Future Fibonaccis

So, we are contemplating putting together a Dodge Nature Preschool Newsletter edition devoted to math.

Wow!

Math is a really big topic.

Math is everywhere, every day, all day long.  As far as I can figure, math is a human construct, a sort of handy/cumbersome language for describing phenomenon in the world and making good predictions about that phenomenon in the future.  Know what I mean?  Stuff is already here:  vines, atoms, seashells, dirt, air, DNA.  Math is describing the stuff that is here, explaining how it got here maybe, and what you could do with it in the future.  Take an ear for example.  Lot's of animals on this earth have ears.  Scientists can now grow human ears on mice.  Kind of gross, but true.  And, if you happen to be born without ears, or you are a burn victim, you might be glad that scientists are using math to grow ears.  Math is behind finding the cure for cancer and figuring out how to curb greenhouse gas emissions.  Math is even behind making ice cream.  Math, math, math.

So here are my first raw and rangy musings about math in and out of the classroom here at Dodge.

Units have meaning.  My co-teacher, Joey, says that her math teacher used to rail, "Write down the unit or it has no meaning!"  6 milliliters is a lot different than 6 tons.  Family units have a great deal of meaning, especially for preschoolers.  If a child says, "I have six people in my family," that means a lot, especially to the child.  Each number represents an entirely different and complicated person:  "I have a mommy, and a mommy and two dogs and a brother and me."  Perhaps a young child can count to 100.  But can she tell if the friend sitting next to her at the snack table has more pretzels than she does?  Maybe she can guess who has more, just by looking. Or maybe she has to count each pretzel individually in order to really understand who has more.  Math is not only everywhere, but the most important math skills, the big concepts, seem to be acquired and developed rather insidiously very early in life, every day as a kid goes about being a kid.

I think young children naturally acquire and develop early math skills through what Maria Montessori called "practical life” or “daily living” activities.  Here at Dodge, children have many opportunities to practice “daily living” skills.  When we enjoy snack in the classroom, we serve the food family style.  Children are required to portion out their share from the whole.  They often literally count individual foods or scoops of snacks, but the less literal emphasis is on dividing the whole (which is great for developing social and emotional skills like empathy and patience too).  Materials in the classroom must be shared.  Kids are grouped and re-grouped for hiking and other activities.  Even when children are taking turns, they are learning mathematical concepts, seeing themselves in relation to the whole and understanding a basic sequence of events.  

When kids cook with us, they are following a recipe, moving step by step, portioning and, in a sense, working to solve a problem:  how do you make muffins?  Attention to detail, the relationship of the parts to the whole, the transformation of disparate raw ingredients into an entirely different single product, the dawning concept of an outcome—these aspects are all fundamental to understanding numerical ideas and they serve to help children develop an approach to problem-solving, don't they?  When children work on dressing themselves for outdoor play, they are working through a set of tasks which they must order to be successful:  snow pants, boots, jacket, hat, mittens.  Mess with that order, and your process will be downright frustrating, so frustrating that you might not reach your objective of getting dressed and playing in the snow.  

At a very young age, through experience, children learn to discriminate between all the objects in their life.  They learn to tell a cat from a dog, a shoe from a spoon, a finger from a toe and so on.  Differentiation is not only a part of basic survival (you can eat this, but you can’t eat that) and of language development, it is also a huge component of math skill development.  Categorizing and grouping things and concepts are important in mathematical problem-solving.  The very spookiness of Einstein’s “spooky action at a distance” theory seems to find its foundation and importance in math’s reliance on logical relationships, or correlations:  if I do something to X, then Y will be impacted.  Here is a bastardization of Brian Greene's metaphorical explanation of this theory:  Let’s say I had a pair of gloves and I gave one glove to you, and you flew to the moon with it.  If, while I am here on earth, I spin my glove around in a circle, your glove, the one you have up on the moon would spin too.  Something about the fact that they are a pair makes them linked, and even if they get split up, they are entangled.

This quantum mechanics' theory of entanglement is weird and spooky and, as I understand it, seems to have been proven about particles.  Scientists' growing understanding of entanglement brings us closer to having Star Trek teleporters being a part of daily life (I'm taking their word for it).  As hard as it is for me to try to understand quantum mechanics, I get that most math and science is based on exploring the relationships between stuff, or phenomena.  Scientists are trying to figure out the nature of our day to day existence by examining logical relationships, and your kids are gaining a really fundamental understanding of relationships when they interact with their environment.  We usually call this sort of hypothetical thinking, “cause and effect.”  If I put on my boots first, it’s going to be hard to put my snow pants on after.   Kids may not always get dressed at the speed of light, but when dressing they are practicing skills that will help them understand the speed of light later.  

