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. 

Categorical Variables in Your Feet

The same question has come up three times in the last three months, from three different sources.  Each time I have had no good answer.  When it happened again last week I knew it was absolutely time to figure this out.  The question, you ask?

"What is the difference between variables and attributes?"

Exibit A: The source of the confusion:

I hate giving answers I don't fully understand.  My answer has generally been: "I think I'm using the word 'variable' in the colloquial sense, you know, things that can change around -- I think that, mathematically, these are really what are called 'pattern attributes'."

You see? Totally unhelpful - to the person who doesn't understand math and to the person who does.

In my defense, up until just yesterday I didn't think the use of the word 'variable' in Math in Your Feet was mathematically accurate because I knew that variables are part of algebra and algebra is about inquiry into growing patterns. We make dance steps (pattern units) in our math/dance work but not growing patterns. When we evaluate sameness, similarity, difference and change we are focused more on the, well, attributes, that comprise each individual beat of a four beat percussive dance pattern.

After last week in Minnesota I knew I needed an answer, and I needed it soon! Luckily there was a meeting on the calendar with Gordon Hamilton and Maria Droujkova. We're working on a book that is currently titled, funny enough, "Variables and more". Luckily, I had warned both of my collaborators that this question was coming down the pike.

Maria's answer? Essentially, attributes and variables are almost interchangeable. There's the kind of variable used when looking at change (algebra) and the kind that is used when analyzing something that is set, or static (which we could call an attribute, if we wanted to) like in geometry (or statistics, apparently).

According to Maria, this class of variable is called a "categorical variable" and it is useful for things "that are not ordered".  I think I'm remembering correctly that ordered means, for example, a thermometer. You can compare differences in temperature - it can be hot or it can be cold or it can be in the middle, but the change is measured in a system that is already set.  With categorical variables there is much more freedom to analyze properties and make up your own categories, for example: Movies I Like, Movies I Hate.  Movies My Cat Likes, Movies My Cat Hates. In each of those four cases Movies have been sorted into different equivalent classes, meaning - every movie in the Movies I Hate category will be the same in some way.

The dance equivalent (ha ha!) would be that as students are building their first 4-beat pattern I often have us analyze, as a whole group, individual 'works in progress'.  We do this by focusing on our attention on one movement category at a time (e.g. identify only the directions in this pattern, or only the foot positions). The process of focusing their attention in this way makes their dancing much clearer and much more precise.

Clarifying direction.

But, as I found out, I was right to think that it is mathematical activity too, just a bit beyond my current elementary understanding of sameness, similarity and difference. When Christopher Danielson and I met after my Minnesota dance workshop last week he posited that perhaps a fundamental characteristic of mathematical activity is when you say exactly what it is you want to pay attention to (decompose), focus only on that attribute and ignore everything else, which is really what we are doing when we build and analyze our dance steps.

The only thing still in my mind is that instead of the activity of sorting these movement variables while they create moving patterns they are instead actively choosing them while they compose/design a dance pattern. It think it is probably a similar thinking process, as in "Hmmm, I don't like jumping on all four beats, what other movement can we use?" In that case, students really are focusing on one attribute/category at a time as they choreograph their steps.

This idea of decomposition and equivalence classes are a new conceptualization of 'sameness' for me, something well beyond the idea of congruence which we also use in our dancing. I've had hunches over the last year about how sameness and attributes are the mathematical ideas at the core of our dance work, and I've still got some thinking to do to integrate this new information about categorical variables but, in the end, I am just thrilled that my hunches been have validated in such a spectacularly specific manner.

New Math Game: Change Your Rule

This post is in two parts: The first part outlines the basic flow and structure of a new game idea which, I think, has the potential to support the conceptualization of math concepts such as combinations, permutations, patterns, variables/attributes and reflection symmetry.  

I know that's a tall order so in the second part of this post I briefly discuss my reasoning, pose a few lingering questions and then ask for feedback.

Part One: Basic Flow of the Game
The idea for this new game started with a picture:

Photo via: http://www.fotomat.es/arte-combinatorio/

I immediately thought how cool it would be to have this kind of mathematical art/visual inspriation in a living or learning environment, something to hang out with that evokes noticing and wondering.  In this case, a real-life shelf with spheres and cubes just begs for interaction, something to play around with and ponder. I want one! In the process of thinking about how to get this kind of installation in my own house, a new game was born.  Here's the basic idea:

Make/build a rule.  Find a way to change it.

These are the game pieces. Set 1 includes two shapes, four colors.  Set 2 includes six shapes and six colors.

Rule Change #1: Make a rule/pattern. Using the same shapes/colors how can you make the next iteration different from the first.  Does order matter?  How many different combinations can you make?

Rule Change #2: Make a rule/pattern. Change one element of each shape to make a new pattern.  For example, orange turns into red, or square turns into circle.  For each new design change only one variable for each piece.  How many times can you change the rule? How many new rules/patterns can you make from your starter rule?  Can you ever get back to the original rule/pattern?

Rule Change #3: Build your pattern then reflect the design. What do you have to do differently when you build the reflected pattern?

But what happens if you only use one color and shape?

How many combinations can you make using three of one shape and one other shape?  Would it change things if each of those circles were different colors?

On another line of questioning, what larger design can you make by building your design line by line using the same shapes/colors in each line?

Part Two: Thoughts and Questions

Even though Part One seems like a lesson flow, it's really meant to be more of a general framework for exploration.  My intent was to provide a little structure, some basic 'rules' and a lot of room for inquiry.

I'm wondering if it really feels like a game, or is more of an activity?  If it's going to feel more game-like, does it need more structure?  A timer? Some 'change your rule' cards?  Or, maybe, some cards that say how many different attributes to use in the pattern?  How many different rules would be enough to create a sense of chance? What other 'change rules' would you include?

Based on trying this out with my own kid this morning, if you were to do this more as an activity/lesson perhaps it would be helpful to have some really easy starter examples, like all one color, or two different colors to get the ball rolling in a positive direction.  I asked my daughter to build a pattern with three pieces but then gave her the 'change rule'.  She didn't really like that and we left it there.  The game pieces and board are still out, though, hanging around.  I'll see if she wanders back over.

Questions I still have include: When combining or creating permutations, how much does the second attribute matter?  For example, this first design (two colors, one shape) looks like it combines in exactly the same way as the second design (two colors, two shapes):

Another one of my goals for this game/activity was to also explore the ideas of attributes/variables in design -- how well do you think it does that?  I intentionally did not use pattern blocks -- only one attribute/thing to change -- and instead made my own (Set 1). What would pull the attribute/variable idea out a little further?

I hope it's clear that I am hoping to dig into the brain trust that is my modest but wonderful readership.  That means you by the way, so please if you have any thoughts about all this, I'd love to hear your ideas.  

Things I would love feedback on: key ideas about combinations/permutations at the elementary level, thoughts about the game structure, ideas for other 'rule changers' to add into the mix, and any other thoughts you may have.  Thank you!!

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