On Talking Points, disidentification, meditation, and the need for a structure

One of the things I have noticed with discouraged math learners and Talking Points — or other disidentification techniques — is that students often express a kind of euphoria afterwards. “That was so much fun!” they will often say (or yell).

This is a result of the disidentification process. They are not accustomed to speaking in their own authentic voices in math class. They have become conditioned to attending math class under what Brousseau called “the didactic contract.” Under the didactic contract — the implied contract to which they have become conditioned — they are required to check their authentic self at the door, with all its attendant messiness, blurting-out, hesitant and half-formed ideas. Instead, the didactic contract demands that they conform to very narrow ideas of what a “good math student” is supposed to do: Sit down. Shut up. Pay attention. Get the right answer. Don’t ask why.

There’s no point in our denying that this is what most math students have become acculturated to. They didn’t make up these requirements themselves. Somewhere along the way, everybody encounters a math learning environment in which these are the expectations.

This situation is what the great Swiss psychoanalyst Alice Miller spoke of in her book, The Drama of the Gifted Child. In an adult-centered, authoritarian society (which most societies are), the social constructs are organized to side always with the adult/parent/teacher instead of with the child. And in fact, as Miller’s work showed, not just *instead of* the child, but at the child’s expense.

This is the general situation for discouraged learners in math classes. Many students perceive early on that their authentic selves are not welcome here. They quickly learn to wear a mask in math class and to pretend to be smart, compliant, and “mathematical” — in other words, to adopt a false persona in math class.

The problem for us as math teachers is that it is not easy to crack through this false self. Disidentification is not an easy or a straightforward process. The psyche adopts these masks as defense mechanisms — and for very good, if outdated, psychological and emotional reasons: fear of abandonment, fear of humiliation, fear of shaming, fear of annihilation.

These may be outdated, outmoded fears by the time a student reaches middle school or high school, but that does not make them any less real or active in the present moment. For a traumatized math learner, they are the most real thing in their world during 5th period.

In their own different ways, the psychologists Eugene Gendlin, A.H. Almaas, and Francine Shapiro have all posited that trauma in the past forms a kind of “stuck place” in the human mind/brain/psyche. Whenever we encounter a stimulus that “triggers” that stuck place, we “flash back” to the moment of trauma and our defense mechanisms lock into place.

This is what makes disidentification so difficult to achieve in practice. A defended psyche is not a receptive psyche. And a student may *hear* that s/he needs to adopt a growth mindset in math class, but s/he hears this message from his or her bunker, thirty feet under ground and behind several feet of concrete protective functions.

Raise periscope. Spot the threat. Lower periscope and retreat.

Evolutionary psychologists consider this fight-flight-freeze response and its replay during anxiety dreams as a most ancient form of threat rehearsal. Knowing what they know from their previous experience, the protective functions of the psyche leap into action and do their best to make sure we remain vigilant and safe from incoming threats. They perceive this to be a matter of survival, which is why they go to such great lengths to make sure we perceive it that way too whenever we step over the threshhold into math class.

So the first order of business in the process of disidentification is to establish trust and to form a safe — and sane — alliance with all learners. If math class is to become a growth mindset place for all students, then it must first be established as a safe place in which to remove our masks and to return to being our deeper, authentic, creative selves.

To make any place safe for the authentic self to come out, it helps to have a structure in place. That way, the structure can provide the psychological and emotional safety (and freedom) in which we can drop down into our authentic selves.

In all forms of mindfulness meditation, this structure consists of three things: a posture, an anchor, and a timed period.

In Zen, we sit on a black cushion in the lotus or half-lotus position (or forward on a chair with both feet flat on the floor). We place our hands on our knees or in the cosmic mudra and we face a white wall. We lower our gaze to a 45-degree angle with the floor, and we anchor our attention on our breath.

Whenever our attention wanders — and monkey mind guarantees that it will inevitably wander — we gently redirect it back to our breathing.

