Not to the Test
Posted: October 9, 2026 Filed under: Child Centered Activities Leave a comment
8th grade standardized tests contain questions like these:
A triangle has three angles in the following degrees: 70°, 30°, and x. What is the measure of angle x in the triangle?
A. 30°
B. 60°
C. 80°
D. 150°
My son knows all triangles have 180° so x can only be 80° when the other angles are 70° and 30°. “That wasn’t hard,” he says.
Marty has $80 to spend at a sporting goods store. He will spend $56 on a shirt, and then buy some darts. Each box of darts costs $6. He wants to buy as many boxes as possible. Which equation shows the number of boxes of darts, x, he can buy?
A. 80 = 56 + 6x
B. 80 = 56 * 6x
C. 80 = (56)(6x)
D. 80 = 56/6x
He knows that x can only be 4 if 56 + 6x = 80 but he deadpans, “I made an educated guess.”
Our homeschool ignores standardized tests. There is value in knowing social conventions, a functional literacy. He needs to know how to read a graph, how to reconcile a checkbook, calculate a discount, read fractions on a tape measure, recognize and solve an equation. Like “blocking and tackling” those skills are important, but to learn about number, we take a different path.
We follow the Quadrivium: the mathematical architecture of the Real.
Plato, in Book VII of the Republic, outlined the path away from ignorance up toward the light of eternal truth and the Good. Mathematical knowledge is like a ladder, allowing the student to ascend higher, “it leads the best part of the soul up to the view of the best among the things that are…”
In the Quadrivium, Arithmetic is “pure reason,” the art of number; Geometry is the “Form of the Good,” number in space; Harmonics/Music is “Beauty and Order” which is number in proportion and time; Astronomy is a “Diagram of the Pure” ordered motion in space.
Quadrivium means “a crossing of four roads,” a name used by Boethius, the Roman philosopher, around 500 AD for the four mathematical sciences. Combined with the three language arts – grammar, logic, and rhetoric, referred to as the Trivium – these became the Seven Liberal Arts. The origin was at Plato’s Academy, a private school founded around 387 BCE whose tradition of the mathematical sciences plus logic endured and influenced Christian, Islamic and medieval European scholars. In 529 AD, Emperor Justinian passed laws banning non-orthodox teaching, which effectively disbanded the Neoplatonic Academy.
Consider a 900 year tradition, and some of its students, these just a few among many who studied or were influenced by the Quadrivium: Plato, Aristotle, Euclid, Cicero; Plotinus, Boethius, Dionysius the Areopagite, Bede; St. Augustine, Bernard of Clairvaux, Hildegard of Bingen; Roger Bacon, Saint Thomas Aquinas, Dante, Copernicus, and Johannes Kepler.
My son asks, “Why are so many places named after Greeks and Romans? St. Augustine, Florida. Euclid, Ohio… it’s by Cleveland.” We stop the math lesson and talk about St. Augustine, and Euclid.
The Magister teaches the Trivium, and so we pursue the Quadrivium. But what happened? Why did the Trivium and Quadrivium become obsolete?
The Dark Ages ended. Monasticism ran its course, universities were formed. The Renaissance Humanists arose, then the Scientific Revolution changed the focus from metaphysical truth to observation, experiment, the quantitative. The Enlightenment further emphasized reason, criticism, empirical knowledge: “I think, therefore I am.”
Then the Industrial Revolution brought standardized education; the school bell like the factory whistle training cogs in the machine. The seven liberal arts were neither technical, nor vocational, not the STEM path to employment. The modern world is about measurable and quantifiable output. Quantity, not metaphysical quality.
Our Art Farm Homeschool Academy is focused less on the measurable, more on perception, a quality of discernment, on understanding.
Sacred Number is mystic arithmetic, shared widely among religions, paths to wisdom, among all peoples, across all time. The fascinating question is why this is pervasive.
