Two Sides of the Same Coin

So I’m working on a book.  While I work on another book.  This other book is a project through Moebius Noodles and Maria Droujkova’s publishing company Delta Stream Media. I’m collaborating with Maria and Gordon Hamilton of Math Pickle to create original puzzles, games, and making activities exploring numerical and categorical variables … for young kids!  It’s super awesome extremely cool.

And, today?  Today, our conversations in relation to the variables book helped me clarify something I’ve been thinking about on a LOT of different levels for literally YEARS.

This level
What’s the difference between using the body to illustrate mathematical ideas and using the body to create and express an understanding of mathematical ideas?  

This other level
What’s the difference between using body knowledge (ala Papert and his gears) and creating body knowledge?

And finally
What’s the difference between identifying properties of an object (say, a piece of art, or an insect) and actively choosing from an inventory of attributes to make your own?  

llustrate, use, identify  <======>  Create, express, choose

Each of these questions needs its own, more specific treatment.  My observation today is simply that each pairing seems to create a similar tension in my mind.  These are all active words, but the nature of the activity is qualitatively different depending on which side of the learning process you're on.

Today the words fixed and flexible came up in relation to how we are approaching the activities in the variables book.  

Fixed
Puzzles and games that focus on identifying properties happen within a fixed structure.  Using body knowledge to understand a set of gears assumes that you are using a certain set of body experiences, created at some point in the past.  Illustrating math ideas using the body means there is a predetermined goal for the activity and that the outcome needs to look a specific way.

Flexible
Learning vocabulary and language in context and, in math, using multiple strategies to solve a problem are both process oriented and context dependent.  In making, having a large inventory of ideas/things/skills from which to choose and create your own novel ideas (like a dance step) is an open-ended investigation.

“Fixed” and “Flexible” are not judgments; they are inverses of each other.  They go both ways.  Just like you need to compose and decompose numbers to see the full relationships embedded in those two activities, so do you need to identify and use properties, build and use body knowledge, and illustrate and express mathematical ideas. 

Fixed: The parts of learning (anything, really) that are perhaps learning objectives that are easier to identify in an assessment, but still crucial.  Some call this skill building.

Flexible: Relates to the processes of learning which (as anyone who has tried arts integration, project based learning, or focusing on mathematical practices may have experienced) are much harder to nail down when tasked with assessing such activity.  Some call this fluency.

You can't have one without the other.

“Without skill there is no art. The requisite variety that opens up our expressive possibilities comes from practice, play, exercise, exploration, experiment.”  --Stephen Nachmanovich, Free Play: Improvisation in Life and Art

 
Thoughts, feedback, pushback and conversation are always welcome. 

My Favorite Math Games & Interactives: Online & Off

A friend of mine has a couple kids in kindergarten right now.  He asked if I had any recommendations for "online computer games for arithmetic, patterns, creative math exploration."  Why, yes, I think I do!  Following is an annotated list of some of my favorite interactive online habitats for a wide range of ages, including parents.
 
However, I think I'll start with a list of activities that are not computer based.  I'll be the first to admit that I'm a little biased toward hands-on, not-virtual learning, especially for the primary grades; children's bodies really need to be involved in the  learning process, especially math.

Also, although my kid has some computer skills, she is basically raising herself as a luddite, which is fine with me for now.  She's never been drawn to the computer, even when I specifically show her something I think is fun or cool.  So, many of the games and interactives listed  in the online category, below, are places that I go to get activity ideas to adapt for her off-line lessons or to grow my own math understanding or just for inspiration.

OFFLINE FUN
I don't know where we'd be this year without the endless Shut the Box tournaments of last year.  I attribute at least half her mental addition skills to rolling two dice (which can make up to 12) and finding different ways to make the rolled number.

I attribute the other half of my daughter's mental addition ability to UNO.  It's a classic game that we played speed style all last year.  And, at the end, the winner got to add up all the points she won from her mother opponent who always remembered to chant the UNO Point Tallying Mantra: "Find your tens!"

Peggy Kaye's book Games for Math  has LOTS of easy and inexpensive ideas including Nine Men's Morris which is sort of like a more complex and moveable tic tac toe (strategy, logic).

Variations on "sort and count the change jar" along with "save and spend" from about the age of five means that now, at age seven, the child is a whiz at money math (also known as adding and subtracting 10s, 100s and 1000s).

And, of course, The Game That is Worth 1000 Worksheets from the Let's Play Math Blog -- for the littles all the way through high school.  This fall we used a version that was a combination of sums and differences.


ONLINE GAMES & INTERACTIVES (NOT JUST NUMBERS)
Some of these are for you, lots for them, and some for the whole family (like Scale of the Universe 2 and the triangle interactive!!)

National Library of Virtual Manipulatives: Truly a class act and a challenge for all ages. 

Math Pickle: This link takes you to a large library of interesting videos and ideas organized by grade level.

Kaleidograph:  I first found this toy and played with it online, and then I bought the real thing for Christmas. Both the online and hands-on versions are great but, no surprise, I find more meaning in the actual toy.

From my childhood memory bank, a classic Sesame Street clip with a lovely song: Two Little Girls and Their Little Doll House.  My adult eyes immediately saw one-to-one correspondence but my little girl heart remembered something different.



Scale of the Universe 2: I have no words for this interactive graphic other than just GO.  You can turn off the music if you want.

I've included this link to Planet Seed/Math Puzzles for Kids because I don't want to forget about it.  It was on a long list of bookmarks already.  Puzzle topics include combinatorics, algebraic thinking, topology, probability, etc. 

Games from Freudenthal Institute look great, and they even have a star interactive!

Make your own isometric dot graph paper here.  These and the other dot graph paper you can print out are fun to leave around the house so kids can make designs whenever the mood strikes.

Math cats!  Need I say more?

I personally love Symmetry Artist from Math is Fun and my daughter actually liked playing with it for a while. It's a great way to really experience the differences between line and rotation symmetries and you can print out the designs you make at the end.

And, you really cannot miss this triangle interactive from the Triangulation blog.  Truly.  Run, don't walk.

To close it out, here is a lovely and thought provoking classic Sesame Street animation with music by Phillip Glass called The Geometry of Circles.