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.

And this is why I teach...

It was another crappy Friday in an arithmetic series of crappy Fridays that were running together and threatening to define the limit of my patience for fall trimester as x approaches a mid-sized number that is nowhere near infinity. So I have no idea what possessed me to wake up even earlier than usual to pull together an extra day's practice activity for my right-after-lunch class of rumpled and discouraged algebra students — the ones who believe to their core that California's Algebra 1 requirement is God's own punishment for unremembered karmic crimes they must have committed in previous lifetimes.

But I did it.

The topic was solving and graphing compound inequalities — a skill set that must be mastered in order to have any hope of making sense of and mastering the next topic traditional algebra curricula force-feed to our students: the dreaded topic of absolute value inequalities.

There's really nothing I can say to convince a roomful of skeptical eighth graders that compound inequalities will prove not only useful in business planning (which, after all, is simply algebra writ large across the canvas of the economy) but also amusing and possibly even interesting little puzzles to delight the mind.

To this group of students, they're simply another hoop to be jumped through.

So something in me understood that I needed to reframe the task for them, and to do so using Dan Pink's ideas about intrinsic motivation from his book Drive.

Nothing unlocks the eighth grade mind like an authentic offer of autonomy. As I explained recently to a room of educators at a mindfulness meditation training, middle school students suffer emotionally as much as adults, but they have comparatively little autonomy. A little well-targeted compassion about this can carry you for miles with them, though I usually forget this in the heat of working with them.

For this reason, I like to save practice structures such as Kate Nowak's Solve—Crumple—Toss for a moment when they are desperately needed. I have learned to withhold my Tiny Tykes basketball hoop for moments like this, when students need a little burst of wonder in the math classroom. And so even though I was tired and very crabby about the ever-increasing darkness over these mornings, I pushed myself to pull together a graduated, differentiated set of "solve and graph" practice problems to get this group of students over the hump of their own resistance and into the flow experience of practicing computation and analysis.

And oh, was it worth it, in the end.

The boys who are my most discouraged and resistant learners came alive when they understood that a little athletic silliness was to be their reward for persevering through something they considered too boring to give in to. They suddenly came alive with cries of, "Dr. X— watch this shot!" from halfway across the room. One boy who can rarely be convinced to do the minimum amount of classwork completed every problem I provided, then started tutoring other students in how to graph the solution sets and perform a proper crumpled-paper jump shot.

The girls in the class got into it too, but they seemed more excited about the possibility of using my self-inking date stamp to stamp their score sheets. So I gladly handed over the date stamp to whoever wanted to stamp their own successfully solved and graphed inequalities.

I was far more interested in reviewing their mathematics with them. One of the things I love best about practice structures like this one is that they give me an excuse to engage one on one with discouraged students under a time crunch pressure that adds a different dimension to their motivation. Suddenly they not only want to understand what they have done, but they want to understand it quickly, dammit, so they can move on to another problem, another solution, another graph, another bonus point.

Ultimately, Solve—Crumple—Toss becomes an occasion for conceptual breakthroughs in understanding.

I can't tell you why this happens. I can only tell you that it does happen — often. It makes me feel lighter, more buoyant about teaching them algebra. And it makes them feel happier too.

I wanted to write this down so I could capture it and remember this for a few weeks from now, when it stays darker even longer in the mornings and when I feel crappier and crabbier and more forgetful.

Radio Silence Does Not Mean Nothing Is Happening...

Wow, did I ever fall off the radar.

Plop. That "splat" you might have heard was me, falling off the blogging radar.

But I'm back, baby.

Last night I had the most wonderful dinner with @btwnthenumbers and @woutgeo and @mythagon, who was in town for a conference/collaborative meeting, and I tell you, it pretty much restored my faith in teaching, in mathematics, and probably in all of humanity.

I have been working at a near-frantic pace these last five weeks, prepping, teaching, grading, not grading, having parent conferences, having meetings with parents and the principal, having meetings with parents and principal and superintendent, going to IEP meetings, collaborating with my department members to write goals that will help us to align our curriculum with the Common Core, and generally dealing with all those things that go haywire as soon as you start to nail down some satisfying, finite part of your teaching.

In other words, just like you, life has been kicking my ass.

But between last night and this morning's drive to work something shifted. Something sane and healthy intervened.

That something was my connection with the Twitter- blogo-sphere.

Whenever I'm feeling exhausted and run over with skid marks across my face and body, connection with my tweeps -- any connection -- seems to be the best medicine. I don't know why this is true; I only know that it is so. Remembering this makes me think of a quote I have from von Neumann hanging in the ring of inspiring quotes that encircles my classroom: "In mathematics, you don't understand things; you just get used to them." Some days that's how I feel about things in my classroom or in my school or in my life.

I only know that five or ten minutes of venting to my tweeps about an impossible situation -- even when @woutgeo is only half-listening because (a) the Giants are sucking pretty hard against the Cardinals and because (b) my venting is both predictable and boring -- it helped just to have reconnected with the connection. In Jakobsonian structuralist linguistics, this kind of communicative connection is known as a "phatic utterance" (look it up, Riemann, I have to look up all of your crap).