One is unity. The monad. The origin and point with no dimension. The Great Spirit. God. Permanence. The one name of all names. The shoreless ocean of Unity, it is circle, center, the purest tone, the indivisible All. The unmanifested Divine, the I am that I am.
Two is duality, polarization. Either/or, yin and yang, the two sides of every coin, left and right; shadow, opposite, and object.
Three, the first odd number, makes a triangle which is the first shape to hold space; a three legged stool can stand anywhere, a braid is complete by the third strand, life unfolds across the past, present and future; birth, life, and death. In music the perfect fifth vibrates at a 3:2 ratio, which resolves conflict; singing bowls tuned to 3:2 calm the nervous system. The circle of fifths is considered the cosmic spiral of becoming.
Four is manifestation. The cardinal directions and the elements of Fire, Air, Earth, and Water; in modern physics the building blocks are protons, neutrons, electrons, and neutrinos; the year is divided in four, as animals walk on legs of four.
And so it goes for all the natural numbers. Seven are the days of creation, the chakras, the classical planets, while 12 is the months of the year, signs of the zodiac, tribes of Israel, the disciples; the number of cosmic order, completion and time.
“These are not proofs of a cosmic numerology, but a pattern, a language humans have used to interpret order.” With a surprising confidence, my son says, “That is not too far out. It makes sense and I understand it. I have heard about that. I think I have heard you talk about that.” Somewhat taken aback, I reply, “Then let’s talk about zero.”
The Babylonians, around 300 BCE, and Mayans, around 30 BCE, used a placeholder to show an empty space; 501 is different from 51 because there is no value in the “tens” space. But the placeholder was not a number. It predates zero.
The Dark Ages in Europe were an enlightened time in India. A brilliant mathematician named Aryabhata developed the concept of zero and used it in calculations. In the 7th century the mathematician Brahmagupta defined the rules for zero: 0 – 0 = 0 but any number + 0 remains unchanged and yet 0 multiplied by any number = 0.
The concept traveled along the Silk Road and was embraced by Islamic merchants and mathematicians. An Italian, known as Fibonacci, brought the concept to Europe and popularized the Hindu-Arabic numerals. The merchant guilds favored Roman numerals, while the Catholic Church regarded “nothingness” as demonic and dangerous. Zero was a long process of adoption, resistance, and refinement.
But zero simplified bookkeeping, which merchant bankers embraced. Gutenberg’s printing press made clear that Arabic numerals were more legible than Roman numerals. Linear perspective was discovered, its vanishing point a single point of nothingness, visually equal to the void. Renaissance humanism loosened the Catholic Church’s grip, while Cartesian coordinates helped lay the foundation of calculus.
Zero changed the world.
“But what made them choose that symbol? What is so important about zero?” Flummoxed, I say, “Well, there is no accounting without zero. What is $100?” He counters, “Ten thousand pennies.” “No,” I say, “What is the number?” Dryly he replies, “Oh, a 1 with two zeros.” “Precisely,” I respond, “but it’s more.”
Zero is absence, the whole number of the empty set, what early Indian treatises called “Sunya,” meaning “void,” the abyss, the unknowable, not “nothingness” but the cosmic womb, that from which all other numbers and creation emerge. Zero makes the void calculable.
Pushing further, we discuss Phi, which represents the Golden Ratio, an irrational number equal to approximately 1.6180339887. Any line divided into two, (roughly 5/8 and 3/8) such that the ratio of the entire line to the longer line equals the ratio of the longer line to the shorter line yields Phi. Euclid wrote about this in the Elements.
“Fibonacci,” I explain, “that Italian mathematician, who grew up in North Africa, learned about the pattern where two numbers added together equal the next number in sequence (1, 1, 2, 3, 5, 8, 13, 21, 34…) and when dividing any number by the one before it, the further you go, the closer you get to 1.618. 5/3 =1.667, 13/8 =1.625, 34/21 =1.619. The Fibonacci sequence relates to Phi.”
