(October 28, 2014) The kids began by explaining our continuing dilemma to K (who was absent last week): How should we proceed in the attempt to see if any set-ups in Conway’s Game of Life lead to a pattern of continued growth?
On the one hand, the systematic approach is thorough. OTOH, it’s tedious. I had recopied, more neatly, their results from the prior 2 weeks onto the board for all to see. “What should we do?” I asked.
“Let’s just do our own set-ups,” said pretty much everyone.
H had come to class with something he had been thinking about during the week – starting with a solid rectangle at the corner of the board. He wanted to try that. A wanted to test a 3×3 square that was empty in the middle. Seeing these 2 set-ups triggered M and K to want to test what happens when you set up “half the board” (a 4×8 rectangle). For the past 2 weeks, L had been asking whether it’s possible to fill the whole board, but unfortunately she was absent today. (Try it at home!) J was a bit disappointed to not have a conjecture of her own, but she watched the others test some of these hypotheses. Four kids were working from 2 boards (one a Go board, the other Othello).
As the kids worked, an occasional counter went flying across the table. I kept that in check with some biographical information about the inventor of Life. Over the past few weeks, the kids had been requesting his story. I read to them from an article with one of the most entertaining interviews with a mathematician that I have ever read: “An Interview with John Horton Conway” by Dierk Schleicher0.
“Here’s a photo of him. He’s still alive, and he leaves nearby.”
This news prompted A to nearly leap out of his seat. “Then there’s still a chance that we could get the prize!” he said excitedly. Everyone wondered whether Conway was still offering that $50 (In 1970, he offered this prize to anyone who could prove or disprove that ever-expanding growth was possible). Once again, there was discussion about how measly this prize seemed for such a famous national magazine.
The kids worked and I related a few anecdotes from the article. (I would stop talking when the Life exploration got more intense, and then resume if a counter went flying.) The kids listened to and discussed
- mathematical topics Conway is known for (Fun for the kids to be reminded that math is not simply arithmetic. No one knew that one can be a “knot theorist.” They also loved that Conway created a new number system, “surreal numbers.” Four of the kids in this class created “Pumpkin Numbers” last year in our course last year about named number types.
- a funny quote about Conway’s enjoyment of teaching
- Conway’s Free Will Theorem (I was really taken aback and excited when I read about this; it shattered a bunch of assumptions I didn’t even realize I held. The kids were not fazed by this new idea at all. They do not wear those blinders of “precedent” that we adults do. This is why I love to work with children – the constant reminder of limitless possibility.)
- Conway’s opinion that the Game of Life is really “rather trite… trivial,” especially since the Free Will Theorem may render the applications of Life irrelevant (if I’m understanding the physics in this article correctly).
- Steven Wolfram’s disagreement with Conway about the relevance of Life
- how/why Conway came up with Life (seeking simplicity)
- an expensive arithmetic mistake he made when offering a large cash prize. “And he’s a mathematician!” chided K. (In retrospect, I wonder whether she meant that he should know better, or that everyone makes mistakes.)
- why he suspects he may be a bad influence on people
- his “recipe for success” in a mathematics career
If we had more time, I also would have shared his anecdote about how he “never applied for an academic position in my life.” Also I should have shared his message to “enjoy yourself.” (I’m being pretty vague here about the content of these anecdotes because I’m hoping you too will read the article. I enjoyed it so much that I read it twice – at least – before sharing it with the kids.)
“One more thing I forgot to mention from the article,” I added as an aside. “He went to prison.” Some eyebrows shot up at that.
“Yeah,” said Jo nonchalantly. “He seems like the kind of person who would have gone to prison.”
“For what?” ask the others, of J.
“For something really awesome!” she replied. I quoted the article about Conway’s 11 days in prison for participating in a “ban-the-bomb” protest. This was exactly the type of “crime” that J had in mind.
“No one would get arrested for something like that in our times,” said someone, seemingly assuredly, but with undertones of doubt. Then the kids debated whether it really “counted” as “going to prison” if he spent “only 11 days” there.
