Amelia, the Enduring Math Doll

First, there was Amelia the hand-made doll and her essay "What Infinity Means to Me." That was early in 2013.

Since then, Miss Amelia has been on all sorts of notable journeys.  Most recently she's been the subject of some interesting math questions.

For instance, Amelia needed a new dress. Unable to find the tape measure, my disorganized eight year old devised an interesting new way to measure out a piece of cloth.  She used this work board from a physics activity (investigating levers, pulleys, etc.). 

We have never discussed coordinate grids. She has, however, seen a lot of grids, mostly in the form of multiplication arrays.  She measured Amelia vertically...

...and then horizontally...

...and then cut out two identical square pieces of cloth (front and back of new dress) with which to sew the dolly a new garment.

Amelia was also recently the subject of a conversation about doll years.  I'm pretty sure I didn't have enough math to truly help her, but here's what came of it.

My kid wanted to make sure Amelia was SIX in doll years.  Apparently (after a very heated, frustrating conversation mostly, I assume, because she wasn't really clear about what she was trying to ask) one doll year is equal to two human months.  The only way I could help her figure it out was by writing out her ideas for her, one step at a time. One problem was that my kid was thinking in too many time units at once -- days, months, and years.  

We did finally come to a conclusion that was agreeable to her.  I almost typed 'concussion' because that's how I felt after it was all over.

I wonder what kind of math adventures Amelia will have next?

Fun While it Lasted

The first group (sixth graders) arrived in my classroom after what appeared to be an intense recess. They were, all of them, either drenched in sweat or nursing some kind of injury.

I surveyed the carnage. Turns out half their class was absent this day as well.  I decided that dancing was not in the cards.

"Okay guys," I said, "come sit down near me. Let's talk about our options for today."  I outlined my plan. They had played around with the straws and pipe cleaners last week and loved it.  I had given them a chance to figure out for themselves how the materials worked and they loved the experimentation and play, even going so far as to exclaim "This is better than Xbox!"

Today, I wanted to challenge them.  I showed them a sheet with pretty good but not overly helpful illustrations of the Platonic solids.  I told them they could work individually or in teams - the goal was to see if, as a class, they could make at least one of each solid.  Most chose the octahedron, surprisingly. But by the end of class when the three-person team was finally, after a lot of muddling and helpful argument, finishing up their icosohedron, a bunch of other kids decided they wanted to make one too.

Since class was almost over at that point, it was a race down to the final possible minute.  Because once you make an icosohedron, you also have to spin it!

I love how certain aspects of this solid are made more obvious through...movement!

Later that day the fourth and fifth graders were more up for dancing but they also got a chance to work a second time with the straws and pipe cleaners.  Their particular challenge this time was to build something using an odd number of edges in their starting shape.  This essentially meant three (triangle) and five (pentagon) as the base shapes.  They grumbled. Some made squares anyway.  I reminded them of the challenge.  They grumbled some more but then...

Two More Small Moments of Success

The longer I facilitate interdisciplinary, arts-based learning experiences for children, the more sure I am that success in my temporal, moving classroom is best documented with the observation of small moments that might go unnoticed by standard assessment tools or a final performance. Here are a couple small moment of success from last week.

First Small Moment of Success
Dancers find their center and work from there. In Math in Your Feet, students start their steps with feet together in center. It's a place of control and potential.  I am often heard saying, "Find your center..."

In our dance spaces center is also likened to the intersection of x- and y- axes.  So -- zero, or origin. Where you start, where you move from. I don't spend a lot of time on it because it fades into just part of what we do.  It's something we know and use.

There are a lot of other concepts and vocabulary I bring with me into the classroom and I'm always thrilled when kids start to use new words in the context of their own work.  Thursday, while observing a boy's dance pattern in progress a question came up -- I thought he had five beats instead of four. One too many. He split his feet, one foot on the right side of his square, the other on the left and said, "But this is my zero." It all made sense then - zero is the place you start, 1 is the place you move to.  And he knew this.

I was so proud!

