## Hard Problems and Hints Friday, Jul 11 2014

I have a friend O with a very mathematically engaged son J, who semi-often corresponds with me about his and J’s mathematical experiences together. We had a recent exchange and what I was saying to him I found myself wanting to say to everybody. So, without further ado, here is his email and my reply (my take on Aunt Pythia) -

Dear Ben,

J’s class is learning about volume in math. They’ll be working with cubes, rectangular prisms and possibly cylinders, but that’s all. He asked his teacher if he could work on a “challenge” that has been on his mind, which is to find a formula for the volume of one of his favorite shapes, the dodecahedron. He build a few of these out of paper earlier in the year and really was/is fascinated with them. I think he began this quest to find the volume thinking that it would be pretty much impossible, but he has stuck with it for almost a week now. I am pleased to see that he’s not only sticking with it, but also that he has made a few pretty interesting observations along the way, including coming up with an approach to solving it that involves, as he put it, “breaking it up into equal pieces of some simpler shape and then putting them together.” After trying a few ways to break/slice up the dodecahedron and finding that none of them seemed to make matters simpler, he had an “ah ha” moment in the car and decided that the way to do would be to break it up into 12 “pentagonal pyramids” (that’s what he calls them) that fit together, meeting at the center of rotation of the whole shape. If we can find the volume of one of those things, we’re all set. A few days later, he told me that he realized that “not every pentagonal pyramid could combine to make a dodecahedron” so maybe there was something special about the ones that do, i.e., maybe there is a special relationship between the length of the side of the pentagon and the length of the edge of the pyramid that could be used to form a dodecahedron.

He is still sticking with it, and seems to be having a grand time, so I am definitely going to encourage him and puzzle through it with him if he wants.

But here’s my question for you…

I sneaked a peak on google to see what the formula actually is, and found (as you might know) that it’s pretty complicated. The formula for the volume of the pentagonal pyramid involves $\tan 54$ (or something horrible like that) and the formula for the volume of a dodecahedron involves $15 + 7\sqrt{5}$ or something evil like that. In short, I am doubtful that he will actually be able to solve this problem he’s puzzling through. What does a good teacher do in such a situation? You have a student who is really interested in this problem, but you know that it’s far more likely that he will hit a wall (or many walls) that he really doesn’t have the tools to work through. On the other hand, you really want him to find satisfaction in the process and not measure the joy or the value of the process by whether he ultimately solves it.

I certainly don’t care whether he solves it or not. But I want to help him get value out of hitting the wall. How do you strike a balance so that the challenge is the right level of frustrating? When is it good to “give a hint” (you’ve done that for me a few times in what felt like a good way… not too much, but just enough so that the task was possible).

In this case, he’s at least trying to answer a question that has an answer. I suppose you could find a student working on a problem that you know has NO known answer, or that has been proven to be unsolvable. Although there, at least, after the student throws up his hands after giving it a good go, you can comfort her by saying, “guess what… you’re in good company!” But here, I’d like to help give him some of the tools he might use to actually make some headway, without giving away the store.

I think he’s off to a really good start — learning a lot along the way – getting a lot of out the process, the approach. I can already tell that many of the “ah ha” moments have applicability in all sorts of problems, so that’s wonderful.

Best, O

Dear O,

Wow, okay first of all, I love that you asked me this and it makes me really appreciate your role in this journey J is on, in other words I wish every child had an adult present in their mathematical journey who recognizes the value in their self-driven exploration and is interested in being the guardian of the child’s understanding of that value.

Second: no matter what happens, you have access to the “guess what… you’re in good company” response, because the experience of hitting walls as you try to find your way through the maze of the truth is literally the experience of all research mathematicians, nearly all of the time. If by any chance J ends up being a research mathematician, he will spend literally 99% or more of his working life in this state.

