Primes of Beauty

We started the Club on May 4tth, 2010 with Richard's Monster poster projected on the wall. Erin said she was playing with Alec's Primitives applet at home, and noticed something interesting. We pulled it up so Erin could show us. B. asked to pause at 8, because he was drawing similar pictures last Club, and kids started to draw "fractal tallies" like this. In this picture, you can see the similarity of the first two parts, and then a different third circle. It's very typical of young kids to keep the structure for a while, and then to change it wildly. As one artist said, "The secret to beautiful children art is to take the paper away from them at the right time!" I am adding an item about it to the Math Behavior bingo (more below).
Club_05_04_2010_tallies_1.jpg
Club_05_04_2010_tallies_1.jpg

Anyway, what Erin noticed had to do with the order of primitives in the applet. We could not figure out the rule for the order, and decided to email the creator. Here is the comment I left on the site:

"Thank you for a beautiful applet! We used this in a Math Club today, together with the poster "You can count on monsters" from here: http://www.math.brown.edu/~ res/PosterPrimes/pos... Kids and parents were making up their own ways to "tally" numbers. Erin noticed that there is no consistent rule to how groups are formed. For example:

- 6 is "three groups of twos" but 12 is "two groups of two groups of threes"
14 is "seven groups of twos" but 21 is "three groups of seven"

We were wondering if there is some hidden logic and beauty in this choices - or are they random to show the variety of possibilities?"

Before the day was done, Alec replied. I love the internet. Here are our next two messages:

"Maria,

Thanks for your feedback. I'm really glad that this resource is still being used to teach maths!

The ordering of the primitives is random, but if you click on the bar on the right (that shows a 2 and a 3 in circles for example) and drag them into a different order you can make "three groups of two" into "two groups of three". I made the application very quickly for my own class, and never really finished it, so some of its features are a little bit hidden.

Thank you for showing me the monsters poster. I had no idea it existed; it is lovely to see someone has a similar thought process to me!"

---

"Alec, how cool! I am sending this on to the Club members. Maybe you can just draw a little "hand" icon next to numbers, or some other symbol for "drag me." I really like this feature.

The poster's author, Richard Evan Shwartz, just published a book based on it. I sent him your applet's link yesterday, and he did not know about it - he said it's really neat.

I am working on a similar idea from a different angle, still: finding "essential multiples" in nature or culture. Some examples are here: http://www.naturalmath.com/ multpics/index.php However, this software turned out to be so clanky I disconnected it from the front page of the site. I plan to continue this using a better tool, like Prezi, LiveBinders or Wallwisher. Thank you for the inspiration!"

---
Returning to the Monster Poster, we looked at how monsters worked. Kids loved the monsters and spent a lot of time playing with shadows on top of the poster's projection. They could not figure out why six was the way it was (the monsters for 2 and 3 playing together). Good, I thought! We can investigate! So we sat down and got to work on making our own tallies for six, and then other numbers.

L wanted to add color to D's tally. Her first drawing was bright, big, metaphoric - and the tally was hard to discern in it. I asked her to use a separate color for each tally mark and the horse (which was mark number six). We went through several iterations before producing a tally that was similar to what D did, color-coded, and marked with "counter" dots of a different color. Then L ran away seemingly doing something else - but brought back a small deer toy with six horns!

Here is the report of this part:


Meanwhile, at the next table, parents were working on cutting up the Bingo and sorting cards into categories. In the current unsorted state, it's impossible to use during observations! It led to lovely discussions and overall was a great activity. Check out all the different categories! You can find three out of four of today's creations at the Bingo page http://naturalmath.wikispaces. com/bingo - Sue wanted to work more on hers. The color-coded sorting by Christine came to me yesterday by email, from the invitation at the Living Math group.

While grown-ups were talking about the Bingo, children played with small counters and blocks. They had a complex roleplay world going, which we did not disturb other than giving them some fresh corn and apples for sustenance. However, a war of sorts broke out between the alien city and the animal city, which called for a prolonged peace talks. We used a large shell (if you get the reference) to take turns while speaking. It was beautiful how kids helped one another to get the shell, and to pay attention to the speaker, even when they were arguing. Many good points about war and peace and life were made in this conversation: that aliens don't want to join animals, but animals want to live peacefully and play with aliens; that "you do to others what they do to you" is different from "you do to others what you want them do to you"; that it is hard to stop the cycle of city destruction... Here are pictures of the animal city being built, and the alien city with E. talking about it holding the shell:

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We ended the Club sitting on a blanket, with Heather reading us "How Big is a Foot?" - a lovely book about measurements. Delta brought a bucket of measurement tools, but we did not get to it today - next time! Meanwhile, here is the Club listening to the book - and then answering the titular question!

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Reply from Richard:
Hi Maria,

Thanks for all the feedback! I'm really gratified to see that
children are looking at the poster and enjoying it.

Best,
Rich

Reply from Alec:
Dear All,
I am grateful to be included in this discussion. I am not actively involved in education at the moment, though plan to return to it one day. It has been inspiring reading about your work, and pleasing to see my work being used.
I would like to ask your permission to precis this email in the form of a blog post that I can post on my education blog, ptolemy.co.uk. I would like to use the picture of Richard's poster, as well as a link to the prezi. Please let me know if you are each happy for me to do so.
In case it is of interest, and by way of a plug, I would like to point out that I too have posters for sale, based on the Primitives idea: http://www.atm.org.uk/ shop/products/vis067.html. It's a relatively selfless plug, because all proceeds go to the Association of Teachers of Mathematics, a UK charity which serves as a professional association for mathematics teachers in the United Kingdom, of which I was formerly a general councillor. They are UK-based however, so postage costs to the USA may be prohibitive.
Warm regards,

Alec McEachran

Alec's poster preview:
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Built-in multiplicative system: pentatonic scale


Natural Math email group discussion "Learning like a musician"

Fractal Abacus by David Gibson

Fractal Abacus by David Gibson