Teaching Geometry through Euclid’s Elements

Euclid’s Elements is the most successful and influential mathematics textbook ever written, served as the foundation of mathematical understanding for students for over two millennia, and stands as one of the greatest and most beautiful achievements of the human intellect produced by Western civilization. While it’s obvious that the Elements ought to stand at the center of a classical approach to math, it’s not at all obvious how to teach a class from the Elements. This talk aims to share practical reflections arising from several years of experience teaching geometry from Euclid, as well as some reflections on what this book has to teach us about that nature of mathematics and math instruction across the curriculum.

Arron Kau

Arron Kau is Mathematics Department Chair at Covenant Classical School in Fort Worth, where he lives with his wife and four children. He has previously worked in Mathematics and Humanities at The Ambrose School in Boise, and also as Mathematics Curriculum Director for Brilliant.org, an online math and physics problem-solving community with millions of members.

Reclaiming Mathematics as a Liberal Arts Course

We know that math is supposed to be one of the liberal arts, yet math is one of the least appreciated subjects among students. Let’s consider how treating math as a story rather than as a “fact” might help us reach the souls of our students.

Arron Kau

Arron Kau lives in Meridian, Idaho with his wife and four children. He has a Masters in Arts from St. John’s College and has taught mathematics for over ve years, most recently at The Ambrose School. He is currently employed as mathematics curriculum director and resident teacher at Brilliant.org, a math and physics problem solving community.

Euclid’s Elements and Mathematics as a Liberal Art

If we are to recover what has been lost in education, one of our most challenging tasks will be to reclaim mathematics as a liberal art. In particular, math faces not just the general challenges that confront other liberal arts but also the far more difficult problem that it is eminently useful. While this can and should be a strength, it has led, even in classical schools, to a pedagogy of mathematics that neglects and even undermines what should be math’s most significant contribution to a liberal education.

That mathematics could be misunderstood, some- times even by those teachers who hope to impart it to their pupils, should not be terribly surprising. From at least the time of Pythagoras, mathematics has carried the flavor of the mystical. There is something both terrible and won-derful about the world of ideas in which triangles and theorems and prime numbers contrive to perplex even the brightest among us. Mathematics is at the troubling inter- section of a world that is both so ephemeral that one can never quite pin down its true reality and yet so absolute that it seems to mercilessly break us and our faulty under- standing. What is certain, however, is that the practitioners of mathematics view it as an expression of beauty that touches their very souls; they speak of themselves as artists and explorers. Paul Lockhart said that “to do mathematics is to engage in an act of discovery and conjecture, intuition and inspiration; […] to be awed and overwhelmed by an almost painful beauty.” Carl Friedrich Gauss claimed that “the enchanting charms of this sublime science reveal them- selves in all their beauty only to those who have the courage to go deeply into it.” According to Socrates, mathematics is “the easiest way for [the soul] to pass from becoming to truth and being.” And while these selections could continue, let a statement from G. H. Hardy encapsulate the whole: “A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.” In short, mathematicians view their subject as one filled with wonder, delight, and human flourishing.

We ought to ask then why so few of our students de- light in the study of mathematics. How does a subject that, according to Socrates, is ideally suited to provoke our souls to wonder turn into something unfulfilling and “boring”? I strongly suspect that the dissonance between our students’ perception of the subject and Socrates’ description has much to do with our approach to the subject. It may even be the case that we have fundamentally misidentified mathemat- ics itself. Consider Aristotle’s distinction between the the “servile” arts and the “liberal” arts:

For both a carpenter and a geometrician look for a right angle, but in different ways, for the one seeks it to the extent that it is useful to the work, while the other seeks for what it is or what it is a property of, since he is someone who beholds the truth.

In our textbooks, our testing, and our typical lecture- and-drill classroom structures, we imply to our students that mathematics is about useful answers. As a result, students often come to believe that mathematics requires nothing more than carefully following predefined procedures.
It is remarkable and troubling how deeply they internalize this lesson and how thoroughly it destroys their relation- ship with the beautiful wonder of mathematics.

The impact of what I now call “the liturgy of the math book” can be clearly seen in a story from a seventh-grade course I taught several years ago called “The Logic of Mathematics.” Students in the course were already proficient in the skills of arithmetic, and so the class was aimed at preparing them for 8th-grade algebra through mathematical thinking, introductory logic, and problem solving. Students in the class were routinely assigned very difficult (or impossible) problems to work on for homework: assignments included constructing a map that requires five colors to prevent adjacent color-duplicates (impossible), finding a path across all the seven bridges of Königsburg without crossing any twice (impossible), decrypting secret messages using frequency analysis (very difficult), and many other similar tasks. Students understood that they wouldn’t always reach perfect answers, but they were required to share what they had done and how it had (or had not) worked with their classmates. The homework policy for the class was a 30 minute maximum; students were never required to work more than 30 minutes although they often did so willingly because they enjoyed the work we were doing. For months, I assigned difficult or impossible problems every week with no complaints and with great success.

One day in the middle of the third quarter, however, I made the mistake of assigning students a single page of story problems from our class textbook. While we had done many problems like these before, this was the first time in the year we had actually used the textbook. Like most mathematics textbooks, my students’ books had instructions before the problem section explaining procedures and it also contained answers to odd-numbered exercises in an appendix in the back of the book. To my surprise and consternation, the presence of those instructions and answers fundamentally changed my students’ relationship to their work. The next day, they came to class frustrated and perplexed; many of the students had failed to complete the work or even to attempt many of the problems. They complained that I hadn’t “shown them how to do it” I received concerned calls from over 25% of their parents during the course of the week. All of this came out of relatively easy work that my students had completed successfully many times before, and in a class where impossible problems had become somewhat routine. The presence of the answers and the “right” procedures in the textbook reconfigured the meaning of mathematics for the students. Rather than tasking themselves with investigation and understanding as they had previously done in the class, they reverted to a prior (and deeply ingrained) understanding of their role. The complaints stemmed from their rediscovered belief that their job was to carefully follow instructions, and, in that narrative of math class, I had failed to complete my role as their teacher (which was, of course, to provide the instructions to be followed).

My students so quickly forgot what we had spent the year learning that it was clear they had already been well- trained to think of mathematics in a different, more mechanical way. Our culture exalts STEM education because it is useful to a technocratic society, not because it produces wonder in those who study. If we are to redeem math, then we must relearn an older way of thinking about it, and a different pedagogy that ranks contemplation higher than utility.

While the Greeks recognized that mathematics was useful, they were very careful to distinguish between studying math “like a shopkeeper” and studying in such a way as to enrich one’s own soul. While the two are not mutually exclusive, they believed that we must endeavor to remem- ber that “one ought not to demand a reason in all things alike […], but it is sufficient in some cases for it to be shown beautifully that something is so.” If we allow the pressures of curricular content, standardized testing, and procedural mastery to overwhelm the delight of simply finding and understanding—of recognizing a truth that is not of our own making but nevertheless is true—we run the very real risk of training our students in a great many useful skills but missing the goodness that considered and careful contem- plation of mathematics can provide.

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.”