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. 

Challenging a Literal Approach: Learning Fractions 'through' Music

The last week or so I’ve been thinking quite a bit about what makes real connections between math and dance in a learning setting.  I’ve come to the conclusion that none of those connections can even be attempted without first looking at our assumptions about what math is and what dance is. 

I’ve been sharing my thinking on this blog, to an audience which (I assume) is more familiar with math learning than with arts learning and over at ALT/space, an online writing project I edit and curate for the Teaching Artist Journal.  I’ve not had a huge amount of responses, but the quality of the feedback I have received has been incredibly helpful in moving me forward on this line of inquiry.

Most recently I had a great conversation with friend and colleague Nick Jaffe.  Nick is, among many things, an incredible musician, teacher, thinker, writer and editor.  He’s not a math educator, but he’s definitely not math averse. 

We were talking about how people often try to teach fractions by connecting it to musical notation, which I have always seen as a very literal approach to what people perceive as music (written notation).  Over the course of the conversation I finally understood why I had always viewed that as a very shallow, un-meaningful activity.  Specifically, Nick has a lot of interesting and helpful things to say about discrete and relative quantities when playing music so I thought I’d share our discussion here in case it is helpful to you in any way.
………………………………… 

Me:  I have seen very little evidence that making the correlation between fractions and music have had had any impact. Maybe one study, but they didn't let the kids actually play around with ideas or make their own music -- they just taught them how to play the fractions. And, seriously, do you think like that when you're *playing* music? 

Nick: When people ask me about music and fractions, the first thing we need to establish is that the analogy between musical notation (or the musical reality it describes) is multi-dimensional. Notation denotes both the note as a discrete duration, but also as a division of a larger whole. Which, as I understand it, is also in the dual nature of fractions and an important thing to grasp, and not entirely intuitive.

What seems essential, at least in elementary and middle school math, to understand about fractions is that they have a dual function mathematically--they are a way of signifying two things at once: a discrete quantity, and a relative one. Musical notation functions the same way rhythmically.  In both the abstract, mathematical context, and in the context of music, the same problems arise with regard to this dual nature.  And it is somewhat counterintuitive to students at first to consider this--it's never explained clearly but it's the crux of everything one does with fractions as far as I can see.

I like the question you raise about how one thinks when making music. I think the answer is yes and no. Let's leave aside conscious methods of fractional or theoretically driven music making. Let's consider improvisation which would seem to be the most spontaneous, least theoretical approach to music making. And let's just stick to fractions as a rhythmic concept for this particular argument.

When I improvise I do not consciously think about the duality of a note as a time duration (discreet and part of a larger repeating whole). However I absolutely have to manipulate time, consciously and/or unconsciously as both things.

I think that duality is at the heart of choices about phrasing, for instance. You have to feel a note as a discreet duration, an almost word-like gesture. But you also have to feel it in relation to a pulse, even if there is no pulse, or you are trying to erase any implied pulse--it's still in reference. One does not have to consciously think in terms of fractions to do any of that. However, learning to read notation, and practicing manipulating that duality (playing behind, on, or in front of the beat for instance) can often increase one's control over it.

Finally, analyzing, generalizing and theorizing about that same duality can open up new options, or allow the translation of one dynamic application (actual playing) from one musical context to another.  In that sense theory is both a stimulus to new ideas and means of translating an idea for different contexts. I think theory plays those roles in most disciplines--a generalization that makes possible the prediction of as yet unknown dynamics. 

Me: The problem with using music notation to teach fractions is precisely what you just explained.  The written notation is the quantity, and the actual musicality/music playing is the quality.  What people don't get about mathematics is that 1) it’s not just about the notation (same for music) and 2) it's really about the quality of the quantities.  If you focus just on "how much" you get a dead language or, at the very least, information devoid of meaning. Which is why so many people hate math – for many reasons, and I’m speaking very generally, math education has essentially been bleached of its meaning.  

