Tuesday, August 18, 2015

Math on a Stick: Fold & Cut

This activity is one I am doing at Math on a Stick at the Minnesota State Fair August 27 & 28, 2015. You can find out more about Kirigami, a paper folding and cutting tradition similar to Origami, and more simple designs for kids to experiment with here.


Below are instructions for simple folding and cutting. One additional thing to look out for is that when children begin to cut away paper to create holes they often just make slits in the paper. A "hole" in this sense is at least two cuts that, when you open the paper, create an empty space.


Math on a Stick: Beading Patterns

This activity is one I am doing at Math on a Stick at the Minnesota State Fair August 27 & 28, 2015 and can be found in the upcoming book I co-authored with Dr. Gordon Hamilton titled Socks are Like Pants, Cats are Like Dogs which is available for pre-order.


We often ask children to find and sort everyday objects according to their properties (for example, into piles of white socks and socks that are not white). In early algebra terms, this means sorting items into categories. We can also use this kind of mathematical thinking about properties to create brand new objects.

Beads are full of similarities and differences that can help you create beautiful patterns as simple or complex as you want. All you need to get started are string or pipe cleaners and beads in multiple colors and sizes. You can even make your own beads by cutting up drinking straws! As you create your patterns, you get to ask lots of interesting questions:
“How will you make the pattern interesting?” 
“Will you use all small beads, or a combination of sizes?”
“Will they all be smooth, or will you add a rough textured bead into the mix?”

Whatever you choose to do, it will be yours and it will be beautiful. You will know exactly how to talk about what and how you made this beautiful thing because you’re the one who created it!

Pre-cut lengths of cotton string and/or pipe cleaners
Plastic pony beads, wooden beads, and/or beads made from plastic straws
Bowls or other containers to hold beads

Activity Description
Look at all the beads. Find different ways to describe them (color, shape, size, texture, etc.). Notice the similarities and differences between the beads (for example, same color, different shapes). Create a four-bead pattern unit (four beads in a row) and repeat that unit until satisfied.

Adaptations by Ages
Use very large beads or other objects like balls or blocks. Have baby handle and play with objects. Comment on their texture, shape, and color while baby is playing. Line up objects on the floor in front of baby to create a short pattern; repeat pattern one or two more times. Point to each bead and name one attribute category at a time (for example, “smooth, rough, smooth, rough...” then “red, blue, red, blue...”).

Provide pipe cleaners and a selection of large wooden beads. Let your child experience the beads by touching and stringing them, but don’t worry about patterns for now. Talk through your own making process while your child makes hers alongside you. Talk about why the bead you are using is different from (or same as) the one your child is using or about what comes next in your pattern.

Older Kids
Use an interesting assortment of beads and pipe cleaners (or string), three or more attributes (such as bead shape, color, texture), and three or four beads to make pattern unit. Make your own alongside your child. Take turns investigating each others’ work - how is your child’s pattern similar to yours?

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. 


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