I am looking for as many examples as I can find of contextually rich math and math art learning in elementary education. I put the idea out on Twitter, but I'm hoping readers of this blog will be able to contribute any and all ideas you may have.
So far my list includes:
Froebel Kindergarten
Moebius Noodles (check out their brand new book!)
The work of Catherine Twomey Fosnot. Here's a great video where she explains her approach:
What is not yet clear to me is how and where math learning and math art learning overlap. Math Art, to my mind, can include all artistic media including dance, music and visual forms. If you are able to help me with links or thoughts of your own, even including specific examples from your own teaching, I would be most grateful.
This is only the beginning of my inquiry to make sense of how math learning can happen in multiple contexts, including artistic and design settings, with all roads leading to real comprehension and mastery of mathematics heading into middle and high school.
Sunday, April 28, 2013
Thursday, April 18, 2013
Dinnertime Division
Scene: Dinnertime
7yo: I'm going to leave that [her dinner] for a while.
Me: Hey, wait a minute. Let me see how much you have left in your bowl...you haven't had much to eat today, you need to eat at least half of what's left.
7yo: [Sitting back down, pause in conversation] I have 34 noodles left....
Me: How do you know that?
7yo: I counted them. I'm going to eat 17, okay?
Me: Hey! How'd you do that so fast?
7yo: Well, I took half of 30 and half of four and added them together...
Me: You are super awesome!
7yo: Well, I knew that half of 30 was 15 and half of 4 was 2 so it was easy...
__________________
A question, full disclosure, and some context:
At what point did mental division of 2-digit numbers get "easy" for her? I'm afraid I don't have an answer for that but I'm always curious and thrilled when she makes a leap past "hard" to "easy".
We haven't done much what I call "sit-down math" for weeks and weeks now. There. I said it. We'd been going strong with daily math lessons for most of the school year yet somehow it has gone by the wayside the last month or so. But, in its wake...
...I've been very aware of watching math happen throughout the day: her daily tally of savings for an American Girl doll (both how much she has and how much more she needs); her daily interest in telling analog time (and becoming stronger at base 60 as a result, hence the division of 30 tonight); continued interest in investigating and analyzing the weather report in the daily newspaper; her easy and quite enthusiastic use of the tape measure to compare heights and lengths and just about anything that enters her mind. She's also taken to maps and geography and is now paying attention to the scale to get a sense of distance.
We're steady math learners, no bells and whistles and, obviously, not a lot a drill. She's a good math student but not brilliant. And yet, I find the development of these mental math skills fascinating, bordering on magic especially considering how number-phobic I've been all my life. Apparently I was able to swallow my fear and do a pretty good job with her numeracy development (so far), and mine own as well.
What I've done in the last one and a half years or so really feels like some kind of super power, truth be told. I can tell she probably feels the same about being able to compose and decompose numbers. But I know it isn't really magic. There are incredible resources out there for anyone to do what I did. In terms of 1st and 2nd grade numeracy, here are what my resources and basic blocks of activity have looked like:
- Hundreds of Shut the Box games
-Just as many games of UNO (by adding up the loser's points)
- Lots and lots of hands on interactions with money (earning, saving, counting the change jar)
- Connecting geometry to number (and lots of Sidewalk Math -- walks outside to find structure and pattern all around us including triangles and other polygons, arrays, spirals, etc.)
- Having lots of hands-on maniupulatives around (tangrams, pattern blocks, pentominoes, Cuisenaire rods) to just play around with and occasionally use in lessons
- Reading Constance Kamii and making the conscious decision to focus on mental math and not worry about procedures (regrouping, borrowing, carrying the one, whatever)
- Playing games from Let's Play Math and Peggy Kaye's Games for Math
- Reading Living Math Books
- Investigating wholeness, halfness, doubles, even-ness, even and odd numbers
- The first two chapters of Beast Academy 3A (polygons and skip counting)
- Investigating the Sierpinski triangle to get a sense of three-ness and multiples of three
- Keeping the big picture of math in the picture, so to speak, with books like Penrose the Mathematical Cat and G is for Googol.