Nick: I’m actually not that interested in the idea of using musical notation to teach fractions. That seems boring and highly inefficient. What I am interested in, and I think kids might be as well, is working with music and fractional ideas at the same time to make interesting things. Perhaps those things would be musical, perhaps mathematical, perhaps visual, perhaps both.   Undoubtedly one would learn various things in doing such work, but I think it is the making that is what is appealing to me (and perhaps students) and that any real insights and learning depend on the impulse to explore in order to make. 
…………………
 To sum things up, Nick’s thinking about the nature of a mathematical idea (in this case fractions) in relation to his art form is exactly the kind of thinking process I engaged in to find a meaningful overlap between math and percussive dance in Math in Your Feet. It is also a perfect example of how it is possible to teach parts of math starting with your own experiences inside another discipline.  

Where to go next? One of the best music, math and movement resources I’ve run across is from Ellen Booth Church. The activities described in this article describe perfectly how the core concepts of mathematical activity can be connected to musical and movement activity.  Ellen is an early childhood specialist, so her ideas are based on that age group and applicable up to about grade 2, but I think this is the perfect starting place for anyone interested in making these kinds of connections in their own classrooms. 

Interesting Intersections: Math & Map Edition

Peter Greenaway: A Walk Through H: Cross Route, 1976-78

I think one of my favorite places to be is at an intersection (well, except for maybe the West Coast of Ireland or on a ferry crossing the North Sea).  Among other things, an intersection is a crossroads, a place where people and ideas merge and diverge.  It's also a place where two seemingly unrelated topics find they have something in common or, even better, find that together they produce something much more than the sum of their parts.  Like math and dance!  Or, strawberries and chocolate.  Or, ummm....history and math and science and art all arriving from different directions to paint the dynamic landscape of human creativity and thinking.  Whatever kind of intersection it ends up being, to me it always seems like an exciting place to hang out.

Adam Dant: Shoreditch as Globe, 1999

These days, an intersection is the place where my daughter, who has recently taken on the navigational duties during our walkabouts, has to make a decision.  Left, right or forward?  South, north or west?  To determine which way, she has to consider things like: Which way is the quickest route downtown?  Which way has the most shade or the most interesting houses and gardens for us to look at? Where do the outside cats live so we can say hi to them?

Here's an intersection we recently passed through:

You can also find intersections on maps -- they're full of them.  My daughter loves maps.  Me?  I will use a map, but only half-heartedly.  I am barely patient for that time when I will just know the route and can leave the paper behind.  I want to be free to pay attention to other things (like traffic!) or to just think my own thoughts.  Even when I was touring a lot, and every day brought a new theater and a new town, I'd head out on foot to explore new territory without the benefit of a map.  (Well, except when I was trying to get somewhere on the London Underground or something-- I'd use a map then.) 

What I am more interested in these days are the kinds of maps that are less about getting around and more about representing 'what is known'.  For example, we've been reading a little bit out of this book:

It's a gorgeous display of maps over the centuries which illustrate how our knowledge of the world (what is/was known) and our world view (what we believe) have shifted over time.  I also recently ran into a really interesting post along the same lines over at Brainpickings called Magnificent Maps: Cartography as Power, Propaganda and Art.

My kid loves maps so much she has started collecting them -- recently she found a gorgeous old atlas from the 1920's at our library's book store for $2.00!  It has color maps and all sorts of interesting geographic information and diagrams.  It now sits next to another new find, a $10.00 globe from the 1980s that we found at Goodwill.  These are old, outdated maps but they are also historical artifacts and I think they'll be of great use to us over time. 

Anyhow, as much as she likes to look at maps, my daughter is really more interested in making her own.   And, as you would expect, they reflect her world, her knowledge and her experiences.  Here's one she drew last summer (you might really enjoy the story that goes with it, too).  This map is sort of a zoomed out view of everywhere we go in the car:

Here's one from February, after a walk around our neighborhood (with a little help from me):

She also makes maps of her ideas, like in this dress pattern:

And, earlier this spring, we used a map to stake out our 'cat territory' using dice rolls, our knowledge of our town, and a lot of math.