Second Small Moment of Success
In my summer programming and in my family math night I bring out the math craft making supplies.  A couple days ago is was straws and pipe cleaners, a few instructions and away you go.  A certain group of sixth grade boys were really into it.  I overheard one say, at the end of class,

"This is better than Xbox!"  Total win.

Baby's First Scratch Project

Well, okay, she's almost eight, but it's still a huge milestone.

Today I introduced my daughter to Scratch, a visual programming environment from MIT.  If you've been following my blog for a while, you may be a little surprised at this.  I am generally an advocate for hands-on math making and somewhat of a skeptic when it comes to kids learning from computers, especially those in the pre-school and primary grades.  We need to involve the whole body in learning -- children are already being required to focus primarily on the 2D visual field on top of having limited opportunities for movement throughout the day.  They do not need  more time sitting and looking.

That being said, it's been obvious that my girl was ready for some new challenges.  And, reading has finally become easy.  Words are chunked, weird English language pronunciation and spelling rules are generalized in new contexts, and willingness abounds when there is reason to read. It's clear she was ready for something completely new and different.

Also, I've become a huge fan of Seymour Papert and his work introducing children to computer programming and making math, although I have been skeptical about the computer part until just this week.  His book The Children's Machine is truly the best book of educational/learning theory I have ever read, specifically because Papert himself is such an honest learner and an incredibly astute observer of others' learning.  I could keep gushing, but that will be for another time. In general, though, I am completely smitten with his view of learning as a series of personally relevant connections and his specific descriptions of a variety of children learning Logo, including some who seemed just like mine.

This morning I got some time by myself to poke around the online version of Scratch 2.0.  I knew I was hopeless, but somehow managed to grasp the outlines of how it works.  I wasn't sure if this would be something she would be interested in, but I somehow managed to get her to take a look.  Because it was my idea and I really thought, after reading Papert's accounts of Logo in the classroom, that it needed to be all her from the start, I put her in the driver's seat, literally.  I sat her down in the chair, gave her the mouse, and talked her through the tools. We tried some stuff out, looked at some example projects and, after about 30 minutes, took a break.

At this point I was pretty positive about it all.  She seemed to like a lot about Scratch, specifically the movement/animation, the potential to make her own music, the cat sprite, a chance to play others' games, and the design/paint program.

In the afternoon she wanted to try again.  Since it was my work time and her quiet time the stage was set for her to "have" to work independently.  In the past, learning something new like this would require more attention than I am willing and able to give during this time.  So, imagine my surprise when all I heard for 30 minutes was silence, peppered by a few puzzled-sounding exclamations.  And then I heard:

"Mama, come up and see what I did!" 

She had programmed her 'sprite' (a cat) to walk forward 10 and then meow.  "Believe it or not," she said happily, "THAT took a lot of work."

She continued to experiment for a total of about an hour and then it was time to meet some friends at the park.

When we came home and the well-past-dinner-time blood sugar crash was appeased, she got back on for another 45 minutes.  She was in some kind of zone and it was clear she had outpaced me in one day.  I did sit beside her this time, while she worked, but it was only to ask her how or why she was doing something.  I watched as she moved through the various script categories and made the idea in her head come to life.

Here's what she said in the description of her finished project, which can be found here: Cat Dance

"This was my first day with Scratch. I made it by experimenting at least five times and then just getting it.  It took a couple hours and some hard work"

The best thing is that this is real making to her and she is truly proud of herself!  Computer games have always bored her, but this is real making, real inquiry, and she needed no convincing of that.  She closed the computer at 6:00pm, satisfied with her work but full of ideas for tomorrow. 

A Flood of Self-Initiated Math

So, the seven-year-old has been having a spate of self-initiated math lately.  First there was her 'map of angles' and then having her dolly write an essay on 'What Infinity Means to Me'.  So, I guess I wasn't too surprised to hear her from the other room giving her dolly another math lesson:

In her most patient, teacherly voice:

"I'm drawing a simple house, Amelia.  Everything in the house is mostly 90 degrees...you don't have to be exactly accurate but it just has to be good...a triangle window and here's the front porch....compare these two houses.  Can you fix this one?  Good job!"