In fact, I would want to tweak the message a bit; I find the “guess what… you’re in good company” a tad consolation-prize-y (as also expressed by the fact that you described it as a “comfort”). It implies that there was an underlying defeat whose pain this message is designed to ameliorate. I want to encourage you and J both to see this situation as one in which a defeat is not even possible, because the goal is to deepen understanding, and that is definitely happening, regardless of the outcome. The specific question (“what’s the volume of a dodecahedron?”) is a tool that’s being used to give the mind focus and drive in exploring the jungle of mathematical reality, but the real value is the journey, not the answer to the question. The question is just a tool to help the mind focus.

In fairness, questing for a goal such as finding the answer to a question and then not meeting the goal is always a little disappointing, and I’m not trying to act like that disappointment can be escaped through some sort of mental jiu-jitsu. What I am trying to say is that it is possible to experience this disappointment as superficial, because the goal-quest is an exciting and focusing activity that expresses your curiosity, but the goal is not the container of the quest’s value.

So, that’s what you tell the kid. Way before they hit any walls. More than that, that’s how you should see it, and encourage them to see it that way by modeling.

Third. A hard thing about being in J’s position in life (speaking from experience) is that the excitement generated in adults by his mathematical interests and corresponding “advancement” is exciting and heady, but can have the negative impact of encouraging him to see the value of what he’s doing in terms of it making him awesome rather than the exploration itself being the awesome thing, and this puts him in the position where it is possible for an unsuccessful mathematical expedition to be very ego-challenging. This is something that’s been behind a lot of the conversations we’ve had, but I want to highlight it here, to connect the dots in the concrete situation we’re discussing. To the extent that there are adults invested in J’s mathematical precociousness per se, and to the extent that J may experience an unsuccessful quest as a major defeat, these two things are connected.

Fourth, to respond to your request for concrete advice regarding when it is a good idea to give a hint. Well, there is an art to this, but here are some basic principles:

* Hints that are minimally obtrusive allow the learner to preserve their sense of ownership over the final result. The big dangers with a hint are (a) that you steal the opportunity to learn by removing a part of the task that would have been important to the learning experience, and (b) that you steal the experience of success because the learner doesn’t feel like they really did it. These dangers are related but distinct.

* How do you give a minimally obtrusive hint?

(a) Hints that direct the learner’s attention to a potentially fruitful avenue of thought are superior to hints that are designed to give the learner a new tool.

(b) Hints that are designed to facilitate movement in the direction of thought the learner already has going on are generally better than hints that attempt to steer the learner in a completely new direction.

* If the learner does need a new tool, this should be addressed explicitly. It’s kind of disingenuous to think of it as a “hint” – looking up “hint” in the dictionary just now, I’m seeing words like “indirect / suggestion / covert indication”. If the learner is missing a key tool, they need something direct. The best scenario is if they can actually ask for what they need:

Learner: If I only had a way to find the length of this side using this angle…
Teacher: oh yes, there’s a whole body of techniques for that, it’s called trigonometry.

This is rare but that’s okay because it’s not necessary. If the teacher sees that the learner is up against the lack of a certain tool, they can also elicit the need for it from the learner:

Teacher: It seems like you’re stuck because you know this angle but you don’t know this side.
Learner: Yeah.
Teacher: What if I told you there was a whole body of techniques for that?

Okay, those are my four cents. Keep me posted on this journey, it sounds like a really rich learning experience for J.

All the best, Ben

## Deborah Ball and Lucy West are F*cking Masters Sunday, Mar 31 2013

I recently saw some video from Deborah Ball’s Elementary Mathematics Laboratory. I actually didn’t know what she looked like so I didn’t find out till afterward that the teacher in the video was, y’know, THE Deborah Ball, but already from watching, I was thinking,

THAT IS A F*CKING MASTER. I F*CKING LOVE HER.

It put me in mind of a professional development workshop I attended 2 years ago which was run by Lucy West. Both Ball and West displayed a level of adeptness at getting students to engage with one another’s reasoning that blew me away.

