The Old Schoolhouse “Spotlight” for the month of April, 2012, is on Classical education. Stephen and Kellyann Walker from Charleston, SC, share a bit on an often overlooked part of Classical education, the quadrivium. This is thought provoking reading if you’ve ever wondered where math and the sciences fit in a classical education.
As passionate supporters of a classical education, we feel that sometimes a portion of the classical curriculum is overlooked–the quadrivium. So we would like to explore this area a bit more.
If you have read anything about educating classically you have probably heard the term “trivium” because that is greatly discussed within the confines of a classical curriculum. However, grammar, logic, and rhetoric are but the first of three stages of education as laid out by Plato in The Republic. The second stage, the quadrivium, is composed of arithmetic, geometry, astronomy, and music (or harmonics). The final stage is both dialectic and philosophy, where dialogue between two or more people with different views on a specific subject use reasoned arguments to establish the truth of the matter.
Though each of these subjects has practical value, Plato espoused their study in a pure sense, to help the learner be more thoughtful in all his studies, making him a better leader. In looking at Plato’s own words about the mathematical sciences, it is clear that these subjects and a theoretical approach to teaching them is a part of a classical curriculum.
“‘You see therefore,’ I pointed out to him, ‘that this study looks as if it were really necessary to us, since it so obviously compels the mind to use pure thought in order to get at the truth.’” ~ Plato
The study of arithmetic is not simply the learning of fundamental operations with numbers. It is the study of the rules and relations of numbers to one another. This includes the study of number patterns and sequences, equations and their methods for solution, and all the proofs that go along with these ideas. A basic example of this is our multiplication tables. Beyond just memorizing them, it is important to understand that multiplication is just repeated addition. ex. 3X4 = 4+4+4. In a similar manner, exponents are repeated multiplication as in 52 = 5×5. Most people today would consider these “honors” courses, yet Plato found this to be foundational to a good education.
Geometry is considered in a likewise fashion. Though the study of geometry can center on memorizing numerous formulas for all kinds of problems, Plato’s approach is quite different. His approach looks at definitions and patterns to rigorously prove any conjectures put forth by the student, again making the student more pure or rigorous in his thoughts. An example would be proving the similarity of different figures.
“‘We shall therefore treat astronomy, like geometry, as setting us problems for solution,’ I said, ‘and ignore the visible heavens, if we want to make a genuine study of the subject and convert the mind’s natural intelligence to a useful purpose.’” ~ Plato
Astronomy at this point is not a study of the constellations, but is instead an examination of the equations for the movement of the heavenly bodies. Music is also approached mathematically and not as performance. Here it is looking at patterns and ratio. The purpose of a student’s study is, again, to find patterns, develop proofs, and discover truth.
The study of algebra, geometry, physics, trigonometry, and calculus forces us to be careful in our thoughts and greatly appreciate the beauty and complexity of creation. Plato would likely agree with a great many scientists who say that “mathematics is the language God has written the universe in.”
It is much easier to solve real world problems when you understand how mathematics works and not just have a long list of methods to solve a myriad of problems. Using this approach we begin to see the interconnectedness of all the mathematical sciences. Studying the quadrivium would be perfect for high school students who are ready for a more in-depth look at all these subjects. There are many good programs available to us as homeschoolers that emphasize a conceptual/theoretic understanding of mathematics. Pairing one of these with a thorough great books study will help round out your classical curriculum.