## Friday, October 24, 2014

### On the importance of scaffolding within the zone of proximal development

I don't know about you, but my experiences with even the best Shell Centre tasks (their Formative Assessment Lessons, or "FALs") have been hit or miss. These are among the best anywhere, and while I generally love the ideas of their tasks, in practice, I almost always need to tweak their implementations to make them work in my classroom and with my students.

I wonder if this is not inevitable, given the highly customized nature of scaffolding.

I experienced this phenomenon yet again today when I used their Ferris Wheel task, which I brought out for my extremely able but easily discouraged Precalculus students:

http://map.mathshell.org/materials/download.php?fileid=1252

The purpose of this task is to jump-start students' understanding of modeling and graphing trigonometric functions, using the movement of a ferris wheel cart together with some scaffolded information. The pre-assessment task scaffolded the process well, except for the crazy way they laid out the equations. But then when I ran the card sort task in my 1st period classroom, it was a complete trainwreck. The changes in set-up seemed to yank the problem out of the zone of proximal understanding and launched it into the stratosphere. When something is new, my students are easily thrown (and discouraged), so when they shifted from providing the period to providing the number of revolutions in some arbitrary number of minutes, my students' heads exploded. There was a domino effect of cascading fear and consequences — the equations are set up strangely and are given in degrees rather than radians (after all our hectoring!), the vocabulary changed from revolutions to rotations, etc etc etc.

So during my prep, I made new versions of the cards — ones that would allow my 5th and 7th period classes to recognize the new material and to connect it to things that were still strange, but at least strange in ways they could notice and recognize.

The differences were dramatic. 1st period groups were upset because they were unable to make what they considered meaningful progress on the task (in spite of excellent mathematical conversations) and left feeling unsteady and discouraged. Here is the work from an especially confident and capable 1st period group:

My two later classes, on the other hand, were able to attack the task, making meaningful connections, spot patterns, and really solidify their understanding. Here is the work from an often-cautious and un-confident (but still very capable) end-of-day group:

All in all, it was a good reminder to me to avoid changing horses in mid-stream when I am trying to help learners build conceptual understanding through good scaffolding!

### Slogging through the middle – in praise of wallowing

in praise of wallowing in the middle of stuff -- marinating, versus just discovering

As much as I love discovery learning, there is a dirty little secret I really hate to have to admit: discovery is, in some ways, a cheap thrill.

It's easy to get excited about something new. It's harder to get enthusiastic for the middle. And this creates problems when it comes to developing fluency and deep conceptual understanding. I think that's why my students (who are all superb students, probably in the top top tier of all public high school students) run into troubles in math.

Sustaining enthusiasm through the MIDDLE of mastering something that is conceptually very difficult is what's really hard. But it's also where there is the greatest payoff.  Getting students over their resistance and into flow is where situational motivation and projects can work wonders.

Also wallowing.

Sometimes this means helping students learn how to practice — how to develop what I tell them is called "mental motor skills." With my Geometry students this week, that amounted to  giving everybody an index card first thing every day and playing "Let's Pretend" (i.e., let's pretend this is a quiz) and write down the five ways we know of proving lines parallel. I set the timer for 90 seconds and called out, "GO!." We all wrote down the five techniques we now know — corresponding angles, alternate interior angles, same-side interior supplementary angles, both lines parallel to a third line, and both lines perpendicular to a third line.

We traded and "graded" each other's index cards, remembering to find something positive that others did and circling their errors.

Then I yelled, "Flip over your card and do it again."

The first day we actually did this four times. I handed out new index cards and offered to destroy the evidence of any previous mistakes. And then we did it again — two more times.

With my Precalculus students, I had them flesh out Quadrant 1 of the unit circle on their index card.

And again 3 more times.

From years and years of piano training, I'm used to having to practice a new figure multiple times. I know how to repeat it multiple time across multiple different days so I can burn it into my mind and body.  But many students seem to have missed that learning episode.

So I am giving it to them now.

With my Precalculus students, we then practiced mental event rehearsal through guided imagery afterwards. "Close your eyes and visualize Quadrant I. Find pi over 4 and visualize a point there on the circle. Picture its cosine and sine. Now go back to (1, 0) and travel to pi over 4 in the NEGATIVE direction. Where is the traveling point now? What are its cosine and sine? What is sine of pi over 4?"

"New point: now visualize 2 pi over 3. What is the cosine of 2 pi over 3?" I pause. "Now go back to (1, 0) and travel to NEGATIVE 2 pi over 3."

Over and over, to help them learn how to engage their will in the service of their own best interests.

### The Secret of Flowchart Proofs #officesupplies

Two words for you:

Tiny. Post-Its:

UPDATE —

OK, so maybe there is still a tiny bit more to say about this.

One of my Geometry colleagues and I have students who are still struggling with two-column proofs (which are required for us), so we decided to add in Flowchart Proofs. He went first, introducing them in his 1st Block class, and I introduced them in my 2nd Block class, except... being obsessed with office supplies, I decided to chop up some tiny Post-It notes into REALLY tiny Post-It notes (i.e., I cut them into halves). This way they fit better into students' notes and really gave the idea of representing a single idea.

They also dramatically simplified the process of rearranging steps if you realized you had not ordered them in the most effective way.

At first I had students write their "Reason" underneath the tiny Post-It with their "Statement." But then I realized it would be better if the "Reason" were written at the bottom of the tiny Post-It. That way, if you have to move anything around, there is much less to erase.

Once students are satisfied with their flowchart, they can draw in arrows and add any Latinate flourishes from which they believe their proof will benefit.

## Sunday, October 19, 2014

### Surfacing and studying studying misconceptions via Talking Points

In a class of 36 students, where it can be, shall we say, difficult for me to do formative assessment on every student every day, the Talking Points structure gives me a great way to surface and deal with student misconceptions by getting students to surface, discuss, and correct them.

Par example...

Our sequencing for Geometry has gotten totally screwy this year because of some new district requirements around the Common Core. Having learned how to do all of the basic constructions, we are now finally approaching the unit test on parallel lines and their angles. There are so many possible errors in understanding that can happen around these, I wanted to create a group work activity to address them. I also want to change groups up this week, so I am using this as community-building as well.

I've discovered it is a good idea to weave together community-building statements, growth mindset statements, metacognitive self-monitoring statements, and Always / Sometimes / Never statements. The Talking Points structure lets me accomplish multiple goals simultaneously, which is something I need to do with such big classes.

An editable version of this set of Talking Points is on the Group Work Working Group (#GWWG14) wiki page on the TMC14 wiki.

## Thursday, October 9, 2014

### Constructions Castles

Einstein was right — imagination IS more important than knowledge.

