What are the numbers in the Fibonacci sequence? What’s the difference between a sequence and a series? Is it still the Fibonacci sequence if you start with a different first number (or two)? What phenomenon in nature prompted Fibonacci to study these numbers? How is the first number in a sequence determined? What is that first number called? How do you draw the Fibonacci sequence spiral? How do you draw the crossing-over spirals in Fibonacci quantities to represent a pine cone? Does the number of spirals in a pine cone have to be consecutive Fibonacci numbers? What did Fibonacci have to do with the switch from Roman to Arabic numerals? How does Vi Hart get her concentric “blobby” spirals to look like optical illusions? What was Richard Wilbur talking about in his poem “In Trackless Woods,” and what does it have to do with math? What is a poet laureate, anyway? How can you generate a Sierpinski Triangle from Pascal’s Triangle?
When Math Circle started today, kids went immediately to the table and started drawing geometric creations. Once everyone had arrived (or the 9 who were coming; 4 were absent), they watched Vi Hart’s short film “Spirals, Fibonacci, and Being a Plant.” When it ended, they returned to the table, and R asked, “Do you have graph paper so I can do a Fibonacci Spiral?”
“How do you do a Fibonacci Spiral?” asked P. I started to draw a grid on the board so that we could discuss it, and X walked to the front of the class and led to discussion. Then people started making their own. At the same time, we discussed the above questions. A few were posed by me, but most by the kids. As we discussed, people drew different things: J, X, and R drew Fibonacci spirals; N, X, and Z made pinecone patterns; P and G invented their own patterns inspired by Fibonacci spirals; L listed the numbers in sequences he was creating; and M was writing his name with a compass only. (He had been in our earlier course on Euclidian constructions, and now was applying what he learned earlier to a new project.) While it may seem that different folks were perusing different goals, the really weren’t. All were conversing together. And all were developing the same skills:
• collaboration in the pursuit of revelations about structure (“It would be the Fibonacci rule but not the Fibonacci sequence,” said R in response to a question above.)
• comfort in contributing to class (A spontaneous discussion erupted over number games people play in their minds when they see clocks, elevator buttons, etc.)
• learning what questions to ask so that students can figure relationships and patterns out for themselves (“Can we draw on the pine cones?” asked N.)
• exposure to the etiquette of intellectual engagement, in which students learn to work in community to explore interesting questions. (X suggested a different sequence that is similar to Fibonacci; all worked together to generate the list and try to figure out where the second number came from.)
I heard groans of disappointment when I announced that we have a week off from Math Circle next week. If your kids want to do something on their own on Tuesday afternoon, have them watch Vi Hart’s short film “Open Letter to Nickelodeon, Re: SpongeBob’s Pineapple Under the Sea.” We won’t be viewing this in class, but it has been requested. Watch with your sketchbook and notebook and a pencil, and see what happens.
When we return in 2 weeks, on April 10, families are encouraged to come a bit early and/or stay during Math Circle to peruse the Talking Stick Book Sale. See you then!
NOTE: For those of you with younger children, we will also be looking at the Fibonacci sequence in our upcoming Math Circle for 6-8 year olds