The Butter Knife and the Infinity Knife

Recently, my eight year old spontaneously discovered an activity I've seen in the Moebius Noodles book and yet another version of infinity (read about others here, here, and here).

Scene 1: The Butter Knife

Waiting for the bread to toast. She picks up the butter knife off the counter and places it vertically over a design on the lid to the butter container.

"It turns into an arrow!  What a cool design."

She continues to play around with "cutting" the lid's design in half, one side the real the other the reflection.

Scene 2: The Infinity Knife

The kid calls to me from the other room to tell me she's cutting triangles in half into smaller and smaller pieces, trying to see how many times she can cut them.  When I ask her to show me she picks up another piece of paper and starts again.  She cuts one big triangle in half equally, then one of the halves in half again...

"It has something to do with infinity," she says, as the triangles get smaller and smaller and smaller. "I need something different [than the scissors she's using] -- a tiny knife like scientists have -- to cut infinity stuff like this."

................................

I've been reading books by former colleagues and students of Jean Piaget who have carried his work forward and made it accessible to the rest of us. My take away so far, beyond the idea that all of us construct our knowledge by assimilating new information into what we already know or think we know, is that children think when they have something real, some phenomenon, to think about. Whether it's a butter knife or an infinity knife or whatever else, it's the interaction between the child and the object/idea that inspires the child to think -- what Eleanor Duckworth calls "the having of wonderful ideas." 

"There are two aspects to providing occasions for wonderful ideas," she writes in her book of essays The Having of Wonderful Ideas. "One is being willing to accept children's ideas. The other is providing a setting that suggests wonderful ideas to children --different ideas to different children --as they are caught up in intellectual problems that are real to them." [page 7]  

Whatever else we do as teachers and parents, I think we need to find multiple ways to allow kids the freedom to discover the world on their own terms, in their own ways. At the same time we need to create the time and space in our teacher/parent brains to catch our children and students in the middle of discovering something brand new (to them). Having wonderful ideas, as Duckworth says, is "the essence of intellectual development."  Honestly, being able to observe and even, sometimes, to interact with my kid when she's in the middle of a new thought is pretty much the prize of parenthood -- it is such a gift to see this happen up close.

New Math Game: Change Your Rule

This post is in two parts: The first part outlines the basic flow and structure of a new game idea which, I think, has the potential to support the conceptualization of math concepts such as combinations, permutations, patterns, variables/attributes and reflection symmetry.  

I know that's a tall order so in the second part of this post I briefly discuss my reasoning, pose a few lingering questions and then ask for feedback.

Part One: Basic Flow of the Game
The idea for this new game started with a picture:

Photo via: http://www.fotomat.es/arte-combinatorio/

I immediately thought how cool it would be to have this kind of mathematical art/visual inspriation in a living or learning environment, something to hang out with that evokes noticing and wondering.  In this case, a real-life shelf with spheres and cubes just begs for interaction, something to play around with and ponder. I want one! In the process of thinking about how to get this kind of installation in my own house, a new game was born.  Here's the basic idea:

Make/build a rule.  Find a way to change it.

These are the game pieces. Set 1 includes two shapes, four colors.  Set 2 includes six shapes and six colors.

Rule Change #1: Make a rule/pattern. Using the same shapes/colors how can you make the next iteration different from the first.  Does order matter?  How many different combinations can you make?

Rule Change #2: Make a rule/pattern. Change one element of each shape to make a new pattern.  For example, orange turns into red, or square turns into circle.  For each new design change only one variable for each piece.  How many times can you change the rule? How many new rules/patterns can you make from your starter rule?  Can you ever get back to the original rule/pattern?

Rule Change #3: Build your pattern then reflect the design. What do you have to do differently when you build the reflected pattern?

But what happens if you only use one color and shape?

How many combinations can you make using three of one shape and one other shape?  Would it change things if each of those circles were different colors?

On another line of questioning, what larger design can you make by building your design line by line using the same shapes/colors in each line?

