Play Power

Although I know it already, it never ceases to surprise me just how true it is.
 
Learners of all ages need the opportunity to experiment with a new medium before putting it to its more formal or expected use.  Often this kind of activity is called 'playing around' which I often perceive as a derogatory term in relation to learning.  But, in my experience, if you observe children at play long enough and really pay attention you will be astounded by the myriad of ways they are representing their knowledge, understanding and mastery of a subject.  Play and exploration are not wasted time.  In fact, I think it is exactly the kind of activity that builds the foundation of real understanding.

Here is a case in point from a recent Math in Your Feet Family Night.  Having finally found success using straws and pipe cleaners as a math toy and building material with my own first grader, I decided to include it in the Family Night for the first time.  I made some models of polygons and polyhedra, gave the station volunteer a quick orientation, and left the materials to be discovered.

Immediately, it was the most popular station of the eight offered that night.  As the children descended, the adults followed, providing lots of helpful advice and some modeling...

...which the kids politely and assiduously ignored as they confidently forged ahead.

This initial inclination to explore the materials on their own terms was fortified by the fact that this was not officially 'school time'.  There was no pressure to do things 'right', or follow the rules, or learn and use proper technique. 

As a result, most kids cheerfully ignored the formula for folding a pipe cleaner in half and making a nice right angle before inserting it as a connector between two straws, and instead found their own twisty or unequal ways to make it work. 

Most also ignored the nice models I had made and created their own.  I had never heard of a hexagon cube, for example, but there it was! 

They were having a grand time 'playing around' when I noticed something amazing happening.  After a very focused exploration period, they started discovering the rules on their own!

This little one, two years old according to her brother who sat beside her, had been methodically putting pipe cleaners into the straws, one after another.  It looked like a little gallery of Q-tips, someone joked.

She was working on her own.  She must have been at it for thirty minutes and then...she started connecting straws together!

Voila!  A hexagon.  No one, I suspect, expected much out of a girl so young.  And yet, there she was discovering the materials and watching others around her, ultimately creating something for herself.  I'd wager that if someone had insisted on sitting her down and showing her how to make a hexagon, she might have been less interested, engaged, focused and, ultimately, successful. 

Children much older also experienced this same progression.  Check out what her brother was building, below. 

Kids kept coming up to me wanting to know if they could have the dodecahedron I made as model for the night.  Sort of like a door prize?  I said, "Well, no, that one's mine.  But you could make your own!"

Only one girl decided to make one for herself; she also really wanted me to sit next to her while she figured it out.  I provided moral support for about five minutes, and then had to 'go do something...'  A few minutes later, she came and found me with a question and, still later, enlisted support from another adult so she could finally finish it.  But you know what?  She did all the work, she just needed help 'seeing' the structure and pattern.  If we had had more time she and I could have talked  how to make all the angles congruent so it would be more regular but, still...what a prize!

So, what kind of learning was happening during all this 'playing'?

I heard a teacher mention that this activity reinforced the learning they were doing in class about corners and sides.  Yes, and so much more.  

The side of the shape becomes a shared edge.   You only need one straw for each edge.  The more you build on to your initial shape, the more this aspect of intersection and sharing is apparent.

A vertex can be created from the intersection of two, three, sometimes even five different lines/edges.

Depending on what polygon or polyhedron you're making, the pipe cleaners need to be bent at different angles.  An equilateral triangle's angles are different from a square's which are different yet again when you create a hexagon, or a pentagon.  These are properties you might not truly understand unless you had to make them yourself.   And, when every angle in a shape has to be the same, and you're the one who has to make them that way, you truly build a new understanding of 'sameness'.

And that's just the math stuff and just what I noticed while watching them build.  I'm sure there's more.

All in all, a good evening's work.  I think my new definition of success is when my project idea is just the starting point and, over the course of the 'lesson' not only do multiple right answers emerge but the children are satisfied with their efforts. If the resulting mess is any indication, I'd say it was an entirely satisfying evening. 

Our New Math Studio

We've had a bunch of visitors in the last few days.  We show them around our house.  They see the ten million stuffed kitties.  The see the kid's collection of old books, including the 120 year old speller and the 80 year old (very cool) atlas.  They see the fantastic, never-ending headstand (hint: not mine).  And then they walk into our sun room.  "What is that?" they ask.

Why, it's our new math studio, of course!  Here is the inside view for those lucky few who can fit in the door:

This is what you'd look like from the outside if you were sitting inside it.

Here is the new math studio in use:

And, here is an example of something that might be made inside.  In this case, start with a cube and, um...square it?  Cube it?  Make a cube out of cubes?  (We're both still learning about this math stuff.) 

"I want the inside to be a different color, Mama, so you can see where the middle is."

We have what we call our 'making studio' in another part of our house, and we make a lot of math these days, so it makes sense the kid would think of our new installation as a math studio.

How did we make this wonderful structure, you ask?  Lots of newspaper, tape and pipe cleaners. You can read all the details here.

