We spent this morning in self-selected groups, working on Exeter Math 1, Math 2, or Math 3 problems. I worked in the Math 1 group. We agreed to work in a free-form way on the first four pages of the Exeter Math 1 problems, with discussion, collaboration, and analysis in any way we wanted.
I am still digesting and processing my experience, but here are some of the things I noticed.
My general feeling is that the Math 1 problems are all about cultivating independence. The sequence begins with an investigation into rates, but the work requires the learner to actively use what you know.
No spoon-feeding, no spectators.
As a learner, I found I had to focus on reading, interpreting, and decoding problems, listing information (what do I know? what do I need to know?), organizing it, and identifying my objective in the process. I also noticed that everything went more smoothly when I gave names to things and actively identified equivalences. Naming things and identifying parts of the problems gave me a way into organizing my information and my thinking.
I enjoyed working through problems and brainstorming about methods/strategies. I am now wondering, how I can use problems like these to cultivate independence and problem-solving with my students? How much scaffolding would middle-schoolers need to get started?
One of the things I am really interested in is how @k8nowak uses this kind of method to set up some "productive struggle" for a lesson. Oo, I think I'll go over to the other table and ask her!
cheesemonkey wonders
Thursday, July 19, 2012
Sunday, July 15, 2012
On being a non-native speaker in search of autonomy in the math classroom
I have a confession to make — I'm less like you than you might have thought.
I was never one of the students for whom mathematics came naturally. If I worked really hard and did twice as much practice as the worst "good" math students in my classes, then I was able to keep up. But I never experienced math the way so many of you describe your experiences of it. Don't get me wrong — I've always envied you. But I always seemed to need many more hours of privacy and "think time" and lots of step-by-step review of my own notes and rewriting what seemed to me to have happened in class. Slowly the ideas and understanding would grind their way into my body and mind, punctuated by multiple nights of dream-time reorganization of my learning efforts, until I could clumsily work along with the rest of you top students.
Carlos Castaneda makes a distinction between "stalkers" and "dreamers" — those who learn by pursuing their learning like hunters, finding and identifying and following their learning clues like predators hunting their prey, patiently tracking ideas through the jungle until their arrow meets the target and it becomes theirs. And then there are the "dreamers" — like me — who stumble along in mortal terror and confusion until we become one with the mud and the mystery.
"Wallowing" is a better description of how my primary learning state in mathematics proceeds. Day after day, I'm confused, mute, and incapable, unaware that I'm holding my breath like a deer in the "fight or flight" state, and praying that the teacher will not discover the secret shame that my body and my unconscious mind are desperately trying to conceal — the fact that I don't learn mathematics the way you are trying to teach me, the way you yourself learn mathematics. I wallow and I continue to wallow until the mysterious alchemy of osmotic transfer has occurred, until I have absorbed the mathematics as the mathematics have absorbed me.
I've always worried that my sense-making-reason-making-understanding mechanisms are defective because they don't accept the inputs being provided as the sole ingredients required for my learning. My body-mind cannot seem to identify them as nutrition. Sometimes it reacts to them as if they were pathogens I must be protected against. And that reaction triggers the fight-or-flight response, and I find myself once again holding very, very still to avoid your displeasure, disappointment, and frustration — and your discovery.
This is the curious thing for me about walking among you these days — I am a non-native speaker of this language of mathematics and mathematics teaching that we share. You speak to me as if I too grew up with this language, played with it from infancy, organizing my toys and numbers and ideas in my nest as you did. You speak to me as if I too were a native speaker, as if numbers were my first language. The other day Kate tweeted that her mom teases her that, when she starts teaching in the Southern Hemisphere, she may begin to swirl in the opposite direction from the way she has always operated up here, that, say, she will go from being someone who is naturally brilliant at learning and teaching mathematics to someone who is suddenly "good at poetry and relationships." When I heard that, I found myself wondering if I would experience the same thing if I moved that far south, but in reverse — if I would suddenly become someone who is naturally fluent in mathematics and does not speak it with a funny accent.
As a learner, I listen, I tinker, I do, I experiment, I reverse my thinking, I try to find a pattern, plus I take copious notes along the way so I can use them later to wallow in the material privately until I begin to feel the signs of absorption. I know that I will need time and space to live with my confusion, to wrestle with it, to struggle, and to hand off the baton of responsibility to my subconscious mind to rearrange the learning while my conscious mind and body take a break. Like the student volunteers in Robert Stickgold's sleep research lab at Harvard, playing a downhill skiing simulation video game before napping and playing again to assess their improved results, I need time and space for my unconscious mind to reorganize my daily fragments of learning. I need time and space to sink down into that primal, mysterious, adaptive soup before I will be able to notice that I am holding my breath once again in a fight-or-flight reaction.
