Thanks to a misunderstood statement made by Saint Augustine in De Genesi ad Litteram, which condemned, not mathematics as we define it today, but rather astrology (the popular application of math in his day), many early Christians were strongly opposed to the study of mathematics. But Augustine, in fact, had the highest view of mathematics. For Augustine, nature is inherently quantifiable and mathematical number theory is a creational given. We live in a mathematical universe. Being a former Manichaean and, thus, an expert in Platonic thought, Augustine was no doubt following in the footsteps of Plato, who situated the study of mathematics—the quadrivium of arithmetic, geometry, astronomy (not astrology), and music—as an integral part of what we today call the classical liberal arts. Thus, in Augustine we find, not a condemnation of mathematics, but the essential marriage of classical and Christian thought regarding mathematics. And according to contemporary Christian philosopher James Nickel, “Because man is made in the image of God, he is gifted with the ability to observe the physical Creation and formulate relationships and consequences that both explain and predict.” The study of mathematics is precisely this: observation and formulation. To study mathematics, then, is to pursue humanity’s divine mandate to order and manage Creation. Studying mathematics as part of a liberal arts education and studying mathematics as a feature of Creation are complementary pursuits, and so mathematics ought to play a vital role in our classical Christian education.
The Why: Mathematics Is Indispensable in Classical Christian Education
When I ask my students if they like studying mathematics, invariably the class splits evenly. If anything, there are more students who would rather not study math than those who would choose to do so if it were not required. When I ask why, the unanimous response is that math is hard. And the truth is, mathematics is hard. The discipline required to master arithmetic facts and develop a number sense in the lower school years does not relent in the upper school where students must engage in abstract and logical reasoning. And while the difficulty of the subject matter is not what makes mathematics indispensable, a classical education is predicated upon the mastery of difficult content because this requires the very thing that serves as a formal principle of classical education: character. One of the things that makes math indispensable in a classical education is that it contributes to virtue formation. Students who discipline themselves to learn mathematics become virtuous people, people who are able and willing to do hard things even when they may not want to.
Additionally, Morris Kline, mathematician and author of Why Johnny Can’t Add, asserts that mathematics has shaped the course of modern history. This is best demonstrated in the impact of the Scientific Revolution and Sir Isaac Newton’s development of his calculus, a development which revolutionized science and philosophy along with mathematics, and which married description to causation in a way that had eluded philosophers since Plato and Aristotle. We cannot properly call our education “classical” unless the discipline that shaped the course of history plays an integral role. If mathematics is, as Kline has argued, “a body of knowledge with content that serves scientists, philosophers, logicians, artists, musicians, statesmen, and theologians,” then students who will one day become one of these must comprehend mathematics.
Finally, returning to Nickel’s assertion that to study mathematics is to, in part, pursue the human vocation, educated people must be able to observe, analyze, and model the universe that the Lord created. A cursory look around us makes plain the reality that not only is this world highly ordered, but also that quantity plays an essential role in everyday life. In fact, quantity is an intrinsic quality which the Lord has woven through Creation. If we are to exercise dominion over this world, we must understand this world. And this world is quantifiable. It is mathematical.
The What: Let No One Ignorant of Calculus Graduate
So, what specific mathematical disciplines must we include in a classical education? And what progression through those disciplines makes the most sense? Kindergarten through sixth grade students must study and master arithmetic, the foundation for all further mathematical pursuits. Building on mastery of arithmetic, students must “tame the unknown” through the study of algebra. Moving logically from a premise to a conclusion based on established axioms and properties to determine the value of an unknown quantity—solving a relation—is both a significant logical leap from arithmetic and a necessary step in the development of critical reasoning skills. Thus, the best progression through algebra for teachers and students alike is a consistent movement from concepts and computation skills that are less complex to those that are more complex, achieving mastery along the way. The development of an efficient, symbolic system of notation and the two-dimensional Cartesian coordinate system by René Descartes in the seventeenth century enabled the conceptual coupling of abstract algebraic reasoning with the concreteness of geometric objects such as points, lines, and planes. Thus, algebra both forms the foundation for further study in higher-order mathematics and serves as a bridge to geometry.
While geometry preceded algebra in historical mathematical development, we reverse the order in our instruction precisely because of Descartes’ work. While perennially helpful, the content of Euclid’s Elements by itself becomes much more concrete when taught to students who have mastered algebra and analytical geometry. Thus, a substantive study of geometry, including trigonometry, follows algebra to further polish logical reasoning through proofs and to prepare students to model complex real-world realities with calculus. The value and necessity of Descartes’ analytical geometry cannot be overstated.
The study of mathematics in the high school years should culminate with calculus. Many will balk at this assertion, but as I argued above, it is impossible to comprehend Western history apart from comprehending calculus. Newton’s achievement was the culmination of the thread in Western science and philosophy known as the problem of change raised by Zeno’s paradox. While not all students may possess the aptitude to understand complicated calculus computations in the high school years, all can master the concepts of the limit, derivative, and integral along with their historical and philosophical significance. In my humble opinion, no student ignorant of calculus has completed a classical education.
The How: Pursuing Mastery Through Conceptual Integration
One of the hallmarks of classical education, as opposed to more modern pedagogies, is mastery. We do not merely introduce a broad spectrum of topics and concepts in a perfunctory way as if to say, “Look at how much material this course covers!” Rather, we carefully select topics and ensure that students have mastered them—that is, that students have achieved true comprehension or understanding—before moving on to other topics. And mastery in mathematics entails much more than robotically reproducing canned computational methods. A student who has mastered solving first-degree equations in two variables, for instance, can do much more than transform those equations from standard linear form to slope-intercept form or substitute values for one or more variables and evaluate the result. That student can execute those computations easily because he comprehends the relationship between a first-degree equation in two variables and its corresponding graph in the Cartesian coordinate system: a line. He can move seamlessly from the information in slope-intercept form to the same in the graph. He understands what that equation in algebraic notation represents, what the linear graph represents, and how the two are related. That student has mastered algebra.
In the interest of computational proficiency, too much of mathematical instruction today has adopted the aim of producing human “calculators.” But, to properly orient mathematical study in classical education, we need to complement computational proficiency with conceptual proficiency. The result is an ability to utilize mathematics as a tool to model the Lord’s world. While the study of mathematics is a virtuous end, applying it to the physical world acknowledges that as creatures we discover the mathematics the Lord has woven into his Creation—the wisdom of the Lord. When we graduate students who have mastered a robust and integrated mathematical curriculum culminating in Newton’s calculus, we are realizing the goal of classical education: the cultivation of wisdom and virtue through meditation on the truth and beauty of the Lord’s mathematical universe. Augustine argued that “numbers have fixed laws which were not made by man, but which the acuteness of ingenious men brought to light.” May we continue in the tradition of such ingenious men by making mathematics a pillar of our classical Christian education.