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?

Lessons From A Rafter

I love where I work.

I was just preparing for a couple of homeschool classes I'm teaching as part of a little outreach program for Dodge.  My next classes focus on seeds, so my prep work consists of going out to collect as many seeds as I can.  My job requires that I go on a hike here at the Nature Center.  Don't you feel sorry for me?  While I'm out on the trail prepping, I am quickly stirred from my quiet reverie and reminded that I am not the only forager in the forest.  In short order, I encounter three deer, five squirrels and sixteen turkeys; did you know that a group of turkeys is called, "a rafter" of turkeys?

Well, the small herd of deer, the drey of squirrels and the rafter of turkeys are all too busy eating to give me more than a glance.  Passing through the orchard, I reach up to grab an apple and nearly step in a pile of fresh coyote scat, full of apple chunks, and seeds.  Not only is my job really enjoyable, but it reminds me of my relationship to the rest of the planet.

I get to hike or be outside every day with my students, so I'd have to be pretty thick not to get the message about my relationship to the web of life; we humans are, of course, inextricably linked to everything else, whether we remember it or not.  But, today, on my little solo expedition, my place in the world, really hit me over the head.  I'm just doing what everything else is doing:  living.  Of course, when I'm out with my classes, we see animals frequently (wildlife is so available here at Dodge, so non-plussed by our non-threatening presence, it's really quite silly), but crossing paths with wild things when you are alone is different.

I think highly of solitude, especially in the woods.  A walk in the woods alone, with your phone turned off, is like nothing else.  There is time and space to notice stuff with fewer distractions or concerns (especially when all the leaves have blown off the trees!).

While I was out there, following/harassing the rafter of gobblers for a few photos, I got to thinking about solitude and companionship.  I seem to need quite a bit of both; everyone is different.  Most early childhood people think that socialization, learning through companionship, is of vital importance for children.  I'm in the socialization camp.  Attending school and learning to get a long with others is, as we reminded families during our recent Curriculum Night, curriculum.  Now that I'm teaching a couple of classes for homeschoolers, I'm thinking about the importance of companionship a bit differently.  Parachuting in to a rented space, where homeschooling families gather one day a week and share a little array of classes, is a really interesting contrast to teaching in my classroom at Dodge.  Lively community can certainly be found in that rented church on Thursdays, but it is of a different stripe.  Here at Dodge, I'm in danger of taking my embeddedness, my membership in the school community for granted (thanks to the turkeys for the wake-up call).  At the homeschoolers' site, there are elements of school, as kids move from Spanish to Music and Science (that's me), but there is a lot that is very unlike school as I have known it here at Dodge or in my own past.  That is neither here nor there though; this is no indictment of home-schooling.

Mostly, what I am struck by is the fact that both homeschoolers and schoolers rejoice in the same two things:  companionship and play (and play outside in particular).  Kids find joy with friends in play.  The schoolers and homeschoolers may need companionship and play in different concentrations, and they might even have different motivations to play socially, but they all seem drawn to it, whether they find it fulfilling, engaging or challenging.  In my experience, even children who are hesitant to join social play seem to spend a lot of time observing it and, I therefore presume, considering it.  The homeschool classes, and the turkeys, have served as a reminder of how important a sense of relationship to the world is.  The homeschoolers are engaged in the content of my classes, but their interest in socialization and play reminds me that certain needs are universal.

The turkeys seem to like companionship.  All sixteen of them were hanging together, following a leader, chortling and chuckling under their turkey breaths to each other.  They tolerated me as long as I stayed back at least twenty feet.  Many animals find safety and benefit in numbers.  So do plants for that matter.  The seeds I'm collecting wouldn't be so abundant right now if plants didn't proliferate and "rub shoulders" with each other and the animal kingdom as they do.  Even lone wolves are driven to approach a pack from time to time, if only to fulfill some instinctual drive.

Turkeys, deer, squirrels, kids-- they all remind me that we're in this boat together.  The world may be vast, but we sure do have a lot in common.  If you can, I urge you to take a walk in the woods by yourself; in your solitude, you might discover how much you appreciate being part of the pack.

