Until I chose ‘Discovering Mathematics’ as my elective this year I was totally unaware just how much mathematics affects us in our everyday life’s during absolutely everything we do. One of the recent things that I have learnt is that maths is used just as much in a maths class as it is in an art class. Yes that’s right, an art class.

I was a little sceptical at first as to what Anna was going to speak about for an hour but I can truly say that this was probably one of the most interesting inputs I had attended for a while. Before I go onto speak about the more interesting points I learnt, I realised that almost every little thing in art can be easily linked to maths and when you actually think about it, it is so easy to see why. From things such as the shapes they are drawing, the time they have to complete their art piece, asking the children to collect a certain amount of resources e.g. brushes, paints etc. This is art relating to maths in its most basic form however, this links nicely with Liping Ma’s ‘basic ideas principle’. As without being able to form these simple tasks then the pupil would then not be able to go on to complete their masterpiece – because everything the child creates is fab!

Going back to the more interesting points from the art meets maths input, I am now aware that a lot of artists use specific sequences in order to plan and actually create their art. We looked at the Fibonacci sequence, the Golden Spiral and the Golden Ratio. To begin with each concept absolutely blew my mind however the further the lesson went on the more I began to grasp the idea.

Starting with the Fibonacci sequence, I learnt that this is a series of numbers. 0, 1, 1, 2, 3, 5, 8, 13, 21, 34… It is a very simple sequence that when broken down is just adding the first number to the second to give you the third and then the second to the third to give you the fourth and so on… This sequence was named after mathematician Fibonacci or formally known, Leonard of Pisa. He found this pattern by noticing a recurring simple numerical series found commonly in nature. As well as being found in nature, this exact sequence has also been used by artists when creating their images. For example, Piet Mondrian has been known to have used it within his art work.

As the Fibonacci sequence is found in natural objects and can be seen when drawing the ‘golden spiral’. We drew this spiral on the Fibonacci sequence following prewritten instructions and pictograms. The outcome was ultimately this spiral. This spiral appears over and over in many natural occurrences and this picture shows just a small percentage.

Moving away from Fibonacci, we then looked at mathematician Luca Pacioli who published an article in 1509 on the ‘Golden Ratio’. The Golden Ratio is when we take any two successive Fibonacci Numbers, and divide the larger number by the smaller. The answer will always be this special number approximately around 1.618. This ratio, symbolised by Phi (Φ) appears within mathematics, art, architecture and other areas. It was also used to design the Notre Dame in Paris. The ratio features also in the United Nations building and the pyramids in Egypt (Boaler, 2016.)

Renaissance artists also used this ratio to inspire their beautiful and balanced artwork. The Last Supper painted by Leonardo Da Vinci is also associated with the Golden Ratio and Fibonacci sequence. This painting has clear examples through the design and architectural features to be said that the golden ratio was used. Some also believe that Da Vinci even positioned the disciples around the table in proportion to Jesus using the ratio.

It excites me to now go into a primary classroom and explain, in a more child friendly way, everything I have learnt. I feel that showing the connections between different subjects, particularly ones that seem more appealing than others will have a great impact on the way some children think about mathematics overall. Well… That’s the wish I guess.

Ma, L. (2010) Knowing and teaching mathematics: Teacher’s understanding of fundamental mathematics in China and the United States. 2nd edn. New York: Taylor & Francis

Fibonacci Sequence (2016) Available at: https://www.mathsisfun.com/numbers/fibonacci-sequence.html [Accessed on 28 November 2016]

Design in art: Repetition, pattern and rhythm (2006) Available at: https://www.sophia.org/tutorials/design-in-art-repetition-pattern-and-rhythm [Accessed on 28 November 2016]

The question can animals count arose during one of our mathematics inputs and it got me to thinking, well can they?

We had been looking at Von Osten and his clever horse, Hans. Osten had a keen interest in animal intelligence and with the help of Hans, this would ultimately win him some degree of fame.

Osten believed firmly that animals had skills and a degree of intelligence that humans as a race had dismissed fully. To show the world that actually, animals can understand maths, he started to tutor a cat, who was unresponsive to his work. He then moved onto a bear, who tried to attack him (like a normal, wild bear would) and finally he tutored Hans the horse. Hans learned to work with Osten and when a number under 10 was written on a board, Hans would be able to tap out the correct number with his hoof.

Osten toured ‘Clever Hans’ all over Germany to show off the horse’s mathematical abilities. More and more people heard about this clever horse and the crowds grew larger and larger. The curious onlookers were seldom disappointed.

However, there were the sceptics. Oskar Pfungst, believed he could unravel the mystery behind this ‘clever horse’. Pfungst created a large tent to house his experiments, in order to eliminate the effects of outside visual stimuli. The outcome was simple; Hans performed very well when questions were asked by his owner, Von Osten. However, when the questioner was not his owner and was made to stand out of sight of the horse, something interesting happened: the horse’s accuracy weakened, though it wasn’t immediately clear why. (Bellows, 2015)

In the years that followed, it has been found that many animals are responsive to subtle and unintentional cues from their human masters. To prevent prejudgments and foreknowledge from contaminating experimental results, modern science employs the double-blind method where researchers and subjects are unaware of many details of the experiment until after the results are recorded. For instance, when drug-sniffing dogs undergo training, none of the people present know which containers have drugs in them; otherwise their body language might betray the location and render the exercise useless.

Even though, Hans may not have been able to understand the fundamentals of mathematics this don’t not stop my curiosity so I then done more research into Ayumu the chimpanzee.

Ayumu the chimp, son of Ai, a chimpanzee whose intelligence has been studied for over 30 years by Professor Tetsuro Matsuzawa, can remember the location and order of a set of numbers in less time than it takes the average human to blink. (BBC Nature, 2012) The test was undertaken at Kyoto University in Japan and the chimp managed to solve the puzzle in a remarkable 60 milliseconds. The test was more a test of short term memory which required him to remember the sequential order of numbers which 9/10 he was able to get correct first time. There is no hesitation of Ayumu either, he knows straight away where all the numbers are and completes the task at a mind blowing speed. Regardless of the undoubtable facts that Ayumu can complete this task, does this mean he has a fundamental understanding of mathematics? Does this mean that Ayumu is numerate?

As a class, we tried to do the simplified game and our results were somewhat poor… We were unable as a class of 20 to most times get past the first few numbers. However, does this mean we are not good at maths? Or that our reactions times aren’t as developed as Ayumu’s? Or does it simply mean, our memories are not as advanced as a chimp like Ayumu?

As well as these two well documented experiments there are scientists everywhere trying to prove that animals can understand mathematics and can count. A lot of people are very sceptical of the whole theory and believe that it is just the ability to use their memory better than that of a human memory. Others believe that animals are able to associate action to words or numbers like Hans the horse. Personally, I am undecided on the answer wither or not animals can count but I do believe that all animals are very clever in many different ways and I am sure that we will find out in the near future if they are as clever as we think they are.

BBC Nature. (2012). Chimp sets memory puzzle record. [online] Available at: http://www.bbc.co.uk/nature/16832379 [Accessed 28 Nov. 2016].

Bellows, A. (2015). Clever Hans the Math Horse. [online] Damninteresting.com. Available at: https://www.damninteresting.com/clever-hans-the-math-horse/ [Accessed 28 Nov. 2016].