One trick both of them used was to consistently ask students to summarize one another’s train of thought. This set up a classroom norm that you are expected to follow and be able to recapitulate the last thoughts that were said, no matter who they are coming from. Both Ball and West explicitly articulated this norm as well as implicitly backing it up by asking students (or in West’s case, teachers in a professional development setting) to do it all the time. In both cases, the effect was immediate and powerful: everybody was paying attention to everybody else.

The benefit wasn’t just from a management standpoint. There’s something both very democratic and very mathematically sound about this. In the first place, it says that everybody’s thoughts matter. In the second, it says that reasoning is the heart of what we’re doing here.

I resolve to start employing this technique whenever I have classroom opportunities. I know that it’ll come out choppy at first, but I’ve seen the payoff and it’s worth it.

A nuance of the technique is to distinguish summarizing from evaluating. In the Ball video, the first student to summarize what another student said also wanted to say why he thought it was wrong; Ball intercepted this and kept him focused on articulating the reasoning, saving the evaluation step until after the original train of thought had been clearly explicated. Which brings me to a second beautiful thing she did.

Here was the problem:

What fraction of the big rectangle is blue?

The first student to speak argued that the blue triangle represents half because there are two equal wholes in the little rectangle at the top right.

He is, of course, wrong.

On the other hand, he is also, of course, onto something.

It was with breathtaking deftness that Deborah Ball proceeded to facilitate a conversation that both

(a) clearly acknowledged the sound reasoning behind his answer

and

(b) clarified that he missed something key.

It went something like this. I’m reconstructing this from memory so of course it’s wrong in the details, but in overall outline this is what happened –

Ball: Who can summarize what [Kid A] said?

Kid B: He said it’s half, but he’s just looking at the, he’s just…

Ball: It’s not time to say what you think of his reasoning yet, first we have to understand what he said.

Kid B: Oh.

Kid C: He’s saying that the little rectangle has 2 equal parts and the blue is one of them.

Ball [to Kid A]: Is that what you’re saying?

Kid A: Yeah.

Ball: So, what was the whole you were looking at?

Kid A [points to the smaller rectangle in the upper right hand corner]

Ball: And what were the two parts?

Kid A [points to the blue triangle and its complement in the smaller rectangle]

Ball: And are they equal?

Kid A: Yes.

Ball [to the rest of the class]: So if this is the whole [pointing at the smaller rectangle Kid A highlighted], is he right that it’s 1/2?

Many students: Yes.

Ball: The question was asking something a little different from that. Who can say what the whole in the question was?

Kid D [comes to the board and outlines the large rectangle with her finger]

Kid A: Oh.

I loved this. This is how you do it! Right reasoning has been brought to the fore, wrong reasoning has been brought to the fore, nobody feels dumb, and the class stays focused on trying to understand, which is what matters anyway.

## My Former Students Are Grown-*ss Folks Thursday, Mar 29 2012

When I started out, veteran teachers at my school said to me, “you won’t really understand what you’re contributing until your students grow up and come back as adults.” I didn’t really understand what they were saying because it sounded unfathomably distant in the future.

But I am beginning to find out.

It’s just starting to sink in that the kids I taught in 2001-2, who were then high school freshmen, are now about 1 year older than I was when I taught them. The ones I taught in 2002-3 are 1 year younger.

BANANAS.

Example: I just had an email exchange with one of my 2002-3 students, who is now involved in math education (!) working for the Young People’s Project. This is the second former student I know of to get involved in math education. *Proud.* I would go so far as to say, *kvelling.*

## Dispatches from the Learning Lab: Yup, Time Pressure Sucks Friday, Mar 2 2012

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.

## Vader Wednesday, Oct 6 2010

I’m working with a new tutoring client, and therefore starting at the beginning in training her to stay engaged with the math rather than getting frustrated or trying to read the answer from my reactions to her guesses. The other night, I came up with a new metaphor to help her with this as she was trying to calculate the area of a circle section. I can’t believe I never thought of it before, it’s so obvious.