There's more, I'm sure.
I'm not sure why I'm wrapping up her second grade year in April, but it really feels like my often hands-off approach has created a great base. Best of all, it is very apparent to me is that my daughter is really digging into the math she's learned this year and is now making it her very own. She's using it to answer her own questions, express her own ideas, and find her own way. Joy!
7yo: I'm going to leave that [her dinner] for a while.
Me: Hey, wait a minute. Let me see how much you have left in your bowl...you haven't had much to eat today, you need to eat at least half of what's left.
7yo: [Sitting back down, pause in conversation] I have 34 noodles left....
Me: How do you know that?
7yo: I counted them. I'm going to eat 17, okay?
Me: Hey! How'd you do that so fast?
7yo: Well, I took half of 30 and half of four and added them together...
Me: You are super awesome!
7yo: Well, I knew that half of 30 was 15 and half of 4 was 2 so it was easy...
__________________
A question, full disclosure, and some context:
At what point did mental division of 2-digit numbers get "easy" for her? I'm afraid I don't have an answer for that but I'm always curious and thrilled when she makes a leap past "hard" to "easy".
We haven't done much what I call "sit-down math" for weeks and weeks now. There. I said it. We'd been going strong with daily math lessons for most of the school year yet somehow it has gone by the wayside the last month or so. But, in its wake...
...I've been very aware of watching math happen throughout the day: her daily tally of savings for an American Girl doll (both how much she has and how much more she needs); her daily interest in telling analog time (and becoming stronger at base 60 as a result, hence the division of 30 tonight); continued interest in investigating and analyzing the weather report in the daily newspaper; her easy and quite enthusiastic use of the tape measure to compare heights and lengths and just about anything that enters her mind. She's also taken to maps and geography and is now paying attention to the scale to get a sense of distance.
We're steady math learners, no bells and whistles and, obviously, not a lot a drill. She's a good math student but not brilliant. And yet, I find the development of these mental math skills fascinating, bordering on magic especially considering how number-phobic I've been all my life. Apparently I was able to swallow my fear and do a pretty good job with her numeracy development (so far), and mine own as well.
What I've done in the last one and a half years or so really feels like some kind of super power, truth be told. I can tell she probably feels the same about being able to compose and decompose numbers. But I know it isn't really magic. There are incredible resources out there for anyone to do what I did. In terms of 1st and 2nd grade numeracy, here are what my resources and basic blocks of activity have looked like:
- Hundreds of Shut the Box games
-Just as many games of UNO (by adding up the loser's points)
- Lots and lots of hands on interactions with money (earning, saving, counting the change jar)
- Connecting geometry to number (and lots of Sidewalk Math -- walks outside to find structure and pattern all around us including triangles and other polygons, arrays, spirals, etc.)
- Having lots of hands-on maniupulatives around (tangrams, pattern blocks, pentominoes, Cuisenaire rods) to just play around with and occasionally use in lessons
- Reading Constance Kamii and making the conscious decision to focus on mental math and not worry about procedures (regrouping, borrowing, carrying the one, whatever)
- Playing games from Let's Play Math and Peggy Kaye's Games for Math
- Reading Living Math Books
- Investigating wholeness, halfness, doubles, even-ness, even and odd numbers
- The first two chapters of Beast Academy 3A (polygons and skip counting)
- Investigating the Sierpinski triangle to get a sense of three-ness and multiples of three
- Keeping the big picture of math in the picture, so to speak, with books like Penrose the Mathematical Cat and G is for Googol.
There's more, I'm sure.
I'm not sure why I'm wrapping up her second grade year in April, but it really feels like my often hands-off approach has created a great base. Best of all, it is very apparent to me is that my daughter is really digging into the math she's learned this year and is now making it her very own. She's using it to answer her own questions, express her own ideas, and find her own way. Joy!
