Some work I’m proud of

T was a third grader when all this went down.

At a previous session, I had asked T what she knew about multiplication, and she had told me, among other things, that 4\times 6 is four sixes, and because that’s 24, she also knew that six fours would be 24. I asked why she said so and she didn’t know why. I asked her if she thought this would always be true for bigger numbers, or could it be possible that there were some big numbers like 30,001 and 5,775 for which 30,001 5775’s was different than 5,775 30,001’s. She wasn’t sure. I asked her if she thought it was a good question and she said she thought it was.

So this session I reminded her of this conversation. I forget the details of how we got going on it; I remember inviting her to wonder about the question and note that there is something surprising about the equality between four sixes and six fours. She could count up to 24 by 4’s and by 6’s and mostly you hit different numbers on the way up, so why do the answers match? And would it be true for any pair of numbers? But the place where I really remember the conversation is when we started to get into the nuts and bolts:

Me: Maybe to help study it we should try to visualize it. Can you draw a picture of four sixes?

T draws this –

Tessa 4 6's cropped

Me: Cool! Okay I have a very interesting question for you. You know how many dots are here –

T: 24 –

Me: and you also told me that six 4’s is also 24, right?

T: yes.

Me: – so that means that there must be six 4’s in this picture! Can you find them?

T: I don’t understand.

Me [writing it down as well as saying it this time]: You drew a picture of four 6’s here, yes?

T: Yes.

Me: And that’s 24 dots, yes?

T: Yes.

Me: And you told me before that 24 is also six 4’s, yes?

T: Yes.

Me: So it must be that right here in this picture there are six 4’s!

[It clicks.] T: Yes!

Me: See if you can find them!

At this point, I go wash my hands. An essential tool that has developed in my tutoring practice is to give the student the social space to feel not-watched while they work on something requiring a little creativity or mental looseness, or just anything where the student needs to relax and sink into the problem or question. The feeling of being watched, even by a benevolent helper adult, is inhibitive for generating thoughts. Trips to wash hands or to the bathroom are a great excuse, and I can come back and watch for a minute before I make a decision about whether to alert the student to my return. I also often just look out the window and pretend to be lost in thought. Anyway, on this particular occasion, when I come back, T has drawn this:

Tess 4 6's recircled cropped

T: I found them, but it’s not… It doesn’t…

I am interrupting because I have to make sure you notice how rad she’s being. The child has a sense of mathematical aesthetics! The partition into six 4’s is uglifying a pretty picture; breaking up the symmetry it had before. It’s a kind of a truth, but she isn’t satisfied with it. She senses that there is a more elegant and more revealing truth out there.

This sense of taste is the device that allows the lesson to move forward without me doing the work for her. Her displeasure with this picture is like a wall we can pivot off of to get somewhere awesome. Watch:

Me: I totally know what you mean. It’s there but it doesn’t feel quite natural. The picture doesn’t really want to show the six 4’s.

T: Yeah.

Me: You know what though. You had a lot of choice in how you drew the four 6’s at the beginning. You chose to do it this way, with the two rows of three plus two rows of three and like that. Maybe you could make some other choice of how to draw four 6’s that would also show the six 4’s more clearly? What do you think? You wanna try to find something like that?

T: Yes! [She is totally in.]

At this point I go to the bathroom. I hang out in the hall for a bit when I get back because she seems to still be drawing. Finally,

Me: Did you find out anything?

T: I drew it a lot of different ways, but none of them show me the six 4’s…

She’s got six or seven pictures. One of them is this –

4x6 cropped

Me: Hey wait I think I can see it in this one! (T: Really??) But I can’t tell because I think you might be missing one but I’m not sure because I can’t see if they are all the same.

T immediately starts redrawing the picture, putting one x in each column, carefully lined up horizontally, and then a second x in each column. As she starts to put a third x in the first column, like this,

neat one cropped

she gasps. Then she slides her eyes sideways to me, and with a mischievous smile, adds this to her previous picture:

final cropped

The pieces just fell into place from there. Again I don’t remember the details, but I do remember I asked her what would happen with much bigger numbers – might 30,001 5,775’s and 5,775 30,001’s come out different? And she was able to say no, and why not. Commutativity of multiplication QED, snitches!


