Lynda Barry Math: "Back-Story Problems"

It's sort of a convoluted story* about how I came upon this cartoon but when I first read it I burst out laughing and thought how perfect it was for what I'm doing these days. (Meaning, trying to teach math to my child who creates a back story or narrative for pretty much any any endeavor.)  So, I thought I'd share.  The image came from Lynda Barry's blog The Near Sighted Monkey.



*The long story, if you're interested, is that Lynda Barry went to my alma mater The Evergreen State College in Olympia, WA.  She is a contemporary of Matt Groening (The Simpsons creator and Evergreen grad) and is, by all accounts, an incredible writing teacher.  I have been working up the courage to contact her to see if she might be interested in writing  for ALT/space, the online project of the Teaching Artist Journal (which I edit).  While I was thinking about all this I found her blog and saw that she would be teaching at a summer writing institute in my town.  That was a weird coincidence.  I heard through the grapevine that she is as wonderful as everyone says.  Although I have not yet worked up enough courage to get in touch, I have really been enjoying her blog!

Think Like a Straight Line

It's been a loooong time since the kid has ridden her bike.  So long it seemed like the first time again today.

She felt wobbly.  Steering was a challenge.  So, she gave herself a pep talk as she worked to reacquaint herself with the activity.

"Okay, all I have to do is think like a straight line in geometry..."

She rode back and forth across the basketball courts chanting her new her mantra.

"Think like a straight line, think like a straight line, think like a straight line in geometry."

When she'd get to the end of the court, she'd get off the bike and turn it around.  Then she figured she could make the turn without getting off.

"All I have to do when I get to the end is think like a circle...."

I'm sure she'll be back in the swing of things in no time.  Plus, I love the thought that pathways have specific intentions.  She's in the math, man.  Totally in it.

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.

My Guest Post Over At Moebius Noodles!

I'm beyond thrilled and honored to have been asked to contribute a guest post at the Moebius Noodles blog.  If you've not heard of Moebius Noodles, here's their wonderful description:

Moebius Noodles is an off-the-beaten-path travel guide to the Math Universe for adventurous families. A snowflake is an invitation to explore symmetry. Cookies offer combinatorics and calculus games. Floor tiles form tessellations. The games in “Moebius Noodles” draw on these rich properties of everyday objects in ways accessible to parents and kids, even babies. As the world turns into a mathematical playground, it transforms, one family at a time.

Moebius Noodles is the brainchild of Maria Droujkova of Natural Math, both of whom/which have been instrumental to my growing understanding of mathematics and what math really is all about. Although I am quite comfortable with the math topics that connect to the math and dance we do in my program Math in Your Feet, I knew that there was a way of thinking and seeing math that I did not fully understand.

Over the last year or so following Maria's accounts of math clubs, math treks and, especially, the Moebius Noodles project I have found my way to math.  And, I am not ashamed to admit that, as an adult who is remediating herself from years of math anxiety, I have learned more about real math from an approach created for small children than I ever did in high school!  My new confidence and understanding is largely due to how Moebius Noodles models math and math learning for parents and families -- one small observation, question, inquiry or game at a time.

Here's the link to my Moebius Noodles post!  And, while you're there, don't forget to follow or subscribe to their blog.

Not 'Just Shapes'

So....we've been doing Sidewalk Math on and off since, hmmm, February?  Three months or so.  It's been me, mostly, getting all excited about what we're finding when we walk around town, taking pictures, oooh-ing and ahhh-ing.  I have been having me some serious fun, let me tell you.

The kid (now a freshly minted seven) on the other hand...  She's been there with me, she's been interested, participatory, but now?  Now, when we go out, I am not allowed to pull out my camera.  Ever.  I am not allowed to initiate any math conversation whatsoever.  Sounds horrid, except for the fact that now it's the kid who is finding math everywhere.

She ooohs over every single triangle she sees -- a scrap of fabric, a crack in the sidewalk, something on a wall.  In negative space, she sees a triangle of ceiling made by a partially closed door. She even made me this triangle love note.  Even the message inside is written to take the form of a triangle.

"Look Mama!  When I folded the square [paper] I got two triangles.  Look!  Look inside!"

"Looks like four triangles to me," I replied.  I've never seen someone so excited over four triangles.

She ahhhhs over concentric circles.  She finds them everywhere -- even inside her cucumbers (where she also sees little stars, which, I manage to add over the adulation, are constructed with nature's numbers, Fibonacci, right?  Three, five, eight...the cucumber star is three....)  Seriously, this child has her eyes wide open.  We're in a rest room.  The baby changing fold out thing on the wall has circles, with circles inside them!  Concentric circles on the sidewalk.  At the bottom of the public pool.  Everywhere.

And let's not forget spirals.  I was making an extra effort a month ago to look for them and wasn't too successful, to tell the truth.  Also, the kid kept getting them mixed up with concentric circles.  It took a week or two of clarifying, on and off, to help her see the difference and... SNAP!  The light went on and now she sees them everywhere -- in fences, on light posts, on clothing, all around the house.  She draws them.  She finds small ones, big ones, mostly man made but also a vine here and there, or when we look really closely at how a plant is growing.  It's crazy.  Spirals are everywhere.

Oh, and composite shapes -- rectangles made out of squares, squares made out of triangles, squares made out of squares, trapezoids made out of squares and triangles.  And curves!  Let's not forget them.  And parallel lines!  Right angles!  The plaid fabric is a grid!  You get the picture, right?  I'm exhausted!

Today she said, "I'm going to make a book of all these shapes.  I'm going to take pictures with my camera and Papa will print them out for me."  She wanted me to have nothing to do with it.  Sniff.  But, still.  Isn't it cool that this whole thing has become completely, totally hers?! 

In the back of my head I am always observing my own mind-noise that says, "What's the big deal?  They're just shapes.  Kids learn about shapes in preschool."  But my answer always circles (spirals?) back to the fact that the longer we keep our eyes open for it, the more familiar she has become with the possible variations and anomalies (size, rotation, tilings, etc.) that can occur in 'basic' shapes.

(An aside: It's been hard to verbally explain the difference between a rhombus and a square rotated from its traditional position, but sidewalk chalk saved the day!)

Despite the doubts, my sense is that all the conversations we've been having over the last few months will eventually make geometry on the page that much more meaningful, and maybe even easy.  (I can also imagine that this kind of activity also serves the larger goal of pattern recognition in any area of math she attempts.)  It will all make sense because she will have seen those shapes in their natural habitats, and by identifying them and talking about them she will have made each shape hers and hers alone.  And, as the structure of the universe continues to emerge in front of her very own (and open) eyes, how much more fun will her world be to play in, explore, put together, and then take apart again?  A living math book, our world.  Not 'just shapes', y'all.  No indeed.

Check back with us in three or four years, maybe seven, and we'll see if I had it right.  Something deep inside me, though, says that the more we actively do and see and investigate the structures and forms math takes (like the curvy wall constructed with straight bricks?  Or, the sidewalk lines that look vaguely algebraic?) the more we can actually know math when we meet it in a more formal setting.

What I do know for sure right now is that the kid already considers herself 'good at math' and has added mathematician to her future career choices along with actress, singer, dancer, geologist, archaeologist, astronomer, writer, illustrator and cartographer. 

Have fun everyone!  If you haven't heard yet, math is everywhere!