The Fibonacci sequence is a classical example of a purely mathematical concept which pops up unexpectedly from many fields. It is also a classical example of boy- vs girl-friendly presentation (although I would prefer in this case theoretician- vs applicative-friendly distinction): a possible way to explain them can be found in "Why Gender Matters", by dr.Leonard Sax.
The book is "interesting", I admit, as dealing of gender difference in childhood and their application to life and, what more interests me in this specific case, learning. The book advocates separate-gender classes and other practical measures you may not agree with (I do on some, but definitely don't agree on some others). The account of gender differences is up-to-date however, and in my view is the most valuable part of dr.Sax work.
But let's go back to Fibonacci numbers, and the girl-friendly way of teaching them.
First of all, what Fibonacci numbers are? They can be defined by three very simple rule:
1) The first element of Fibonacci sequence is 1
2) The second element of Fibonacci sequence is another 1
3) All elements following the first two are constructed as the sum of the preceding two terms.
Applying rule 3) once, we get 2 as the third element of the Fibonacci sequence. Applying it once more we have the fourth number is 2+1 = 3. The fifth is 3+2 = 5. And on, forever. The first elements of the sequence are:
1, 1, 2, 3, 5, 8, 13, 21, 34, ...
On pages 104 to 106, dr. Sax gives a possible "girl friendly" presentation of the subject.
Actually, this is a sort of laboratory. First, the bare definition is given, just to introduce the subject (and make it clear is something very abstract).
Then, you jump immediately to example. For example, you may count the number of petals of flowers (the laboratory may be prepared by asking the gals to get Fibonacci-prone objects: sunflowers, pinecones, daisies, other flowers, ..., and take them to the class). Very often, this count turns out to be a Fibonacci number. Then you turn to something more difficult, like counting the number of rightwards and leftwards spirals you can see at center of a sunflower, or on a pinecone.
Dr. Sax writes "Now these girls will start asking questions" (end of page 105)
On beginning of page 106: "And you will have accomplished something that some of the gender experts ... (my omission) ... have deemed impossible: you've got a classroom of twelve-years-old girls excited about number theory".
I agree: they ask questions (finding Fibonacci numbers in nature is something extremely fascinating - I say this also to date, at my age of 47, not exactly 12 years old.
But unfortunately, finding sensible answers is by far not simple.
My first contact with Fibonacci sequence was in secondary school, and the teacher's approach was quite similar to the one suggested by dr. Sax. And in fact we (me included) asked many questions.
I may say this teacher created a monster, as I tried (well, very occasionally to be sincere) to find if anyone had discovered an answer for something like 35 years, without any success. On reading dr. Sax book, this early imprinting popped up once again, and I interpreted it as a provocation ;-)
This triggered a bit of almost-systematic scan of scientific literature, the way I was able to do. Eventually, I realized the developmental events at the basis of the Fibonacci ubiquity among plants are still quite unclear. But also, they have been modeled using an electromagnetic analogy (you may find an account in "Mathematical Biology, 1:An Introduction", by prof. J.D. Murray).
If not the "why", this model gave me some highlight of the "how". A good starter!
The experimental setting, on the other side, is not the best to set up in a class. You need a Petri dish (we may find), some oil with high viscosity (this too, with a lot of caution, may be made). And then, a device able to create a tubular magnetic field with intensity increasing towards the border.
An alternative is, simulating the reaction-diffusion process using a computer.
Given the type of problem, an appropriate simulation method is by "multi-agents".
To date, many multi-agent simulation systems can be downloaded from the Internet free of charge. The one I explored, NetLogo, is very well supported by a web site (http://ccl.northwestern.edu/netlogo/) and easy to install and operate. It can be programmed by us, the end-users, even without a professional programmer background (I've made, being precisely that, an occasional programmer who uses Fortran and has no idea of what Java, C# or other modern languages are - my limit, sooner or later I'll close the gap). Honestly, I can't say writing the program was "trivial": it took me some effort to both conceptualize the problem and coding it in NetLogo (to my successful completion, the extensive help and examples did prove more than essential).
And now, it is!
I've included as a text file, named "GermGrowth.nlogo.txt". To run it, you need to (OK, download, then) strip off the ".txt" extension, renaming it to "GermGrowth.nlogo". And then, download and install NetLogo, and run the model.
The model has been conceived as an experimental platform, allowing to see the relation between flower shape and control parameters among which population size and reaction-diffusion coefficients.
Using the model I hope in the following questions:
- Do Fibonacci numbers appear also in this toy-system? Do my artificial daisies resemble the real ones?
- Are spirals visible? Or, is there some other structure?
- Given the number of germs, is their equilibrium arrangement unique? Meaningful?
- How may a complicate equilibrium pattern emerge from germ rules four line of code long?
- Is the computer analogy of the electro-mechanical analogy realistic? May be improved?
There is enough to think... Waiting for someone to solve the real Problem, plant development.