What qualities should a teacher have in order to use A Brief Quadrivium?
Here are five.
Let's spell out what is meant in each case.
Ordinary Intelligence
People in the United States often talk about math in a self-deprecating way. They say "I was never good at..." and make other statements of that form. The frequency of such assertions makes it clear that something is awry in mathematics education. You might be asking yourself, then: "Can I do this?" In brief, yes. Ordinary intelligence, and an ordinary mathematics background, is all that you need.
Self-Discipline
It is not enough to have a good intention. You must follow through. The textbook A Brief Quadrivium gives students a clear path, but you must make sure it is followed. Clear steps are given in Teaching the Quadrivium. This will require some personal sacrifice and independent study.
Ideological Neutrality
You might have a strong opinion about algebra, or your own high school teacher, or Descartes, or modernity, or any number of things. Perhaps that opinion is part of what has drawn you to the quadrivium. In any case, the study of mathematics should not become an occasion for you to impose your specific vision about those matters. There is a deep, noble, wise view of the world within the quadrivium, but whatever it is, it arises through the study of the mathematics itself.
Reserve
This point extends the previous one. Avoid talking too much. This goes both for math itself and the bigger picture. Sometimes students need to struggle with something for a while. Don't rush in. Let the words be theirs. And don't insist that what you are doing is significant, classical, special, unique, ... Be confident that it is so, but don't say it.
Directness
The books A Brief Quadrivium and Teaching the Quadrivium offer a clear plan for studying the quadrivium and acquiring real mastery of the content. Teachers must direct students with simple, clear, brief instructions. Then, students must be given space and silence to work accordingly.