Nobody argues that schools ought to teach Shakespeare because it helps students to write office memos. Yet many argue that schools ought to teach Pythagoras because it helps students to become engineers. While engineering and other STEM fields are certainly noble endeavors, would it not be a shame to say that any student who becomes a lawyer, librarian, poet, or plumber cannot benefit from a mathematics education? We believe that the study of mathematics is good universally. Through its study, students will discover the order of our cosmos and the mysterious beauty of patterns in numbers, shapes, and arguments. They will discipline their minds to abstract ideas from the concrete. The teacher will lead students through the discoveries of arithmetic, algebra, geometry, and calculus. Mathematics is good not because it tells us merely what we already know, nor because it attempts to tell us things utterly foreign to our nature. Like Shakespeare, it tells us the very thing that we yearn to be true: this world of ours is awesome in wonder and ordered in beauty.
Such lofty goals take years, even a lifetime, to achieve. It starts with arithmetic. For the first six or seven years of schooling, the student must learn about numbers, decimals, and fractions; addition, subtraction, multiplication, and division; and the basic geometric shapes. These concepts establish the fundamental images and relationships for all future mathematical thinking and as such should not be hurried nor trivialized. In order to nurture the natural sense of wonder in a child, it is imperative to supplement the rote learning of arithmetic with playful study of patterns and numbers. This includes riddles, puzzles, games like Sudoku, bar modeling, and other manipulatives. The Singapore method of teaching does this well. Setting up a firm foundation while opening the child’s mathematical imagination, by both providing the fundamental images and increasing his sense of wonder, is the primary goal of arithmetic.
“The scientist does not study nature because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful.”
– Henri Poincaré
The next step comes in algebra and geometry classes. In the early teenage years, many students long to understand how reality works. They argue naturally, and it is the purpose of this stage to teach the student to argue well. By studying the definite procedures of algebra and the power of using abstract variables, the student learns to make general statements about mathematics. No longer confined to a fifteen-by-fifteen times table of facts, she begins making claims about multiplication itself. This is why we focus on Euclidean rather than analytic geometry. Euclid’s proofs provide excellent insight into the power of rational thinking. Using a dozen or so fundamental statements, he is able to create an entire world of perfect images and shapes, interwoven by the thread of deductive reasoning. The primary goal of the algebra sequence and geometry is to show the student the proper way to make truthful, consistent arguments.
Finally, the student studies trigonometry and calculus. Now that the student is able to reason logically, it is time to reason imaginatively. The student will grapple with the idea of infinity, an idea which has no cognate in our five senses. She will use the tools learned so far to make new discoveries and new proofs in trigonometry, appreciating the simple and straightforward nature of the very best proofs. This stage is when the student becomes a true mathematician, more than a human calculator. The primary goal of this stage is to cultivate a mathematical imagination that binds truth and beauty together in elegant expression.
Like all true liberal arts, an education in mathematics aims for the cultivation of intellectual and moral virtue. Through the processes of memorization, repetition, abstraction, and deduction, the student habituates himself to the moral virtues of perseverance and steadfastness. Hard work produces discipline, and discipline produces good character. Likewise, there is perhaps no other study that disciplines the mind as well as mathematics. Indeed, the very words discipline and mathematics share the same root in Greek, μαθητής (mathētēs), and Plato insists in his Republic that “geometry will draw the soul towards truth, and create the spirit of philosophy.” As we learn about the order of mathematics, our hearts and minds become attuned to the order in the world, and we learn to seek and love that order in our daily lives. It becomes beautiful; we begin to stand in awe and wonder of the world around us, searching for meaning and beauty around every corner. In short, a mathematics education prepares the mind for the love of wisdom and the soul for the labor necessary in securing it.
