Over the past two weeks, I have written a series on the philosophy of Classical Education based largely on Clark and Jain’s book The Liberal Arts Tradition. Week one we discussed the preschool years of learning Piety, Gymnastic, and Music. Week two we went over the Trivium, which is the language arts portion of the Seven Liberal Arts learned in the elementary/middle/high school years. Today I’m going to discuss the Quadrivium (mathematics), which encompasses the remaining four of the Seven Liberal Arts also taught in the elementary/middle/high school years.
The Six Curricular Categories of Classical Education
- Seven Liberal Arts
- The Trivium
- The Quadrivium
- The Trivium
The Quadrivium is a set of the Liberal Arts that coincide with the Trivium – meaning that they are experienced simultaneously. The Trivium and Quadrivium are two branches of the same tree. Just as the Trivium sequentially teaches linguistics or “language arts” (Grammar is first, Dialectics are built upon Grammar, and Rhetoric is built upon Dialectics), so does the Quadrivium sequentially teach the study of numbers (mathematics).
Why Study Mathematics?
The Greek philosopher Pythagoras grouped the study of arithmetic, geometry, astronomy, and music into the Quadrivium, but it was Plato who insisted that mathematics be studied so that students would develop the ability to reason and cultivate a love of wisdom. (The utilitarianism of mathematics was not the top priority back in ancient Greece.) Math was seen as valuable because it transcends popular culture and opinions – it has objective truth. Charlotte Mason also viewed math as valuable for this same purpose. In fact, she warned against using reason in judging ideas (the idea being that a person can justify any action) but commended mathematics for demonstrating truth through reasoning. In her 18th principle, she said:
Children must learn not to lean too heavily on their own reasoning. Reasoning is good for logically demonstrating mathematical truth, but unreliable when judging ideas because our reasoning will justify all kinds of erroneous ideas if we really want to believe them. (L. N. Laurio’s 2004 paraphrasing)
So we see that math helps us learn to think in a logical, sensible way. Of course, today we can tack on all kinds of benefits to math skills, not the least of which is getting a job, but in ancient Greece as well as 19th century England, math was valued for its own sake. Today we also know that logic and reasoning help promote healthy brain functions, so dust off those Sudoku puzzle books!
Mathematics and Christianity
We may not always think of math as a religious experience (unless you count praying before taking the math portion of the ACT), but several big names in math history were also expressly Christian and helped bring math into the Christian classical tradition. Italian 16th century mathematician Galileo Galilei claimed to discover the “language of God” everywhere in math. Seventeenth century German mathematician Johannes Kepler was an early scientist to believe that the material world obeyed the mathematical laws of its Creator, which lead to his discovery that planets orbit in an elliptical pattern. And, 17th century English physicist Isaac Newton extensively studied and wrote about the created world using math. Mathematics says nature can be orderly and still be spiritual, and the role of the Quadrivium is to lead our minds to eternal and unchanging truths.
Pieces of the Quadrivium: Arithmetic
The first stair step in the sequential phases of the Quadrivium is Arithmetic. Another ancient Greek guy – Nicomachus – wrote the book on arithmetic in the first century BC, and our understanding of basic math hasn’t changed since then. Nicomachus emphasized the relationships among numbers (not just memorizing “math facts”) and said – relating back to Plato – that mathematics is the path to true wisdom. 20 centuries later, Charlotte Mason threw in her two cents:
The main value of arithmetic and higher math is the way it trains reasoning powers, habits of understanding, quickness, accuracy, and being truthful intellectually. (Volume 1, Home Education – Paraphrased by L. N. Laurio)
My takeaway idea here is that we should not study arithmetic for the sake of memorizing math facts; however, the only way to truly understand the relationship among numbers is to practice math daily. And, the reason we want to understand mathematical relationships is because it sharpens our brains. For my young students, that means Math happens formally every single school day, often dovetailing right off of Circle Time (when we pray, sing, and do other “together” activities). And, even on days when we don’t have Circle Time (co-op days), the boys still do their math. It’s short (one worksheet) and it incorporates a lot of review, but the daily practice is important. I find when they are in good practice with math, they see the relationship among numbers throughout their day (and even on the weekends). That’s when math is fun.
Pieces of the Quadrivium: Geometry
Nicomachus described arithmetic as dealing with the relationship among isolated quantities, but geometry (and astronomy and music) deals with extended, continuous quantities. Before a student can begin dealing with continuous quantities, he or she must first master the relationship among numbers (arithmetic). That is why arithmetic is the first step of the Quadrivium.
In 3rd century BC, the Greek mathematician Euclid wrote his book “Elements” that is the basis of what we know as geometry. Euclid’s theorems define what we think of as deductive proof. (I think Euclid and Sherlock Holmes would have been friends if they had A) lived at the same time and B) both been real people.) In his theorems, Euclid defined and systemized elements of geometry we would recognize today, such as point, line, ray, circle, right angle, parallel, and perpendicular. He also defined ideas like transitive property, addition / subtraction property of equality, and reflexive property (which are the basis of algebra).
