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.

TMC #14 Group Work Working Group Morning Session – Annotated References & Framework

I'm having a lot of fun planning the Group Work Working Group morning session for Twitter Math Camp 2014, and it's time to start sharing.

Here is the background material I'm using for developing the group work morning sessions. Please note that this is NOT required reading!  Recreational reading only! So please don't freak out!  :)

I wanted to give people a sense of the framework and background I'd like us to start from so attendees can decide whether this morning session will be right for them. I also wanted to provide links and titles to valuable materials.

These are listed in order of relevance to the Group Work Working Group morning session — they are not in formal bibliographical form.

National Academies Press, How People Learn (downloadable PDF here)
This amazing free book provides the framework within which we'll consider the use of group work. I am especially keen for us to explore how we can develop and implement tasks that fit within their (approximately) four-stage cycle for optimizing learning with understanding while also fitting with our own individual school and district requirements. In a nutshell, the four stages are as follows:

STAGE 1 - a hands-on introductory task designed to uncover & organize prior knowledge (in which collaboration cultivates exploratory talk to uncover and organize existing knowledge)

STAGE 2 - initial provision of a new expert model (with scaffolding & metacognitive practices) to help students organize, scaffold, & develop new knowledge (in which collaboration provides a setting to externalize mental processes and to negotiate understanding)

STAGE 3 - what HPL refers to as "'deliberate practice' with metacognitive self-monitoring" (in which collaboration provides a context for advancing through the 3 stages of fluency with metacognitive practices)

STAGE 4 - transfer tasks to extend and apply this new knowledge & understanding in new and unfamiliar non-routine contexts

Malcolm Swan, "Collaborative Learning in Mathematics" (downloadable PDF here)
A short and highly readable summary of Swan's instructional design strategy for collaborative tasks, including notes on his five types of mathematical activities that constitute the bulk of the Shell Centre's formative assessment MAP tasks and lessons.

Malcolm Swan, Improving learning in mathematics: challenges and strategies (downloadable PDF here)
An in-depth introduction to Swan's approach to designing and using the kind of rich tasks offered by the Shell Centre and the MARS and MAP tasks.

Chris Bills, Liz Bills, Anne Watson, & John Mason, Thinkers (can be purchased from ATM here)
The richest source book imaginable for ideas for activities to stimulate mathematical thinking. Often credited by Malcolm Swan and Dylan Wiliam.

Anne Watson & John Mason, Questions and Prompts for Mathematical Thinking (can be purchased from ATM here)
The richest source book imaginable for variations on questioning and prompting strategies.

Dylan Wiliam, Embedded Formative Assessment
This book is a gold mine. Don't leave home without it.

Arithmetic of Complex Numbers Placemat Activity - Algebra 2 + Complex Instruction (CI)

Just because you have an all-groupwork and all-Complex Instruction (CI) format doesn't mean you don't need practice activities too.

Our Algebra 2 kids were getting the concepts of complex numbers and complex conjugates, but were still kinda shaky in terms of fluency in working with them.

Based on ideas I stole borrowed a long time ago from the fabulous Kate Nowak (@k8nowak, http://function-of-time.blogspot.com) and the equally fabulous Rachel Kernodle (@rdkpickle, http://sonatamathematique.wordpress.com ), I proposed a placemat activity to my ever-game Algebra 2 teaching team and they dove right in.

Set-Up
We have typical CI four-person table teams set up in each of our rooms, with each person assigned a specific role based on where they're seated at the table. Our roles are Facilitator, Resource Manager, Recorder/Reporter, and Team Captain, although of course, your mileage may vary. Each role has specific tasks they are expected to perform; for example, only the Resource Manager may call the teacher over for a group check-in or a group question (in our program, teachers only accept and answer group questions).