I used to think of that quote a lot as I walked the Princeton campus, through quadrangles and past trees that — legend had it — he had crashed into on his bicycle while he was lost in his thoughts.

I passed his house on Mercer Street every week on my way to music lessons, and I wondered if his sister had not had the door painted it bright red so that he might notice it and not bump into it.

When I realized I would have to teach both constructions AND proof this year in Geometry, I first thought of them as a Big Obstacle. But seven weeks in and I have fallen in love with constructions. I created this Constructions Castle project to give students plenty of practice doing constructions while also giving them a chance to develop their understanding of how shapes and angles fit together.

I think their work speaks for itself.

Files are on here (zip file with all tools).

UPDATE (10/22/14):

The more students do this project, the more awesomeness is revealed:

## Friday, October 3, 2014

### Formative Assessment PLUS Practice Activity: "Lightning Rounds" Review With Whiteboards

I've been looking for a way to do formative assessment that also doubles as a practice activity, and I seem to have stumbled on something good with my new "Lightning Rounds Review With Whiteboards activity. It uses some of what I appreciate most from both Steve Leinwand and Dylan Wiliam.

We have large-ish whiteboards — enough for one per table group — in our classrooms. Since I've also got around 36 students in each of my classes, I've been drowning with my older methods of checking for understanding. With 175 total students, even a 1-minute-per-exit-ticket assessment can take ages to go through — plus all the time it takes me to make it up.

This activity has been working much better. It also takes less prep up front and allows me to engage more deeply with the students and/or groups who need more attention. Assessing 9 tables' work at a glance is much more manageable and less exhausting than trying to flip through 36 pieces of student work. And of course, that is better for everybody.

Here is how it worked today.

In Precalculus, we're working on evaluating trig functions of real numbers on the unit circle. Students and groups are strongly encouraged to use their unit circles and notes to help them gain confidence and fluency.

Each table group gets a board, three markers, and an eraser. On the document camera, covering the non-current rows with a folder, I reveal one row of problems, which contains three problems: (a), (b), and (c).

Students discuss and work the problems, and when they are done, they hold up their table's whiteboard for a "check-in."

I glance around the room and check three answers at a time, calling out, "Table 2, you are checked in and correct!" "Table 6, you are checked in and correct!" "Table 1, you need to reexamine part (b)."

Because there are three problems in each "round," there's plenty of time for groups to make an error, assess their work and reconsider, and re-present their findings. Because there are only nine tables reporting in, it is manageable from both a teacher and a student perspective.

Plus, of course, since there is a group whiteboard and markers to play with, those groups who finished quickly and received their checkpoint are happy to doodle or play tic-tac-toe or offer funny editorial comments or cartoons while other groups receive some attention or extra time.

Last night was our Back To School Night, and student clubs were selling food as fundraisers for their clubs. Two of my students discovered that my classes' parents and I were helpless in the face of their delicious goodies. At one point, I had told them, Look, I don't have cash; my purse is locked upstairs in my desk.

Around the 6th or 7th round, Table 7 included an editorial comment on the side of their board: "We heart Dr. S! Also, you owe us \$2. :)" While everybody started working the next round of problems, I unlocked my purse out of my closet and paid my debts. Some good laughs were had by all, and it was a nice piece of mathematical community-building.

The files I used are on the Math Teacher Wiki.

All in all, an easy, effective practice activity, with embedded formative assessment and community-building built right in. Not bad for a too-hot Friday!

## Sunday, September 21, 2014

### Talking Points and Classroom Community

I am still drowning in papers to grade, but I wanted to write a quick post before the week got too far away from me. I had my first-ever bomb threat this past week, and it rattled us all.  I grabbed my iPhone, my MacBook Pro, and my math pencil and escorted my class out of the building.

I had to make my way home with no keys, no purse, and no money. Just carrying my iPhone and my laptop and my trusty math pencil.

As I battled public transit to make my way home (with no money – just explaining where I'd been and what had happened), it occurred to me that everybody was going to be nuts the next day. So I started writing some new Talking Points for us to do about the events of the previous day.

One of the things that becomes clear with training and time is that as a teacher, my job is to provide an emotional "container" for the experiences we have in my classroom. I decided to use Talking Points as a way of enabling students to process their experience within the structure we've been building for several weeks now.

My students' responses and respect for the structure blew my mind. And they made it possible for us to lose only one day rather than several. And by Thursday, our classroom community had returned to a steady enough state to move ahead.

## Sunday, September 7, 2014

### GEOMETRY – a Jane Schaffer-ish approach to teaching students' first two-column proof

I'm required to teach two-column proofs in Geometry, but having also been trained as an English teacher, this has never seemed like a problem to me. In fact, if anything, the activity I used on Friday seemed to scaffold the process using students' existing knowledge better than anything else I've tried before.

My design process was as follows: because of their ELA and writing backgrounds, students already know far more about constructing an argument in words and statements than we math teachers often give them credit for knowing. All the major writing curricula, such as Jane Schaffer and Six Traits, provide scaffolded methods for teaching students to make claims and to support their assertions with evidence and interpretations that connect that evidence to their claims through interpretive statements. Indeed, the Jane Schaffer method, in particular, has a very lovely scaffolded process (which I've extended in the past) to bridge students' metacognitive processes about their writing, taking them from a place of very concrete thinking to one of considerable abstraction.

So why not use this same kind of process for proof?

Instead of having students merely "fill in" the reasons for the statements in their first proof (in our curriculum, that's the Midpoint Theorem), I created a task card with instructions and materials for creating a "working poster" (an idea I have adopted from Malcolm Swan) of a two-column proof. They needed to set it up the way we'd done it the day before (two columns, Statements and Reasons), and then they would need to (a) sort their cut-out statements from the task card into a correct order (more than one order is possible), and (b) use their notes and discussions to give the justification or reason that permitted them to make each of these assertions in turn.

The richness of their conversations blew me away. They also confirmed my intuitions that (1) math conversations and projects can indeed draw on students' existing competencies in argumentation that they have developed in their English and Social Studies classes (indeed, many relished the opportunity!), and (b) it is indeed possible to create intellectual need (see Guershon Harel and Dan Meyer) for definitions, postulates, previous theorems, and propositions from algebra through situational motivation.

This activity turned two-column proof into a reasoning and sense-making activity that exposed and built on prior knowledge instead of invalidating it; created what Swan calls "realistic obstacles to be overcome"; turned students' notes into a valued and valuable learning resource; and used higher-order questioning, as opposed to mere recall.
I realize I have not referenced the van Hiele levels here, but that is, in part, because I think I may be kind of bypassing some of their assumptions. I'm not at all sure about this, though, and I would welcome better-informed thoughts and thinking about this in the comments.

