Bright, brave, open minds: A problem solving kaleidoscope led by Julia Brodsky and Maria Droujkova is a two-week long open online course in problem solving for parents and teachers of 8 and 9 year old children.

Initially, I wanted to figure out how to use polyominoes in a game-like 3rd/4th grade setting.  Got some help from my Twitter feed and the brainstorming brought me to this:


I'll write more when I have a chance to do the full activity sequence with kids (including building the 1-through 4-ominoes with unifix cubes) but, as always, one thing leads to another and I started wondering if I could make something similar using pattern blocks.

Turns out I can. And, it turns out it was an enjoyable challenge for my daughter, who is somewhat of a skeptic when it comes to trying out my new ideas.  Here's what I did:

I made an isometric dot grid.

I created some closed figures, some very open and balanced, some less regular. The goal of these puzzles is to fill in the entire shape using as few pieces as possible. My daughter did it by herself, but I could see it being an enjoyable, conversation-inducing project for a team of two.



The first puzzle resulted in a very interesting sequence.  I showed her how the hexagon, trapezoid, rhombus and triangle all fit onto the isomorphic dot grid.  I indicated the goal of using as few pieces as possible.  Here was her first attempt.  She used sixteen pieces.



To initiate a second try I wondered if she could fill in the space using less pieces.  As she looked at her first solution I wondered aloud if there were any places where a larger shape could take the place of the smaller ones.  This time she reduced the number of shapes to eleven!


Again I wondered if she could use less shapes, and on her third try she brought it down to nine!


I wondered how she would do with a less balanced, less regular shape. Here's the first try, eleven pieces.  In the process she observed that using a trapezoid and a triangle together covers the same area as two rhombi and uses the same number of pieces.


 Still motivated. Third try, eleven pieces. "Wait, I think I see something!"


In the end, this may be a system with a short shelf life -- once you 'get it' you get it, but I love that it's very open ended and allows for multiple right answers (11 pieces, three ways, for example).  It's also a really friendly way to encourage perseverance in problem solving, spark discussions about composition and decomposition of units, and, heavy on my mind these days, a great way to ease into fractions.

I know this looks somewhat like a traditional pattern block activity but hopefully you can see that it has a different focus. At the very least it's telling that my kid (who has never been into 'solving' the already designed pattern block images) was really into this puzzle format. The only structure here is the outline of the puzzle shape. No need to give any hints by subdividing it in any way.  That would ruin the fun!

If you make puzzles of your own, I'd love to see them.  I'm going to keep working on a few more outlines and if I come up with some good ones, I'll be sure to share them.

I've just finished a five-day residency up in Indianapolis as the second part of Young Audiences' Signature Core Service pilot.  Most of the time it seems the dance and rhythm aspect of this program brings kids closer to the page by motivating them to write, sometimes for the very first time, multiple meaningful sentences about their experiences creating original dance steps.  The program can also help kids think of math as a friend, not a foe, also for the very first time.

The kids I just worked with, however, were already very comfortable in the symbolic realm of mathematics and also possessed strong verbal-linguistic skills and positive problem solving attitudes.  They were, however, still fourth graders in their bodies and feet (and I mean that in the best possible way).  In addition, they were good at following directions, and really dug into the activities I outlined but...

....upon reflection, I realize now they might have been bored. 

They might have been bored because they really knew their math, but at the same time they couldn't really do more than fourth graders normally can do with their bodies.  I think that if I had had more time with them, or could do it over, I would have found a way to get their brains more engaged while their bodies worked at age-level.

I would have challenged them to really play with their patterns.  For example, instead of just making a Pattern A and B and combining them into a third, 8-beat pattern (which is a great amount of play in itself), I would ask them how many different ways they could recombine the four beats that made up each pattern, and then encourage them to ask more questions along that line of inquiry: What if we reflected the pattern to itself, beats 1, 2, 3, 4, 4, 3, 2, 1?  What if we took our two favorite beats from the two patterns and traded them with someone else?  "What if...?" is one of my favorite questions, after all, because you never know where it will take you.

