I. Prelude

One of the great challenges of teaching math is the fact that many students walk into class with trauma surrounding the subject. One way or another they have absorbed the idea that the difficulties they have had solving math problems say something important and damning about their intellect.

Trying to do math makes them feel stupid.

J, whom I taught as a junior in Algebra I, was a very developed writer and poet. He would talk about math as a mythical dragonlike beast waiting at the end of his quest to destroy him after he had surmounted every other obstacle. A, whom I ran into on the street two years after teaching her, told me that her life would be great if she could just understand math. O, a professional adult in the financial industry who took a workshop with me, looked like she wasn’t making progress by herself at one point during the workshop, so I asked another participant to join her. She ran out of the room. I found her in tears in the hall. She had fled rather than let someone else “find out how stupid she was.”

If they are going to learn anything, the this tragic association needs to be disrupted, and as quickly as possible. I know you have all already read Dan’s lyrical description of the problem and one part of how to take it on. For now, what I want to call attention to is the mechanism by which this association renders it impossible to learn.

The mechanism is this: when you feel stupid, you are not thinking about math. Like driving a car and playing basketball, it is not possible to think about math and feel stupid at the same time.

I am using “thinking about math” in a strong sense here. It is possible to execute an already-known algorithm like the multiplication algorithm while feeling like the biggest dumb*ss in the world, although it is harder than doing it when you’re feeling better about yourself. What it’s not possible to do is solve a problem new to you, think creatively or resourcefully, see a surprising connection or a pattern, notice your own curiosity, or any other type of thinking that would cause you to grow mathematically. What I am claiming, in short, is that the activity of feeling stupid excludes all activities that allow you to grow.

To make this concrete:

In the workshop for adults I mentioned above, I had posed the sums of consecutive integers problem in a fairly open-ended way. (What numbers can and can’t be represented as sums of at least 2 consecutive natural numbers? Why? What else do you notice?) Most of the participants in the workshop were having conversations with themselves and each other along the lines of:

“What’s going on here?”
“Is there a pattern in the numbers I can/can’t get?”
“If you give me a number is there a system I can use to represent it [as a sum of consecutive naturals]?”
“What patterns are there in the representations I’ve found so far?”
Etc.

Here are the conversations O was having with herself before I asked someone else to join her and she ran out of the room:

“Everybody else is having all these insights. Why am I not?”
“What’s wrong with me that I’m not?”
“What will they think of me?”
Etc.

I didn’t realize this by looking at her, although perhaps I should have. I thought maybe she just wasn’t making progress for whatever reason. She is a pro at hiding it (along with all other people who have this type of conversation with themselves). Lots of practice. But the point I’m making here is this:

The conversation that O was having with herself was of a totally different character than the other participants. The thoughts she was having, and the work that she needed to be doing in order to grow mathematically, live on different planets. When students begin to have this conversation with themselves, they have gone to Mars as far as learning math is concerned.

I listened to O talk about how she was feeling, gave her a hug and told her something to the effect that it made me mad to think anyone had ever made her feel bad for taking her own time to explore something. I brought her back to the workshop and partnered her with another participant who hadn’t come up with a whole lot yet (and who was also very empathetic). She let O explain herself and vent a bit; I let this happen for a few minutes and then said I thought it was time to get back to the math.

Maybe you have seen this miracle yourself: when that traumatized person unloads their pain and finds it accepted and not judged, or just plain has the cycle of self-doubt/paralysis/self-doubt interrupted in any way at all, and then takes a fresh look at the problem… the natural dynamics of the process of problem solving take hold and they instantly become a frickin genius.1 Not by everyone’s standard but by the only standard that ought to count: they start to see the problem from new angles. This amazes them. I’ve lost count of the number of times I’ve seen this happen and it’s breathtaking every time. They then often invalidate their accomplishment through an unfair comparison with others, but that first moment of seeing-the-problem-in-new-light is there, available, and needs to be highlighted. “When you said, ‘oh, I could simplify the other side first’ and that opened up a path to make progress… that’s what being a mathematician is. That’s the whole game right there. Looking at what’s there and playing with it and working with it till you get a new angle. There is one secret to ‘being good at math': do that as much as possible.”

O, a reflective grownup, got the lesson in a powerful way. How much of her mathematical paralysis was really entrapment in the self-doubt cycle. How much she was capable of, that she didn’t even know about, whenever she could switch off that cycle and be present to the problem.

The key word there is presence. If you are present to a mathematical question, and the reality it is asking about; in other words if the question and its reality are available to you, vivid for you, there before you to touch and probe; then doing math is the most natural thing in the world, and growth is inevitable.

But you can’t be present to the math when you are busy thinking/worrying/stressing that you suck. This takes your attention away from the actual problem and the process of looking for a solution stays shrouded in mystery.

