Ten Times Better, Longer, Faster, Farther: Understanding Scale

I just found the most excellent book!  I was at the library looking for two math related books, one of which was in the multiplication section, when I ran across Ten Times Better.  On first glance it looks like it's just an introduction to your tens times tables, albeit a very engaging one in poem format.  Essentially the animals try to best each other -- one animal has an amazing attribute but there's always another who is 'ten times better' (more, bigger, faster) than the one who started the bragging.

Fun, but as my seven year old daughter said "I know my tens times tables," and I think that would be most folks' assessment of the book.  But...

...if you read all the way to the end into the section just past the poems you will find a gold mine!

Multiplication to me has generally always been about memorization.  I've been trying really hard lately to push through my wall of resistance about numbers toward something that approximates actual understanding -- I want my daughter, as well as myself, to understand what multiplication really is.

What I've come to learn is that multiplication can be used in a number of different contexts and that there are some basic models that can help a child experience, visualize and conceptualize the meaning behind the operation. Through some reading and a great post from Let's Play Math on multiplication models I've learned that these models include:

Sets or groupings, which we've done a lot of.  Arrays are ubiquitous.  But the third concept, measurement, is the one I am least comfortable about.  I mean, when I was in third grade I memorized my times tables and that was that.  But I had no earthly idea what they were for or even how to use them outside the context of an equation.  Working through the back pages of Ten Times Better this morning I finally get it.  And, what's even better, my daughter got to pull out the 30' tape measure, run laps in the sun room and do some long jumps all in the name of mathematical inquiry.  Here's are the highlights:

The back of the book holds amazing information about the twenty or so animals from the poems in the preceding pages. The ant is TEN TIMES STRONGER than humans.  If an ant weighed fifty pounds (the weight of a human child) how many pounds could it lift?  My girl counted it up on her fingers and immediately sprang up and ran around the living room trying to lift up all the chairs.  I nixed that idea, but it was such an immediate reaction that it sparked the idea that this needed to be an interactive experience.

Nine-banded armadillos are one foot long.  They are descended from glyptodons who were TEN TIMES LONGER.  I must admit I initially tried to get her to just answer the question using her knowledge of the tens times tables.  I mean, they're easy, right?  But as this activity proceeded I realized very clearly how memorization of facts does not assure an understanding of either how to apply the operation in context and especially not when size and scale are involved.  My kid, at least was not getting it, even with an 'easy' problem of 1x10.  So, I said, how about using measuring tape?  She ran to retrieve the 30' measuring tape and we found the one foot mark.  Then we counted out a total of ten feet.  Wow!  The glyptodon was not huge, but definitely MUCH bigger than it's modern descendent the armadillo.

Some centipedes have 100 legs, but the garden variety has 30.  "With so many legs," the text reads "the centipede really is TEN TIMES FASTER than most insects it catches for food ... If a centipede were as long as a six foot adult is tall, it could run twelve feet in one second.  How far could it run in ten seconds?" Well.  On this rainy day we couldn't go outside, but we measured the sun room at 24' in length.  Five lengths would equal approximately 120'.  Could she run it in ten seconds? (It took her only twelve!)

Laughing, we went back to the living room. Did you know that elephants are TEN TIMES HUNGRIER than you and eat as much in one day as you eat in a month?  If you eat 40 pounds of food every month how much does an elephant eat daily?  I started skip counting by 40 (which is really the same as 4, right?) and she joined in.  400 pounds!?!  Wow.  That one impressed both of us.

Next, the frog. "Most people can jump as far as they are tall," says the text "but a frog can jump TEN TIMES FARTHER."  Before I knew it we had jumped up, taped down a line to jump from, measured the length of the girl and she was off!  It's true!  She could jump the length of her body, and sometimes a bit longer.  Can you imagine TEN times further?   From the back of the house to the front of the house.  Wow.

A baby giraffe is six feet tall when it is born.  The kid jumped up on the couch to measure how tall that was.  And it is TEN TIMES HEAVIER than a huge human baby.  "If that baby weighs eleven pounds at birth...how much might a small baby giraffe weigh?"  By this point, she really had the concept and jumped in to start counting by 11, fast, marking on her fingers...110 pounds!  She ran off to the bathroom scale.  At the age of seven she's about half the weight of a baby giraffe at birth.  Awesome!

The goldfish question is great -- if kept in a small and/or crowded bowl it only gets to be about 2" long.  If allowed more room, TEN TIMES LONGER!  We pull out the measure tape.  Here is 2" let's count that ten times... Again, the numbers are one thing, and 20" isn't that long, but seeing the tape/fish get longer, and longer, and longer is another.  A living number line!

Our last one was the giant squid that can be TEN TIMES LONGER than the tallest basketball players are tall.  "If a basketball player is seven feet tall, how long would a giant squid be?"  70 feet?!?  How long is THAT?!  Our measuring tape was only 30' long.  Our sun room is 24' long.  So that means almost three sun rooms long??  Wow.

