cheesemonkey wonders

cheesemonkey wonders

Tuesday, December 6, 2011

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.


  1. What a beautiful description of learning! I'm a coach and am working with teachers whose schools and curricula are encouraging them to relax into wallowing. It's a struggle! Teachers want to guide, instruct, encourage, and ultimately, swoop in and solve. It's also my experience that wallowing looks, to the outsider, less organized & purposeful, but is in fact more purposeful. I would love to read about the ways you planned and organized work for students, what you listened for to know when to push individuals or the whole group, how you allotted time for wallowing and when you knew it was over, what students needed to practice for fluency and what they needed to make sense of... (I could go on, but really, I'd love to read anything else you wrote on this topic!)

  2. @Max- Thanks for your affirming comments about wallowing. I think it is not just teachers who want to swoop in, guide, instruct, encourage, and solve. Administrators too are eager for teachers to jump in, assess and immediately remediate -- even before the student has had to fully process their mistaken understanding. I had my principal come in to observe during a Math Basketball review session and one of her questions afterwards was, how do you know who has achieved understanding and who has not. I had to explain to her how much formative assessment takes place in the many, many activities that precede math basketball review -- during Solve-Crumple-Toss (where I see, assess, and stamp their individual work before they can crumple and toss, giving me time to take inventory of individual understanding), during individual or pair-wallowing, and during other during-the-lessons activities.

    I learned so many new techniques for teaching through wallowing from Henri Picciotto at The Urban School/Center for Innovative Teaching here in SF this past summer (see his ). His version of wallowing activities include discovery-based processes such as students working in pairs to gather pretty large sets of data from which they can extract patterns. Also Kate Nowak ( ) is a wizard at creating new ways for students to experience and therefore wallow in new mathematical processes (I am thinking here of her Treasure Hunt activities and her Add 'Em Up collaborative activities).

    What has become clearer than ever to me this year is the fact that wallowing activities need to (a) harness intrinsic motivation (à la our #drivebook book chat) and (b) help students slip into an unconscious water-slide on-ramp into their own state of "flow." That way they lose track of their conscious/cognitive "figuring out" processes and tap into their subconscious, deeper intuitive wisdom to connect with the patterns they need to grasp and internalize. The way I see it, the conscious mind that "tries" (i.e., that strives and efforts to control things) is the biggest obstacle to intrinsic motivation and tapping into intuitive, sub-conscious understandings. The harder you try, the more elusive understanding becomes.

    Only by becoming gently open, curious, and self-aware of the messages our bodies give us about what we are actually unconsciously reacting to can we become free of the conditioned habits that block us from allowing ourselves to learn. The learning I wrote about above that seemed to be going to smoothly and easily ran straight off the road into a ditch as soon as students tried to grab hold of the right answers instead of wallowing in the mud of the questions. So now I am looking into ways I can create new activities to help students reconnect with the original curiosity and openness they had at the beginning. I'm sure I'm going to use some of what David Cox over at has shared about games he uses to help students grasp the idea of multiple representations of linear equations. These are terrific ideas for cultivating curiosity by dropping a trail of breadcrumbs that lead to an interesting path.

    Wallowing can be a tough sell to some, but it is a powerful way of fostering engagement and positive dispositions toward mathematics as well as motivating patience, persistence, and problem-solving. I find that "walowing days" are messy-looking, classroom-management-wise, since I actively WANT students to lose track of time. Success on wallowing days is measured in percentage of students actively doing math, talking math with each other, and laughing together. Not everyone can grasp math through physics. Wallowing offers another path.