This post is going to be related again to mathematics and art and how they are linked. Whilst writing my last post about maths and art in the classroom I realised I was going on and on into a completely different topic – although still to do with maths and art. So this post is going to be about maths and art but a more complex example. The more I am blogging the more I realise I just keep typing and typing and typing until I read it back and nothing really makes sense because I’m writing away at a hundred miles per hour. So here’s to the second post about maths and art… discussing Fibonacci and the Golden Spiral.

Fibonacci

The Fibonacci sequence is as follows: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55…

So to get the subsequent number all you do is find the sum of the previous two numbers. This can go on for ever and ever but all we need just now is the first bunch of numbers. The Fibonacci numbers also link in with The Golden Ratio, this is a value of 1.61803399 and is represented by the Greek letter Phi.

The Golden Spiral

The golden spiral is then built from Fibonacci’s sequence numbers, we had our own chance in class to try out the process to come up with our own golden ratios. In order to do this we used the Fibonacci sequence numbers which were then our dimensions to create adjacent squares.

Here is my example, which was ok I must say for my first attempt (using a compass for the first time in a few years was harder than I remembered). Already you can see just how much everything adds up – the Fibonacci sequence numbers give us the widths of our squares, which are then placed perfectly next to each other making rectangles(which widths are also always in the Fibonacci sequence) and then when we use a compass we can create a spiral that passes through the points accurately. Amazing yet confusing!

Here is a better example of how the Fibonacci sequence numbers make up the golden spiral. Because the Fibonacci numbers give you the length of the squares, which are then put next to one another and everything fits and adds up perfectly.

So you may be thinking how is this art? But the golden spiral is in a lot more places than you can imagine – even within nature itself. Here are some examples of the Golden Spiral in both art and nature, it truly does show us again that mathematics is everywhere.

This highlights to me the link with Ma’s idea of connectedness. Mathematics is not only the one subject but goes way beyond this linking with other curricular areas such as art, but even further than that it is in our everyday life and nature. The galaxy includes the golden spiral, shells contain the golden spiral – mathematics is truly connected to almost everything and we just need to look beyond the basics to realise this.

As a teacher I think this is something I would definitely have a go at teaching because for us at university it was really interesting to learn about however I also think children would be amazed by this idea. Although this could be a maths or art lesson the importance to me would be to highlight to the children how mathematics connects with so many things we may not automatically think about. This will ensure the children are aware of the fact that mathematics is everywhere – and not just a subject they need to do in school.

In one of our inputs we had the chance to come up with our own number system, we had the freedom to do whatever we liked – as long as it made sense to us and we could explain our thinking.

Although when you sit down to begin a task like this you try to come up with something totally original that makes sense, the reality is you probably end up creating something that either already exists or is similar to an existing number system. But never the less it was a challenging task and something that we all became engaged in, and many people came up with some really interesting ideas.

This was my partner and I’s number system that we came up with.

Our number system was very much similar to the one in which we all know – apart from the fact it looks completely bizarre. We went along the lines of having a base of 10 number system like ours and had numerals that represented each number. So each number from 0-9 had a numeral which represented it, and then like our own when we get to 11 we just put together the numeral for 1 and put 2 of them. At the time we never quite realised that we had stuck so much to our number system but when we came to discuss it we realised it was very much the same. One thing we didn’t quite get to was coming up with the language to represent our number system, although we had it all planned for how it would be written, we had no idea what our numerals would sound like.

So where did our number system today come from? We use Hindu-Arabic numbers, which were created by Hindu’s in India years and years ago. Before this we used roman numbers which we are all familiar with – I, II, III, IV, V. However when European countries seen the Hindu-Arabic number system they quickly realised it was a much easier and simpler way to work with numbers and perform calculations. Fibonacci supported this (Bellos, 2011, page124) as when he was first exposed to Indian numerals he realised that they were much better than Roman ones, he then went on to write a book about the decimal place – value system which demonstrated how much faster using Indian numerals allowed you to perform calculations. And since then we have stuck to this number system which is engraved in our minds in a way which makes it difficult to forget or think of something new – as proved when I tried to make a ‘new’ number system.

Our number system is crucial in our everyday lives, from using mathematical language to completing equations – it is everywhere. But it isn’t often we really think about our number system in any depth, we just accept the fact that we understand numbers because it is part of our everyday life. Whereas Ma (2010, page 122) highlights that with a profound understanding of fundamental mathematics you should look within mathematical concepts and be able to make connections beyond what you know, and that is what we done in class when we went beyond the existing number system we know and tried to come up with something new – but still be able to find the connections within it. Therefore Ma’s idea of connectedness proves that we as teachers need to not only use the mathematics we know, but go beyond it and discover basic and complex connections within every day mathematics – for example looking at our basic number system and asking questions like where did it come from? who invented it? when did it come around? what did we use before this? And the answers give you connections within mathematics that you didn’t think about before.