Monday, April 15, 2013
Math: Any Place, Any Time
I've been away, teaching other kids and their teachers. When I came home I discovered my seven year old had discovered triangular numbers.
I think, as a kid, my thinking was very similar and I have always wished that someone would have helped me move this kind of thinking and seeing forward into more formal mathematics. That's why its really important to keep an eye open to what our students do in their 'off time'. Math can happen anywhere, any time and when it does happen it's the perfect opening toward more conscious math making.
Not only that, she had created two nearly identical representations (slide symmetry) and ordered the cat chess pieces by size. Not pictured are her collection of old books, displayed in order of publication date, oldest to newest. None of this echoes any of our sit-down math lessons this year so far.
I think, as a kid, my thinking was very similar and I have always wished that someone would have helped me move this kind of thinking and seeing forward into more formal mathematics. That's why its really important to keep an eye open to what our students do in their 'off time'. Math can happen anywhere, any time and when it does happen it's the perfect opening toward more conscious math making.
Saturday, April 13, 2013
Time to Learn
This spring I've been doing a lot of the one-visit, 45-minute version of the Math in Your Feet program. I was a little proud of myself when I figured out, finally, how to work with the time constraint and make this a worthwhile experience. In a single workshop the kids have enough time to learn and practice four, 4-beat percussive dance patterns (instead of choreographing their own, which is at the heart of the full program). They also have some exposure to how those patterns are constructed, and get to explore the idea of sameness/unison/congruence.
The problem is that, although they can reproduce/dance the patterns pretty well considering the short amount of time we have, they don't have enough time and experience to build the vocabulary they need to analyze the patterns. To build familiarity you need a chance to enact your own agency on the process. That is why having them make up their own dance steps is so important. Not only is it one of my favorite things about my art form but, also, by the end of the process they know their pattern really well -- because they made it and because they had to spend a lot of time talking about it while they collaborate with their partner. Doing and talking lead to understanding, not only of the percussive dance genre but of the nature of patterns, how they are built, and how they are different or similar or the same as other patterns in the room.
Being able to discern whether a two-person team is dancing 'the same' or 'almost the same' and give reasons why is dependent on understanding the aesthetic and vocabulary of the Jump Patterns' variables/attributes. On top of all that they need to analyze the four beats/moves in quick succession. There's always a point during the course of a one-time workshop where I am reminded, yet again, about the complexity of this whole endeavor we call 'learning math and dance at the same time'.
I still think the one-shot-deal is better than no Math in Your Feet at all. Almost every workshop this spring has been filled with kids who willingly jump up and fully engage in the entire 45 minutes with smiles and excitement. Wonderfully, even though I generally tell them to pick just one pattern to work with, some kids end up rearranging the patterns or combining them into longer patterns. I love it when this happens. Kids instinctively know what to do with the Jump Patterns tool and it's great they are so inspired to make their own work even within the constraints of 45 minutes.
But, I have to say it, their thinking and analysis are much more sophisticated when they have multiple days to experiment and practice and be in the process; giving them more time gives them a chance to fully understand how those patterns are built. When they have some space and time to tinker, ask questions, have little disagreements, find common ground, be confused, etc. they become better dancers, choreographers and mathematicians.
This kind of learning is sometimes seen as supplemental or enriching to the 'real' learning that happens on paper or at the desk. I respectfully disagree. It may look different from what we know as 'learning in school' but, honestly, there are many ways to learn and even more ways to harness our children's innate humanity and inclinations to explore, create, build, think, engage, and contribute to their world. It's worth the effort, and the time, to figure out how to do this.
Monday, April 8, 2013
Found Math: My 1970's Kindergarten Worksheets!
Here's what I remember about Kindergarten: A sweet boy named David with funny taped glasses, big ears and plaid button down shirts. Round tables. A rug. My teacher. Being reprimanded for walking on the crosswalk lines as we crossed a street instead of inside or between them. (I was trying to follow directions but spatial concepts are still a little fuzzy at that age.) A math worksheet with bumblebees that had to be matched to other things that I can't remember -- I can still clearly recall how utterly frustrated and humiliated I was about my confusion.