By this morning, I was feeling reasonably happy driving to work for a 7:30 a.m. meeting. I was not totally thrilled about the hour or having to buy gas at that hour or the price of gas for that matter, but I felt pretty great about car-dancing in the dark to Ace of Base's "The Sign" and remembering car-dancing at #TMC12 with @mgolding and @samjshah and @jreulbach and @ bowmanimal on the way to do Exeter problem sets. And I felt great when @rdkpickle's sweet soprano voice was joined by @SweenWSweens and @jreulbach singing "Tweet Me Maybe." And I even laughed when the theme from Sesame Street came on. iPod's "shuffle" feature has a somewhat perverse sense of humor.

OK, and one other thing I have learned is that my dog always knows when it's time for me to end a blog post. Just now he jumped up on my lap and pounded the laptop keyboard with his giant panda bear paw:

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So that's my cue to wrap this up.

I just want to say, if you are feeling alone or frustrated or exasperated and you are reading this, then for the sake of everything we hold dear, please reach out to someone else who is of like mind. "It's hard to teach right... in isolaaaaaaaaation.... So here's some PD.... just like vacation!"

Tweet me maybe, tweeps. Over and out for now.

On choosing sanity, and on modeling this choice for our students

I want to talk a bit about sanity — not because I think I'm an expert (I'm not; no one is) but because I have come to understand sanity as a choice we make moment by moment, and because, in so doing, I have seen how we either model — or do not model — it for our students in our classrooms and in our lives. And this is an enormous part of the social and emotional life training they either receive or do not receive in this crucial part of their lives.

I teach middle schoolers, or rather, I should confess that they teach me. Every student and every class is a mirror in which I can see what I am teaching them. Perhaps because their access to it is new, middle school students have a finely tuned hypocrisy detector. More so than any of the high school or university students I've ever taught, middle school students want to see who "walks their talk." Do so, I have learned, and they will follow you anywhere. Fail to do so at your own peril.

So a big part of this year's learning, for me, has been learning how to tune in to what I am actually doing and checking in on whether this is consistent with the social and emotional lessons on appropriate self-care I am trying to teach. Sometimes that has meant being the adult in their lives who tells them to stop the madness. The only way to stop the war is to stop fighting, stop struggling, stop efforting. There really are only so many hours of the day, and for middle schoolers, about eight of those need to be spent sleeping. That means every piece of homework can't always get done every day all the time. Sometimes a person has to choose sanity. So I try to understand that and allow for it, because learning to allow space for all of life is something they are going to have to learn if they are going to do better in running this world than we have done so far.

I've also had to learn how to trust my training and my gut. I was blessed to study for over ten years with a pioneer in integrating social and emotional intelligence and mindfulness into learning environments ranging from special education to mainstream classrooms to therapy situations. You probably haven't heard of him because he has spent his life being what I think of as a guerrilla bodhisattva — a pioneering educational psychologist and an undercover evangelist for social and emotional health in daily life. His name is Dr. Fred Joseph Orr, and I am blessed to count myself among his students. Actually, we think of ourselves as his disciples, though he would undoubtedly discourage that characterization. But it's a fair one. He taught us to integrate teachings from whatever sources might be beneficial to ourselves and our communities — teachings from Adlerian psychology, spiritual development, meditation, yoga, Buddhism, his own mischievous sense of humor and spirit of adventure, writing as a practice, the practice of joy and creativity in whatever form they might take, and environmental restoration and ecological rebalancing. I came to him as a writer, writing teacher, and longtime meditation practitioner, but I quickly became much more than that under his mentorship. And it was the kind of mentorship that is a true spiritual gift — the kind you can only repay by sharing it with others. In his life, he suffered in ways that most of us would find unimaginable, and yet he remains the most radiant and joyful person I have ever known. And although his active teaching practice has been cut short by a medical condition that has become the focus of his own personal life practice, his students carry on his great efforts, sharing the learning and the gifts we received from his teachings. His teaching of us was an investment in the future and a labor of love. I wake up every morning determined to be worthy of the effort, love, and energy he poured into teaching me.

There's an urban fable Fred used as a teaching tale. It's known as The Hundredth Monkey principle. It began in the 1950s as a call to end the escalating nuclear arms race, but Fred believes it has broader applicability as a model for how human awareness and sanity can be activated too, and I have come to believe this too. The idea is simple, yet profound: when a critical mass of individuals' consciousness gets raised, it inevitably triggers a paradigm shift in the dominant culture. So this turns out to be a more leveraged model for social change than the conventional wisdom tends to think, for when we direct the focus of our daily efforts on helping individuals and small groups to shift their energies, awareness, and attention, we are doing our part to put our culture on the path that leads toward greater sanity, in addition to greater achievement.

So what does all this squishy-sounding woo-woo stuff have to do with teaching and learning mathematics?

