One way to view this different pedagogy in action is to examine one of the most successful textbooks of all time: Euclid’s Elements. It is somewhat surprising that in all of the enthusiasm for ad fontes in classical education, there has not been more of a public cry for the study of the great works of mathematics in our schools; nevertheless, whatever justifi- cation there might be for neglecting Newton and his flux- ions or Dedekind’s “cuts” in the rational numbers, it is hard to imagine a good reason to withhold from our students the book that, short of the Bible, has been printed more than any other and that has stood for over two thousand years
as a mark and milepost of education. While the problems in modern mathematics education are deeper than any single book can solve, considering the approach and pedagogy of the Elements can certainly help to cast the study of mathematics in a freer and more humane light.
It is striking that such an important and long-lived textbook should be so inscrutable in a modern classroom. There are no exercises, no answers in the back, no helpful mnemonics, and not even any explanations, save the propositions that constitute the body of the text itself. It is utterly incompatible with a “lecture and practice” pedagogy. The text, however, is unquestionably beautiful: “A point is that which has no part,” may not be the clearest definition ever given, but it certainly invites the students to consider carefully what is the nature of the objects they are setting out to study, and it demands of them their attention and care. The Elements invites the reader to participate and to wonder.
Euclid’s work also demands a conceptual, rather than a procedural, attentiveness from the student. The focus of the text is not a set of algorithms to be memorized but instead a rising tower of mathematical claims, each building upon the last and carefully demonstrated. The student’s task in reading Euclid is not, primarily, to be calculating or remembering but, rather, understanding. In my experience with the text, both as student and teacher, the surest way to bring the mathematics present in the Elements to life is to enact it in the classroom through presentations and conversation. As students labor at the blackboard to prove and convince their peers that the claim made in the text is, in fact, true, they embody and dramatize the mathematical tension of the propositions themselves. Even in stepping through Euclid’s own demonstrations, students make the thoughts their own as they speak them and defend them to their classmates. To bring the Elements into the classroom requires that we invest the students in the life of the math- ematics under consideration. If we were instead to present the “material” of the text as something to be memorized and merely repeated, the nature of the Elements means that we would almost certainly fail. The excessive detail and pre- cision of a geometric demonstration only makes sense in the context of its conceptual aims, and none but the most eidetic of students could successfully memorize the proofs without understanding them. Thus the Elements invites us to a more participatory and living mathematics classroom, where students are not just an audience to the work of thinking through patterns and procedures but must themselves be active contributors.
The most important lesson we can take from Euclid’s work is the idea that student participation in mathematical thinking must take precedence over the delivery of the relevant content knowledge (useful though that material may be). The Elements certainly provides a great deal of important mathematical insight, but always with the aim of demonstrating “beautifully that something is so.” How else could we explain the doubled proof of the Pythagorean theorem in the text, once as the capstone of Book I, and then again, with more generality and greater meaning in Book VI? Not content to show merely the truth of this great theorem, Euclid proves it once and then spends hundreds of pages teaching about ratios so that he can present it again in a more compelling and beautiful way. The fact of the theorem, that a2+b2=c2, where a and b are the sides of a right triangle and c is the hypotenuse, is known to almost any- one with mathematical experience. What Euclid presents, however, is a call to us to understand and contemplate the relationships that give rise to this fact and a carefully considered path that can lead us to that understanding.
Our students are rational, worshipping beings created in the image of the triune God. Insofar as our approach to mathematics requires them to slavishly follow instructions, chasing what is useful rather than joining in the pursuit of truth that mathematics can and should be, we have dishonored them as well as the One who created the universe and its order. However, if we ignore the clamor of a culture that exalts utility above all other consideration and instead allow our students to enter the study of mathematics as participants rather than just recipients, we might sacrifice some efficiency, but in exchange, they might someday be able to say in agreement with Edna St. Vincent Millay, in wonder and delight, that “Euclid alone has looked on Beauty bare.”