So, imagine my very special surprise and pleasure in finding five Kindergarten math worksheets my mother had saved with some of my early artwork. And, yes, the sheets are mimeographed.
The first sheet, titled "Discriminating Difference" shows me not quite getting this concept, at first, but figuring it out by the last two problems.
In the second sheet it's clear I knew how to count. (Too bad they couldn't figure out a way to close that star polygon.)
This sheet is titled "Visual Perception - Pattern Comparison" with a great quote that certainly shows how much things have changed in the last 40 years:
"Kindergartners are not ready to read. They are ready to remember and reproduce visual forms."
These last two are my very favorites because they prove (to me anyways) how my brain has been wired from the start. Ignore the numbers. Instead, focus on the color and patterns I employed and how my young brain saw the world.
I sincerely believe that children show you every day how they know and understand the world and that their 'output' can give you some incredible clues as to their strengths and their growth areas. In this case, I seemed to have been very good at what I would consider math skills -- exploring permutations, design, symmetry, part/whole, and pattern. And, what's more, I seem to have been enjoying myself!
I guess my biggest reaction to all this, especially the last two pictures, is that of recognizing myself in the work. I see my five year old self as having some strengths, but like many other kids, these were not supported or given outlet and guidance past kindergarten or first grade. Given more support, a variety opportunities to explore mathematical ideas in multiple contexts, and a guide to help me name and analyze structural observations I made about my world every day, I probably could have been a great math student. So could a lot of kids.
After finding these sheets it's apparent to me that I've found my way back to math after all these years using the mathematical strength I've had inside me all along. Seriously, I am really touched to have had a chance to visit with my young self. I'm also glad to know that, despite the bumblebee trauma, I seemed to have enjoyed myself during kindergarten math time!
.......................
p.s. Here are a few images of how color and math still go together these days as we create and learn math at our house.
My daughters self-discovered "Map of Angles"
Exploring mathematical stars
Factor trees
Scissor Stories: Tales of Transformation
My kid thinking
Weaving inverses and multiples
Weaving Fibonacci
So, imagine my very special surprise and pleasure in finding five Kindergarten math worksheets my mother had saved with some of my early artwork. And, yes, the sheets are mimeographed.
The first sheet, titled "Discriminating Difference" shows me not quite getting this concept, at first, but figuring it out by the last two problems.
In the second sheet it's clear I knew how to count. (Too bad they couldn't figure out a way to close that star polygon.)
This sheet is titled "Visual Perception - Pattern Comparison" with a great quote that certainly shows how much things have changed in the last 40 years:
"Kindergartners are not ready to read. They are ready to remember and reproduce visual forms."
These last two are my very favorites because they prove (to me anyways) how my brain has been wired from the start. Ignore the numbers. Instead, focus on the color and patterns I employed and how my young brain saw the world.
I sincerely believe that children show you every day how they know and understand the world and that their 'output' can give you some incredible clues as to their strengths and their growth areas. In this case, I seemed to have been very good at what I would consider math skills -- exploring permutations, design, symmetry, part/whole, and pattern. And, what's more, I seem to have been enjoying myself!
I guess my biggest reaction to all this, especially the last two pictures, is that of recognizing myself in the work. I see my five year old self as having some strengths, but like many other kids, these were not supported or given outlet and guidance past kindergarten or first grade. Given more support, a variety opportunities to explore mathematical ideas in multiple contexts, and a guide to help me name and analyze structural observations I made about my world every day, I probably could have been a great math student. So could a lot of kids.
After finding these sheets it's apparent to me that I've found my way back to math after all these years using the mathematical strength I've had inside me all along. Seriously, I am really touched to have had a chance to visit with my young self. I'm also glad to know that, despite the bumblebee trauma, I seemed to have enjoyed myself during kindergarten math time!
