The Map is Not the Territory

Baby's First Scratch Project

2013-05-14T20:48:00.001-04:00

Well, okay, she's almost eight, but it's still a huge milestone.

Today I introduced my daughter to Scratch, a visual programming environment from MIT. If you've been following my blog for a while, you may be a little surprised at this. I am generally an advocate for hands-on math making and somewhat of a skeptic when it comes to kids learning from computers, especially those in the pre-school and primary grades. We need to involve the whole body in learning -- children are already being required to focus primarily on the 2D visual field on top of having limited opportunities for movement throughout the day. They do not need more time sitting and looking.

That being said, it's been obvious that my girl was ready for some new challenges. And, reading has finally become easy. Words are chunked, weird English language pronunciation and spelling rules are generalized in new contexts, and willingness abounds when there is reason to read. It's clear she was ready for something completely new and different.

Also, I've become a huge fan of Seymour Papert and his work introducing children to computer programming and making math, although I have been skeptical about the computer part until just this week. His book The Children's Machine is truly the best book of educational/learning theory I have ever read, specifically because Papert himself is such an honest learner and an incredibly astute observer of others' learning. I could keep gushing, but that will be for another time. In general, though, I am completely smitten with his view of learning as a series of personally relevant connections and his specific descriptions of a variety of children learning Logo, including some who seemed just like mine.

This morning I got some time by myself to poke around the online version of Scratch 2.0. I knew I was hopeless, but somehow managed to grasp the outlines of how it works. I wasn't sure if this would be something she would be interested in, but I somehow managed to get her to take a look. Because it was my idea and I really thought, after reading Papert's accounts of Logo in the classroom, that it needed to be all her from the start, I put her in the driver's seat, literally. I sat her down in the chair, gave her the mouse, and talked her through the tools. We tried some stuff out, looked at some example projects and, after about 30 minutes, took a break.

At this point I was pretty positive about it all. She seemed to like a lot about Scratch, specifically the movement/animation, the potential to make her own music, the cat sprite, a chance to play others' games, and the design/paint program.

In the afternoon she wanted to try again. Since it was my work time and her quiet time the stage was set for her to "have" to work independently. In the past, learning something new like this would require more attention than I am willing and able to give during this time. So, imagine my surprise when all I heard for 30 minutes was silence, peppered by a few puzzled-sounding exclamations. And then I heard:

"Mama, come up and see what I did!"

She had programmed her 'sprite' (a cat) to walk forward 10 and then meow. "Believe it or not," she said happily, "THAT took a lot of work."

She continued to experiment for a total of about an hour and then it was time to meet some friends at the park.

When we came home and the well-past-dinner-time blood sugar crash was appeased, she got back on for another 45 minutes. She was in some kind of zone and it was clear she had outpaced me in one day. I did sit beside her this time, while she worked, but it was only to ask her how or why she was doing something. I watched as she moved through the various script categories and made the idea in her head come to life.

Here's what she said in the description of her finished project, which can be found here: Cat Dance

"This was my first day with Scratch. I made it by experimenting at least five times and then just getting it. It took a couple hours and some hard work"

The best thing is that this is real making to her and she is truly proud of herself! Computer games have always bored her, but this is real making, real inquiry, and she needed no convincing of that. She closed the computer at 6:00pm, satisfied with her work but full of ideas for tomorrow.

"Dear Malke..." 4th Graders' Letters of Learning

2013-05-12T12:26:00.000-04:00

I had an incredible week of dance making and math making with 160 fourth graders. Yes, that's five classes of 32 students. Every day for five days. And, yes, I was tired, but it was totally worth it.

It was worth it even before I got a packet of incredible letters from one of the classes, but what I found written there showed me just how impactful this week really was.

For just a little context, this was the ideal Math in Your Feet residency. The teachers were all on board and supportive during the dance classes, which makes a huge difference in students' learning. And, they also made time to have the kids work in their residency journals, with special attention to the daily reflection prompts and word studies which also makes a huge difference. The classes were filled with enthusiasm for making and learning. By the end of the five days it seemed that almost every student had moved forward in their understanding of and skills in both the dance and math.

There was one class, though, that seemed to struggle more than the rest. Their attention would wander and, when I talked, they seemed to need lots of time to process my words. It took me until the final day to feel like I was making a connection with them. So, it really was success when, on that final day, almost all the students in that class were able to perform their final original 8-beat pattern.