RESOURCES:

Task card for intro to two-column proof: 1-5 intro task Sorti…o 2-Column Proof.pdf

Editable Word doc: 1-5 intro task Sorti…o 2-Column Proof.doc

Original editable Pages doc: 1-5 intro task Sorti…2-Column Proof.pages

## Monday, September 1, 2014

### What do you do after Formative Assessment reveals a gaping hole in understanding? More Talking Points, of course. :)

My Geometers took the opportunity to inform me through their Chapter 1 exams that they really don't get how angles are named. So this seemed like a perfect opportunity for more Talking Points, of course. :)

This time I'm giving everybody a diagram of a figure that the Talking Points refer to. They will have to do some reasoning about naming angles in order to do the Talking Points. They love doing Talking Points, but they mostly like coming to immediate consensus. Hopefully this will throw a monkey wrench (so to speak) into those works.

Here is the Talking Points file (they print 2-UP) and here is the set of diagrams (they print 6-UP) to use together for this lesson/activity.

More news as it happens!

## Sunday, August 31, 2014

### PRECALCULUS: Transformations of Functions Speed Dating

I know I am going to take some grief for this statement, but I have learned it to be the truth. There are some things that, once discovered, you simply need to memorize — and practice. This is about mental (and often physical) motor skills. Becoming fluent at some things requires more or less practice for different people, but for most of us, it does require something.

I come by this knowledge honestly.

I have played the piano since I was three or four. In growing as a pianist, there are conceptual learnings, procedural learnings, and contextual learnings. Technique, concepts, and contexts. To play well, you need first to become proficient and fluent technically. That is why there are so many established sets of technical figure exercises: Hanon, Czerny, and many others. The same is true for the violin (Schradieck, Kreutzer, Hrimaly) and every other instrument under the sun. If you want to be capable of playing the advanced works in your instrument's repertoire, your fingers and body need to know how to flow over the keys or strings in every known and familiar situation (scales, arpeggios, finger-crossings) so that you can summon them automatically in your pursuit of unknown and unfamiliar territory in the literature of your instrument.

But working your way through all of the possible finger movements will only make you a master technician. Doing only finger exercises will not give you a very deep understanding of music. It simply builds the finger and muscle memory you will need to face different and difficult technical situations and requirements as you dive deeper and deeper into the piano repertoire.

In other words, technical proficiency is necessary, but not sufficient to become a strong and capable musician. Without technique, you cannot progress.

Beyond technique, there are concepts to master — harmony, counterpoint, and composition, as well as music history. If you don't understand the basics of scales and harmonies and chord progressions in Western music, you will lack the basic musicianship that is required to make sense of the piano repertoire. You need to understand how melodic lines or voices can be woven together using harmonies and rhythms and chord progressions to build a piece of music with coherent and reproducible grammar and syntax that can be both encoded and decoded by others. Without the foundational concepts, your attempts to interpret and play the repertoire will be incoherent.

Finally, there are contexts — historical, cultural, interpretive. The ultimate context is your instrument's repertoire. Music, like mathematics, is a cultural act. As a musical learner, you need to be mentored into the repertoire. You need to experience other musicians' interpretations and experiences to learn what competent, coherent, and ultimately subtle musical communication sounds, feels, and looks like. And while you are doing this, you also need to be able to explore the repertoire for yourself so that you can find your own way.

It seems to me that this is a similar situation to what we face in mathematics education. Technical proficiency is boring but somewhat mindless. If you had to listen to anyone (including yourself) only playing Hanon's same 60 exercises day in and day out, you would undoubtedly lose your mind. As music, the technical patterns are boring — up and down, back and forth, crossing and uncrossing, stretching and shifting. But they're necessary to develop a foundation of muscle memory and motor skills, as well as the habits of mind and of practice you will need as you gain proficiency and advance to building the finer and finer skills of musicianship.

So there was a need for teachers to create pieces that are very rich musically but very restricted technically so that they are accessible to new learners. This is why Bach, for example, wrote the pieces in his Anna Magdalena Notebook. They created an on-ramp for his new wife to be included in the family's deep musical conversations. She needed a low barrier-to-entry, high-ceiling on-ramp to sophisticated music. These are also the motivation behind Bach's Two- and Three-Part Inventions. Bach's life as a composer was inextricably bound up in his life as a teacher, and these accessible works are the "rich problems" of piano education. They are accessible pieces that even the greatest virtuosi find beautiful.

This is why I love Glenn Gould's recordings of Bach's Two- and Three-Part Inventions. They are the music of his childhood, but from an advanced standpoint. The recordings are filled with joy and delight. To the dismay of critics, he often hums along with himself as he plays. The critics find this annoying. Personally, I find it inspiring. If Glenn Gould can still delight in playing these pieces, then so can I. It gives me great permission. The recordings and the playing are magical.

In my view, this is how we should be looking at the balance we need to strike in math education. There are technique and concepts and repertoire that learners need. Without strong technique, your understanding of concepts will be shallow. Without context/repertoire, your understanding and practice of mathematics will be joyless and without wonder.

 The long-range purpose of our practice was clearly stated on the board
So this is the basis from which I used Kate Nowak's Speed Dating structure to solidify my Precalculus students' early understanding and integration of transformations of functions on Friday. As I keep reminding them, the parent functions and their graphs, as well as the transformations of these basic function graphs, are the essential vocabulary development work for calculus. This is the Hanon and Czerny mindset shift on which we are focusing this year: elementary things that we consider from an advanced standpoint. The order of operations becomes more sophisticated. "Groupings" replace "parentheses" in your thinking about where to start. Groupings, I warn them, are masters of disguise. Sometimes they show up as parentheses, but much of the time they show up wearing a moustache or another costume. They might show up in the form of the absolute value bars or the fraction bar. Don't be fooled, I warn them. Keep your focus on whether you are dealing with transformations of inputs or outputs.

We started simply, using only a single function du jour. On Friday, that was the square root of x. We also restricted our investigation to horizontal and vertical shifts as well as reflections across the x- and y-axis. We were considering the impact on the graphs of basic parent functions as we operated on either the input or the output of the function. We did not multiply by any value other than –1 to start. The Day 1 problem cards are here.  The Day 2 problem cards are also available now (formatted by the amazing Meg Craig - thank you!).