[N is visibly struggling to unify the geometric and algebraic information. I love it, I feel like I can literally see her brain growing, but she’s getting frustrated.]

Me: You’re going to grow from this.

N [skeptical]: Really?

Me: Yeah. You’re going to figure this out, and then you’re going to understand that you already had everything you need to figure it out. Have you seen Star Wars?

N: No, but I know it.

Me: You’re Luke, this is Vader. You face Vader and then you become a Jedi.

[Long pause while N thinks about the problem, punctuated by occasional exchanges like, “if I divide the circle area by $\theta$, does that give me the area of the wedge?” “I don’t know, make up numbers.”]

N: Oh! 360 divided by $\theta$ will give me the fraction of the circle that’s the wedge.

[4 or 5 second pause]

N: Right?

[N has evidently been watching my face intently for the last 4 or 5 seconds, trying to get external confirmation of her insight. When it isn’t forthcoming, she begins to doubt.]

Me: The reason I’ve been looking out the window is so you won’t try to get a cue off of my face. [Stands up] I’m going to do the more radical version of this and get out of your visual field. [Starts to head out of the room.]

N: You’re not going to tell me if I’m right?

Me: You’re trying to get me to fight Vader for you.

## The Talent Lie Monday, Aug 9 2010

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.

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.”
* 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. 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.

## Talking Openly about How to Do It Better Friday, Jul 30 2010

Last week I somewhat impulsively picked up and read cover-to-cover the new book of an important mentor of mine.

Linda is the principal of the Boston Arts Academy, where I did my student teaching a decade ago, in what I believe was the school’s 3rd year of existence. The book is largely a collection of vignettes from the BAA’s 12-year history. The vignettes have a theme:

Education involves facing difficult dilemmas. The thing that needs to be done is to bring together the people involved, open up the lines of communication, and try to figure out jointly what to do.

Some of these dilemmas are pedagogical, some pragmatic, some political, and some interpersonal. Some are a combination. The community of people involved may be administration, teachers, students, parents, or a combination. But however configured, Nathan is saying this process is at the heart of education: put the hard choice to the community, and keep everyone engaged with each other as you undertake to work it out.

This book was a very refreshing read for me. We are deep in the days of Arne Duncan, Michelle Rhee, Race to the Top, the Common Core Standards, and the tendency among journalists1 to regard the KIPP schools as the greatest thing that have ever happened anywhere in the universe because they have high test scores. Now I have some nice things to say about some of these things. The Common Core Standards in 6th to 8th grade math are an order of magnitude better (i.e. shorter and less concerned with trivia) than the New York State standards have been, and while I have no firsthand knowledge of KIPP schools, I’ve been curious about them in a good way since my student teaching year at BAA, when a fellow student teacher came back from a visit to a KIPP school very excited about SLANT. But what this list is meant to capture is that I can’t escape the feeling that the highest-profile conversations about education in this country, in their frenzy regarding accountability and competition, have totally lost sight of the following facts:

a) Students are people and they have cares and values.
b) Teachers are people and they have cares and values.
c) Everybody involved has cares and values.
d) Education takes place in a community. (Corollary: improving education involves improving community.)

Reading The Hardest Questions… felt like walking into a room full of people who had never lost touch with any of this. Nathan is talking about thinking through educational dilemmas with her staff and students and being guided by what all the people involved value. Stating and working for what matters to her, and asking her teachers and her students what matters to them. It’s absurd that this should feel like a refreshing notion, but to me right now, it does. The Race to the Top funding criteria include a lot about assessments and data that will be used to measure teacher and principal effectiveness, and no encouragement whatsoever for students, teachers, principals or even state superintendents to reflect on what they value.