[Her explanation to me when I asked her later what she was doing: "I drew a house without angles and with angles and Amelia had to fix the house without angles up!"]

After Amelia's success, she continued the lesson with this explanation:

"There are even angles in nature -- straight up and down trees, but some are even 80 degrees, slanting.  The old ones are 50 or 40 degrees."

Later, I got a look at her drawings:

In the larger house I see her thinking through the angles all starting from the bottom left vertex/corner of the house, which is forward movement from her original representations in the Map of Angles post.  Below the big house is the 'house with angles' at the bottom and what I think is the 'house without angles' (all wonky looking) above that (I thought I saw a different drawing with the same ideas but that, apparently, has gotten lost in the shuffle.)

Another quiet moment found her exploring the structure of an isosceles triangle. 

"See, there are eight of these triangles on each edge [above] and fifteen squares on the bottom edge," she told me.  She also called the line she drew from the top vertex to the center of the bottom edge a "diameter" which she knows is how you divide a circle in half.

In addition to all our sidewalk math adventures over the last year, we've learned more about identifying and classifying geometric shapes in the Beast Academy 3A series but it's been a while since we did the polygons chapter.  We got through skip counting which was perfect and, after entering the perimeter chapter decided to take a break.  This drawing really shows me she's thinking very specifically about the length of each edge.

And, finally, although this may seem more in the 'art' category, I know for sure that drawing three-dimensionally has all kinds of math involved in it, I just don't know what kind, lol!  Six or nine months ago she tried to sketch Platonic solids and really didn't do it very successfully.  I think her eye has come a long way:

Her milk box:

An Asian ceramic bowl with some paper flowers in it:

Her electric pencil sharpener:

I love seeing (and hearing) the world through her eyes.

Look How Our Number Wall Grows!

As a little update to my recent Seeing Numbers post, I wanted to show you what an effect these re-imagined factor trees are having on my child.

Here is the first day, focusing on the factors of 20.  As detailed in the Seeing Numbers post, my seven year old used the tree metaphor wonderfully to illustrate different ways to add and multiply numbers to make 20:

The next day she chose 40.  As before, this tree utilized addition and multiplication as she built its branches.  We also noticed, when we made the preliminary wall of factors with our Cuisenaire rods, that we used the same color rods as when we factored 20, just twice as many.

I had other math things planned for today but this morning she asked, "Can we make another number tree?"  Yes, of course!

We settled on 30 and this is when her understanding of what we were doing really took a jump.  After we created wall of C. rods and after I sketched out all the different factor trees for 30 (like I did both times before) she, unbidden, picked which version she wanted to make.  Huge.  Both times prior to today I walked her through the structure of my own tree (more in line with factoring) but did not define what exactly a number tree should be.  As long as she was composing or decomposing numbers somehow on her tree I was set to be happy which is why I was surprised that she moved over to the factoring camp so quickly.

In her 30 tree her artistic vision is grand but her numbers are tiny, lol!  As we looked at her tree closely she showed me the first level of 5 and 6 branches, and then the additional 2 and 3 branches off the 6.  And, as with the other two trees, she added a crowning flower showing the day's number.

These pictures are all on the wall next to our dining table.  She has been admiring them and examining them closely every time she sits down to a meal at which time we also admire each others handiwork. I have invested in some new markers and paper because I never want this to end!

I will be forever grateful to Simon Gregg and his 5th grade students at Toulouse International School for providing us with such a perfect vehicle for exploring numbers in a way that makes sense to us.  I had been looking for a way to look at and explore numbers that involved an expressive visual element and am excited to see where this will take us. We've grown so much already through the process of observing and using the commutative property with the C. rods, applying understanding of groupings to our design, and conceptualizing division in a completely different, very exciting way.

Seeing Numbers

I've been thinking on and off about ways to help my seven year old daughter come to know, appreciate, and maybe even love numbers.  Geometry is easy for both of us because it's visual.  But because my math schooling was of the memorize and drill variety I have really had to stretch myself in order to make my daughter's early experiences with numbers more meaningful than mine were.  As it turns out, I'm learning a lot in the process and, as a bonus, I am finding that numbers are quickly becoming an unexpected but welcome gang of amusing little friends. 

