There’s maths in art… No i’m not going crazy!

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.

013I 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.

010As 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.

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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.)

014Renaissance 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]

Profound Understanding of Fundamental Mathematics

 “Teachers must have a profound understanding of fundamental mathematics.” – Liping Ma, 2010

009To teach maths, you must first understand maths. To understand maths you have to be fully engaged and willing to put yourself in your pupils’ shoes and learn. Liping Ma states that this is one of the most important factors in terms of enhancing teachers’ knowledge of, and ability to better teach, primary mathematics.

However, having a concrete profound understanding of the fundamentals of mathematics is much more than being able to understand primary mathematics. You have to make sure you have a sound knowledge of the overall theoretical structure and basic attitudes of mathematics. By having this, you will then be able to use this as a foundation in order to teach your children effectively and inspire them to want to learn maths for themselves and not because they have to. It is very important to have a conceptual and procedural understanding – to know how and why we do something – that is deep, broad and thorough.

In order for yourself and your pupils to meet what Liping Ma states as a profound understanding of the fundamentals of mathematics you have to make sure that you can identify and understand the four principles that are essential in being able to meet this ultimate goal.

The first of these principles are (inter) connectedness. This refers to being able to see connections between concepts and procedures. Meaning that children can see why there is a need to have elements of maths that connect together in order to be able to use at other points of your mathematical education. This could be as simple as letting the children understand that in order to move on to understand and describe two dimensional shapes they first have to know the properties and characteristics of two dimensional shapes. This will then help to ensure learning is not fragmented, but viewed instead as a unified body of knowledge.

The second principle is multiple perspectives. This is having the ability to comprehend and also appreciate the different approaches you can have to one specific mathematical problem. This therefore encourages a more adaptable way of thinking and is therefore not restricting any child as it is not focused on one learning style. Teachers who have the ability to develop multiple perspectives for every topic within mathematics will have a better sound knowledge of the fundamentals of maths overall.

Thirdly, Liping Ma talks about basic concepts and having a full awareness of all the central ideas that surrounds primary mathematics. It is important that the basic ideas that recur throughout maths are constantly revisited until they are fully reinforced and have created a solid foundation in order to move forward onto future concepts. Without basic concepts, we would not be able to move forward and enhance our mathematical ability.

The last feature is longitudinal coherence. This is having a full awareness of the entire mathematical curriculum and how one basic idea or principle can be built upon another. What is taught today becomes the base for future knowledge, just as current mathematics teaching builds upon students’ previous knowledge, however fragmented that knowledge may be. This allows for there to be much more understanding and flexibility in terms of where learning is headed as lessons can be tailored with this in mind.
With this profound understanding of fundamental mathematics, we as teachers will be able to teach students more successfully. I fully agree with everything Liping Ma says and believe that I will definitely work to this model in order to teach mathematics to the best of my ability to my pupils in future.

008

Ma, L. (2010) ‘Knowing and teaching elementary mathematics’. London: Routledge.

 

Lets have fun, lets do maths!

One of the most important factors for me to have children engage with mathematics at a level that they want to learn and more important understand what they are learning is to make sure they enjoy what they are learning. In order for this to happen it is up to us, as teachers, parents and careers to make mathematics enjoyable and allow them to see that maths can be used whilst playing.

As an adult being able to play during a maths input put a smile on my face. I loved it. Being able to physically get to grips with what we were learning through the use of building blocks, games and other various maths resources that can typically be found in an everyday classroom and most home environments was so much more refreshing than a boring PowerPoint.

007Allowing children to play is an extremely central factor in their overall holistic development. It allows them to make connections to their learning, especially when they are in a relaxed environment. It empowers them to experiment and encourages creative and flexible thinking.

The National Scientific Council on the Developing Child recognises that child development is the key to the future success of a society.  They define the core concepts of development as including “cognitive skills, emotional well-being, social competence, and sound physical and mental health” They also stress that if these areas of development are nurtured in the early years through quality learning environments, positive relationships, and engaging social interactions, a foundation will be laid for future successes that everyone wants for every individual child. Some of these successes that the NSCDC describe as things such as; positive school achievement, future economic productivity, as well as responsible citizenship.

005A simple thing such as the building blocks we used within our input are small but so effective. They can help a child start there numerical experiences by counting one by one, moving onto addition and subtraction and then can be used further up the school for more challenging lessons such as cubic centimetres. We take for granted as adults how easy counting actually is. We do not remember the stress and anxiety that we had to go through when learning the simplest fundamentals of mathematics. This step is one that has to happen though; Liping Ma states that we have to be aware of the basic concepts of maths and these concepts have to be reinforced and revisited before children have the foundation to deal with future concepts and can then move onto more complex mathematical problems.

My initial concept of maths has completely changed in the few weeks that I have being doing the Discovering Mathematics elective and I feel this is due to the fun that has been incorporated into lessons and inputs. By being able to work with friends to solve problems or to play maths based games brings out feelings towards maths that I never knew I had. As a student teacher, this is one of my main goals when teaching maths to my pupils. I personally feel that I will move away from the old games and resources, as good as they were, I used them 17 years ago so they are a bit outdated. But with all the technology we have now a days and how easy it is to access different resources I hope to let all my pupils know that maths does not need to be hard, stressful or boring… Maths can be fun. 006

Haylock, D. (2010) Mathematics explained for primary teachers. 4th edn. London: SAGE Publications

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

Han’s The Magical Horse

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 Vo001n 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.

