By Robin Lancaster
I vividly remember sitting in my first Freshman Math on the first day. In high school I had never counted math among my interests nor among my strengths. I often had more questions than my teachers thought were relevant or wanted to answer. For this reason and others I was very worried that math at St. John’s be new and difficult. And, to some degree it was as foreign and challenging as I expected. Gone were the familiar theorems to memorize, tests to study for, and problems to solve. In their place were the great and terrible geometric proofs which I had encountered (and hated) in high school Geometry. In addition to all of this I would have to face these great and terrible beasts standing at a chalkboard in front of the entire class. But, as would come to be a common experience in that first year, the things which I worried about the most proved to be the most rewarding and eventually the most enjoyable. The proofs I ran into in high school were almost exclusively memorization based. It was all about memorizing steps and seeing how closely you could replicate what the teacher had said. I was pleasantly surprised to find that the experience of proofs at St. John’s was completely different.
Beginning with the very first definition it became very clear that we would be doing something else entirely. All of Euclidian Geometry begins with the point. The first definition in the Elements reads: a point is that which has no part. I had an army of questions about this definition. What sort of part does it not have? Are all things that have no part points? Are all points the same because they have no part? I didn’t have high hopes that I would get to ask let alone answer these questions. In that first class my tutor sat down, introduced herself and then asked “so what’s up with that first definition? What is a point?” Three class meetings later we didn’t exactly have an answer but a great many of the classes’ questions had been asked and addressed by their classmates.
Not long after we started on the proofs and that was when I truly fell in love with mathematics and how it is practiced at St. John’s. Seeing how each proof was built upon the definitions and propositions which precede it gave me the sense of building a world from the ground up. I had learned the Pythagorean Theorem along with everyone else, I knew the mantra: C2=A2+B2. What I didn’t know was why and this is exactly what studying Euclid gave me the chance to explore, Euclid not only allowed me to prove this theorem, but also to prove the tools which make such proof possible. You cannot prove that in right triangles the square on the hypotenuse is equal to the squares on the sides unless one first demonstrates the ability to construct a square, or a right triangle for that matter. You can see all of this for yourself in the video below.
When experienced this way math no longer seemed like a series of unconnected points to be memorized but an elegant and complex world which I was building. My classmates and I were masons, creating math for the very first time, our tutor a slightly less confused mason, and Euclid the man who laid the plans for this world. Working at the board, rather than being the worst part of the experience became the best. My classmates and I not only learned math but put it together from the ground up. Some students from the Santa Fe campus took this method of studying and turned it into a YouTube series called Project Euclid. If you are interested in seeing how we do math at St. John’s or just want to reminisce about the Euclid days, check out the Project Euclid channel.
It has been a year and a half since that first class, but I doubt I shall forget how I learned to love mathematics again. Once experienced, the joy of this construction never really leaves.
Whether we are constructing a model of the heavens with Ptolemy or trying to figure out what is up with Cones with Apollonius, I still feel the same engagement and joy which I found in that first discussion.