Friday, June 12, 2015

Socks are Like Pants, Cats are Like Dogs, Really!!

Do you want your children and young students to feel like algebra is beautiful, playful, and intuitive? Come play, solve, talk, and make math with us! Support our book, reserve your copy, and make these math adventures available to children, parents, and teachers all over the world.

I've co-authored the book Socks are Like Pants, Cats are Like Dogs with Gordon Hamilton of Math Pickle. It's filled with a diverse collection of math games, puzzles, and activities exploring the mathematics of choosing, identifying and sorting. Teachers and parents have tested all activities in real classrooms and living rooms. The activities are easy to start and require little preparation.

There's even a pdf of sample activities from the book for you to try!
We’re almost done with our book; all that’s left is a few finishing touches. We’ve estimated the crowdfunding goal for this project to be $4,000. Any amount will help us reach our goal. Please help make this book a reality! Visit the crowdfunding site over at Natural Math for more images, and information about our project!

Thank you for your support!

Saturday, June 6, 2015

Summer Math Photo Challenge [#mathphoto15]

Don't miss the Summer Math Photo Challenge! Week 1 is underway in English, Spanish and French with images shared from around the world. There are are eleven more weeks of fun ahead and I hope you and your friends, kids and/or students will play along this summer! 

Here's how to play: 
Check the weekly challenge information every Monday. 

Keep your eyes open.  

Take pictures of what you find.

Share to Twitter using the #mathphoto15 hashtag. 

Feel free to add your own hashtags as well.

Encourage your friends to play! You can do this by copying/saving the weekly schedule in the language of your choice (below) and share it via social media.

You can also view all the math-y gorgeousness on Twitter or Flickr.

Wednesday, April 1, 2015

Some thoughts on "Hands-On" Math Learning

Last night on Twitter Michael Pershan asked me to weigh in on hands-on math learning. The request stemmed from a conversation/debate about the various merits of different ways to learn math.  

The minute I read the question I knew that my answer was going to be more detailed than a response on Twitter would allow. Here are some of my thoughts on the matter.

1. The discussion reminded me of the "concrete to abstract" conversations which, to me, seem like an especially frustrating example of recursion. They go round and round but we never really get anywhere new.

I think many connect the word "concrete" to Piaget and his discussions about children's thinking moving from the concrete to the abstract. This in turn has led to many assumptions that take the term "concrete" quite literally. But, as Deborah Ball wrote in her article Magical Hopes, 
“Although kinesthetic experience can enhance perception and thinking, understanding does not travel through the fingertips and up the arm.  And children also clearly learn from many other sources—even from highly verbal and abstract, imaginary contexts."  
 The best treatment of the concrete/abstract dichotomy comes from Uri Wilensky:
"The more connections we make between an object and other objects, the more concrete it becomes for us. The richer the set of representations of the object, the more ways we have of interacting with it, the more concrete it is for us. Concreteness, then, is that property which measures the degree of our relatedness to the object, (the richness of our representations, interactions, connections with the object), how close we are to it, or, if you will, the quality of our relationship with the object."
I LOVE this treatment of "concrete" as simply the quality of your relationship to an idea. Seriously, read the whole piece. You'll be glad you did.

2. Professional mathematicians utilize a multi-sensory approach to their work. Here is some perspective from researcher Susan Gerofsky:
“Movement, colour, sound, touch and other physical modalities for the exploration of the world of mathematical relationships were scorned ... as primitive, course, noisy and not sufficiently elevated or abstract.  This disembodied approach to mathematics education was encouraged despite the documented fact that professional research mathematicians actually do make extensive use of sensory representations (including visual, verbal and sonic imagery and kinesthetic gesture and movement) and sensory models (drawings, physical models and computer models), both in their own research work and in their communication of their findings to colleagues in formal and informal settings.  These bodily experiences ground the abstractions of language and mathematical symbolism.”
3. Children think and learn through their bodies. We should use children’s bodies in math learning.

Known in the research world as embodied cognition (thinking and learning with one’s body) is something we begin developing from birth. Developmental psychologists have shown that in babies “cognition is literally acquired from the outside in." This means that the way babies physically interact with their surroundings “enables the developing system [the baby!] to educate [herself]—without defined external tasks or teachers—just by perceiving and acting in the world.” Ultimately, “starting as a baby [as we all did!] grounded in a physical, social, and linguistic world is crucial to the development of the flexible and inventive intelligence that characterizes humankind.”

Understanding what embodied cognition and embodied learning looks like is the focus of a multidisciplinary group of cognitive scientists, psychologists, gesture researchers, artificial intelligence scientists, and math education researchers, all of whom are working to develop a picture of what it means to think and learn with a moving body.  

