’m being called to dinner : )