As these stories were being told and dissected, people were making ground with their Life progress. With these new set-ups, things were neither stabilizing nor dying after just a few moves. The shapes of the populations were changing dramatically. A new conjecture arose: “If you fill the whole board or larger (rectangular) area, all but the corners will die and a new row will be born.” The half-board and open square that M, K, and A had started at the beginning had morphed into new shapes. H’s smaller corner rectangle had gone extinct, so he started testing a new set-up. (H’s new set-up is hard to describe. Look at the photo of the board work – there’s a drawing of it at the bottom. It’s labeled “3”)
More work, more stories. Then came an excited interjection from M and K. Their half a board had morphed into two “steady squares.” The conjecture morphed into “we’re done, it won’t die, but won’t grow anymore either.”
Then M asked hopefully, “Should we name it?” The others nodded. I asked A for a name for his set-up; he called it “The Three-Square.” I asked M and K, and with a giggle M named it “Half the Board.” H didn’t name his.
“Giving a name to something is a way of knowing it,” says Elaine Brooks.1 In all my years of teaching young children, I’ve felt this. Naming a thing – either learning its formal name (as Brooks means) or assigning it a name – gives us a little bit more ownership of the thing, invests us more heavily, gives us more responsibility to understand and care for it. Also, to quote Michael Pollan, “Proper names have a way of making visible things we don’t easily see or simply take for granted. (p43, In Defense of Food, Michael Pollan)2
But there’s a flip side of this coin. There are freedom and infinite possibilities in the unnamed. “The need to label (is) the need to diminish,” says novelist Ilie Ruby.3 This too is something I’ve always felt. Names can limit – preconceived notions, self-fulfilling prophecies, and all that. So the challenge seems to be finding the balance, knowing where to draw the line.
GETTING DEEPER INTO LIFE
While our kids were naming formations, someone noticed that the two steady squares on M/K’s board were just one diagonal unit away from each other. Therefore a birth could occur! The end was not near, maybe. Everyone went back to work, and I resumed storytelling.
Then, sadly, the Half-Board set-up died after many moves. M and K decided to restart, from the position where 2 steady squares were one diagonal unit apart. As they worked that, we documented our findings so far on the board:
The Three-Squre: about 20 moves in, it was still growing
Half-the-Board: died after 20-30 moves
H’s second set-up: still growing with 10 moves
Then something really interesting happened. (For four of the students, all of this was clearly interesting, clearly. I overheard the comment “this is fun” more than once. For J, it was not very interesting – probably because she didn’t have her own conjecture to test, and therefore didn’t feel the ownership in the work that the others did. Fortunately, she is a huge fan of biographies so thoroughly enjoyed the running narrative about Conway.)
Any, here’s the interesting thing I was about to mention: M and K’s revised work with the 2 steady squares kept growing past the point where it did before. How could this be? Conjectures included (1) there was a mistake, and (2) it matters where you place your set-up on the board, and this set-up started in a different place relative to the edge. Kids commented that a larger board would be useful. Someone mentioned how an infinite board, or a computer program, would be very useful.
I asked whether we could assume that any of these patterns that were growing at this point would continue to grow. Particularly, kids posited conjectures on how many moves would be enough to be sure:
- 30 moves
- 50 moves
- 1,000 moves
- No such number/you can never be sure. (After the group discussed this for about half a minute, everyone agreed with this conjecture.)
Since today was our last class, we discussed how kids could continue their exploration of Life at home. H said, “I want a Go board.”*
As he mentioned last week, and the week before, A said “I want to learn to play Go.”
“I can teach you,” said J, as she did every week. Sadly for them, I didn’t make time in class for this to happen. Then we talked about other ways to play Life at home if you don’t have a Go board: Othello, Scrabble, possibly checkers, home-made with paper. (Some of these options, as the kids know, don’t offer enough squares to progress very far.) Also, Josh Jacob’s blog has some interesting-looking links.
“BESTING” GARDNER AGAIN
We had about 10 minutes left. The kids requested another of Gardner’s “Tricky Mysteries.” So I read them Funny Business at the Fountain.4
“At a hotel in Las Vegas, a lady rushed out of the manager’s office to get a long drink at the water fountain in the lobby. A few minutes later she came out for another drink. This time she was followed by a man. There was a mirror behind the fountain. When the lady raised her head, she saw that the man behind her had a knife in his upraised fist. She screamed. The man lowered his knife, and then both of them began to laugh. What on earth is going on?”
