The Quadrivium

Part 3 of My Series on Classical Education

Welcome back to The Burnett Breakdown. I am continuing my series on classical education this time talking about the quadrivium.

The Quadrivium

The first piece in my series on classical education addressed the “why” of education by saying that the purpose of education was to “instill wisdom and virtue” in students. (If you missed that newsletter then you can read it here). Last week, I began to talk about what I believe makes classical education unique and that is in the “how” classical education seeks to instill wisdom and virtue. That is, classical education does this by equipping students in the liberal arts.

In particular, last week I talked about the first half the liberal arts: the trivium. (If you missed that, then you can read it here). The trivium consists of the three linguistic skills, grammar, logic, and rhetoric, that equip students to use language well so that they can think and communicate well.

This week I am going to discuss the second half of the liberal arts: the quadrivium. While the trivium deals with language, which takes place in the human mind, the quadrivium deals with numbers and their relationship with the material world.

The quadrivium can also be a little more confusing than the trivium largely because it uses terms in a particular way that we no longer use them. The key thing to know about the quadrivium though is that, as the trivium deals with language, the quadrivium deals with numbers. The quadrivium teaches students how to think in numbers as the trivium teaches them how to think in words.  

So, the four quadrivium arts are arithmetic, geometry, astronomy, and music. 

The basis of the quadrivium is arithmetic which is the skill of manipulating discrete numbers. This is essentially learning the numerical operations of addition, subtraction, multiplication, and division.  

By learning, I don’t necessarily mean students are capable of understanding the larger mathematical concepts when they are in first grade. This is part of the problem with much math education today. They want to teach students why 2 + 2 = 4 when it’s more important students know that 2 + 2 = 4.  

That’s not to say that it’s bad if students know the “why” behind arithmetic, but it is more important that the students can do the various arithmetic operations accurately and quickly. Once they are fluent with their numbers, they will be able to understand abstract principles, like in algebra, better because they aren’t expending brain power trying to do the computations. They can focus on learning the abstract principles.  

This is why elementary school math curriculum should have students spend pretty much their entire time teaching arithmetic and then drilling and drilling and drilling to increase fluency.  

If arithmetic is the skill of manipulating discrete numbers, music is the application of discrete numbers to the natural world, particularly regarding time and proportion. When the quadrivium speaks of music, it is concerned with music theory and learning the various elements such as rhythm, scales, pitch, harmony, intervals, etc.  

The association of music and math goes back to Pythagoras. He discovered that if you had a taut string (think a violin string) that played let’s say a “C” note, then you could change the note by holding the string down at any spot. However, if you hold the string at exactly the halfway point, it plays another “C” note but this one is an octave higher than the original. Pythagoras realized that we could use whole numbers in ratios to evaluate the world around us. 

This application of discrete numbers to the world is the skill that is referred to when we speak of the “art” of music.  

It will be helpful for me to draw a modern-day equivalent of the same skill of music. In music, you can take a measure and divide it into four notes (1 2 3 4) or you can take that same measure and divide it into eight notes (1 and 2 and 3 and 4). This is just taking the concept of division and applying it to nature, in this case, dividing a certain amount of time into parts.  

Now, imagine you are an engineer for a diaper company (shout out Paul). You have a target goal of 100 million diapers that need to be produced in a year. You can use arithmetic to figure out how many diapers need to be produced per day. In other words, you take the principles of arithmetic and apply them to the natural world. This application of arithmetic is the skill of music. 

Geometry, on the other hand, is dealing with space. Arithmetic deals with numbers that answer the question “How many?” Geometry deals with numbers that answer the question “How much?” It involves distance, shape, size, and relative position of figures in space.  

This art is closely related to geometry as it is taught by almost everyone today. If you have learned about rays, area, perimeter, angles, and geometric proofs, then you have pretty much learned geometry.  

Finally, astronomy is the application of geometric concepts in space. This is what the ancient and medieval men did when they tracked and calculated the movements of the planets and stars in the sky. They applied geometric principles to the natural world so that they could better understand the universe.  

Today, we don’t necessarily spend our time studying the planets and stars (though, there are some who certainly still do that), but the same skill is used to track the movements of objects through space. This is essentially what physics and calculus are.

When engineers design an airplane that they can be sure stays in the air, they are using the plane’s position and change in space and time to make calculations. This is merely taking geometric concepts, albeit more advanced ones, and applying them to the material world.

So, when we teach subjects such as physics and calculus, we are teaching the liberal art of astronomy.    

Now, notice that arithmetic is necessary for the other three quantitative arts, music, geometry, and astronomy. This is why it is so vital that students have a complete and total mastery of arithmetic before they move on to more complex concepts.  

There is certainly more that can be said about each of these liberal arts, but I hope this provides a general explanation of what each one is. By understanding the liberal arts, we understand the standard so to speak that classical education holds itself to. Classical education seeks to teach students to: 

- invent and combine symbols (grammar) 

- properly structure arguments and think through the content of arguments (logic) 

- effectively persuade (rhetoric) 

- manipulate discrete numbers (arithmetic) 

- apply discrete numbers to the natural world (music) 

- understand space and how figures relate to it (geometry) 

- and, finally, apply those geometric concepts to the natural world (astronomy). 

By teaching these specific skills, classical education provides students with the “critical thinking tools” that will best equip them to discern that which is good, true, and beautiful.  

To use the liberal arts is what it means to “think critically.”  

All schools will claim that they want to teach their students how to “critically think,” but they can never provide a specific definition for what they mean by this. Or, as in the case when I studied education in college, critical thinking simply means criticizing existing power structures.  The imparting of the liberal arts to students is what classical education means by “teaching students how to think critically.”

God Bless,

Hunter Burnett