But when I read the 32 beautifully hand-written letters from the students in this class I knew it was more than success, I knew it was an out and out victory. I mean, just listen to their reflections! They are filled with descriptive details of personally relevant learning and understanding of program topics.

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“I really want to thank you for helping us with dance and math. I really enjoyed when you danced for us it was awesome. It surprised me when you taught us about reflection. I would never have thought about you doing that. I learned that you can make math fun while dancing.”

“I like (sic) all the things you taught us when we were in there, but what I like best was that you were allways (sic) excited with what we had did in our patterns. I am happy we learnd (sic) this and thank you.”

“Thanks for every day leting (sic) all of our bodies stretch out every day at 9:15…I loved how on the last day you left the tape all mest (sic) up and you said you can do your dance step without the tape. I’m still kind of confused with how we did A + B together, but it was still fun because you were there to help us.” [I love this comment about getting to move/stretch. This came up in verbal reflection in a different class too.]

“I like how you teach everybody you meet that you can learn to clog and learn math at the same time.” [Well, not everyone....]

“Thank you for coming in and teaching us clogging, patterns and tap dancing. I learned that a pattern is a rhythm or beat that repeats. I also learned congruent means all the same. Also reflection means the same but oppisite (sic) rights and lefts. I enjoyed and was surprised we got to make and perform our own Pattern A and B. My partner and I are still struggling to combine and reflect our pattern. Thank you!” [Kids are often surprised that they can make a dance step.]

“I was surprised because I didn’t know how much fun math and dancing together was. Thank you for helping me realize that. I realy (sic) enjoyed Math in Your Feet.”

“You have taught me and my class so well. You taught me about movement and direction. I enjoyed when we got to make our own dance move.”

“Thank you so much for teaching me about percussive dancing and math. It helped me on my math test and I got to have fun too. The best thing about Math in Your Feet was sharing my dance with my classmates. I had a lot of fun.” [Just for the record, this is the first time a kid has mentioned a test in this kind of reflection.]

“Thank you for the math, dance and patterns. I really learned a lot. What I learned was that your dance moves has to be all the same. I also learned how to combine my dance moves together, although it was hard but I got it. I had a great time with you and I’m pretty sure the rest of the class did to (sic). [The idea of 'patterns' is introduced and carried on from the very first day.]

“I really enjoyed the part where you got to find a partner and make up a 4 step dance. I also enjoyed the warm-ups when we got in the room. I really enjoyed learning and dancing with you. You have taught me things I have never knew (sic) about dancing.” [Kids often mention liking our warm ups!]

“Thank you for teaching my class some more about math and angles with degrees. I really injoyed (sic) you dancing for us and I liked how you put music on and you were singing the directions [to the warm ups]. I was surprised how fun and easy it was to dance and learn angles at the same time.”

“You taught me a lot of things like patterns new math vocabulary words that I didn't know and I am really really greatful (sic) for that you don’t even know how much I needed thows (sic) lessons.”

“I don’t like dancing but I really like it this time.”

“The games we played were really fun. I learned all my degrees and angles because of you. You helped me so much by helping me with my pattern A and B. You are really paticent (sic).” [We play some games I developed to help train our eyes to watch the moving patterns and discern whether both partners are dancing congruently or with a reflection.]

“What surprised me was that I can do a lot of dance steps with my feet. I also learned that patterns can be different.” [To me, this is a huge revelation. Patterns in elementary math are usually of the linear, single attribute variety: red, blue, blue, red, blue, blue, for example. Our dance patterns combine a number of attributes on each beat and change from beat to beat.]

“The thing I liked was when we all got to do the two games. My partner and I got the hang of combination with pattern A and pattern B. Thank you for teaching me about the turns. The turns were fun and hard at first. When you keep practicing you could get the hang of it.”

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I loved that the program was hard for them and, at the same time, a challenge that they wanted to meet. Most of all, I loved getting their letters and being able to hear so clearly what was important to them about this experience. As a visiting artist, here one week, gone the next, there isn't always a chance to get this kind of feedback. And for that, I am am completely grateful to their teacher. Thank you Mrs. Trent!!

