When we last sat with my eighth grade algebra students in Part 1 of this post, my pragmatic lesson on radical addition had just been hijacked by the Spiral of Theodorus:
An investigation of this beautiful mathematical shape had quickly stirred the imaginations of my students, leading to some thoughtful follow-up questions and more than a few excellent attempts at spiral constructions. Class ended before the “planned” lesson had been taught, but no teacher would have complained; that spiral had sent us into a mathematical Narnia.
But what happened the next day was even more inspiring. That same student who had asked about the cyclist problem, “Why do we have to know this?” walked up to me during study hall and asked if she could have a sheet of my big flipchart graph paper. When I asked why, she explained that she and her friend wanted to try to make a much bigger square root spiral. You can imagine my reaction and my response.
Soon the three of us were strategizing together on what size to make the first triangle, how to determine where on the paper to start the spiral so that the area of the paper would be maximized, etc. The two students quickly became absorbed into their little project, despite the fact that it was not assigned, nor was I offering any extra credit for a completed spiral. And no, there was nothing “real world” about this—well, that is to say, creating a spiral of square roots was not going to help them get a job some day. But this little mathematical engagement may have been the most “realistic” math (read: most true to the nature of math) that these students had engaged all year long.
They worked meticulously on their spiral for 15 minutes of their daily study hall over the next week and a half. On one of these days, when both students were immersed in their work, a fellow classmate walked over and asked them, “Why are you still working on that?” The student who was not too fond of the cyclist problem answered quickly, “Because it’s cool.”
I do not think she found the square root spiral “cool” because it represents a useful application of mathematics to a practical problem likely to be encountered later in her life (although those instances can be very cool). Rather, I think that her soul craves and is naturally attuned to that attribute we call “beauty,” something that is inherent in the very nature of the spiral.
Consider a parallel example. I can introduce ratios to students by giving them a scale drawing of my backyard and asking them to determine how much planting area I have in my raised beds by telling them that one inch equals four feet. Or, I can play a middle C on the piano, record the frequency of that note, then play a C one octave higher, record that frequency, and watch the looks on my students’ faces when they discover that the ratio of the two seemingly random frequencies is exactly two.
Let’s look at Berry’s words again:
“The difference between a path and a road is not only the obvious one. A path is little more than a habit that comes with knowledge of a place . . . As a form, it is a form of contact with a known landscape . . . It is the perfect adaptation, through experience and familiarity, of movement to place; it obeys the natural contours . . . A road, on the other hand . . . embodies a resistance against the landscape. Its reason is not simply the necessity of movement, but haste. Its wish is to avoid contact with the landscape; it seeks so far as possible to go over the country, rather than through it; its aspiration, as we see clearly in the example of our modern freeways, is to be a bridge . . . [Interstate 71, for example,] was built, not according to the lay of the land, but according to a blueprint. Its form is the form of speed, dissatisfaction, and anxiety.”
Interacting with a square root spiral or octaves on a piano puts students into the natural landscape of math, to borrow from Berry’s metaphor. These soul-provoking ideas necessitate a slow pace and a contemplative posture, not unlike a fall hike on a path through a mountain forest.[i] I often find that our discussions about such formative and beautiful ideas never really end; like a truly breathtaking view of creation, those spirals and harmonious notes take root in the students’ imaginations and change how they view the world around them. The backyard planting area problem, on the other hand, while both realistic and extremely practical (I myself did the quick calculation before heading to the nursery), ends as soon as the students find the answer (and this path to the answer is usually an anxious race to see who gets there first, and certainly no one wants to be last).
The methodologies of classical education exhort us to put things that are true, good, and beautiful in front of our students and then to pretty much get out of the way. I think we all agree in theory that this approach works, but sometimes it is helpful and inspiring to see the evidence. Well, here it is:
The finished artwork of these two students, now and possibly forever hanging on my wall, helps to instantiate for me the aims of a classical mode of teaching. That is, when we cast off the shackles of pragmatism and put in front of our students interesting ideas—true, good, and beautiful ideas—the conversation shifts from a scripted exchange between instructor and pupil to a communion among souls, a communion that is fueled by a shared longing for the Giver of all good and perfect gifts. In such a community, transformation transcends information.
Now the reader may rightfully respond at this point, “This is a good story and all, but did those students ever learn how to solve radical equations, or did they just draw spirals for the next two weeks?” The answer is: both. But perhaps the better question is, “What will these students remember about this lesson? What will they take with them for the rest of their lives?” I, personally, cannot remember learning how to solve radical equations, despite being a straight-A math student throughout my academic history. Nor can I recall a time when I had to solve a radical equation during my six years of “real world” life as a petrochemical engineer. However, I do somehow have fond memories of a tessellation project I did in 9th grade geometry—the alternating triangles and hexagons in purple and orange . . .
Sure, there will be times when we just need to jump on the interstate and race to where we are going. But I think that the way home will always be along a meandering path that follows the contours of our soul.
I do not ever see myself reconciling this tension between pragmatic requirements and soul formation in education. Perhaps this push and pull is necessary; perhaps it need not exist at all. Maybe this tension is simply a result of my sinful posture toward knowing. But sometimes I find myself daydreaming about a different sort of education, an education where students will probably still ask, “Why do I need to know this?” but one where a teacher could confidently answer, “Because it will change you. Because it will help form your soul.” What kind of education would that be?
Or, what if a student never felt compelled to ask this question in the first place. What if the transcendent value in what he or she was learning was readily apparent at the very core of his or her being, like when an arduous hike up a mountain path ends in the splendor of a waterfall. What if every day was inspired, because learning cohered with the natural landscape of the soul. What kind of education would that be?
[i] I have also been learning over the past couple of years something that I was never taught in math but that most of you already know: the history of mathematics is incredibly interesting and important. I am finding that “math as narrative” provides a similar appeal to the imaginations of my students as an octave played on the piano. Everyone loves a good story, after all (like the legend of Hippasus’ unfortunate fate after finding the first irrational number).