Recently, I've nudged her mapping activities towards the abstract with some graphing, which I consider a kind of map.  It may not be the kind of map she'd make on her own, but it was worth a try and I was curious to see how she'd respond.

So, a few days after doing our sidewalk chalk functions game at a park, I tried the same game but this time with graph paper and dice.  Each person got two rolls.  The first roll determined the rule for the x-axis, the second roll for the y-axis.  My result was four 'over' and two 'up'.  The kid's was two 'over' and six 'up'.  Here's what it ended up looking like:

As she graphed her rule there was some confusion about what exactly we were counting and where to put the graphed points.  "Well," I said, "where those lines meet is called an intersection, like on our walks where we have to make a decision about which way to go next."  It was helpful to have that real experience crossing streets to refer back to, and she had no trouble plotting points after that.

She went along with the game good naturedly enough, but I couldn't convince her to play it more than once.  What she really wanted to do was make a map of our walk through campus and downtown on the way to Sunday bacon at our local co-op cafe, which is where we were at the time.

The resulting map is all her doing. She used the new graph paper I had printed out for our graphing game and as she drew she described to me how each color, line and picture symbolized a landmark along our route downtown.  She even threw in a fractal tree to represent the campus woods.

I've thought a lot about how focused physical experiences in the world can support children's emerging understanding of symbolic systems in math.  From many years of observation and experience (creating and teaching the program Math in Your Feet and, more recently, from observing my daughter learn math) I've come to believe that 'real world' and/or kinesthetic activity really does help kids build a useful bridge between the meaning of math and the representations of that meaning.

In our house, math is conversational and game based and, so far, my daughter hasn't had a whole lot of exposure to standard written mathematical expressions or symbols.  But, all the same, she seems really drawn to visual representation as a way to communicate to others what she knows, especially math concepts.  Over the last year she has shown me what she thinks and understands through spontaneous, unprompted creation of charts, maps, and diagrams; often these are private ruminations that I happen to unearth while tidying up after bedtime.  I view these self-initiated efforts to symbolize and quantify (in a way that makes sense to her) as a kind of bridge between the experience and her eventual use of standard mathematical notation and representation.

I am also always tickled to find another piece of evidence that she is fully engaged in constructing her mathematical understanding.  I don't necessarily understand why maps, in particular, hold so much interest for my daughter.  But, ultimately, an intersection is the place where things come together and maps seem to be that common ground where her experiences in the world intersect with her need (drive?) to communicate that experience.  And that, my friends, is my favorite kind of intersection of all -- the one where learning happens.

Exploring the Nature of Numbers

I've written elsewhere in this blog about how I am remediating myself by learning math along with my currently five year old daughter.  I am a classic case of someone who probably would have LOVED math but instead got confused by a worksheet in kindergarten (something about bumblebees) and never fully recovered.  In high school I loved geometry so much that I was motivated to memorize all the theroms.  I was never able to apply what I learned, but I think it was the challenge and process of understanding interesting rules that caught my attention.  Plus, geometry is visual, which is my thing too. 

I really had to address my math phobia when I came up with the bright idea to integrate math with percussive dance, but what I realize now is that my love of geometry (which some people, I've heard, call the only 'real' math, but that's another argument conversation) has really served me in that effort.  And, the effort I've put in over many years to make sure I really understand the math the fits with percussive dance has also restored some of that natural mathematical curiousity which I lost back in kindergarten in the '70's.  I am really starting to see how math can be playful and full of interesting ideas and questions and it's own form of inquiry unto itself, just like being playful in the process of creating percussive patterns.  It's okay to ask questions that have more than one answer.  Not only that, it's FUN!

So, anyhow, this morning my daughter was teaching her toy cat Matilda her letters and some basic arithmetic.  They do other kinds of math in her kindergarten, but she seems taken with what numbers are and how they get bigger and smaller.  Here is a snapshot of our conversation this morning.  First she tells me:

"I'll write 100 really BIG because it's a BIG number and 1000 even BIGGER. And then a quadrillion."