On Friday, each of my 36 students started out with his or her own problem card that contained two related transformations — a shift and a reflection. We organized the speed dating structure, moved our backpacks along the two empty walls, and established our rules of movement (the students along the window side of each row travel, while the hallway-side students stay where they are). If you run into problems, ask the expert in the room on that problem. He or she is sitting directly across from you.
 36 precalc students in a state of flow

Trade, analyze, investigate, sketch, discuss. For forty minutes, my Precalculus students lost themselves in analyzing, investigating, sketching, and discussing functional operations on inputs and outputs. Every two minutes, my iPhone timer would go off and I would call out, "Shift!" And all 36 students would trade back their problem cards, while half of them stood up and moved one seat to their right. Then I would reset the timer and they would lose their ego-selves in each new immersion. I loved the hush that fell over the room after everybody settled down into the next round of problems. I eavesdropped on moving and purposeful conversation about inputs and outputs of functions, shifting and reflecting, as they worked collaboratively to help each other and to help themselves attain proficiency. The preciousness of each minute of mathematical conversation was not lost on me.

And I tell you, it was glorious.

## Friday, August 22, 2014

### WEEK 1 - PROJECT 'MAD MEN' -- Classroom Rules PSA Skits

Leonard Bernstein once said, "To achieve great things, two things are needed — a plan and not quite enough time." I decided to put that principle to the test this first week at my new school by assigning a project on Day 1 that thew together strangers with an absurd but achievable goal: given a particular classroom rule or guideline, create a Public Service Announcement  (i.e., a 30-second "TV commercial" in the form of a skit) whose purpose was to motivate viewers to follow the rules/guidelines for the good of the group.

I created a set-up, instructions, and a rubric for the group project. And my students did not disappoint.

The idea was to get students to think about the consequences of their actions and choices, but their ideas for implementation exceeded even my wildest dreams. Most skits followed a "slice of life" strategy, but the ones that really blew us all away were the ones that parodied existing campaigns.

Two brilliant PSAs started from already-iconic Geico insurance commercials, but the one that left me with tears running down my cheeks was a take-off on Sarah McLachlan's ASPCA spots. The song, "In the arms of the angels..." began playing, and student "Sarah" appears onstage making the exact same kind of appeal she makes in those ads. They had the tone, cadence, and music exactly right, and they clearly understood the emotionally manipulative rhetorical strategy — the seemingly endless list of forms of ignorance designed to eventually provoke self-recognition in almost everyone. Their "mathematical justification" was as follows: the narrator enters and says, "In the past year alone, texting in class tragically cost 5 of Doctor X's students their lives. Remember, think twice before texting in class — there may be fatal consequences for your grade, and for you!

It was pure and inspired genius.

I also loved the spot-on impressions of my teacher persona. One student gave a pitch-perfect parody of my "Function Basics" talk that made me both cringe and laugh my ass off simultaneously.

The best thing about this assignment was that it really pushed the voice of authority downward, into the student community itself. Whatever they made of the experience, they owned it.

I am going to try and remember this for later in the semester, when we've become too routinized.

This is definitely going to be an ongoing part of my repertoire of Day 1 activities. I got through what I neded to,  then gave them the rest of the abbreviated period to collaborate. The time pressure was a thing of art.

It was perfectly imperfect — exactly the way first days ought to be.

-----------
UPDATE:

Here is the link to a generified Word document that you can customize for your own class:

PROJECT MAD MEN- classroom rules PSA generic.doc

And here are the three sample 30-second PSAs I showed my classes to give them ideas:

'You Lost Your Life!' – game show hosted by the Crash Test Dummies (Since Vince & Larry, the Crash Test Dummies, were introduced to the American public in 1985, safety belt usage has increased from 14% to 79%, saving an estimated 85,000 lives, and \$3.2 billion in costs to society)
'
What could you buy with the money you save?' - throwing things over a cliff (You could purchase TVs, bicycles, and computers with the money most families spend on wasted electricity)
'Five Seconds' – at highway speeds, the average text takes your eyes off the road for 5 seconds (Five seconds is the average time your eyes are off the road while texting. When traveling at 55mph, that's enough time to cover the length of a football field)

## Tuesday, August 5, 2014

### Edwin Moise on postulates considered as self-evident truths

My favorite book I've read this summer has hands-down been Edwin E. Moise's, Elementary Geometry From An Advanced Standpoint (yes, I know, I'm a nerd). I'm nowhere near finished with it, but in planning for teaching Geometry this year, I've used it a lot to help me think through what we assert and why we assert it.

This section is a beautiful summary of the many wonders and problems with an axiomatic system such as the one we continue to teach. I don't know exactly how I'm going to use this, but it is definitely going to inform my presentation of something.

pp. 282-3

27.2 POSTULATES CONSIDERED AS SELF-EVIDENT TRUTHS

In the time of Euclid, and for over two thousand years thereafter, the postulates of geometry were thought of as self-evident truths about physical space; and geometry was thought of as a kind of purely deductive physics. Starting with the truths that were self-evident, geometers considered that they were deducing other and more obscure truths without the possibility of error. (Here, of course, we are not counting the casual errors of individuals, which in mathematics are nearly always corrected rather promptly.) This conception of the enterprise in which geometers were engaged appeared to rest on firmer and firmer ground as the centuries wore on. As the other sciences developed, it became plain that in their earlier stages they had fallen into fundamental errors. Meanwhile the “self-evident truths” of geometry continued to look like truths, and also continued to seem self-evident.

With the development of hyperbolic geometry, however, this view became untenable. We then had two different, and mutually incompatible, systems of geometry. Each of them was mathematically self-consistent, and each of them was compatible with our observations of the physical world. From this point on, the whole discussion of the relation between geometry and physical space was carried on in quiet different terms. We now thing not of a unique, physically “true” geometry, but of a number of mathematical geometries, each of which may be a good or bad approximation of physical space, and each of which may be useful in various physical investigations. Thus we have lost our faith not only in the idea that simple and fundamental truths can be relied upon to be self-evident, but also in the idea that geometry is an aspect of physics.

This philosophical revolution is reflected, oddly enough, in the differences between the early passages of the Declaration of Independence and the Gettysburg Address. Thomas Jefferson wrote:

“ . . . We hold these truths to be self-evident, that all men are created equal, that they are endowed by their creator with certain unalienable rights, that among these are Life, Liberty and the pursuit of Happiness . . . . “

The spirit of these remarks is Euclidean. From his postulates, Jefferson went on to deduce a nontrivial theorem, to the effect that the American colonies had the right to establish their independence by force of arms.

Lincoln spoke in a quite different style:

“Fourscore and seven years ago our fathers brought forth on this continent a new nation, conceived in liberty and dedicated to the proposition that all men are created equal.”

