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.

SBG, Intrinsic Motivation, and the "Grading" of "Homework"

One of the surprising parts of this latest round of parent conferences was the number of parents who wanted to talk to me about why their child is suddenly interested and engaged in learning mathematics when — as I gathered — this was not previously always the case.

I teach in a district which places a very high value on school, teachers, and academic achievement, so this conversation in and of itself was not the surprising thing.

I explained about using Standards-Based Grading, frequent formative assessment, and the remediation and reassessment method I first stole learned about from Sam Shah and others in my Twitterverse/blogosphere orbit, but two things came up again and again during this round of conversations which really caught me by surprise: my emphasis on in-class autonomy as a mode of differentiation and my approach to grading homework.

In-Class Autonomy
My Algebra 1 classes are unusual for a middle school in that they contain a mixture of 7th and 8th graders. I find there are huge benefits to this kind of heterogeneous grouping. For one thing, the students in one grade tend not to have met the students in the other grade, so there are fewer preexisting status issues to contend with among math learners (for an excellent discussion of working with status issues in the math classroom, see Between the Numbers' presentation on this issue from the Creating Balance conference on Math & Social Justice in an Unjust World). For another, it creates a healthy competitive atmosphere in which neither age range wants to be shown up by the other. 8th graders do not want to have their clocks cleaned by a bunch of 7th-grade whippersnappers, and this is an excellent antidote to the problem of 8th grade "senioritis." At the same time, 7th graders are somewhat intimidated by being around the older kids, and that motivates them to bring their A game to class to help them compensate for any feelings of insecurity. The mixing of students encourages everybody to notice and value what others bring to the situation and to stay focused on their own work.

I am pretty much tied to the curriculum, our pacing guide, and the state testing schedule, with minor variations allowed to deal with large-group (or whole-group) lostness as need be. But that means that there are times when the most with-it students could get frustrated or bored if I did not provide them with some differentiated alternatives to keep them engaged while I work with the 75% of the class who are catching up to them.

So I allow students who are ahead of others to either "work ahead" or "dive deeper" during these times. I see no reason to bore them when I can challenge them and call them back to work with the whole group when I need everybody (or when there is a whole-class activity they do not wish to miss out on). I provide them with self-selectable options and I find that it works out really well.

This fits well with Dan Pink's thesis in his book Drive that intrinsic motivation arises from our basic human desires for autonomy, mastery, and purpose. Middle-schoolers do not have much autonomy in their own lives, so giving them a little bit in the classroom goes a long way towards both motivation and harmony (which is how I prefer to think of "classroom management").

Apparently this has been my unwitting secret to getting many of my students' cooperation. Students who would be bored or frustrated at being tethered to a whole-class pace that is either too fast or too slow feel happy and engaged because I try to make it possible for them to work at a pace and a depth that is meaningful for them. I did not realize this was such a giant change for so many of them.

The Grading of Homework
The other thing that seems to be working for my students is the change in emphasis on the "grading" of "homework," in that I do not actually grade their homework.

I was convinced long ago that Sam Shah's approach to Binder Checks is the best way to place an appropriate value on homework — namely, that homework is work one does at home to improve one's own learning. In an SBG world, mastery is measured by the student's performance on assessments — not by the teacher assessing each of 40 problems that one has worked on at home. The purpose of homework is to provide practice and investigation time, in addition to exposure to different kinds of problems and issues that may come up. The purpose of "assessing" homework is to assist the student in developing good study habits and organizational habits so that homework becomes a meaningful part of their school lives.

When I moved from high school to middle school, I discovered that the full binder check approach was a recipe for discouragement. It seems to be a developmental issue. So instead, I have modified the program into a system of "mini-binder checks," in which I check the corresponding homework "chunk" while they work on the test/assessment on that particular chapter/chunk. The "grade" or "score" they receive for "homework" is merely a completion score. It is not a problem-by-problem assessment of their thought process on each homework assignment.

Apparently it's a novel approach to trust motivated middle school students to do their homework and check it all at the assessment point in one fell swoop. At conferences this week, I heard some pretty upsetting stories about students staying up until midnight or one o'clock in the morning, trying to get all their math homework done so they would not get punished and graded down. I heard stories of students I think of as super-mathletes breaking down into tears and meltdowns because they couldn't get their homework all done and they got punished (and shamed) in class because of this failure. So I heard a lot of appreciation that I assign a reasonable amount of homework and expect them to take ownership of getting it all done in time for a reasonable assessment of completion.

It makes me kind of sad to hear these stories because I think of the students in my Algebra classes as pretty joyful learners. And it also saddens me because I do not see these practices as leading toward the "positive dispositions toward mathematics" that we are supposed to be building.

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."

Something's happening here... what it is ain't exactly clear...

A funny thing happened today on the way to basic linear equation and function skills today in Algebra 1:

students started using precise mathematical language without my having to prod them.

First, a student demonstrating a problem at the board announced matter-of-factly, "Well, first you use the distributive property, making sure not to drop any negative signs. Then you add '2x' to both sides of the equation."

I almost swallowed my own tongue.

