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!

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.

Fun with Functions: To Infinity and...Back!

What is a function?  Before yesterday I had no earthly idea except that it's clearly an important mathematical concept and one that young children can easily experience at the concept level.  Based on my search for function machine activities last night, it seems that functions are essentially about rules and change, both of which seemed to dominate my little experiment with the kid today.

It all started yesterday when I was looking for math games at the library and found the book Anno's Math Games II.  The first chapter is about a marvelous, magical machine that turns one thing into something else.  A chicken into a chick, a butterfly into a caterpillar, and...can you guess the rule?   Later in the chapter Anno moves the rules toward transforming numbers but when I suggested to my six year old that we make a magical machine of our own she jumped up and literally shouted:

"Yay!  Let's make one!  I'll go get Lucy [our cat] and we'll turn her back into a kitten!"

Um.  Well, not really a real machine, right?  Sort of a machine in our heads?  With numbers?  We left it there for the day.

Last night I got to searching.  It seems that a function machine is actually pretty straightforward.  You put a bunch of numbers in and try and figure out from a number of results what 'the rule' (function) is.  So, I made my own function machine using clip art of a washing machine and a dryer (dirty in and clean out...) and some arrows.  I figured that we could play around with it a little and see what happens. 

What happened was that it basically flopped.  I made the first rule, we put a number in and wrote down what came out.  We did it a couple times and then she successfully guessed what the rule was.  Not exciting.  Then she got to make up a rule, and the whole thing collapsed into an activity that essentially boiled down to practicing familiar math facts.  Plus, she was really, really disappointed about the machine itself.  I think partially because it was a picture, and we all know a picture of a machine is not a real machine, right?

No worries, though.  At that point she took matters literally into her own hands and made her own...

There were buttons for adding, subtracting, multiplying and dividing.  There were buttons for stop, go and compute. 

There was a little door where you could put in a slip of paper with the input written on it.  There was an emergency brake. (If you can't tell by now, these were all her ideas.  I was just grateful she had salvaged the activity -- even if it meant being demoted to the position of scribe as she dictated the rules of how to operate the thing.)

There was also a screen: "Numbers can be a bit dangerous," she said. "The screen is for if you don't want to go on into infinity." There was a steering wheel "in case we go into infinity, or a really, really hard problem.  Press the steering wheel to pause, so you can think about it.  Or, if you get into infinity steer the wheel backward and you'll be back where you started."

Apparently, the transformation of numbers is not only tricky but a bit dangerous as well.  

When it was time to try out the machine I encouraged her to put her results on my nice little worksheets, but her energy was really more about the slips of paper she wrote on and put into her machine.  Fair enough.  Here are some of the rules we played around with:

One round of multiplying by three.  Four rounds of 'multiply by 2, take away 1' (on another sheet, not pictured).

We also put the machine on 'geometry setting' and steered "the wheel backward to get it to change the number of sides" of a shape.  We started with a triangle and the rule of 'add one side'.  Interestingly, the kid turned the triangle into a rhombus, not the square I expected, which made a lot of sense given the angles.  And, when she tried to add a side to the square she initially just divided the square with a diagonal line.  A nice moment to reinforce 'side' and 'edge'. 

Also, because we recently explored the concept of half-ness and double-ness we did some doubling using the legos.  I suppose we were on the 'geometry' setting then, as well, because of the 3D shapes.

Reading this post over, I realize I forgot one really important idea: that you can run the 'rule' both forward and backward through the function machine.  Except that...the machine she built does have a steering wheel.  As she did say, "...you can steer the wheel backward and you'll be back where you started."  I guess that's an inverse operation right there!

I've been thinking about how we might do this next time.  Harnessing my kid's imagination and her need for the concrete/physical seems like the key.  Maybe we'll figure out how to make bigger machines, some toy-sized, some human-sized.  The little plastic cats could go in one door and a different amount of bigger stuffed cats could come out the other?  Or, a dress-up function where you go into the machine (a tent?) dressed normally and come out with hats, scarves or necklaces added?  Hmmm...  Maybe we could make up simple Jump Patterns (ala Math in Your Feet) and change them somehow? 

It's easy enough for me to come up with ideas for physical/concrete functions, but harder to think of ones that create some complexity and/or interesting results. I also can't tell which is more important -- doing the computation of the rule, or figuring out what the rule is.  I've obviously got some thinking and learning to do here so if you are able to elucidate this topic for me, I'd be really grateful!

In the end, it's clear from watching her make her own machine that she's basically got the concept: you can change something into something else using different kinds of rules.  More importantly (for her, anyhow) is that you don't have to settle for a representation of a machine -- you can make your own real one!