Here Lincoln is referring to one of the propositions mentioned by Jefferson, but he is not claiming, as Jefferson did, that this proposition is self-evidently true, or even that it is true at all. He refers to it merely as a proposition to which a certain nation was dedicated. Thus, to Lincoln, this proposition is a description of a certain aspect of the United States (and, of course, an aspect of himself). (I am indebted for this observation to Lipman Bers.)

This is not to say that Lincoln was a reader of Lobachevsky, Bolyai or Gauss, or that he was influenced, even at several removes, by people who were. It seems more likely that a shift in philosophy had been developing independently  of the mathematicians, and that this helped to give mathematicians the courage to undertake non-Euclidean investigations and publish the results.

At any rate, modern mathematicians use postulates in the spirit of Lincoln. The question where the postulates are “true” does not even arise. Sets of postulates are regarded merely as descriptions of mathematical structures. Their value consists in the fact that they are practical aids in the study of the mathematical structures that they describe.

## Wednesday, July 30, 2014

### "The organism moves towards health" — reflections on TMC14

Everybody is writing blog posts about feeling like a fraud after an amazing experience at Twitter Math Camp 2014. Impostor syndrome. I feel like a fraud too, at least, most of the time, but I am trying to practice refraining from my conditioned habits of reacting automatically and giving in in response to that defense mechanism. I am practicing not-reacting. I am trying to notice the positive energy that is there and to just allow it. I am trying to allow myself to experience myself as a competent, good-enough teacher I have respect for and want to continue to be.
 me practicing accepting myself as a competent,good-enough teacher, seen here with supportive tweeps & a giant margarita

What worked in the Group Work Working Group session was setting up a structure to sustain that positive energy of presence. Having learned how to do that is a huge gift I have given myself over the past 25 years of dharma practice. It's a "gift" that comes from working very hard at being present and practicing every day, rain or shine, whether I feel like it or not, whether I do it well or do it badly. I follow the three teachings my teacher Natalie Goldberg learned from her teacher Katagiri Roshi: Continue under all circumstances. Don't be tossed away. Make positive effort for the good. I have done that in my practice every single day for over 25 years. It's the one thing I know in my feet that I am good at.
 starting with restorative classroom circles

So I decided to bring THAT to Twitter Math Camp this year.

The structure works because it is a structure for teaching and sustaining presence — learning to be present with an open heart. I have dharma sisters and brothers all over the world, but when we practice together online, we practice asynchronously — each of us on our own, in our own lives, in our own homes. When we come together using the asynchronous forum of online communication, we maintain that same structure of presence. Write when you write. Read when you read. Listen when you listen.

No comment.

 using the Talking Points structure; no comment
"No comment" is the most important part of the structure, and the hardest part to implement online in a forum like Twitter, which is designed to support comments. "No comment" is about allow there to be space for everyone. It is about all being present together authentically and about staying present with whatever arises. THAT is the thing that most adolescents don't get exposed to in their lives, and it is the thing that can make the greatest possible difference in the quality of their experience — both in the math classroom and everywhere else in their lives.

Most people in our culture don't have a lot of experience in being present and staying present. It takes an enormous amount of energy to learn how to stay present and not flinch. But do that with anything you love and you will have a magical experience. Do that with math, and you'll unlock the treasures of your amazing human mind. But being present with others in a big open space is hard. At first it can be scary. It's very naked. That is why the structure of "no comment" is so important. It helps to create a shared space of emotional safety. It gets everybody focused on their own stuff and supports dropping the "act" you bring to most in-person interactions. That's why it's good to do so right from the start. It's about reframing our conditioned habits of personality.

Very quickly, the timed structure and the practice of "no comment" makes the practice of presence very freeing. You begin to relax into that big open space. You become curious. Your defenses soften. You begin to notice the interesting patterns of your own mind. Best self and worst self. Curious self and bored self. Zen mind and monkey mind. Defense mechanisms, such as snark.

 collaborative mathematics using the Talking Points structure; no comment
The practice of "no comment" creates a space in which the authentic thoughts of your own amazing human mind can arise and step forward. And we honor that process by persisting in not-commenting as we continue.

Natalie describes this process as stepping forward with your own mind.

Once you get a taste for being present, you'll naturally begin to crave it more. That is something I count on in my classroom management practice. Fred always said, "The organism moves towards health." That is one of his greatest teachings for me. "The organism moves towards health" means that, in the process of growing up, we all fall away from the naturally sane and healthy patterns of our organism. "Fight or flight" is a falling away from the natural discharge cycle of "rest and digest" we experienced as infants. When you're hungry, you eat. When you're tired, you sleep. Fred said there is a deeper wisdom inside us that is always available for us to tap back into. It's like an underground stream that is part of our psychological and emotional water table. When we practice being present through structures like Talking Points or meditation or writing practice, it feels like a homecoming — a homecoming to a natural state that is healthy and inquisitive and curious to see what will happen next. It is a natural reconnection with our own inner growth mindset that is our birthright — not some artificial fantasy state we impose on students from without by telling them to have one.

 assigning competence after group work &  observation; still no comment
A growth mindset is just the psyche's way of attuning to the fundamental idea that our organism moves naturally in the direction of health if we will let it — if we can get out of its way and allow it to unfold as it needs to. Allowing means learning to refrain from interfering with that natural movement, and so we use structures that make it manageable for ordinary human beings like us to access the extraordinary ocean of intellectual and creative possibility that is mathematics.

Ten minutes at a time is about what I can muster, I have learned over the years.
 Kate test-driving a geometry task using the Talking Points structure; still no comment
In my experience teaching meditation and writing practice and other structures that cultivate presence, I have found it is about what most people can handle. Ten minutes of Talking Points, no comment — GO. Ten minutes of writing practice — keep your hand moving, no comment, GO. Ten minutes of mathematical conversation, no comment, GO. Learning how to be present with the big, scary openness of not-knowing is no small thing. That is why we zone out, check our phones a hundred times an hour, play video games, watch TV, assault-eat, numb out, zone out, distract ourselves. We all crave the real stuff, but connecting with it feels like sticking a butter knife into the electrical socket. So we break it into more manageable chunks. We set a limit for ourselves and dive in for a limited period. We practice being present for ten minutes at a time. And then we give ourselves and our students a break. It helps us build our tolerance for the intensity of presence and it builds our courage to come back and try it again the next time.

Natalie says that monkey mind is the guardian at the gate, protecting the treasures of our heart and strengthening us for the challenge of opening ourselves to presence and to Big Mind. The structure of "no comment" makes it feel safe for us to touch in to that fire at the center of our being. It helps us to close the gap between what we THINK we've been doing and what we have ACTUALLY done. For me, it's about strengthening students' courage to open their hearts to contact with their amazing mathematical minds — with what my friend Max Ray of The Math Forum at Drexel calls their "mathematical imaginations." We math teachers know the secret that everyone has this mathematical imagination. Our greatest challenge is to get students to trust that they have it too and can access it safely and reliably.

