Thinking in Tens

Saturdays we walk to the farmers market.  It's two miles from our house.  We visit our favorite booths, visit with our favorite market friends and, every summer, make new friends too.  And, we go to see the kittens the animal shelter brings every week.

The kitties (and the assorted teenagers who hang out with them) are all housed inside the lobby of City Hall.  Today, the two grey kittens were darling and the young woman who my almost eight-year-old has dubbed "the dramatic teenager' was teaching my kid how to sing.

I found myself with some (incredibly rare) time to think my own thoughts.  My phone got boring quite quickly, but I noticed some art on the walls so I wandered over to take a look.  Turns out they were prints of rubbings made from sewer covers, one of my very favorite places to find fascinating visual math.

The first picture has a fun math-y overlay in the reflection...see it?

So much to look at and take in.

I went and found a comfy bench and pulled out my notebook and a not-very-sharp pencil.  From across the lobby I could see my favorite image, the one with squares in a circle.  Satisfied with sitting I squinted across the space and tried to figure out how many squares made the inner circle.

And then I started doodling.

At first I tried to freehand some squares in a circle.  I thought about angles.  I thought about how I didn't want there to be eight, because that's too easy.  So, I picked ten.  I thought about what it meant to freehand a circle.  I thought about how I liked that using an un-sharp pencil made me really think about precision.  I thought about how I was getting better at spacing ten dots around each circle.  I asked myself, how many different ways can I connect these dots to make something interesting?

On the walk home, things literally came full circle when we saw this:

I love Saturdays.

Baby's First Scratch Project

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

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

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!

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!.