Stop, children, what's that sound... everybody look what's going down... 

Another Turning Point in Algebra 1: from larva to butterfly

When things go right in Algebra 1 -- and that is by no means a given for all students in all Algebra 1 classes out in the real world -- there are some truly breathtaking paradigm shifts that take place in student thinking: the moment when a critical mass of students begin to understand that absolute value and inequalities have to do with a relationship with zero; the moment when students begin to grasp the higher concept of groupings (parentheses, brackets, braces, the occasional fraction bar) as having meaning for operations; the light bulb moment they have when they begin to occupy the distributive property.

I admit it -- I'm a sucker when it comes to witnessing a student having a really intense light bulb moment. My own mathematical light bulb moments were very hard-won, so perhaps that gives me a special appreciation for them in others. Still, there's something deeply  moving about the courage it takes to let go of your old frame of reference when you know it's worn out but you don't yet know -- or trust -- what will come in its place. It's a frightening emotional moment in one's learning process, and I know that from first-hand experience in the math classroom.

The closest description I know of how it feels to undergo this transformation comes from A.H. Almaas:

  So there is a need for an attitude of allowing, allowing things to emerge, to change, to transform, without anticipating how this should happen. You can direct things only according to the way you are now. You can conceive of the future only according to the blueprints you already know. But real change means that the blueprint will change.

  The only thing you can do is to be open and allow things to happen, allow the butterfly to emerge out of the larva and be a different being. You might be amazed, saying, "All this time I thought I had to crawl faster! I didn't know it was possible to fly." It is possible to fly, but if you want to remain a larva, you can learn to crawl a little faster. You can even learn to crawl sideways. But it will never occur to you that you can fly. You see things flying around you but don't think of flying because you haven't got wings. If you allow things to happen, you might find that you do have wings and that you are flying around. (Diamond Heart: Book One,  page 153). 

I was privileged to experience the first sparks of such a turning point today in class, as students began to grasp the relationships between and among all the different elements and aspects of linear equations, graphs, and functions they need to be able to take apart and recombine in dozens, if not hundreds, of different ways. Given a linear equation, find its intercepts. Given the intercepts, find the equation of the line. Given the slope of a line and one intercept, find the equation of the line. Given the slope-intercept form of a line, find the standard form. Given the standard form, find the slope-intercept form. Given the slope and some non-intercept point on the line, find me the equation of the line and write it in slope-intercept form. The whole quest involves a collection of movable parts, a juggling act at at first strikes some discouraged students as ludicrous bordering on the impossible. I'll never be able to manipulate all those moving parts, the discouraged student despairs. I'm a larva -- not a butterfly! What kind of crazy-ass thinking are you asking me to engage in here? This is insane! Absurd! The best I can hope for it to crawl a little faster, maybe to be able to crawl sideways and someday do The Twist. But fly in the air like that? Are you totally nuts?

And so it takes a certain amount of what I like to call wallowing. Wallowing in the confusion and the array of perplexing terminology and movable pieces that have to be taken apart and put back together like so many parts of a clock.

I have a friend who is one of those people who can fix literally anything. The fastest way to get something broken fixed is to tell him it's hopelessly broken and can't be fixed. He doesn't know the meaning of the words "can't be fixed." He doesn't trust that as an existential state of being. For him, hearing the words "it can't be fixed" is like somebody double-dog-daring him to prove them wrong. He can't stop himself. He sits with the problem and the pieces and the brokenness until he has resolved it. To him there is no other way.

For me, this situation is exactly the opposite. I look at the broken alarm clock and think to myself, Well, that's clearly hopeless. It must be time to move.

Many of my students feel this way when confronted with the point-slope, point-point, slope-and-intercept, and other linear equation/function/graph skills that are central to Algebra 1. They throw up their hands and yell, "What do you want from me? I'm just a larva, trying to crawl a little faster here? What are you trying to throw me into???"

And so we simply wallow in the mess.

I remind them of my friend Sam Shah's motto, "Take what you don't know, and turn it into what you do know." I suggest some of my friend Avery's habits of mind ideas. I encourage them to tinker with things on scratch paper or graph paper. Make a table. Plug in values. Draw a graph. Try rearranging the elements. See if you can find the x- and y-intercepts.

It doesn't really matter what they do, so long as they do something. And since I won't just give them the answers, they have to practice tinkering and struggling. I believe that learning to struggle with problems is one of the most essential skills a person can develop in life, much less in math.

And a curious thing began to happen in Algebra 1 today. Students began to articulate different distinct patterns in and among the equations and elements. Like, if you are given the equation in slope-intercept form, you don't actually have to DO anything extra to find the slope of the line it represents. It's already there. It's a freebie. There's knowledge there and you get both to use it and to keep it.

All afternoon students were figuring little things out like this in class, and it reminded me of the importance of having time and space -- and support -- to wallow in the beautiful, beautiful logic and relationships of algebraic thinking. I realized that I sometimes get convinced that I have to add something external -- something "extra" -- to bring the mathematics to life.

But sometimes all they need is their problems, their minds, and the friction that comes from a little push to help you get to the next level.