The Vietnamese Zen teacher Thich Nhat Hanh teaches the use of a gatha, or mindfulness verse, as an attentional aid during meditation. With each in-breath or out-breath, one thinks a line of a simple verse:

Breathing in, I calm my mind.
Breathing out, I smile.
Dwelling in the present moment,
I know that this is a wonderful moment.

Which reduces to:

In,
Out.
Present moment,
Wonderful moment.

I’ll say this about Thich Nhat Hanh: you have to be a pretty evolved being to be able to teach this kind of clarity and sanity to the very countries that launched your own into chaos.

We do all of this for the whole timed period, whether it is ten minutes or 45 or an hour. Gradually, with patience and lovingkindness, we learn how to do this for longer and longer periods, until the timed period we are working with is every day for the rest of our lives.

We do this because this is our structure.

To the uninitiated, a structure might seem to be a rigid thing, but that is a misunderstanding, and I will tell you the secret: it is actually the essence of freedom.

It gives our defense mechanisms and our wounded child ego-self-psyche something important to do while we drop down into the vulnerable place where our authentic self is kept safe — beneath all those layers of protective functions, social masks, people-pleasing, snark, and our “on-stage” personas.

The structure makes it safe for a human being to reconnect with that deeper, authentic self.

So it is natural to experience a kind of euphoria afterwards. Our culture generally doesn’t encourage us to connect with our authentic selves, so when we do, many people experience it as a kind of homecoming. Intuitively, we know that it is the source of all our greatest ideas and energy and creative fire. Finally, it is a relief to drop the masks we wear and to just be fully and authentically ourselves.

The Enlightenment poet Friedrich Schiller described this experience of flow as arising from the competing impulses toward being present and toward thinking, which operate in a kind of luminous reciprocity, with their harmonious interaction producing a third impulse which he terms the Spieltrieb (or 'play impulse'):

Irresistibly seized and attracted by the one quality, and held at a distance by the other, we find ourselves at the same time in a condition of utter rest and extreme movement, and the result is that wonderful emotion for which reason has no conception and language no name.

                       — Friedrich Schiller, Twelfth Letter on the Aesthetic Education of Man

When the mind is both fully at play and fully at rest in this way, it is at home. 

And when this experience happens in math class, students are growing and truly experiencing mathematics.

This is the sanest, healthiest, richest, most creative human state I know — and I want all of my students to experience it in my math class. Only then can they connect with the growth mindset and the mathematics that are their birthright.

But the key to unlocking that moment is through structure. And for me, in my math classes, that structure is Talking Points.

Models of exploratory talk from my youth — the NeXT years

In planning the group work morning session, I keep asking myself what I want group work to look like — and more importantly, to feel like — for my students. So far, the best description I have found in the literature comes from Douglas Barnes, by way of Neil Mercer (of Cambridge University) and Malcolm Swan and the Thinking Together project in the UK.

So far, Barnes’ conception of exploratory talk, as fleshed out by Mercer and Swan in their research, has come closer than anything else to what I first experienced in the most creative and effective engineering cultures in my adult life.

Lately I have come to the realization that what I really want to prepare my students for is the kind of passionate, creative, and incredibly effective exploratory talk culture that first electrified me during the three years I worked for Steve Jobs at NeXT.

Steve was a master of exploratory talk skills, though he was definitely stronger on the concept development side of things than he was on the social and emotional skills. But more than anybody else I have ever known, Steve valued exploratory talk. In many ways large and small, he worshipped it. And so did we. That was a big part of how I — and many others of us — justified putting up with the craziness we endured while working for him during that period. In search of the “insanely great,” Steve was open to crossing over into the extreme. You had to really want to be there.

