Marx Brothers Math: Transformation & Reflection

I'll wager that each one of us looks into a mirror at least once a day. Surprisingly, what we see in the mirror is up for some debate; at least that's been my experience when talking with fourth graders about the subject of reflection.  The result of these conversations is that I now firmly believe that when we use movement to explore the concepts of transformation and reflection, we gain a truly three-dimensional understanding of the subject.  Here's a little peek into how it all goes down:

Me: "What do you see when you look in the mirror?"
4th Grader: "Myself."
Me: "But is it really you? There's only one of you!  There's no one else like you in all the world.  You are an original!"
Different 4th grader: "It's your reflection!"

Later, after my 'magic wand of transformation' has turned the entire class into multiple reflections of me and they've had a chance to experience what it's like to exist on the other side of the mirror, I ask:

Me: "How many of you think it would it be fair to say that your reflection is doing the same thing as you?"
Half the class raises their hands.
Me: "Or, is your reflection doing the opposite of you?"
One third of the class raises their hands.
Me: "Or, how many of you think it might be both, the same and the opposite?"
One or two hands shoot up, other hands raise and lower tentatively.

We work through answering this question in class using our creative dance work.  In lieu of this experience here is a video clip for you from the Marx Brothers movie 'Duck Soup' (below).  I find this video to be simultaneously fun, highly entertaining, and instructive about the process of reflection.  Remember that transformation is essentially about change, and I assert that movement is a particularly effective way to make the process of change visible.

A few things to consider before watching the video, below:

Most of the time we are looking into a mirror straight-on. We brush our teeth, wash our faces, or comb our hair, all while looking at our faces and the fronts of our bodies. In this orientation is easy to think that the reflection is doing the same thing as us.

But remember, the mirror can reflect all sides of our bodies.  As you watch this video you will see Groucho and Harpo directly facing the "mirror" but also walking along the length of the mirror (shoulders to the mirror line) and turning toward and away from the mirror. There's even a fun bit where their bottoms are closer to the mirror than their heads!

In Math in Your Feet, children reflect their dance patterns by deciding who will dance the original pattern and who will reflect that pattern; the reflection changes the original pattern in small but very important ways.  Based on the narrative arc in this particular video, Groucho is the homeowner (original) and Harpo an interloper (reflection).  As you watch, ask yourself:

When is the reflection doing the same thing as the original?

When is the reflection doing the opposite of the original?

I'll give you a couple examples to get you started. When Groucho first sees his 'reflection' in the 'mirror' they both move in toward the mirror and then away from the mirror. In this case they are doing the same thing. Then, still facing each other, Groucho's right hand goes to his chin, but it is his reflection's left hand that goes up. Both hands go up to the chins, but they are using opposite hands.

One more example: At 0:35 Groucho turns away from the mirror over his right shoulder, for a total distance of 180°. Harpo also turns 180°, but over his left shoulder.

How many examples of same and opposite can you find? Can you find any mistakes? I had a hard time tracking if they were using opposite rights and lefts in their footwork, for example. Have fun and don't forget to try out some of the activities listed below when you're done watching!



How'd you do? Ready for a little application of the concepts?

Try this at home:
Put a line of tape on the floor. This is your mirror, otherwise known as a line of reflection.
Decide who will be the original and who will be the reflection.
To start, the reflection has to be the same distance from the mirror line as the original.
Move slowly at first so the reflection has a better chance of accuracy.
Most important: don't forget to experiment with having different sides of your body be 'reflected' in the mirror.

Extra challenge:
Make up a short piece of choreography with a variety of moves and levels (high, medium and low).  In Math in Your Feet, the foot based patterns are units of four steady beats.  See if you can make a four- or eight-beat combination of moves using your whole body. 
Both people practice doing this choreography congruently (everything the same).
Then, do the choreography with the line between you. The original needs to move slowly while the reflection figures out what parts of the choreography needs to change (hint: everything is the same except the reflection uses opposite rights and lefts).
When you're well-practiced and have it a tempo that both people can do comfortably, show off your work!

Extra, extra challenge:
Perform your choreography with your partner first congruently (everything the same) and then reflected (opposite rights and lefts).
If you want a triple challenge, change roles and have the other person become the reflection.