And speaking of measurement and time, preschoolers are fascinated by both, and they naturally seem driven to form questions and pursue ideas around these concepts:

“In five sleeps, I will go to grandma’s.”  

“Two yesterdays ago, I found a frog.” 

“I am as long as this stick.”  

“My dad is bigger than you.”  

“I eat supper when it’s dark out.”  

“After Halloween, comes Christmas.”  

“How many minutes until my mom comes?”  

Kids' comments and observations tell us that they are nearly obsessed with these big mathematical concepts.  At Dodge, teachers work to highlight concepts or extend inquiry as they follow the lead of interested kids.  Many years ago, during my first visit to Dodge, I hiked with a group of kids for an afternoon.  It was springtime and we came across a swollen stream.  The children fanned out on either side of the Farm Road and began to rummage through the brush.  They were looking for sticks.  Sticks acquired, the kids eagerly stepped into the stream:

“It’s up to here.”  

“Up to here on my stick.”  

“This is how high it is today.”  

"Lot's of water in the creek."

"More water in the creek today."

Not only were these threes, fours and fives measuring the stream with their sticks, and comparing their findings, they had clearly been measuring it routinely:  it was a practice that a teacher cultivated and worked to continue, an inquiry she had tried to extend.  Teachers help children notice phenomena, make observations and form hypotheses all day long here at Dodge.  Our daily routine itself—outside play, snack, group time, free choice—not only manages expectations and provides for learning, it teaches children to notice time, and to think about how to work with it.  Children know that if they hike all morning, there will be little “inside time.”  If a teacher has given a “five” or a five minute reminder, then a child knows she might not finish a journal story she has just begun.  And of course, at Dodge, we are extra aware of the larger schedule informing our days:  the seasons.  Experiencing seasonal phenomenon is a powerful way to understand time and change.  When a child rakes autumn leaves, finds a shed antler, hears a nesting owl, smells an apple blossom or eats a ripe apple, she is not only having a terrific experience, she is coming to understand a concept and a process.  Through these experiences, she will begin to grasp relationships in the world:  flower-bee-fruit-me!  If the bee visits the flower…

Of course, we see and encourage math in our classrooms in all sorts of more explicit ways.  We ask for help when we count.  Children build with blocks, practice geometry and form hypotheses as they build.  Kids play board games, taking turns and counting spaces.  They add and subtract physical items.  They match symbol to object in sorting games.  They play dominoes.  They decide how many “bad guys” will be in their journal story.  They weigh items on a set of scales.  They experiment with objects that sink or float (and discover that bigger does not necessarily mean heavier).  They trace numbers at the writing table.  They work on all sorts of math-centric activities that we provide for them, but more importantly, they come at the world with a bigger, more dominant curiosity about the way things work.  And math is a useful way to try describe the way things work and to make sense of the world.

There are moments when children move beyond math for survival and happiness, like sharing food or toys, and they embrace the more abstract or metaphorical aspects of math itself.  Joey recalled that we had a student who sat in his cubby, amidst a row of other cubbies, and mused to himself, "I am one.  One is one.  I am one.  I am one.  None is none.  None is nothing.  But I am one."  It was like having an audience with the Buddha.  Kids frequently reject our routine "head counts."  We might be counting children, lightly touching them as they walk through the door; we touch one head and say, "Six."  And then we hear, "Six!  I'm not six, I'm four!"  For me that begins to say it all:  Math starts with experience, and that experience has to be personal.

The child sees her personal relationship to the rest of the world:

-I am me.
-One.
-Different, but the same too.
-Part of the whole.

From there, math knowledge and skill seems to spiral out.  The concepts get bigger, there are more parts to the whole, the spiral gets broader, but it is still connected to that one stone in the center.  The spiral metaphor is no mistake.  There is an architecture to physical existence, isn't there?  A pattern to nature?  The shape of the universe, the rotation of the earth, the stream of the air and seas, the pattern of seeds in the flower's head, the twist of DNA... like Fibonacci, young children see, learn to look for, talk about and make patterns and relationships through daily, hands-on experiences with the world.  This looking for patterns and correlations, talking about them and making them is developmentally appropriate math practice, isn't it?