This is really interesting to me, actually.  I do, despite what I've said so far, understand the concept of scale and size and amounts, etc.  But, I read somewhere that the whole 100 days of school project (which practically every kindergartener experiences) started back in the 1980's and I'm sorry I missed out!  On the 100th day of school this year she brought in 100 dried chickpeas in a little baggie.  I was surprised that it actually didn't look like very much.  She also, as a joke (and this was totally her idea) brought:

"One hundred pennies in the form of a dollar bill." 

My daughter has also had some questions about if the numbers ever end.  I think she was happy when we read a book called The Cat in Numberland which was about the Hotel Infinity where numbers keep moving in and there are always enough rooms for them, and it seemed to provide the answer she was looking for.  She still tells me she loves me 'to the end of the numbers' which is very sweet, even if it's not accurate!

In her kindergarten class she's been learning basic addition and subtraction.  She comes home and writes down little equations.  This morning she said, "Tell Matilda [her toy cat] what subtraction is."  I said, "What can you tell her?"  She answered, "When you take away a number to make a smaller one." 

Also on the subject of subtraction, at some point recently she started trying to take away a larger number from a smaller one, which led me to comment, "Did you know there are numbers smaller than zero?"  I tried to explain but I didn't know if I made sense or if she understood, until this morning.  She was writing down an equation for her cat.  This is what she said while she wrote it out:

"Three minus eight equals....zero!  And I'll put a minus sign on the zero because it's less than zero."

My response?

"I think I'll go out and get you a number line!"

Multiplication Models Poster

Even though I don't do anything with multiplication in my program, I still thought this multiplication models poster from Natural Math was too good (not to mention too beautiful) to not share!

Update August 2013: This poster is available for purchase here. :-)

Perfect Reasoning: The Right Brain Initiative (Manifesto)

Yep

From Spark: The Revolutionary New Science of Exercise and the Brain, by John J. Ratey, MD, a wonderful gift last week from Templeton ES PE teacher Monica Chapin:

In the Introduction:

"In today's technology-driven, plasma-screened-in world, it's easy to forget that we are born movers - animals, in fact - because we've engineered movement right out of our lives.  Ironically, the human capacity to dream and plan and create the very society that shields us from our biological imperative to move is rooted in the areas of the brain that govern movement.  As we adapted to an ever-changing environment over the past half million years, our thinking brain evolved from the need to hone motor skills.  We envision our hunter-gatherer ancestors as brutes who relied primarily on physical prowess, but to survive over the long haul they had to use their smarts to find and store food.  The relationship between food, physical activity, and learning is hardwired into the brain's circuitry." Page 3

In the chapter Learning: Grow Your Brain Cells:

"The body was designed to be pushed, and in pushing our bodies we push our brains too.  Learning and memory evolved in concert with the motor functions that allowed our ancestors to track down food [the reason we learned how to learn in the first place], so as far as our brains are concerned, if we're not moving, there's no need to learn anything." Page 53

So...if you are not moving you are not learning?

I think that's it in a nutshell.

By the Book

I have to stop beating around the bush. 

You know, I know, and everyone knows that a chance to reflect on one's learning is where the learning actually happens.  That's why, when we were first developing the Math in Your Feet program, my wonderful co-creator Jane Cooney said, "We have to have a workbook to go along with the dance classes."

Yes, I know the label of 'workbook' has a lot of connotations, including busy work, and if you can think of a better name, please let me know.  Residency Journal?  That might be better.

Anyhow, Jane's idea was that the workbook would be the place for written reflection, word study (math and dance words/concepts), a record of the students' creative work (including mapping the patterns), and extension activities that look more like recognizable math problems related to the residency work.  For instance: "Make a scale drawing of the dance space you use in dance class."  As I've discussed elsewhere in this blog, I build the bridge between math and dance in my workshops, but it is up to the classroom teacher to make time to use the workbook and bring this learning back to the page.