Since we're working on conceptualizing multiplication and division these days through various explorations, I thought it might be fun to make 'number cards' to visualize one number at a time.  Using coins (and clarifying their intent as reconciled in my recent post on units: When is a 10 Not a 10?) we looked at all the different ways to split 25 and found lots of remainders.  I suggested 60 after we read that base 60 was popular early on because it can be divided in so many ways. 

As you can see, the results are less than inspiring and it's not clear to me that the child really learned anything more from making the cards in addition to the array work with the coins.

These cards are still taped up hopefully on the wall, but I had basically abandoned the project...until I saw this fantastic post  from an international school in France that re-imagined factor trees in the most awesome way.  Isn't this the most lovely factor tree you've ever seen?  And there are more -- all as beautiful as this one.  Their project in France has inspired us here at home and has become a perfect and flexible solution for our nascent number wall idea:

I showed the girl the post that inspired me and she was entranced and eager to get started.  She is doing well with hands-on and mental multiplication and division, but factoring as an activity is new to both of us and so I decided 20 would be a nice easy number to start with. 

We took out the Cuisenaire rods and she found all the factors of 20 herself.  She double checked every rod from 1 to 10 and noticed a lot during the process, skip counting as she went.  "If I can use fours, I can use twos, too!"  "Threes...no that makes twenty one...."  

And then we wrote them out as equations and sketched a preliminary tree.  And here is where this activity became as important for me as it was for her.  I do not recall in all of my fourteen years of school mathematics (including those awful matching bumblebees in kindergarten, my tenth grade algebra D, my tenth grade geometry A, and a deliriously confusing semester of college algebra) ever learning about factors.  Ever.  So, it took me a minute, but I made it work.  (Oh, the things you do for your children!)

Then we got to work.  We drew and we drew and she whistled a happy tune (literally) for over twenty minutes.  She would not let me look at her work.  She was in heaven.  This is what she finally unveiled to me:

To me, it's glorious.  Even though it's not a factor tree, per se, it shows so much of how her 7 year old self is thinking about numbers, including an emerging understanding of factors.  All the numbers she wrote are obscured by the coloring, so I had to ask her about them.  At its base  are the numbers 10, 10 and 20.  Notice the overall symmetry of the tree.  The flowers at the base are 5+5+5+5 (do you see the 'plus' sign written on the trunk?) and, even better, the colors are inverted in each pair.  Moving up the tree, the next two longer branches on the outside are 2 and 10 (so 10x2) and the two small branches inside them are both 10 (so 10+10) and the crowning flower is 20!

This tree motif is doing so much more for her visualization of numbers than those little cards ever could!!  I'm so excited.

And, as a little bridge to future trees, after she walked me through her drawing I showed her mine.  I asked her what she noticed and what the differences were between the trees. 

She noticed a lot, but not the numbers so I pointed her attention to the fact that although each tree split in different ways (10 and 2, 4 and 5), they still had the same prime flowers -- two 2s and a 3. (Did you notice I made each prime it's own color?  And I wonder if there are other ways to bring out the number properties beside just writing the numeral?)

There is no need for us to rush into full-out factoring, but it is an interesting exploration and variation on the division we've been doing.  And, the introduction of another type of number, the primes, is an interesting development as well.  I think this is an approach that we will be able to use time and time again: create, observe, discuss, posit, repeat. 

Let's see: trees as metaphor.  I've seen fractal trees, factor trees, and trees whose branches are the positive numbers and roots the negative ones.  There is so much potential in this kind of image/metaphor for number explorations through all elementary grades.  And, you do know that part of this approach is giving kids an empty piece of paper, some awesome pens and letting them create their own trees from the ground up, right?

I know one thing for sure: our number wall is going to be gorgeous!!  [Addendum: Here's what happened the next day.]

Messin' Around with the Commutative Property

Malke Rosenfeld delights in creating rich environments in which children and their adults can explore, make, play, and talk math based on their own questions and inclinations. Her upcoming book, Math on the Move: Engaging Students in Whole Body Learning, will be published by Heinemann in Fall 2016.