I believe it provides us with an on-ramp, a way to reach the hearts and minds of the students who are most discouraged and shut down in the math classroom. It's differentiated for the most capable students because most of them have never been given tools to cultivate self-awareness and other-awareness as a part of their learning. And by learning — and by teaching these discouraged students how — to cultivate their social and emotional intelligence in how they engage in mathematical practice in our classrooms, we can change their relationship with mathematical studies to one based on what the American Tibetan Buddhist teacher Pema Chodron describes as "unconditional friendliness."

Over the last couple of days, as I've begun my yearly summer rituals of cooking and storing school lunches in the freezer for days when I need the loving self-care of hand-made soup in the middle of my day, I have realized that this is the heart of my Twitter Math Camp talk. My topic is Dan Pink's book, Drive: The Surprising Truth About What Motivates Us, and how to use his ideas about intrinsic motivation to reach the discouraged math learners in our classrooms. I've begun to understand how Fred's work with me is an arrow that flies straight to the bullseye of this target, and it's given me some hopefully valuable insight into how to create the toughest of Pink's three pillars of intrinsic motivation. That pillar is autonomy, and I'll write more about it in an upcoming post as I flesh out my talk with ideas and activities.

In the meantime, I'm going to let my subconscious mind work on the problems while my conscious mind takes a nap.

On Meeting Sam, or How Twitter and Blogging Refreshes My Teaching Practice... and My Life

So Sam Shah was in town for a visit, and a bunch of us got together Friday night for drinks and dinner in his honor.

Before anything, I should tell you that everything magical you've heard about Sam is true, including that elusive quality that Jason of Always Formative captured so well when he said that "hugging him is like being wrapped up in a freshly laundered rainbow."

If you teach, then you also understand that there is something both improbable and heroic about going out on a Friday night. Especially for a late dinner.

But these are my math teacher-blogger tweeps, the people who restore my faith in the power of teaching and learning, and nothing recharges my teaching batteries like connecting with them IRL (In Real Life). So I'm glad I got my butt off the couch and met up with them at Bar Tartine for an evening of conversation, laughter, and understanding.

There's something precious about having a circle of teacher-blogger tweeps that is hard to explain to teachers who don't use Twitter or blogs. Don't get me wrong, my colleagues at school are amazing and I love teaching with them. But my math teacher-blogger tweeps are the ones who really "get" me. They inspire me. They know me at a surprisingly deep level. They are the colleagues who are trying to improve as teachers in the same ways I am trying to improve as a teacher. They are the ones who respond to my Twitter distress calls with lesson ideas and foldables and encouragement and energy of their own. They share my love of office supplies and unicorns and my outrage at stupid copier breakdowns the suggestion that Khan Academy videos are the solution to all of American education's problems.

Sam, of course, is the grand wizard of intellectual generosity in the Twittersphere, the keeper of the Virtual Filing Cabinet and well as the frequently hilarious "Favorite Tweets," which is why I follow him around like a duckling.

And that is why it was such a gift for all of us to get to meet him IRL.

The Big Ideas inside the 'big ideas'

As I've been organizing my job search materials, I've been reflecting on some of the Big Ideas I have learned are the most important among the too-many "big ideas" our textbook and state standards emphasize for Algebra 1.

One of the problems with the state standards is that they can't let go of anything as being less important than anything else. Which is why there are 26 overall standards, plus embedded sub-standards inside the standards, and the whole thing is a nasty ball of yarn to try and untangle.

One thing I've tried this year, which seems to be working well, is to choose my emphases based on what developmental psychologists have discovered about children's mathematical development. I am particularly grateful for the work of the British psychologists Terezinha Nunes and Peter Bryant, who also do a lot of work with Anne Watson of Oxford's math education program. Nunes and Bryant's book, Children Doing Mathematics, has really blown my mind open to what they call the "generative" quality of children's mathematical development and number sense — that is to say, kids develop their sense of number and of mathematics in layers, the way some inkjet printers work, with each pass of the printhead setting down another layer that completely transforms the image that is emerging on the paper.

In Nunes and Bryant's synthesis, as well as in their summary of others' research, kids' understanding of quantity is revealed to be an extremely fluid, dynamic, and multi-faceted set of tools. As they put it, "a successfully developed understanding of number comes from four distinct developmental threads" which they summarize as:

1. the ability to COUNT discrete OBJECTS
2. a deep familiarity with a wide range of QUANTITIES OF QUALITATIVELY DIFFERENT KINDS (such as both countable and uncountable quantities)
3. the ability to COMPARE QUANTITIES OR COLLECTIONS of objects, assessing both similarities and differences regardless of their qualitative kind(s)
4. the ability to use established notation for all of these (Nunes and Bryant, pp. 1-20)

Yet, as they have observed, our current curriculum tends to address only #1 and #4.

This helped me understand something I had been struggling with for a long time — namely, the fact that there are many incomplete understandings children develop that are sufficient for them in their context, BUT that are insufficient over the long term as a foundation for mature mathematical understanding. These are the mathematical versions of ideas like "the tooth fairy" or "Santa Claus." They are enabling fictions that are developmentally appropriate in their time and place, though they are not at all what we want our young adults to rely on by the time we release them out into the grown-up world of mathematics.