Steve’s primary mode of exploratory talk was what could best be described as “gladiatorial.” You had to be willing to die in the arena — and die over, and over, and over again over weeks or months or even years. If you knew what you were talking about — and were prepared to defend your ideas to the death — then you were equipped to step into the arena. However, you also had to be prepared to get bloodied. The emotional toll was tremendous, and many of the most brilliant thinkers I knew at NeXT were simply not willing or able to go into the ring. They stayed as long as they could and made amazing contributions to the experience while still preserving their souls and their sanity. As I grew up, I began to understand that the price of Steve’s mode of exploratory talk was exclusion. Like him, most of the people who were willing to engage in that exchange were white men. I was unusual in that regard because I was not. Most of the leaders of Apple are still primarily white men.

One of the most powerful things about Steve’s engagement in exploratory talk was they when you were right about something, he would eventually come back and give credit (or take credit himself while in proximity to you). As many others have said, he did not do this with a tremendous amount of grace. He could be awkward and blunt and cruel and manipulative. But he could also be deeply and sincerely celebratory of your best work, and a big part of his genius was in being able to bring together some of the brightest, most intensely creative people in the business — the ones with the best ideas and the most flexible skills and the ability to get shit done. And he was a genius at launching us all into combat.

When I joined NeXT, I knew that I was going there to connect with the people I would be starting other companies with and working with for the rest of my life. That belief proved to be true. To this day, the ex-NeXT network remains my most active and cherished alumni group. I started other software companies with exNeXTers, and I worked with some of those who later took over Apple. We shared (and continue to share) a common framework — a common way of engaging in exploratory talk that is recognizable by us all. It’s a sixth sense about a kind of passionate and engaged exploratory talk in which the participants are fully present, and totally bringing their ‘A game’ to the conversation.

In the years after leaving NeXT, most of us refined our processes of exploratory talk in ways that made the process gentler and more generous, more nurturing. Steve’s way was just too damaging. It also left too many brilliant minds and voices out of too many conversations — conversations that would have benefited from the contributions of people who were less combat-averse than the rest of us.

For my own part, I found that mindfulness, restorative practices and good therapy really helped.

But none of us were ever willing to give up the electric quality of those product development conversations. They were incandescent. They left you hungry for more. After the meetings ended, we would all crawl back to our offices, drained and exhausted. But under the surface, we were all making notes, sketching ideas, and plotting our next pitches.

Hours or days later, somebody would pull you into their office to show you something they’d hacked together on their own time, working through some unresolved part of the central idea. That was how you prepared for combat in the arena — you tested your ideas against the best minds you knew. You forged alliances.

Some parts of this process were hilarious. My friend Henry hacked together a UI (user interface) component out of the AppKit to demonstrate some point he’d been trying to convey. In the last piece of his model, there was a pulldown menu of possible actions this one modal dialog allowed you to select. The last of the possible action options in the menu was often, “Drive an 18-inch spike through my brain.” The standard buttons at the bottom right of the dialog window were ‘Cancel’ and “OK.”

For me, this is the ideal of the kind of exploratory talk conversation I want my students to taste in my classroom. I want them to experience that process of brainstorming that takes you out of your own skin — and even out of your own mind — into a kind of magical space that Neil Mercer has termed “interthinking.” It’s that experience of being part of a Bigger Mind than your own individual, cognitive awareness. Brainstorming your way into truly great ideas takes a lot more commitment to flow and to “allowing” than most cognitive psychologists and theorists are comfortable talking about.

But that’s where all the payoff is.

A dented patchwork circle: new school, new impressions

This was my first week in my new school, which means I've been going through a few simultaneous transitions: (1) from middle schoolers to 11th and 12th graders, (2) from a 15-mile commute to a 1.5-mile commute, and (3) from a high-performing to a very diverse, high-need school.

I could not be more excited.

This first week was challenging because my partner-teacher and I were making a transition we could not inform them about fully until the end of the week. Also, he is beloved, which makes him a tough act to follow. But he is also my friend, so it was good, I think, for the kids to see that even math teachers have math teacher friends and that we are working hard to support them in a difficult transition. We did a restorative circle with Advisory so that everyone could be heard in the process of leave-taking, and we will do a round of circles with everybody tomorrow, Monday, to acknowledge the transition and to embody the process of support.