Each table was given:

• two, double-sided "placemat" sheets for doing work in the center of the table
• a set of problem cards (there are four sets, one for each round of play; to simplify clean-up and organization, I printed each round of cards single-sided on a different color of paper, one set per table group. I've got 7 tables in my room, so I made seven sets of cards. I laminated them and clipped them together, but hey, that's just me)
• the sum to which all four answers for any given round should add up

The sum for each round was written on the whiteboard, though it could have been projected via document camera or Keynote/Powerpoint slide.

Objectives
We had mathematical objectives for the activity as well as CI or norms-based, group work objectives. My students in particular needed reinforcement in group work norms and collaboration. Our objectives were:

     Math Objectives

• achieve greater fluency in the arithmetic of complex numbers (including the distributive property)
• deepen understanding of and fluency with the powers of i
• deepen understanding of and fluency with complex conjugates

     Group Work Objectives

• work in the middle of the table
•  same problem, same time (no one moves on until everyone moves on)
• using table group members as resources

The next time I run this activity, I will definitely give a Participation Quiz because the group work norms are so beautifully reinforced in this activity.

How We Ran It
Recorder/Reporter writes the sum in the central oval of the first side of the placemat. Each group member gets a problem card for round 1 (problem a) and works his or her problem on his or her quadrant of the placemat.

When everybody is finished with their problem, the Facilitator facilitates the addition of all four answers. If they add up to the given sum for that round, the Resource Manager calls the teacher over for a "checkpoint" and the next set of cards for the subsequent round of work.

If their answers don't add up to the given sum, they need to work together through everybody's work on the placemat to diagnose what went wrong and where, as well as how to fix it. Then when they've fixed it, they call the teacher over for a checkpoint and the next set of cards for the subsequent round.

Group Work Benefits — Reinforcing Norms
For my classes, the greatest benefits of this activity came from the fact that it forced students to work in the middle of the table, to use each other as resources, and to talk mathematics. Getting kids to work in the middle of the table is the hardest part of CI, in my view, because it goes against the grain of most of their in-school conditioning. The placemat format makes it nearly impossible NOT to work in the middle of the table. And once they're doing that, it seemed like everything else ran pretty smoothly.

I especially liked the fact that this activity created a context in which students experienced an intellectual need for the using the rules of arithmetic for complex numbers and for the powers of i. It was situationally motivated, but extremely targeted.

Sums for Each Round 
The sums for each round are as follows (if you find an error, please speak up):

• Round 1 (problem a):   26-73i
• Round 2 (problem b):  0
• Round 3 (problem c):  165
• Round 4 (problem d):  2 – 48i

PDF Files for the activity
These are available also on the Math Teacher Wiki on the Algebra 2 page. If you haven't visited the Math Teacher Wiki, you don't know what you're missing.

• Complex Numbers Arithmetic – Problem Cards
• Complex Numbers Arithmetic – Placemat Master

Intro to Quadratics — from "drab" to "fab" (or at least, to something less drab)

Recently, I created a new anchor lesson for my Algebra 1 quadratics unit. I found that, while I really liked the sequencing of activities and questioning in the NCTM Illuminations lesson on "Patterns and Functions," I found their situation and set-up simultaneously boring, contrived, and inane.

Actual photograph of San Francisco monkeys

hosting a tea party in the wild

As is so often the case, I find that a certain, judicious sprinkling of silliness and fun in the set-up can really liven up the lesson. A certain amount of contrivance is necessary in many activities, even those that are based on "real-world situations." So why not stretch the real world to make it conform to the needs of my algebra students?

The Made To Stick elements are all here: multiple access points are provided through manipulatives, storytelling, and humor.

My student investigation sheet, Table for Eighteen... Monkeys is available on Box.com. A PDF of the Table Tiles master is available here on Box.com
here.

Tiny plastic monkeys sold separately. :)

UPDATE: Worksheets now also on the Math Teacher's Wiki, at http://msmathwiki.pbworks.com/w/page/55614036/Algebra%201#view=page

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?