My third-grade teacher had to bribe us in order to get us to memorize the multiplication tables. For every table you mastered, you won one part of an ice cream party. The ones got you a bowl, the twos a spoon. If you proved proficient in the threes you won a scoop of cheap vanilla ice cream, the fours got you two, and so on until, once you had demonstrated perfect proficiency of the twelve times, you had amassed a topping laden sundae that was the stuff ten year old's dreams were made of. I coveted that sundae. Unfortunately, my sugar-coated fever dreams were dashed. I believe I earned up to chocolate syrup but was unable to progress satisfactorily past the sixes. If I am being perfectly honest, I still can not.
As I progressed in my mathematical education, I found myself becoming increasingly bored and frustrated in turns. Long division was difficult, algebra, with its insidious introduction of abstract alphabetical symbols and values on either side of the equal sign, frequently reduced me to tears. I discovered my spatial reasoning issues in geometry, and I can boast that I never passed a single calculus test.
My father, a nuclear engineer who crafted new equations for his doctoral thesis, would attempt to help me. These tutoring sessions would nearly always end, not in new understanding, but in shouting and tears. The best I could expect from my teachers was enough help to get me to pass classes with a B. It was perhaps inevitable that I would come to the conclusion that I was 'not a math person,' and that 'math was stupid anyway."
Far from being an isolated incident, I find as an educator that many of my students are undergoing similar mathematical experiences. I teach an honors section of eighth-grade students. They're bright children. They are incredibly gifted in nearly every single class, taking to new subjects, information, and skills with relative ease. And yet, mathematically they struggle. One girl came to class with a literal spring in her step. As she flounced through the door she greeted her peers and gaily announced that she and her father had engaged in a shouting match the previous evening as he attempted to help her with her mathematics. The others laughed. Far from being horrified or confused at her struggle, they revealed that they were also part of the culture of mathematical ineptitude.
Not every student struggles to grasp mathematics, but those who show a natural talent for the subject have become increasingly rare. It's tempting to look at these academic unicorns as opposed to the rest of the unwashed masses and conclude that there are some people who are simply 'mathematically minded.' The best that the rest of us can hope for is to learn our basic sums and rely on calculators to do anything more complicated than single-digit multiplication for us.
However, if one looks beyond the borders of the U.S. it becomes clear that the issue is not necessarily that there are only a very few men and women who are capable of mastering the mathematical discipline with any sort of competence. Students who live in countries that are not nearly as prosperous as the United States routinely outperform our students in mathematics by an alarming margin. A 2018 PISA study showed that United States (37/78) children test well below children from countries such as China (1/78), Canada (12/78), and even Hungary (34/78) 1. Perhaps the issue is not that there are 'math people' and everyone else. Perhaps the issue is that American educators have forgotten what math is and other cultures haven't.
I identify myself as part of the American classical education movement. In classical education circles there's a lot of talk about returning to a 'liberal arts' education. What people usually mean when they say this is that children should read old books that comprise a made-up entity called the Western Canon and learn Latin. It is almost entirely centered on a medieval concept called the Trivium.
Because the pedagogical tools modern classical educators use are unfamiliar to the vast majority of the citizenry, I'll attempt to offer some basic definitions of the relevant terms. When classical educators reference a liberal arts education they are referring to the traditional seven liberal arts classed into two groups - the Trivium and the Quadrivium. The Trivium refers to the three arts Grammar (the art of symbol creation and combination), Logic (the art of reason), and Rhetoric (the art of expressing oneself persuasively). The Quadrivium refers to the four arts of Arithmetic (the art of number theory), Music (the art of number theory applied to time), Geometry (the art of spatial theory), and Astronomy (the art of spatial theory practically applied). These are not definitions that are set in stone. There is a messy and dizzying debate within the classical community about the precise nature of each of these arts, their importance, how they should be understood and taught, if they should be taught, etc.
The point I'm attempting to make, however, is that classical schools tend to focus on the Trivium - Grammar, Logic, and Rhetoric, placing a larger emphasis on the subjects that seem to apply to these easily - namely Literature, History, Art, and Latin. The Quadrivial arts tend to be underemphasized with mathematics and the sciences treated like embarrassing bastard children. Many classical schools simply seem not to know what to do with them.