Her Own Math, Not Mine

2013-05-09T13:33:00.000-04:00

Here are a few examples of how the math is going these days:

Scene 1: The grocery store

Seven-year-old is pushing cart around the store, narrating as she goes: "Go forward, now one quarter turn to the right, now go forward, parallel park. Okay, now turn half way around, go straight, one quarter turn..."

She sounds like what I imagine kids are doing when they programmed their Logo turtles in Seymour Paper's classrooms. We've never discussed quarter turns or half turns but there they were, helping her guide our cart.

Scene 2: Rest room at City Hall

Her: Mama! Look at the math on the wall!"

Me: What kind of math do you see?

Her: Look at the designs...let's figure out the perimeter!

Me: What kind of designs do you see?

Her: [silence, moving away from the scene]

Me: When I look at it I see a large square with green tiles in each corner and one in the center.

Her: Why do you always have to take pictures?

Scene 3: The Park

My kid is playing with a younger kid. Both are on a 1970's era climbing structure. The other kid's mom calls to the younger child to be careful and climb down. My daughter replies, "Don't worry, it's one hundred percent safe!"

We read a little about probability in G is for Googol which is where (I think) she first heard about this idea. She's also been reading the daily weather reports in the newspaper which are full of both percentages and probability. It's clear to me she's playing with these two ideas and trying to figure out how they work.

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I am always a little tickled when I overhear or observe my daughter applying or identifying math in new ways. But over the last year I have come to wonder why she so clearly wants math to be all her own, separate from me. After thinking about this on and off for the last year (over which time she has really come to see herself as "good" at math) I think it is partly that she is such a fiercely independent learner. But I also think there is something more than personality at play here.

Based on my own math learning experiences these last few years, I can tell you that learning math is personal. I'm reading Seymour Papert's The Children's Machine and he is brilliant at specifically and concretely illustrating how real learning is a series of personally relevant connections. I think his is a theory that can be applied to many subjects, but it totally makes sense within the context of learning math. I've also read in different places that we all have what I would call differently constructed math schemas -- we all see math differently. The challenge of the math teacher is to teach from where the student is rather than require the student to assimilate the teacher's mind map of whatever math topic is being explored.

So, what I've ended up doing is creating situations for her to explore math within an actively hands-on, visual, and often narrative-based context. This approach started out as my way of dealing with a learner who was resistant to instruction, but quickly became a wonderful opportunity for me to re-conceive what math is, as well as where and how it can be learned. Basically, I had to rearrange my concept of math to fit my particular learner.

In our first grade year the learning happened during conversations about the math we saw walking around town (which I started calling "sidewalk math"), reading living math books, playing lots of math-y games, and strategically placing math manipulatives around the house.  (I'm still sort of in awe at the independent work that went into this shape study using tangram pieces when she was six.)

In her second grade year we've done more math sitting down at a table using more recognizeable math manipulatives, but with the same approach as last year; our math has been hands-on, very visual, with a lot of room for personal aesthetic and narrative contexts.

I made a conscious choice to pause our math progression this year in about mid-March. I knew we could have kept going, but my instinct (and my work schedule) told me she'd be fine. And I was right.

What I've observed in my daughter over the last seven weeks or so is that within this 'void' of math lessons she has begun processing her learning by applying and using the math we've done together since mid-August.  There's been a veritable flood of daily self-initiated math activity, thinking and conversation. The three examples above are just a fraction of how she has been playing with the math she knows (or is trying to figure out) by applying and using it in a variety of different settings.

I did have some thoughts in March about seeing how far we could get into third grade math but now I'm glad I didn't push it. It's been more than worth it to make the space and time for assimilation, that special kind of deep learning that happens unconsciously, below-the-surface. You have to be patient for this kind of 'proof'' to bubble to the surface, but if you keep your eyes open, kids will show you every day what they know. Even though we could have gone further in math this year, it's clear that the math we did encounter and explore is now truly all her own.  And, I think this is the best possible outcome for my particular second grader.

My TEDxBloomington Talk: Jump Into Math!

2013-05-06T08:19:00.003-04:00

Okay, so they say you're your own worst critic, yes? Well, after watching this video a couple times it's clear to me I said most of what I'd planned to say, which is actually quite a victory. There's no way to practice the special kind of adrenaline surge that you get when, not only are you in front of large audience, but you know that your work will be archived online in perpetuity. No pressure. Really.