Along with all of these theoretical properties, Euclid’s Elements also contained “constructions” or drawings to help us visualize and reproduce them using only a compass and a ruler. Because geometry is both an abstract and visual step of the Quadrivium, the classical model of education suggests that students should study geometry before algebra so that they understand concretely what they are doing when they get around to algebra. Charlotte Mason agreed that math must be demonstrated when possible. She said about math:
A child can learn his multiplication tables and do a subtraction problem without ever understanding the reason for doing either one. He may even become good at figuring and applying the rules but never understand when or why to use them. Arithmetic becomes the first step in doing real math only when every process is clear in the child’s mind. 2+2=4 is pretty obvious even without proving it. But 4×7=28 can be proved by demonstrating with manipulatives. (Volume 1, Home Education – Paraphrased by L. N. Laurio, emphasis mine)
I don’t know about your experience in math curriculum, but the particular brand of curriculum we use in our home (Math U See) sandwiches its geometry book in between Algebra 1 and Algebra 2. My oldest isn’t to this point yet, so I don’t know how well Mr. Demme addresses Euclid and geometry in beginning algebra. But, if my sons don’t understand the relationship between geometry and algebra, I will certainly supplement. We do that already sometimes with Khan Academy, which I know directly addresses Euclid.
Pieces of the Quadrivium: Astronomy
According to Clark and Jain, “Astronomy was the centerpiece of ancient science,” dating back in Egyptian records as far as 3500 BC. It has been used for such utilitarian purposes as timekeeping, navigation, and recording of history. But, it wasn’t until the time of ancient Greece that people began understanding the mathematical relationships of planets, moons, and stars. In the 3rd century BC, Greek mathematician Aristarchus proposed the Heliocentric System, which placed our sun at the center of the known universe with the Earth revolving around it. Living in the same century, Greek philosopher Aristotle considered astronomy – the study of the stars – as the middle science that joined mathematics with natural philosophy (what we now call “natural science”).
Just as geometry represents endless points on an abstract plane, astronomy is the study of infinite time and space based on a cosmic amount of observable data. The hard part about working with infinite data is coming up with a system with which to handle it all. Astronomy, then, according to the classical tradition, is about understanding observable data (mathematical empiricism). Isaac Newton, in his book Principia Mathematica, develops his famous three laws of motion in order to prove an astronomical truth (law).
I am no astronomer. I have a hard time seeing and appreciating constellations, and I’m much more apt to appreciate the stories about the Greek gods than the images of them seen in the heavens. But, I do understand that the study of astronomy is the next logical step in learning mathematics – it’s the concrete (although infinite) representation of the abstract facts and figures put on paper. And, as Clark and Jain point out, the history of astronomy and its part in the rise of modern science is worth studying as well.
Pieces of the Quadrivium: Music
At the top of the Quadrivium staircase is music. Early mathematicians were fascinated by music because they believed all reality contained mathematical relationships. In 4th century BC (predating all of these other ancient Greeks aforementioned), Greek philosopher Philolaus said, “Nature in the cosmos is harmoniously composed of the limited and the unlimited, both the entire cosmos and everything in it.” Perhaps music is the pinnacle of the Quadrivium because music is the beauty in math. In fact, music is considered a forerunner of contemporary theoretical physics.
The Liberal Art of Music has three divisions: (1) musical instrumentalis, (2) musical humana, and (3) musical mundana. The first refers to music in the sense we think – harmonies made by choirs, orchestras, and bands. Because harmony refers to a mathematical proportion (not just those found in sound), the latter two division of music refer to mathematical proportions found in human society and the natural world (respectively). Pythagoreans would say the universe is alive with music and came up with the phrase “the music of the spheres” to describe the proportions in the movements of celestial bodies.
It’s no secret that Charlotte Mason did not stress math. Her beef was with how literature was being taught in her day with dry textbooks. In many ways, Mason’s philosophy was created in reaction to her contemporaries. It wasn’t that she didn’t see math as important – she did! She just wasn’t as worried with how it was being taught:
I don’t need to discuss mathematics. It already receives enough attention, and is quickly becoming a subject that’s taught with living methods. (Volume 3, School Education, Paraphrased by L. N. Laurio)
Truth, Goodness, and Beauty
As you have seen, the Quadrivium fits right into the classical model of education because it leads us to eternal and unchanging truths. There is beauty in the order of the created world. In her 6th volume, Philosophy of Education, Mason says:
One strong case for giving math a central place in our curriculum is because of its truth and beauty. … It’s a great thing to come face to face with a law, with a whole natural law system that exists and is true whether we agree with it or not. Two straight lines can never enclose a space. That’s a true fact that we can grasp, say, and act upon–but there’s nothing we can do to change or alter it. This kind of truth helps children have a healthy sense of living with limitations, and inspires a reverent respect for natural law. (Paraphrased by L. N. Laurio)
Until we meet again next Friday, be well.
I’m still reading these:
- The Count of Monte Cristo by Alexander Dumas (audio book)
Moderately challenging books:
- Last Child in the Woods: Saving Our Children from Nature Defect Disorder by Richard Louv
- Personal Reflections of Joan of Arc by Mark Twain
- Mind to Mind: An Essay Towards a Philosophy of Education by Karen Glass
- For the Children’s Sake: Foundations of Education for Home and School by Susan Schaeffer Macaulay (plan to begin Brandy Vencel’s Start Here study)
- Henry V by William Shakespeare (studying this with A8)
- The Normans: The History of a Dynasty by David Crouch
Still reading about asynchronous learning. Read a related book called The Whole-Brain Child by Daniel J. Siegel and Tina Payne Bryson this week. Gave excellent strategies for dealing with high emotions.