Over the years, I've notice teachers becoming more and more stressed by the push of their district's curriculum and by testing, so much so that just bringing their kids to me for an hour a day for a week is now about all they can handle.  Maybe they get their kids to complete the daily reflection journal prompt and the word study, but that's about it.  Math in Your Feet, unfortunately, sometimes ends up being just a 'fun' event (and don't even get me started on 'fun'...).  There's also sometimes a perception that teachers carry that we're not doing any real learning while we're moving, the least of which would be math.  I once actually heard a teacher say to his students, when leaving my class, "When we get back to the classroom it's time for math."

An obvious sign that students are skipping the written work and reflection is when I start to hear the kids say, "Where's the math, anyway?"  Or, "Why are we doing this?"  Not every kid says that, and more often than not, kids still love dancing and making up their dance steps.  But I can tell they don't have the same kind of facility with the math and dance terminology.  I can tell they don't remember what we did the day before or why.  I can tell that they don't really fully understand the process and concept of transforming their patterns.

It's an opportunity lost, if you ask me.

Last week I was at Templeton Elementary School in Bloomington, IN where I live.  (Thanks to an Indiana University ArtsWeek 2011 grant!)  As the week went on I got lots of fantastic reports about kids who had never wanted to pick up a pencil who were excited to go back to their classrooms and write about their work in Math in Your Feet.  I've heard this before, actually.   Engage a kid's emotions, give them something real and personal to write about, and they'll respond.  And, along with that, REQUIRE that they complete their residency workbook assignments.  Even if it takes longer than expected.  Even if other things have to be pushed aside for the week.  Even if the math topics in this program don't align with your lesson plans for the week I'm at your school.   

There's the potential for deep, thoughtful engagement with both the math and dance content in this program, but it's all water through a sieve if kids don't get a chance to reflect on their learning and make it their own.

Learning to Listen

Are kids smarter than they were five years ago, or is it just that I've stopped looking for the 'right' answer?

Are my students more perceptive than they were, or is it just that I am more receptive to what they have to say, even if it's not 'time' for it?

Are fourth graders smarter than they used to be, or is it just that I am less interested in 'moving the lesson forward' and more interested in listening to their reasoning?

Are kids understanding the program better now or is it just that I'm finally really hearing what they have to say?

Have they always had insightful observations, or is it just that I am more open to multiple interpretations of the same event?

I think we know where this is going.  Kids have always been smart and I haven't always known what to listen for.  It's been really exciting this week to hear my students' observations, thoughts and insights.  I am, for the first time, really enjoying the process of teaching this program. 

Representing Math Concepts Through Percussive Patterns

Not quite congruent.  One partner has landed while the

other is still up in the air.

This week I'm working at Christel House Academy, a charter school up in Indianapolis.  This is part of a grant-funded pilot project for Young Audiences' Signature Core Programs.  The fifth graders are fantastic!  They are perfectly perfect in all their 11-year-old-ness, and quite observant and thoughtful to boot.  They make connections easily and ask interesting questions that show they are really thinking about how this all works.

This is an interesting situation for me.  I am usually invited to schools where kids are at least a grade level or more behind in math and my role is to assist in catching them up.  At this school, the fifth graders know and understand quite a bit so we are in the position of applying what they know to a new situation instead of learning it for the first time.  But the really fascinating thing for me is that, although they 'know their math' they are still challenged by representing it physically.

In my reading about mathematics education, I've come across an idea called 'the power of three'.  Essentially, the idea is that to really understand a math concept a child needs to represent it in at least three different ways.   This would be through pictures or some other means.  I'm just beginning to realize that one of the strengths of Math in Your Feet is that it provides an opportunity to experience and represent math concepts in the kinesthetic realm.  Part of this challenge lies in the fact that these patterns are not static, but require students to literally be 'in' the pattern.  Just today I had an interesting conversation with some boys about whether to record a turn as being on the third beat or on the fourth.  We eventually came to the agreement that the turn was actually happening between the third and fourth beat, but that since third beat ended in one position and the fourth beat was in the new position, we had to record it as being on the fourth beat.  My system may not be perfect, but it does create a structure to ask these kinds of questions. 

