Bright, Brave, Open Minds: A Problem Solving Kaleidoscope | An Open Online Course for Parents and Teachers!

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.

About the Course:

Why: Preserve children’s divergent thinking. Develop their critical thinking and problem solving skills. Contribute to making a book about young problem solving.

How: Provide a variety of insight problems to young children. Introduce complex, open-ended, and ill-defined problems as a way to teach problem solving skills. Recognize the feeling of being stuck as a necessary step in problem solving. Let children face deep, multi-dimensional problems. Meet other adventurous parents and educators. Contribute to research in math and problem-solving education.

Who: The course organizers are Julia Brodsky, former NASA astronaut instructor and math and science teacher, and Dr. Maria Droujkova, math education consultant. The course participants are families, adventurous teachers, math clubs, playgroups, and other leaders of groups doing problem solving with young kids.

What: In the first week of the course, we will discuss settings and practices for introducing problem solving to young children. You will look at topics from Julia’s math circle, and discuss how to teach them. In the second week, you will gather your kids and their friends in a casual math circle, try out the topics, and then answer a few questions about your experience.

When: Sign up via the Moebius Noodles website by December 2nd. The main course activities will happen December 2nd through 16th. Expect to spend several hours a week doing course tasks.

Where: Main course activities will happen at our online ask and tell hub. The page you are on now will be updated with major links and news. Organizers will send you summary emails. We will have two live meetings online, at the beginning and at the end of the course, and post their recordings to this page.

Growing Triangles, Growing Skills

"You know what I noticed, Malke?" says one of the kids during this week's math groups in my daughter's 3rd/4th grade class, "Everything you've done with us is about triangles."

You know what I've noticed?  I've noticed that making math with the kids in my daughter's 3rd/4th grade classroom once a week is giving me empathy for learning how to teach something brand new.  I'm thinking specifically of the teachers who attend my Math in Your Feet workshops to learn how the program works -- it's not lost on me how much newness there is to manage even in simply getting kids up to learn and think with their bodies, not to mention learning math in a completely new way.

The newness for me is the fact that I am not teaching dance. And, even though I am focusing mostly on having kids explore numbers and number patterns visually, which seems easy enough, the newness is in orchestrating the experience so that we are not just noticing visual patterns but that we also move those observations forward with analyzing the number patterns we see. It's a fun and engrossing challenge, but I'm glad I have four groups to work with because it gives me multiple opportunities to smooth out how I present and sequence the lesson. 

Today we investigated how triangles grow. We got out the pennies and started building.  "You know what I see?" said one girl, "Each new triangle has the old one inside it."

Then we started recording our observations on dot paper, one triangle at a time, including how many pennies it took to make each triangle.

Every minute or two I'd check in with them and get the consensus about the number of pennies in each triangle.  Once most kids got to the fifth triangle we put down our pencils and pennies and started looking for patterns.

During construction/hands-on time, many kids noticed while they worked that each new triangle was built by adding on a new row, and that the new row was one more penny than the row before.  The picture above shows quite clearly the pattern we found in the growth between each triangle and that it confirmed our observations about the growing rows.   

At this point I stopped all penny action and asked them to take "what you know about how they grow" and skip ahead to the 10th, 11th or 12th triangle using only the numbers to help them calculate the answer. Some kids did this easily.  Whether it was easy or less easy, most kids found it an enjoyable challenge, which I was very happy to see.

Other kids seemed less interested in solving the final challenge of skipping ahead and instead enjoyed finding patterns within the triangles they had recorded or by playing with the pennies themselves.  I consider this all good grist for the mill and great thinking for many reasons.  

If we think of learning and understanding as something that builds, one experience at a time, then I'm happy that kids are engaged in the activity in some way. If we consider that children are investigating their own ideas based in some way on the investigation I presented, then I'm all for it -- we want students to ask new questions and find new patterns.  If we always push toward a certain kind of answer we might miss out on really noticing thinking like this:

Math groups with Malke happen on Friday math game day.  One of the teachers remarked that he thought kids might not want to leave their games to come to math group. The reality has been, though, that when their group is called, they all troop willingly into my little room, often a little curious. This, coupled with their intent concentration and thoughtful work, shows me that though I may be new to teaching sit-down math (!) I am orchestrating investigations that are engaging to these eight- and nine-year old learners and flexible enough to challenge a range of math abilities. Their skills grow, my skills grow. We all enjoy math. Win-win-WIN.

