How will the quadrivium fit in with the rest your curriculum?
As a practical matter, A Brief Quadrivium can be used in the place of a one-year course of geometry in the first year of high school. It can also be used as a one-year elective course for seniors looking for something other than calculus or statistics.
The Quadrivium Builds On and Supports Other Studies
It isn't enough for the quadrivium to be a novel addition. It should also extend and develop the teaching and learning that are already underway. It certainly does this.
We expect that as students mature, they become capable of giving clear accounts. Early on, they are asked simple questions about naming and distinguishing things. That is grammar. Later, they are asked to repeat certain forms of reasoning, to say 'why' and not just 'what.' That is logic. Finally, they are asked to create persuasive forms of reasoning themselves. That is rhetoric.
This is exactly the intellectual development that the quadrivium fosters, making it a wonderful way to link mathematical learning with the rest of a curriculum. The quadrivium is all about mathematical proof. That means it is about giving clear reasons. The objects involved are simple: triangles, natural numbers, sounds, a sunrise. Students grasp these with confidence, and so they are able to provide arguments that they and others find compelling.
The quadrivium, which is focused on proof and deals with the simplest, most familiar objects, is an excellent way for mathematical study to be integrated with the broader educational goal of thinking and speaking clearly and persuasively.