Thursday, December 12, 2013

Two Sides of the Same Coin

So I’m working on a book.  While I work on another book.  This other book is a project through Moebius Noodles and Maria Droujkova’s publishing company Delta Stream Media. I’m collaborating with Maria and Gordon Hamilton of Math Pickle to create original puzzles, games, and making activities exploring numerical and categorical variables … for young kids!  It’s super awesome extremely cool.

And, today?  Today, our conversations in relation to the variables book helped me clarify something I’ve been thinking about on a LOT of different levels for literally YEARS.

This level
What’s the difference between using the body to illustrate mathematical ideas and using the body to create and express an understanding of mathematical ideas?  

This other level
What’s the difference between using body knowledge (ala Papert and his gears) and creating body knowledge?

And finally
What’s the difference between identifying properties of an object (say, a piece of art, or an insect) and actively choosing from an inventory of attributes to make your own?  

llustrate, use, identify  <======>  Create, express, choose

Each of these questions needs its own, more specific treatment.  My observation today is simply that each pairing seems to create a similar tension in my mind.  These are all active words, but the nature of the activity is qualitatively different depending on which side of the learning process you're on.

Today the words fixed and flexible came up in relation to how we are approaching the activities in the variables book.  

Puzzles and games that focus on identifying properties happen within a fixed structure.  Using body knowledge to understand a set of gears assumes that you are using a certain set of body experiences, created at some point in the past.  Illustrating math ideas using the body means there is a predetermined goal for the activity and that the outcome needs to look a specific way.

Learning vocabulary and language in context and, in math, using multiple strategies to solve a problem are both process oriented and context dependent.  In making, having a large inventory of ideas/things/skills from which to choose and create your own novel ideas (like a dance step) is an open-ended investigation.

“Fixed” and “Flexible” are not judgments; they are inverses of each other.  They go both ways.  Just like you need to compose and decompose numbers to see the full relationships embedded in those two activities, so do you need to identify and use properties, build and use body knowledge, and illustrate and express mathematical ideas. 

Fixed: The parts of learning (anything, really) that are perhaps learning objectives that are easier to identify in an assessment, but still crucial.  Some call this skill building.

Flexible: Relates to the processes of learning which (as anyone who has tried arts integration, project based learning, or focusing on mathematical practices may have experienced) are much harder to nail down when tasked with assessing such activity.  Some call this fluency.

You can't have one without the other.

“Without skill there is no art. The requisite variety that opens up our expressive possibilities comes from practice, play, exercise, exploration, experiment.”  --Stephen Nachmanovich, Free Play: Improvisation in Life and Art
Thoughts, feedback, pushback and conversation are always welcome. 


  1. Your post, as usual, has me thinking quite a bit. I find some of the distinctions here between fixed and flexible (for me, at least) are bound up in a particular view on which gives an ontological status or existence to "mathematical ideas." I prefer to think of mathematical ideas (or relationships, objects, etc.) as existing ONLY in the mind. Because of that, some of your statements are problematic for me. For instance, you write:

    "Illustrating math ideas using the body means there is a predetermined goal for the activity and that the outcome needs to look a specific way."

    This is only a statement that an observer, or knower, could make. It basically says that a child is moving in a way YOU recognize as matching your own mathematical knowing. So, to me, a "fixed" objective means that a teacher/person wants to recognize their own thinking in the actions of another.

    In contrast, you wrote about "flexible":

    "In making, having a large inventory of ideas/things/skills from which to choose and create your own novel ideas (like a dance step) is an open-ended investigation."

    To me, this sounds very much like how Piaget described the functioning of the mind. We build schema that are basically ways of acting/thinking that lead to viable outcomes in our experience. When we are confronted with a novel situation, we try to assimilate that experience into an existing conceptual structure. If that does not produce a viable or expected outcome, we are forced to make an accommodation (essentially, learning). But, that does not mean the accommodation will be one that the observer would identify as "correct" or matching their own way of knowing.

    So, to wrap up this little rambling, I'm reading the distinction between fixed and flexible to essentially refer to the outcome the observer will be content with. Must they require one that matches their own way of knowing (fixed) or is it open to individual ways of knowing (flexible).

    Thanks, Malke!

  2. Hi Bryan -- great thoughts. What if "fixed" meant working with something that has already been made, something tangible, and "flexible" meant that the learner had some impact on how that something is to be formed? What then?

    Also, honestly, I get a little itchy when I think of all knowing/understanding as lodged in the mind. Everything I speak of above is something that children can touch, hear, do, see or do with their own bodies -- which is where children do much of their thinking. There's a really interesting study about children's gestures in math learning that concludes "children think and learn through their bodies." I'm not denying mental constructs, but I believe there to be a much larger role of the body in the knowing/learning process *especially* for children.

    I suppose I have some bias against the fixed side of things -- I am known for being pretty open minded to a wide range of 'answers' within children's choreography and even in the math groups I run at my daughter's school. So, in that way I am definitely, as you say, "open to individual ways of knowing (flexible)" and I really like that phrase "the outcome the observer will be content with." That's definitely something to think about a little more. What I am trying to do here, though, is to pin down an answer to what I saw as an inverse to Papert's body knowledge (and also make sense of the difference between identifying attributes vs. using an inventory of attributes to make something. I'm still not completely sure I have it or, even, if it's really that necessary to figure out.

    Now I'm at the risk of rambling. Thanks again!


Thanks for reading. I would love to hear your thoughts and comments!


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