We were delighted to get to talk to Peter Ulrickson about his new math textbook/teaching companions, A Brief Quadrivium and Teaching the Quadrivium: A Guide for Instructors (both Catholic Education Press, 2023). Ulrickson is assistant professor of mathematics at The Catholic University of America.
Q: How does your approach to mathematics through the quadrivium differ from most modern math instruction, even methods with a liberal arts mindset?
A: In comparison with more standard, computationally-focused mathematics curricula, the quadrivium gives greater emphasis to proof. Consider geometry, for example. Following Euclid, we set out certain simple postulates, first principles that correspond to the intuition we obtain with our senses. From these principles we derive conclusions about triangles and circles and lines, giving clear verbal accounts known as proofs. A different approach, something that can be called “analytic geometry,” emphasizes working with circles and lines and other curves in a plane with coordinates, the Cartesian plane. One can then ask questions like “what is the x-coordinate of the intersection of the two curves?” The familiar quadratic formula falls in this category.
Both approaches to geometry have their place. In practice, though, computation often gets to be too much for the typical student, and the explanatory character of mathematical speech gets lost in rules and procedures. A focus on proving, with very simple objects, means that students engage in fully mathematical activity, the sort of thing professionals do, but at a level appropriate for their ability and experience.
In comparison to specifically liberal arts minded math instruction, I think my books offer a couple things. One is a balance. While Euclidean geometry has an important place, the other disciplines of the quadrivium also have great value and deserve attention as well. Another thing my books offer is the perspective of a practicing mathematician. The classic sources of mathematical learning—we can call them great books—have historical, philosophical, even theological value. First, though, they are mathematical texts.
Q: The classical quadrivium is comprised of four subjects: arithmetic, geometry, music, and astronomy. To most people, though, music stands out as the black sheep here. How is music mathematical, and how does it work within a mathematics curriculum?
A: The sounds that we make with our voices and with musical instruments can be described in many ways. One of the ways we describe these sounds is with pitch, how high or low a sound is. Pitch is comparative; one is higher or lower than another. Beyond mere comparison, though, we can consider whether two different pitches sound pleasant or complementary together. Such consonant pitches turn out to correspond to ratios of whole numbers.
Someone who plays a violin or guitar can play the whole string. That musician can also find the middle of the string and stop it there with a finger so that only half the string vibrates. The pitch made then turns our to be higher in a natural, pleasant, consonant way. It is the octave. The musical interval of the octave corresponds to the numerical ratio 2:1. Other intervals also correspond to simple ratios.
The story is more complicated, though. When building a scale, if certain intervals are taken to be simple, others will necessarily be more complex. This inescapable fact can be explored and explained mathematically, considering properties of prime numbers.
There is more to music than pitch. St. Augustine wrote a treatise on music. His focus was not on pitch, but on rhythm in poetic meter. He had hoped to supplement his rhythmic study with a study of pitch, but other duties intruded and he set the project aside.
Q: How does writing a textbook for students differ from writing an instruction guide for teachers? What was it like to juggle both projects at once?
A: It was quite natural to write the two books at the same time. The textbook, A Brief Quadrivium, is significant because of how the material has been selected and arranged, rather than for any new technical result. As I studied and wrote, I was often thinking of how I would explain what I was doing to various people: to fellow mathematicians, to teachers and administrators of schools, to parents teaching their children at home. The instructor’s guide, Teaching the Quadrivium, is a way to share some of my thoughts and to help promote a pedagogical culture capable of transmitting the quadrivium today.
One wishes to avoid extremes. Old mathematics is not simply a museum piece, wholly separate from us, effectively dead. On the other hand, it is not so similar to current mathematics (and here I speak especially of secondary school and college pedagogy rather than research mathematics) as to be subsumed by it in a mere change of notation. To teach the quadrivium well one needs to find the virtuous mean between these two. I am trying to help teachers do that.
Q: What do you think is the greatest benefit of a liberal arts education?
A: We experience rest and joy when our minds grasp some portion of the order that fills our world. This refreshment is a gift that can encourage and strengthen us as we tend towards an even greater End.
To learn more about Peter Ulrickson and his books on the quadrivium, visit his website.