________________________________________

TRY THIS
Read Taming Your Gremlin by Rick Carson.

## Tuesday, July 22, 2014

### TMC #14 GWWG – Annotated References: the research on explicit teaching of exploratory talk

OK, last summarizing post before I have to get packing.

Here is the background research on ‘exploratory talk. Once again, please note that this is NOT required reading!  Recreational reading only! So please don't freak out!  :)

I 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.

Shell Centre,  MAP PD: Students Working Collaboratively (PD Module 5)
An adequate introduction, with pointers to some of the major reseach on exploratory talk in the math classroom. PD module on using talk in the math classroom. There are two parts to download: the overview and the “handouts for teachers.” The Handouts for Teachers is more useful than the overview, but you kind of need both to get started.

Neil Mercer and Steve Hodgkinson, editors, Exploring Talk in School
The richest single source of research on cultivating exploratory talk, though many of the chapters I found most useful are available on the internet.

Douglas Barnes, “Exploratory Talk for Learning” (chapter 1 in Exploring Talk in School, but downloadable PDF here)
Barnes is the research who originally pioneered the concept of exploratory talk as distinguished from cumulative talk and disputational talk. Everybody else’s work builds on his.

Neil Mercer and Lyn Dawes, “The value of exploratory talk” (chapter 4 in Exploring Talk in School, but downloadable PDF here)
An explanation of the need for differing kinds of talk in the classroom (asymmetrical teacher-student talk as well as symmetric student-student or peer talk). Makes a strong case for doing the work to intentionally raise the level of symmetrical talk to make group work effective and meaningful.

Lyn Dawes, The Essential Speaking and Listening,  Chapter 2: “Talking Points” (downloadable PDF here).
Overview of her original strategy for using the “Talking Points” activity structure to encourage students to develop what Barnes referred to as “an open and hypothetical style of learning.” This is what I used as the basis for developing my own version of the Talking Points activity.

Thinking Together
https://thinkingtogether.educ.cam.ac.uk
How can children be explicitly enabled to use talk more effectively as a tool for reasoning? The Thinking Together program was founded to address this specific question. A freely available, evidence-based interventional program of three talk lessons plus supporting materials, developed by University of Cambridge Faculty of Education researchers (led by Mercer and Dawes) as “a dialogue-based approach to the development of children’s thinking and learning.” It has been widely implemented in the U.K. as a means of cultivating exploratory talk in the classroom. Their materials are targeted at elementary children, but can be adapted for older students as well.

Sylvia Rojas-Drummond and Neil Mercer, “Scaffolding the development of effective collaboration and learning”
http://www.sciencedirect.com/science/article/pii/S0883035503000752
Summary of a collaboration between Mexican and British researchers that documents the effectiveness of using exploratory talk strategies to improve collaborative reasoning and learning in the group work-centered classroom.

Neil Mercer and Claire Sams, “Teaching children how to use language to solve maths problems”
https://thinkingtogether.educ.cam.ac.uk/publications/journals/MercerandSams2006.pdf
Summary of the research behind the interventional program called Thinking Together  and how to proactively make talk-based group activities more effective in developing students’ mathematical reasoning, understanding, and problem-solving.

## Monday, July 21, 2014

### #TMC14 GWWG: Talking Points Activity – cultivating exploratory talk through a growth mindset activity

This activity is the one I am most excited about bringing to #TMC14 and to the Group Work Working Group. My intention is to blog more about how this goes during the morning sessions. I also hope that participants will blog more about this too and contribute resources to the wiki.

Exploratory talk is the greatest single predictor of whether group work is effective or not, yet most symmetrical classroom talk (peer talk) is either cumulative (positive but uncritical) or disputational (merely trading uncritical disagreements back and forth).

This activity is based on Lyn Dawes’ Talking Points activity but has been adapted for use within a restorative practices framework. It’s a great way to practice circle skills (i.e., respecting the talking piece) and get students to practice NO COMMENT (i.e., trying to score social points rather than focusing on the task at hand).

PURPOSE
• to support students’ exploratory talk skills by pushing them out of cumulative and disputational modes and into a more exploratory talk mode (i.e., speaking, listening, justifying with NO COMMENT)
• to reveal student thinking about speaking, listening, justifying and about having a growth mindset
• to cultivate a growth mindset community
PROCEDURE
17 minutes

Get students into groups of three.

Talking Points is a timed activity. Groups will have exactly ten minutes to do as many rounds as they can do. For the whole-class debrief at the end, try to keep track of who thought what and why.

Like classroom circles, Talking Points proceeds in rounds. Each “talking point” statement on the list receives three rounds of attention. One person reads the first statement aloud with NO COMMENT. There are then three rounds of speaking and listening. You want these to be “lightning rounds” rather than plodding or deliberate rounds. The Talking Point statements are provocative and designed to stimulate reactions that can be worked with.

TALKING POINTS ACTIVITY
10 minutes

ROUND 1 – Go around the group, with each person saying in turn whether they AGREE, DISAGREE, or are UNSURE about the statement AND WHY. Even if you are unsure, you must state a reason WHY you are unsure. NO COMMENT. You’ll be free to change your mind during your turn in the next round.

ROUND 2 – Go around the group, with each person saying whether they AGREE, DISAGREE, or are UNSURE about their own original statement OR about someone else’s statement they just heard AND SAY WHY. NO COMMENT. You are free to change your mind during your turn in the next round.

ROUND 3 – Take a tally of AGREE / DISAGREE / UNSURE and make notes on your sheet. NO COMMENT

Groups should then move on to the next Talking Point.

At the end of ten minutes ring the bell and have groups finish that round. Don’t let the process go on — they need to stop and move on.

GROUP SELF-ASSESSMENT
2 minutes

At the end of the ten minutes, give students exactly two minutes to fill out the Group Self-Assessment. They will use this during the whole-class debrief and will hand this in for a group grade.

WHOLE-CLASS DEBRIEF
5 minutes

Ask each group to report out about specifics, such as:

Who in your group asked a helpful question and what was it?
Who in your group changed their mind about a Talking Point? How did that occur?
Who in your group encouraged someone else? How did that benefit the conversation?
Who in your group provided an interesting additional idea and what what is?
Who did your group disagree about and why?

I generally choose two or three of these questions and then move on to the actual mathematical group work while the energy level is still high.
_____________________________

Here are links to the first two documents for this activity:

Talking Points handout 1 – talking about talking
Talking Points – Group Self-Assessment

I'll post links to a whole barrelful of the research as soon as I can!

