When I started planning for our study into Fibonacci, I really struggled. Not with the maths of his number sequence, but with the why’s of teaching them. The whole rabbit problem seemed convoluted fantasy, based on assumptions. Maths, in the main, isn’t based on assumptions but fact. I read our selection of books before handing them over to the children, and at the end of each reading my immediate reaction was ‘eh?’. I didn’t get it. I didn’t see its significance. And if I didn’t, how on earth could I teach the children?
I had been really looking forward to exploring Fibonacci with the children, but now I was growing nervous. What could I teach them, when I didn’t get it myself? I guess, though, the key word here is explore. Maybe I didn’t need to know in advance. Maybe we could explore together. It didn’t sit well, this ‘winging it’ with maths. I have two daughters for whom maths is very hard. I needed to understand, surely, to help them understand?
I decided to begin from a place I was comfortable with teaching – facts about the mathematician known to us as Fibonacci. Hopefully, I’d figure it out.
Fibonacci the Man
Fibonacci, known to his friends and family as Leonardo Pisano, is most well-known for a sequence of numbers he demonstrated using a rabbit’s mating pattern. Bizarrely, it is this sequence of numbers which shot him to fame even though it was he who first wrote about the replacement of the Roman-numeral system with our Hindu-Arabic place value, decimal system in his book, Liber Abaci. He was also the man responsible for the placement of a line in between two numbers to denote a fraction. These two ‘inventions’ seem to be far more important than a series of numbers.
Blockhead is all about the life of Fibonacci, from his childhood onwards. It is a beautifully illustrated book, and to be honest, probably the only book you need to read with regards to studying Fibonacci’s numbers. However, I always work under the premise of why read one book when you can read many. So we did.
Blockhead was the last book to arrive in the post and I think had I read this one first I maybe would have ‘got it’ quicker. Unfortunately, it literally arrived this morning. The thing I liked most about this book was seeing how, from the time he was a young lad, Fibonacci saw the world in terms of numbers. He questioned everything. I think if you want to see what ‘living maths’ really looks like, it is shown through his life. The inquisitiveness, the learning from those more knowledgeable than he, learning from the past, from nature and from the patterns all around him.
Fibonacci’s number sequence
The above book also explains very well how this sequence is obtained using Fibonacci’s own method of breeding rabbits and both my younger children and my older children enjoyed this one.
The sequence is as follows:
0, 1, 1, 2, 3, 5, 8, 13……..and it continues with the next number being the sum of the two preceding numbers.
The following is an extract from Fibonacci’s book, Liber Abaci, where he sets himself a problem and then solves it:
How Many Pairs of Rabbits Are Created by One Pair in One Year
A certain man had one pair of rabbits together in a certain enclosed place, and one wishes to know how many are created from the pair in one year when it is the nature of them in a single month to bear another pair, and in the second month those born to bear also.
He then goes on to solve and explain the solution:
Because the above written pair in the first month bore, you will double it; there will be two pairs in one month.
One of these, namely the first, bears in the second month, and thus there are in the second month 3 pairs;
of these in one month two are pregnant and in the third month 2 pairs of rabbits are born, and thus there are 5 pairs in the month;
there will be 144 pairs in this [the tenth] month;
to these are added again the 89 pairs that are born in the eleventh month; there will be 233 pairs in this month.
To these are still added the 144 pairs that are born in the last month; there will be 377 pairs, and this many pairs are produced from the above written pair in the mentioned place at the end of the one year. You can indeed see in the margin how we operated, namely that we added the first number to the second, namely the 1 to the 2, and the second to the third, and the third to the fourth and the fourth to the fifth, and thus one after another until we added the tenth to the eleventh, namely the 144 to the 233, and we had the above written sum of rabbits, namely 377, and thus you can in order find it for an unending number of months. Source
Assumptions are many concerning deaths and number of baby rabbits made and so forth. I understand the maths, the adding the previous two numbers together. What I struggled with was the point of this arbitrary (to me) sequence.
Fibonacci Numbers and Spirals
Swirl by swirl is a lovely introduction to spirals in nature. Not strictly a Fibonacci book, it nevertheless demonstrates the strengths and therefore the importance spirals have in the natural world. I think this is a very good picture book which extolls and explains why spirals are such a strong and prevalent shape in nature and therefore a great book to read before moving on to the next book which starts to link the spirals with Fibonacci number sequence.
How Fibonacci Numbers Make a Spiral
This book, page by page, lays out very simply in words and pictures how the Fibonacci numbers create some of the spirals found in nature. It also demonstrates clearly how to count the spirals in cones, pineapples and the like.
We tried to look at some of the examples accessible to us which had been mentioned in the book:
Fibonacci Numbers and spirals and curves
This was the only book (bar Blockhead) to mention curves in relation to Fibonacci. Did you know that most of the important curves in nature align themselves to some part of the spiral which Fibonacci’s sequence of numbers produce? The book gives examples of walrus tusks, pelican beaks, eagle talons, sea-horse tails…
It is now understood that the Fibonacci sequence has a special significance. It is a ‘blueprint that describes how living things grow in an orderly and harmonious way’ (Blockhead p39)
It wouldn’t be until we researched the area more fully that we would come to fully appreciate how beautiful these numbers really were, just how aesthetically pleasing they naturally were to the human eye and how artists such as Da Vinci have used the geometry of the sequence to create art which some would say had a touch of the ‘golden ratio’ about them!