This is my contribution to Sam Shah’s Virtual Conference on Mathematical Flavors. I was honored, shocked, and thrilled that he asked me to be a keynote blogger, as this is something I’ve been thinking about, on and off, for years. The seed was planted with a comment on Dan Meyer’s blog, pointing out the difference between teaching math and teaching about math. One might reword Sam’s prompt as: We all teach our students math. But what are we teaching them—either explicitly or implicitly—about math?
I teach math because I can’t not teach it.
I have a passion for my subject that borders on obsession—which is both an asset and a liability as a math teacher. What makes it an asset is probably clear. (My highest marks on my student feedback surveys are always from the item, “My teacher is passionate about his/her subject area.”) The liability comes when I am prepping for a class, and casting about for interesting questions to ask. Invariably I will hit upon an idea or question that hooks me, and I am compelled to chase that rabbit down its hole, and then somehow it’s an hour or so later, and I am no closer to having something for my students to do.
Forget anti-drug: Math is my straight-up drug.
I started teaching because I wanted to do whatever I could to help students experience the same beauty and joy of doing mathematics that I do. I have learned a lot in the almost twenty years since, and I have also had time to reflect on what it is about mathematics that attracts me to it so strongly. The answer I keep coming back to is this: Math is freedom and power.
Or, put less tersely: Being able to think mathematically, meditate and ruminate on difficult questions, and persevere through challenges maximizes your freedom, empowering you to do things in your life you might not have thought possible.
(I daresay this is not unique to math. Blake Boles, in trying to define what “education” is in this piece, paraphrases John Taylor Gatto: “An education is the capacity to author your own life instead of merely accepting the one handed to you.” Love it.)
So how do I try to bring this to my classroom? How do I try to maximize my students’ freedom while still accomplishing my goals for them? How do I show them what learning mathematics enables them to accomplish? It is not easy. My students come to me with an already deeply-ingrained notion of what math is, so by the end of the school year, the best I can hope for is—as Sam says in his prompt for this conference—”moving the needle” a degree or two. But Lord knows I try.
The ideas I’m going to share here are by no means ground-breaking. They have been discussed and dissected in the #MTBoS since its inception, but for me they have been the keys to moving that needle over the last few years.
Take students’ ideas seriously.
I get it. We are given only so many minutes with our students each day, and we tend to have very specific ideas and plans for every one of those minutes. So when a student lobs something at us we are not expecting, it is tempting to treat it as an aberration to be dealt with in order to proceed with the plan, rather than an opportunity to explore and consider a new idea. Choosing to engage in the latter does three things. First, it shows the students that they are not mere receivers of information, they are generators of original ideas. Second, it shows them that I value, and am genuinely curious about, those ideas, which in turn helps them be more comfortable sharing them. Thirdly, it models more closely the true nature of doing mathematics: Playing around with new ideas, wondering things about them, and seeing where they lead us.
Doing this well requires not only an excellent pokerface (as Jenna Laib wrote about last week), but a pokervoice. When a student offers an answer to a question I’ve posed, nine times out of ten my response is simply, “What makes you say that?” in as judgment-neutral a tone as I can muster. Try as they might, they are unable to suss out my assessment of their contribution, and are forced to dissect it, carefully consider it, and justify it themselves.
Which leads us nicely into…
Give students authority to consider and evaluate both their own ideas and those proffered by others.
When I refuse to simply give students answers, and instead ask them to explain their reasoning, they learn very quickly that I am not the answer key. When this begins to dawn on them at the beginning of the year, there is usually a moment of panic, as when one is still learning to swim, and discovers that the water they’re in is a little deeper than they were anticipating. They complain. They get frustrated. But once they discover that the waters are in fact safe, and that they can in fact doggy-paddle, they begin doggy-paddling furiously.
My students are not used to being pushed to explain their reasoning. It is not a simple thing to learn—or to teach. It is not something that we can cover in a unit, have a test on, and move on. It is infused into all of our daily interactions in class. It’s the water we swim in. They practice their reasoning constantly without being told that that’s what they’re doing. And while I assess their reasoning formally fairly regularly¹, I am kind in my grading for most of the year. I’m playing a long game, and it’s usually not until the spring that some students will start to openly admit that, as difficult as explaining reasoning and proving a claim is, they actually kind of enjoy it.
Empower students to persevere through challenges.
I’ve found this to be the hardest. To abuse another metaphor, my students tend to pull the ripcord on their problem parachutes moments after they jump out of the plane. There are many factors causing them to do this, but most of them involve pressure of some sort. And the pressure that most negatively affects their perseverance potential is that of time. They have been taught—usually implicitly, via expectations of past teachers—that speed is important in doing mathematics. That being “good at math” means answering questions quickly. I try to change this just by slowing things down in general. Increasing my “wait time” before soliciting student thoughts or questions. Taking a few minutes, when a student wonders something aloud that I hadn’t considered, to go down that road a ways and see where it leads. Yes, sometimes this comes back to bite me toward the end of the year, when I have to worry about “covering” the requisite material. But in the end, it is completely worth it to me. My students are stressed out enough as it is; I want math class to be a place where they feel they can both relax and think deeply.
To help drive home the fact that speed is not king in mathematics, I regularly assign my students problems that are meant to be worked on over the course of a week. They can ask me for help with them, but my “help” typically comes in question form. “Well, what are your thoughts?” “What might you try?” “What else could you try?” “Forget what you’re being asked to find. What can you figure out, using the information you have?” “What do you need to know in order to answer the question?” By the end of the year, the hope is that I become so predictable in the questions I ask them when they’re “stuck” that they will eventually begin to ask them of themselves.
Even when I use these strategies well, it’s not always clear that they are having the desired effect. But every once in a while I get a clear indication that my messages are actually sinking in.
It was March or April of this year. We were two minutes shy of the opening bell ringing, and students were filing into my classroom for Geometry. Mostly freshmen, a handful of sophomores. There was some light conversation about which classes everyone had come from, and how they were feeling about entering this one. One freshman piped up with, “I really like this class, because I can say whatever I want.”
Now, this particular student had already developed a reputation (and not just in my class) for not only saying whatever she wanted, but saying it whenever she wanted. So I gave her a sidelong glance and said, “Could you… maybe… expand upon that a bit?”
She went on to say—though not in these exact words—that she felt comfortable sharing her ideas in my class, even when they were only partially formed, because she knew that they would at least be taken seriously and given careful consideration. It was the clearest indication that I’d had in a long time that the needle was, in fact, moving.
Postscript. Only after writing the bulk of this post did I realize that I came very close to simply writing the blog-post version of Dan Finkel‘s spectacular TEDx talk, Five Principles of Extraordinary Math Teaching. If you haven’t watched it, you must. Regularly.
¹ … and he lands the Triple Adverb!