Did I mention I really, really love this book?  Scale is such an elusive concept for me, and I'm sure for kids too.  Ten times bigger, longer, faster and smaller is a large enough amount to make an impact, psychologically speaking, on kids who know intuitively that they are small creatures in an adult-sized world. I read somewhere that intelligence isn't the biggest factor in being 'good' at math -- it's actually personal motivation (and, I would add, personal relevance) that motivates someone to engage in mathematical activity.  This book provides motivation in spades.  I think that kids from preschool to middle school could all get something out of physically measuring out 'ten times longer/bigger/faster' to figure out the answer instead of just calculating it.  Just because you've memorized something does not mean you 'know' it.  I am living proof!

p.s. I just found this book.  No one paid me for a review.  But, if asked I'll say it's TEN TIMES BETTER than any other math book I've read in a long, long time!

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.

Big Math: Kid Sized Geometric Structures

Oh my gosh, look what we built!   This was not a small moment of math, no indeed.  This was a BIG math moment, one that took nearly all day (mostly because we had to figure it out for ourselves from start to finish).  And, it cost next to nothing and yet we gained so much.  In these next two pictures it's only about a third of the way done, but isn't it wonderful? 

You can run around the outside of it, of course...

Or you can hang out inside it but, at this point, it's still not all that stable and there's a LOT more building to do.  But I'm getting ahead of myself -- here's how it all started...

Having just finished the entire Hunger Games series in one week I was left with nothing to read while the kid fell asleep except this.  It's a catalog that came in the mail and I was flipping through it when there it was!  Some kind of structure involving tubes of newspaper, making some kind of geometric form...hey, I could make that!  What's this?  $40.00 for some connectors?  I don't need those connectors -- I'll figure out some other way!

That evening I figured out how to make those paper tubes and found out it's much easier and faster if you roll a section of newspaper around a dowel rod.  After it's rolled I tape down the long edge to the roll with clear tape and then slide the dowel out. 

After making four or five of them I left it there for the night.  As I often do, I left the tubes lying around to be 'discovered'.  It didn't take long.  At about 8:30am the next morning the questions started coming in from the girl.  All I had to do was show her the catalog picture and we were ON!

So, we got started building.  I couldn't tell all that much from the catalog picture, but I figured it was at least a hexagonal base, and those triangles were reminiscent of the dodecahedron and icosahedron I made from straws and pipe cleaners this winter.  I figured this was pretty much the same process, just bigger.  This was our first big lesson -- the issue of scale.  Turns out, the bigger your structure is, the more effort it takes to make sure it doesn't fall down.

Anyhow, at least I had the tube figured out.  My rolling method leaves a pretty sturdy paper tube but it is somewhat time consuming to make a bunch and pretty much for the mama to do and the six year old to watch.  Luckily, my kid likes hanging around, soaking in the process, while making things like this are happening.  You can learn a lot by listening to your parents mutter to themselves!

But what to do for the connector?  Now that was the perfect opportunity to do some brainstorming with the kid.  After some experimentation we settled on a small amount of folded newspaper that made a nice connector, but only when two empty tubes meet.

For a connector that attaches to an empty tube on one end and an already-connected tube on the other, the kid and I finally figured out that a combination of paper and folded pipe cleaner would do the trick.  And it did, sort of. 

It turns out that the points where five tubes meet need much more reinforcement.  In that case, use lots and lots of tape.  But I'm getting ahead of myself.

After finishing the hexagonal base and the first level of triangles we needed a break and luckily it was time for lunch.  How did it get to be lunch time so fast?!  We left it there for an hour or two but eventually both of us felt compelled to return to the project.  During our break I took a minute to look at the catalog picture and description more closely and discovered the following:

It was $40 for connectors and 120 tubes around which to roll your newspaper...???  Why did they even mention newspaper in the first place?  With all those tubes, no wonder that structure in the catalog looked so sturdy! 

I was pretty much clueless as to how to proceed past the first level.  I experimented with adding triangles to make a second layer, but it didn't seem stable or look right.  I said to my girl, "Hey, I think we need to make a model so we can figure out what we're going to do next.  I'll need help with that."

We pulled out the straws, I cut up some pipe cleaners, she connected the straws to make triangles, folded angles and helped me build the model.  Turns out this was a very helpful process.  What we figured out was that the second row of triangles needed to be connected by shorter edges (half of a straw length) if it was to have any chance of being useful.

Isn't it pretty?  Not sure what to call it, but at this angle it has the look of an icosahedron about it.

We applied what we had learned  in the model making process to the big structure and...it was still really hard.  Completely unwieldy.  Things falling, sagging, coming apart.  If those connectors had been close by for purchase, I  might have caved in and bought the things.  Since this was not the case, I had no other choice but to persevere.  I noticed the connectors had started failing and so I brought in the big guns -- tape.  

It was actually an incredible lesson in physics.  When even one side of one triangle lost its connection to the structure, the whole thing would start to tumble.  The kid was in the center helping to hold everything up so that it wouldn't completely collapse.  I'll spare you the gory details but, short story, I was quite liberal with the tape and we did eventually get it to the point where it felt fairly sturdy.

At some point in the middle of this circus act I said to the child, "It's a good thing we're doing this ourselves.  If we had bought the connectors I don't think we would be learning as much as we are." 