References

Bellos, A. (2011) Alex’s Adventures in numberland. London: Bloomsbury Publishing Plc.

Ma, L. (2010) Knowing and teaching elementary mathematics: Teachers’ understanding of fundamental mathematics in China and the United States. Anniversary Edition. New York: Routledge.

From my experience of placement earlier this year I definitely noticed that the general opinion from my class was that maths was a boring subject and art was a fun subject. I done a few art lessons with my class and they were always fully engaged and were eager to do the work, I think this is because their teacher done a lot of art with them and her love of art passed on to the children. This is an example of how it is crucial for us as teachers to be enthusiastic about every curricular area we teach because it definitely reflects on the child’s opinion of a subject. Anyway, my class loved art and were really enthusiastic about it whereas maths they just got on with but they didn’t look forward to that part of the day. So in my class the children didn’t seem to link the two subjects together as they seemed to have an overall different opinion on them, however they didn’t realise just how much mathematics they were using – and to be honest neither did I.

Every little thing near enough links back to mathematics, this is when you realise just how much maths is literally everywhere. Here are some examples thinking back to my art lessons:

• Asking the children to collect 2 pieces of paper and 1 paint brush – they need to understand quantity and how to do basic counting.
• Asking the children to half their piece of paper – they have to understand the mathematical language of ‘half’ and how to do this.
• Drawing shapes – they need to have an understanding of different shapes and how to draw these.
• Working with a time limit – basic skills of how to tell the time and also how to manage their time.
• Writing the date on the back of their work.
• Working in pairs – understanding the language again, automatically knowing what a pair is.

These are just very basic examples of how the children and I used mathematics without even thinking it. These all link very closely with one of Liping Ma’s four properties – basic ideas. All the children in my class could do all of the above and understood what they were doing, but it all started when they were much younger learning the simple but powerful basic concepts and principles of mathematics. (Ma, page 122) When children grasp these basic concepts and principles such as simple equations, it then allows them to build on top of these foundations to then become more confident in mathematics and explore into much more challenging concepts. The fact that any child in my class could half a piece of paper or collect 2 paintbrushes highlights that they already have engraved in them the basic ideas within mathematics.

That is a simple example of how mathematics links with art – and to be honest any other subject area. Because like I said maths really is EVERYWHERE!

References

Ma, L. (2010) Knowing and teaching elementary mathematics. Teachers understanding of fundamental mathematics in China and the United States. Anniversary Edition. Routledge: New York.

Some people think this question is ridiculous and that of course animals can’t understand numbers in any way, but is that really the case?

After leaving our input discussing this I was left confused to say the least, in this module you are constantly thinking deeper into the mathematics behind things and I often feel that one minute I understand something and then a minute later I am back at square one. So I am going to try come to terms with the question of animals being numerate through the example’s of Clever Hans the horse and Ayumu the chimpanzee.

Clever Hans is an example in the topic of animals being ‘numerate’, Clever Hans was a horse whom became extremely famous for being able to ‘count’. But was that really the case?

When a number was called out Clever Hans would tap his hoof the exact number of times and people were truly amazed – a horse could understand mathematics. For years the horse was taken into crowds and it would get asked questions in which he would respond by tapping his hoof to represent the number he meant. For example if Clever Hans was asked the question “What is 2 + 4?” he would tap his hoof 6 times. People couldn’t believe that this horse could actually understand mathematical questions and could then represent his answer with his hoofs. However it wasn’t long before scientists got involved and began to question what was really going on, they came to the conclusion that the horse couldn’t actually understand the mathematics at all but was being ‘signalled’ in some way by the owner/person asking the questions. So when the horse was put behind a closed door with no one to look at he no longer got the answers correct. Also if the owner/person asking the questions didn’t know the answer then the horse would answer incorrect. To me this proves that the horse was definitely not numerate because he could not understand the meaning of numbers at all, he was purely given visual aids on when to start and stop tapping his hoofs.

However we then looked at Ayumu the chimpanzee, he is very well knows for taking part in memory tasks and getting results which are outstanding – and a lot better than humans. The chimpanzee took part in a short term memory task which requires him to remember the sequential order of numbers, and the results he gets are amazing. Ayumu can do it with no difficulty at all, he gets it correct most of the time and if he makes a mistake he will get it correct the next time. As well as that, the speed he completes the task in is simply mind blowing. When we tried this in our input our results were not quite the same as Ayumu’s, with around 20 of us in class we struggled to remember the order of the numbers past 4 or 5. So what does this mean? Is Ayumu numerate or does he just have a good memory?