II. An Analogy

All of this is set-up for what I really wanted to talk about.

In my six years as a full-time public school classroom teacher, I spent a lot of time and emotional energy thinking about and struggling with classroom management. I was, of course, not alone here. It’s a major issue for beginning teachers.2 Everybody knows this.

I learned a fair amount about classroom management in that time, but there’s something important that I don’t think I ever understood, till this year when I worked as a teacher trainer. I feel like I could have accelerated my learning curve immensely and spared myself and my students a lot of pain if I’d understood it earlier. Consider this true statement:

Struggling with classroom management made me feel like sh*t as a person.

My intention is for this sentence to have landed with some echoes in the background, but just in case:

… Trying to do math makes them feel stupid.

Like math itself for so many of our students, classroom management struggles have left many teachers traumatized. And with reason. Math’s power to hurt is based on the perverse culturally taught belief that accomplishment in math is a manifestation of some important inborn intellectual attribute and struggle to understand is evidence you don’t have it. The power of struggles with classroom management to make you feel bad are likewise amplified by the current cultural milieu, in which the idea that teachers need to be more minutely and exhaustively judged is the coin of the political realm. But the fact is that the experience of being treated rudely by a room full of children pretty much bites, whatever the cultural context.

Reasonableness aside, though, just as math trauma paralyzes the growth of the math learner, feeling bad about yourself because your kids aren’t listening to you is an activity essentially different from, and incompatible with, in fact on a different planet than, growing as a classroom manager.

Let me make this point more concrete. This year, as a supervisor for an MAT program and as the math coach at a high school, I had the privilege of witnessing a lot of different people teach and thinking with them about how to improve their teaching. Frequently this role called upon me to help them think about their classroom management. I found myself, to my surprise, with lots of advice. What was happening was that it was much easier to perceive the dynamics of the classroom as a third-party observer who knows what they look and feel like but is not presently involved. If you’ve got at least a few years experience but have never stepped into the classroom of a fellow teacher with the intent to give management advice, do it – you’ll be surprised how useful you are. It’s the essential awesomeness of what not actually being caught up in it lets you see.

What really threw me, in a good way, is that the suggestions I was making were things that by and large

a) I was sure I would have benefited from during my own full-time classroom practice; and yet

b) most of them were in areas I had never thought about. They were like a whole new angle on the classroom. More specifically, they were smaller and more concrete than most of what I had thought about in all those years of stressing about management.

When things went badly in my classroom, and I thought about what to do about it, my questions were most often like:

“How do I convey strength?”
“What’s the appropriate response to insubordination?”

“What’s wrong with me that they don’t listen to me? (and is it possible to fix? probably not…)”

These are big, abstract questions. What I’ve come to understand this year is that this abstract level is not where the answers live. They live in the minute-to-minute, real-time interactions that constitute a class period. They are solid, tangible, low to the ground. A discipline problem would develop, and then boil over, so that I found myself furious with a student or multiple students, and feeling like a failure. I would then ask myself these big abstract questions. In so doing, I would totally divert my attention from the tiny incremental steps by which the problem had built itself, and from the tiny, concrete things I could have done to head it off before the axe fell. I would also make myself feel horrible for no reason. I felt weak, as though the difficulty I was having had been caused by my fundamental inadequacy as a human. In reality, it was caused by a chain of extremely small and concrete failures of technique. These techniques can be taken on and learned one at a time. They are all individually too small to be worth feeling bad about.

To get specific. Here are some of the suggestions I found myself giving to teachers repeatedly this year. They may be individually useful to you if you are struggling with management and recognize your classroom in the situations they are designed to address. But the big thing I am trying to communicate is that these suggestions do not relate to anything it makes any sense for a teacher to feel bad about. They’re just bits of technique. If your class is messing up because you’re not doing one of them, all this means about you is that you haven’t learned this bit of technique yet.

* In the 1-3 minutes following a transition in which you issue an instruction to the whole class, do not converse with any individual kids. Keep your attention on the whole class. Make it your only job to see that your instruction gets implemented.

(I gave this advice, for example, when I saw teachers give an instruction and then immediately begin to help or reprimand an individual kid, while the rest of the class implemented the instruction inconsistently or not at all.)

* If you have assigned classwork and are trying to help the whole class through it one desk or table at a time, stop the work and call the class back together. The work wasn’t ripe for doing yet it turns out.

* Do not communicate disappointment when a student fails to do something you didn’t communicate a clear expectation about. Communicate your vision of how the class should behave before they have an opportunity to fulfill or disappoint that vision.

(This piece of advice was usually coupled with specifics.)

