“But the effect of her being on those around her was incalculably diffusive: for the growing good of the world is partly dependent on unhistoric acts; and that things are not so ill with you and me as they might have been, is half owing to the number who lived faithfully a hidden life, and rest in unvisited tombs.”– George Eliot, Middlemarch
To learn calculus is, at its core, to learn the rules of playing with infinity: the infinitely big and the infinitely small, or infinitesimal. Take the drawing below of a 1×1 square. Its area must be 1.
If I cut it in half a few times, I get the following pieces with the following areas.
I could, in theory, keep cutting my square in half forever. That would give me an infinite amount of boxes, yet my area will remain 1. Each new box gets smaller, and they become so infinitesimal, that I can add them up without my area exploding to infinity (and beyond).
I write this sequence out as ½ + ¼ + ⅛ + … = 1. In a curious twist that takes a bit of calculus to explain, it turns out that the slightly different sequence ½ + ⅓ + ¼ + … ≠ 1, but instead does snowball up to infinity. Even though the terms get smaller and smaller, this second sequence spirals out of control because the pieces do not get quite small enough quickly enough. The tiniest differences in the tiniest of numbers lead to the difference between a sum of one and a sum of infinity. At the infinitesimal level, every one of those terms matters, no matter how meaningless one billionth of a billionth may seem on its own. This is the tension between the infinite and infinitesimal with which we wrestle in my calculus classes.
I think that our roles as teachers can be beautifully captured by this mathematical tension and truth. On the one hand, everything matters. Several years ago, I saw one of my seventh graders crying during study hall after an award ceremony. I took her into the hall (choosing to leave the other twenty-some girls unsupervised for a few minutes at the end of the day), told her she was more than her grades, I was proud of her, then walked back into the room. We didn’t talk about that day for five years, until at her graduation last spring, when she tearfully told me how that little gesture changed her entire outlook on her education and her worth. At the same time I also know that some flippant comments of mine have had an equally detrimental effect on other students. The pressure to make the right decision at the right time can be crippling when I consider how important every moment could be.
On the other hand, it can feel like all the effort in the world leads to no change in students’ hearts and minds. I’ve told students that a negative times a negative equals a positive more times than I can count, and still students forget the rule. Teachers at our school chase down students at lunch endlessly, just trying to get students to sit down and do a ten minute homework assignment, and it is often the same students week after week that fail to turn things in. Every spring, my heart breaks as students choose to attend other schools despite all of the love and attention we have poured into them for years. It feels like we have failed to train their affections to love the truth and the beauty of the education and relationships they have here, and the sting of goodbye mixes with the sting of failure. In short, it feels like all the effort was for nothing.
So which is it? Are teachers the life-changing demi-gods we see on social media posts, or are we Sisyphus toiling away at a meaningless task to be repeated every fall? Neither. Teachers are human beings with an important part to play in the timeless dance between the infinite and the infinitesimal.
Everything a teacher says and does has an effect on his or her students, which means that nearly an infinite number of interactions occur between the teacher and the student over the course of the year. Each of those acts, however, is infinitesimally small, merely a nudge in one direction or another. When the teacher makes mistakes, when he or she brings the student down in some way, there are infinitely many opportunities to bring the student back up. It may take a few days, months, even years, but the power of small, steady words and acts of charity will begin to correct even the gravest of wrongs. The one thing we must never do is despair and give up.
The progress, though, may only be billionth by billionth. Each act feels trivial, but the sum effect can be profound, even if it only occurs years down the road when we cannot see the fruits of our labor. For better and for worse, no act and no word is definitive. Each is merely a part of an unending sequence in the cultivation of the child. A few years ago a student was out of uniform, and I wrote him up, highlighted the part of the student handbook he had violated, and pulled him from study hall to show him how he was wrong. It was foolish, petty, and mean. For at least the next year, he rightfully looked glum and guilty every time he saw me in the halls. A few well-timed jokes, a kind note to his parents, and consistent praise when he does well have changed our relationship, though, and now he greets me with a smile every day.
Every single billionth is needed to complete the square in our picture above. So, too, every billionth of good we can do in “unhistoric acts” is necessary to make our students whole people.