One of these incomplete understandings — one that drives university-level mathematicians like Keith Devlin head-banging mad — is MIRA, or the idea that Multiplication Is Repeated Addition. Their argument is that this is such a stunted understanding of multiplicative reasoning that it threatens to undermine the very foundations of civilization, dammit.

But in truth, like the idea of the tooth fairy or Santa Claus, MIRA does have a legitimate place in a child's generative mathematical development — as long as his or her teachers understand that, like the idea of the tooth fairy, it is an incomplete understanding that is meant to be expanded upon into a much richer and more scalar concept of multiplication.

The place where I am finding MIRA to be an extremely useful tool is with Algebra 1 students making their first forays into the abstractions of algebraic reasoning, which is to say, in dealing with polynomial arithmetic. I say this because young adolescents are such intensely concrete thinkers. When I ask them to consider combining like terms such as 5 elephants and 3 elephants, they can easily understand what I am asking for. But the moment we start investigating the idea of combining 5x^2 and 3x^2, their heads explode. Things only get worse when they are asked to combine 5x^2, 3x^2, and  6x^3. It seems like they forget everything they have ever known about the combining of like terms, and they start adding or multiplying exponents and or worse things than most of you can imagine.

This is the place where some teachers find algebra tiles to be helpful. But I find that algebra tiles have a grammar and a rhetoric of their own that is not easily extensible into polynomial arithmetic beyond quadratic thinking. Also with their color-coding, they also add in moving parts I find my students are not yet ready to think about.

But guide their attention away from abstraction for a moment, and ask them what happens if they combine 5 dogs and 6 apples. They understand the logic of ConcreteLand completely. In this case, I have found, getting them to think about 5x^2 the way they think about 5 dogs and about 6x^3 the way they think about 6 apples, and their conceptual understanding shoots through the roof. I can even use Brahmagupta's idea of "fortunes, debts, and ciphers" (positive numbers, negative numbers, and zero) to help students think about what happens in a trading economy where I might "owe them" 3 dogs (or 3 x^3) as we barter our algebraic quantities away for each other's lunch components.

Students still need a lot of practice and experience with this whole crazy abstract insanity to cement their understanding in place, and they can be expected to relapse several times into believing that they need to add exponents instead of thinking about coefficients as quantifiers. But eventually the idea of quantifying (and if need be, combining) x^2s the way they count and quantify dogs or apples gives them a surer footing as they begin to construct a new and deeper understanding of multiplying variables. And that is something I can eventually build a rich and scalar concept of multiplication on top of — on that would be appropriate to eventually deliver to Professor Devlin's lecture hall.

I do this knowing that even within a few months, this conceptual framework will be revised and replaced with other incomplete understandings many times over. But I do so knowing that I am teaching my students how to learn by giving them tools for understanding how to build tools that help them understand what the heck they are doing.

Put some big, ironic air quotes around that word "winning"

My school is located in a sports-crazy town. It's not just that we are located very close to our major professional sports teams or that many of the players and owners live in our town. It's also that it's a very outdoors-oriented, sports-minded place. Monday-morning conversations about what students and their families did over the weekend always revolve around a list of soccer matches, lacrosse games, swim meets, softball and/or tennis and/or golf games, basketball or touch football games, long recreational or competitive runs, and even some hilarious made-up sports.

Although I am the child of a terrific amateur athlete and major sports fan, I did not inherit the sports gene. In fact, if anything, I inherited the opposite of the sports gene. I love walking my dog and hiking, and I'll do yoga or other health-oriented activities, but I don't care about organized competitive sports. This is probably because I was such an unsuccessful and discouraged participant in sports as a child — always the last kid picked, usually humiliated, never celebrated on the blacktop or the athletic field.

The places where I could compete were always in the classroom or on the musical stage. At our schools, I was always considered a "brain" or a "music kid," which had its own kind of competitive aspects, but not the kind that sports-minded people think of.

So when Sports came up as a Spirit Day theme, I mentally waved it away as something irrelevant to me, something I feel too defended against to participate in. But I knew that the kids would hound me about why I wasn't wearing some sports team's paraphernalia, which meant I had to think about what to wear instead.

On the day of the Spirit Day/Class Competition, I wore my NASA sweatshirt.

When students asked me why I wasn't wearing a sports team shirt, I spoke to them honestly. I told them I'd been a terrible athlete in school and that I had always been made to feel ashamed on the blacktop and on the playing fields. But, I said, there are other kinds of teams and many other kinds of "winning" in this world. And one of the teams I admire most in our country is NASA because, if what they do every day and every year — with little money and constant attacks — isn't winning, then I can't imagine what is.

It seemed to gladden the hearts of my fellow nerds and non-athletes tremendously to have a faculty advocate and fellow traveler in this regard.

NCTM Standard 7: fostering "positive dispositions toward mathematics"

Standard 7 for the National Council of Teachers of Mathematics measures whether well-qualified math teachers "support a positive disposition toward mathematical processes and mathematical learning." But their criteria and performance indicators don't exactly reach through the screen, grab me by my shoulders, and inspire me to inspire my kids to love mathematics. They seem more like a floor than a ceiling, and personally, my desire is to aim higher than that.