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We were at our second classroom (aka local co-op cafe) and just starting our project when a guy comes over and enthusiastically asked, "Can you tell me what you're doing?"

"Oh," said I, "we're building a multiplication tower."

"A multiplication tower?  What's that?"

"Well, we've got these posts on a grid, and we are going to use different colored beads in different amounts to..."

And then he says, I kid you not, "Oh, yes, multiplication is just repeated addition, really."  (Seriously, I'm not making this up!)

I was flummoxed, because after some great conversation about multiplication in the comments section of a post of mine from last March, and the aha! moment I had about scale while reading the Ten Times Better book, I know that's not all there is to multiplication.  Not at all.  I was so flummoxed it was all I could do to reply, "Well, it's also about understanding scale and measurement and rate..."

And just like that he cuts me off and says, "Well, this seems to be a good concrete way to learn addition..." (or something like that) and leaves, obviously much less enthused and impressed than when he came up to us.

And that about sums up the collective view of elementary mathematics, doesn't it?  Essentially, the message we get is: "There's nothing much to it, just learn your facts and when you get a little older you can do real math [pat on the head]." 

No, I'm not bitter, just perplexed, because when we started building this tower the questions started flooding in.  I started wondering about a LOT of things, most of all the commutative property (which I inadvertently keep calling a process, and I think I might be on to something, see what you think.)  Here is what we did:

I measured out a grid and my daughter and I inserted bamboo skewers into the intersections to make a total of 25 posts.  Our first try at the tower grid was to start with one bead at the origin and then move outward on both the x- and the y-axes with two beads on the next posts, three beads on the next, etc.  Here's a picture (isn't it pretty!?): 

First, it's such a nice three-dimensional gradient, don't you think?  And the different colors show the multiples of each amount clearly.   But the questions started when we got through with 2 x 2 = 2 green + 2 red.  Moving up to (1, 2) I suddenly thought:

"What colors should we use?  Is that two 3s or three 2s???"  It helped to see that the square numbers 4, 9, 16 and 25 moved up the diagonal and it calmed me a little to know that, whatever happened, we weren't completely on the wrong track.

It also helped to turn the model and look at it from different perspectives.  This is a nice view because you get a linear progression of the colors.  On the far left the posts increase by five: 1 five, 2 fives, 3 fives... and on the far right the posts increase by one: 1 one, 2 ones, 3 ones... This, incidentally, is the view my 7 year old prefers, perhaps because it is so orderly.

But then I started wondering, what if every column from (1, 0) onward stayed the same amount all the way up to (1,4) and so on for (2,0), (3,0) etc.?  If you look closely at the picture below you will see that that means instead of two 3s there are actually three 2s and instead of two 5s, there are actually five 2s.  It changes the look....

...but nothing else changes!!!  Is that a surprise to you?  I am not ashamed to admit that it surprised me.  I initially thought I might have to change the posts lengths to accommodate my new approach to the beads.  Here's the thing:  

I do absolutely understand in my head that 3x2 is the same as 2x3, but after re-beading the grid/tower it does not seem exactly the same.  I mean, look!  The progression of colors and total number of beads are the same, but the distribution of each color is completely different than it was the first time.

Not surprisingly, I had more questions.  What would happen if each x column increases by one and was the number rule for the multiples?  Essentially, there are only 1s on the first row and the 2s row goes up by 2s, the 3s row goes up by 3s, etc.

It's a similar result to attempt #2 but the progression to purple now occurs on the x axis!  My daughter likes this version a lot, but I'm not so sure.  The whole thing is less symmetrical than the others but, on the other hand, there are other things to observe and learn from it that you don't see in the first two.

I'm sure there is more analysis I could do, including comparing the total number of beads and distribution of color used in each  attempt but that's for another day.  My point is, yes, multiplication and addition can be done in any order, but our response should not be 'so what, that's easy, don't forget, let's move on' but rather, "Wow, that is SO amazing!!!"

