This started as a paragraph in my “About Me” section, and perhaps reveals the philosophical mush I found myself in while I spent an embarrassingly long time choosing the title of this blog. But as a first blog post, it’s a decent attempt at my current, honest – if wildly philosophical and abstract – feelings toward teaching and math. I’ll get more specific in future posts.
Why “Arbitrarily Close”?
Some of you might already know what this phrase means. Some of you may have never heard it before. It is a phrase in common use in some mathematical circles, and not at all in others. It is a phrase that annoyed the snot out of me when I first learned it. I didn’t understand it, and it seemed infuriatingly self-evident to everyone else in my classes. This made me feel like an idiot. After a LONG TIME and much repeated exposure to the phrase in context, it gradually began to make sense to me. Now, when I hear it, a pleasantly ambiguous yet clearly defined image floats through my head, and the elegance of the idea pleases me. “Arbitrarily close” carries a beautiful freedom of choice, while being constrained by the architecture of mathematics.
For those not familiar with it, here is how I think of it:
In a mathematical sense, arbitrarily close is a phrase used when describing the boundaries of a set of numbers. I have seen it most often used when talking about inequalities, sequences, sets, and infinity. What follows is not a formal definition, mind you. It is the image I have in my head when I hear it.
Let’s say we’re working with the numbers between 0 and 1. Not zero. Not 1. Just the numbers between them. There’s an infinite number of them. That fact is gosh-darn beautiful. It’s also REALLY, REALLY hard to wrap your head around. (This is why I love math. Simple statements, mind-boggling complexity.) Anyhow, here’s our number line:
and we want the number closest to 1. That open circle means we can’t use 1. So we’ll try to pick the thing right next to it. Well… there isn’t such a number. Because if we choose 0.9, we realize 0.91 is closer. Fine, we say, choose: 0.999999. Well, my friend, what about 0.9999991? And down the rabbit hole we go.
This brings us to another mathematical beauty. 1 is our limit. A finite, definite barrier.We can’t get there. By definition of what we’re working with – the numbers between zero and 1 – we are not allowed to use 1. But we can get close to 1! Super close. And then, we can get closer. And after that, we can get closer still! We can get as close as we want. Choice constrained by the architecture of the problem. Beautiful.
Maybe today 0.9 suits our purposes. Maybe tomorrow we’ll need 5 billion nines in a row to suit our needs. How close do we need to be today? It’s our decision.
Sometimes I’m driven mad by a desire to reach 1 – I’m so close, how can I not be there!? It’s exasperating to never reach it. Infuriating.
Some days, though, I’m fine with 0.9. “Close enough”, I decide, satisfied with the approximation.
To me, this is exactly how I perceive teaching. Not being omnipotent, I cannot control all of the “things”. The constraints are already fixed. I don’t pick who is in my class and what mood they are in. I don’t pick their history, nor how they feel about math. I don’t choose the standards I teach. I don’t pick the amount of time I get with students. I don’t choose the physical space I work in. I am constrained. The limit is 1.
Sometimes this lack of control weighs me down so I can barely breathe. I fall into the holes of whining and blaming students or administration. I snap and hurtful things come out of my mouth and through my body language. Students can always tell. I’m not fooling them with a fake smile. It’s not that I don’t love them, but can’t they see how hard my job is?! Gimme a break, kid!
Then I remember that even in these constraints, I have a huge amount of choice in my days. I choose how to interact with students. I choose what questions to ask. I choose how long to let students think, and how long to think myself. I choose what problems are assigned. I choose who I bounce ideas off of. I choose when to admit to students that I screwed up. I choose what questions to answer. I choose what to put on the walls. I choose what debates to foster, and which ones to squelch. I choose when to listen to my students. I choose what space they have to express themselves. I choose how close we need to be to 1 for that particular day, hour, minute, moment.
Sometimes so much choice is crushing. The weight of wrong decisions keeps me up at night. Only 3 years in and I have so many regrets about choices taken and choices not taken. But then someone pulls me back. Or some thing. Or some student. I remember all of the ways that I can help out. All of the reasons I’m here to begin with. The infinite space between the number I chose and 1. So much possibility for redemption. So much possibility to begin with.
I’ll never get there. There’s not a time or a place when any of us get to announce, “I’m the best teacher in the world! I finally did it!” In the last few days I’ve been so heartened by the number of teachers sharing their vulnerabilities, their fears, but also their hopes. I’m not alone. We’re all stuck in the space between – working on being the best we can for our students. To make the space for our students to grow the most, to express themselves most purely.
It’s summer. I know I’ve forgotten that I’m not actually in as much control as I suggest here. I might choose that 0.95 is what we need today, but the class, the weather, or the whatever forces us closer to 0.85. Part of my job is accepting and flexing with our approximation. Though our goal is 0.95, we’re just not yet there. The best teachers I know are those who keep faith and build plans to head back toward 1. I’m probably stretching the analogy here, but you get the idea.
Teaching is a constant balance of a million different things. (And I don’t think that a million is much of an exaggeration.) I am imperfect, and I make mistakes. I will continue to be imperfect, and I will continue to make mistakes. But I also have good days, and make good choices. I am so glad to have a community of teachers who thoughtfully reflect on their craft, share their wisdom, and welcome me into their ranks.
To me, arbitrarily close, encapsulates the things I can control, the things I cannot, the impossibility of this craft, and the beauty of the exploration despite the challenges.