* Do not make capricious decisions about your students’ attention. For example, if you set them to work 3 minutes ago and someone asks you a question that you think deserves the class’s attention, don’t take lightly the decision to interrupt the work to share the question. If you want to be able to direct students’ attention you need to be willing not to ask too much of it.

(This is a piece of advice I could really have used myself.)

Again, the point is not about these specific suggestions, which I gave to particular teachers facing particular challenges that may or may not be yours. The point is that each suggestion connects to a bit of learnable classroom technique that can be taken on one at a time; that there’s nothing here to feel bad about, since each bit of technique is nothing more than that; and lastly that the big heavy questions of self-worth that plague so many teachers struggling with management are really distractions from these techniques. They pull your attention up and out, to the broad and abstract, and carry you away from what is actually happening in your room between you and your students.

Now I want to be clear: it’s not that the individual techniques are easy, and it’s not that you can just learn them by deciding to. Sometimes, the techniques involved get deep into your being. One of the deepest: communicate the intention that your instructions be followed. This bit of technique is totally natural to some teachers before they walk into the classroom. Others (I’ve been one) need to learn and sometimes relearn it, and learning it may not be as simple and external as the other techniques I’ve listed.

The point is that in spite of this, it’s still just a technique. You just learn how to look, sound and feel like you mean it when you tell your students to do something. This skill can be broken down into smaller components that also can be worked on individually: relaxation and confidence in the tone of voice; relaxed posture; steadiness in the body; a steady gaze. Follow-through: the maintenance of all this personal force in the second and the minute following your instruction. Doug Lemov’s “stand still when you’re giving directions” is the same thing. You can get better at each of these components. Because they have to do with deep habits of your body and social M.O., they may be hard to work on. It may help to videotape yourself or work with a coach, mentor or colleague. But the point is just this: there is nothing mysterious in improving these skills. They are nothing more than techniques. Underdevelopment of any one of them, or many of them, is simply something too small and concrete to feel bad about. That heavy burden of self-doubt is ironic because it’s simultaneously an awful experience and an obvious gambit by the lazy-bum part of our brains to distract us from the real job of learning these techniques. (Isn’t being a lazy bum supposed to be kind of pleasant?)

So: the kids are battered by self-doubt because they think struggle with math impugns their intellectual worth. This cycle distracts them from the math. Free them from this cycle and they grow. The teachers are battered by self-doubt because we think struggle with classroom management impugns our worth as people/professionals. This cycle distracts us from the real job of getting better at the techniques that comprise classroom management. Free us from this cycle and we grow.

I hope if you’ve been there that this post can be part of helping you stay free.

* I started to put together the thoughts in this post in some comments I wrote in response to a beautiful post from Jesse Johnson.

* Jesse and Sam Shah, who have been at PCMI the last 3 weeks, have both been writing about teacher moves, and distinguishing teaching from teachers. Meaning, learning to focus on the actions that are being taken in the classroom, rather than on judging a person. This distinction seems to have been introduced at PCMI in the context of looking at video of other teachers, but both Jesse and Sam recognize you can use it on yourself as well. This is closely related to what I’ve talked about here: the realization that just like a kid learning math, getting present to the real, actual, concrete process of teaching both empowers and is empowered by letting go of judging yourself.

* Here’s a 3-year-old post from Dan Meyer drawing an analogy between the process of subdividing our job into small, concrete bits that can be worked on one at a time, and integration (as in, $\int$). Closely related and very cool.

* When I looked up that New York Times Magazine article about Doug Lemov to link to it, I realized that some of the same issues are being dealt with there. Maybe this is part of why (except for acting like Lemov is the first person to wonder how good teachers do their job) that article was so refreshing to me.

IV. In Other News…

The New York Math Circle has their Summer Workshop coming up in a week! It’s about the Pythagorean Theorem and based on talking to the organizer, Japheth Wood, I think it’s going to be both mathematically and pedagogically interesting. (Y’all know this theorem is the greatest single fact in K-8 education. If I may.) The program is housed at Bard College and doesn’t cost very much for a week-long residential thing. (\$375.) Clearly the place to be.

Notes:

[1] Assuming that the problem is at an appropriate level of challenge. Another way to put this is, assuming that the reality the problem is asking about is available to the student. (This could be a physical reality or a purely mathematical one.)

[2] Totally unnecessary citation: “A significant body of research also attests to the fact that classroom organization and behavior management competencies significantly influence the persistence of new teachers in teaching careers.” Effective Classroom Management: Teacher Preparation and Professional Development, p. 1 (issue paper of the National Comprehensive Center for Teacher Quality, 2007), citing Ingersoll & Smith (2003), The Wrong Solution to the Teacher Shortage. Educational Leadership, 60(8), 30-33.