J immediately announced, “I have an answer!” Here’s a paraphrase: A person working in the office was trying to sell a bunch of stuff to the couple at the water-fountain. It was really annoying. They wanted to get away from the person who kept trying to sell them stuff. They couldn’t pull themselves away. So they faked the knifing attempt to distract the salesperson from the sales pitch. And it worked.”
Other conjectures followed, most involving some sort of prank. Since last week the consensus was reached by checking and rechecking the exact wording of the problem from the book, the kids immediately compared their conjectures with the stated information to avoid conflicts. Finally, the consensus seemed to be that the group “liked” J’s story best. (Love it that they are forming preferences for solutions to problems.) Of course, they then demanded Gardner’s explanation. (See footnote if you want – it is a spoiler.)5
Everyone appreciated the logic in his solution, but as happened last week, appreciated their own explanations more. Then it was time to go.
Oh, how I I wish we had more weeks to further explore Life! If I were to do this course again, I would introduce Life right in session 1, and allocate 20 minutes per session to work on it for the whole course. Fortunately, we’re doing another math circle in the spring for this group, so it was just a see-you-again, à bientôt, Shihemi më vonë, lihitraot, or རྗེས་མ་མཇལ་ཡོང་། (jema jay-yong) .6
One final note: if you know someone with younger kids who would enjoy math circle, please help me spread the word about our upcoming circle for approximate ages 5-6. It’s around this age when many kids make the shift from appreciating mathematics from a deep conceptual place to seeing math as a set of instructions to be followed. One goal of this math circle is to prevent, stop, or at least slow this shift. In this course, We’ll explore the idea of infinity using drama (puppets!) and embodied mathematics. The kids will use their imaginations and physical movements to play with patterns that have limits and patterns that don’t. And we’ll work on verbalizing our mathematical ideas as we try to figure out what patterns exactly are. More details are here: http://talkingsticklearningcenter.org/math-circle/
0 Notices of the AMS, May 2013: http://www.ams.org/notices/201305/rnoti-p567.pdf (Do footnotes always have to begin with number one? I suspect Conway would be okay with departing from ALA conventions.)
1 Elaine Brooks, quoted by Richard Louv, in Last Child in the Woods, p41. Having this course outdoors in a beautiful natural setting inspired me to finally read this modern classic. I’m about halfway through. The first half is about what Louv calls “nature-deficit disorder,” a topic that’s been on the forefront of educational thought in our part of Philadelphia (and many other places in the country) for the past decade. The second part, which I’m very excited to get to, is about how “reducing that deficit – healing the broken bond beterrn our young and nature – is in our self-interest, not only because aesthetics or justice demands it, but also because our mental, physical, and spiritual health depends on it. The health of the earth is at stake at well. How the young respond to nature, and how they raise their own children, will shape the configurations and conditions of our cities, homes – our daily lives. The following pages explore an alternative path to the future.” (Louv, p3, introduction to second edition of this book, 2008, in which he then states hopefully that transformation is occurring.)
2 p43, In Defense of Food, Michael Pollan
3 The Salt God’s Daughter, p135 Ilie Ruby
4 Gardner, PPTT, p41
5 (SPOILER, SO I’M MAKING YOU WORK TO READ IT BY MAKING THE FONT TINY) “The lady had the hiccups. Her boss was trying to stop them by frightening her.” Gardner, p97
6 http://www.omniglot.com/language/phrases/goodbye.htm See here for translations of good-bye into many languages. What a fun site. I never knew until researching this right now that “goodbye” actually literally means “God be with you.” But according to a discussion on wordreference.com, that meaning has not been retained into modern usage. Oh how I do enjoy Google!
*Barnes and Noble sells a nice Go board. You can learn to play it online here: http://playgo.to/iwtg/en/. I found, though, that it was helpful to get an overview of the game and some cultural context first from the Manga serial book Hikaru No Go.