Dispatches from the Learning Lab: Yup, Time Pressure Sucks

Continuing the series I began here and here, about snippets of new-feeling insight about the learning process coming from my new role on the student side of the desk…

This one is funny, because I knew it, I mean I knew it in my bones, from a decade working with students; but yet it’s totally different to learn it from the student side. I’m a little late to the blogosphere with this insight; I’ve been thinking about it since December, because it kind of freaked me out. Even though, like I keep saying, I already knew it.

Learning math under time pressure sucks. It sucks.

It sucks so much that I ACTUALLY STOPPED LIKING MATH for about 5 days in December.

I didn’t know this was possible, and I don’t think anyone who’s ever worked closely with me in a mathematical context (neither my students, colleagues, or teachers) will really believe it. But it’s true. It was utterly, completely unfun. There was too much of it and too little time. It was like stuffing a really delicious meal down your throat too quickly to chew, or running up the Grand Canyon so fast you puke. Beautiful ideas were everywhere around me and I was pushing them in, or pushing past them, so hard I couldn’t enjoy them; instead they turned my stomach, and I had the feeling that the ones I pushed past in a hurry were gone forever, and the ones I shoved in weren’t going to stay down.

I had some independent study projects to work on during winter break, and what was incredible was the way the day after my last final exam, math suddenly became delicious again. Engaging on my own time and on my own terms, that familiar sense of wonder was back instantly. All I had to do was not be required to understand any specific thing by any specific date, and I was a delighted, voracious learner again.

Now part of the significance of this story for me is just the personal challenge: most of the grad students I know are stressed out, and I entered grad school with the intention of not being like them in this respect. I was confident that, having handled adult responsibilities for a decade (including the motherf*cking classroom, thank you), I would be able to engage grad school without allowing it to stress me out too much. So the point of this part of the story is just, “okay Grad Program, I see you, I won’t take you for granted, you are capable of stressing me out if I let you.” And then regroup, figure out how to adjust my approach, and see how the new approach plays out in the spring semester.

But the part of the story I want to highlight is the opposite part, the policy implication. Look, I frickin love math. If you’ve ever read this blog before, you know this. I love it so much that most of my close friends sort of don’t feel that they understand me completely. So if piling on too much of it too quickly, with some big tests bearing down, gets me to dislike math, if only for 5 days, then the last decade of public education policy initiatives – i.e. more math, higher stakes – is nothing if not a recipe for EVERYONE TO HATE IT.

And, not learn it. Instead, disgorge it like a meal they didn’t know was delicious because it was shoved down their throat too fast.

In short. The idea of strict, ambitious, tested benchmarks in math to which all students are subject is crazy. It’s CRAZY. The more required math there is, and the stricter the timeline, the crazier. I mean, I already knew this ish was crazy, I’ve been saying this for years, but in light of my recent experience I’m beside myself. If you actually care about math, if you have ever had the profound pleasure of watching a child or an adult think for herself in a numerical, spatial or otherwise abstract or structural context, you know this but I have to say it: the test pressure is killing the thing you love. Its only function is to murder something beautiful.

If you teach, but especially if you are a school leader, and especially if you are involved in policy, I beg you: defend the space in which students can learn at their own pace. Fight for that space.

The Talent Lie

Back in the fall when I was a baby blogger I wrote a discussion of Carol Dweck’s research about intelligence praise. I did this because I think this research is intensely important. However, I didn’t really let loose on the subject with the full force of what I have to say about it. The truth is I was shy, because a) I’d just had a kind of frustrating conversation on the subject with Unapologetic at Jesse Johnson’s blog, so I was wary of being misunderstood, and b) more embarrassingly, I was excited by the positive response to my previous post about Clever Hans and I didn’t want to alienate any of my new audience.

Now I am a toddler blogger. My godson, with whom I spent the day a few weeks ago, is an actual toddler.
My Godson
He is profoundly unconcerned with anybody’s opinion of him, and just blazes forth expressing himself (climbing on things; coveting whatever his big sister is playing with; being turned upside down as much as possible) all day long. I am going to take this as inspiration, and commence a series of posts about the idea of “math smarts” and talent and intelligence more broadly. These posts have two central contentions:

1) People constantly interpret mathematical accomplishment through the lens of math talent or giftedness.

2) This is both factually misleading and horrible for everyone.