Another refreshing aspect of The Hardest Questions… is that it doesn’t uniformly make Linda or the BAA look good. (Often – and from firsthand experience they are good – but not uniformly.) The book narrates some play-by-play encounters with some difficult conundrums that don’t have clear resolutions, so it airs some missteps. (Different readers will probably count different moves as missteps.) One of the most pernicious elements of the accountability-and-results orientation in the national conversation about education is that it gives everybody (states, schools, teachers and students) a great reason to hide every mistake. You can’t learn math while you’re trying to hide your mistakes and you can’t learn to teach that way and you can’t learn to run a school that way. You can’t learn that way, period.2

Some specific themes and highlights:

* Schools need to develop a “unifying framework” – what the school stands for educationally. This is not a mission statement that collects dust in an administrative folder but a vision articulated frequently to students of the most important themes in their education. The faculty needs to be involved in developing it. The administration needs to be willing to commit to it in a long-term way. The school community needs to periodically revisit whether and how the school is implementing this shared educational vision. At the BAA, the unifying framework the faculty eventually came to, after 2 years of discussion and debate, is a list of four “habits of the graduate” – refine, invent, connect, own. The idea is that these words are the faculty’s answer to the question, “what we are committed to cultivating in every student?” and that this goal defines the school. Nathan makes a point that she initially tried to have faculty sign on to other lists of words (that to an outsider now don’t look so different), but it turned out to be necessary for the faculty to go through the intense and time-expensive process of answering this question for themselves.

I am suspicious of statements that begin with “All schools should…” But this is one I truly stand behind: all schools should develop and use a unifying framework. The “new initiative every year” model doesn’t work. Teachers need to be involved in articulating the framework, and a school must be willing to commit to the implementation of the framework over the long haul. Finally, I would argue that schools without a unifying framework still have an unspoken one – a de facto assumption of what this school is about. If it were expressed in posters on the wall, these frameworks might be “We Are Failing: Who Should We Blame?” or “High Scores and College Admissions – Everything Else Be Damned!” To honestly answer the question “What does your school stand for?” takes a willingness to ask again and again how your practices are improving, what students know and can do, and how day-to-day realities in the classroom match the ideals you have articulated. pp. 30-1

* Developing a school’s commitment to social and moral values also takes a community-wide process, and this one has to go beyond the faculty to the students. And it needs to be continually recreated, because new kids come every year. Chapter 2 of Nathan’s book describes how the BAA faculty first articulated a group of “Shared Values” in response to a community crisis (a “white power” graffito in the bathroom), and then slowly learned more and more, over the course of a series of other community crises (involving theft, homophobia, alcohol…), about what it would take to make these shared values a part of student culture. Some highlights:

As Shared Values became a way to talk about what was important in our community, and even the way to address some of our rules, a few students suggested that we change our quarterly honor-roll assemblies to be called Honor Roll/Shared Values assemblies. They wanted the school to recognize students when they were “Caught in the Act of Shared Values,” a phrase they coined. Students or faculty could nominate students who had done something to exemplify a shared value. The action wouldn’t have to be a big deal, but it had to be something that everyone could applaud. We have, for instance, acknowledged students “caught in the act” of putting up posters that someone had ripped down, staying behind to help clean up a classroom, bringing in doughnuts for everyone in the class after a strenuous day of testing… pp. 38-9

In the spring of 2005, some BAA music students performed at a local music club… It was a wonderful concert; the house was packed… However, the next day the owner of the club called to report that alcohol had been stolen from her establishment.

Ms. Torres [the assistant principal] gathered all the musicians together, and initially had an awful time getting any of the students to say they had seen anything. Finally, one of the young musicians, Martin, a leader in the band, said to the whole group, “Hey, listen, someone saw something. It will be terrible for our school and our reputation if we don’t figure out who did it and make sure it doesn’t happen again….” Martin spoke fervently, but still nobody talked, not for another few days. During these days, the entire school was buzzing with talk about expulsions and rumors that the music department would never be able to perform outside of the school again. In the meantime, Ms. Torres and security personnel managed to uncover the truth: which students had actually stolen the alcohol, which had looked away but knew what was going on… They were all suspended and the ringleader… was expelled…

Even though this incident only directly involved one group of students, so many students were talking about it that Ms. Torres decided to hold another whole school assembly. She also decided to have students talk to students rather than… expect administrators to chastise everyone. Ms. Torres asked Martin if he would address the student body and explain why this was such a big deal… Ms. Torres explained, “I need you to talk about the larger issues, Martin…” He agreed.

