Tuesday, August 27, 2013

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.

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