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.

Another Infinity 'Discovered'

It's always awesome when the math ideas we read about pop up weeks, months or even years later in interesting new ways.  Take infinity, for example:

First there was her hand-made dolly Amelia writing an essay about 'what infinity means to me.'

Then there was her recent theory of multiple universes which was buoyed by set theory and the illustration of an 'infinite number of infinities of different sizes.'

And, this morning, there was this:

Her: Mama! I discovered another infinity!  Half of two is one.  Half of one is one-half.  Half of one-half is one-half of one-half.  Half of one-half of one-half is half of...

Me: Hey, I totally get it, but you wanna hear an easier way to say it?

Her: Yeah.

Me: What's half of two?

Her: One!

Me: What's half of one?

Her: One half.

Me: What's half of one half?

Her: I don't know.

Me: Ummm...you know how when we bake muffins?

Her: Yeah.

Me: Well, when we use the measuring spoons...[I talk a little more but it becomes obvious she's not getting it and I'm not equipped at 6:30 in the morning to think clearly. I abort the mission]...okay, here's the easier way to say it: Half of one is one-half.  Half of one-half is one-fourth.  Half of one-fourth is one-eighth, half of one-eight is one-sixteenth...

Her: So it gets smaller as it gets bigger!

What's remarkable to me about this particular conceptualization is that one, it mirrors thinking of done thousands of years ago by philosophers -- but she has never heard those paradoxes.  The second thing is that, obviously, she's not done much with fractions and yet, the idea of half-ness and size is fully there.

And all this before 7:00am.  Any time is math time, right?

A Shift in the Universe

Something shifted in my newly eight-year-old in the last week. It shifted in terms of how she understands math and science and in how she sees herself.

It started with a morning thought she had about the universe. It was such a big idea I knew I needed my smart friends on Facebook to help us out.  At first, she was not interested.  "No one cares what kids think," she said.  My reply? "No way! That is not true. I have tons of friends who really value what kids have to say and who are interested in what kids think.  Why don't you tell me your idea again and we'll see what they have to say?"  She finally agreed.  This is what she dictated and what I posted on my personal Facebook page:

Here’s how I came up with this theory. I saw the globe and was wanting to go on vacation and then I thought they should have a map of the universe but then I thought it would be impossible. But then this theory came to me. Since the universe doesn't have an end it must be a sphere and that is why no one has ever reached the end of it.

You know a tube of toothpaste with toothpaste in it? Well, it might be like the universe with all the planets in it and the universe might be shaped like a sphere and part of something bigger. Just like the tube of toothpaste is part of our bathroom which is part of our house which is part of our town which is part of our city which is part of our country which is part of our world which is part of outer space. So, those images get bigger and bigger just like the universe is in something bigger.

Seventy (70!) comments and twelve hours later, many of my wonderful math, science and artist friends had shared their thoughts, conceptualizations, and experience with us.  Frankly, it was mostly over my head, but I did start to understand 4D geometry a little better.  As for my daughter, she noticed that there can be different well-conceived theories out there that do not necessarily agree with one another.  Also, that adults don't always know everything, but if you want to know more about something you keep asking questions.  And that some of the reputable theories out there on the universe match her own thinking and visualizing.  The fact she accepted the uncertainty of it all was a HUGE leap forward.  But that's not all...

She had been working on and off for a few days to fully understand her own ideas about the shape and boundaries of our universe.  Unfortunately, a lot of the physics concepts offered in the FB comments were hard to conceptualize at the elementary level.  We were at the library a day or two later and she wanted to do some research - she found a book about the universe we hadn't read before and we started to read it.  It was at that point I had a thought: thinking about the structure of the universe is also thinking about infinity.

We've done some reading and thinking about infinity in the past few years, on and off but, until now it's been sort of a fuzzy concept. Here is one of her musings from this winter titled "What infinity means to me" written by her little dolly. "It is sewing that has never been done. It is also cloth that can never be done. And a mountain that reaches on forever."

At that moment in the library I remembered the TED-Ed video called "How Big is Infinity?" and thought it might help her in her quest to clarify her questions and theories.  The video utilizes set theory to help visualize the idea that "there are an infinite number of infinities of different sizes."  In the end, I think that being able to visualize infinity via numbers set in a really fabulous animation helped her settle into her own thinking about infinite universes (the core of her personal theory).  See what you think:


After she watched it she said, "I want to solve one of those unsolved math problems!  But I'll work on my theories [about the universe(s)] first."

During all this Maria from Moebius Noodles asked Isobel if she would allow her thoughts to be published on the Moebius Noodles blog.  My kid had to think about that for a little while.  After all, these were her ideas in question. But, I reminded her about our conversations over the preceding few days that many good ideas stand on the shoulders of the work someone else has done.  That's how good ideas are born -- from the seeds of past ideas and discoveries. "Not only that," I said, "but Maria is really interested in what kids think especially about math and science. She wants more kids to be able to share their interesting ideas."  So, my kid consented, and even drew a picture of her idea.  Her post is up now over at Moebius Noodles.