Tentatively, here is the table of contents for this series. I may edit these titles, add or remove some, and I’ll add links when I’ve got the posts up. But here’s the plan for now:

I. Why the talent lie is a lie; how to understand math accomplishment outside of it
II. How the talent lie is spread (in pop culture, and inside the discipline of mathematics)
III. How the talent lie hurts people who are “good at math”
IV. How the talent lie hurts people who are “bad at math”
V. How to train students to understand math accomplishment outside of the talent lie
VI. Why the talent lie is so entrenched, even though it is stupid and harmful

I should make more precise what I mean by “the talent lie.” It’s really several variants on a fundamental idea. People who are really good at math must have been born with a gift, for example. That they must be extra smart. That being good at math (or not) is something that doesn’t change over time. That being smart (or not) doesn’t change. In short, that your intellectual worth, and the worth of your engagement with the field of mathematics in particular, is an already-determined quantity that’s not up to you. That’s the talent lie.

Some examples of the talent lie at work:
* Any time anyone has ever said, “I’m bad at math.”
* The “gifted” in gifted education.
* Just about any time anybody makes a big deal about the age by which a young person does something intellectual. (Starts talking, starts reading, starts learning calculus…)

(In that last bullet, the “just about” is there only because of the theoretical possibility that a big deal might get made for a reason other than to prognosticate about the person’s ultimate intellectual worth.)

I give you these examples to show that I am not talking about a fringe, outmoded idea but something very mainstream. I will have much more to say about how the talent lie is manifested in the forthcoming posts.

I expect to spend a long time writing them. This project may take all fall year the next several years decade. I believe the message I’m communicating is vital for our field and important more broadly as well. It’s also a very personal message. Like all urban educators and all math teachers, I have a lot of first-hand experience with the damage that the labels “not smart” and “not good at math” can inflict. But I am also speaking as someone who spent my early years being seen by others, and regarding myself, as mathematically gifted. This was a heady and thrilling thing when I was in middle school, but I became vaguely aware of the complications by the end of high school, and with hindsight it’s clear that it left me with baggage that took a decade of teaching, learning and introspection to shake. So my own journey is a big part of the story I’m telling here.

I will save the detailed analysis for the forthcoming posts, which means that I am going to defer a lot of clarification and answering-questions-you-might-have for later. But I would like now to articulate in broad terms what I believe needs to change.

According to the Calvinist doctrine of unconditional election, God already decided whether you are going to be damned or saved, and did this way before you were born. Nothing you can do – not a life of good acts, not a wholehearted and humble commitment to acceptance or faith – can have any effect. The most you can do is scan your life for signs of God’s favor, and read the clues like tea-leaves to see if you are chosen or cast away. Modern American culture doesn’t buy this doctrine from a theological point of view, but is 100% bought in when it comes to math. When a person performs mathematically, we obsessively look at the performance, not on its own terms, but as a sign one way or the other on the person’s underlying mathematical worth, a quantity we imagine was fixed long ago.

We need, as a culture, to gut-renovate our understanding of what’s going on when we see people accomplish impressive mathematical feats. Likewise, when people fail at mathematical tasks. We need to stop seeing people’s mathematical performance as nothing more than the surface manifestation of a well-spring of mathematical gifts or talent they may or may not have. Relatedly but even more importantly, we need to stop reading the tea-leaves of this performance to determine these gifts’ presence or absence. This whole game is bunk.

Not only is it bunk but it’s a crippling distraction, for everyone – teachers, students, parents, and our culture as a whole – from the real job of studying, wandering through, becoming intimate with and standing in awe of the magnificent edifice known as the discipline of mathematics.

When you step to the gate and present yourself before it, math doesn’t give a sh*t about the particular profile of cognitive tasks that are easy and hard for you at this moment in time, and you shouldn’t either. There are institutions that are very keen to divine from this profile your worthiness to enter, but this is the curtain they hide behind to make themselves look bigger than they are. It’s time to tear that curtain down.

More on its way. In the meantime here is some related reading:

* I Speak Math recently tackled this same subject. I plan on drawing on some of the research she links.

* Jesse Johnson and I had a conversation about this stuff close to a year ago, and she wrote about it here and here. I’ll go into much more detail on these themes in the coming posts.

* While not as credentialed, the Wizard of Oz nonetheless has a fair amount in common with wolverine wranglers. See if you see what I mean.

Despairing vs. Working: Learning Classroom Management and Learning Math

[Virtual Conference on Soft Skills]

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?”
“Can I get this number [as a sum of consecutive naturals]? How about this one?”
“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?”

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?”

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?”

On bad days,

“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.

III. Related Reading

* 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.


[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.