Fraction A Day - Day 33+ Half a Spongebob

"Stencils allow children to explore the math behind the patterns. It is difficult for them to perceive the difference between 1/8 and 3/8 without visual aids in the beginning. Unlike charts, Etoys is interactive. Unlike paper or cardboard, it allows children to repeat any number of times without waste."

- K.K. Subramaniam (who created the Stencil code in squeak to make this possible) 

So I used Stencils (which hopefully will be in an upcoming version of Etoys) to allow kids to explore fractions.  See the example below:

Fractionator  (2011-Jul-30 1:29 EDT) by Mr. Steve <mrsteve>  Tool for kids to create and explore Fractions.

Using this Etoy, kids, can create and explore their own fractions.  Fractions can be made using the "Fractionator" (using the Stencil functionality) or by drawing lines to separate parts of the set.

The line drawing tool, allows you to create multiple segment lines by hitting the key to add a new segment (actually it simply adds a new vertice). If you hold down the a, you can get a behavior a bit like drawing (would be interesting to see how this behaves on a touch screen).

The "Fractionator" lets kids specify the number of parts.  I would like to extend this so kids create their own fractionator tools by writing scripts to set the size of the Fractionator.

The last pages have an explanation of how the the "line drawing" scripts work.  I used a magnifier and speech balloons to explain what was happening, it would be good to add step by step actions as you highlight each scripting tile.

What do you think?  How can this be used? How can it be improved?  Leave a comment.

Another great use of stencils can be found here: http://community.ofset.org/index.php/Grab_area_button This was done by Susanne Guyader, an 80+ year old artist who used Squeak/Etoys to guide art teachers in France. 

http://community.ofset.org/index.php/Grab_area_button

Fraction A Day - Day 32 - Cutting Cookies, Construction, Instruction and Perceptual Learning

Been working on Cookie Cutter program hopefully by Monday I will have something to post.
Also been thinking about when we should use Construction vs Instruction and had some interesting conversations with a few folks.

So the question I am asking is: Given that we can't expect kids to construct everything themselves (who has the time, or the appropriate facilitators)  How should we balance instruction in skills and learning to construct and test things for yourself?  The article Brain Calisthenics for Abstract Ideas talks about Perceptual learning and perhaps using that technique could free up some time in classrooms for kids to learn to construct and test things themselves.

Well let me get back to making my playthinks, then maybe I'll have some tools to test things out and see what we can learn.

Fraction A Day - Day 31 - Introduction into Infinity

Maria Droujkova tells the story in a upcoming book: "Playing With Math: Stories from Math Circles, Homeschoolers, and the Internet"

We did another fun activity where several kids laid down head-to-foot to form a line. If we kept doing this, would we get an infinite line? Something interesting happened: in both clubs, kids answered, "Yes, the line will circle the Earth and meet itself and this ring will be infinite, because there is no end." The idea of a cycle as an example of infinity is powerful with children, and apparently the image of Earth being round goes with this infinity representation in their cosmologies. We will need to devote at least one club meeting to cycles.  

Then I asked, but what if the line of kids could go on and on like this in a straight line, would the length of that line be infinite? Which led to a nice discussion about running out of kids, running out of space in the Galaxy, and then distinguishing what exists in math and in our abstract imagination from what exists in the physical reality. This discussion has to happen again and again around every calculus idea!

then later ...

After doing the "infinite line of kids" activity, I asked the question, "Do infinitely many pieces always make an infinitely long line?" Everybody but Yasmin said, "Yes!" and we did a paper cutting activity next. Cut a strip of paper in two and lay down one piece. Cut the remainder in two and lay down that piece next. Cut the remainder in two, etc. After going, again, into the all-important discussion of distinguishing the physical paper from our mathematical ideals, kids could see how adding infinitely many pieces can result in a finite length. That was a big surprise! I will definitely do more of this in both clubs, with paper folding and cutting. The idea of a finite limit of an infinite sum is extremely powerful. It resolves several infinity paradoxes and provides important metaphors for understanding the world. 

One thing I really like about what Maria did is how it sets up the kids and then gives them a big surprise and does it all with concrete tasks that kids can understand and easily do.

I also like this as an introduction to one of the problems from Don The Math Man's Calculus By and For Young People, where he asks them to add 1/2 + 1/4 + 1/8 + ...  and then his Cookie Cutter Problem.

I have stated working on a "Cookie Cutter" in Etoys (using Polygons filled with a Cookie Bitmap, hmmm, I may try embedding instead, always more than one way to solve a problem).  I have it to the point where it cuts the polygon in half (vertically or horizontally).  Hope to share something soon.

Fraction A Day - Day 18-30 Vacation Thoughts

I was on Cape Cod with the family and went on a  "Computer/Internet diet"  but I did think about fractions and kids and learning.  Checked out all the books on fractions from the local library and watched some PBS shows with the kids (we don't have TV at home,  don't wan't the kids to "distract themselves to death", but PBS does some wonderful things we can learn from and use).  Here is a brief summary:

Sid the Science Kid, had an episode, where they were talking about measurement and they used their bodies to measure a room, at first laying down, marking the spot, the laying down again.  They then made a paper trace of Sid's body and used the paper tracing to measure.  Then I thought you could use this and when you get to the end of the room and lets say there's 1/4 of a Sid left.  How could you measure 1/4 of Sid (well fold in half and then in half again).  But what if it was 1/5 of a Sid? Or they wanted to measure items smaller than their unit Sid?  We could come up with some lessons where everyone uses their own units (with paper tracing).  Also if you taped papers together you could use two units "Sid Head to Heel" and "Sid finger tip to finger tip" (with arms spread).  Then measure again at the end of the year (to drive home the importance of standard units and have some fun.

Don the MathMan Cookie Cutter Problem - Don actually raised a good question, in his comment on the Day 4 post: "I think starting with just coloring in without the goal of infinite series is a mistake- as my parent said-boring, irrelevant." Which got me thinking how do we introduce and grab kids.  Don's cookie cutter problem is a good one.


Math Circles: I am helping edit (more accurately reviewing and making comments as time permits) on Math Circles one take aways from this wonderful book is that kids need time to play and digest after a lesson.  Also Maria's chapter got me thinking about how we can provide kids with concrete fraction experiences in the kitchen and elsewhere.  


And that is the question I ask, how can we provide kids with concrete experiences with fractions?