Favorite Places to Walk:

Dodge Nature Center
Murphy-Hanrehan Park Preserve
Ritter Farm Park
Lebanon Hills Regional Park
Lake Harriet Peace Garden
Lakewood Cemetery (lot's of solitude!)
Afton State Park
Carpenter Nature Center

Tradition.....Tradition!

My kids love the opening number from, "Fiddler on the Roof,"-- the one Tevye sings, "Tradition."  Tradition is the central theme of the musical.  The villagers are trying to hold on to their cultural practices in the face of pogroms and social unrest...the father doesn't want his daughter to marry someone who isn't Jewish...Tevye and his wife have the classic ironic spousal relationship:  he is supposed to wear the pants, but she actually does...  Of course our traditions tell us a lot about who we are.

Yesterday, my girls and I played hookie.  Every autumn, we take a day off together.  A day of hookie is our tradition (I'm not sure what this tradition says about us; absenteeism is not supposed to be a teacher thing).  Fall is our favorite season, so we usually try to coordinate with Mother Nature.  We shoot for a day in early October, in order to take full advantage of autumn color.  My first season in Minnesota taught me that if you blink, you'll miss the leaves.  Come the second week in October, the wind whips every last leaf from the trees.  New England, where I grew up, has a longer and gentler autumn; it hangs around long enough to make an impression.  New England is famous for fall, right?  Anyway, this fall day off has become a habit for us, and part of the tradition is that I try to surprise the girls.  They are ten now, but a whole year passes between these days off and they tend to forget.  Yesterday, I woke them up and told them my alarm had not gone off (I'm usually up and off before they get up in the morning) and so I would be the one to drop them at school.  We hustled around and jumped in the car.  They were crabby about not riding their bikes to school, and when I drove right by their school, they were totally bewildered.  When I laughed, they looked down right worried:  "Oh Dear God," their faces said as they exchanged glances, "she's off her nut!"  Then JJ got it:  "It's our day off!"  Wahooo!

Our day off, ironically, is a day off in the Big City, usually Minneapolis.  When you work at a nature center, you don't necessarily want to hike on your day off.  Our annual urban safari has some tried and true pit stops.  After seven years, there is very little change in our itinerary.  We've got it down.  If I forget something, the girls remind me, and they aren't quite satisfied if we skip a stop, or if something changes dramatically.  Little changes can be a delight, like when we discovered a new animal at Wild Rumpus.  Big changes can be disappointing, like when we learned that a favorite window-shopping destination was relocating.  And yesterday, Turtle Bread was all out of chocolate rolls.  That was almost a crisis for me.  If my almond croissant default hadn't been available, there's no telling what may have transpired.  Lunch is our only wild card really and I think we're destined to make a long term commitment there too.  Part of what makes the day so great is remembering what we always do, and anticipating returning to the same activity or place (or food); it's tradition.

The bittersweetness of this season is likely highlighting the phenomena of building tradition, but I've been noticing emerging, developing and continuing traditions here at Dodge lately.  This is my third season here, and it seems only natural that I can now see some of the Preschool's traditions.  There are the obvious habits and practices.  Every September, we go see the bee demo out at the apiary.  A couple weeks later, we all get a tractor ride out to Mrs. Dodge's apple orchard and press cider at the Cider House.  Every October, we host a Curriculum Night.  We always have a Friday night Halloween party.  November comes and it's time for parent/teacher conferences.  These are activities that happen each fall.  But there are less noticeable traditions too, those that perhaps have a bigger impact on the children, and are a part of the true fabric of our classrooms, and the school.

