A gorgeous example of mathematical art.
And a more in depth article from the Smithsonian on his process.
And...quick edit....A couple hours after I posted this I found an interview with Erik Demaine's father Martin. In it he describes their process of making the the art and the math indistinguishable from each other. He says:
"I was interested in the art and he was interested in the mathematics and we started exploring both together using the art as a visual idea for understanding many of the problems we were studying in mathematics. And before we knew it I became hooked on the mathematics and he became hooked on the art and our goal today is to try and combine them so that they're kind of indistinguishable. So you can't tell whether we're doing art or math but they're both there together."
That's what I'm trying to do in Math in Your Feet. If you're interested, here is a video that I made recently describing how I combined percussive dance and elementary mathematics into the same learning and creating process.
Friday, February 1, 2013
Thursday, January 31, 2013
Accumulated Math Moments
Do you ever have days, or even weeks, where things just don't go the way you imagine they should? It's been one of those weeks for us. When things got challenging this week I started to despair about whether we were doing 'enough' school and I'd still be feeling that way if I hadn't remembered to take a look at the photos on my camera roll.
Somehow, along the way, small moments of math started adding up...
First, I happened upon this book at the library over the weekend. G is for Googol by David M. Schwartz is the absolute best math overview book I have read. Ever. I think it's the combination of humor and the way he works in multiple, connected concepts into each entry.
The entry on Moebius strips, for example, included a little lesson on topology and inspired us to try out a few things, with my daughter exclaiming "right now!" while I ran to get the paper, tape and scissors. On the left is the Moebius strip and, moving clockwise, what the strip looks like after being cut down the center, then cut in half again, and then the result of not cutting around the strip down the center and instead cutting one third from the edge.
"R is for Rhombicosidodecahedron" (kid loved learning how to say that) "N is for Nature" (a great re-introduction to Fibonacci numbers) "K is for Konigsberg". We've been reading three or four entries a day at meals. Each one has something interesting to think about or do or introduces a concept with a lot of passion and verve. My recent favorite is the entry under V, which is perhaps the only math book I've found so far to include 'book readers', 't.v. watchers' and 'underwear poem-writers' in the same Venn Diagram. I think Mr. Schwartz really knows the mind of a child.
Then there was a day when we just had to head outside, take a trip to the children's science museum, whatever, just get out of the house. While we were waiting for it to open we hung around outside. "Hey!" I said, "What's the very fastest way you can figure out how many small squares there are?" She wasted no time counting ten across, ten down and skip counting by tens. "100!"
"How many squares make the diagonal?"
This one she initiated all on her own.
While we were at the science museum I found an introductory set of Tantrix for about $6.00. It said for ages 8 and up, but my seven-year-old (who is not a puzzle person, or so I thought) really loved the structured challenge of creating loops of increasing size and turned out to be a whiz! The instructions said the approximate time for solving the 10-piece red loop was 20+ minutes, but she did it in under five. I love the hexagon pieces ("T is for Tessellate" thank you very much!).
Have you ever seen Vi Hart's videos on spirals? I'd seen them before but, on a whim, showed them to the kid who was impressed enough to sit through all three on the subject. Afterwards, she rustled up some permanent markers, drew the spirals first (which inverts Vi's process) and drew in the petals following that pattern. I think it approximated pretty well what Vi Hart was trying to do.
So, for a week where we were at loose ends and lost causes, it sure seems like we actually have something to show for showing up. Thank goodness for my camera.
Somehow, along the way, small moments of math started adding up...
First, I happened upon this book at the library over the weekend. G is for Googol by David M. Schwartz is the absolute best math overview book I have read. Ever. I think it's the combination of humor and the way he works in multiple, connected concepts into each entry.
The entry on Moebius strips, for example, included a little lesson on topology and inspired us to try out a few things, with my daughter exclaiming "right now!" while I ran to get the paper, tape and scissors. On the left is the Moebius strip and, moving clockwise, what the strip looks like after being cut down the center, then cut in half again, and then the result of not cutting around the strip down the center and instead cutting one third from the edge.
"R is for Rhombicosidodecahedron" (kid loved learning how to say that) "N is for Nature" (a great re-introduction to Fibonacci numbers) "K is for Konigsberg". We've been reading three or four entries a day at meals. Each one has something interesting to think about or do or introduces a concept with a lot of passion and verve. My recent favorite is the entry under V, which is perhaps the only math book I've found so far to include 'book readers', 't.v. watchers' and 'underwear poem-writers' in the same Venn Diagram. I think Mr. Schwartz really knows the mind of a child.
