Amelia, the Enduring Math Doll

First, there was Amelia the hand-made doll and her essay "What Infinity Means to Me." That was early in 2013.

Since then, Miss Amelia has been on all sorts of notable journeys.  Most recently she's been the subject of some interesting math questions.

For instance, Amelia needed a new dress. Unable to find the tape measure, my disorganized eight year old devised an interesting new way to measure out a piece of cloth.  She used this work board from a physics activity (investigating levers, pulleys, etc.). 

We have never discussed coordinate grids. She has, however, seen a lot of grids, mostly in the form of multiplication arrays.  She measured Amelia vertically...

...and then horizontally...

...and then cut out two identical square pieces of cloth (front and back of new dress) with which to sew the dolly a new garment.

Amelia was also recently the subject of a conversation about doll years.  I'm pretty sure I didn't have enough math to truly help her, but here's what came of it.

My kid wanted to make sure Amelia was SIX in doll years.  Apparently (after a very heated, frustrating conversation mostly, I assume, because she wasn't really clear about what she was trying to ask) one doll year is equal to two human months.  The only way I could help her figure it out was by writing out her ideas for her, one step at a time. One problem was that my kid was thinking in too many time units at once -- days, months, and years.  

We did finally come to a conclusion that was agreeable to her.  I almost typed 'concussion' because that's how I felt after it was all over.

I wonder what kind of math adventures Amelia will have next?

"Its math skills are the wonder of the land"

My eight year old gave me a lovely Christmas present.

By way of explanation, long-time readers are well aware of her affinity for cats. In addition, we have taken a very inclusive approach to learning math by which I mean that we endeavor to find it in all parts of our daily life.  This is why math appears so frequently in her expressive moments and is featured prominently here in the small piece of prose presented to me on Christmas morning.

(I've standardized her spelling...)

A cat is graceful and beautiful but most of all its math skills 

are the wonder of the land.

A cat needs to understand the agile speed and how much 

force to run to catch a squirrel!

Otherwise it will not stop. It also has to understand the
geometry and agility of jumping!

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.  

Fixed
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.

Flexible
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. 

Does "Body Knowledge" Have an Inverse?

I have some questions which may not be fully formed.  If you know of or have read the work of Seymour Papert, I could really use your thoughts here.

First, here is a famous excerpt from Seymour Papert's seminal book Mindstorms.  In the preface Papert introduces the idea of body knowledge in a story about his childhood fascination with a set of gears.  He says:

“Piaget’s work gave me a new framework for looking at the gears of my childhood.  The gear can be used to illustrate many powerful ‘advanced’ mathematical ideas, such as groups or relative motion.  But it does more than this. As well as connecting with the formal knowledge of mathematics, it also connects with the ‘body knowledge,’ the sensormotor schemata of a child. You can be the gear, you can understand how it turns by projecting yourself into its place and turning with it. It is this double relationship—both abstract and sensory—that gives the gear the power to carry powerful mathematics into the mind.”

My question is around this idea of 'body knowledge.'  It seems Papert uses that phrase to mean that body experience in the world can, at a later time, be brought to bear on understanding mathematical ideas.  His Logo geometry (which used a programmable object called a turtle) is said to be something you can “walk” through.  From my perspective, programming a moving object is not actual walking or actual movement; it is, like Papert mentioned in his gears story, a projected walking—projecting body knowledge into the object itself.  

Here's my question:

When thinking about the body's role in math learning, I wonder if Papert ever considered the inverse to such a process, specifically, how a body might participate in actively creating body knowledge of mathematics. The double relationship he mentions ("both abstract and sensory") seems to relate only to the sensory memory, not to the using the senses in the moment.

Any thoughts will be incredibly helpful. If I'm barking up the wrong tree, tell me.  If you want me to clarify anything, let me know.

Thanks in advance. (Edit 12/9/13 -- check out the comments!!)

Learning without a Body

My third-grade daughter goes to a school where they have a whole hour for lunch. Not only that, although lunch time is monitored, children are free to move around both inside and outside of the school within boundaries determined by their grade level. From the reports my daughter gives me about this time of day it seems like lunch is not just for eating (as evidenced by half-eaten sandwiches in the lunch box at the end of the day).  Lunch is for making up skits, finding interesting properties in the rocks you are pounding, for having arguments and making up, for exploring the narrow (but long) strip of trees that line one side of the school's property, called 'the woods', and for creating clubs.  At a school with a no-exclusion rule, a club can be pretty much any combination of kids at any one time.

One day, my daughter told me the club had made a fort in the woods. At first glance it looked rather like a wall:

A very well constructed, sturdy wall, a wall built with a lot of thought and insight. There is a base of bricks and concrete and then a layer of sticks.

