I would like to make a confession: as a teacher, there are some days when I am less inspired than others. Okay, let me be more honest: there are some days when I find that I am simply going through the motions. Blame it on weariness. Blame it on the tyranny of the urgent. Blame it on “a difficult group of students.” There are plenty of excuses.
But almost always the only legitimate cause for blame is my own perspective. Thankfully, when I fall into this type of funk, my students are typically quick to snap me out of it.
On one such uninspired day last school year, I borrowed a word problem from our textbook to introduce the concept of radical equations to my eighth grade algebra students. We read aloud in class:
“The greatest speed at which a cyclist can safely take a corner is given by the formula “s=4√r” in which s is the speed in miles per hour and r is the radius of the corner in feet. What is the radius of the sharpest corner that a cyclist can safely turn if riding at a speed of 30 miles per hour?”
At this point in our reading, I interjected, “So tell me, what should we be thinking right now?” I was hoping for an answer like, “The variable we need to solve for is under a radical sign.” I would have even invited a question like, “Where did the ‘4’ come from?”
Instead, I received this response: “Why do I need to know this?”
I laughed. But the student wasn’t trying to be cute or funny; it was an honest, sincere response, said with a straight face. I believe what she meant specifically was, “Why do I need to know the sharpest corner a cyclist can safely turn at a speed of 30 miles per hour?” And, let’s be honest, that’s a great question.
As the weight of her question began to diffuse into the room, I suddenly realized that—much like that cyclist turning a familiar corner—I had just been going through the motions.
I cannot put the blame of this particular uninspired moment on the textbook—I actually quite like our algebra book. The culpability is rather completely my own, because on days when I am weary, or feeling the pressure of “finishing the curriculum,” or forgetting that math is first beautiful, I tend to slide back into teaching the way that I was taught: we must didactically communicate to our students that math is useful and necessary for survival in the real world.
Dan Meyer, a math teacher and prominent voice in the online math world, has been a refreshing inspiration to me on my journey in education. He encourages teachers to give students math problems that are compelling for being exploratory, rather than pragmatic for being some sort of investment into the “real world.” Meyer has helped me recognize that, more often than not, those “real world” problems either are not actually realistic (and the students are not being fooled into thinking that they are) or, even if they are plausible, the students still don’t find them interesting. (Who cares how two separate investments are going to grow or what percent of my home value my property taxes come out to be? My dad doesn’t even pay me to cut the backyard!)
And there’s the rub, right? When an idea is naturally interesting, the idea itself inspires the student, without the teacher or textbook writer needing to dress it up. And really rich or compelling ideas can be more than just an exercise for the mind; they can stir the student’s soul. Is not the field of mathematics (and even the “real world”) chock full of such ideas? When I catch myself doing it, I must ask myself: why are you fighting against the very nature of math instead of teaching with the grain of the subject?
This tension of pragmatic versus soul-forming education leads me to one of Wendell Berry’s richly instructive metaphors:
“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.”
Berry wasn’t writing about education, but I don’t think he would mind the connection. Does not that last sentence describe the modern classroom?
We shall invoke Berry’s metaphor again later, but for now let’s return to my algebra class, just a few days after our cyclist turned that corner. On this particular day we started a lesson on the addition of square roots by again referring to the textbook. The Spiral of Theodorus (or, square root spiral) appeared at the top of the page:
At this point my deep-rooted tendencies would allow me to pause for maybe a minute or so to comment on the spiral of square roots before delving into the practical part of the lesson. After all, this particular lesson comes near the end of the year, when it is not unusual for me to feel “behind.” But this particular day, for some reason, I felt compelled to pause a bit longer and engage my students in a collective contemplation of this mathematical structure. I asked questions like, “What do you see? What do you like? What is interesting?” To my surprise, just a few simple questions led to a pretty lively dialogue about the figure. My students’ engagement suddenly became self-sustaining, so I thought I would run with it. “Hey, let’s all try to recreate this figure on our paper.” This led to more discussion–questions like: “How shall I start?” and “What units should I use?” and “Does it even matter what length we choose for the base of the first triangle?” Before we knew it, class was over before we had a chance to continue with the planned lesson. But I didn’t care—this little spiral had captivated my students’ imaginations.
And that was only the beginning . . .
(to be continued in Part 2)
 In fact, it may be the best question. Reading between the lines, is she not asking, “What is the telos of an education in mathematics?” Forcing myself to give that question an honest answer has compelled me to challenge some bad assumptions I have carried—if unwittingly—for many years as a math teacher.
 Wendell Berry, A Native Hill (1969), in The Art of the Commonplace: The Agrarian Essays of Wendell Berry