 My colleague, Joey, began making a list of all the songs our class knows.  She talked about this list with the students and posted it in the classroom.  Kids were invited to illustrate the song titles and to contribute more suggestions.  Many of these songs are ditties certain teachers sing from year to year-- favorites.  Our co-teacher, Luzia, who is from Brazil, loves to sing and often solicits song ideas from us (well, Joey mostly; I'm really not a songbird, or even a warbler).  Luzia's enthusiasm made me realize that without practice, memory, or documentation (in the form of that list), these songs are in danger of dying out in our room.  Teachers (like Joey) and kids keep them alive.  This is like any other oral tradition-- practitioners breathe life into the stuff.  Just the other day, we were on a windy hike with a continuing student, a child who was with us in the Spruce Room last year.  When we stopped to rest, she requested, "The Wind Song."  I asked if she would lead us and she did.  A small handful of other returning students took up the tune too.  Pretty soon, the new students had caught on and the entire class was belting out the song.  Now it is a favorite in this year's class.  Returning children are responsible for carrying a lot of our traditions forward; this is one of the beauties of the mixed-age classroom.

Last week, our afternoon class hiked up to Tipi Hill (aka, "Princess Mountain"-- see previous post).  It was a return trip.  We had happily picnicked up there just days before.  And during that previous visit, a child had lamented over the fact that the "Stone City" (a "Roxaboxen" of sorts which we had created up there last year), was no more.  We chatted about this and then made some loose plans to return to Tipi Hill and "do something" about the Stone City.  So, we kept our promise and returned last Friday.  We milled about, admiring the blaze of Sugar Maple leaves and absent-mindedly kicking at the hundreds of fist-sized stones now scattered far and wide.  Then the same kid, the returning kid with the great memory, said, "Remember the stone spiral?  Marlais, do you think we could build another one?"  And before I could answer, she scampered away shouting over her shoulder, "Let's build it right here!"  And there were just enough kids who were with her last year.  You could see the memory jolt through them.  Their reaction was actually visible; it was physical-- a catalyst!  Suddenly they were bending and picking up rocks, following the excited directions of their new foreman.  And you know what?  These kids were tiny last year!  The youngest in the class.  Last year, they were recruited by older kids to help out in the spiral project.  This year, the Dodge "graduates" have gone on to Kindergarten, but they left a legacy behind.  Here were last year's apprentices assuming command, confidently issuing directives and recruiting a whole new crew of eager young assistants.  Viola!  Out of an old Dodge tradition (mixed-age classes) a new Dodge tradition is born:  the stone spiral.

Readers of this blog may recall that the original stone spiral, built last autumn, was inspired by a snow spiral, created that previous spring.  All last year, we loved watching children alter that spiral.  One class' enthusiasm spread to other classes.  All the Dodge kids visited Tipi Hill and played with the spiral.  By winter, everyone was making spirals again, bigger and better.  So Tipi Hill with it's stones joins other Hall of Famers here at Dodge like Challenge Hill, Mini Challenge Hill, the Stick Forts and the Castle Wall.  These places, their names, and the activities that occur there are now learning traditions, kept alive and brought forward by returning teachers, and returning students.  Old teachers (or should I say, "experienced"?) share their practices and knowledge with new staff, just as the returning students share knowledge with newbies.  New teachers and students bring their experiences to the group as well.  Traditions may remain, but they are altered, and improved.

Last Friday, up on Tipi Hill, I watched Kristenza, (we co-taught last year), teach kids how to make an assembly line with their bodies, passing stones from hand to hand, to build the spiral bigger, faster.  I don't recall us doing that last year.  Kristenza had found something new.  And it was beautiful; it was real teamwork, collaboration, motor skill, impulse control, kindness and more all rolled into one.  The perfect mixed-age activity; something for everyone!

And just when I thought it couldn't get much better, I turned around to see that Luzia had thrown a big log over the stone pit.  Kids were trying like you-know-what to dash across without falling off.  A broad spectrum of large motor control was on display.  Threes, fours and fives dashed, wobbled, fell and flew (remember, age does not necessarily correspond to ability; everyone develops at different rates) over that log.  Two great new Tipi Hill traditions in one afternoon!

Together, we get to decide what endures and what falls by the wayside.  We are a community of individuals, after all, and which traditions we choose, year in and year out, tell the story of who we are.  Next year, maybe nobody will like, "The Wind Song."  Maybe they'll be crazy about "The Pirate Song," instead.  Who knows?  But maybe someone will remember...