At the assembly, Martin got out of his seat, twirling his drumsticks in one hand. “We all know this school is pretty amazing,” he began. “Sure, we’ve got beefs and there are things that we all think are stupid and try to change. Sometimes we do. I know all you freshmen want to have lunch off campus, for example. Well, maybe you can change that. But, you know, one thing that keeps us together is that we have these Shared Values. Sure, some of us might laugh when Ms. Torres gets on the intercom every morning and tells us to live one of the Shared Values, but it’s cool. We do believe in diversity with respect. Just look around at how many different kinds of people are in here. And passion with – ” And then he held his mic out to the audience like a DJ as they responded, “Balance!”

“Yeah, that’s right,” Martin continued. “And we believe in community with – ” And again the audience responded, “Responsibility.”

“So, like you’ve heard from Ms. Torres, they’re dealing with the students who did this, but I just think we all have got to think about what this means for our whole community and our reputation out there. We live by our reputation as artists, and if it gets tight out there for us, we won’t be performing…”

We didn’t want students to dismiss the incident as “just something that happened to the music majors.” Dumb, destructive behavior like this is common among adolescents… As sad as I was that BAA students had stolen alcohol, and as disappointed as I was that other students hadn’t turned them in, I was proud of our school’s overall response to the incident. Martin’s leadership meant so much to me. It established a norm that respected student leaders could support school values publicly… pp. 48-51

* Great teachers are empowered to be great by the community they’re a part of. The principal needs to work for the creation and maintenance of this community in order to empower teachers to be great. Building a great school involves “transforming a faculty into a professional learning community.”

Success truly begets success… This plays out in Ms. Chan’s [dance] class, but we see it even more clearly in Mr. Ali’s [humanities class], where students are not all here by choice. Mr. Ali can build on Aleysha’s engaged identity as an artist to encourage in her an engaged identity as a scholar. He has listened to her concerts over the years, and he knows she has a gift and love for music. It is his challenge to create the same set of expectations and joys in his own humanities classroom. p. 78

Teaching at BAA is decidedly not a solitary activity. While I have very little influence on what goes on moment-to-moment in Ms. Chan’s or Mr. Ali’s classroom, I can, and do, work on the schedule (the skeletal system of a school) to ensure that teachers help each other, that worries and questions are shared among team members and the entire faculty. Mr. Ali meets weekly with academic and arts colleagues to discuss students and to develop curriculum. At the end of the year, he will spend two days with his team reviewing and critiquing each other’s units and lessons, and creating notebooks on the year’s courses so that they continue to build a collective archive of work.

Mr. Ali and Ms. Chan are not “one-offs” or “the exceptions” at BAA. I tell their stories here as representative of the ways in which our teachers can be successful. As a leader, it is my job to build a school in which all teachers work in teams, and have the time built into their schedules to talk, to visit each other’s classrooms, and to create curricula as carefully and self-critically as artists create their pieces. pp. 80-1

* A school that wants to make progress on the achievement gap needs to have frank and potentially uncomfortable conversations with faculty and students about race.

There are a lot of really compelling passages to quote on this one but it’s already several hours past the time I told myself I would have finished this post. Read the book.

Here is a video of a half-hour talk that Linda and some BAA students gave. (At Google I guess??) I found it much harder than the book to follow thematically, but it’s cool because the students do a performance based on the unifying framework (refine, invent, connect, own) and talk about it afterward.

Here is a review of the book written by a former BAA student for feministing.com.

[1] I’m thinking of Malcolm Gladwell (in Outliers) and Daniel Coyle (in The Talent Code), for example.

[2] As an aside, one of the reasons I think The Wire is such a significant show is its persistent exploration across different urban institutions (school, law enforcement, city politics) of the way that numerical “accountability” incentivizes maintaining the status quo and hiding the dirt rather than digging into the problems and seeking real improvement.