I wanted to leave it there, but the girl wanted the pointy thing on top, so we persevered.  And, what do you know?  Finally, finally, the thing felt whole, strong, sound.  Plus, the girl can stand up inside it!

"Mama, look!  A pattern!!  Up triangle, down triangle, up triangle.  Rhombuses!  Big triangle, narrow triangle.  And, uh, what do you call it? [running over to get the Cuisenaire rods to make the shape] A trapezoid!"

We even called over to our next door neighbors so they could come and admire our accomplishment, which they did and which is precisely why we like them so much. 

"Mama," she said, "It's beautiful on the outside, but it's even more beautiful when you're inside it.  There's a flower up there."

Ah, the magic of math.  Structure, symmetry, order, strength, beauty.  Ours. 

[Linking to Saturday's Artist at Ordinary Life Magic.  :-]

Sidewalk Math: Functions!

After a mild winter we had a lovely and quick blooming spring which allowed us to get out and about earlier than we might have otherwise.  In March I posted about an outside adventure where we discovered a veritable treasure trove of circles in juxtaposition with other shapes.  I think this might have been the origins of what I've started calling 'sidewalk math'.



Sidewalk math is fun because, generally, all you have to do is keep your eyes open.  If you've got a camera to record your observations, all the better.  This is not necessarily an original idea; the photographer Tana Hoban has a whole series of books with photos of the math all around us.  Her camera is the eye through which we can notice math in the physical world.  There are also the engaging Math Treks developed by Maria Droujkova of Natural Math.

For us, sidewalk math is a combination of these two approaches and has turned into a large percentage of our first grade math classroom.  It capitalizes on my daughter's propensity to notice everything, fulfills her need for movement while she learns, and bypasses her resistance to formal lessons.  It's also an opportunity for us to make observations and pose questions in a collaborative way, which is an approach that works for both of us.  For example, on a recent walk my daughter notice a crack in the sidewalk that initiated an hour-long in-depth conversation and exploration into the nature of triangles as we traveled to the hardware store and back.

(And it's apparently it's sticking with her: As I'm writing this my daughter calls down to me to report that she and her dad saw "seventeen triangles on their way home from the park this morning....did you know that part of an arrow is a triangle?!" )



But, in this story, sidewalk math plays another role, that of salvaging my initial attempt to introduce functions to my young daughter.  You can read about my first attempt here where she was wholly and unequivocally unimpressed with my presentation of the subject and took matters into her own hands.  I ended the post wondering what to do next.

I was understandably thrilled when I came across the book A Game of Functions by Robert Froman.  It's part of the Young Math Series from the 1970's and is out of print.  A quick Google search found copies available for purchase between $17.00 and $115.00!!  Luckily, my husband works at a university with a very comprehensive library and I got my hands on a copy.  I read it to my daughter one morning.  She wasn't having a great day, but she didn't protest, and we got through most of it.  I let the idea sit, waiting patiently for an opportunity to put the ideas into action as the book suggests. 

The book starts out with an introduction to the idea of 'function', as in 'whether we go the park this afternoon is a function of the weather -- if it rains this afternoon we will go shopping, if it is nice we will go to the park' (I'm paraphrasing here).  Or, as in this example below, how long it takes you to run around the outside of your house depends on on whether you crawl, walk or run. How quickly you go is a function of your mode of movement.

At that point, the book introduces the functions 'game'. 

You find a nice big area and draw a line across and a line up.  Lucky for us I had sidewalk chalk on me and we were at a park with a parking lot that looked almost like graph paper!

When I asked the kid what her 'rule' was, she said she wanted to take ten steps over and ten steps up.  We quickly realized that we needed a way to make sure her steps were the same length so we landed on her personal foot length, heel to toe.  She made a little white chalk X at ten steps.

And then, ten steps up from the X, and marked with chalk.

Although I helpfully informed her that she didn't need to go back to the beginning each time to 'add ten' to the last result, it was interesting watch her ignore me completely and then figure it out for herself.  And it didn't take her long -- by the time she was working on 30 steps, she realized she should just add ten to the second X (twenty steps) instead of count 30 from zero.  It wasn't an 'I told you so' kind of moment, just a little bit more proof that if the kid wants to figure something out on her own I should just let her do it. Lesson learned and internalized!  (For her and me.)

When she had used as much space as she could I asked her to stand in the corner and look at all the Xs she had marked up into the space.  "They go in a diagonal!" she observed.  And then she ran from (50,50) all the way to (0,0).  

We also worked on another rule for a little while (nine out and eight up) but she was running out of steam.  That was a lot of thinking for one morning.  It was perfect timing, too.  As we were packing up, four cars drove into the parking lot and covered her work!

At the very least, I feel like I've redeemed this concept for her (or, more likely, myself).  I haven't labeled what we did 'functions' but I did use the word 'rule' a lot, for example "The rule is to 'add ten' so your next move is ten more steps than the last time...let's see what happens when you do that a bunch of times!"

As you can see, above, the book goes on to show how you can do the same work on graph paper.  I'm thinking about how to make it a game...maybe two rolls of the dice determine the rule?  I could do my line and she could do hers and then we could compare?  Steepest line?  Line with the most graphed points?   Which one gets to the top of the paper with the least number of graphed points?  And, maybe include the question: "How steep your line is depends on (is a function of)....?"