One of the ways in which a positive disposition is fostered is by building a strong and positive learning alliance with my students. In a therapeutic situation, the establishment of a therapeutic alliance is a critical step. The client must believe that the therapist believes in them and in their commitment to change. The great Jungian analyst, teacher, and storyteller Clarissa Pinkola Estés says that nobody can truly accomplish great things completely on their own steam. The same is true for students in the math classroom. When they know in their bones that I am rooting for them, they begin to feel that success in mathematics is possible. That doesn't mean I don't "display attention to equity/diversity" or "use stimulating curricula" and "effective teaching strategies," as NCTM standard number 7 demands. It means that every day, in every way, I try to demonstrate my own commitment to being in their corner and cheering for them.

Another way in which I can help foster positive disposition toward mathematics is through sharing my own enjoyment of the processes we investigate. To me, positive disposition is about cultivating curiosity and patience. I find that when I model my own process of enjoyment, I give them a window into what their own relationships to mathematics can look like.

And so I consider it a wise investment of time and energy to cultivate positive dispositions wherever I can, regardless of a student's success or lack of sucess so far in Algebra, and I see it paying off in little and big ways throughout the year.

One experiment in building positive disposition I've been trying lately is something I ripped off borrowed from Sam Shah -- the use of motivational buttons. Inspired by Sam's lead, I made up some motivational buttons for students to wear during tests:

If students wear their Algebra Warrior! button during tests, I give them an extra credit point on the test. I gave these out during our last test and made everybody "take the pledge" to wear their button during tests, including the state standardized tests this year.

Still, I recognize that they are middle school students who can (a) lose anything and (b) easily be distracted even from things that are important to them.

So I was pleased when at least 80% of each class showed up eager to show me that they were wearing their Algebra Warrior! buttons for the test on Friday.

It's a tiny thing, but it's a tangible way of getting them to practice demonstrating their commitment to being impeccable warriors in mathematics. Like warriors putting on armor to do battle, my little Algebra Warriors put on their buttons and remember to form an alliance with themselves — to advocate for themselves and remember that they are connecting with something much bigger than their fear or confidence whenever they do mathematics.

SOLVE - CRUMPLE - TOSS in Algebra 1: hommage à Kate Nowak

Kate Nowak creates some of the most innovative and engaging practice activities anywhere -- especially for those skill/concept areas that are more like scales and arpeggios than like discovery/inquiry lessons. Some skills, like basic math facts, simply need to be practiced. This is true not because students need to be worn down but rather because it takes the mind and body time and first-hand experience to process these as matters of technique. It takes time to get used to the new realities they represent.

Nowhere is this more true than in tinkering with the multiple different forms and components of linear equations in Algebra 1. No sooner have students gotten the hang of finding the intercepts of a line than they're asked to find the slope. They figure out how to find the slope and the y-intercept, and they're given the slope and a non-intercept point. They figure out how to crawl toward slope-intercept form, but fall on their faces when asked to convert to standard form. Standard form, point-slope form, slope-intercept form, two points and no slope, it's a lot of abstraction to juggle. Mastery is part vocabulary work, part detective work, part scales and arpeggios, and part alchemy of different forms. It's a lot to take in.

Enter Kate's Solve - Crumple - Toss activity. I have loved this practice structure since the day I first read about it, but I have struggled with the fact that the most engaging part of the activity destroys the paper trail/evidence. This was less important with high school students, but it is really important with middle schoolers, I find, because they are so much more literal.

For today's linear equation-palooza in class, I created a basic "score sheet" for each student and I numbered each of the quarter sheets on which I glued blocks of problems (4-6 problems per mini-sheet). I also differentiated them from "Basic" level (Basic-1 through -4, Level 2-1 through -4, etc), so that students could choose their own levels. Students were also invited to work in pairs or groups of three because I find it encourages mathematical language use and increases risk-taking. It also seems to be more fun.

After the "Solve" part of the activity, students brought their solved mini-sheets to me to be checked. If they completed the problems correctly, they got a stamp on their score sheet and proceeded to the back of the classroom where I'd set up the Tiny Tykes basketball hoop over the recycling bin. There they completed the "Crumple" and "Toss" stages, awarding themselves a bonus point on the honor system if they made the shot. Then they returned to the buffet table of problems and chose a new mini-sheet.

Because my middle school students like to bank extra credit points toward a test wherever they can, I like to attach these to practice activities such as this one or Dan Meyer's math basketball. Being more literal and concrete than high school students, middle schoolers seem to find great comfort in the idea that they can earn extra credit points ahead of time in case they implode on a quiz or test. What they don't seem to realize -- or maybe they do realize and they just aren't bothered by it -- is that if they participate in the process, they win no matter what. Either they strengthen their skill/concept muscles and perform better and more confidently on the test; or they feel more confident and less pressured because they have banked a few extra-credit points for a rainy day; or both.