Friday Fun! Video of Some Awesome Dancing

Melody Cameron is one of my favorite dancers, and a former teacher of mine. She is from Cape Breton Island, Nova Scotia, Canada and has her own particular, fluid, musical style of Cape Breton step dance.

Here she is improvising to a piece of 'mouth music' sung in Scots Gaelic by Joy Dunlop.

Enjoy!

"Children Think and Learn Through Their Bodies"

Needless to say, as someone who harnesses movement as a teaching and learning tool, I am fascinated every time cognitive  science reveals another link between the body and learning, what is often called 'embodied cognition'.

I'm about to embark on a reading of Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being, by George Lakoff and Rafael Nunez (thanks to a tip by Michael Paul Goldenberg.)  This is a book co-written by a linguist and a psychologist hoping to launch the discipline they would call "the cognitive science of mathematics." 

In a similar line of inquiry, the researchers who recently  published the article Children's Gestures and the Embodied Knowledge of Geometry concluded that "children think and learn through their bodies."

Here's another recent article on a similar topic: Thinking with your hands: A link between gesturing and intelligence:

"My colleagues at Germany’s Humboldt-Universität zu Berlin , Potsdam University and I have discovered differences between people who gesture frequently and those who gesture rarely. Our study shows that gesturing may be a function of, and may even contribute to, brain development... [Emphasis mine]

"We do not know yet whether gesturing facilitates the development of fluid intelligence or whether it is a by-product. But we do know that children who are asked to gesture in certain ways while learning new tasks learn better than children who are asked not to gesture. Considering that gesturing benefits children while learning, it is possible that gesturing plays a role in the development of fluid intelligence, perhaps by simulating action. If this proves to be true, children might be able to literally give themselves a hand in their own development by gesturing more."

...and thoughts about this article, from the Eide Neurolearning Blog:

"It's interesting to think that teaching children to problem solve certain types of problems should involve strategies that take into account that fact that one is trying to train the imagery of the students. Just verbally saying back the steps of a problem or even watching an explanation won't internalize the imagery. To really 'get' certain problems, we have to enter into the simulation and perceive the question and solution in a bodily way." [Emphasis mine.]

From the Edutopia blog of neuroscientist and teacher Dr. Judy Wills:

Students need to be explicitly taught and given opportunities to practice using executive functions to organize, prioritize, compare, contrast, connect to prior knowledge, give new examples of a concept, participate in open-ended discussions, synthesize new learning into concise summaries, and symbolize new learning into new mental constructs, such as through the arts or writing across the curriculum. [Emphasis mine -- one of the reasons 'the arts' are helpful in this context is that most artistic activities employ the body during the art making process.]

My only question is why, with research and thinking like this, many/most schools still insist that keeping kids at their desks, and lengthening the school day behind said desks, is the best way to insure student learning.  I know all schools don't think this.  I know a lot of really smart folks out there are connecting the dots, but our policies don't reflect the very real evidence that our bodies must be engaged and involved in the learning process. 

Kitty Census: Vet Edition

We have a LOT of kitties in our house.  And they're all real, too.  Most are stuffed, some are ceramic or plastic.  Others include a rare bookmark breed and one in a frame.  We also have an amazing example of the exotic puzzle breed, one who is an imaginary friend, an actual breathing, chipmunk-catching cat and, finally, one who is a special kind of human cat with white fur and a brown spot on her head.

When faced with multitudes like this, a mama's mind turns to math: Just how many kitties do we have and can we remember all of their names?  How many different types of kitties are living here?  How old are they?  Who is related to whom?

This morning the girl wanted to play vet.  I wanted to revisit the Kitty Census we started back in the summer.  As with many things, we compromised. 

ALL the cats went to the vet. 

I was the intake coordinator.  Each cat was registered by name, age, type/description, and malady.  Some of the issues included: "a little sad", "lying around, droopy whiskers", "smelly", "bedraggled", "music slows down at end" (that was the music box cat), and "hurt paw".  In the end, twenty eight showed up for well visits and eleven had some serious issues.  After a triage that took about an hour, we attended to the sickest first. 


This is the bunch that were there for their well-checkups.



We will revisit this data later.  For today we were concerned with 'sick' and 'well'.

Aventurine, of the rare 'puzzle breed', was assessed at intake as "needs fresh air and run around time without breaking."