Only then will I be able to breathe again and relax.
I don't know why learning mathematics is this way for me. I only know that it is. And I that gives me the confidence to say out loud that, as a learner, it simply doesn't work for me to be told to relax or to follow someone else's investigative process. It isn't always sufficient for someone like me to be invited into your curiosity about the purpose of mathematics—no matter how engaging your digital media representations are. Likewise, it isn't always sufficient for someone like me to be invited to pave a pathway to confidence through mastery of a checklist of concepts or procedures.
What does give me confidence — as well as engagement and determination — is being trusted and granted the autonomy to trust my own learning process and my own deeper wisdom about my own learning.
Of course, the tricky thing about cultivating autonomy, though, is that autonomy is by definition an all-or-nothing proposition. As a teacher, you can't grant autonomy cautiously or halfway, and you can't assume you know what another person needs in order to feel autonomous in their learning activity. If there is one thing I have learned from twenty-plus years of meditation practice and teaching, it is this:
Reframing strategies are also valuable in encouraging autonomy in capable and curious students, providing variety and texture to their experience as well. But most importantly, by engaging everyone in the room in their own personal and collective pursuit and experience of flow, they help to create the social and emotional conditions under which students can experience lasting and meaningful engagement.
What I am arguing here is that a key to developing a sense of autonomy in our math classrooms is to harness reframing strategies that support the greatest number of students in experiencing the flow state while doing mathematics as much of the time as is possible.
By blending these strategies with those that cultivate a sense of purpose (such as Dan Meyer's digital media + "Any Questions?"), and with those that, like SBG, clean up the tense and often fraught expectations and communication of mastery, I believe we can dramatically boost the percentage of students in the room who feel autonomous and taste a sense of flow while doing mathematics.
And that, it seems to me, should be the goal we are targeting.
I was never one of the students for whom mathematics came naturally. If I worked really hard and did twice as much practice as the worst "good" math students in my classes, then I was able to keep up. But I never experienced math the way so many of you describe your experiences of it. Don't get me wrong — I've always envied you. But I always seemed to need many more hours of privacy and "think time" and lots of step-by-step review of my own notes and rewriting what seemed to me to have happened in class. Slowly the ideas and understanding would grind their way into my body and mind, punctuated by multiple nights of dream-time reorganization of my learning efforts, until I could clumsily work along with the rest of you top students.
Carlos Castaneda makes a distinction between "stalkers" and "dreamers" — those who learn by pursuing their learning like hunters, finding and identifying and following their learning clues like predators hunting their prey, patiently tracking ideas through the jungle until their arrow meets the target and it becomes theirs. And then there are the "dreamers" — like me — who stumble along in mortal terror and confusion until we become one with the mud and the mystery.
"Wallowing" is a better description of how my primary learning state in mathematics proceeds. Day after day, I'm confused, mute, and incapable, unaware that I'm holding my breath like a deer in the "fight or flight" state, and praying that the teacher will not discover the secret shame that my body and my unconscious mind are desperately trying to conceal — the fact that I don't learn mathematics the way you are trying to teach me, the way you yourself learn mathematics. I wallow and I continue to wallow until the mysterious alchemy of osmotic transfer has occurred, until I have absorbed the mathematics as the mathematics have absorbed me.
I've always worried that my sense-making-reason-making-understanding mechanisms are defective because they don't accept the inputs being provided as the sole ingredients required for my learning. My body-mind cannot seem to identify them as nutrition. Sometimes it reacts to them as if they were pathogens I must be protected against. And that reaction triggers the fight-or-flight response, and I find myself once again holding very, very still to avoid your displeasure, disappointment, and frustration — and your discovery.
This is the curious thing for me about walking among you these days — I am a non-native speaker of this language of mathematics and mathematics teaching that we share. You speak to me as if I too grew up with this language, played with it from infancy, organizing my toys and numbers and ideas in my nest as you did. You speak to me as if I too were a native speaker, as if numbers were my first language. The other day Kate tweeted that her mom teases her that, when she starts teaching in the Southern Hemisphere, she may begin to swirl in the opposite direction from the way she has always operated up here, that, say, she will go from being someone who is naturally brilliant at learning and teaching mathematics to someone who is suddenly "good at poetry and relationships." When I heard that, I found myself wondering if I would experience the same thing if I moved that far south, but in reverse — if I would suddenly become someone who is naturally fluent in mathematics and does not speak it with a funny accent.