I mean, sure, you get the same final answer no matter what, but just look at the variation in quality.  For example, a combination of four 2s (2 green, 2 red, 2 blue, 2 yellow) looks completely different from two 4s (4 green, 4 red).

Before I started my personal math remediation, multiplication seemed like nothing more than a table of right answers.  Now, however, it seems like a wonderful opportunity to find as many different right answers to the same question as possible.  My kid got a little bored with the beading process but no matter.  I'm going to leave my favorite version out on display, which will make at least four different multiplication models up in our house right now, for us to ignore or puzzle over as we will.  It's already generated some nice conversation about square numbers. 
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p.s. I got inspired to do this project while reading the proofs of the grids chapter in the new Moebius Noodles book.  They say the project's target age is birth to age 6 but, using today's activity as an example, I really think anyone can benefit from their playful, hands-on, inquiry based approach to math. 

p.p.s. I kept the skewers long during the first beading iteration but quickly decided to trim them because I truly could not see the forest for the trees, lol!  I think the trim helps focus the eye on the gradient and the patterns produced by the color quantity and progression..

Thinking in Threes

My seven year old daughter and I are really quite a team.  One of us will share an idea or an observation and all of a sudden the other of us will be inspired into action.  For example, she found a triangle in this clover last week and I immediately knew what we could do with it.



And, although she sometimes eschews direct participation in projects I think up, my kid is generally always around as I'm making something.  In this particular case, I chatted with her about what I was doing while I glued and pasted, and she made a lot of observations, which is good enough for me.

Isn't it cool?!?

It's a (dried) clover Sierpinski triangle!  We picked clovers, flattened and dried them in a sketchbook, and finally found some time (and a glue stick) to paste them down.  I don't know about my kid, but I am really enjoying all the different types of Sierpinksi triangles we've made over the last few weeks: out of candies, with our straight edge and ruler, with our colored pencils, with money, and now this!

It's fun to find math, wherever we go and it's even more fun to make math out of the things we find. 

Weaving Geometric African Motifs, Part 1

When I found out about a new exhibit of African textiles and baskets I knew I had to go.

I was in the middle of the first flurry of exploring mathematical paper weaving inspired by Patrick Honner's guest post over at Moebius Noodles.  At the time, I was having fun experimenting with warps made from various combinations of multiples as well as the different patterns created by a weaving algorithm and its inverse, and using Fibonacci numbers to create interesting designs.  So, a chance to see traditional, often geometric, weaving designs up close and personal was definitely something I did not want to miss.

The Woven & Constructed exhibit was fabulous.  It was something else entirely to see these incredible, LARGE pieces hung with care, side by side, ceiling to floor. Photos are one thing, but these pieces had a presence that is hard to describe, but felt by everyone in the gallery.

Over the next few weeks I will be working on woven paper designs inspired by the motifs in some of my favorite pieces from this exhibit.  Here is my first design which is, I realized after the fact, inspired by a design that is pieced rather than woven.  This may have been why it was so hard to weave!

Here is a close up view of the picture above, a pieced design from the Democratic Republic of the Congo:

Here is a sketch in my book, as I tried to figure it out (and a sneak peek at some other ideas):

And here I worked to graph it out and figure out the basic pattern.  The weave pattern was 4 and an inverse of half (basically), and the warp was built on multiples of 4.

And here is what the woven piece looks like: 

Oh my gosh, it was SO hard to keep track of the six individual lines of weaving; I had to start over a couple times.  Working with slightly different paper also slowed me down a little. I had decided to use 1/4" strips instead of 1/2" so I could get more iterations of the design in the piece.  I liked the thinner strips but will probably like them better cut from colored copy paper rather than this high-end construction paper used here.

I am really enjoying this dual artistic/mathematical inquiry through paper weaving.  It's easy and satisfying to have an idea and finish a piece in a short amount of time.  And, every time I finish one piece I have questions that lead me right to the next, but different, piece. It's an exhilarating feeling, honestly.  This is exactly the kind of work students and I do in Math in Your Feet and I feel certain that this kind of double inquiry can be similarly structured for paper weaving, even as early as first grade.

