Mathematician Study: Fibonacci and his numbers

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!

29 comments

1. Great post. I agree with you completely that it is astounding how Fibonacci, with all his great mathematical knowledge, came to be famous only for the problem of rabbits. Also, if you are curious to find out more about Fibonacci himself, his encounters with the Emperor Frederick II, survey of his works, or the various pros and cons for the application of the “Golden section” almost universally to every piece of art and nature, may I suggest our little book “Fibonacci, his numbers and his rabbits” (Andriy and Denys Drozdyuk).

Best of luck in your research!

1. Thank you, I’ll check out your book!

2. I’ve enjoyed this post, Claire, and your question of “What’s the point of this?” I think when you get into abstraction, such as you are with the Fibonacci numbers, the same question is asked by many. Just being in awe of such wonderous discovery and the abstract representation of concrete matters might be sufficient to put meaning into this area of human endeavor. I often don’t see the point of astrophysics, but some people spend their whole lives doing research in this area. 🙂

1. LOL! Thank you Hwee. You’re right, of course, sometimes enjoying the beauty of something is enough!

3. I am glad that you continued despite your own misgivings to find the beauty of math with Fibonacci.

1. Sometimes I hit a wall with something that seems so clear to others but I just can’t ‘see’ it. I think this might have been one of those times. I got there in the end though!

4. Brilliant post. I an definitely going to get some of those books now. Thank you!

5. Sounds like a fun unit of math/history study! Here are a couple of videos you may enjoy: Keith Devlin on Fibonacci numbers and Vi Hart’s Fibonacci series, part 1 (with links to part 2 & 3 in the description). And if you have a chance, check out Keith Devlin’s books about Fibonacci, too. [The link goes to my review of Leonardo & Steve.]

I would like to take issue with your concept of math: “Maths, in the main, isn’t based on assumptions but fact.” This is actually the exact opposite of truth. Of all the sciences, math is unique in its total lack of concern for facts.

Math exists only in imagination, and it depends entirely on assumptions. Math says, “If we assume this, then we can conclude that.” But always, we start with an assumption. The assumption defines the imaginary world in which we are going to play the game of mathematics. Will there be straight lines in our math world? Will parallel lines exist? Will numbers be continuous, or are we going to imagine them as discrete? Will shapes hold their form when we move them around, or will we imagine them as able to stretch or compress? and so on…

1. Thanks for the links Denise! And please do feel free to take issue with anything I write! Although it was written as a statement, it was a rather careless one and probably reflects my opinion of maths rather than the truth. From the time we learn maths at school it is taught to us as facts to learn – right or wrong. I have advanced level maths but honestly had never considered it to be anything except a fact based subject which needed to be mastered. Even studying statistics at degree level we weren’t encouraged to question, just to do.
I enjoyed (though didn’t fully comprehend) all the links you left, and it makes perfect sense that maths is based on assumptions, even if that is not what we are exposed to in schools. I suspect this may have been the issue behind me not really grasping the whys behind Fibonacci’s numbers. I appreciate you sharing with me and really wish you lived next door so I could pick your mathematical brain when it came to teaching my girls!

1. I think the statement reflects the majority viewpoint in Western society, so you are far from alone in your opinion. In the U.S., one of the big goals of the Standards for Mathematical Practice is to change that understanding of math, to switch from an emphasis on math as a performance subject to math as exploration, reasoning, and justification. Opinions vary as to whether schools can really change this or whether the heavy emphasis on testing will overwhelm all good intentions, but we can hope.

But also, there really wasn’t any “why” behind Fibonacci’s rabbit story. It was just an excuse to practice with the new Hindu-Arabic number system and to show other Europeans how to calculate stuff. It’s a fun number pattern to play with, but the fact that it connects to a ratio that shows up in nature is an unintended coincidence.

2. By the way, I’ll admit that I don’t understand everything in the links I left, either. My theory on math articles is to read the introductory stuff and keep going as long as I find it interesting, but then to stop when the author goes over my head. I don’t finish much, but I learn a lot that way 🙂

Here’s a less-technical video you might enjoy: Nature by Numbers.

3. Thank you, I’ll check it out!

6. Having never studied Fibonacci for the purpose of math, I just always thought the spiralling sequence was cool 🙂 I’d likely be asking myself the same question about the point of it all were it about math – I do like the way you’ve outlined the books, and best order to read them to better understand/appreciate it.

1. I think just enjoying it as being cool is, well, cool really!!

7. Khan academy have a short video about Fibonacci spirals too. It goes quite fast but made us feel very enthused – we even went out and bought a pineapple for research purposes!

1. Thank you, I’ll check out the video – I like the Khan videos!

8. Elizabeth Hafferty says:

‘Just wanted to say thanks for the post. Such an interesting topic. My 8 yr old enjoys math, so I am going to look for an opportunity to share these books and ideas with him.

9. Thank you for the great resources. As always, you have done such a great job of presenting the material you are studying. I appreciate your honesty in not always getting it right away. I have those issues myself. It is a joy to learn something right along with the children.
Let me just say again, I L O V E your blog. You have been such a blessing to me over this past year. I always look forward to seeing what is going on on your side of the world. I have been running days behind, thus some of the short comments, although I am sure you have enough to read without my “books.”
Hugs to you, my friend:)

1. Aw, thank you Donna! I always miss you when you don’t comment and was wondering if you were okay this week. I’m glad it was just busy-ness rather than illness.
Hugs back to you!!

10. Let’s see if this time it likes me and lets me leave a comment.

I remember learning about Fibonacci in 6th grade, this would be the same teacher who taught me math is fun, so she showed us all the ways the sequence shows up in nature, and so because of that I’ve alwauys found it rather cool.

1. I’m thinking this probably wasn’t my finest mathematical moment! Ah well, you can’t win them all, I suppose!

11. My eldest learnt about the sequence a while ago but I never thought about fining additional literature to support it. You have given me test another fab idea and I must thank you.

Thanks for linking up with the #homeedlinkup

1. Thank you Prudence. I’m so pleased you found something useful!

12. We think Fibonacci is pretty cool and have enjoyed some of the books you mention. He did so much! I’m familiar with the Fibonacci sequence through art as well as math and nature study and find it fascinating. (The rabbit story is new to me.)

13. Wow. Just had a chance to read this properly. What a great post and set of comments you inspired. I feel fortunate we stumbled upon Blockhead first. We read Wild Fibonacci later but didn’t find it nearly as inspiring. I too will benefit from all the links – thank you!

14. Ooo! I just found something new about Fibonacci: He’s got a math game named after him. But his name kinda gives away the strategy, so you might want to call the game “double-Nim” or “super-Nim” or something when you teach it to your kids. (You have already taught them regular Nim, right?)

15. We enjoyed several of these books as well. Fibonacci is so fun for visual math.

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