As a learner, I listen, I tinker, I do, I experiment, I reverse my thinking, I try to find a pattern, plus I take copious notes along the way so I can use them later to wallow in the material privately until I begin to feel the signs of absorption. I know that I will need time and space to live with my confusion, to wrestle with it, to struggle, and to hand off the baton of responsibility to my subconscious mind to rearrange the learning while my conscious mind and body take a break. Like the student volunteers in Robert Stickgold's sleep research lab at Harvard, playing a downhill skiing simulation video game before napping and playing again to assess their improved results, I need time and space for my unconscious mind to reorganize my daily fragments of learning. I need time and space to sink down into that primal, mysterious, adaptive soup before I will be able to notice that I am holding my breath once again in a fight-or-flight reaction.
Only then will I be able to breathe again and relax.
I don't know why learning mathematics is this way for me. I only know that it is. And I that gives me the confidence to say out loud that, as a learner, it simply doesn't work for me to be told to relax or to follow someone else's investigative process. It isn't always sufficient for someone like me to be invited into your curiosity about the purpose of mathematics—no matter how engaging your digital media representations are. Likewise, it isn't always sufficient for someone like me to be invited to pave a pathway to confidence through mastery of a checklist of concepts or procedures.
What does give me confidence — as well as engagement and determination — is being trusted and granted the autonomy to trust my own learning process and my own deeper wisdom about my own learning.
Of course, the tricky thing about cultivating autonomy, though, is that autonomy is by definition an all-or-nothing proposition. As a teacher, you can't grant autonomy cautiously or halfway, and you can't assume you know what another person needs in order to feel autonomous in their learning activity. If there is one thing I have learned from twenty-plus years of meditation practice and teaching, it is this:
You can't control somebody else's autonomy. You can only cease to interfere with their learning process.This is why reframing strategies — such as games and game-like activity structures — can be such powerful additions to the learning environment. They free learners to choose a different focus for their conscious attention — something other than their ability to understand or not understand the learning target. For the especially shut-down or anxious learners, such reframing activities give our conscious minds a displacement activity (finding the treasure, completing a turn in a game) that keeps it busy and out of our hair while our unconscious minds and bodies can help us organize and reorganize the material before us into understanding.
Reframing strategies are also valuable in encouraging autonomy in capable and curious students, providing variety and texture to their experience as well. But most importantly, by engaging everyone in the room in their own personal and collective pursuit and experience of flow, they help to create the social and emotional conditions under which students can experience lasting and meaningful engagement.
What I am arguing here is that a key to developing a sense of autonomy in our math classrooms is to harness reframing strategies that support the greatest number of students in experiencing the flow state while doing mathematics as much of the time as is possible.
By blending these strategies with those that cultivate a sense of purpose (such as Dan Meyer's digital media + "Any Questions?"), and with those that, like SBG, clean up the tense and often fraught expectations and communication of mastery, I believe we can dramatically boost the percentage of students in the room who feel autonomous and taste a sense of flow while doing mathematics.
And that, it seems to me, should be the goal we are targeting.
Saturday, July 7, 2012
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.
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.
Wednesday, April 18, 2012
Rational Expressions Treasure Hunt (or, Intrinsic Motivation for the Intrinsically Unmotivat-able Bits)
We are coming up on our state testing window (STAR testing in California, ugh), which means it's time to finish "covering" all the so-called key standards material in preparation for the last-minute test prep and hand-wringing.
In our Algebra 1 curriculum, the last unit before state testing is the Rational Expressions unit, which I find to be one of those inherently frustrating and procedural sections.
The good news of the rational expressions unit is that it gives Algebra 1 students an opportunity to see how some of the foundational skills and concepts they've been learning can come together to provide some very sophisticated conceptual and computational tools.
The bad news is, students are tired -- tired of the routine of observation - guided practice - independent practice - followed by a chapter test. And they're just plain tired in the sense that they've absorbed a lot in every class they're in and basically their brains are kind of full.
So I reviewed my notes on Dan Pink's book Drive to see how I could weave together his ideas of autonomy, mastery, and purpose. And that gave me the following ideas:
It occurred to me that I could use a treasure hunt idea (organized more as a geocaching or letterboxing hunt) as an intrinsically self-motivating reward system -- coupled with a series of self-check-able problem sets/worksheets -- to give students a little break from routine and a reason to work together purposefully to practice simplifying rational expressions and finding the excluded values.