Classical schools may produce better test scores in mathematics than other public educational models, I am not sure. If they do, however, I would not be convinced that this was not a happy accident - that by teaching certain subjects well, the mental acumen needed to have some success in the quadrivial arts bleeds over.
I have recently experienced a kind of revelation that has made me reevaluate my juvenile attitude toward mathematics. I have long suspected that the mathematical arts are not a waste of time, and may, perhaps, even have some value inherent in themselves. I have also come to believe that the liberal arts, all of the liberal arts, are necessary in order to create men and women who have free minds. I have struggled, however, to understand how math fits into the creation of a free and happy man. The arts of the Trivium seem obvious. Understanding how to speak well and think well can obviously help a person to avoid being taken in by jargon and propaganda. Written and verbal language can also be used to persuade one of the truth, to debate and explore the subjects that define our civilization and to some extent out humanity. They can be used to communicate deep and meaningful truths about the nature of reality and our experience of it. My understanding of the Quadrivium, on the other hand, has been hampered, not only by my deep ignorance of the subjects themselves, but by my inability to understand what they truly are.
I cannot, therefore, credit myself with this insight, if indeed it is any sort of insight at all. If there is any truth in it, it did not come from me. The Biblical picture of reality, given in the creation account, then Genesis 1:3 is the most profound statement in the whole of human history as regards our understanding of the nature of reality: "And God said, Let there be light, and there was light." Creation was accomplished through Language.
Now, no one should suppose that the language God used in the act of creation was anything like the verbal language we use to speak with one another, to express our thoughts, ideas, and feelings. How did God communicate Being? I do not think that we will ever truly know what exactly constitutes divine language. However, God in His graciousness allows us to understand something of what it must have been like through mathematics. I do not think that this is a new idea. I am well aware that as far back as Pythagoras number was associated with the divine. Men like Kepler and Newton knew that God spoke about His universe through the language of mathematics.
What I want to offer is this: The classical movement, in its attempts to reunify knowledge, has struggled to articulate a philosophy of education that truly unites learning into a cohesive unit. Their issue is that, for many, they still see the arts of the Trivium and Quadrivium as functionally different rather than facets of one Master Art.
The Trivium and Quadrivium are both one half of the way human beings are able to understand and interact with Truth. Reality can only be understood by human beings through language, because creation was, at bottom, a linguistic act. It is our view of what language is that is far too narrow. The Trivium and Quadrivium cannot be divided into the linguistic arts and the mathematical arts. Written expression, verbal expression, and mathematical expression are all aspects of one divine Language. Our troubles understanding this stem from our finite natures. We are not God. We are contrained by the fall, by sin, by the inherently limited nature of our own being. Even still, we are inherently linguistic creatures. We can only understand reality through language.
All of the Liberal Arts represent ways in which human beings can master language. The Trivium contains the arts that give humans mastery over moral and relational reason and by extension human speech. Although it seems limited now, it is possible that human speech once had godlike power. In the wake of Babel and the intervening linguistic corruptions of thousands of years, we will never know for certain. However, there is still great power in human speech for good. God used human speech when he sent his prophets to declare His kingdom and to call men to repentance. He demonstrated this power in His own person through the incarnation of Christ. There is also, however, power in the spoken language for great evil. The devil used speech to tempt Eve to sin. History is littered with tyrants and demagogues who seduced whole countries to great evil using spoken language. The arts of the Trivium are truly human arts. They belong to us. They are limited in the ways that we are limited and they are corruptible in the ways that we are corruptible.
The language of the Quadrivium, however, is of a different kind. This mathematical language seems less corrupted and more universal than the linguistic arts of the Trivium. Grammar, Logic, and Rhetoric give us mastery of the language of man. Arithmetic, Geometry, Music, and Astronomy allow us to speak the language of the universe.
I do not know to what extent this language, in the hands of man, is corrupted and corruptible, but it seems to me mathematical language is not inherent to man. Instead, man must learn to discern and understand mathematical language as if it were a language one had known as a child but had forgotten through years of disuse.
If, then, this is the case, let the student master the spoken, human word, but do not allow him to neglect the language of the stars. Mathematics is not a mere series of numbers and equations and theorems but is a linguistic expression of the mind of God- a reverberating echo from the beginning of time. The Word crying "Let there be Light."
1https://factsmaps.com/pisa-2018-worldwide-ranking-average-score-of-mathematics-science-reading/