.......................
p.s. Here are a few images of how color and math still go together these days as we create and learn math at our house.
My daughters self-discovered "Map of Angles"
Exploring mathematical stars
Factor trees
Scissor Stories: Tales of Transformation
My kid thinking
Weaving inverses and multiples
Weaving Fibonacci
Sunday, April 7, 2013
Sidewalk Math: Circle Fractions, Division and Multiples
Okay, so we've walked past these two covers about a million times but today I saw them in a completely different light.
This winter/spring we've been doing a LOT with number multiples and conceptualizing multiplication and division. Last week my mind moved toward the inevitable: fractions. Although shivers go down my spine every time I think about fractions I'm still resolved to figure it all out for the sake of my seven year old, if not myself. It's been sitting in the back of my mind so I guess that's why this cover caught my eye and brought me to a dead stop.
Can you see it? Fractions and multiples!
And, a little further on, this beauty: an 8-star and some fractions!
What I really want to know is who designs these? I want that job. To see some other cool round things we found on a walk last spring, go read my post Channeling Tana Hoban: Juxtaposition Edition. Now that was one amazing day for circles!
This winter/spring we've been doing a LOT with number multiples and conceptualizing multiplication and division. Last week my mind moved toward the inevitable: fractions. Although shivers go down my spine every time I think about fractions I'm still resolved to figure it all out for the sake of my seven year old, if not myself. It's been sitting in the back of my mind so I guess that's why this cover caught my eye and brought me to a dead stop.
Can you see it? Fractions and multiples!
And, a little further on, this beauty: an 8-star and some fractions!
What I really want to know is who designs these? I want that job. To see some other cool round things we found on a walk last spring, go read my post Channeling Tana Hoban: Juxtaposition Edition. Now that was one amazing day for circles!
Thursday, April 4, 2013
Colorful Math: Area, Multiplication and Square Numbers Edition
Two things our house has a lot of: math and color.
We eat colorful, mathy breakfasts. (Isn't this hexagon egg-citing?)
And then we work on our colorful and original attributes matching game. The attributes include a multitude of choices in the following categories: shape, color and pattern. We ask: How can I make the pair exactly the same? How can I make the next pair different? How are my designs similar to each other?
After a bath and second breakfast, we learn about square units and determine the size and area of each picture. But why stop there when there are colored pencils around? I wanted to move on but quickly concede that not only is color is a great addition for highlighting structure within the areas in question, but basic math gets a whole lot more fun to do!
At which point I think: Wow, I've got some graph paper...how about tomorrow we look into square numbers?
And how do I start that lesson? Pick your seven favorite colors!
We do the first six numbers and the child, unbidden, suddenly looks over her work and says "Wow, they get bigger! Oh, and there's a pattern!" We figure out the specifics and then use that observation to predict the seven square. I know there is lingering confusion about how exactly that number is made but, as you know, tomorrow is another day and I think I have an idea...involving colored pencils, of course.
We eat colorful, mathy breakfasts. (Isn't this hexagon egg-citing?)
And then we work on our colorful and original attributes matching game. The attributes include a multitude of choices in the following categories: shape, color and pattern. We ask: How can I make the pair exactly the same? How can I make the next pair different? How are my designs similar to each other?
After a bath and second breakfast, we learn about square units and determine the size and area of each picture. But why stop there when there are colored pencils around? I wanted to move on but quickly concede that not only is color is a great addition for highlighting structure within the areas in question, but basic math gets a whole lot more fun to do!
At which point I think: Wow, I've got some graph paper...how about tomorrow we look into square numbers?
And how do I start that lesson? Pick your seven favorite colors!
We do the first six numbers and the child, unbidden, suddenly looks over her work and says "Wow, they get bigger! Oh, and there's a pattern!" We figure out the specifics and then use that observation to predict the seven square. I know there is lingering confusion about how exactly that number is made but, as you know, tomorrow is another day and I think I have an idea...involving colored pencils, of course.