When is a Line not a Line?

"Mama, is a curve a line?"

"What do you think?"

"I don't knowwwww...."

"Oh, come on, you asked the question, I bet you have some thoughts about it."

Showing me her Etch-a-Sketch and turning the knobs: "Okay, this is a line [drawing a straight line horizontally to the left] and this is a line [drawing a line vertically upward] but this is not a line [squiggling the line back and forth]."

"Why isn't that back and forth drawing a line?"

"Because it has to go in one direction. This curve is a line, but not when it goes back and forth..."

I left it there because the kid was sick, it was time to start resting and I know answers often rely on more questions about and interactions with the idea at hand. I'm sure this'll come up again at some point in the future. 

Approximation of original by the mother.  As users of Etch-A-Sketches
will understand, the original got destroyed with an inadvertent shake.

Word-Mind, Body-Mind: Toward a More Balanced View of Learning and Knowing

I've just started a book project.  What's it about?  Well, for now, I think it's about digging into the whys and hows of learning math and dance at the same time. This may change, but that's where I'm at right now.

To organize and clarify my thoughts I've been doing quite a bit of background reading, having e-mail/online conversations with various wonderful people, and drafting some preliminary chapters in the blissfully serene Silent Reading Room at our local public library.  Tons of natural light, total silence except for the hum of the HVAC and the occasional cough.

One day last week I sat down to read and felt that old feeling coming on that I used to get in college. It signifies I am completely saturated and overwhelmed with thinking, reading and writing. It makes me a bit crazy, honestly. To combat it, I've learned to skim text and only read closely when I need specific information. Even better, go for a long, long walk with far away vistas.

I call that part of my brain my 'word-mind' and it is very clearly located in my head. 

Using my word-mind is a completely different experience from thinking with my body-mind.  I know, because after college I started dancing and was introduced to another part of the thinking equation -- the body thinks too, a phenomenon that is defined and described by studies in embodied cognition (a branch of cognitive science).

The body thinks, too.  This was truly a revelation to me.

Ultimately, there were years (and years) where I was required to use only my word-mind to learn. And then there were bunches of years (as an adult) where I learned and expressed myself solely through my body mind (dance and music performance, never wanted to get on the mic to talk). My body-mind thinks differently than my word-mind, but what I have come to realize is that, in addition to needing to find a balance between the two in my daily life, as a learner I learn best when I am using both at the same time. 

..........................

I was in Minnesota this summer, sharing my work with a group of teachers and teaching artists. That's when I met Christopher Danielson in person. Yes, an educator of math educators came to my workshop. I was a wee bit nervous but I needn't have worried.  Even though in my workshops we make dance and math for a large proportion of the time, Christopher was game to dance and dance he did.  He also took copious notes by hand at various intervals.  I was again a wee bit worried, but it turned out our 90 minutes of dance and math making had gotten him thinking.  In a good way. By engaging with the work of Math in Your Feet from the inside of the experience instead of simply watching, he generated many new questions that moved his thinking forward in new ways.

The next day we met for a breakfast conversation during which, in relation to one of the questions he had while dancing, Christopher mentioned a now defunct unit in the Connected Mathematics series that investigates transforming squares in a series of combinations and is related to the thinking in modern algebra.  Later, he sent me the book.  I meant to get started on it but life took over.  And then my book took over.

...........................

This week my word-mind gave out. I was literally a little dizzy from thinking in my head and was completely DONE with words. I do a lot with language these days.  I keep a blog, I'm active on Facebook and Twitter, and I edit a year-round, international online writing project for the Teaching Artist Journal.  So to have reached my limit with words is saying something.

I needed something completely different to do.

I pulled the Connected Mathematics book off my desk, sharpened my pencil and headed for the library.  I made my little square out of nice stiff paper. I labeled the vertices.  I combined series of turns and flips, filled in the table and investigated the patterns I saw.  For a good portion of the time I was completely frustrated and confused -- turns out I was turning clockwise when I should have been turning counter.  Apparently there is some convention among mathematicians that all rotations go counterclockwise.  They must have all been left handed and never learned analog (clockwise) time because this right dominated person kept seeing the "R" for rotate as meaning "turn RIGHT."