So, here's how it works.  Kids make up a four-beat dance pattern using the elements of percussive dance that I've outlined for them.  They learn to make their dancing congruent by producing (with pre-teen bodies!) the same tempo, foot placement, movement, and direction as their partner.  After that, we start transforming these patterns using different symmetries, starting with reflection.  At that point, all the pathways forged between the body and the brain have to be shuffled around as one partner dances the original pattern and the other (on the opposite side of the line of reflection) has to change the pattern by dancing the opposite lefts and rights.  For example, a turn to the right would be reversed to go left, or a right foot would be switched to a left foot.  This all sounds rather straightforward as I'm writing about it, but after observing the CHA fifth graders this morning, I realize that no matter how well they understand it in their heads, and no matter how 'smart' their bodies might be, it's still a challenge!  There's quite a bit of thinking going on here, in both body and brain, and it takes a lot of practice to remember a sequence of the four moves that make up their pattern.

This is only the third day and we have a couple more to go.  Things do get more interesting and more challenging when we start combining individual patterns into larger ones (i.e. start the second pattern where you ended the first, not at your original starting point and then try the reverse) and also when we transform the patterns using turn symmetry which seems rather straightforward in a static representation on paper, but is absolutely spectacular when you see it in motion. 

I'll keep you posted!

You Can Count on Monsters

You Can Count on Monsters, by Richard Evan Schwartz
Paperback, 244 pages, A K Peters

I happened to be listening to NPR's Weekend Edition recently and I heard a conversation between Scott Simon and NPR's math guy Keith Devlin about this new book You Can Count on Monsters by Richard Even Schwartz.

From Schwartz's website:

"The book starts with a 20 page introduction, written at an elementary school level. After explaining multiplication, prime numbers, and factoring, the introduction lays out the general idea for the rest of the book, as I'll now describe. To each prime number, we associate a pattern of dots and a monster.

"There is something about each monster that has to do with the prime. Part of the fun of the book is figuring out how the monster is related to its prime. For each composite number, we factor the number into primes and then draw a scene that involves those primes. We also show an arrangement of dots and a factoring tree that helps explain the picture. (A factoring tree is a kind of diagram that shows one way to factor the number into primes.)"

Near the end NPR piece, Devlin said about the book:

"The thing that distinguishes mathematicians is that we, at some stage in our development, we develop this understanding that numbers do have personalities, they have structures, they have relationships.  We form that, but most people don't manage to get it.  What Schwartz has managed to do is use his own skill as an artist to bring out some of the personalities, and the point is that what he brings out through his art is actually the structure and the personality that those of us in the business have always seen, we just haven't got the tools and the ability to make it accessible the way Schwartz [has].  It's his skill as an artist that makes this work [emphasis mine]." 
-- Keith Devlin on NPR's Weekend Edition, Saturday, January 23, 2011

As a dancer who integrates percussive dance and elementary math, I am in the business of making math accessible. I work to illustrate math concepts through a thoughtful sequence of activities; the children build original percussive dance patterns and learn and apply the math that arises naturally from this creative process.  I have spent many years learning and building my own understanding of the math content and practices that relate to this work.  And, I have carefully built a learning bridge that makes meaningful connections between the two subjects.

Now that I have built my bridge and my curriculum is where I want it, for now, I have become fascinated with searching for and finding examples of other kinds of bridges to math.  I am also trying to figure out just what it is that mathematicians see that the rest of us can't.  I'm coming at this task from a couple angles (no pun intended). 

First, going on some information I heard recently that it is most effective to learn a new language like a baby does (there's been some research findings about this, but I can't locate them right now), I'm working on (re)learning math myself alongside my five year old daughter by exploring math concepts through hands-on experience.  And, because I'm not five, I'm also looking ahead to where we might go next.  A few years down the line we might both be ready for You Can Count on Monsters. 