## Sunday, July 20, 2014

### TMC #14 Group Work Working Group Morning Session – The Game Plan

There is so much that we could focus on that I felt a strong need to narrow things down. TMC morning sessions seem to work best when they are more “master class” than “introduction,” so I am going to keep us aimed in that direction.

So this three-day morning session will NOT be an introduction to using group work. Rather, we will use the three days as an “Advanced Topics in Improving Group Work” session.

I’ll kick things off with a brief overview presentation on the background, research, and framework for these two topics. Then, we will spend the three days we are going to focus our work in two key areas:
1 - EXPLORATORY TALK
We will explore the need to cultivate a culture of ‘exploratory talk’ in our classrooms (as opposed to ‘cumulative’ or ‘disputational’ talk) and practice ways to develop and deploy exploratory talk tasks. Questions we’ll investigate include — What is ‘exploratory talk’? Who has studied it? Why is it important? How does it improve learning? How is it measured? How does it fit into the learning cycle? What is the evidence-based work that’s been done in this area? How can I harness this work in my classroom?
2 - IMPROVISING GROUP TASKS
We will work together on getting better at improvising group task development and deployment Questions we’ll investigate include — What are the major types of group-worthy tasks? What fits where in the learning cycle? Which types are effective for which processes? How can I develop fluency in developing and deploying the different kinds of tasks in my course area and classroom?
We’ll try out different tasks and task types, analyze them, and split up into sub-groups to sketch out new tasks of the various types that fit what we ourselves will be teaching. Then the whole group can act as a “beta test site” for the tasks we are sketching out.

Each day, we’ll begin with some exploratory talk tasks and structures I’ve researched and will share on the web.

I think it would be amazing if we could also come up with a way of categorizing/labeling different types of tasks to give ourselves more helpful information about what tasks are useful for what purposes.

Looking forward to working with you!

## Thursday, July 17, 2014

### DIY Geometry Vocabulary Game, courtesy of the MTBoS (a collaborative effort)

Through an amazing collaborative effort on Twitter that took, like, all of 15 minutes, the collective hivemind of the #MTBoS came up with a great way to teach/reinforce vocabulary using Maria Anderson's tic-tac-toe style of Block games.

Most of the needed resources are referenced here on my Words into Math blog post.

I just added a bunch of new files to the Math Teacher wiki, including blank vocab cards so the kids can make up their own practice cards.  I make cards in Pages, so I'm also including the PDF and an exported Word doc version.

BLANK TEMPLATE Words into Math Block game cards.pages

BLANK TEMPLATE Words into Math Block game cards.pdf

1-3 Words into Math Block game cards LEVEL 1 SIDE A.doc

1-3 Words into Math Block game cards LEVEL 1 SIDE B.doc

1-3 Words into Math Block game cards LEVEL 1 SIDE B.pages

1-3 Words into Math Block game cards LEVEL 1 SIDE A.pages

Here's a link to the gameboard.

And voilÃ ! A vocab activity for Geometry is born.

When we receive the Oscar for this production, credit should go to Teresa Ryan (@geometrywiz), Julie Reulbach (@jreulbach), Kate Nowak (@k8nowak , aka The High Priestess), Sam Shah (@samjshah), Tina Cardone (@crstn85), Michael Pershan (@mpershan), and if there's room left in the credits, me too.

-------
UPDATE (08-Aug-14):

Two things:

Thing 1 - I made an actual set of Words into Geometry cards (double-sided) because I want to be ready to have students actively practice. You can find the file here on the Math Teacher's Wiki: Words into Geometry game cards.

Thing 2 - I'm decided to use lima beans and pinto beans as counters for block games in Geometry because I am going to have a lot more students than I've had in the past! These will never go out of style and will always be replaceable as needed.

## Monday, July 14, 2014

### Life on the Unit Circle - Board Game for Trig Functions

I suppose it was predictable that I would turn the unit circle into a board game as a practice activity, but somehow this still caught me by surprise last night as I was drifting off to sleep. Thank you, deeper wisdom of the unconscious.  :)

This uses the same basic idea as Life on the Number Line, but you can use whatever trig or inverse trig or other practice cards or problems you want to use. Choose your game piece, take turns, and everybody solves whatever problem comes up. Rolling a positive-positive-number or a negative-negative-number on the dice moves your game piece in the counter-clockwise direction (positive angle in standard position) from (0, 1) to start; rolling a positive-negative-number combination moves you in the direction of a negative angle in standard position (clockwise around the circle from wherever you are).

I like this as a practice structure because I try to avoid games that have a strong win-lose association. It also communicates my group work norms of "same problem, same time" and "work in the middle."

You can use the game cards from Trig War (on Sam's blog) or Inverse Trig War (on Jonathan's blog) or from anyone else. I know I plan to. Have kids shuffle them into two piles, one for positive moves and one for negative moves. The point is to do a lot of problems and to get kids used to living on the unit circle. Print the problem cards on colored paper for variety.

I have one new rule, though: whatever point you land on, you have to declare the cosine and sine of that function.

I "score" this as a tournament, requiring each group to do a minimum number of problems, usually a pretty huge number. I keep group scores on a poster or on the white board. I also offer extra credit points for groups that exceed an even more outrageous number of problems. For some reason, the words "extra credit" seem to add a magical flair. I say, who cares as long as kids practice — they're only points!

I buy my plus-minus dice from this seller on eBay and have never had a problem.

## Friday, July 11, 2014

### Beginner's Mind

Few of us succeed in truly bringing a beginner’s mind to our experience of mathematics. By the time we reach middle school, it is already impossible. We have experiences, beliefs, and opinions about what it takes to do math that have been formed by other people’s experiences, beliefs, and opinions. Students show up in my classes believing — as early as age eleven — that they are good at math or bad at math, that they love math or hate it, prefer decimals over fractions, use a calculator as a defense mechanism and a talisman, and even those who love books believe that word problems are evil. These attitudes get locked in early, and they continue to shape our experiences with math for the rest of our lives.

When I was in first grade, Mrs. Williams gave us each a little muslin drawstring bag of colored wooden blocks and rods. There were little ivory cubes and stubby red rods that were about the same scale as the houses and hotels in a Monopoly set. The apple green rods were just a little bit longer. One, two, three. Each rod got one unit longer, but not wider, than its predecessor. All the way from one to ten. We each kept our little bag in our desk all year long.