The hilarious thing is that my almost eight-year-old daughter, who coined the phrase "math mommy" as spoken in a derisive tone (don't worry, she likes math more than she lets on) was actually quite generous with her critique. She overheard my husband and I talking about my reaction to seeing and hearing myself in this format and she said:

"Are you kidding?!? It came out great. It was actually a little interesting for a change."

Oh my gosh, life as a parent is never dull.

Anyhow, I was thrilled to present at TEDxBloomington. It was an amazing day with an array of talks on many different subjects all connected to the theme Jump IN. This meant there were a lot of doers and solution finders sharing their ideas that day, from a doctor whose efforts have literally saved thousands of AIDS patients in Kenya, to a variety of realistic solutions for re framing our views on mental health issues, to frank and hopeful discussions about race and poverty. I was in fine company and it was an honor to share my work in this way.

One important thing that did not come out in the video editing is a really important slide that correlates with my description of the Jump Patterns tool (starting at about 3:48 in the video). I've put a version of the slide below the video. Speaking of which...enjoy!

More information about the Math in Your Feet program can be found at www.mathinyourfeet.com.

While you're there you might be interested in a video I made this winter about What Success Looks Like in my Teaching.

Or, you might really dig the Math by Design page. It's the place where the Math in Your Feet approach is translated to all the best making supplies: straws, pipe cleaners, paper, glue and tape.

There's also the extension of this blog on Facebook where I share the amazing, cool, creative math that people make, in all different forms.

I'd be pleased to see you on Twitter, too!.

Rich Contexts for Learning Math & Math Art

2013-04-28T11:43:00.002-04:00

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.

Dinnertime Division

2013-04-18T17:36:00.000-04:00

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!

Math: Any Place, Any Time

2013-04-15T20:49:00.000-04:00

I've been away, teaching other kids and their teachers. When I came home I discovered my seven year old had discovered triangular numbers.

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.

Time to Learn

2013-04-13T14:08:00.000-04:00

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.

Found Math: My 1970's Kindergarten Worksheets!

2013-04-08T15:05:00.000-04:00

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!

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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

Sidewalk Math: Circle Fractions, Division and Multiples

2013-04-07T20:05:00.000-04:00

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!

Colorful Math: Area, Multiplication and Square Numbers Edition

2013-04-04T14:50:00.001-04:00

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.

Embracing the Unknown: Adventurous Professional Learning

2013-04-01T17:38:00.001-04:00

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:

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:

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.

A success on so many levels -- well done!

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:

A New Math Song Before Bed (Video)

2013-03-30T08:37:00.000-04:00

So...who likes bedtime? I, for one, am not generally enamored with detours from the normal bedtime routine.

But when my kid said, "Hey Mama! Want to hear this song I made up that helps you with your three times table?!" I decided to give her a little leeway so I could capture the moment. What can I say, I'm a sucker for math!

The song starts: "There were three ice cream trucks at the corner of Circle Drive..."

My favorite part is 'on the corner of Circle Drive' since, of course, circles have no corners! But, I'm pretty sure this was not intentional on her part.

Did see her looking off to her right to silently skip count the answers in her head? We've been doing a lot with conceptualization of multiplication/division (arrays, multiplication towers, factor dominoes, scale, re-imagined factor trees, exploring the concept of units) but almost nothing with memorization.

I'm happy to have this unexpected piece of evidence that she is thinking about and internalizing these concepts. I'm also pleased to report a happy conclusion to bedtime!

Full to the Brim

2013-03-28T15:10:00.000-04:00

I've got so many amazing things happening these days. Not only am I facilitating my daughter's path to math and bringing the Math in Your Feet program to students and teachers, but I've got a lot of other fun projects going on as well, all related to the theme of math and making. Take a look!

My TEDxBloomington Talk: Jump into Math!

It was an amazing process and, finally, an amazing day at TEDxBloomington. (There I am in the green/yellow shirt at the end of the day when all the speakers, crew and organizers made their way onto the stage.)

My talk will be online in the next couple months alongside the talks from Drew Ramsey, MD (one of psychiatry’s leading proponents of dietary change to balance mood, sharpen brain function and improve mental health), Ryan Germick (Google Doodle team lead), Eric Deggans (TV/Media Critic for the Tampa Bay Times and author of Race-Baiter: How the Media Wields Dangerous Words to Divide a Nation), Robert Einterz, MD (whose efforts have literally saved thousands of AIDS inflicted people in Kenya) and many other incredible people with incredible stories. I was in fine company and it was an honor to share my my work in this way.