Despite these musings of mine, I think things went well.   Here are their perceptions and reflections of the first three days of the residency...

Day One: "You have a friend who was absent today and missed the first day of Math in Your Feet.  Tell that person about all the different ways you made patterns with your feet."

First, we had vocab words, such as hop, slide, clog, etc. then, we made and [sic] pizza, and used all of the dances.  For example, we did sausage as slide.  As it says, Math in Your Feet!

Today Ms. Malke came in and we learned all about dance steps.  This is called clogging.  Clogging and step-dancing are actually cousins!  When we were dancing sometimes we put our feet to a 90°angle and other times she told us to turn our feet to an acute angle.  We also used a lot of math vocabulary words.  We all had a great day full of dancing and math.

Today for an hour we did math in your feet.  We did clogging, and the chug.  We made a pizza in our mind and did different dances to represent what we put on the pizza.

We made patterns by dancing.  When we danced we also made rhythims [sic].  We also pretended to make pizza.

Day Two: "What did you have to do to dance congruently with your partner?  Using complete sentences, name at least three things that had to be the same.  What kind of challenges did you and your partner face to make your dancing congruent?"

To dance congruently with my partner we had to count out the four beats.  The timing, dance steps, and the beats all had to be the same.  One challenge we had was timing at first, but then we practiced and practiced and we finally got it at the same time.

My partner and I had to take one step with our right ft. 1st, then left, turn 180° right, then 180° left.  The kind of challenges we had to face are having to make sure we both knew what order our steps were in then we counted 1, 2, 3 to know when to start.

Do the same dance you have to keep a steady beat, go at the speed of the slower person, and practice.

My partner and myself really didn't have any problems, but if I had to pick three I would pick that 1. was that I could not get the steps right.  2. We could not stay together.  3. We could not figur [sic] out how we wanted to do are [sic] steps like for example speed fast, mideam [sic] or slow.

Me and my partner faced lots of challenges.  We did a 270° turn wich [sic] was hard to be congruent while doing.  The speed, movement, and moves had to be congruent.

To dance congruently, I had to say something to signal us to start dancing.  The 2 diagonal splits & 1 side split had to be the same so it would look good.

Day Three: "Write a friendly letter to your partner.  First, tell this person about what you had to do to reflect your dance pattern across the line of reflection.  Then, tell your partner what your favorite MIYF dance move is and why!"

I had to turn right instead of left to mirror your moves.  I love our Pattern A!  It's casual, but interesting.  I can't wait to see Pattern B.

To reflect our dance pattern across the line of reflection you had to do it in the opposite sides and I did it originaly [sic].  My favorite MIYF move is a slide, with our feet together, and back just because it's fun to do!

In Math in Your Feet we did a line of reflection and how we did it was we had to pick a person to be the original and the reflector.  The reflector had to do oppsite [sic] lefts and rights.  My favorite dance move was the cross because it is fun and it makes me feel like a real dancer!

I had to instead of going right diagonal at first I had to go left to right Diagonal.   I don't have a favorite dance move because I don't like to dance.

I love having you as a partner because you don't get mad when I mess up and you agree with anything we do.  I could go on with the tanes [tons] of things good about you but I'll stop there.

First, we had to pretend we were looking in a mirror. We had to  switch our rights with lefts and lefts with rights.  My favorite dance step is turn because I like getting dizzy.

I love fourth graders!

Not quite congruent.  One partner has landed while the other is still up in the air.

This is only the third day and we have a couple more to go.  Things do get more interesting and more challenging when we start combining individual patterns into larger ones (i.e. start the second pattern where you ended the first, not at your original starting point and then try the reverse) and also when we transform the patterns using turn symmetry which seems rather straightforward in a static representation on paper, but is absolutely spectacular when you see it in motion. 

I'll keep you posted!