What I find fascinating about this little project is that I used a different set of skills today than in previous weavings.  Compared to my previous efforts, there was a lot more analysis and a lot less "I wonder what would happen if...?" kind of thinking.  If I had to choose, I'd probably pick the "I wonder.." approach, but I can appreciate how reproducing other people's designs can inform both my design process and my mathematical understanding.  It was certainly a good challenge to move a design from the inspiration piece to the paper version. 

More to come!

p.s. Join us at the Math in Your Feet Facebook page where I share fun stuff like the math my daughter and I find all around us as well as interesting links to things I find around the web relating to elementary math learning...

Mathematical Weaving, Part 1: Young Children & Grid Games

Patrick Honner's Moebius Noodles guest post about mathematical weaving has been in the back of my head for over a month and a half.  Mathematical weaving employs one of my favorite making materials - colored paper! - and I thought it would be fun to try with my seven year old.  I know the rudiments of weaving, but I wasn't sure how to get started, so yesterday I played around to try and figure out a few things.  It was actually sort of challenging, but I landed on some solutions and new grid games, so I thought I'd share.

I'm not done with my exploration, but what I have discovered so far is a perfect little unit for young children.  I am imagining that the weaving and the games would be completed in an enjoyable collaboration between adult and child over the course of a day or two.

I started by experimenting with loose 1" strips of paper as the warp (vertical strips) but soon found that much too unwieldy for even my adult hands.  The pieces were not connected to each other so they slipped all over the place and I had to use a lot of tape to keep the weft (horizontal strips) connected to the warp, which wasn't ideal.  So, I searched for some advice on how others have made paper weavings.  A quick Google search and I found this video (which is cool, but still too persnickety for the young ones) and this video which, although the cutting is somewhat haphazard, led me to a solution for how to weave paper without tape...

I first decided that a 3/4" width for vertical and horizontal strips made a more pleasing final product to my eyes than 1".  To make the vertical strips I folded a piece of paper in half and used my paper cutter to cut 3/4" strips from folded edge to about 3/4" away from the open edges closest to me.  Essentially, I was creating a paper warp that was still basically one piece of paper.

As you can see, below, the horizontal strips weave in very nicely and don't need any glue or tape to keep them in place if you focus on pushing them gently, but snugly, downward.  For the young ones, at least, a basic over/under/over/under weave is challenging enough.  Using two (or more?) horizontal colors creates visual interest and perhaps even a conversation about the patterns you see: alternating colors both vertically, horizontally and diagonally.  You can also make a connection to odd and even numbers.  Yellow squares in the design show up 2nd, 4th, 6th... places.  Green squares are 1st, 3rd, 5th...

The minute I finished the piece I thought - A GRID!  It's a grid!  The Moebius Noodles blog is very inspirational and a great source of grid games (my favorite so far is Mr. Potato Head is Good at Math) and I always have grids at the back of my mind these days because of them!  Here are some of the ideas I came up with using a newly woven paper math and one of my favorite math manipulatives -- pennies!

Adult: Oh look!  There are three different colors of squares in our woven grid.  I've got some pennies -- I wonder if we could make a square by putting pennies down on only one of the colors?

Adult: That does look like a square. Let's count and see if there are the same number of little squares (yellow, blue, yellow, blue...) that make up each side?  There are!  How many little squares are there on each side?

Adult: But, wait! Look what happens when I push a corner penny in toward the center!  Yep, it lands on a green square!  Let's do it with the rest of the corners and see what we get.  Oh, lovely.  A rhombus.

Adult:  The corners on the rhombus are on the yellow squares.  I wonder what would happen if we pushed them one square toward the middle?  Ooooh, look!  We have another square.  Is it bigger or smaller than our first square?  Each side on our first square was six little squares long.  This square has sides that are...three little squares long.  Cool.

Another exploration:

Adult: Here's a little story about a tiny X who wanted to get bigger.  Can you help him figure out how to help the X get bigger?

Or, how about the tale of some square numbers who also wanted to get bigger?  What little kid doesn't want to grow up?