This ratio seems to show up everywhere: in the proportions of the Parthenon in Greece, Egyptian pyramids, Leonardo da Vinci’s Mona Lisa; the spiral of sunflower seeds, the shape of pinecones, hurricane clouds and the shells of nautiluses appear based on Phi. The seeds and pinecones are undisputed, the art and architecture are disputed, the nautilus and hurricanes follow different ratios, but appear similar.
Everyday items, like credit cards and laptop screens are shaped closely like “Golden Rectangles.” To be precise, I get out the ruler and measure my credit card, which measures 1.586, not quite 1.618 but he sees the basic point. That Phi is a real, recurring ratio is beyond dispute, reason enough for our lesson.
My son and I go into the garden to look at the coneflowers. I show him images of the Parthenon, the Mona Lisa, The Great Wave by Hokusai, other paintings by Rembrandt, Raphael, Dali and Mondrian. He asks, “So this is not philosophical, but mathematical?” I reply, “It is a ratio, found throughout nature, art, architecture, in all cultures.” “I think I get this,” he quietly says.
We go to the piano and I play a perfect fifth, which is the ratio 3/2. 3 and 2 are Fibonacci numbers, but the ratio 3/2 equals only 1.5, too early in the sequence to equal 1.618. Phi is an irrational dynamic, spatial geometry, while the perfect fifth is harmonic symmetry, a clean pure resonance, number in time. I play the Prelude of Also sprach Zarathustra by Richard Strauss, whose fanfare to the sunrise opens with a rising perfect fifth. He hears the sound, I show him the math.
The symbols in Phi ( Φ or φ ) look like a 1 passing through 0, curiously, but that may be a coincidence. We began by talking about natural numbers and zero, which lead us to void and unity, which lead to ratios and patterns, which are geometry, music, and planetary motion. The Quadrivium. The path we follow teaching number in the 8th grade.
But what practical application could this have? Here are sample Quadrivium test questions:
In classical arithmetic, two distinct positive integers a and b are called ‘amicable numbers’ if the sum of the proper divisors of a equals b, and the sum of the proper divisors of b equals a. Which of the following pairs satisfies this condition?
- 220 and 284
- 496 and 8128
- 118 and 182
- 140 and 200
In Pythagorean musical tuning, intervals are constructed using ratios of fundamental frequencies based on pure harmonics. If a reference pitch has frequency f, what is the frequency of the note that is an octave and a perfect fifth higher?
- 4 f
- 5/2 f
- 3 f
- 7/2 f
“Whoa! Dad, are you serious? This is hard!”
To go forward to his future, we go back 2,000 years.
___________________________
(NB: Quadrivium answers are below).
The gardens still produce, though late season. The tomatoes’ last harvest, while an abundance of pole beans hang still on the vine. Garlic shall be planted later this month, after the first frost.
A “proper divisor” of a number is any positive factor excluding the number itself.
Proper divisors of 220: 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, and 110. Sum=1+2+4+5+10+11+20+22+44+55+110=284.
Proper divisors of 284: 1, 2, 4, 71, and 142. Sum=1+2+4+71+142=220.
Because the sum of 220’s divisors equals 284, and the sum of 284’s divisors equals 220, they “love” one another in the Pythagorean tradition. Of note, 220 and 284 was the very first pair of amicable numbers discovered by the Pythagoreans, who viewed them as a symbol of friendship.
An Octave corresponds to a ratio of 2:1. Raising a frequency f by an octave doubles it: Octave=2×f=2f.
A Perfect Fifth corresponds to a ratio of 3:2. Raising a pitch by a perfect fifth multiplies its frequency by 3/2 =1.5f.
Combining them: To go up an octave and then a perfect fifth higher, multiply the frequencies sequentially: Frequency=f×2×3/2 =3f.
The interval of an octave plus a fifth (a twelfth) vibrates at exactly triple the frequency (3f) of the fundamental tone. This embodies the Monad (1) expanding through the Trinity (3).