I am always on the lookout for connections and commonalities between mathematics and the arts, dance in particular.  In my on-going survey of math/movement/dance approaches out there, most of what I've found seems focused on numeracy and procedural concerns; these are not necessarily wasted efforts, but why stop there?  There is also a lot of illustration of math concepts using dance, but the two subjects stay on parallel tracks with no real connections between the two.  Overall, approaches like these never get to the deeper commonalities between math and dance because they are often more focused on memorization or performance of prescribed movements than understanding and application.

The main focus of Math in Your Feet is a thorough exploration of patterns.  Since my dance form is based in making foot-based dance patterns, it is a natural and meaningful connection, especially if the goal is to develop mathematical thinking and not simply working to improve skills or memorization of procedures or formulas.  After patterns, another commonality between math and dance is certain habits of mind and flexibility of spirit in the process of finding solutions.  With that in mind, I submit to you two diagrams, one from a math point of view, the other from an arts perspective. 

When I look at this diagram from the blog Keeping Mathematics Simple each one of the intersecting circles could describe what I do as a dancer including, I think, even the theoretical.    


From the blog Keeping Mathematics Simple

Coming from the other perspective, when I look at the next diagram, from the article Defining Arts Integration, I think about how mathematicians go about posing and solving problems and definitely see connections to this arts-based diagram of the creative process.  The creative process is a hallmark of artistic activity and, I'll argue, a hallmark of mathematical thinking as well, just not labeled as such. 

From the article Defining Arts Integration

Be you mather or dancer, what say ye?  (Alternative and/or opposing viewpoints welcome!)

I started this post a few weeks ago.  I guess I needed some time, space and perspective to finally finish it.  Here you go: 

I just finished a big week with four classes of fourth graders.  Their teachers told me they were a tough group on the whole, but of course it's only through experience that I figured out what that meant, exactly. 

I've had tough groups before, but these kids confounded me, actually.  There were bright spots in each class, and there was also some good thinking and creating, when it happened.  But, on the whole, they could barely stay in their squares.  They would forget any verbal instructions I gave.  I had to stay within two feet proximity to get any attention at all.  Most of them could not consistently find the center of their dance spaces.  They took every gentle reminder as a power struggle.  Transitions such as moving from sitting to standing or dancing to not dancing with control was a herculean effort for them.  

Don't get me wrong.  These were not 'bad' kids, just kids with a lot of challenges.  I've spent a lot of time over the years figuring out how to help kids control their bodies while moving and then again when the moving is done.  I've also spent many years helping kids become more aware of where they are in space and where in space they are intending to go, with good results.  But this particular week, none of my strategies seemed to work.

I often compare my week-long residency to the first week of school.  By the end of our time together I've figured out exactly what each particular class needs to be successful, but by then it's time for me to go on to the next school, or the next group of kids.  Not really understanding who I'm teaching and what they need as individuals and as a class is my least favorite part of the job.  If I had the week to do over, this is what I'd do:

