To those unfamiliar with Classical education, the terms Trivium and Quadrivium are sure to sound foreign. Technically, they are foreign because they are Latin. Yes, proponents of Classical education love Latin words, and the teaching of Latin is part of a Classical curriculum. After all, Latin is a Classical language that opens the door to a wide variety of literature, not to mention the rich history of Latin in the Church. Believe it or not, but it used to be a requirement for entering students at Harvard to be fully versed in Latin; but I digress.
Susan Wise Bauer, a well-known proponent of and author on the subject of Classical education, succinctly lays out the three-stage process of the Trivium (hence the “Tri” in Trivium): “The early years of school are spent in absorbing facts, systematically laying the foundations for advanced study. In the middle grades, students learn to think through arguments. In the high school years, they learn to express themselves.” These three stages are roughly structured around the developmental stages of children. That is, in the early years (elementary school grades), students enjoy absorbing information (e.g. names of plants and animals) as they are being introduced to the world. Classical education takes advantage of this by having students engage in songs, chants, copying, and memorization. This early stage is known as the grammar stage because it is fundamentally concerned with speaking, hearing, reading, and writing, especially through imitating the great literary works throughout history. This emphasis on language, written and spoken, is carried throughout the entire Classical education, as language-based learning, as opposed to image-based learning, is simultaneously rigorous, vigorous, and restful, and feeds the child’s imagination.
In the middle years (middle school grades), children become more inquisitive, asking questions like “Why?”. They want to understand the relationship between things, such as cause and effect. This stage consists of “the art of following questions and finding arguments,” which is why it “has often been equated with the skill of logic.” This essentially means to engage in a logical flow of thought, and it is why this stage is known as the dialectic or logic stage. Students take the knowledge gained from the grammar stage and expand on it by taking notice of its symbiotic relationship. Dialogue and debate characterize this stage of learning as students begin to engage more critically with the world around them. They learn how to think, not just what to think.
In the later years (high school grades), students continue to build on the previous stages by becoming more expressive of their views and convictions. This is known as the rhetoric stage, or “the art of making creative, persuasive, and productive use of language in public speech.” Originality and specialization mark this stage of education.
The Quadrivium is taught alongside of the Trivium. Whereas the Trivium is linguistic in nature, the Quadrivium is mathematical in nature. The Quadrivium consists of arithmetic, geometry, astronomy, and music. While we typically think of mathematics in a highly practical manner today (e.g. to build a house from blueprint to construction), Classical education aims at the contemplative, as well as the practical. The contemplative, or philosophical, aspect of mathematics is especially rooted in Plato. Both aspects, however, are important. Therefore, “the study of mathematics ought to strike a balance between wonder, work, wisdom, and worship.”
I’ll avoid going into detail about these four subjects that make up the Quadrivium. However, I will briefly comment on the place of music, as we don’t typically think of music as falling in the category of mathematics. If you think about it, however, steps between notes constitute ratios. Melody, harmony, and rhythm are all mathematical in nature. But music goes beyond such instrumentality. The harmony of the world and of humanity is included in this Classical understanding of music: “The musica mundana [music of the world] represented the proportionalities woven into the world, and the musica humana [music of humanity] reflected the mathematical proportionalities that exist in the person and in human society.” We can therefore think of musica instrumentalis [instrumental music] as a reflection of the music that already exists in creation. Now we have a better understanding as to why wonder ought to be joined to work when speaking of mathematics, as the world and all that we do is mathematical in nature. This harmony is certainly cause for worship of our Creator, the one who brings order out of chaos.
In conclusion, the Trivium and Quadrivium make up the seven liberal arts. These liberal arts serve as the ancient paths of wisdom and knowledge, training the student to be pious (have respect for the past and for authority) and virtuous (live a principled or well-ordered life in community). For a fuller, though still brief, introduction to Classical education, I recommend Christopher Perrin’s book, An Introduction to Classical Education.
 Susan Wise Bauer, “What Is Classical Education?”, Last accessed on January 20, 2021. https://welltrainedmind.com/a/classical-education/?v=7516fd43adaa
 Kevin Clark and Ravi Scott Jain, The Liberal Arts Tradition: A Philosophy of Christian Classical Education, Classical Academic Press, 2019, 49-54.
 Ibid., 55.
 Ibid., 60.
 Clark and Jain, 69.
 Ibid., 89.