And, there is also a latch (the wire) which is lifted by the "door" (the stick), "Although we don't usually go in this way because it's not sturdy.  We usually just go in over the low wall."

The space to the left side of the photo is the, I guess, living space of the fort.

Hearing about and seeing this fort I immediately thought about the article Ophelia's Fort by fourth grade teacher and artist David Rufo which I edited last year for the Teaching Artist Journal's online writing community ALT/space. In it he writes:

"During our conversations it became evident that Ophelia was focused on making for herself a “special place” [4] rather than a special structure with four walls, a roof, and a door. As David Sobel emphasized in his book Children’s Special Places: Exploring the Role of Forts, Dens, and Bush Houses in Middle Childhood: 'Through making special places, children are experiencing themselves as shapers and makers of small worlds. This experience contributes to making them active shapers of the world in their adult lives.'" 

In sharing this story about my daughter's lunchtime adventures, I am aware that it is not about math learning, per se, but it does relate to a book I am writing about the body learning math. In past writings I have focused very closely in on the specifics of the Math in Your Feet program; this new book is a chance to step back and look at the broader issues involved in making math and dance at the same time.  

One of the ideas coming into view as I zoom out is the necessity of agency in learning, and it is clear that issues of learner agency begin with the body.  As I was finishing up a chapter today, it became clear to me that when thinking about using dance or movement as a partner in learning we must start by identifying how the body has historically been employed during school hours.  That is to say, how the body has not been employed (bolding emphasis mine):

“The embodied experience of traditional schooling is often, as educational philosopher John Dewey might suggest, an anaesthetic experience, devoid of any heightened sensory experience or perception. In school, our bodies are still, serving primarily a utilitarian function.  We learn to from an early age not to squirm or leave our desk chairs in classrooms. We learn to sit up straight, raise our hands to be called upon, or walk single file to lunch.  By the time we reach high school our bodies are often reserved for gym class…or for moving from one class to another. In a sense, we educate from the neck up, leaving the rest of the body to act largely as physical support rather than as actively involved in our quest for knowledge, thinking, and understanding … implicated in this analysis is the importance of agency in relation to activity. Providing curricular opportunities that are experience-based, that encourage the use of the body and engage the senses in learning could create a different kind of [structure] for schooling if learners are encouraged to explore connections between learning, self and the broader social and cultural frameworks of meaning in which they are situated.” 

Source: Powell, K. The apprenticeship of embodied knowledge in a taiko drumming ensemble. In L. Bresler (Ed.), Knowing bodies, moving minds: Embodied knowledge in education (pp. 183-195). Dordrecht, The Netherlands: Klewar Press.

The body is not simply a vehicle toward realizing the perceived pinnacle of abstracted knowledge housed in the mind.  The body is where learning originates. Living in a body is also the way children learn personal agency as they make decisions about how their bodies will move and act and how that power can influence and shape their world. And, in the process, learning that there are obvious consequences and responses in relation to their actions. This is literally and viscerally democracy in action. 

Perhaps most importantly, despite the incredible change of pace and screen-focused activity in modern life, children still have brains that learn best by moving and pulling sensory input in through all parts of the body.  Hundreds of years of thoughtful analysis, research, and observation of children learning and growing has shown this to be true and yet the body is still being marginalized in favor of knowledge as something gold and shiny to be won and placed on a high shelf for viewing, far removed from any experience and personal understanding. 

What is a body without agency?  What is learning without a body? Thinking about these questions is the important first step in understanding the inherent worth of children using their bodies to make math and dance at the same time.  Onward!

How Are We the Same?

At my daughter's school, middle of the morning, to pick her up for a dentist appointment.  Ran into the restroom before I went to get her.

In the restroom I inadvertently opened a stall door with a little person inside. Apologizing, I moved on to the next stall.  A darling conversation ensued along the lines that the latches weren't that tricky, but were very tricky if you were a five-year-old child. And then the delightful conversation continued when I noticed that we had both opened our respective doors at exactly the same time...

Me: "Hey look! We left at the same time!  ... And now we're washing our hands at the same time..."

Child: "And we're both using soap at the same time."

Me: "And now we're both rinsing..."

Child: "Do you like to do things at the same time?"

Me: "I don't know, do you?"

Child: "Maybe."

Me: "... and now we're both drying our hands at the same time."

Child (walking toward the closet door): "Oh, I thought that was the door..."

Me: "They do look the same, don't they?  Hey, I know, let's go out the restroom door together at the same time!"

And out we go, into the hall, and on with our respective days. 

At the same time. She is little, I am big.  She is young and I am ... not young.  She was at one sink, I the other.  And yet, we did the same actions at the same moment.  And in that moment we were the same.  And we knew it.  And that was fun!