The Quadrivium Explained: Applying Classical Education in Catholic Homeschooling
In a recent post, “The Trivium Explained,” we reviewed the concept of the Trivium, within classical education and how it can be applied in a homeschool setting. Today we will continue the discussion with a review of the Quadrivium, the second tier of classical education.
What is the Quadrivium?
In the Middle Ages, the trivium and quadrivium were two study tracks for young students that laid the groundwork for further university studies. Quadrivium comes from two Latin words, quadri and via, meaning “four roads,” signifying its four disciplines. Whereas the trivium focused on language and literary skills, the quadrivium focused on quantitative reasoning, i.e., mathematics. The trivium was known as the “Art of the Word.” Conversely, the quadrivium was known as the “Art of Numbers” because of its strong mathematical foundation.
The four disciplines of the quadrivium are arithmetic, geometry, music, and astronomy. The quadrivium was taught after the trivium; the skills of speech and reasoning developed in the trivium were considered necessary prerequisites for the concepts developed in the quadrivium.
As you may have noticed, the medievals had a broader definition of mathematics. Few people today would consider music or astronomy to be branches of math. However, in the classical tradition, math was more than solving equations; any quantitative system was considered mathematical. Music and astronomy were, therefore, considered part of mathematics because both disciplines involved the study of patterns, ratios, and numerical relationships. Arithmetic is numbers; geometry is numbers in space; music is numbers in time; astronomy is numbers in time and space.
The Quadrivium in Homeschooling
In classical home education models, students typically move into the quadrivium in high school, beginning with arithmetic and culminating in astronomy at the end of high school. For most families, beginning arithmetic in high school is not feasible. Besides, many states require arithmetic from a much earlier age. But that’s fine because, as we noted last time, you don’t have to literally replicate the medieval quadrivium model to apply the principles of classical education to your homeschool. The quadrivium is fundamentally about applying quantitative reasoning in an increasingly complex manner, a concept that can be incorporated into your homeschooling even if you don’t follow the classical model beat for beat.
Arithmetic
In the abstract, arithmetic is the study of numbers. Why should we study arithmetic? This is an important question, as students laboring over an algebraic equation are prone to huff and say, “Why do I need to learn this? When will I ever use this in real life?”
The question supposes that math isn’t “real life.” In fact, arithmetical reasoning is something profoundly connected to “real life.” Arithmetic is ultimately about understanding the complex nature of cause and effect in objective entities. Given A, what happens if I add B to A? What result will I get? What if I divide A by B or multiply it? Besides developing the ability to reason quantitatively, arithmetic grounds a student in the objective nature of reality. Numerical relations (like the Fibonacci sequence) are not inventions of Man. They simply exist as part of nature, left for us to discover.
Geometry
Moving on from the study of numbers, geometry is the study of numbers in space. In Plato’s philosophical Academy in Athens, there was reputedly a sign over the door that read “None But Geometers Enter Here.” But what does geometry have to do with philosophy? For Plato, the two were intimately connected: geometry is how we form connections between the abstract knowledge of arithmetic and ground it in three-dimensional reality. Geometry “incarnates” the principles of arithmetic into actual space, helping students to perceive their real-world application.
Music
In classical education, music was sometimes referred to as “harmony” or “harmonics.” Harmony consists of a right relationship between notes. These relationships form the building blocks of music, which the medievals considered as a number in time. In classical education, music was viewed as a mathematical study of ratios and proportions, essentially “applied numbers,” where students learned to understand numerical relationships through the concrete experience of harmony and musical intervals. Besides being a very enriching discipline for a child to master, the study of music imparts a keen insight into the harmonious structures behind our experience of beauty.
Astronomy
Astronomy—number in time and space—is the pinnacle of the quadrivium and of the entire septet of the liberal arts. We might be surprised that something we tend to treat as an ancillary scientific discipline was once considered the culmination of a young person’s studies. If we think this, we forget what the study of the heavens meant to pre-modern people. “The heavens declare the glory of God; and the firmament proclaims his handiwork,” the Psalmist says (Ps. 19:1). It is in the heavens that the wisdom, splendor, and harmony of God’s creation are most admirably seen. Done properly, astronomy need not be dry scientific data. It can be a subject that elicits a young person’s natural wonder at the beauty, power, and rationality of the cosmos. To the medievals, to study the heavens was to behold the handiwork of God directly.
Conclusion
As we’ve explored in these articles on the seven liberal arts, the principles of the trivium and quadrivium offer a robust foundation for classical education, emphasizing the development of critical thinking and quantitative reasoning. By integrating seven liberal arts into your homeschool curriculum, you’re not just teaching subjects—you’re equipping your children with the tools to think deeply, express themselves clearly, and comprehend the world around them.
If you want to learn more about classical education, feel free to explore our other blog posts on Catholic classical homeschooling.
What are your thoughts on this topic? To continue the discussion, join me and other homeschooling parents at our Homeschool Connections Community or our Facebook group!