I did not know this then, but these are called manipulatives. Mathematical manipulatives. Cuisenaire rods, more specifically. They were invented in the 1920s by Georges Cuisinaire, an imaginative Belgian teacher, and were popularized in the 1950s by the great mathematical educator Caleb Gattegno. As a teacher and as a learner, I have always had a crush on Gattegno. His math pedagogy mirrors the inner development teachings of my root teacher, Fred Orr. Gattegno said, “Only awareness is educable.” Fred said, “Noticing shifts the energy.” When I first encountered those Cuisenaire rods, I could not have known that these two currents of awareness would flow together into a stream that would subtend every aspect of my life.

Those little rods became my friends. I loved the way they felt between my fingers. I loved the smell of the muslin bag and the way they clicked together inside the bag when I took them out during arithmetic lessons or tucked them back into into the darkness of my desk’s inner compartment.

I loved the sound they made when they clattered onto the desk in preparation for our daily lessons. We used them for counting, equivalence, and for modeling mathematical operations. I also used them for personal puppet shows. I traced their shapes on paper with a fat yellow pencil, and I colored in my outlines according to their shapes — white, red, green, blue. Those little blocks cast a long, cool shadow in my mathematical memory. They kept me connected to mathematical wonder even when I felt certain I had no clue about anything mathematical.

Years later, I realized you could buy them on Amazon, and I bought the traditional wooden set. They no longer come with the muslin bag. I cleared off the old oak dining table so I could hear that long-ago sound of the blocks on my desk. It was still exactly the same. I picked up one block or rod at a time and turned it between my fingers. I inspected them. I smelled them. I lined them up: white, red, green. It was one of my madeleine moments.

Before my parents moved away from that old town, I borrowed my mother’s Jeep and drove back to my old school. It was exactly as I had remembered it, only a little bit uglier and smaller. It was after school, and the door was open. I followed the corridor around, past my kindergarten classroom at the first corner that had long ago become the school library. I walked down the linoleum-floored, painted cinder block corridor to the room that had once been Mrs. Williams’ classroom and my own. It was unchanged, except that the blond spinet piano was no longer in the corner. That spot now housed a small cluster of computers.

I wanted to lift the lid of one of the desks to see if there was a drawstring bag of Cuisenaire rods inside.

The rods make me think about beginner’s mind in mathematics. Our minds and emotions get so tangled up and cluttered, it’s not easy to approach math without preconceptions. My first Zen teacher was Keido Les Kaye of Kannon Do. He was the thirteenth monk ordained in America by Shunryu Suzuki, known by most American dharma practitioners as Suzuki Roshi. Suzuki Roshi said, “In the beginner’s mind, there are many possibilities, while in the expert’s mind, there are few.”

This is as true in school mathematics as it is in the study of mindfulness.

As learners, we quickly become experts at either embracing or avoiding math. Our defense mechanisms against shame, humiliation, and confusion become fierce. They harden our hearts. Those of us who suffer from trauma and anxiety around  math come by it honestly. In response to years of competition and sorting and embarrassment, our unconscious minds develop powerful and uniquely personal sets of defense mechanisms. These arise as protective functions in the Self, emerging to shield us against intolerable feelings of panic and confusion and shame.

But there is a different kind of relationship with mathematics that is possible, and it is not only possible for anyone to enter it — it is also essential. It begins with setting aside what we think we know about math, and allowing ourselves to experience it through beginner’s mind — through a mind that is open and self-aware and unselfconscious. It is about setting aside our adult preconceptions for a few moments and allowing ourselves to sink down into the flow of curiosity — the kind of curiosity I remember feeling when I sat on the curb and watched a millipede shuffling along in the ninety-degree New Jersey heat and in the ninety-degree angle between the curb and the gutter, its black legs fluttering along like little windmills, flipping over and over in an articulated lumber, like a deck of cards being shuffled before the magic begins.

### Growth Mindset Quote of the Day — 11-Jul-14

"To achieve great things, two things are needed: a plan and not quite enough time."
— Leonard Bernstein

## Thursday, July 3, 2014

### Rooting Out Blind Spots in the Language of Group Roles in Complex Instruction-based Group Work

There are a number of blind spots in the original roles and norms from Complex Instruction (CI) that amount to a form of self-sabotage in practice. These have always bugged me.

It seems to me that if you are using CI to address status problems, then you need to ruthlessly root out any unconscious status implications from the language you use for roles and norms. In both the short term and in the longer term, these can severely undermine the effectiveness of any status treatments you may try to implement.

The most self-sabotaging label in my experience is that of “Team Captain.” Two things bother me about this label. The first is that the word “Captain” implies a social hierarchy within the group that is antithetical to the goal of developing shared responsibility and accountability. The other is that, when used in conjunction with the word “Team,” the word “Captain” cannot help but imply competition within the classroom rather than collaboration. For these reasons, I have banished these words from my vocabulary for roles.

Having managed many, many groups and projects in the corporate world, I have always been concerned that the language of Complex Instruction roles is so imprecise. In my experience, imprecise language about roles leads to a lack of clarity in response to expectations. It also leads to charter conflict and “turf wars.”

Here’s how I have tweaked the language of group roles for my classroom this year to make them more precise and more equitable.

NAVIGATORfocus on GROUP INTENTION
Keep the group pointed in the direction of reaching a shared understanding:

• read instructions aloud
• enforce norms
• resolve conflicts, find compromises
• substitute for absent group members
• lead the process of filling out the group self-assessment questionnaire to turn in

FACILITATORfocus on GROUP PROCESS
Keep the process of collaboration running smoothly:

• get the work off to a fast start
• watch progress & the time
• make sure everyone participates (turn-taking)
• assign sub-roles
• make sure the group meets its deadline

RESOURCE MANAGERfocus on GROUP RESOURCES
Get what your group needs:

• get & return all needed supplies
• organize the group to ask a group question (make sure that anyone in the group can state the shared question)
• call the teacher over for group questions
• organize clean-up

RECORDER/REPORTERfocus on GROUP RECORD
Manage group production of notes and deliverables:

• take notes for the group
• make sure everybody takes good personal notes
• organize production of high-quality group deliverables to turn in

As part of the infrastructure for group metacognition and self-monitoring, I am going to implement a group self-assessment questionnaire this year. Right from the start, I want to be deliberate in training students and groups to think reflectively about how they are participating individually and how their participation is being perceived by others as well. I also want them to actively perceive and reflect on what others have done that they could try themselves in the future.

I think of this as part of a larger process of assigning competence in the classroom community.

Here’s my draft of a Collaboration Self-Assessment questionnaire. We’ll test-drive this at the Group Work Working Group morning session at Twitter Math Camp 14.

Collaboration Self-Assessment Questionnaire

## Monday, June 30, 2014

### 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.

## Sunday, June 22, 2014

### 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.