Who knows?  This journey is full of adventure and surprises.  It's not always smooth sailing, but we're learning a lot, her and I.  And, one thing's for certain, there is more sidewalk math in our future.

Out & About: Tricky Triangles

It all started with a lovely morning walk to the hardware store.  We needed a long tape measure for our 'measure the house' project that's been brewing for a while.

"Look, Mama!  A triangle!  Except...I didn't know triangles could do that.  I knew that two triangles make a square..."

And therein lies the rub, doesn't it? For a kid who has played with tangrams (right triangles) and pattern blocks (equilateral triangles) for years, those are what triangles 'are' aren't they?  Already shaped and scaled to make a whole of some other shape (rhombi, squares, rectangles, hexagons).  But there this triangle was, obviously not dividing the space evenly, just a modest, unbalanced slice out of this square-ish rectangle portion of the sidewalk.

As we walked further I wondered about how to respond to her observation.  As luck would have it, I had a stray piece of sidewalk chalk in my bag!

"There are a lot of ways to divide a square or rectangle into triangles.  Let's see how many we can find!"

A little later on, "Look!  More triangles!"

And here, a triangle?  She thought it was at first, but what does a triangle need to have to be a triangle?

Our eyes were open at that point, and we found other triangles on our journey.  Here are two with a square -- an almost- trapezoid in the wild!  (I say almost because of the curved bottom edge -- yes, there are shapes all around us but some of them are truly geometric and the others are not really geometric, like the 'triangular' yield sign with it's curved corners, and still others are 'natural' shapes which have their own wonderful rules.)

As a bonus, at the hardware store we found a section that specializes in tools used in real-life geometry -- things that help builders measure lengths, widths, diameters and angles! 

In closing, and in honor of triangles, take a look at this triangle interactive from the Triangulation Blog.  All you need to do is move your cursor/mouse.  I swear, it'll be worth your time.   Who knew triangles could be so funny!?

You may also be interested in a previous post, Channeling Tana Hoban: Juxtaposition Edition, where we discovered many, many more shapes on a similar walk, especially an incredible number of circles in juxtaposition with other shapes.  

Open Ended

I just realized that a lot of my posts lately have been about making mathy things out of open ended materials: straws, pipe cleaners, craft sticks, rubberbands, paper, glue.  It's made me think about when I teach Math in Your Feet, how fourth and fifth graders almost can't believe it when they've made up their own eight-count pattern out of nothing more than three categories of movement-based attributes.  One child even said to me, during our end-of-week reflection time: "I didn't know I could make anything."   Heavy sigh.

This ad from 1981 reads:

"Have you ever seen anything like it?  Not just what she's made but how proud it's made her?  It's a look you'll see whenever children build something all by themselves.  No matter what they've created."

There's a lot this ad says to me, but the very most important thing is that:

It is imperative that we give our children, and our students, a chance to explore and discover at least part of what they learn ON THEIR OWN, ideally with open-ended materials and perhaps a very casually introduced question.  This doesn't mean that the learning environment becomes chaotic, or that teachers aren't needed.  Structure and forethought are still useful but, in this case, they become the background against which children employ their own desires to the materials at hand. 

We need to trust that kids can figure things out for themselves sometimes, and that the best learning often happens as a result of a question and the self-motivation to find an answer that suits them in the moment. 

Playing Math Every Day from Moebius Noodles

If you haven't heard of Moebius Noodles, I highly recommend you check it out!

From the Moebius Noodles blog:
We are creating an advanced and accessible math book for young kids and their parents, called “Moebius Noodles” and an online knowledge exchange hub to support it. It’s 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.

The most recent post has a fabulous menu for 'playing math every day' for the week of November 28 through December 4.  Activities include playing ball outside and then exploring a type of fractal called Apollonian gasket, fun subitizing activities, and starting a math journal. 

Try it out and let me know what your favorites are!

Marx Brothers Math: Transformation & Reflection

I'll wager that each one of us looks into a mirror at least once a day. Surprisingly, what we see in the mirror is up for some debate; at least that's been my experience when talking with fourth graders about the subject of reflection.  The result of these conversations is that I now firmly believe that when we use movement to explore the concepts of transformation and reflection, we gain a truly three-dimensional understanding of the subject.  Here's a little peek into how it all goes down:

Me: "What do you see when you look in the mirror?"
4th Grader: "Myself."
Me: "But is it really you? There's only one of you!  There's no one else like you in all the world.  You are an original!"
Different 4th grader: "It's your reflection!"

Later, after my 'magic wand of transformation' has turned the entire class into multiple reflections of me and they've had a chance to experience what it's like to exist on the other side of the mirror, I ask:

Me: "How many of you think it would it be fair to say that your reflection is doing the same thing as you?"
Half the class raises their hands.
Me: "Or, is your reflection doing the opposite of you?"
One third of the class raises their hands.
Me: "Or, how many of you think it might be both, the same and the opposite?"
One or two hands shoot up, other hands raise and lower tentatively.