## Despairing vs. Working: Learning Classroom Management and Learning Math Tuesday, Jul 13 2010

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.

## Reflections, Part II: Try It vs. Hear How You Did It Friday, Jul 2 2010

This is a followup to my last post, processing a provocative conversation I had with my awesome friend and colleague Justin Lanier.

I want to try it vs. I want to hear how you did it

In my previous post I was saying I’d kind of like to take an Algebra I class and alternate units between my profound homespun curriculum design (if I may) and carefully group-digesting the textbook, and then at some point start letting the class choose which approach to follow. This gets to something that Justin brought up, which broadens this question beyond textbooks:

You’re presented with a mathematical problem. Unless you’re on the research frontier (and often even then), there are always two things you can do:
1) Try to solve the problem
2) Find out how somebody else solved the problem
Both things are, to paraphrase Justin, essential parts of mathematical experience. Both are options that the two of us sometimes go for.1

When, if ever, do we give our students this choice?

What percentage of the time do we make it for them?

Justin was struggling with this question (what percentage of the time am I making this choice for my students and consequently what percentage of the time am I letting them make it?). In particular, with the sense that the former percentage is quite high, and not feeling entirely right about that. Having now had a month to mull on it,
a) I dig that Justin raised it as a question to be concerned with because now we get to choose deliberately what we want the answer to be and then try to implement that; and
b) depending on the class, it’s fine to make the choice for your students a very high percentage of the time.

More detail about (b) – this is related to another thing on my mind a lot lately, partly due to the conversation with Justin and partly due to its obviousness as a thing to think about:

In what (pedagogically relevant) ways am I similar to my students as a math learner and in what ways am I different?

I have a partial answer to this question, in the form of a principle:

The most important difference among math learners is not who is visual and who is auditory, or who is linguistically-oriented and who is nonverbal, or who is algebraic and who is geometric, or anything like that. These are interesting, useful differences, but they’re not the most important.

It is certainly not who is “fast” and “slow” or any such meaningless and pernicious bullsh*t. (More to come on this topic.)

The most important difference among math learners is, when faced with a problem to solve, an idea to grasp, a theorem to prove, who believes that they can they can solve/grasp/prove it and who doesn’t.

Category I: People who believe they can solve/grasp/prove it.
Category II: People who do not believe they can.

I believe that the best thing math teachers can do for the world is take people in Category II and move them to Category I.

To repeat my question: in what (pedagogically relevant) ways am I similar to my students, and in what ways different?

My partial answer: One difference is that I am in Category I and most of my students are in Category II. I think this is probably the main difference.2

Now, when faced with a math problem and thus the choice above between option (1) (try to solve it) and option (2) (find out how somebody else did), people in Category II always choose option (2). (Why? Clear: because they don’t really believe (1) is an option.) Meanwhile, in order to learn math, you need to be taking option (1) a big percentage of the time. This is true of everybody but especially true of the folks in Category II. How could you ever come to believe you can solve it except by solving it, a lot of times? And how could you do this if you don’t first try to solve it?

Now as I write this I can’t escape the feeling that I’m writing stuff down that is obvious to the point of being tautological. But I think it adds up to something useful in considering Justin’s question. (How often do we choose for our students between (1) and (2) and how often do we let our students make this choice?)

What it adds up to is this: if you teach a lot of Category II folks, you need to do a lot of choosing (1) for them, since they need it desperately and won’t be choosing it on their own. In fact, you need to be all up in their sh*t, not just at the moment they are faced with the choice between (1) and (2) but
*When they have a sound thought you need to point it out to them
*When they have a wrong thought that leads to a productive conversation you have to call their attention toward the productive conversation it led to
*When they see something from a new angle that illuminates it in a new way, you have to make sure the reflexive feeling of exhilaration doesn’t go unnoticed
*When they jointly solve a problem you need to make sure they noticed their own contribution to it
*Etc.
In other words you need to be in control of quite a few elements of their math learning process. Their choice between (1) and (2) is only one of these elements.