Our talking piece for circle practice is The Batman Ball — a small, inflated rubber ball with Batman on it that moved around the circle as each participant expressed his or her feelings about our shared situation.

What really struck me was their honesty and their authenticity. They honored the circle and each other. And they were willing to give me a chance. I know I will probably receive some of their displaced frustration and feelings of abandonment over the next few weeks, but they were making positive, honest effort that was moving to witness. For the guys in the class, it was especially hard. Most of them have at least one strong female authority figure in their lives, but for many of them, Mr. T was it — their one adult male role model: a young, whip-smart, kind, funny, warm, math-wizardy hipster with oversized glasses, a ready smile, and a heart the size of the ocean.

"Meetings end in departures," the Buddha said, but the fact that it's true doesn't make it any easier. They're still here, and now with me, but their hearts are going to be hurting for a little while. Plus we have finals coming up.

The other thing that made me happy to see was that they are incredibly capable math learners — more capable than they realize. Our department uses complex instruction pretty much exclusively, which was one of the reasons I really wanted to teach there. These gum-cracking wiseacres some of whom live in situations which are hard for most of us to imagine will sit their butts down in their table groups and do group work. I mean serious, collaborative mathematics.

The fact that they don't yet believe in themselves is a different problem. But that is a workable problem too.

My classroom is across the hall from the Special Ed department's special day class, and they are generous with their chilled filtered water and holiday cheer.

So tomorrow is another new beginning. I am trying to stay open and to notice and not to hesitate as I jump in. I am dressing warmly, drinking lots of water, and making effort to be present with an open heart. Looking forward to seeing what happens next.

Noticing and Wondering as a practice with my 6th graders

When I'm using MARS tasks with my 6th graders, I have found no structure to be more effective in aligning their attentions and energies to the task than the Math Forum's Noticing and Wondering structure.

We kind of go into a "noticing and wondering" mode, in which we are choosing to limit our our monkey mind attention to just plain noticing and we let go of any other kind of attention that comes up during that cycle.

Noticing, in particular, is a quiet and nurturing structure for kids to simply be present with what they notice. We are not privileging noticings or knocking down noticings, we are simply welcoming them as valued and arriving guests.

6th graders love having a structure, so they loved the structure of noticing. Then once we'd heard from everybody,we did a round of noticing. It's powerful when space is allowed to sit with this first round of work.

Look at all these amazing insights they had:

We took a little time to admire this collection. It's a great list!

They even did great a great job of thinking about the value of doing noticing and wondering at the start. The first item from the link is from @fnoschese, whose wisdom even my middle schoolers can grasp.

Doing this task together and then talking about it made students reflect in deep ways about what kinds of growth processes were going on for them.

I just wanted to share this one implementation for anybody who is interested in ways you can use this.

"Wow, what a great question!" or Repeat After Me: A Unit Test is Still a Learning Opportunity"

I have met teachers who refuse to answer questions of any kind at all during tests, and I admit that this puzzles me because from where I sit, those are some of the highest-leverage teaching and learning opportunities I will ever have.

I just hate to waste them.

For they are the moments when I have my students' complete and undivided attention.

And that means they are my best hope for encouraging and guiding students in the process of productive struggle.

During a test, I will happily entertain anybody's question about anything. Really. Ask me anything. I will gladly offer encouragement and encourage their courage because for many students, THAT is the moment at which they are most deeply engaged and present in the process of struggling with their learning.

But I am apparently the most frustrating person in the world because the best answer I will ever give students is to say, "THAT is a GREAT QUESTION!"

I smile and nod and encourage them and urge them to keep going. And at first, they really think they hate me for it.

During today's Math 8 test, kids kept asking questions and I kept answering, "That is a GREAT question! What a terrific insight!" and leaving to move on to the next questioner. "But is this RIGHT?" They would ask, sounding wounded. And I would say, "You are asking a FANTASTIC question! Keep going!" and move right along.

