Bittersweet Math

It is somewhat ironic, not to mention bittersweet, that on the day we got the call offering my eight year old a last minute spot at the local independent school my kid asked me if I wanted to play Shut the Box. And then Blokus. (Both of which had been lingering on the shelf for months, if not a year.)

We've been homeschooling the last two years and, as regular readers of my blog will know, we have had some incredibly amazing math explorations together. Some of my favorite stories about math she and I have experienced together include:

Too amazing for words, but I managed to write about it anyway: A Game of Her Own: Discovering Division

The very first post in what became four more incredible months of Sidewalk Math during the very mild winter of 2012.

The one and only time we looked at the 100s chart, during which she said: "I love this. I feel like I'm inside the chart!"

In first grade, especially, homeschooling allowed my resistant learner the space and time to ask her own questions, wonder, and initiate her own visual proofs.

How I simultaneously quelled some interpersonal conflict and clarified the concept of units for both of us in When is a 10 not a 10? 

Watching her deepening thoughts about the nature of infinity.  This post is her discovering Zeno's Paradox out of the blue but also links to other stories about infinity.

Perhaps not always apparent, the last couple years have been both the biggest challenge and the greatest joy I have known in terms of teaching and learning and I'm a little sad to see it end.  But, the math joy is only part of the picture and my kid is SO ready to head out on new adventures that do not include her mother.

My biggest sorrow is not getting to try out all the cool math explorations I had in the queue for third grade homeschool. But, although I'll not be the one in charge of her learning now, I still know a few things:

- Since we've been focusing on building strong mental math skills (influenced by all the wonderful resources at Let's Play Math and also Peggy Kaye's book Games for Math) her new teachers have agreed to let her continue to solve problems in a way that makes sense to her.  I asked this because research has shown that learning algorithms can significantly reduce number sense in children.

- I am required to volunteer 45 hours during the school year and, after the parent meeting tonight, I think I may have a chance to be involved in some way with enriching the math learning in her 3rd/4th grade classroom. It's hard to tell right now what form that will take, but I'm excited about the possibility to contribute -- I've got a TON of ideas already.

- We'll have to see how my kid is with having me present in her classroom. At the very least, I may offer a play-with-math class option during the all-school afternoon project time. However I end up contributing my Parent Involvement hours I'm pretty sure it'll be math and making related and that I'll learn lots and have many stories to tell here.

As bittersweet as this moment is, at least I know that every day she will be headed off to school with math on her brand new lunch box. Just look at that six-around-one gorgeousness!

 

It's another chapter, but definitely not the end of the story.

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.

Cross-Disciplinary Pollination

Collaborations around teaching and learning have been the theme of my summer and, speaking as someone who thrives in this kind of give-and-take intellectual interaction, this has created an ideal learning situation for me.

Most recently, I was in Minneapolis presenting a Math in Your Feet teacher workshop as part of the Artist to Artist network  and the Teaching Artist Journal Design Team meetings (read more about the event here).  I have been meaning to write about my experiences with the A2A protocols, the brains behind their implementation (my brilliant colleagues Barbara Hacket Cox and Becca Barniskis) and how this collaboration has helped my teaching practice but am not quite ready to do that yet.

What I can say at present is that a wild hope became a reality when I invited one of my math education heroes, Christopher Danielson (who resides in the MSP area) to the Minnesota workshop and he accepted.  Because I respect his deep thinking and clarity of thought, especially when evaluating others' ideas, practices and assumptions, I was both excited and nervous to have him experience Math in Your Feet first hand. 

Harvard Recap

Last week I attended a three day institute at the Harvard Graduate School of Education (HGSE) called The Arts and Passion-Driven Learning led by Steve Seidel and members of Yo-Yo Ma’s Silk Road Ensemble.  I signed up after receiving a very generous invitation-only grant award that encouraged me to attend one of HGSE's summer professional learning programs with Project Zero staff. 

Before I went, I wrote a post about my workshop choices for the institute and was feeling quite hopeful about the potential of the overall experience - I mean, it's Harvard, right?!  In this post I want to outline some of my personal highlights, but also raise some questions about what it means to teach and learn in the arts, especially within a professional learning context.

Personal Highlight

My very favorite thing was the Scratch workshop led by Karen Brennan who conceived of and runs the Scratch Ed initiative.  Scratch is an online visual (and very creative) programming environment, developed by the brilliant folks in the Lifelong Kindergarten Lab at MIT.  We spent the WHOLE time working on our own projects, totally engrossed – Scratch is a miracle. Truly brilliant.  I also got a personal tour of the MIT Media Lab from Amos Blanton, who supports the Scratch online community, before the Harvard institute started.  

Another fabulous workshop I attended was about ‘powerful partnerships’ led by the program director of KID smART in New Orleans, a really sharp, interesting, organized woman named Elise Galinot Goldman.  She gave us a very useful experience with some of the tools they've developed to address the inevitable conflicts/issues that arise when an artist comes into a school to teach in collaboration with classroom teachers.  In their model, an artist is placed for a full year of work in a school (amazing!) so the need to build and maintain strong partnerships/collaborations with the school team is paramount. I won't go into detail here, but it was a very interactive and productive few hours of exploring ideas and discussion which I really appreciated.

I also loved having a chance to talk with other artists who teach. Normally, when asked what it is I do, I often have to first provide an explanation about the teaching artist profession.  Not here!  Being able to talk to and hear more about the kind of arts activities/learning people are doing in their own schools and communities was time well-spent.  On the whole, the people I met and talked to were really interesting, smart, thoughtful educators and/or artists and I really enjoyed those conversations.   

I’m happy to have taken away some wonderful moments like these, but I do still have some big questions about the institute as a whole…

Conversational Math in the Neighborhood

Our cat is a straight out cat-cat.

My daughter and I were out for a walk this lovely Sunday morning.  We ran into a family sitting on a bench, apparently on a break from a bike ride around the park.  Mom, three curly headed boys, a dog and...a cat.

My daughter immediately started cooing - she loves cats.  "She's actually more like a cat-dog," the Mom mentioned. "She follows us wherever we go."

"Oh!" I said, "We've run into her on other walks. We love hanging out with her."

My daughter noticed the dog. "I don't like dogs. I don't like their licking and their jumping."

"Well," said the Mom, "She doesn't do any of those things."

"So, your dog is more of a dog-cat, then?" I asked.

"I guess that's true," said the Mom.

I looked at the Mom and said, "It's a real Venn diagram of family pets, isn't it?  What if you had more pets, like a..."

"Like a bird?" said the Mom, "Who was also a fish?"

"Like a penguin!" said one of the boys.

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