At some point, though, I was looking at my hands, turning, flipping, holding corners to orient them as I turned and flipped, and I realized I was in some kind of zone. I was talking to myself in words: specific conscious reminders to turn left when rotating, murmuring less conscious words while filling in the chart, a gentle narration as I talked myself through each new sequence of flips and turns and, eventually, the body and words merged and I was just doing.  I had fallen into a zone of concentration while making math that mirrored what often happens when I am learning something new in dance.

After making numerous combinations of transformations with my hands, I now have a new feel for what a square is and is capable of.  Both the word-mind and the body-mind had a part to play in this new understanding.

I hope it is obvious that this is not an either/or kind of story -- it's about both.  Understanding cognition and thinking and knowing is a vast endeavor and I don't know if we will ever fully understand the complexity of it all.  My focus is on what it means and looks like to learn math with the body as an equal partner and to do this I need to keep the body's way of knowing and thinking firmly in my sights.

Ultimately, what I am wrestling with is this:

We can see the body moving, but most of us have not yet learned how to look at and understand the knowing and learning happening while the body moves.  It's an inside process and not one that a child will necessarily be able to tell you about.  As more experienced learners the onus is on the adults to look for how children show us what they know and think with all their one hundred languages, including the body.

I'll leave you with a powerful paragraph found in the conclusion of the recent study Children's Gestures and the Embodied Knowledge of Geometry (bolding emphasis mine):

"Despite two decades of research on the embodied nature of cognition, constructivist perspectives continue to emphasize abstraction of knowledge from the physical engagement with the world ... In this way, constructivism orients our attention away from the body, concerned as it is with the construction of mental entities and representations...  

Once we acknowledge the body as the seat of knowledge itself rather than as a stepping stone to abstractions, it is possible to organize teaching differently.  In doing so, [this] brings forth the possibility of inquiring into such questions as, 'What might we do to recognize and understand children's knowledge expressed in modes other than speech or writing?'"

One thing you can do is to try learning math and dance at the same time yourself. I'm available to talk about how I can help, if you like.  Feel free to get in touch.

The Butter Knife and the Infinity Knife

Recently, my eight year old spontaneously discovered an activity I've seen in the Moebius Noodles book and yet another version of infinity (read about others here, here, and here).

Scene 1: The Butter Knife

Waiting for the bread to toast. She picks up the butter knife off the counter and places it vertically over a design on the lid to the butter container.

"It turns into an arrow!  What a cool design."

She continues to play around with "cutting" the lid's design in half, one side the real the other the reflection.

Scene 2: The Infinity Knife

The kid calls to me from the other room to tell me she's cutting triangles in half into smaller and smaller pieces, trying to see how many times she can cut them.  When I ask her to show me she picks up another piece of paper and starts again.  She cuts one big triangle in half equally, then one of the halves in half again...

"It has something to do with infinity," she says, as the triangles get smaller and smaller and smaller. "I need something different [than the scissors she's using] -- a tiny knife like scientists have -- to cut infinity stuff like this."

................................

I've been reading books by former colleagues and students of Jean Piaget who have carried his work forward and made it accessible to the rest of us. My take away so far, beyond the idea that all of us construct our knowledge by assimilating new information into what we already know or think we know, is that children think when they have something real, some phenomenon, to think about. Whether it's a butter knife or an infinity knife or whatever else, it's the interaction between the child and the object/idea that inspires the child to think -- what Eleanor Duckworth calls "the having of wonderful ideas." 

"There are two aspects to providing occasions for wonderful ideas," she writes in her book of essays The Having of Wonderful Ideas. "One is being willing to accept children's ideas. The other is providing a setting that suggests wonderful ideas to children --different ideas to different children --as they are caught up in intellectual problems that are real to them." [page 7]  

Whatever else we do as teachers and parents, I think we need to find multiple ways to allow kids the freedom to discover the world on their own terms, in their own ways. At the same time we need to create the time and space in our teacher/parent brains to catch our children and students in the middle of discovering something brand new (to them). Having wonderful ideas, as Duckworth says, is "the essence of intellectual development."  Honestly, being able to observe and even, sometimes, to interact with my kid when she's in the middle of a new thought is pretty much the prize of parenthood -- it is such a gift to see this happen up close.