Part Two: Thoughts and Questions

Even though Part One seems like a lesson flow, it's really meant to be more of a general framework for exploration.  My intent was to provide a little structure, some basic 'rules' and a lot of room for inquiry.

I'm wondering if it really feels like a game, or is more of an activity?  If it's going to feel more game-like, does it need more structure?  A timer? Some 'change your rule' cards?  Or, maybe, some cards that say how many different attributes to use in the pattern?  How many different rules would be enough to create a sense of chance? What other 'change rules' would you include?

Based on trying this out with my own kid this morning, if you were to do this more as an activity/lesson perhaps it would be helpful to have some really easy starter examples, like all one color, or two different colors to get the ball rolling in a positive direction.  I asked my daughter to build a pattern with three pieces but then gave her the 'change rule'.  She didn't really like that and we left it there.  The game pieces and board are still out, though, hanging around.  I'll see if she wanders back over.

Questions I still have include: When combining or creating permutations, how much does the second attribute matter?  For example, this first design (two colors, one shape) looks like it combines in exactly the same way as the second design (two colors, two shapes):

Another one of my goals for this game/activity was to also explore the ideas of attributes/variables in design -- how well do you think it does that?  I intentionally did not use pattern blocks -- only one attribute/thing to change -- and instead made my own (Set 1). What would pull the attribute/variable idea out a little further?

I hope it's clear that I am hoping to dig into the brain trust that is my modest but wonderful readership.  That means you by the way, so please if you have any thoughts about all this, I'd love to hear your ideas.  

Things I would love feedback on: key ideas about combinations/permutations at the elementary level, thoughts about the game structure, ideas for other 'rule changers' to add into the mix, and any other thoughts you may have.  Thank you!!

p.s. If you want to leave a comment but don't see the comment section below, please consider refreshing this post or closing the post altogether and come back.  This template has some bugs. If you don't see a green-lettered header and/or the page menu at the top, then chances are the template did not fully load.  Sorry for the hassle. 

Small Moments of Math

I often write about the times when mathematical inspiration hits but, most of the time, our daily lives are made up of smaller, less dramatic math moments. It's during these lovely transient events where I really get a good glimpse at how my daughter is thinking, mathematically speaking, and how she is applying her understanding in a number of different contexts.

My approach to math exploration at home has been hands-off, necessitated by a child who likes to captain her own ship.  This basically all boils down to the fact that math generally happens in bite-sized pieces around here.  It doesn't mean I am not influencing the process but it does mean I hardly ever make formal plans; instead, I am always looking for new games, thinking about what she might need or want to learn next and also how to introduce new things in a way that has the appearance of being at least 50% her idea.  I also leave stuff lying around to be 'discovered' or engage in my own pursuits, which inevitably leads to some curious inquiries from the wee bystander. 

In addition to all this stealth planning, these days I'm also preparing for some summer arts programming up in the city.  In this case, I'll have three groups of students (ages 6-7, 8-9, 10-12) an hour each day for a week or two at a time.  It'll probably be harder to orchestrate the kinds of gorgeous, in-the-moment discoveries we have at home in a windowless basement room of a run-down church building with lots of kids and no air conditioning, but I'm going to try.

Here are some examples of the small moments of math that have been happening around here lately, including some of my thinking and experimenting about the summer.  I'll start it off with half a lunch which actually started as a whole lunch, but when the kid saw the design with a whole tomato in the center she insisted in cutting it in half before eating:

A spontaneous, self-initiated design from the kid after breakfast, utilizing her rock and mineral collection.  I guess she understands reflection symmetry after all!

The kid was designing/drawing mazes while we were at the library but then got interested in a 2001 NCTM magazine on mathematics and culture I was reading, specifically an article was about a game played by the Fulbe children in Cameroon.  The game is essentially about creating designs by drawing line segments through a series of dots lined up in a grid. 

After a quick look my kid drew her own version  ("The same design equals two different things!") and then also started drawing the library ceiling (square tiles and long rectangles of lights).  "There are patterns everywhere, Mama...Oh, look a book about cats!"  And so it goes...

It's clear the kid likes functions, but the function machine she built is totally off limits to well-meaning but meddlesome adults.  The photo below is a bit blurry, but you'll see she really just likes the process of figuring out the equations using the same rule and different inputs.  The minute I said something about looking for a pattern in the answers the whole thing went south.  I've learned the hard way that, in this particular instance, there is no room for me to ask any questions about anything whatsoever.  However, I'm confident that she'll figure it out at some point like she did when made peace with the times tables last week.

She'd been multiplying on her fingers but finally realized how much time and effort that takes. "The multiplication table is like sight words," she told me, "you just have to memorize them."   Reading a book called Piaget for Teachers from the 1970's makes me realize that my approach is probably on target given her developmental age (newly operational) -- telling her that particular 'fact' myself will not help her thinking process.  The way to help her thinking process is, basically, to just let her figure it out through a variety of experiences (cleverly orchestrated in secrecy by her scheming mother, ha!).  

I made the kid a puzzle/game last week called "Make Them the Same" using elements of line, shape, design and color.  I didn't take a picture of the starting designs but if I had you would have seen that both designs in each pair were incomplete in different ways -- you had to compare the two to make them both a complete design.  I'm still playing around with this idea.  This version was too easy and my second try was too hard but I think I'll be able to make it work.  I really want to have it figured out by summer.

I spent a couple hours over the weekend looking through the math book lists at Love2Learn2Day and Living Math.  The theme for my summer program is tentatively called "What can you do with a square?" (or a pattern unit, I'm not sure which yet) and, in Three Pigs, One Wolf, Seven Shapes the answer is obviously 'tell a story with it'!  I made a sheet the kids could use while listening to the story, but when my kid discovered it she tried to put the tans on the outlines which are not to scale.  For the youngers I may have to give them each one or two real-size tan puzzle sheets instead (which I found at Mathwire), and use this sheet for the olders.

Chessboards are made up of lots of squares.  When I read this book to my daughter a few days ago to make sure it was something I wanted to use in the summer, I had the thought that I could make paper chess boards and give each kid a baggie of rice to follow along with the story. 

It might get messy with all that rice, but part of my thinking is that Math in Your Feet is about order and structure and pattern.  A chess board is so very orderly and there's a growing pattern in the story (doubling, my current favorite topic).  And, having the real rice at hand will hopefully make the story, well, more real.  Also, I'm thinking about doing more with paper quilts and tilings and the image of the chessboard would be great to refer back to when bringing up tessellations with the older kids.

Not surprisingly, more small math moments have happened during the in between times of writing and editing this post.  It seems that little jewels of math are all around us just waiting to be discovered at unexpected moments!  

As a postscript of sorts, I want to say that Maria Droujkova at Natural Math has been a huge inspiration to me over the last year and one of the main reasons I have learned to see math all around me.  I would not have grown my 'math eyes' if not for her Natural Math forum or reading about the math clubs and Math Treks she has orchestrated as well as the new Moebius Noodles project for young children. 

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.

Reflection is Good for Everyone (Even the Teacher!)

I've just finished a five-day residency up in Indianapolis as the second part of Young Audiences' Signature Core Service pilot.  Most of the time it seems the dance and rhythm aspect of this program brings kids closer to the page by motivating them to write, sometimes for the very first time, multiple meaningful sentences about their experiences creating original dance steps.  The program can also help kids think of math as a friend, not a foe, also for the very first time.

The kids I just worked with, however, were already very comfortable in the symbolic realm of mathematics and also possessed strong verbal-linguistic skills and positive problem solving attitudes.  They were, however, still fourth graders in their bodies and feet (and I mean that in the best possible way).  In addition, they were good at following directions, and really dug into the activities I outlined but...

....upon reflection, I realize now they might have been bored. 

They might have been bored because they really knew their math, but at the same time they couldn't really do more than fourth graders normally can do with their bodies.  I think that if I had had more time with them, or could do it over, I would have found a way to get their brains more engaged while their bodies worked at age-level.

I would have challenged them to really play with their patterns.  For example, instead of just making a Pattern A and B and combining them into a third, 8-beat pattern (which is a great amount of play in itself), I would ask them how many different ways they could recombine the four beats that made up each pattern, and then encourage them to ask more questions along that line of inquiry: What if we reflected the pattern to itself, beats 1, 2, 3, 4, 4, 3, 2, 1?  What if we took our two favorite beats from the two patterns and traded them with someone else?  "What if...?" is one of my favorite questions, after all, because you never know where it will take you.

Despite these musings of mine, I think things went well.   Here are their perceptions and reflections of the first three days of the residency...

Day One: "You have a friend who was absent today and missed the first day of Math in Your Feet.  Tell that person about all the different ways you made patterns with your feet."

First, we had vocab words, such as hop, slide, clog, etc. then, we made and [sic] pizza, and used all of the dances.  For example, we did sausage as slide.  As it says, Math in Your Feet!

Today Ms. Malke came in and we learned all about dance steps.  This is called clogging.  Clogging and step-dancing are actually cousins!  When we were dancing sometimes we put our feet to a 90°angle and other times she told us to turn our feet to an acute angle.  We also used a lot of math vocabulary words.  We all had a great day full of dancing and math.

Today for an hour we did math in your feet.  We did clogging, and the chug.  We made a pizza in our mind and did different dances to represent what we put on the pizza.

We made patterns by dancing.  When we danced we also made rhythims [sic].  We also pretended to make pizza.


Day Two: "What did you have to do to dance congruently with your partner?  Using complete sentences, name at least three things that had to be the same.  What kind of challenges did you and your partner face to make your dancing congruent?"

To dance congruently with my partner we had to count out the four beats.  The timing, dance steps, and the beats all had to be the same.  One challenge we had was timing at first, but then we practiced and practiced and we finally got it at the same time.

My partner and I had to take one step with our right ft. 1st, then left, turn 180° right, then 180° left.  The kind of challenges we had to face are having to make sure we both knew what order our steps were in then we counted 1, 2, 3 to know when to start.

Do the same dance you have to keep a steady beat, go at the speed of the slower person, and practice.

My partner and myself really didn't have any problems, but if I had to pick three I would pick that 1. was that I could not get the steps right.  2. We could not stay together.  3. We could not figur [sic] out how we wanted to do are [sic] steps like for example speed fast, mideam [sic] or slow.

Me and my partner faced lots of challenges.  We did a 270° turn wich [sic] was hard to be congruent while doing.  The speed, movement, and moves had to be congruent.

To dance congruently, I had to say something to signal us to start dancing.  The 2 diagonal splits & 1 side split had to be the same so it would look good.

Day Three: "Write a friendly letter to your partner.  First, tell this person about what you had to do to reflect your dance pattern across the line of reflection.  Then, tell your partner what your favorite MIYF dance move is and why!"

I had to turn right instead of left to mirror your moves.  I love our Pattern A!  It's casual, but interesting.  I can't wait to see Pattern B.

To reflect our dance pattern across the line of reflection you had to do it in the opposite sides and I did it originaly [sic].  My favorite MIYF move is a slide, with our feet together, and back just because it's fun to do!

In Math in Your Feet we did a line of reflection and how we did it was we had to pick a person to be the original and the reflector.  The reflector had to do oppsite [sic] lefts and rights.  My favorite dance move was the cross because it is fun and it makes me feel like a real dancer!

I had to instead of going right diagonal at first I had to go left to right Diagonal.   I don't have a favorite dance move because I don't like to dance.

I love having you as a partner because you don't get mad when I mess up and you agree with anything we do.  I could go on with the tanes [tons] of things good about you but I'll stop there.

First, we had to pretend we were looking in a mirror. We had to  switch our rights with lefts and lefts with rights.  My favorite dance step is turn because I like getting dizzy.

I love fourth graders!