So under this circumstance (namely a room full of Category II folks), it’s really actually quite appropriate to be choosing for them – in particular, choosing option (1) for them.

What you don’t want to be doing is choosing option (2) for them when they were ready to choose option (1). I think one of the important sensitivities to develop as a math teacher, along with the ability to recognize at a glance the “I don’t understand but I don’t want to ask a question” face and the difference between somebody thinking about it and somebody being totally stuck, is a sharp ear for the quiet, tentative, tinkly sound of a Category II student who is about to choose option (1). If you hear it, do not intervene.

This all said, one of the things I took from the conversation with Justin was a reminder that although Category I is always a better place to be than Category II, option (1) is not essentially better than option (2). They are both legitimate choices. Category II people need a lot of option (1) and won’t choose it for themselves, so you gotta take control there, but in general there is a time and place for each kind of mathematical experience. Each one’s got something going for it the other one doesn’t:

By far, all the most exhilarating mathematical experiences of my life have followed the selection of option (1). These were the times there was adrenaline involved, so that my hand was shaking as I put pen3 to paper. Nothing in mathematical life can top the peak experience of realizing something.

By the same token, option (1) immediately preceded all my most profound experiences of mathematical power. (Empowerment, if you like.) I pretty much walk through life feeling like I can accomplish anything with enough patience, and solving mathematical problems myself has been maybe the biggest thing that’s made me feel that way.

However, by definition, I had chosen option (2) prior to every time I’ve had the privilege of seeing something beautiful that I would never have thought of myself.

Relatedly, most of those experiences of empowerment I mentioned above involved some option (2) that came before the option (1) and empowered my actually solving something. In an abstract algebra class last year, my professor said, “now I’m going to solve a cubic… with my bare hands.” I didn’t get that sense of power till I did it myself later that summer, but I probably wouldn’t have done it myself if I hadn’t watched him do it.

So Justin’s train of thought feels like it’s raised my consciousness a bit. This is something to be deliberate about. For students to appreciate both options (1) and (2) and have the choice as much of the time as possible feels like a beautiful ultimate goal. But the main thing is just to keep in mind that not only [option (1) vs. option (2)], but also [our choice vs. students’ choice], are pedagogically siginficant arenas, and we should stay aware of this as we navigate them.

One other thing I recollect from the conversation with Justin that I want to record before wrapping this up:

I really love the act of explaining things I think are awesome. But this is not a reason to do it in a teaching situation. The idea is to be pedagogically deliberate about making the choice between options (1) and (2) for your students. If I decide you’re going to listen to me share a solution, I should be doing it because I think it’s what’s best for you, not because I love doing it. I will repeat myself:

The impulse to share something awesome needs to be entirely repressed while teaching. Sharing something awesome should come from a judgement of the pedagogical need, not a desire to share.

(We are allowed to love it once we’ve decided it’s pedagogically called for; but the desire to do it can’t be driving this call.)

Notes:

[1] We didn’t talk about it at the time, but there is a third very important option, which is to work on the problem with somebody else. This is sort of “between” the other two and also sort of qualitatively different. But the two choices above work to define the extremes of a spectrum, so I’ll stick with them for the purpose of this discussion.

[2] It is not the subject of this post but I have to acknowledge what I think is the second most important difference among math learners: when faced with a problem to solve, an idea to grasp, a theorem to prove, who sees value in solving/grasping/proving it and who doesn’t.

[3] Yes, pen. I do all my math in pen. I am not going to tell you to stop telling your kids they have to use a pencil, but honestly I don’t like this pencil-only doctrine. “Put that pen away right this minute! If you don’t use pencil, you won’t be able to hide your mistakes and pretend they never happened!!” This is the message we’re going for?

(If you want to tell me you have a particular kid who really should be using pencil, I believe you. It’s the general principle I’m objecting to.)