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.