We work through answering this question in class using our creative dance work.  In lieu of this experience here is a video clip for you from the Marx Brothers movie 'Duck Soup' (below).  I find this video to be simultaneously fun, highly entertaining, and instructive about the process of reflection.  Remember that transformation is essentially about change, and I assert that movement is a particularly effective way to make the process of change visible.

A few things to consider before watching the video, below:

Most of the time we are looking into a mirror straight-on. We brush our teeth, wash our faces, or comb our hair, all while looking at our faces and the fronts of our bodies. In this orientation is easy to think that the reflection is doing the same thing as us.

But remember, the mirror can reflect all sides of our bodies.  As you watch this video you will see Groucho and Harpo directly facing the "mirror" but also walking along the length of the mirror (shoulders to the mirror line) and turning toward and away from the mirror. There's even a fun bit where their bottoms are closer to the mirror than their heads!

In Math in Your Feet, children reflect their dance patterns by deciding who will dance the original pattern and who will reflect that pattern; the reflection changes the original pattern in small but very important ways.  Based on the narrative arc in this particular video, Groucho is the homeowner (original) and Harpo an interloper (reflection).  As you watch, ask yourself:

When is the reflection doing the same thing as the original?

When is the reflection doing the opposite of the original?

I'll give you a couple examples to get you started. When Groucho first sees his 'reflection' in the 'mirror' they both move in toward the mirror and then away from the mirror. In this case they are doing the same thing. Then, still facing each other, Groucho's right hand goes to his chin, but it is his reflection's left hand that goes up. Both hands go up to the chins, but they are using opposite hands.

One more example: At 0:35 Groucho turns away from the mirror over his right shoulder, for a total distance of 180°. Harpo also turns 180°, but over his left shoulder.

How many examples of same and opposite can you find? Can you find any mistakes? I had a hard time tracking if they were using opposite rights and lefts in their footwork, for example. Have fun and don't forget to try out some of the activities listed below when you're done watching!



How'd you do? Ready for a little application of the concepts?

Try this at home:
Put a line of tape on the floor. This is your mirror, otherwise known as a line of reflection.
Decide who will be the original and who will be the reflection.
To start, the reflection has to be the same distance from the mirror line as the original.
Move slowly at first so the reflection has a better chance of accuracy.
Most important: don't forget to experiment with having different sides of your body be 'reflected' in the mirror.

Extra challenge:
Make up a short piece of choreography with a variety of moves and levels (high, medium and low).  In Math in Your Feet, the foot based patterns are units of four steady beats.  See if you can make a four- or eight-beat combination of moves using your whole body. 
Both people practice doing this choreography congruently (everything the same).
Then, do the choreography with the line between you. The original needs to move slowly while the reflection figures out what parts of the choreography needs to change (hint: everything is the same except the reflection uses opposite rights and lefts).
When you're well-practiced and have it a tempo that both people can do comfortably, show off your work!

Extra, extra challenge:
Perform your choreography with your partner first congruently (everything the same) and then reflected (opposite rights and lefts).
If you want a triple challenge, change roles and have the other person become the reflection.

Kitty Census: Vet Edition

We have a LOT of kitties in our house.  And they're all real, too.  Most are stuffed, some are ceramic or plastic.  Others include a rare bookmark breed and one in a frame.  We also have an amazing example of the exotic puzzle breed, one who is an imaginary friend, an actual breathing, chipmunk-catching cat and, finally, one who is a special kind of human cat with white fur and a brown spot on her head.

When faced with multitudes like this, a mama's mind turns to math: Just how many kitties do we have and can we remember all of their names?  How many different types of kitties are living here?  How old are they?  Who is related to whom?

This morning the girl wanted to play vet.  I wanted to revisit the Kitty Census we started back in the summer.  As with many things, we compromised. 

ALL the cats went to the vet. 

I was the intake coordinator.  Each cat was registered by name, age, type/description, and malady.  Some of the issues included: "a little sad", "lying around, droopy whiskers", "smelly", "bedraggled", "music slows down at end" (that was the music box cat), and "hurt paw".  In the end, twenty eight showed up for well visits and eleven had some serious issues.  After a triage that took about an hour, we attended to the sickest first. 


This is the bunch that were there for their well-checkups.



We will revisit this data later.  For today we were concerned with 'sick' and 'well'.

Aventurine, of the rare 'puzzle breed', was assessed at intake as "needs fresh air and run around time without breaking."

First, the sick kitties, pictured above, all had a thorough examination, including temperature taking and whisker inspection.  Of the group, the Webkins family all had strep, two were afflicted by a 'fever' of undetermined origin, and two needed immediate whisker surgery.  Two others, originally thought to be sick ('lying around'), were determined to simply be acting like cats.

In the end, I think we simply set the stage for more questions about our brood.  I had her chart out the number of sick kitties compared to the well.  She was curious about how many kitties there were all together, and initiated a count starting at 28 (well) and counting up the sick column to find the sum total.

We'll revisit the intake records and mine it for more data in the near future.  How many family groups do we have?  How many different breeds and how many in each category?  What else do we need to find out about our household of kitties?

Until then, one thing's for certain: We've got a whole lotta' kitty loving going on 'round here.

"Mama!  Lucy's purring!"