So here are the components I created:
A couple of observations. First of all, your "clue cards" will be different from mine based on your school and where you wish to send your students on their searching. Being a California indoor-outdoor type of school that is built into a hillside, we have a lot of beautifully xeriscaped/landscaped pocket gardens that made for good treasure box hiding places. So my clues said stuff like, "Facing room ELEVEN, TWO TIMES TEN paces or so to your right, you'll find a bathtub fit for a bird. Look below the bushes behind..." Or "Head toward LUCKY 13 and look for the LAVENDER HEDGE. The treasure box is tucked behind..."
Students responded positively to this shift toward autonomy. They delight in completing a practice task that grants them permission to snuffle around a little outside in the fresh air. And they seem to love unraveling the clues, maybe because they love reading and thinking about themselves and their own environment. The Hall Pass requirement gave them a clear set of expectations and accountability, and even my least accountable students have been taking it seriously, which is a refreshing change of pace for all of us.
Their assessment will come from my review of their group project folder -- its tokens, its completed and corrected worksheets, and the presence or absence of worksheets for each student in the group.
The worksheets and sample hall passes and clues are up on Box.net
In our Algebra 1 curriculum, the last unit before state testing is the Rational Expressions unit, which I find to be one of those inherently frustrating and procedural sections.
The good news of the rational expressions unit is that it gives Algebra 1 students an opportunity to see how some of the foundational skills and concepts they've been learning can come together to provide some very sophisticated conceptual and computational tools.
The bad news is, students are tired -- tired of the routine of observation - guided practice - independent practice - followed by a chapter test. And they're just plain tired in the sense that they've absorbed a lot in every class they're in and basically their brains are kind of full.
So I reviewed my notes on Dan Pink's book Drive to see how I could weave together his ideas of autonomy, mastery, and purpose. And that gave me the following ideas:
- AUTONOMY - students needed a unit which relies more on self-checking than on getting a teacher seal of approval (or a rubber stamp);
- MASTERY - students seemed to need a unit which would emphasize tracking their mastery of skills and concepts, which meant giving them lots of opportunities both to do and to spot patterns in their doing;
- PURPOSE - students needed a sense of purpose to the overall activity, rather than the "bigger picture" sense of Dan Meyer's WCYDWT (What Can You Do With This?) - to provide some light-hearted gratification or, as some might put it, a cheap thrill. :-)
It occurred to me that I could use a treasure hunt idea (organized more as a geocaching or letterboxing hunt) as an intrinsically self-motivating reward system -- coupled with a series of self-check-able problem sets/worksheets -- to give students a little break from routine and a reason to work together purposefully to practice simplifying rational expressions and finding the excluded values.
So here are the components I created:
- a set of graduated worksheets (one for each day) for them to use as practice problems
- a handwritten Answer Key that I slipped into sheet protectors and tacked up on the front bulletin board for students to check their work against
- a "rubber-stamping station" where I or they would stamp their completed AND checked worksheets to place in their...
- group folder
- a Ziploc bag or sheet protector stapled to the inside of the group folder where they could store their treasure tokens
- a set of "clue" cards, one for each worksheet's treasure token, that students would take together with a...
- Hall Passes (which clearly states that I have given them permission to snuffle around in the designated area to find their treasure tokens and come back to my classroom)
- one weatherproof plastic Gladware container PER treasure box/worksheet to contain the treasure tokens for each worksheet/level (I used Woodsies stars that I bought at Michael's and labeled them 11-2a or 11-2b to correspond to their worksheets, but you could also use punched-out paper or cardboard shapes of any type); treasure boxes were labeled with masking tape and also contained a marker and an index card on which students would write their group's initials when they removed a token
A couple of observations. First of all, your "clue cards" will be different from mine based on your school and where you wish to send your students on their searching. Being a California indoor-outdoor type of school that is built into a hillside, we have a lot of beautifully xeriscaped/landscaped pocket gardens that made for good treasure box hiding places. So my clues said stuff like, "Facing room ELEVEN, TWO TIMES TEN paces or so to your right, you'll find a bathtub fit for a bird. Look below the bushes behind..." Or "Head toward LUCKY 13 and look for the LAVENDER HEDGE. The treasure box is tucked behind..."
Students responded positively to this shift toward autonomy. They delight in completing a practice task that grants them permission to snuffle around a little outside in the fresh air. And they seem to love unraveling the clues, maybe because they love reading and thinking about themselves and their own environment. The Hall Pass requirement gave them a clear set of expectations and accountability, and even my least accountable students have been taking it seriously, which is a refreshing change of pace for all of us.