I'm also finding articles and online communities that are focused on how to teach math concepts for comprehension (not just for memorization of procedures) and learning from others' descriptions of how they teach and the kinds of questions they ask students.  I'm also on the lookout for quality examples of how art in general can help build a bridge to real comprehension of math concepts.  Schwartz's monster book seems fit perfectly into the bridge category, in a big way!  By the way, not only does Schwartz appear to be a working artist he is also a Chancellor's Professor of Mathematics and Director of Undergraduate Studies, Department of Mathematics, at Brown University.

So, happy reading and happy learning!  I'm off to the library to find myself a copy!

Play

"The artist is accustomed to working in the open spaces of creativity, ambiguity, uncertainty, opinion, and personal story.  Woven into this relationship with the discipline is a sense of play.  Play may be a key to understanding how people learn and how artistry and scientific thinking are linked. [...] The playfulness of artistry can be absorbing, exuberant, intense, and transcendent.  It can provide temporary perfection and sanctuary of mind that is a refuge from the mundane banalities of ordinary existence. Playing often involves rules, but it also includes freedom, imagination, risk unanticipated outcomes, and the possibility for  participants to become deeply immersed in the activity."

From Mark A. Graham's article "How the Teaching Artist Can Change the Dynamics of Teaching and Learning."  Teaching Artist Journal 7.2 (2009): 89.

A List

Mathematical aesthetics. 
I had heard about the beauty of mathematics, but I never really understood what that meant.  Now, I think I'm 'getting it'; not with numbers, but definitely on a visual level.  Sue VanHattum from Math Mama Writes posted this video called Doodling in Math Class: Infinity Elephants.  When I watched it something just clicked for me. 

More pictures of tape in action.
I've got this idea in my head to find as many examples of tape being put on the floor (or wherever) to further a child's learning, or to change an environment to promote exploration of space.  Send me yours!

Heading toward the front left diagonal!

Find new music and dance to it. 
Always on the look out for a great tune.

Gestures and embodied cognition in mathematics learning. 
After reading research findings about this, I've been more aware of how people move their hands while talking, especially when they're trying to describe a procedure or a design.  A friend was describing a plaid shirt in her closet; her hands moved across the front of her body horizontally and vertically while she said the word 'plaid'.   

Clear the mind.
Find new music and dance to it.  Better yet, I'm teaching kids next week!  There's nothing like a class of moving fifth graders to keep one in the present moment.

Building an icosahedron by folding paper plates. 

How many paper plates will it take?  I've made a two-frequency tetrahedron so far, which is four plates.  The reason I'm interested is that I've read that Labanotation (a method for notating dance movements and choreography) was created by visualizing the human body inside an icosahedron. 

This is actually an open icosahedron
from http://www.wholemovement.com/

Van Hiele Levels of Geometric Reasoning. 
Is this useful to me as a dance teacher teaching math?  I teach a lot of geometry.  I think the kinds of questioning employed that are intended to help move children from level 0 to level 2 might be helpful.  Need to look into it more.
 
Learn how to teach math with Cuisenaire Rods using daughter as guinea pig.  

I wish my math education had consisted of these, but I can re-learn math as I teach my own daughter.  These unit blocks are great for developing a real sense of what numbers mean, but when when you grow tired of that focus, you can use them in other ways!  I drew a line of symmetry and made up a game where one person puts down a rod, and the other person 'reflects' it on the other side of the line.  In this case my daughter led and I followed, but you could take turns in any number of ways.  Maria Droujkova from Natural Math posted some very interesting, videos of kids using these rods.

Featured NCTM Workshop!

Hey, check this out!  My workshop at the NCTM 2011 Annual Meeting and Exposition is featured in their promotion of hands-on workshops at the conference this coming April.  I see a few there I'd like to attend myself.  Maybe next year I can do one on tape as the ultimate math manipulative?  Big talk, I know, but I'm on a mission to prove my theory.

Will you be there?

Scary Thought

I recently read an article in the New York Times about the use/overuse of technology in teenagers to the point that their brains never get a chance to rest.  That rest period is crucial to cognitive growth and making connections between ideas, so there's a chance of an entire generation may not be able to access the very human experiences of intuition and instinct, both of which are crucial for creative and divergent thinking. Hmmm...