Conversational Math: Part Two

In trying to capitalize on the kid's penchant for 'talking math' I recently decided to try a game with her that I found in the booklet that came with our set of Cuisenaire Rods. 

The game is called Build What I Have.  One person describes a design they are making with their rods and others try and reproduce that design by listening closely.  One of the main points in this game is to introduce and/or reinforce math vocabulary.

The suggested age range for this activity is 2nd-8th grade; even though the kid is a young six I knew we could still get something out of it.  I decided that, to start, I would capitalize on concepts she already knew (parallel, points, edges, top, bottom, sides, etc.) and introduce some new ideas (perpendicular, horizontal, vertical). 

The rest we'd muddle through somehow, I figured, but she did surprise me by knowing her lefts and rights.  "We've been doing that in ballet class, Mama," she stated mater-of-factly.  Fabulous.

To start, we hid our designs from each other.


This is the first design.  I led and she followed, trying to make her design match mine by following my instructions. I started by saying: "Lay your blue rod parallel to the bottom of our work surface."  She already knows the concept of parallel really well, often times finding and noting examples of parallel lines when we are out and about.  "Then," I continued, "take your green rod and place in perpendicular [holding the rod in the air] up and down like this, and place the end in the middle of the blue rod."  Success!  Our designs matched!

This is the game she led.  To start she told me to put my orange rod parallel to the bottom of the workspace, but about an inch up.  The second orange rod was to be 'a couple' inches above the first one, but when she told me to put the blue rods on the sides to 'make a rectangle' I clarified the distance.  "Looks more like three or four inches, to me," I said.  I asked her to clarify the placement of the blue rods -- do they go on the outside ends of the orange rods, or inside?  Notice that this design is mostly made up of parallel lines, a concept she is most familiar with.

This is the second design I led.  I said, "Take your three light green rods and put them so they are together and vertical, up and down, in your workspace...Oh look!  They make a nice little cube!"  At first she thought she needed a fourth one to make it a square, but I clarified and said we're not making the outline of a square, but a solid shape.  When we revealed our designs to each other we saw some differences! 

This is how she recreated my instructions.  The white cubes are essentially in the right areas, but I had actually challenged her to put each white block 'point to point' with each corner of the light green square.  The dark green rods are essentially in the correct place; I knew that was somewhat complicated to execute.  And, I just noticed, the light green rods are horizontal, not vertical.

This is the last design in our session, which she led.  Perfect!  She wanted to use a bunch of different rods, but everything is still parallel here.
 
Here is what I find fascinating:  

My daughter's designs were much simpler today than normal and I think it might be because she had to describe what she was doing as she built them.  There is an equivalent experience that I find to be true in my work with 4th and 5th graders as well.  Often times I tell those kids that they are doing complex mathematics in their bodies and grade-level math on the page; they understand more math in their bodies than they can communicate through words or symbols.  Sometimes it is impossible for them to notate their Jump Patterns because they are just too complex for their current stage of symbolic mastery.

Often kids can do, know, and understand way more than they can communicate symbolically.  If we only judge a kid by her output on paper, we're not really seeing the whole child.  There are many ways represent comprehension: we need to listen and watch carefully for other indications of understanding as well. 

It wasn't too long ago when I brought the word 'parallel' into my daughter's universe.  It will be exciting to observe her body and conversations show me she's 'got' the concepts of perpendicular, horizontal and vertical. 

To Each its Own: Targeting My Professional Development Workshops

In the last couple months I've had whole bunches of fun presenting professional development workshops in a variety of settings, to a variety of people.  Let's see...math teachers from all over the U.S., PE teachers from across Indiana, classroom teachers from Indianapolis, fellow Teaching Artists from a variety of disciplines, and arts education administrators from Young Audiences affiliates from around the country. 

Each session bore the title 'Math in Your Feet' and was a combination of big picture information and hands-on experience, but that is where the similarity ended and my job got really interesting!

For the classroom teachers I started by focusing on the challenges of using movement in a classroom setting.  As they started to move and experiment with foot-based percussive patterns they became more comfortable and sure in their own movement.  This approach usually leads to a greater willingness to embrace, sometimes for the first time, the possibility of leading their own students in movement-based learning.  To some extent I am also encouraging them to have fun with math, many for the first time.  I consider the 'doing and making' of percussive dance patterns in this program the same as the 'doing and making' of math so, in every teacher workshop I do, I walk them step by step through the intersection where math and dance meet.  We're so used to focusing on the symbolic, static realm of mathematics that we don't always recognize when we see math happening in front of our eyes.  It helps to have a guide.

For the self-identified math teachers at the NCTM annual meeting I also started with a message of 'anyone can lead movement in the classroom and here are some tools' but then quickly moved toward 'here is an opportunity for your students to represent their math understanding in a new way within the kinesthetic realm'.  I also drew their attention to the fact that the processes of solving a problem in both math and dance (choreography) are often similar -- question, understand what tools it might take to answer the question, experiment with ideas, use your resources, find an answer that seems to work, evaluate and then ask more questions. 