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.  :-]

Small Moments of Geometry

Sometimes, all you have to do is ask the question.

I was looking for more geometry to do with the kid, so I headed over to the Living Math Forum for some inspiration.  Needless to say, I got great suggestions which jump started my thinking process in a big way.  Not all of the suggestions worked for us at this particular moment; my daughter is not all that enamoured with computer games, for instance, even though there are a ton of great online math game resources.  I also got some recommendations for building sets, but historically she hasn't been drawn to that kind of thing.  Also, most of them are pretty pricey and I'm just not sure I'd want to risk the money at this point.

One of the best suggestions for my particular learner was the book String, Straight Edge, & Shadow: The Story of Geometry by Julia E. Diggins.  We found a copy at the library and are taking it a couple of pages at a time.  It's a wonderful story but a bit dense, in the best possible way.  There's a lot to ponder and it's worth taking a few days to process things between readings.  Every time we read it we immediately find connections in our daily life.

My favorite thing about the book so far is the writing which weaves in and out of descriptive suppositions about how early humans discovered the circle or the right angle interspersed with conversational observations about how a child might also observe those same ideas in his or her daily activity and environment.  After reading about right angles today in a chapter about early earth measurers, a simple folded piece of paper has now become our 'right angle checker.'  We use it to observe how this helpful angle maintains order within our physical world allowing us to move easily through life without worry of wonky walls or telephone poles. I never knew a right angle was so interesting, useful or ubiquitous!

Anyhow, it seems geometry is everywhere, just waiting for us to find it.  Such is the nature of small math moments.

For instance, the kid has really taken to what I'm calling her 'geometer training'.  She's a whiz with the compass.  We had an initial, and very successful, exploration of circles one Sunday and, a few days later our second round found me exploring concentric, intersecting circles.  The kid tried making her own. 

In our first round of exploring circles, the kid saw a triangle when she placed three circles in this position.  This time, as we sat working together, I took a minute to illustrate for her how I saw those triangles (below) -- by connecting the center points, and also by extending straight lines on the outside.

After pondering what else besides a compass might energize our study of geometry, I decided it was time to go out and purchase the game Blokus.  The kid was thrilled when she found that over half of the pieces are actually mini, multi-colored pentominoes.  (I also found a great pentomino puzzle generator for up to six pieces here.)  The rules are not hard (all pieces of a certain color need to be touching corners to play) but the kid keeps noticing the open spaces and wanting to fill them in with the shapes instead of playing the 'touch corners' rule. 

The picture below is when we finally decided to throw out the rules and see how many different sized squares and rectangles we could make out of the entire inventory of game pieces.  For some reason, this was a lot more fun for her.  I modeled counting the squares around the perimeter to determine whether it's a square or a rectangle and figuring area from multiplying length and width.  I think that an adult modeling his or her thinking and problem solving process in a real-life context is one of the most potent teaching tools at one's disposal.

Speaking of modeling, the best way to get my kid interested in something is to be doing something interesting myself (like coloring in triangular grids) and having some 'extra' on hand for her to do as well.  Sneaky Math Mama! 

(My grid designs are on the left, hers on the right.  I had my mind focused on rotation symmetry, but just let the kid explore as she would.  Her second design, on the right side of her page shows some reflection in her design -- a bit hard to see in this picture.)

I always love finding something lovely after quiet time...

Sunday morning walks have become our routine and look at this beauty we found last weekend!  This morning she suffered through half of a fun (to me!) book on polygons.  The first half was about triangles, so next time we go out we'll be able to use our new words -- scalene, isosceles, equilateral, right angle, obtuse angle and acute angle.

Or, how about this?  I love the stair step rectangular blocks.  The girl noticed the circle first, but there is also a little square in the middle of it, and look!  Small triangular blocks creating a square around the circle.  Awesome!

And neither of us could resist the big red circular table with cool circular cutout decorations in the middle.  It may be hard to see but the inner circle is made of little squares rotated 45 degrees and the outside ring I can't quite describe but is so lovely.

I'm in the middle of prepping for a Math in Your Feet Family Night for next week.  The kid really wants to go, but it's far away and she has a ridiculously early bedtime.  But, I said, I could bring up a few activities from the basement.  What would she like to do?

Turns out, the rotational paper pizza designs were quite engrossing.

We talked about how each pizza slice had to have the same toppings in the same positions on each piece, a point to which she paid special attention.  And, after that, it was time to put them into position so could see the overall rotation design (which might also have some reflection in it too!)

And, it's been months and months since I was obsessed with making the Platonic solids out of straws and pipe cleaners, a process she basically only observed.  But, when she found out they were part of the family night she suddenly got interested in making one all by herself!

A really tall tower of cubes, that is.  My previous experience showed me that six inch straws do not hold up well under scrutiny from children, but three inch straws actually create a comparatively stable structure. 

Maybe I should go for a building set after all.  Seems like she might be ready for it!

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.