First, the sick kitties, pictured above, all had a thorough examination, including temperature taking and whisker inspection.  Of the group, the Webkins family all had strep, two were afflicted by a 'fever' of undetermined origin, and two needed immediate whisker surgery.  Two others, originally thought to be sick ('lying around'), were determined to simply be acting like cats.

In the end, I think we simply set the stage for more questions about our brood.  I had her chart out the number of sick kitties compared to the well.  She was curious about how many kitties there were all together, and initiated a count starting at 28 (well) and counting up the sick column to find the sum total.

We'll revisit the intake records and mine it for more data in the near future.  How many family groups do we have?  How many different breeds and how many in each category?  What else do we need to find out about our household of kitties?

Until then, one thing's for certain: We've got a whole lotta' kitty loving going on 'round here.

"Mama!  Lucy's purring!"

One Awesome List

The participants of the Math in Your Feet teacher workshop at Clowes Memorial Hall (Butler University) last Thursday night created this list at the start of our three hours together:


Smiley faces indicate that point is no longer an issue.  A check mark means that this
will always be an issue, but not insurmountable. 



























This was the most comprehensive list thus far compiled in one of my teacher workshops.  By the end of the workshop, it was collectively agreed that most of the items on the list were no longer viewed as challenges.  Here's why:

• Large class size:  Taping down fifteen pairs of boxes onto the floor takes some spatial problem solving in limited space, but it is doable.  Students do all their work in the little boxes, there is no need to move around.  So, if you have the space for the boxes, you've got space for this program.
• Room size/obstacles: This one got a check mark because there are always going to be issues with room size and obstacles, but they are not insurmountable.  One idea that teachers have had is to have kids push their chairs under their desks and tape the squares directly behind their chairs.
• Behavior management/self control:  The rules are clearly laid out -- stay in your square and do whatever you like, except for 360 degree turns on one beat.  Because kids are focused on self-initiated work and collaborate with partners, there is always something for them to do, including simply sitting down and taking a break.  If things do start going off track, then it's 'group practice time,' time for teams to demo their work, time to add some new math, or even just time to stop.
• The teachers were initially worried about how to keep order and whether adding movement would lead to chaos.  They collectively agreed this would not be a problem using the room layout and my suggested formula of 3-5 minutes of moving, creative work and/or practice, followed by a few minutes of new information or observation, followed by a little more movement, etc.
• Overloaded kids: This is a really active, often noisy program, but there are regular times where we stop and quiet our bodies.  During the workshop I modeled a number of ways for kids to move and maintain self-regulation at the same time; some of this I already knew I was doing, others strategies were pointed out by the participants.  I purposefully use my 'count down to silence' to bring us back to a group focus.  It ends with smooth quiet movements and voices, by which time we are all on the same page and ready to focus on new information.  There are also times I communicate without talking (hand gestures, pointing to words on a poster, etc.)  Someone pointed out that everything in my class is really predictable, including the flow of events -- stand/sit, move/quiet, etc. as well as my expectations for staying focused on creative work.  This is an on-task program with a creative process broken down into manageable bits, one bit at a time. 
• They were initially worried that not every child would want to participate -- shy kids, kids who are normally disinclined to participate, kids who don't like to move, etc.  There are always slow starters, ones who need to observe before jumping in.  As long as they don't get in others' way, that's totally okay.  There are also always a few teams of kids who never volunteer to show their work.  I tell those kids privately that it's okay if they don't want to come up now, but by the end of the week they will be expected to show what they've been working on.  Final presentations are never an issue on the last day because of this.  A participant noticed that 'everything is a positive' in this program meaning no matter a person's skill level we can always find success.  This aids participation because as long as they've moved forward in some way from the first day, mathematically, choreographically or socially, it is time to celebrate!
• Another big worry was about the attitudes of school administrators: "Why are the students out of their seats?!?"  By the end of this workshop the teachers said that all you need to do is show the list of vocabulary and concepts used in this program to those who might doubt the value of such an approach to illustrate the relevant learning happening while the kids are moving. 

The teachers in this particular workshop had between eight and 37 years of full-time teaching experience.  What I love is that, based on evaluations, they all found this three-hour workshop to be a positive learning experience and that the ideas presented would be useful in their teaching.