Finally somebody thought to ask me in a tiny and supplicating voice, "Dr. S, is THIS a good question to ask?" And I peered over and looked at their paper and exclaimed, "Yes! That is a super-fantastic question!"

I was certainly the most annoying person in the room, but they are starting to catch on to this whole productive struggle business. Eventually it became a humorous trope. "Oh, yeah — don't bother asking. I'm sure that's a really GREAT question."

To which I would chime in, "Yes — it really IS a super-great question!"

Don't get me wrong — this is NOT an easy thing to do. It takes strength and practice and intestinal fortitude. It will never be featured as "great classroom action." But it is the most precious and valuable thing I know how to offer my students.

I can say this because I have also been on the receiving end of this kind of teaching. It is a teaching about the value of struggle. It is an incredibly precious gift, but nobody can ever explain it to you. You actually have to experience it in order to understand what a profound act of respect it is for the primacy and centrality of your own personal experience to your own personal learning.

I spent plenty of years complaining bitterly about meditation teachers who practiced this kind of bounded containment. But I sat with it. I stayed with it. I learned first to accept it, and then to embrace it. After a lot of struggle, it humbled me. It changed me in ways big and small. It opened my heart and empowered me to discover the value of struggling within my own life's journey, as well as in subjects like mathematics. That kind of preciousness and wholeheartedness is all too rare in America, but I hope that all human beings will at least taste it at some point in their lives.

So even though it was in some ways a crappy day and a frustrating day and an exasperating day, it was also a kind of gift day too. I want to remember that.

Collaboration Literacy Part 2 — DRAFT Rubric: essential skills for mathematical learning groups

I have said this before: middle schoolers are extremely concrete thinkers. This is why I find it so helpful to have a clear and concrete rubric I can use to help them to understand assessment of their work as specifically as possible. I'm reasonably happy with the rubric I've revised over the years for problem-solving, as it seems to help students diagnose and understand what went wrong in their individual work and where they need to head. But I've realized I also needed a new rubric — one for what I've been calling "collaboration literacy" in this blog. My students need help naming and understanding the various component skills that make up being a healthy and valuable collaborator.

My draft of this rubric for collaboration, which is grounded in restorative practices, can be found on the MS Math Teacher's wiki. I would very much value your input and feedback on this tool and its ideas.

I don't want to spend a lot of time talking about how and why Complex Instruction does not work for me. Suffice it to say that the rigid assignment of individual roles is a deal breaker. If CI works for you, please accept that I am happy that you have something that works well for you in your teaching practice.

This rubric incorporates a lot of great ideas from a lot of sources I admire deeply, including the restorative practices people everywhere, Dr. Fred Joseph Orr, Max Ray and The Math Forum, Malcolm Swan, Judy Kysh/CPM, Brian R. Lawler, Dan Pink's book Drive, Sam J. Shah, Kate Nowak, Jason Buell, Megan Hayes-Golding, Ashli Black, Grace A. Chen, Breedeen Murray, Avery Pickford, "Sophie Germain," and yes, also the Complex Instruction folks. I hope it is worthy of all that they have taught me.

Allegory, iambic pentameter, and 8th graders

In 8th grade English we have just started our poetry unit, which is probably my favorite literature unit, and today was probably my favorite lesson of my favorite literature unit.

I had to start by finishing up what I think of as the "poetry bootcamp" section. There are all the basic terms, the mandatory vocabulary, bleep, blorp, bleep, blorp, and a yada yada yada. BO-RING. That is no way to engage 8th graders.

So I took my opening when I got to allegory, which, as I explained to them, is what we call an "extended metaphor," or as I like to think of it, a "story-length metaphor."

Like the fable of The Ugly Duckling.

I am a believer in the power of storytelling and poetry to save lives. They've saved my life many, many times over, and I know many others who've been saved by them as well.