It was fun to hear my previously less-engaged students infused with a rush of sudden, unanticipated motivation to tease apart a tangled ball of yarn they have previously been unmotivated -- or uncurious -- to unravel. And something about the arbitrary time pressure of trying to complete as many problem sheets as possible in a short period was also fun for them. I'm feeling a little ambivalent about not having found the secret ingredient of intrinsic motivation in this required blob of material. But I am grateful that, once again, an unexpected game structure generated what the late Gillian Hatch called "an unreasonable amount of practice."

The last word goes to the one student who put it best: "The crumpling is definitely the most satisfying part."

Beautiful, fluid chaos -- or what "learning in flow" actually looks like to the trained eye

One of the things I like best about my new school are my colleagues. In fact, I don't believe I could stand to teach English (instead of math) if I did not happen to have my particular grade-level team of amazing, insightful, reflective, open-minded collaborative English-teacher colleagues.

I'm not saying that teaching math is one big continuous picnic of sparkly rainbows, unicorns, and effortless class periods of absorptive learning, but in English Language Arts -- particularly at the middle school level -- you have to teach some of the most thought-numbing, soul-dissolving parts of the curriculum ever to torture the human mind. Spelling. Grammar. Vocabulary. I throw up in my mouth a little bit each time I have to schedule time for them on a new weekly assignments calendar. But they're a part of the curriculum that's mandated, and so they have to be taught.  Some things you just have to pinch your nose and swallow as quickly as possible.

On the other hand, or possibly as my reward for fulfilling these less satisfying obligations of my curriculum each week, I get to work with kids on one of the deepest and dearest endeavors of my heart. I get to teach them writing.

Now, one of the things I have learned in my many decades on this planet is that writing -- and learning how to write -- is MESSY. Learning to write a first draft is more about learning how to tolerate the waves of revulsion that come over you as you confront the your own feelings of inadequacy at what you put down on the page than it is about about learning how to structure a proper academic paragraph. In fact, I'm convinced that I could teach a goat to structure a proper academic paragraph. What takes genuine human maturity and emotional/psychological courage is learning to get a first draft down on paper.

For that, you have to gain the willingness to produce what writer Anne Lamott calls "a shitty first draft."

Since I'm not permitted to use that kind of language in a public school classroom with middle schoolers, I use the methods I first learned from, and later taught with, celebrated writing teacher Natalie Goldberg. Her system of "writing practice" emphasizes "separating the creator from the editor," and basically involves a small number of inviolable rules for producing your first draft of an idea. These are:

• keep your hand moving
• don't cross out
• don't worry about spelling, punctuation, grammar, or other rules
• lose control
• don't think
• go for the jugular
• follow and trust where your mind takes you
• give yourself permission to write the worst poop in America / on Planet Earth / in The Milky Way Galaxy

So for first drafts in my classroom, this is how we practice.

We close the door and the windows (so we don't bother anybody trying to do more traditional learning), I make earplugs available to anybody who needs to block out noise in order to think, and I tell my students to let it rip.

As soon as they finish a draft, they can come find me or a peer-editing partner and they or we look at what they have done. First we do so aloud but with no comment until the draft has gotten a first hearing. Natalie says, you can't know what you've written until after you've written it, so first we give it a hearing, then we go to town giving it a quick edit. We look for what our rubric tells us to look for. Then they go back to their desk (or to the floor, or to the back table -- wherever they want to be writing) and they bang out another draft.

The beauty of this system is that it gives them a lot of practice compressed into a very short space of time. Everyone can get a fast, free, immediate edit from a published working writer without time for judgment, shame, or a sense of disgrace to take hold. The kids have very quickly grasped how to use writing practice to harness the flow state, get their juices flowing, and not become too attached to what they've put down on paper. It gives them a wonderful experience of the feedback loop in writing and it gives them immersive time in the flow state that has rapidly improved everyone's basic writing skills noticeably and quickly.

The, um, downside of this system is that while we are having one of these in-class writing workshops, my classroom looks like a chaotic free-for-all. Or so I thought until the other day.

See, when we're doing this, even though I am not actually writing, I fall into the flow state too. I get absorbed in reading, writing, listening, editing, coaching, and cheerleading and I completely lose track of time. In a good way.

But the other day, one of my English colleagues and I were scheduled to trade classes halfway through the period to teach each other's classes part of a jigsaw lesson we were doing. I knew we were doing so, and I had everything ready and prepared, I just lost track of time. So when she arrived in my classroom to trade, she got treated to the sight of me on my knees next to somebody's desk giving one quick edit after another, kids reading aloud and giving each other peer edits, one kid sitting under the phone table next to the flag because he concentrates best down there with earplugs in using an Algebra textbook as his lap desk, and other kids writing (with earplugs in) either alone or together, but in what I imagined to look like total unfettered chaos.

Fortunately, this teacher is both an enlightened person and a reflective practitioner in her teaching, so she was absorbed for a few minutes just watching what was going on, taking it all in and finding it extremely effective and engaging.