The Tape Chronicles Project
Oh Happy Day! The Tape Chronicles Project has finally launched! I've had this idea for a couple years and am so excited to see what we can build. Here's the basic description:

Tape! The ultimate open-ended, the world is your oyster, creative, hands-on learning and making supply. Check out the endless ways tape can be employed in the interest of math, art, kinesthetic exploration, invention and education.

Check out the Tape Chronicles page on my website and please consider submitting examples of your own!!

Teaching Artist Tool Shop
As part of my participation in the new collective, Teaching Artist Tool Shop, I created a video about what success looks like in my teaching of dance and math and in a moving classroom. You can watch the video What Success Looks Like in My Teaching on my website. From the feedback I've received, the ideas in this video can be applied across disciplines.

Moebius Noodles
In January I was invited to contribute to the Moebius Noodles blog! I'm so honored! Read my first two contributions Thinking in Threes and Hidden Math: Book Edition and then spend some time over there in math adventure land!

Math in Your Feet Facebook Page
Oh, we're having so much fun over there. It's the place where I share all sorts of cool percussive dance videos, incredible visual math art, links to interesting ideas about teaching and learning math....all of which would never really make a great blog post on their own. We all hope you'll join us!

Math: In, Out, Through, Between and All Around...Everything

2013-03-26T20:48:00.000-04:00

I really do think that the way to make math more enjoyable for everyone is to find a million tiny examples of how math plays into our interests and daily lives. Because then? Then it's OURS. And, as any math teacher will tell you, math isn't really that hard, but you do have to be motivated to take it on.

I get re-motivated every time my daughter and I enter a new cycle of discovery and exploration. I've determined that even though I can't always predict when it will happen I do know for sure that it'll come around again...

The latest cycle started Saturday with a trip to the Lotus Education and Arts Foundation's annual spring outreach program, the Lotus Blossoms Family Day Bazaar.

Cultural traditions from all over the world were represented there and my seven-year-old jumped for joy when she discovered a table with Native American activities, specifically Navaho. She got to make a beaded cord by twisting fiber in a specific way and then finished it with her own beads at home. Her final product is below:

The other part of the booth was learning a basic traditional string figure of a Navaho rug design, which I did while my kid was working on her cord. On Sunday she found the string I had brought home and played around with it. "Hey look! A triangle!"

Lucky for me I had been sitting on a Cat's Cradle book, ready to pull out at just the right moment, and now she knows three basic figures. It reminded me of James Murphy's work; he used Native American string figures (all of which he learned as a child) to teach reluctant high school math students a few decades ago in New York City. His book is on my wish list now and I can't wait to read more about his work, but I did notice that, among other things, string figures require the ability to follow an algorithm/sequence/recipe which was the perfect challenge for my second grader.

More than anything the Native American booth reminded my daughter that she knows a lot about that subject and she decided to create her own Native American museum. She wanted to start right in on a mural but I encouraged her to do a draft first. On Monday she got started on the real thing:

While she was drawing she pointed out the color patterns she was using in the bead and quill work detail on the person's clothing. The ability to create pattern units and be conscious of the patterns she designs as specific entities is something that popped into view just about the time her reading really took off this past December. Based on my work last summer I'm convinced that the development of this kind of "chunking" skill in both math and reading is completely related. I'm even more sure about this after her facility with multiplication and division concepts (understanding units and groups) really zoomed ahead this January and February.

Also on Monday I chose a book to read at lunch which I had found at the local library's book sale for a quarter: Fun With Numbers by Massin. The girl often rolls her eyes when I bring out a book like this outside our normal math time, but the first page had some fun stories about the Mayans and base 20 and she was way more open when I pointed out that the Mayans are native peoples. Then came the Sumerians and base 60 which, as the book says, is how we tell time today.

"Go get me the clock!" the girl commanded, "so I can count like a Sumerian!" She was eager to show, yet again, that waiting until the learner is ready makes learning easy -- last week she learned how to tell analog time in one day.

Culture, math, art, math, stories about math history, culture, math. And that is our story of how math is in and out and all around us for the moment. I'm excited to see how it all connects when it comes around again!