Clearly the chimpanzee has amazing memory because he only see’s the numbers for a small amount of time and can then immediately recall them, but to me he must understand numerals to be able to do this task because without this how would he know which number to tap.

Matsuzawa (2010, page 1 ) highlighted that Ayumu had mastered the ability to touch the nine numerals in the correct order as they appeared randomly arranged on the screen. A numeral is a symbol which represents a number so surely the chimpanzee understands this concept? He may not be able to actually apply mathematical skills with the numerals or numbers, such as if there were 2 phones beside each other he wouldn’t associate that with the numeral 2. However , he does understand the concept of numerals as he remembers the symbols which represent the numbers. Ayumu must also grasp the idea of ordinal and cardinal numbers, because although he understands numerals, how does he know which numeral comes after the other? He understands cardinal numbers (or at least the numerals that present these – 1, 2, 3, 4…) but he must also have knowledge of ordinal numbers to understand the idea that 2 comes after 1 and 4 comes after 3 etc. There are many questions and theories about how the chimpanzee really thinks and acts but to me it is very clear that Ayumu is numerate in some way. He understands numerals and the order in which they come (ordinal and cardinal numbers), and also understands what he has to do in order to be correct each time – tap the numbers/numerals in order.

Animals surprise us day by day but is it likely one day we will see them taking part in a maths exam? Who knows.

References

Matsuzawa, T. (2010) The Mind of the Chimpanzee: Ecological and Experimental Perspectives. United States of America: The University of Chicago.

After a long 3 and a half months it was finally time to get back to Dundee and start my second year of primary education at university. I think like everyone you feel as if your brain has shut down and you forget how to get back into the swing of doing work, but after a few days back at uni I feel as if I had never been away. Blogging is part of our discovering mathematics module this year which made me feel a bit nervous. Although I was blogging throughout first year I wouldn’t say blogging is something I am confident in doing so hopefully by keeping up to date with it this year I will become more confident within my writing.

Anyway what this blog post is really aimed at is my previous experience of mathematics and why I chose the Discovering Mathematics module.

At primary school I loved maths as a subject, it was definitely my favourite and something I was very confident in. I would always be that annoying child who would ask if we could play the game sparkle when we were allowed a class game (it was based on times tables), as you can imagine most people in the class went in a huff and said it was a stupid idea. After being out on my placement earlier this year I definitely seen the same thing happening, if a child wanted to play a mathematical game the majority of children would be huffing and suggesting other games such as ‘who stole my pencil’ and ‘head down thumbs up’ – games which I didn’t realise would still be so popular now as they were when I was at primary school. Therefore in primary school I definitely had a very positive experience with maths and went onto high school with a positive attitude.

When I got to high school I still really enjoyed maths, up until the end of 4^th year when I was sitting my standard grades I loved it. It was a subject that I enjoyed going to class and even when I was revising for my exams I wouldn’t mind sitting doing past papers for maths over and over again, I really did enjoy maths and ended up getting a 1 in my exam. So after so many years of enjoying maths and being really confident in it I didn’t think twice about taking higher maths when I went into 5^th year. Maths quickly turned from my favourite subject to my worst subject. A few weeks into higher maths I realised it wasn’t anything like standard grade, it was a huge jump and I quickly fell behind. My best friend and I sat next to each other in class which probably didn’t help things as we chatted thinking we could easily do work at the same time – we were wrong. As the year went on I ended up with a tutor however did terribly in my prelim and my teacher decided it was best if I dropped out and continued it in 6^th year. Although I was disappointed in myself I concentrated on my other Highers and forgot about maths for the time being. 6^th year came and I started maths again… I HATED IT. I dreaded going to class and wasn’t motivated at all. History repeated itself and I ended up getting to the stage where I knew I needed to drop out, this time was definitely gutting as I knew it was my last chance. I left school with good results and I was proud of myself however I just knew in the back of my mind I should have left with my higher maths. Therefore in high school I went from loving maths in 4^th year to leaving school in 6^th year absolutely hating it.

When we had the chance of choosing our modules for second year I seen Discovering Mathematics and was on/off about it, but I soon decided that I would go for it. For a long time maths was something I really loved and I was simply put off it because of higher maths, so I thought this will give me the chance to hopefully look at maths in a positive way again because I knew we would not only be looking at sums and equations but going much deeper into what the meaning of mathematics is and where it came from. So here’s hoping to finishing this module and feeling positive about mathematics again.