And, today, I noticed the shift.  Subtle, but in a 'who is this child?' kind of way. All of a sudden, out of the blue, my daughter, the one who coined the term "math mommy" spoken in a derisive tone, says: "You know, I think math is actually pretty interesting."  Not only that, today the math conversations seemed to flow much more freely:

Me: "Oh look, the Venus fly trap has caught some bugs!"
Her: "I know.  I've seen it multiple times."

Her, looking at the pile of library books I brought home: "At dinner will you read me The Cat in Numberland?  It's such a puzzling book." (And we read it all the way through together tonight.  We've read it a couple times before -- it's amazing how much more we both understood now, after going through first and second grade math together.)

Her: "The picture of the [hexagonal] chip on the package is bigger than the chip in the bag."
Me: "How much bigger?"
Her: "Oh, about two times bigger."
(She even helped me as I worked to figure it out for myself on graph paper. More on this in a future post.)

I know this is a long post, but it's been a pretty remarkable week where my kid had an idea and the adults took it and ran with it.  That's got to be super empowering if you're newly eight and realize your thoughts and ideas have weight within the larger world. I thank all those who participated in this pretty astounding adventure with us.

She's not the only kid to have had these kinds of thoughts and ideas, but right now what I'm celebrating is just how great my friends were to have taken her ideas seriously which, in turn, showed her just how relevant she is to the larger picture of life.  If you're a parent I think you'll understand just how grateful I am.

p.s. If you want to comment just be aware they don't seem to always show up in this template, so try either a) adding a '?' at the end of the address and hitting return to reload, or b) just clicking out and trying again.  Sorry for the hassle -- some weird Blogger bug I think.  

"What Infinity Means to Me"

From the other room came:

"C'mon Amelia, time to write!"
"No, you're right handed!"
"Don't scribble!"

Curious, I checked it out.  "What's going on in here?" I inquired.

To which my seven year old replied, "Amelia [the doll she sewed] is writing an essay called 'what infinity means to me.'"

"Oh cool, what did she write?"

"It is sewing that has never been done [finished].  It is also a cloth that can never be done [finished].  And a mountain that reaches on forever."

"Wow, that's cool!  But, I'm curious why Amelia is thinking about infinity on a Saturday morning..."

"Well, she was in math class and this was her assignment."

Now there's a math class I'd love to attend!  Lucky dolly. 

Small Moments of Infinity

[Edit: This post was originally titled Vignettes. About two seconds after I posted it, I came up with a title I liked better.]

#1

On Monday I read The Cat in Numberland by Ivar Ekeland with my daughter at the library (thanks to a reminder about it from Sue VanHattum at Math Mama Writes).  We read it last summer when she was a new six and the kid was really excited to see it again.  She's at a better math comprehension level for it now and we consume it whole in a quiet, sunny corner of the children's section.  This time around we notice the title page with the cat looking at itself looking at the title page looking at itself looking at the title page...recursion!

Anyhow, the hotel owner is griping about Zero and how there's no room for Zero at the Hotel Infinity.  And, to top things off, the owner doesn't think it's a good idea for Zero to room with Number One at the Hotel Infinity because how will he tell them apart from Number 10? 

Luckily the numbers are personified in the illustrations.  Unfortunately the hotel owner doesn't seem to notice this.  My kid does, though.  She says [referring to the illustration of 1 and 0 next to each other, compared to the number 10]: "There are two heads [one for 1 and one for 0].  Ten only has one head." (The head on 10 was on the numeral one.)  Not that this has any direct math 'value' per se, but it was still interesting to me that she noticed this.

#2
I forget exactly which part of the book we were in at the time, but at some point the cat, who is always thinking and "trying to figure things out", wonders why, if the hotel is full now, there are always more rooms available. (Or something like that.)  The kid notices the infinity sign in the illustration (figure 8 on its side) and she starts tracing it in the air saying to herself: "It goes on, and on, and on, and on..."

(Here's an interesting review of the Cat in Numberland by James Propp which includes a fascinating discussion of math pedagogy in relation to this story.)

#3Still at the library, the kid is chatting up the librarians about the resident frog 'Henri'.  She asks how old the frog and it turns out she came to the library as a tadpole, in March a couple years ago.  The kids says, unprompted:

"Well, human months are actually frog years.  There are twelve months in every year, and twelve plus twelve is twenty-four, plus two more months, so that makes Henri twenty-six years old!"  [Never mind it's July, not May...the shocking thing for me is that we were so busy playing UNO and Shut the Box all last year that I completely neglected calendars.  Turns out she's figured it out by osmosis.]

#4
This morning, a summer virus with a fever (after a winter of no illness) has the kid on the sofa listening to the Charlotte's Web audio book.  At some point even that isn't enough to distract her from her discomfort so I head to the couch to entertain her.  She asks me to draw pictures for her (and graciously tells me I can use any of her sketch books that I want!).  I'm really not inspired to do draw but I remember something I saw recently on the Math Munch blog -- Vi Hart doodling musically/visual frieze patterns.  I've always been a doodler, so I decide to experiment with my own frieze patterns.  I'm not sure if they're exactly what they're supposed to be but they're fun to make.  I erase a few, especially the ones that look like they have eyes.  Too much for a kid with a fever!  The kid is absolutely thrilled with the ones we kept, especially the last two. 

I think the frieze patterns/designs have something to do with infinity, and so I end this post now where I began it, discussing recursion and infinity.  I love it when that happens!