The group of 80 or so PE teachers was a new one for me simply because there was not one bit of trepidation or reluctance to get up and move!  Not all of them were comfortable with the idea of dance, at least initially, but they were definitely game.  I was only with them for about an hour, and I couldn't go very deep, so I stuck with active modeling of the bridge between my particular brand of movement with an academic content area.  If I had had more time with them, I would have focused on the  process for moving the dance to the page -- speaking the words that describe aspects of our movement as we move, writing those words down, turning these words into symbols, and graphing foot positions on a coordinate grid.  I did the point that Math in Your Feet can be a collaboration between classroom and specials teachers, just like it is when I lead my residency.  The concrete movement and math activities can be done in PE or music class which then build the bridge to the formal, written, symbolic realm of math back in the regular classroom.

At their conference the arts education administrators were focusing on how to add the A in arts to STEM topics (STEM to STEAM).  I gave a general overview of the program and laid out my process for building the program and integrating the dance with the math.  The most important issue for me is that when you are thinking about integrating any art form with another content area you really need to be honest with yourself and ask 'is it a good fit?'  If the answer is no then it is not worth forcing the issue.  If you think 'maybe' then do a little more work to explore the connections.  In the end, though, the connections need to be more than skin deep.  Just because we count our beats in this program doesn't mean I consider that a good example of what math and dance have in common.  I also gave a similar account of how I combined math and dance to my fellow Teaching Artists.

My favorite moments while teaching teachers are when they ask me questions that show me they are imagining how they will do this work with their own students.  It's similar to house hunting, I suppose.  The minute you start imagining where you're going to put your furniture the realtor knows you might really be serious!  I love hearing all the different ways engaged and caring education professionals imagine tailoring my ideas for their own particular learning environments. 

Perfect Reasoning: The Right Brain Initiative (Manifesto)

Yep

From Spark: The Revolutionary New Science of Exercise and the Brain, by John J. Ratey, MD, a wonderful gift last week from Templeton ES PE teacher Monica Chapin:

In the Introduction:

"In today's technology-driven, plasma-screened-in world, it's easy to forget that we are born movers - animals, in fact - because we've engineered movement right out of our lives.  Ironically, the human capacity to dream and plan and create the very society that shields us from our biological imperative to move is rooted in the areas of the brain that govern movement.  As we adapted to an ever-changing environment over the past half million years, our thinking brain evolved from the need to hone motor skills.  We envision our hunter-gatherer ancestors as brutes who relied primarily on physical prowess, but to survive over the long haul they had to use their smarts to find and store food.  The relationship between food, physical activity, and learning is hardwired into the brain's circuitry." Page 3

In the chapter Learning: Grow Your Brain Cells:

"The body was designed to be pushed, and in pushing our bodies we push our brains too.  Learning and memory evolved in concert with the motor functions that allowed our ancestors to track down food [the reason we learned how to learn in the first place], so as far as our brains are concerned, if we're not moving, there's no need to learn anything." Page 53

So...if you are not moving you are not learning?

I think that's it in a nutshell.

Representing Math Concepts Through Percussive Patterns

Not quite congruent.  One partner has landed while the

other is still up in the air.

This week I'm working at Christel House Academy, a charter school up in Indianapolis.  This is part of a grant-funded pilot project for Young Audiences' Signature Core Programs.  The fifth graders are fantastic!  They are perfectly perfect in all their 11-year-old-ness, and quite observant and thoughtful to boot.  They make connections easily and ask interesting questions that show they are really thinking about how this all works.

This is an interesting situation for me.  I am usually invited to schools where kids are at least a grade level or more behind in math and my role is to assist in catching them up.  At this school, the fifth graders know and understand quite a bit so we are in the position of applying what they know to a new situation instead of learning it for the first time.  But the really fascinating thing for me is that, although they 'know their math' they are still challenged by representing it physically.

In my reading about mathematics education, I've come across an idea called 'the power of three'.  Essentially, the idea is that to really understand a math concept a child needs to represent it in at least three different ways.   This would be through pictures or some other means.  I'm just beginning to realize that one of the strengths of Math in Your Feet is that it provides an opportunity to experience and represent math concepts in the kinesthetic realm.  Part of this challenge lies in the fact that these patterns are not static, but require students to literally be 'in' the pattern.  Just today I had an interesting conversation with some boys about whether to record a turn as being on the third beat or on the fourth.  We eventually came to the agreement that the turn was actually happening between the third and fourth beat, but that since third beat ended in one position and the fourth beat was in the new position, we had to record it as being on the fourth beat.  My system may not be perfect, but it does create a structure to ask these kinds of questions. 

So, here's how it works.  Kids make up a four-beat dance pattern using the elements of percussive dance that I've outlined for them.  They learn to make their dancing congruent by producing (with pre-teen bodies!) the same tempo, foot placement, movement, and direction as their partner.  After that, we start transforming these patterns using different symmetries, starting with reflection.  At that point, all the pathways forged between the body and the brain have to be shuffled around as one partner dances the original pattern and the other (on the opposite side of the line of reflection) has to change the pattern by dancing the opposite lefts and rights.  For example, a turn to the right would be reversed to go left, or a right foot would be switched to a left foot.  This all sounds rather straightforward as I'm writing about it, but after observing the CHA fifth graders this morning, I realize that no matter how well they understand it in their heads, and no matter how 'smart' their bodies might be, it's still a challenge!  There's quite a bit of thinking going on here, in both body and brain, and it takes a lot of practice to remember a sequence of the four moves that make up their pattern.