Then there was a day when we just had to head outside, take a trip to the children's science museum, whatever, just get out of the house. While we were waiting for it to open we hung around outside. "Hey!" I said, "What's the very fastest way you can figure out how many small squares there are?" She wasted no time counting ten across, ten down and skip counting by tens. "100!"
"How many squares make the diagonal?"
This one she initiated all on her own.
While we were at the science museum I found an introductory set of Tantrix for about $6.00. It said for ages 8 and up, but my seven-year-old (who is not a puzzle person, or so I thought) really loved the structured challenge of creating loops of increasing size and turned out to be a whiz! The instructions said the approximate time for solving the 10-piece red loop was 20+ minutes, but she did it in under five. I love the hexagon pieces ("T is for Tessellate" thank you very much!).
Have you ever seen Vi Hart's videos on spirals? I'd seen them before but, on a whim, showed them to the kid who was impressed enough to sit through all three on the subject. Afterwards, she rustled up some permanent markers, drew the spirals first (which inverts Vi's process) and drew in the petals following that pattern. I think it approximated pretty well what Vi Hart was trying to do.
So, for a week where we were at loose ends and lost causes, it sure seems like we actually have something to show for showing up. Thank goodness for my camera.
Sunday, January 27, 2013
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.
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....
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.
_____________________________________
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..
____________________
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.
_____________________________________
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..
Wednesday, January 23, 2013
My Favorite Math Games & Interactives: Online & Off
A friend of mine has a couple kids in kindergarten right now. He asked if I had any recommendations for "online computer games for arithmetic, patterns, creative math exploration." Why, yes, I think I do! Following is an annotated list of some of my favorite interactive online habitats for a wide range of ages, including parents.
However, I think I'll start with a list of activities that are not computer based. I'll be the first to admit that I'm a little biased toward hands-on, not-virtual learning, especially for the primary grades; children's bodies really need to be involved in the learning process, especially math.
Also, although my kid has some computer skills, she is basically raising herself as a luddite, which is fine with me for now. She's never been drawn to the computer, even when I specifically show her something I think is fun or cool. So, many of the games and interactives listed in the online category, below, are places that I go to get activity ideas to adapt for her off-line lessons or to grow my own math understanding or just for inspiration.
OFFLINE FUN
I don't know where we'd be this year without the endless Shut the Box tournaments of last year. I attribute at least half her mental addition skills to rolling two dice (which can make up to 12) and finding different ways to make the rolled number.
I attribute the other half of my daughter's mental addition ability to UNO. It's a classic game that we played speed style all last year. And, at the end, the winner got to add up all the points she won from hermother opponent who always remembered to chant the UNO Point Tallying Mantra: "Find your tens!"
Peggy Kaye's book Games for Math has LOTS of easy and inexpensive ideas including Nine Men's Morris which is sort of like a more complex and moveable tic tac toe (strategy, logic).
Variations on "sort and count the change jar" along with "save and spend" from about the age of five means that now, at age seven, the child is a whiz at money math (also known as adding and subtracting 10s, 100s and 1000s).
And, of course, The Game That is Worth 1000 Worksheets from the Let's Play Math Blog -- for the littles all the way through high school. This fall we used a version that was a combination of sums and differences.
ONLINE GAMES & INTERACTIVES (NOT JUST NUMBERS)
Some of these are for you, lots for them, and some for the whole family (like Scale of the Universe 2 and the triangle interactive!!)
National Library of Virtual Manipulatives: Truly a class act and a challenge for all ages.
Math Pickle: This link takes you to a large library of interesting videos and ideas organized by grade level.
Kaleidograph: I first found this toy and played with it online, and then I bought the real thing for Christmas. Both the online and hands-on versions are great but, no surprise, I find more meaning in the actual toy.
From my childhood memory bank, a classic Sesame Street clip with a lovely song: Two Little Girls and Their Little Doll House. My adult eyes immediately saw one-to-one correspondence but my little girl heart remembered something different.
Scale of the Universe 2: I have no words for this interactive graphic other than just GO. You can turn off the music if you want.
I've included this link to Planet Seed/Math Puzzles for Kids because I don't want to forget about it. It was on a long list of bookmarks already. Puzzle topics include combinatorics, algebraic thinking, topology, probability, etc.
Games from Freudenthal Institute look great, and they even have a star interactive!
Make your own isometric dot graph paper here. These and the other dot graph paper you can print out are fun to leave around the house so kids can make designs whenever the mood strikes.
Math cats! Need I say more?
I personally love Symmetry Artist from Math is Fun and my daughter actually liked playing with it for a while. It's a great way to really experience the differences between line and rotation symmetries and you can print out the designs you make at the end.