Their assessment will come from my review of their group project folder -- its tokens, its completed and corrected worksheets, and the presence or absence of worksheets for each student in the group.
The worksheets and sample hall passes and clues are up on Box.net
Sunday, March 25, 2012
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.
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.
Sunday, March 18, 2012
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:
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.
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:
- the ability to COUNT discrete OBJECTS
- a deep familiarity with a wide range of QUANTITIES OF QUALITATIVELY DIFFERENT KINDS (such as both countable and uncountable quantities)
- the ability to COMPARE QUANTITIES OR COLLECTIONS of objects, assessing both similarities and differences regardless of their qualitative kind(s)
- the ability to use established notation for all of these (Nunes and Bryant, pp. 1-20)
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.
Thursday, March 15, 2012
Waiting for Gratitude: a reflection on pink slip day (or, Beware the Ides of March)
I got my pink slip early this year, and I'm finding that waiting for gratitude is a bit like waiting for Godot. But my dharma practice teaches me that waiting is just another word for trying to find a doctor's note that will excuse me from this human experience of groundlessness.
So as long as I keep waking up early anyway, I've been getting my butt out of bed and onto the couch to do writing practice on how this particular episode of groundlessness really feels -- trying to capture on the page what I am experiencing as I keep running out of runway.
Wile E. Coyote is my patron saint of groundlessness. I keep an enameled pendant depicting him hanging over my desk. He is nose-down, hanging by his left foot, having chased the Road Runner over the cliff yet again.
Like me, he really ought to know better, but he is a slow study. Like me, each time it happens, he looks out at the camera and blinks twice, before he crashes to the canyon bottom.
The hardest part of today was the fact that my students remained so bloody happy to see me and to spend time with me. My Algebra students wanted to wrestle with factoring nonmonic quadratic trinomials, while my English students wanted to brainstorm on their "Product of the Future" ideas for our science fiction unit. Being eighth-graders, most of their ideas for outstanding products of the future revolved around bathroom components, clothing/shoe/makeup accessories, or variations on teleporting devices.
My only product idea was for a Recess-Extender -- one that would stop time and allow me to take a nap during recess after I bolt my yogurt.
The best part of today was doing math with students -- finding patterns as we factored nonmonic quadratic trinomials, and saying "nonmonic quadratic trinomials." They love the words of mathematics, as much as the language of algebra. Anything they can use to stun their parents at the dinner table is a good day's work.
At the end of class in English (as we were cleaning up from the product of the future brainstorming), two of my 8th-graders who are in Geometry asked me about a problem they were struggling with. For about three minutes, I lost myself in the Pythagorean Theorem and in wondering how -- or whether -- we could prove that the area of the black region of a hexagon was equal to the white region of the hexagon.
This led to a quick discussion about equality, equivalence, and proof. And that made me feel sad as I remembered that I had just been laid off.
So as long as I keep waking up early anyway, I've been getting my butt out of bed and onto the couch to do writing practice on how this particular episode of groundlessness really feels -- trying to capture on the page what I am experiencing as I keep running out of runway.
Wile E. Coyote is my patron saint of groundlessness. I keep an enameled pendant depicting him hanging over my desk. He is nose-down, hanging by his left foot, having chased the Road Runner over the cliff yet again.
Like me, he really ought to know better, but he is a slow study. Like me, each time it happens, he looks out at the camera and blinks twice, before he crashes to the canyon bottom.
The hardest part of today was the fact that my students remained so bloody happy to see me and to spend time with me. My Algebra students wanted to wrestle with factoring nonmonic quadratic trinomials, while my English students wanted to brainstorm on their "Product of the Future" ideas for our science fiction unit. Being eighth-graders, most of their ideas for outstanding products of the future revolved around bathroom components, clothing/shoe/makeup accessories, or variations on teleporting devices.
My only product idea was for a Recess-Extender -- one that would stop time and allow me to take a nap during recess after I bolt my yogurt.
The best part of today was doing math with students -- finding patterns as we factored nonmonic quadratic trinomials, and saying "nonmonic quadratic trinomials." They love the words of mathematics, as much as the language of algebra. Anything they can use to stun their parents at the dinner table is a good day's work.
At the end of class in English (as we were cleaning up from the product of the future brainstorming), two of my 8th-graders who are in Geometry asked me about a problem they were struggling with. For about three minutes, I lost myself in the Pythagorean Theorem and in wondering how -- or whether -- we could prove that the area of the black region of a hexagon was equal to the white region of the hexagon.
This led to a quick discussion about equality, equivalence, and proof. And that made me feel sad as I remembered that I had just been laid off.
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