This is only the third day and we have a couple more to go.  Things do get more interesting and more challenging when we start combining individual patterns into larger ones (i.e. start the second pattern where you ended the first, not at your original starting point and then try the reverse) and also when we transform the patterns using turn symmetry which seems rather straightforward in a static representation on paper, but is absolutely spectacular when you see it in motion. 

I'll keep you posted!

Making Things

Curried red lentil soup with sweet potatoes and greens.

I know a lot of people who make things -- dinner, for example.  I know people who grow their own food, make their own music, build their own rock walls, design and build their own houses, decorate their homes with their own artwork, sew their own quilts, and bake their own bread, all as a matter of course.  I know people who knit their own socks!

Then again, when I go into schools, I meet legions of children who never even considered the possibility that they could make something, let alone their own percussive dance patterns.  Here the thing, though -- most kids, even if they don't regularly engage in making things, get excited when it's their turn to make their own dance patterns in class.  It's amazing and energizing to watch, honestly.  It's like I'm giving them permission to take their rightful place in the human race.

I have been reading back issues of the Teaching Artist Journal and finding common themes emerging in what I read.  One of the things that comes forward is that many Teaching Artists (like myself) teach with less of a focus on the technical aspects of their art and put more emphasis on helping their students make things within the structure of an artistic practice and process.  When a Teaching Artist takes their art into the classroom (or community center, or after-school program, or a nursing home, or wherever people are)  they work with people who may have never taken a dance class or made music or used paint in a meaningful way in their entire lives.  They usually have limited amounts of time to work with a group of students and so they become masters at helping people make personally relevant art with a limited vocabulary of skills.  

This is what I do in Math in Your Feet.  In an upcoming article to be published in April 2011 in the Teaching Artist Journal, I go into detail about a teaching tool I created which I call 'Jump Patterns.'  Essentially, Jump Patterns are to percussive dance what creative movement is to ballet or modern dance.  Jump Patterns reflect the elements of percussive dance and allow children to be creative and successful in this particular dance style, but without great need for technical development. 



Don't get me wrong, skill development is important, but sometimes a focus on technical issues creates a huge obstacle to experiencing the incredible joy and importance of just making...things.  Art.  Ideas.  Music.  Shoes out of paper plates.  Even having this particular discussion is part of the cultural baggage we carry, a particular mindset that gets activated and reactivated whenever we think of art solely as something to be 'good' at  instead of an opportunity to make things.  And that's a pity, because I believe that making things is one of the most meaningful activities in which a person can engage. 

An 'old fashioned dress' the creation
of which was facilitated by safety pins and
some adult modeling.

Right about now you might be asking yourself, 'Just how can someone make something without skills?'  The answer is facilitated making.  Kids are like everyone else, you do have to give them something to work with before they can begin to create.  But, in the end, all you really have to do is help kids explore the medium, find a way for them to  investigate the processes of the art form, help them formulate some questions or just wait until one is asked, and, when it's time, add instruction on specific skills when the student is ready for them.

Here are some examples of what I mean by facilitated making:

Example I:  In Math in Your Feet, Jump Patterns are the tool that facilitate the making of foot-based dance patterns.  Once a kid learns three of the four elements of percussive dance (leaving out the most technical aspect, parts of the feet) they are then ready to experiment and create their work.  They become participants in the traditional aesthetic -- everyone has something to offer, everyone has their own style or take on the dance, everyone is welcome.  Not everyone is good but, more importantly, everyone participates.  Without participation there is no form.

Example II: A Teaching Artist friend of mine works with kids who, more often that not, have no formal music training.  When he sets up digital recording studios in their schools they are able to get immediate feedback on their own ideas and find ways to evaluate and adjust their ideas to make real, and often interesting, music and lyrics in a collaborative way with their friends.  He spends parts of each class talking about the science of sound or new techniques for using the equipment but mostly the kids experiment on their own and come to him if they have questions.  In this case, the facilitation is the TA's approach to using the equipment with his students. 

Mermaid's Tail

Example III: My daughter, who is five, has always had many ideas and many of them get expressed through her art supplies.  When she was two, she wanted paper cut into shapes so she could make something with them.  She had an idea but still did not yet have the skill to cut the paper the way she wanted.  Did I say, "Sorry darlin'.  You'll have to wait until your hand muscles are stronger and more coordinated before you can get that idea out of your head"?  Nope.  I cut the shapes for her and she did the rest.  Did it change the fact that it was her artwork?  Nope.  A few years later, when she was a few months shy of her fourth birthday, she had an idea for a mermaid's tail.  By then, facilitating her making process simply consisted of making sure the tape was secure so all those little pieces stayed together.

Making things is not rocket science...well, unless you're trying to build a rocket.  But what comes before that rocket?   All the times when kids get a chance to ask questions, experiment with materials, find their own answers and get some help and guidance from adults along the way.  We can sometimes get wrapped up in giving kids what we think they need to learn, but it's in the doing where the meaning is made.