Then she came over, tapped me on the shoulder, and reminded me that we needed to switch classrooms and finish the jigsaw activity.

At lunch later, she told me how much she had enjoyed getting the chance to watch our process unnoticed because it gave her a totally different vision of what an in-class writing workshop could be.

That made me feel both grateful and relieved, since I had felt certain that doing what I was doing was the straightest path to getting myself fired (except for the part about my kids' writing improving more and faster than many other teachers' classes).

Anyway, it was really interesting to have been observed while I myself was in the flow state.

Renegade Math Teacher Brings SBG into the English Classroom: Film at 11

SBG works so well in my Algebra classroom I've been looking for ways it could transfer into my eighth-grade English classroom -- especially with regard to writing. One of the hardest things about teaching persuasive or expository writing is that each "skill" is composed of multiple, discrete sub-skills, each of which could itself be broken down further into tinier and tinier (i.e., more refined) sub-skills. In many ways, it's an M.C. Escher-like process -- a mise en abîme of nested skills.

BACKGROUND

Our school and district use our own combination of two methods that work for us, adapted through our own collaborative practice, reflection, and research over many years to fit our district goals and population. As a starting point and foundation, we use the Jane Schaffer method for teaching the actual composition process and a slightly modified version of 6 Traits program for assessing the finished product (or the work-in-progress). 

What I like about the 6 Traits assessment system is that it has a strong SBG orientation. It is a rubric-based system that assesses idea development, organization, voice, word choice, sentence fluency, and conventions -- the main basic categories that young writers need to master to produce both competent and coherent arguments, paragraphs, and essays. In our district, we have whittled it down to a four-point scale, which gives most middle schoolers a fighting chance of making sense of their scores.

What I like less about the 6 Traits assessment system is the complexity and denseness of its rubric. As Edward Tufte, the infographics pioneer might say, its information density approaches near-total opacity.

While I appreciate that its authors are trying to be comprehensive, my experience is that, for a middle school or high school student trying to juggle the many skills that come into play in writing a persuasive paragraph, it's just too damned complicated.

MY IDEA

One of the things I've noticed early on in this school year is that even our strongest writing students tend to have only a tenuous grasp of what makes an effective topic sentence. And having taught literature at the university level, I have seen how this confusion tends to persist and worsen over time.

So my goal was to come up with an activity that integrated two tools I've found useful in SBG in the math classroom: (1) a clear, simple, compelling four-point rubric for judging the effectiveness of a topic sentence, and (2) an activity to give students practice in judging a wide range of topic sentences, along with practice in using the rubric as a basic for analyzing, debating, and justifying their assessments of each one.

With that in mind, I created the two tools which are attached here: a Topic Sentence Rubric and a "Judging Topic Sentences" activity for use in pairs or small groups. The "Judging Topic Sentences" activity sheet includes twenty topic sentences I wrote based on a recent writing prompt for that staple of the eighth-grade English curriculum, "Flowers for Algernon." The writing prompt (which was deliberately broadly written) asked the student to compose a persuasive paragraph regarding the author's message in the story about cruelty toward people with mental disabilities. 

I gave them 30 minutes in class to work together on the assessing activity before we came back together as a whole class to discuss and give closure to the process.

RESULTS

What was fascinating as I circulated among the groups was how quickly everybody grabbed hold of the idea of using criteria from the rubric as the basis of their judgments. Suddenly I was hearing arguments about how, yes, a certain claim was definitely true and supportable but was basically pretty trivial! I was also hearing students argue that another example made an "original and juicy claim," but that it was awfully long and wordy and could easily be improved with better word choice and sentence construction.

When we came back together as a class, I asked for examples of the worst topic sentence on the list and the best. The discussion was productive in that it brought students to an understanding that an "OK" topic sentence could kick off a really great paragraph if the writer used all the tools at his or her disposal. It also made them realize that a truly outstanding topic sentence could launch a truly mediocre paragraph if it was followed by weak use of evidence from the text and lame or badly written analysis and interpretation.

My fellow eighth-grade English teachers used this activity in their own way over the next few days and found it to be very helpful in getting students to think about what makes a strong and effective topic sentence.

So now it looks as though it will become a regular part of our writing curriculum. 

Another triumph for SBG!

Favorite Tweets #18 - Special Unicorns Edition

druinok Developing Mathematical Thinking with Effective Questions... http://to.pbs.org/mVya4R 7/15/11 9:52 AM

cheesemonkeysf @druinok Two similar resources I plan to blog about next week like this are: http://t.co/MGrbJIy AND http://t.co/R473yfN 7/15/11 10:07 AM

sarcasymptote Just got a text from a student: "I bought a wallet that has the first 2840 digits of pi printed on it." Success! 7/14/11 10:55 AM

druinok Yeah, so in case you ever wondered, a Dyson will suck up a thumb drive... the things you learn when hubby vacuums... #sigh #men 7/16/11 10:05 AM

park_star @jybuell @sarcasymptote @persidac @jackieb @druinok seriously Jason, it's like YOU'RE IN MY MIND. STOP IT. #ordont #dontneedtotweetanymore 7/11/11 9:56 PM

park_star @cheesemonkeysf @k8nowak @fnoschese @joshgiesbrecht I *LOVE* that Winnipeg is the slurpee capital of the world. #wearesoweirdhere 7/11/11 10:16 PM

someecards Teacher attempts to get class to stop playing Angry Birds by giving exam question about Angry Birds. http://t.co/O6PS4i5 7/11/11 3:05 PM

