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:

• 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. :-)

As these ideas whirled around, I felt envious of Kate Nowak's wonderful circumcenters lesson (from Geometry) that harnessed fun clues with the need to draw circumcenters on a map of the school as a motivation for hunting down numbered markers which she had tucked around the school.

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:

1. a set of graduated worksheets (one for each day) for them to use as practice problems
2. 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
3. a "rubber-stamping station" where I or they would stamp their completed AND checked worksheets to place in their...
4. group folder
5. a Ziploc bag or sheet protector stapled to the inside of the group folder where they could store their treasure tokens
6. a set of "clue" cards, one for each worksheet's treasure token, that students would take together with a...
7. 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)
8. 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

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.

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.

Cherry Blossom Season in the Classroom

On a morning like this, when I have too many papers to grade, lessons to plan, and comments to write, I try to remember the essential sweetness of this moment in the school year.

It's the season of cherry blossoms in the Bay Area, and I see them coming out everywhere. At first it's just a hint of pink fuzz. Each day on my way home, I marvel at three beautifully restored Victorians that are fronted by a trio of flowering plum trees. On my drive to school each morning, I pass a long hilly driveway lined with cherry trees that form an ephemeral pink outline. Each day this month it has gotten pinker and pinker, and I think to myself, What a brave act to plan a planting that erupts in color for such a brief display. So much care and tending to make sure every tree stays healthy at at the same stage of growth to create this fragile outline that lasts for only a few weeks of the year year.

This is the same feeling I get about my classes right about now. They are no longer forming. They are formed. They just are.

My students and I know each other. And each class has formed a community. Expectations are clear, even when they are not being met. All the roles are in place, and we've been able to loosen up on enforcement of the rules a little bit. Students are now allowed to talk softly during morning announcements and to sit wherever they please while I take attendance. They know my blind spots and view morning attendance as something we are all responsible for together. When I call out, "Where is So-and-So?" I get an immediate response. "He's not here today" or "He's in the Bat Cave," which is our code for hanging out on the floor under the table in the front corner of the classroom where the phone lives.

We have our own dialect and in-jokes that no one else would get. When one of the quietest students says something without first raising a hand, one of the most boisterous students will bellow, "GIVE ___ A DETENTION!!!" And we all laugh. We have a shared history. We all know why it is funny.

Students also know me well enough now to tease me about my idiosyncracies. A student furrows his brow at an unexpected result and another student will call out, "TRUST THE FRACTIONS!" 

They know me. They get me. And I get them.

That is how I know we are entering the beginning of the end. This moment is fleeting. We are passing through it just as it is passing through us. And like the cherry blossoms, sooner than I expect, it will be gone, never to be experienced again.

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.