And, you really cannot miss this triangle interactive from the Triangulation blog. Truly. Run, don't walk.
To close it out, here is a lovely and thought provoking classic Sesame Street animation with music by Phillip Glass called The Geometry of Circles.
However, I think I'll start with a list of activities that are not computer based. I'll be the first to admit that I'm a little biased toward hands-on, not-virtual learning, especially for the primary grades; children's bodies really need to be involved in the learning process, especially math.
Also, although my kid has some computer skills, she is basically raising herself as a luddite, which is fine with me for now. She's never been drawn to the computer, even when I specifically show her something I think is fun or cool. So, many of the games and interactives listed in the online category, below, are places that I go to get activity ideas to adapt for her off-line lessons or to grow my own math understanding or just for inspiration.
OFFLINE FUN
I don't know where we'd be this year without the endless Shut the Box tournaments of last year. I attribute at least half her mental addition skills to rolling two dice (which can make up to 12) and finding different ways to make the rolled number.
I attribute the other half of my daughter's mental addition ability to UNO. It's a classic game that we played speed style all last year. And, at the end, the winner got to add up all the points she won from her
Peggy Kaye's book Games for Math has LOTS of easy and inexpensive ideas including Nine Men's Morris which is sort of like a more complex and moveable tic tac toe (strategy, logic).
Variations on "sort and count the change jar" along with "save and spend" from about the age of five means that now, at age seven, the child is a whiz at money math (also known as adding and subtracting 10s, 100s and 1000s).
And, of course, The Game That is Worth 1000 Worksheets from the Let's Play Math Blog -- for the littles all the way through high school. This fall we used a version that was a combination of sums and differences.
ONLINE GAMES & INTERACTIVES (NOT JUST NUMBERS)
Some of these are for you, lots for them, and some for the whole family (like Scale of the Universe 2 and the triangle interactive!!)
National Library of Virtual Manipulatives: Truly a class act and a challenge for all ages.
Math Pickle: This link takes you to a large library of interesting videos and ideas organized by grade level.
Kaleidograph: I first found this toy and played with it online, and then I bought the real thing for Christmas. Both the online and hands-on versions are great but, no surprise, I find more meaning in the actual toy.
From my childhood memory bank, a classic Sesame Street clip with a lovely song: Two Little Girls and Their Little Doll House. My adult eyes immediately saw one-to-one correspondence but my little girl heart remembered something different.
Scale of the Universe 2: I have no words for this interactive graphic other than just GO. You can turn off the music if you want.
I've included this link to Planet Seed/Math Puzzles for Kids because I don't want to forget about it. It was on a long list of bookmarks already. Puzzle topics include combinatorics, algebraic thinking, topology, probability, etc.
Games from Freudenthal Institute look great, and they even have a star interactive!
Make your own isometric dot graph paper here. These and the other dot graph paper you can print out are fun to leave around the house so kids can make designs whenever the mood strikes.
Math cats! Need I say more?
I personally love Symmetry Artist from Math is Fun and my daughter actually liked playing with it for a while. It's a great way to really experience the differences between line and rotation symmetries and you can print out the designs you make at the end.
And, you really cannot miss this triangle interactive from the Triangulation blog. Truly. Run, don't walk.
To close it out, here is a lovely and thought provoking classic Sesame Street animation with music by Phillip Glass called The Geometry of Circles.
Tuesday, January 22, 2013
A New Venture for a New Year!
My first post Hidden Math: Book Edition is up now! It includes two of my favorite children's books that have way more math in them than I first realized. There's also a new post just today by Yelena on a similar theme. Her post is called Math Goggles #1: Math-y Librarian and it highlights all the cool math she and her son found inside a beautiful library in her city.
Truly, I don't know where I'd be without Moebius Noodles (and other great resources like Let's Play Math and Living Math). In the last year or so I would never have grown my math eyes or moving past a life-long math insecurity that started back in kindergarten with a bumblebee matching worksheet. It was so utterly alarming and confusing to my five-year-old self that I can still remember that moment like it was yesterday.
Today, however, relearning math slowly but surely along with my seven-year-old daughter has softened the edges of that memory. And, even better, I know that my strategy of focusing on first-order experiences, hands-on making, math games, living math books and sidewalk math adventures is bearing fruit. Just today my seven year old (who is currently building competency with number multiples) stated confidently that not only does she "likes math" but that she is "good at it, too!
Yay for having fun with math and thanks to Maria and Yelena at Moebius Noodles for inviting me to join the adventure!
Subscribe to:
Posts (Atom)
LinkWithin