RT by @jazzbo51 of all people

k8nowak Just realized the technique for making pyth triples from squaring complex numbers thing is really just m^2-n^2, 2mn, m^2+n^2. #nowicansleep 7/12/11 9:18 PM

samjshah Math teacher bloggers and twitters! Huzzah! #loveit #TdF #pcmi http://twitpic.com/5pcztp 7/12/11 9:17 PM

mikeklonsky CapitolFax: Jonah Edelman’s blunt talk freaks out almost everybody. http://bit.ly/pK7oaO 7/13/11 9:36 AM

jybuell #cubelady "Governments should really be banned. Well some of them" Me: So who would ban them? ..... .(I blew her mind) 7/13/11 11:07 AM

fnoschese STAY AWAY FROM MY SCHOOL. http://www.khanacademyschools.com/home 7/13/11 10:32 AM

mctownsley Albert Einstein just added me to his Google+ circle. Not sure I have much to add there, guy. 7/13/11 12:13 PM

sophgermain last night in my first apartment. it was seriously the best. 7/13/11 9:25 PM

OMGFacts According to NASA's FAQ page, "There are no plans at this time to send children into space." 7/13/11 9:26 PM

sockington DISASTER PREPAREDNESS DRILL note mouth is open and no food in it WHAT DID WE LEARN IN THE DRILL 7/16/11 4:34 AM

samjshah Do some topology or read? I'm leaning towards the former but I'll listen to arguments for the latter. Go. 7/13/11 9:11 PM

samjshah http://samjshah.com/2011/07/13/make-it-better-drawing-with-geogebra/ @bowmanimal made a genius post. READ IT NOW. that is all. 7/13/11 2:47 PM

park_star @rdkpickle oh em gee, can we be best friends? 7/13/11 8:42 PM

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ThinkThankThunk @k8nowak Are you in Utah? With @samjshah? 7/13/11 8:22 PM

Mythagon @ThinkThankThunk @k8nowak @samjshah yes, but she's looking at a problem like she wants to light it on fire so I wouldn't disturb her... 7/13/11 8:28 PM

k8nowak @Mythagon @ThinkThankThunk @samjshah CARE BEAR STARE! 7/13/11 8:30 PM

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jybuell What's a good gift (book) for a new 4th grade teacher? Something practical that doesn't include the names Harry or Wong. 7/14/11 9:19 AM

sarcasymptote @jybuell i always get new teachers a bottle of whiskey 7/14/11 9:26 AM

jybuell @sarcasymptote that's more of a midyear gift when they really will appreciate it. 7/14/11 9:57 AM

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sarcasymptote @rdkpickle have fun. pretend i know enough about harry potter to make some witty remark here. 7/14/11 9:12 PM

druinok I don't think I had ever connected the equation of a circle (x2+y2=r2) to the pyth theorem... #ahamoments 7/15/11 8:43 AM

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cheesemonkeysf I wonder if any of these mathematical questions could be adapted to incorporate unicorns. 7/15/11 11:18 AM

RaisingUnicorns @cheesemonkeysf Yes. Everything can - and should - be adapted to incorporate unicorns. #MathMadeAwesome 7/15/11 11:33 AM

cheesemonkeysf FINALLY, A SOUL MATE! “@RaisingUnicorns: @cheesemonkeysf Yes. Everything can - & should - be adapted to incorp. unicorns. #MathMadeAwesome” 7/15/11 11:43 AM

RaisingUnicorns @cheesemonkeysf <:-) 7/15/11 11:58 AM

sarcasymptote @cheflincoln man, i was hoping for an image generator too 7/16/11 8:07 AM

cheflincoln @sarcasymptote Here you go!: http://jwebgen.com/generate There really is everything on the internet... 7/17/11 5:22 PM

sarcasymptote @cheflincoln @cheesemonkeysf jesus christ. the internet is weird. BUT AWESOME. 7/17/11 5:22 PM

cheesemonkeysf @sarcasymptote @cheflincoln Truly so! http://t.co/136GZqs 7/17/11 6:44 PM

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samjshah I just watched the last harry potter. It was a #TdF for sure. It was sooooo good. 7/15/11 10:40 PM

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jybuell Answer: Joanne Jacobs, David Brooks, Jay Mathews ....Question: Name 3 highly intelligent people who I want to slap with a trout 7/15/11 9:22 PM

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sophgermain watching my cat meet my parents' dogs is like watching animal planet. 7/15/11 11:18 PM

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woutgeo Getting drunk with your family can be a beautiful thing 7/16/11 1:01 AM

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DalaiLama We must generate courage equal to the size of the difficulties we face. 7/16/11 2:28 AM