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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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