Their research findings and theory building over the past few decades have resulted in a general acceptance that it is impossible to ignore the body’s role in the creation of “mind” and “thought”, going so far as to agree that that there would likely be no “mind” or “thinking” or “memory” without the reality of our human form living in and interacting in the world around us. 

4. Finally, instead of sorting out the various merits of individual teaching/learning strategies what we really need to do is look at the bigger picture: Most student learn math best when provided with multiple contexts in which to explore a math idea.

A learner needs time and opportunity to experience a math idea in multiple ways before being able to generalize it and how it can be applied.  An idea, any idea, becomes “concrete” for the learner when the learner has had an opportunity to get to know it. Uri Wilensky said it best:
“It is only through use and acquaintance in multiple contexts, through coming into relationship with other words/concepts/experiences, that the word has meaning for the learner and in our sense becomes concrete for him or her.
Pamela Liebeck, author of How Children Learn Mathematics, developed a useful and accessible learning sequence to help bridge the gap between a math idea and a meaningful relationship with that idea.  Based on the learning theories of psychologists such as Piaget, Dienes and Bruner, Liebeck’s progression is similar to how babies and young children learn to recognize the meaning of words, begin to speak, and then to first write and then read. It includes four different learning modes in which to interact and express mathematical ideas and includes:

a) experience with physical objects (hand- or body-based),

b) spoken language that describes the experience,

c) pictures that represent the experience and, finally,

d) written symbols that generalize the experience.

This sequence illustrates what many math educators already believe, whether or not they use this exact outline – that elementary students need active and interactive experiences with math ideas in multiple learning modes to make sense of math.  

After a recent and particularly robust online discussion on the many different ways to support primary students in making sense of number lines, including a moving-scale line taped on the floor, Graham Fletcher said, “At the end of the day, it's all about providing [students] the opportunity to make connections.” 

Graham's statement points to the importance of focusing on the child's relationship to the math and the environment in which she learns that math. Hopefully it's an environment where many different ways of thinking, expressing and applying mathematics are celebrated and nurtured. 

Tuesday, February 10, 2015

Building Competence [#miyfeet Primary Project]

I've had some questions running around in my head about the Math in Your (little) Feet primary project/experiment. The one I'm pondering the most is: Why am I focusing more on the mapping aspect with the primary kids than in the upper ES version? After today, I think I might be closer to an answer.

Today, in all three of my sessions, the kids seemed to have settled into the dancing and by that I mean, compared to what I saw in the beginning, their bodies executed the (deceptively simple) dance moves with ease and accuracy, and we were moving more as a group. It truly was a beautiful, beautiful sight.

I believe this new-found ease with precision footwork has come about because I have also made a point to have them read and represent their foot work visually, even if it's challenging or not always "correct."  

Last week, instead of having the K-2 class build dance steps and then map them, I wanted to see what would happen if they could figure out how to dance a pattern starting with the mapIt was clear at that point that reading dance maps and making dance maps are two different processes. I wanted to see if I could move those processes closer together. I decided to make a game!

The "starter kit" included yellow cards for the movements Jump and Step. Blue cards were direction cues; arrows for split feet (2 feet on the ground at one time), dots for individual feet. You could also use the backs of the blue cards to draw in foot positions as needed.

It created an interesting interaction between what you want to do and how it's mapped out in front of you.  Were you doing what was on the card? No? Do you want to change the card to match what you were doing? Instead of committing your ideas to paper (which sometimes needs erasing) the cards were semi-permanent, flexible, and interactive. 

I also was hoping that the cards would scaffold teams of kids toward real collaboration, maybe help them share ideas more fluidly, in a way that, up to this point, hadn't really been happening. I think part of the issue is that the dance goes by so quickly they weren't catching each others' ideas. I'm hopeful that these cards can and will support the development of some teamwork and idea sharing in the next three weeks.

The class was as focused as I could have hoped for given everything. It was an incredibly pleasant 45 minutes. 

Just like everything else I've tried so far, some kids loved the cards and some didn't. Upper ES kids don't need this kind of reinforcement to be fully oriented to the math/dance making. For the primary kids, though, my efforts to help create a visible connection between the dancing and the maps seems to be supporting them in building competence in both realms.

Friday, February 6, 2015

Intent on Creation [Family Math Night]

Everywhere I looked last night I saw people intent on making math. It was glorious!

I mean, just look at these messes!

Just look at the focus!

Just look at all the ideas!


As a finale, I present three views of the interlocking cubes created by one participant. I also hasten to add that while there are lots and lots of materials provided, kids and adults are free to do with the materials what they will. This often leads to unexpected and marvelous personal discoveries and creations, like these:


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