I told them a version of Clarissa Pinkola Estès' version of The Ugly Duckling. I wove the story from the perspective of the bewildered, misfit duckling who cannot belong but who tries so hard to belong until he JUST. CANNOT. EVEN. At which point, he gets driven out of the flock into the landscape of despair.

He wanders through the landscape of despair — through the forest of his fears — until he has reached the end of all that he knows.

Finally, exhausted and hungry, he paddles out on the lake in search of solace and food. As he is paddling around, lost and spent, a pair of magnificent swans paddle up alongside him and ask if they can swim with him.

He looks over his shoulder to see if there is somebody else behind to whom they must be talking. The water is empty.

After many backs and forths, he relents and allows himself to swim with them. And as the sun peeks through the thick cloud cover, the glassy surface of the water turns into a giant reflecting glass, into which he looks, expecting to see his familiar, unlovable image.

But instead, he sees quite another image looking back at him — the reflected image of a third, equally magnificent swan on the lake.

I told them, we all wander lost at some point in our lives, but if we hold on and remain clear about what we are searching for, we will all eventually find our flock, our tribe, our true pack. The people with whom we can be authentic and with whom we belong. Estès talks about "belonging as blessing" as a promise, and I have learned that this is true, even though I always find the needle on my gas gauge quivering around the "E" end of the spectrum by this point in my journey.

On my own path right now, I'm not "there" yet. I don't know where I'll be teaching this time next year, but I do know the shape of this journey, and I understand that now is the moment when I need to redouble my faith in the archetype — even though every fiber of my being is ready to just lie down and allow myself to be eaten by whatever hungry ghosts are passing my way.

I told my students that there are patterns to our experience, just as there are patterns in mathematics and the natural world and in human history. And I think that I told them what I needed to hear for myself, namely, that education and growing up is the process of discovering and learning to trust the patterns that are bigger and greater than our own, fidgety little monkey minds.

Reflection on wallowing after the "Two Faces of 'Smartness'" workshop at the Creating Balance in an Unjust World conference

So yesterday I was at the Creating Balance in an Unjust World conference on math and social justice in San Francisco with Jason Buell (@jybuell) and Grace Chen (@graceachen), and I finally got to meet Brian Lawler of CSU San Marcos (@blaw0013) and Bryan Meyer (@doingmath) in person. They are (of course!) both terrific. I came away so impressed with Brian Lawler — a wonderful math education teacher and researcher as well as a fun guy and a total mensch, in addition to being my friend Sophie (@sophgermain) Germain's mentor. You should definitely follow him on Twitter if you're not already.

He and Jason and I crashed the "Two Faces of 'Smartness'" workshop session yesterday right after lunch, which was beautifully given by Nicole Louie and Evra Baldinger.  It was glorious to do math on the floor with Brian and Jason at the back of a classroom where about fifty math teachers from around the country had crammed ourselves in because we wanted to learn about this and well, honestly, we just didn't care about having to sit on the floor.

What was wonderful about it?

Well, first they are both such amazing, caring, reflective teachers and mathematicians. We were given an Algebra 1-style complex instruction group task and we worked on it deeply, in our own ways, for 15 or 20 minutes. There was so much respect for the others in the group, along with deep listening and amazing mathematical and teacherly thinking.

Some of that wonderfulness was wonderful for me was because of my own issues over the years with shame about my own mathematical thinking processes, which are usually quite different from those of other mathematicians and math thinkers I work with. Even after many years of intensive work, I still have a conditioned habit of abandoning my own thinking in favor of somebody else's — anybody else's — especially if they seem confident about their thinking. I have a sense that this is similar to what our discouraged math students often experience, the ones who prefer English class or music or social studies, because I was one of those students myself.

Jason is a middle school science teacher, so he also has a slightly different way of looking at math than Brian or I do. At one point he suggested that we verify our idea by counting the squares. Brian and I looked at each other dumbfounded for a moment, since that had not occurred to either one of us. I exclaimed, "What a great idea! It's so science-y!"