Monday, April 1, 2013
Embracing the Unknown: Adventurous Professional Learning
One of the very best things about keeping a blog, and a Facebook page, and a Twitter account is how many thoughtful, smart, interesting and cool classroom teachers and home educators I have had the chance to 'meet'. And then there's all the listening in I get to do, as well. I am completely inspired by the exchange of ideas -- about specific projects, yes, but also about the process of teaching and learning, for both the students and their teachers.
These kinds of exchanges actually make me feel patient, for once, about not knowing everything I think I should know. I have a new respect and understanding of learning as a continuum and as an adventure with no guarantees, some dead ends, and unknown, occasionally inspiring outcomes. This makes me a better, more flexible teacher and more willing and able to try out new ideas, often to great results.
So that is why I was thrilled to hear from Simon Gregg who recently jumped in with both feet, literally, to try something brand new in his 5th grade math classroom over in France.
He had a goal: "We've just been 'doing' symmetry on paper, and I'd like to see how they get on with doing it physically, plus a four beat rhythm." He read up on Math in Your Feet (which provides background and an overview of the program). And he tried it out!
His work with the students was an awesome approximation of the program in the best possible sense! What I mean is, it takes a while to cross the bridge: it took me a while to grow my own 'math eyes' to identify all the math we are doing in the program, and it will take him some time to grow his 'dance eyes'. (It might be the same for you, one way or the other, if you decide to try it yourself. I'm sure it will easier for everyone when I am finally able to put together a curriculum guide and DVD.) But, all in all, Simon and his students made a fine showing and there was a lot of learning for everyone!
I was especially impressed that his students devised their own notation to record their dance steps/patterns.
I was also impressed that, despite the newness of it all, the overall experience mirrored so much of the really important things that happen when I am working with kids. He writes:
If you're interested in finding out more about the teaching and learning within the Math in Your Feet program, here's a video I created as part of the Teaching Artist Tool Shop collective:
These kinds of exchanges actually make me feel patient, for once, about not knowing everything I think I should know. I have a new respect and understanding of learning as a continuum and as an adventure with no guarantees, some dead ends, and unknown, occasionally inspiring outcomes. This makes me a better, more flexible teacher and more willing and able to try out new ideas, often to great results.
So that is why I was thrilled to hear from Simon Gregg who recently jumped in with both feet, literally, to try something brand new in his 5th grade math classroom over in France.
He had a goal: "We've just been 'doing' symmetry on paper, and I'd like to see how they get on with doing it physically, plus a four beat rhythm." He read up on Math in Your Feet (which provides background and an overview of the program). And he tried it out!
His work with the students was an awesome approximation of the program in the best possible sense! What I mean is, it takes a while to cross the bridge: it took me a while to grow my own 'math eyes' to identify all the math we are doing in the program, and it will take him some time to grow his 'dance eyes'. (It might be the same for you, one way or the other, if you decide to try it yourself. I'm sure it will easier for everyone when I am finally able to put together a curriculum guide and DVD.) But, all in all, Simon and his students made a fine showing and there was a lot of learning for everyone!
I was especially impressed that his students devised their own notation to record their dance steps/patterns.
I was also impressed that, despite the newness of it all, the overall experience mirrored so much of the really important things that happen when I am working with kids. He writes:
The idea and inspiration for this activity came from Malke Rosenfield's brilliant "Math in Your Feet" program, though it should be said that ours is a very rough approximation of what is a much more developed and professional program. That said, there were a number of things that were really impressive about what the class did:A success on so many levels -- well done!
- Their total concentration;
- Their amazing and precise cooperation in pairs;
- Their energy;
- Their ability to devise notation to record positions and moves;
- Their use of mathematical vocabulary to describe their patterns.
If you're interested in finding out more about the teaching and learning within the Math in Your Feet program, here's a video I created as part of the Teaching Artist Tool Shop collective: