Stake Your Claim

My daughter's 3rd/4th grade class has been working on arrays over the last few weeks. (I recently found out they are using the TERC Investigations curriculum, which seems really strong. Arrays are the first investigation for the 4th grade sequence.)  One of their homework assignments was to find, draw and quantify examples of arrays in their environment at home and at school.  Of course, I love that! 

It reminded me of a game I created for my daughter and I to play in the spring of 2012.  The post about the activity is called Totally Territorial: Cats, Maps, Area & Multiplication.

Essentially, I drew a grid over a map of our fair city and, armed with dice and some pens, we staked out our claim.  It was totally fun and my almost-seven-year-old was completely into it.

Since they are in the middle of a world geography unit (meaning maps!) in addition to their array inquiry I thought it might be fun to formalize the game for the class. I searched for 'maps of imaginary places' and found a bunch at The Explorer's Notebook, created some grids and a game structure and....voila!

The game right now has five maps.  The rules are simple, but I think the narrative context provided by the maps is enough to make this fun:

Purpose of the Game:To capture as much area on your map as possible.

Rules of the Game: 

2 players

Roll one dice to determine who starts the game. 

Roll one dice. First roll determines the number of horizontal squares/length. Second roll determines number of vertical squares/width (include the first horizontal row in your count). 

Outline or highlight your full array in your chosen color. Compute total area (total number of squares) for each turn. 

Game is over when all territory is captured or both players have both had two unsuccessful turns (meaning not being able to find enough area to capture based on their roll). Player with the most territory/area wins although you may want to consider the value of land vs. water in the final value of area claimed. 

They played it today (while I was working with small groups working on half-ness, midpoints, congruence and three-ness with this Sierpinski Triangle activity from The Fractal Foundation).  Their teacher said they were all completely into it for the entire math class.  From what he overheard, it was unclear if the kids had developed any over-arching strategy at this point.  More likely, he thinks, they were simply enjoying rolling the dice, computing the area, and capturing territory. 

I know I'm not on the cutting edge of game design, but the fact that 27 kids were happily engaged in mathematical activity for almost an hour probably says a lot. The teacher and I did discuss upping the difficulty by adding a rule about only being able to add territory attached to previously won territory.  I love the idea, but think you'd need a bigger playing area for that.

Wanna' play? Here are the files! Please do let me know how it goes and what modifications/improvements you made to the execution, or any design suggestions you might have.

Between 6 & 8: Learning Math by Ear, Part Two

My daughter may be headed to school for the first time in over two years, but she's apparently still learning at home, too.

A big maker, doer and thinker (but, interestingly, not a big builder) my child has historically loved making maps (math!), drawing diagrams (math!) including sewing patterns, and generally thinking math 'out loud' in a visual way. The first image below is from when she was six (a map to Grandma's house the next state over) and the second two are from when she was seven (a map from our house to our 'second classroom' at the local co-op via the college campus, and a sequential diagram of how to turn a whole piece of cloth into a dress).

She's eight now and just look what she did this weekend!

She had been trying to build a fort on the couch but it kept falling down. "Oh well, I guess I'll make a plan for a new fort," she sighed.  

At that point I left to do other things. When I came back much later there was essentially a large cube in the middle of our living room made out of the PVC piping and connectors that had been lying around unused for the last half year.

I took a closer look at her plans. "Why are there two squares?"

"One is the bottom and the other is the top."

"And how did you get those numbers?"

"I measured the pipe."  (Apparently with an 18" metal ruler)

Here's what I noticed:

Neither drawing is quite to scale (inch to foot) but what stands out to me are her very clear markings and groupings of twelve inches in each segment as well as the fact that each edge is 12, 12 and 6.  And, it is also fascinating that she was measuring in 12" segments even though her ruler was 18" long.

Also, she drew the image of the top of the structure with the ruler (square on right) which, for some reason, I find important as it is indicates a shift toward precision that I have not seen before.

This is so exciting to see because it is the first time she's modeled/planned something before she's made it -- a level of abstraction that is brand new. She also obviously wanted to accurately portray how long each edge really was as well as the fact that there was a top and a bottom to the real space she was creating.

As I put together this post I realized I was getting a bird's eye view of the mathematical development of one particular child over a two-year period, specifically the progression of mapping and modeling between the ages of six and eight.  I suppose it's because I was there for every bit of it that I find the view so compelling. Also, none of this happened in a vacuum; despite my child-led, seemingly hands-off approach I have been quite intentional about the structure and tone of her learning environment, especially in terms of having a chance to play with and talk about mathematical ideas in multiple ways. For that reason, I humbly submit this post as another example of what it means to learn math by ear. 

Sumptuous Saturday | Treasure Map Math

It was the perfect, perfect, perfect Saturday, for math and just about everything else.

After months and months of cruelly hot weather with muggy 80 degrees by 7am every day, one recent Saturday morning we woke up to 60 degrees, a clear blue sky, and even some mist.

The kid and I walked the two-plus miles downtown to the Farmers' Market via a paved trail that runs miles through our little city.  Lucky for us and our walkabout math habits, the lamp posts on the trail are numbered.  I was considering musing aloud what the half of each number would be when the kid spontaneously starts marching and, along with this steady rhythm, she starts to chant: "Eight plus eight is sixteen. Sixteen plus sixteen is thirty-twooooo.  Thirty two plus thirty two is six-ty-four!  Sixty-four and sixty four are...hold on...[pausing, eyes unfocused, obviously concentrating internally] ... one hundred twenty eight! [marching resumes] One hundred twenty eight plus one hundred twenty eight are..."

I don't remember how it went at that point.  We've not really done mental addition that high, but she made some mighty efforts nonetheless.  And, as we still had a ways to go, she started a couple more rounds of doubling, one starting at fourteen.  The other I can't recall.

We finally arrived at our destination and and found our friend L whose parents have a booth at the market.  Have you ever had lemon cucumbers?  Theirs are divine.  Directly after we found him, we also found a banjo!   Here's a quick video of the movement that ensued:



My kid has been wanting a waltz partner for like forever.  I'm not sure L. knew what hit him!  And, as for dance (and math) my kid seems comfortable in the phrasing and ratios of a number of time signatures, 3/4, 4/4 and cut time included. 

We then moved on to other regular market activities, like meeting the kitties from the shelter.  No math there, but I suppose someday we'll find it.



There's a lot of great spiral math in the fountain outside city hall.  No water in it this day, but kids regularly start in the center and splash spiraling outward, or float bark boats in the reverse direction.  The spiral actually opens way up on the left there, meandering in curves around until it meets the actual fountain.

And, although the water in the spiral fountain had been turned off for some reason, it didn't stop the kids from playing.  Apparently my girl and L. had been searching for big and interesting pieces of birch bark on the ground when she ran over to me all excited about finding a map of the fountain drawn in pen on the bark "with an X on it!"   Of course, that had to mean treasure was close by. 

And, of course, they started trying to follow the map (surprisingly, almost to scale) of the fountain area.  They got completely engrossed in figuring out how to read the map to find the location represented on the map by that curious X.

At some point they recruited another girl, maybe older?  It's hard to tell because my kid is pretty short for her age.  Even L. who is a year and half younger is half a head taller than her.  Anyhow, they pondered, and searched, and retraced their steps over and over.  After about fifteen minutes I realized that they were looking for an actual X on the ground.  At that point I stepped in and wondered aloud whether someone who had hidden treasure would want to mark the spot on the ground or not.  I mean, really.  If you've got treasure you really don't want anyone to know about it, right?

The three kids finally decided they knew the exact place where the treasure was buried and I had to agree.  They had done a perfect job at reading the map but couldn't find anything.  The finally decided the treasure was under the cement trough somewhere and inaccessible.  A little downcast, but ever perseverent, they moved away from the spot to talk some more about the whole situation.

At that point I figured that, if they had spent over twenty minutes working together to analyze the map and the territory and actually succeed in pinpointing the exact spot (not to mention collaboratively analyzing the potential motives why someone would leave a map behind) they should be rewarded in some way.  Being short on change I petitioned bystanders who had been watching the proceedings with interest.  Between us I got three dimes and three pennies.  I made sure the treasure hunters weren't looking and poked the change into the dirt.  Then I said:

"Hey, guys!  Weren't you just over here?  Come here, and take a look.  I think I might see something..."

My bystander friends were looking on at a distance and we all just beamed at how excited the kids were to find their treasure.  They figured out how to divide the money equally.  I'm not sure, but they may have wondered why it wasn't more than 33 cents but that didn't stop them from telling their story over and over to anyone who would listen.

You know, the story about The Day -- the day they followed a found map to buried treasure on city hall grounds.  Perfect.

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.

Totally Territorial: Cats, Maps, Area & Multiplication

I know it doesn't look like much, but there was a battle going on.  We rolled the dice (one six sided, the other twelve sided) and staked out our territory.  We're warrior cats, you know.  You didn't?  Have you read those cat clan books?  My kid hasn't -- too many kitties getting hurt makes her crazy -- but her best friend has.  They 'play' those stories every time they meet.  You should see them in their 'warrior training'.  It's crazy.

The minute we started rolling the dice I knew I needed to give this activity a special spin.  We had just finished a 'make fifteen' game where the last person to fill in a combination of fifteen wins.  There was no real strategy involved, hence her lack of buy-in.  In fact, she had been thoroughly unimpressed with it and didn't appear too eager to try something else. 

This new activity was basically built on the same model.  The goal was to see, after lots of dice rolling, who would come out with the most area colored in after the whole page was covered.  (For the life of me I cannot remember where I got this idea, but I know it was from a recent blog post.  If it's you I apologize for my lack of recall -- and please let me know so I can credit you!) 

Since success in this game was mostly left to chance, I knew I had to think quickly. "Let's stake out our clan territories and see who gets control of the most area!" I exclaimed.  Needless to say that's all the motivation she needed.  She decided almost immediately that each new piece of territory had some kind of real-life correspondence -- campus (the scene of our recent Tana Hoban adventure), our favorite hangouts, she wanted them all.  I pointed out that she could be strategic with where she filled in her boxes and, if you look at the first photo again, you'll see that she's surrounded parts of my blue territory so that I have to cross her land to get to other parts of mine.  Nice.

Then I realized that since she was thinking of real places we could probably do the same game on a real map!  Here's what we did later in the day...

We got a free map of our fair city at the bike shop.

With her watching, I took the yard stick and marked off a 1/2" grid over the entire city.  The grid was not to scale but was consistent across the entire map.  The area unit was 'one square'.

We looked at the map and found our landmarks -- hangouts, parks, campus, lakes, our neighborhood -- and at the map's key to find the roads, schools, bike and walking trails, etc.  Then we started strategizing about which areas we'd most like to 'have control over'.

We rolled the twelve sided die first to get our length (or width, whichever, depending on where it was best used strategically).  In those boxes we placed a dot using a permanent marker.  (We wouldn't be able to read the map if we colored everything over.)  Then we drew a line to show the border of that 'territory'.  Using the six-sided die we rolled the other side and figured out the total area of our newly acquired land.

I used the words area, length, and width as well as north, south, east and west during our conversations about what parts of the city she and I wanted to acquire and why.  Some of our reasons were strategic and others were more personal -- I got a lovely lake with the hiking trails, for example, and she got the big park with two awesome playgrounds.  I also modeled a lot of multiplicative thinking.  Even though we did use skip counting to compute the total area, I would point out groupings: "How many rows of seven do you need to fill in?  Second row, two sevens are fourteen.  Third row, three sevens are 21..." 

The strength of this game is that we know our city pretty well.  We've been to a lot of the parks, we move through town frequently on foot, on bikes, on buses, and in the car.  Our personal landmarks relate to where our friends live and work and go to school.  It really is our territory which is why the map was so real to her; it made sense to her despite being an abstract 2D visual representation.  Plus, the cat clan narrative is one she knows intimately, further increasing the emotional connection to the material.  Tomorrow we might play the game again and total up the red and blue areas to see who has the most.  But then again, given the engagement level today, it almost doesn't matter.  I think we've both already won. 

Mapping the Familiar

There was a bright, blue sky up above, and at ground level there was frozen fog.  The result was a morning delicately dressed in sparkles.  Perfect for a walk.

Down the driveway.  Pick a piece of lavender, covered with millions of tiny ice crystals.  Which way do you want to turn, left or right?  Next intersection, right or left? 

We walked through the fog, already burning off.  We could feel the sparkles entering our lungs with each breath.

Busy road and cars zooming.  Three choices, right, left, or straight ahead.  "Mama, I used to be afraid of those trees, but I'm not any more."  We pick tiny pine cones off the tips of the branches. 

Next crossroads, "Mama, that's University Street down there, let's go there."  Thinly frozen puddles alongside thinly disguised reading practice: "What does that P stand for, do you think?  The one with the red circle and the cross over it?  What does the rest of the sign say?" I ask.

Right turn down University and an old dog barking at us across the lawn.  "Mama, I used to be afraid of dogs, but now I just don't like their licking."

Find two sticks, clap them together and then we're marching.  Left turn.  Right turn. Rose street!  Find the stone pig, march on home.  "I wanna make a map!" the girl says.

She wanted red and blue like the roads in the atlas.  We recalled which way we turned, what we saw, remembered the crossroads, one block at a time.  All of a sudden, she notices corners and the geometry of the street layout.  "I always thought the blocks went in a circle.  I didn't know they were squares."  The things she loves go on the map (friends' house, stone pig) as well as the things that 'used to' frighten her.  Her long battle with anxiety has its landmarks as well.

It may not be to scale, but when we go out again this afternoon to follow our map, the issue of scale might come up.  I'm also pretty sure we didn't get all the streets on there especially at the end of our route, but next time we'll bring paper and make some notes as we walk.

Mapping the familiar twists and turns and landmarks of our neighborhood -- bodies first, memories second, paper and pencils third.  I marvel at the human brain inside my six year old daughter's head that is so driven toward representation of her experiences and activities; driven toward it even though she is still just learning to decode print and write using 'the rules' .

She makes maps of other kinds, too.  Sewing "patterns", also not to scale, but clearly a sequence of steps mapped out:

I recently read a fascinating article in the New York Times about teachers taking their young students on walking field trips as a way to develop literacy.  This kind of activity is literally a step in the right direction.  Without concrete, kinesthetic, physical experiences like these, no child can fathom the meaning behind the marks on the page or develop full mastery of the human brain's greatest gifts. The order needs to be sensory experience / memory / symbols, not the other way around.