What sort of student can study the quadrivium?
Here are five things a student should be able to do before studying the quadrivium.
The student should be able to:
Let's go through the list and talk about each briefly.
Explain and Argue
The mathematics of the quadrivium is about giving explanations. Mathematical explanations are known as 'proofs.' To understand proofs, students must have a sense of what it means to give a logically coherent account. This does not mean that they need to have studied logic in a formal way. It does mean that they should have arrived at the point at which they want to give accounts and argue.
Define and Understand Definitions
To give formal arguments we need clearly defined terms. Thinking about definitions requires maturity and a certain capacity for abstraction. We must be willing to concede that a word will be taken to mean exactly what is said in the definition, even if we have a prior inclination towards a different definition. To be ready for the quadrivium, a student should understand that clear definitions are necessary in reasoning.
Work with Discipline
Reading and learning mathematics is not like reading a novel. One cannot do it rapidly, in a single burst. In mathematical study we form habits by which we speak and think in a certain way. These habits are developed over time. To develop the habits we must act repeatedly. This requires self-discipline, a quality that a student should possess in a reasonable degree before beginning the quadrivium.
Remember
Memory plays an important role in the mathematics of the quadrivium, and in all of learning. We cannot hope to construct everything for ourselves, from scratch, every time we need it. Instead, we rely on memory to hold in our minds the vital core of our disciplines. The central parts, the key terms, arguments, and conclusions, are not negotiable, not up for grabs. Other people created them, it is for us simply to remember. Students will succeed in the quadrivium if they are willing and able to apply their memory.
Be Generous
The book A Brief Quadrivium does not go out of its way to be fun, though students will sometimes enjoy it. It does not have dramatic, colorful pictures spanning multiple pages, or links to videos on the web. The book is simple and direct. This is not because of any needless harshness. Instead, the bare style is born of the conviction that students experience satisfaction from gaining competence. That kind of satisfaction endures even when novelty wears off. The goal of the book, then, is to give a clear path to success, success that students can recognize as their own. This is mathematics, after all, a discipline which we find uniquely and profoundly persuasive. Students need not be excited about mathematics, though that enthusiasm is fine and good. What is needed is simply personal generosity, a willingness to follow the steps and to find satisfaction in doing so. At times, perhaps, joy too will come.