• 30 minute class length, or less.  I am contracted to implement my program within certain parameters including the amount of time each class has with me (usually 60 minutes at a time).  Classroom teachers who implement Math in Your Feet in their own classrooms have the flexibility to work on it in ten minute chunks or whatever length they choose.  My 4th graders this week had to hold it together for a whole hour with me each day, which was a loosing proposition. 
• Take it out of the gym and find a more enclosed space.  The gym for this particular week was not a bad gym in terms of sound issues, but it was a gym none-the-less.  I never use the whole gym and always define the limits of our dance space (with tape!), but if we could have put walls around us to enclose the space somehow, it might have helped the kids feel safer and may have helped the focus issues. 
• Reduce transitions.  It's a hard reality, but we have to get up and move, and then we have to sit down and listen.  We do both in short bursts, which these kids needed, so I'm not sure exactly how I would have done this.
• Dance for them more to keep them focused on the reason for why we were together. 
• Teach one two-person team at a time. The only times they were really quiet, focused and, well, not argumentative and actually helpful, was when a team was up in front of the class showing work in progress or demonstrating a math concept through their dancing.  If I could do this over I'd take the plunge and just teach one team at a time while the others watched.  I think we would have gotten a lot more acomplished this way.
• Acknowledge the issue of 'first reaction' sooner and with all the classes.  I think I got this idea from a parenting book I read in the the last year.  Essentially, each one of us reacts in our own way to new situations.  We have either a negative first reaction or a positive first reaction.  Knowing how we operate in this regard can help us overcome the challenges each present.  For those with a negative first reaction (hold back, regard with suspicion, decline to participate, or exhibit unhelpful behavior) it just means making sure that they know that I understand they're not into it at the moment and that's okay with me.  At that point I usually outline what my minimum expectations are for participation-- stand when the rest of the class is standing and sit when the rest of the class is sitting, and that's it.  I've had success in the past acknowledging the reality that kids may not love what we're doing as much as me, but I still expect them to give it a shot.  By the time I realized what I was working with, it was too late in the week for this strategy to have much effect.   That usually gets us over the hump, but not this week. 

So, all in all, we did get through the week.  I suppose the bright side of all this is that I'm a little more ready the next time I encounter a group like this.  And, I'm still a big supporter of using dance and movement in integrated ways for all children.  Just because it didn't work for them the way I wanted this time doesn't mean that it can't.  It just needed to be different.  Too bad my time machine's in the shop, or I'd try again.

Like I said at the beginning, there were some bright spots.  This is what one girl wrote in response to a reflection journal prompt:

"Ten years from now I will remember how hard it was to learn and how much fun it was.  The challenges I had were putting pattern A and pattern B together.  The thing that helped me work through them were because I had D. as my partner and he was always hard working and an on task friendly boy. :)  I loved this program."

I am always thinking about better ways to describe what exactly is happening in Math in Your Feet.  It's actually been quite difficult for me to explain because, in the end, the total experience is more than the sum of it's parts.  Think about it -- this program brings together two subjects which communicate, in their own mystifying language, about space, time, and movement.  Teachers who have been through it once often advise first time teachers that they'll "understand it after they're done," which is not ideal.  Luckily, I'm meeting with some teachers tomorrow to plan for an upcoming residency.  While preparing for the meeting I took the opportunity to update my thinking about what is really going on while a bunch of kids jump around in small boxes taped on the floor.  Here's what I came up with:

Specific Learning Areas in Math in Your Feet (Upper Elementary)

There's a video showing up lately in the different places I'm visiting on the Internet.  It's an RSA talk by Sir Ken Robinson, world-renowned education and creativity expert, called Changing Educational Paradigms.  This is surely a subject that's outside the scope of my experience and this blog, but there are certain things he said during the talk that speak to what I think about when I'm working with or creating programming for children.

"Creativity is the process of having original ideas that have value.  Divergent thinking isn’t a synonym, but it’s an essential capacity for creativity.  It’s the ability to see lots of possible answers to a question, lots of possible ways of interpreting a question, to think laterally, to think not just in linear or convergent way.  To see multiple answers not just one."

There is also another interesting RSA talk I recently watched called The Surprising Truth About What Motivates Us.  It's a talk given by Dan Pink and although it is focused on workplace motivation, I think there are some parallels to a school setting. 
My purpose in Math in Your Feet is to create opportunities for children to develop a level of mastery using the language of percussive dance to solve problems in a creative context. This context is naturally open ended and a place where there is more than one right answer, indeed an infinite number of right answers.  My philosophy is to give children limits (defined work space, four and/or eight beats only) and some tools (elements of percussive dance, a clearly defined process) and then let them work it out from there. 

I have spent most of my dance life performing and teaching on a 3'x3' square dance platform.  For perspective's sake I should say that most cloggers, step dancers or tap dancers do not regularly work in such a small space.  For me, working within the confines of my dance board was born out of necessity and eventually influenced the course of my creative life as both an artist and a teacher.