And, here's my favorite.  It's a 'let's make a rule' kind of game.  The first penny goes in the bottom left hand corner, and you start counting from there.  The first rule here (pennies) was two over, one up.  Each time you repeat the rule, you start counting from the last token on the grid.

You're probably wondering about the buttons?  Well, that's a different rule: one over, one up.  Isn't it cool how they overlap, but not always?  Kids can make up their own rules after a little modeling or you can challenge them to guess a rule you made up and keep it going. 

And then, of course, the final thing would be to leave the pennies and the paper grid mat out to explore at leisure. 

I have some more questions about how to facilitate Patrick Honner's activity with slightly older children (first and second grade-ish).  One of my thoughts is that there is a basic algorithm for weaving that is a combination of overs and ups.  The design in the picture at the top of this post starts on the first line (weaving right to left) as 'two over, one under'.  The next line is different: 'one over, two under' and then the next two lines are actually the inverse of the first two.  Since my seven year old is already a fairly competent weaver, I think giving her some examples of how different combinations of over/under interact with each other would be a good place to start.  I'm also curious whether my daughter would be interested in the mathematical modeling at this stage in the game.  She's still a do-first, map-it-second (maybe) kind of gal.

p.s. I've got a new Facebook page where I'll be sharing links to cool math activities I find and some other things I'm doing with math, making, dance and rhythm.  Hope to see you there!

Prelude: Spiders Who Spin Fancy Webs

Here's an lesson in the making that I'm really excited about!  Last week, when I was working hard to figure out the basics of stars and their relationship to math, I ran across a really helpful description of how to make and notate stars.  One of the images the author uses is that of spiders spinning webs, one line at a time, between or across points.

Part of the reason I'm looking into stars is because I think there's real educational potential in this kind of inquiry for all ages, including young children. Most of the material I've run across seems to target fifth graders and above.  But, after experiencing how helpful the spider analogy was for me I thought, why not tell my seven year old a story about a spider who wanted to weave a really fancy web and have her spin some webs of her own?  The kid is pretty excited to be a spider in the near future, especially since the project has morphed into a lot of hammering and tons of embroidery floss.

Today I was in the middle of creating the framework for each web (six through ten points) when my daughter ran to get her rubber bands out of the math basket and started make designs (hers on the left, mine on the right).  Even thought it wasn't planned, I thought this was a really cool way to explore the circular structures and posts/points before moving into a more formal activity.











Stay tuned for more developments, but in the mean time, I've got a new page on Facebook. I'd love it if you'd pop over for a visit. Check it out here!

The Sound of Scissors

It was the last week of my summer programming up in the city and it was quiet.  Really quiet.  I had never seen such a large group of five- to nine-year-olds so collectively intent on something.

Maybe it was the stories I had just told them about my square friends (documented in detail here).  Those squares were incredibly jealous that the hexagon in the middle of our space had transformed himself overnight with some yellow tape. I mean, come on!  Everyone needs a change of pace once in a while, why couldn't the squares try it too?  After a little folding, a little cutting, each square was thrilled with their newly transformed, and utterly unique, paper selves.  I put them up on display so they could show themselves off.

When it was the kids' turn, they took their scissors, glue sticks, and paper and went for it.  I didn't do much else other than remind them of how I folded the paper during the stories and tell them they could have as much paper as they needed to experiment and make mistakes. 

Once they had a design they liked....

...they could glue the transformed square onto another square of a different color. 

When they came to me with three or four designs I asked them to put them in an order that they liked, or that made sense to them.

Even when their scissor skills were not the greatest, the overall effect was really fun and amazing.  It's obvious that they tried different things each time.

And, my favorite thing about letting kids explore a medium: the activity, but also not the activity.  This boy devised a layering technique that looks incredible.

And then there was the five year old boy who cheerfully adapted the activity for himself.  The red square was the first effort (so lovely) and by the second square he had fashioned something completely different. 

Ultimately, it was probably 35 minutes of focused concentration.  All you could hear was paper being folded and the sound of scissors.  Now that is the kind of quiet you want in math class -- not the stuck, frustrated kind, but the kind where kids are intrinsically motivated to find solutions and discover structure all by themselves, one cut at a time.  Just listen: