The Hundred-Face Challenge [C-rods and Constraints]

It was "Math with Malke" day in my daughter's 3/4 class! This activity was inspired by Simon Gregg's ongoing Cuisenaire Rod work with students; it was one recent conversation in particular about some "faces" that had shown up during one of his classes that got me thinking.

Because the kids' exposure to the C-rods was limited, I wanted to give the third and fourth graders a short intro to the rods before presenting the project.

I built a sequence of simple investigations that led up to the big challenge. Kids were split into (semi) random pairs; each group got a bag filled with a random amount of C-rods. I asked them to open the bags and find one of each color. 

While they were sorting, I noticed that the groups were naturally ordering the rods as they went through their bags. I paused the class to have everyone look around the room at how others had set out their 10 rods. We did a quick list of our noticings:

- 1 cube different between each

- Different sizes

- Each block is 1 number away from each other

- Looks like stairs
- Looks like a graph chart
- Looks like a sail
- Different colors

To give them some tactile experience with the lengths/amounts of the rods I had them pick up white, red, light green and yellow rods and put them behind their backs. They faced their partners and gave a series of commands: "Show me...green!" "Show me...white!"

As a final introduction before the face making I modeled adding the rods together, white to orange. Then came the challenge! 

Make a face that "adds up to 100" or as close as you can get. 

Constraint #1: Use only the rods in you bag. 

That's it. Off they went! Some groups added as they built and found a need for paper and pencil to keep track.

Some kids went immediately for the tens which made for easier counting.

Some kids built first, counted second and added or subtracted rods as needed.

Some kids just made awesome faces. Me: "Hmmm...that looks like it's more than 100. What are you going to do?" Kid: "I guess we'll take off the hair."

Some faces were closed (all rods touching), some were open.

I like this humble little guy:

Then it was time to pause for some more noticing:

- We used 10 rods to make 90 

- Some groups built on top of other blocks

- We changed plans a lot

- We had to add carefully

- Added by 10s

- We counted as we went along

- Everyone used at least one 10 rod

- Some people built and then counted

And the final challenge! Make a second 100 face different from the first. For example, if your first one was open, find a way to have all rods touch each other. If you had a closed face, make an open one.

Constraint #2: No matter what you do, the face needs to be balanced like a human face.  I illustrated this step by step on the board. [And, yes, I know that a human face is naturally balanced, but I wanted to make the idea explicit. It's one thing to see the balance/symmetry, it's another thing altogether to actually make it.]

When everyone was done with face #2 I said: 

"We’ll go around the circle and you can tell the group just a little bit about how you made this particular face.  What changed about your strategy from the first time you did it? Was there something you had to do differently, something you guys talked about that was a challenge while you made it? Or something that came to mind while you were building it."

Here are few of the conversations we had:

Group 1:
Girl: We started by just making a symmetry face and then counted them up and figured out we needed 24 more. So then we basically just added the hair with 24. Then it wasn’t perfectly symmetrical so we kind of made it the same on each side.

Me: And did it add all the way up to the hundred or did you get close?

Girl: It equals to the hundred.

Me (to boy): Is there anything else you want to add?

Boy: Well, no.

Me: I saw you put the pencil down [the middle of the face). Want to show them what you did? [He puts pen down the center.] Do you see that there’s a red on each side, an orange on each side?

Group 2:
Girl 1: At first we had a round face and we had eyes, a nose and mouth in it. But this time we made the background of the face ten each so we have 90 here and then we used twos to make the face.

Group 3:

Girl 1: Our first face was open and then we made another face but then Malke told us to use smaller pieces…

Me: [Laughing] Because why? Why did I give you that challenge?

Girl 1: Because we finished only using 10s, 9s…and the last face wasn’t as symmetrical. So we decided to do four browns in the middle and two blacks on either side. But then if you counted them up there was only 80 on the bottom.  So what we had to do is 4 + 4 is 8 plus 2 is 10 so we could only get up to 90 and if you let us use the 10 we could have made feet.

Me:  [Laughing] But that’s the thing, you have to work within the limits.

Group 4:

Boy 1: Well we started off again with a round face but then…

Boy 2: But then he was already making this other face on the ground and we just added some more to it.

Boy 1: One of the newer things was we took away the triangular nose and  put in these two [white blocks] and we added these red things.

Me: Does it add up to 100?

Boy 2: Mmm hmm. We got hair and you can [to other boy – can you hand me your pen? Thank you…laying the pen vertically down the center] it actually does go down and there are two teeth and two red things on either side.

The conversation reveal thinking about sameness and differences and emergent thinking about symmetry and balance. The teacher and I were both extremely happy about the activity, engagement and conversations in the individual teams and the class as a whole.

Next time I want to be make the constraints well, more constraining, like: "use only rods one through six."  I think this would also be a good to do again with this same group so we can deepen and extend the idea of balance.

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.