Stephen Nachmanovitch, in his book Free Play: The Power of Improvisation in Life and the Arts, includes a chapter titled 'The Power of Limits.'  This post is the first in a series of articles, inspired by his chapter, exploring how limits not only enhance creative problem solving but are actually a requirement of such a process.  Creativity, and all the intense and surprising things that word implies, requires of us resourcefulness, flexibility, ingenuity, and the necessity to think outside the box. 

In Math in Your Feet (elementary version) kids make up their own percussive dance patterns.  While we're engaged in this process I make the point that 1. both problem solving and the creative process usually don't occur with one action, but with a series of actions and choices that bring you to a solution and 2. sometimes, if things aren't going the way you'd like, you have to revisit a previous step of your process to refine or redirect your ideas. 

One of the other things kids experience in Math in Your Feet is the difference between choreography and improvisation.  Choreography is usually the end result of a lot of improvisation -- trial and error and trial again.  There are a lot of choices and decisions you have to make a long the way.  Sounds a lot like problem solving to me.

So, we are all creative, on a daily basis.  We all have multiple opportunities throughout the day to identify a need, make choices, and strategize about what to do next.  Not every problem needs to big and not every solution needs to be a work of art or cure cancer for us to engage daily in this wonderful process.

As an example of daily creativity, please consider Exhibit A: a list of ingredients for a recipe I used to make called Skillet Black Beans:

1 1/2 tbs olive oil
1 med-large onion
2-3 garlic cloves
1 med green bell pepper
1 16oz can diced tomatoes
2 tsp ground cumin
6 6-inch corn tortillas, cut into narrow strips

This is a great recipe, don't get me wrong, but after eating it four or five times it just wasn't working for me.  I didn't like the potatoes or the bell pepper.  I'm not a fan of chiles.  We can't eat corn in our house right now.  So, I used the original recipe as a starting point.  I experimented and made adjustments.  It's still called Skillet Black Beans, but now it looks like this:

Oil (not measured, just eyeballed)
1 sweet onion
Lots of garlic
Half a small head of cauliflower, cut into small pieces
1 big can of fire roasted crushed tomatoes
2 small cans of black beans
2 tsp ground cumin
3 large rice tortillas, cut into small strips

It's still the same recipe, but different, and it suits my tastes and my needs.  I'll never be considered a gourmet (there are many, many better cooks out there than myself) but when I cook I feel a sense of my own mastery.  My food is the direct result of my choices, for better or worse.  If I don't like what I made, I get to try again to make my dish more the way I'd like it.

I was reading my daughter the book The Trumpet of the Swan by E.B. White this week when I found this passage.  In my mind, it makes a strong statement for accepting the fact that there is usually more than one right answer, and that kids need meaningful, real-life experiences to make them effective and enthusiastic problem solvers.

Just as background, the character Sam is a born naturalist who spends many hours in nature, walking and observing the life around him.

Back in his own room, Sam sat down at his desk...their teacher, Miss Annie Snug, greeted Sam with a question. 

"Sam, if a man can walk three miles in one hour, how many miles can he walk in four hours?"

"It would depend on how tired he got after the first hour," replied Sam.

The other pupils roared.  Miss Snug rapped for order.

"Sam is quite right," she said.  "I never looked at the problem that way before.  I always supposed that man could walk twelve miles in four hours, but Sam may be right: that man may not feel so spunky after the first hour.  He may drag his feet.  He may slow up."

Albert Bigelow raised his hand.  "My father knew a man who tried to walk twelve miles, and he died of heart failure," said Albert.

"Goodness!" said the teacher.  "I suppose that could happen, too."

"Anything can happen in four hours," said Sam.  "A man might develop a blister on his heel.  Or he might find some berries growing along the road and stop to pick them.  That would slow him up even if he wasn't tired of didn't have a blister."

"It would indeed," agreed the teacher.  "Well, children, I think we have all learned a great deal about arithmetic this morning thanks to Sam Beaver..."