We laughed, counted, and continued our work together.

There can be such joy in a group task like that, but not if the group's working structure is set up with rigid norms that limit individual students' participation. Frankly, I just hate the "assigned roles" kind of group work because I find that it fails to reflect -- or prepare students for -- the kind of group work that happens in a real-world collaborative setting such as a software design meeting. There, engineers and product marketers and managers come together but are not restricted in their participatory roles. No one is limited to being a "recorder" or a "task manager." One person presents a starting point to kick things off, and from there everybody just jumps in with the best they have.

That's how we were working on the floor together yesterday. It was an organic, free-flowing process, and as a result, both the learning and the mathematical conversation were far more authentic than I've usually experienced in this kind of work.

There were also rat-holes — glorious rat-holes! — we chased down. At one point we had even (temporarily) convinced ourselves of the possibility of a quadratic formulation of the rule, even though we'd been told (by the title of the worksheet) that this was a linear growth function. After a few minutes, we punctured the balloon of that idea, and felt a little deflated ourselves. We'd been working for several minutes straight but unlike most of the other table groups, we had still not called the teacher over for even our first of three check-ins. I hung my head in discouragement. "We are fucked," I said. But then we laughed again, brushed ourselves off, and picked back up where we'd last seen something productive.

The end of the workshop activity involved reflection on what the others in our group had contributed to the process in a method known in complex instruction as "assigning competence." You highlight one of the key competencies that another group member demonstrated and you tie it to a positive learning consequence to which it had led us. For example, I appreciated how Jason had brought in ideas from the real-world thinking of science because it had added rigor and a verification mindset to our process. Brian appreciated how I'd been able to stay with my confusion and keep articulating it in a way that made my process visible and available for investigation.

This pleased me because it is something important I think I have to contribute. I call it my process of "wallowing." I have a deep and self-aware willingness to wallow in my mathematics.

One of the other teachers in the room asked me to say more about what "wallowing" meant, and at first I felt overwhelmingly self-conscious about speaking up. But then I remembered that (a) I was being asked by other math teachers who are passionately interested in understanding different ways of reaching students who have their own unusual relationships to mathematics, and (b) I had Brian Lawler and Jason Buell on either side as my wing men, and come on, who wouldn't find uncharacteristic courage in that situation?

So I told him that for me — and in my classroom — wallowing means learning how to be actively confused and developing a comfort and a willingness to stay present with that confusion while doing mathematics.

In much school mathematics, appearing confused is a frequent and constant source of unspoken shame. And so most people with any amount of self-regard quickly learn how to cover it up and hide it from view. I went through much of high school disguising my confusion and shame as thoughtful reflection. I became a master of avoiding humiliation in class by keeping my confusion well-hidden. If I didn't get *caught* being confused, then I couldn't be humiliated or shamed about it.

Later I would work through my confusion privately so I could show up in class always and only being able to raise my hand about something I felt I understood cold.

So I believe there needs to be a culture of allowing for wallowing in active confusion in our math classrooms. We should not be too quick to dismiss confusion or try to resolve it or spackle over it.  I would even argue we need to consider it a badge of honor and an activity worthy of our time, consideration, and cultivation. The only way to cultivate curiosity is to cultivate an environment that is supportive of wallowing — active engagement and presence in the process of being confused.

It is the deepest form of mathematical engagement I know, and it is thrilling, inspiring, and honoring to be a part of it. When a student experiences that euphoric, lightbulb moment of authentic, personal insight, we can be assured that the understanding it signals is not only deep but also durable. This is true because when you are actively confused about something, you are fully engaged in making your own mathematics. Whether it is a "big" idea or a "little" procedure, these moments of insight are the source of all intrinsic motivation. And isn't this the kind of reflective, metacognitive insight about student learning processes that we are hoping to cultivate?