002Osten 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.

003Ayumu 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].

I HATE MATHEM… Oh actually it’s not that bad!

Primary 5… some dreary day in mid-November. Maths test. The dread I felt when the teacher uttered the words the moment we came back into the classroom from morning break have never left me. I would not say I ‘fear’ mat1110express-student-fearshs however, it genuinely makes me feel uneasy. I think the main reason for this was this specific maths test. It was mental maths. 20 questions. And we had 10 minutes to answer as much as we could (no working allowed – including the use of fingers!!) It’s safe to say I never done very well, 7 out of 20. Although, at first, I was rather pleased with myself for getting that much correct, until my teacher stated that ‘you can only be good at either English or maths; you cannot be good at both.’ So from that moment, I had always considered myself rubbish at maths so really… Why try?! Eastaway and Askey state that people’s mathematics anxiety can develop from a parent or teacher but mainly it is not the fear of maths itself but the fear of being shamed. (Eastaway and Askey, 2013, p15.) Personally, for me, I feel that, this point in my education was one of the main factors as to why I feel anxious about mathematics.

As I have grown the idea that my primary 5 teacher had fixed in my head that you can only be good at maths or English slowly but surely started to vanish. I do not believe that if you are good at English, you cannot be good at maths, or vice versa. I have seen first-hand, many people that have the natural ability to be good at both.

You may prefer one subject to the other, therefore may shine in that said subject but this does not mean that you cannot then excel at the other if you put the work in for it. Eastaway and Askew tell us that, ‘there is no such thing as a maths gene’ (Eastaway and Askew, 2013, p14.)

According to these men, today’s society is much more sophisticated in maths compared to those in medieval times. Showing that over time we have adapted to the different concepts and ways to understand math that categorically there cannot be a gene that has programmed us to ‘be good at maths’ (Eastaway and Askey, 2013, p15.)

One of the main reasons for choosing ‘discovering maths’ was to get over this anxiety I had surrounding maths. I can already categorically say that I am so glad I choose the elective I did.

Our first lecture, ‘What is maths? Why teach it’ was an eye opener for me. I went in to this lecture a little apprehensive and left with a new excitement surrounding mathematics that I had not felt before. This was due to the lecture being made fun and relatable. The main task set to us was to work out how many snaps it would take to break up a bar of chocolate that had 64 squares. By working together in groups for this task with actual physical props we were able to explore maths in a way I don’t think I ever had. It brought out discussion, conversation, sharing language and most importantly, play.

As a learner, being able to relate my maths to real life was very important to me and I feel helped me understand clearer and ultimately enjoy what I was doing. As a teacher working with children that may feel anxious and withdrawn for maths, I will strive to ensure that I will relate it to real-life as much as I possibly can. By doing this and adding in the ‘fun factor’ I feel children will not had this fear of mathematics that most do today. This will hopefully give the children an insight as to why they need maths and I hope that they will never be asking themselves ‘when will I use this again?’

Eastaway, R., Askew, M. (2013) Maths for Mums and Dads. Square Peg. London.

Scientific Literacy

Group One

Lucy Allen, Corrie Donaldson, Eildih Lamont and Skye McLauchlan

Assessment criteria:

  • AC1 – Explanation of the concept of scientific literacy – Skye McLauchlan
  • AC2 – Analysis of an example where a lack of scientific literacy has led to inaccurate media reporting.
  • AC3 – Discussion of how teaching fair testing in school science links to scientific literacy.
  • AC4 – A carefully researched and referenced paper on scientific literacy.

Oxford University Press describes being literate as, “the ability to read and write” (Oxford University Press, 2016). Without the ability of reading and writing we would lead a very sheltered life where the most basic of tasks are problematic for us to carry out.

Therefore, scientific literacy could be broken down and be said that it is simply having the ability to understand different concepts and ideas in order to succeed within science.

Scientific literacy is also be the ability to use your knowledge of science in order to identify questions and to have the ability to come to your own conclusions through the use of looking at different sources of evidence. (OECD, 2003)

Scientific literacy is slowly but surely becoming the focused and primary goal for primary school science. (Millar. R, 2007). However this by no means, is saying that all children learning from the science curriculum are going to become scientific geniuses. In light, it is giving the children the chance to fulfil an “enlightened role” (European Commission, 1995) in being able to make and inform their own choices and ideas throughout their learning within science.

A lack of scientific literacy can lead to inaccurate media reporting’s and false belief from the public. For example, the outrage caused when it was, apparently, scientifically proven that the MMR vaccine caused Autism in babies. This belief has managed to frighten mothers into not allowing their babies to be given the vaccine. However, the evidence for the connection between the MMR vaccine and autism was based on a girl, Hannah Poling, in the USA who experienced a reaction to the vaccine. The author of the study claims that Poling was then diagnosed with autism but this is not true. Poling had a very rare mitochondrial disorder, due to her genetics not the MMR vaccine, which was made worse due to her bad reaction with the MMR which led to the occurrence of a few autism like symptoms, but not an autism disorder. Although this report made about MMR and Autism was not necessarily true, it still managed to make an impact on the public as there has been a revival of diseases such as measles and whooping cough, which could be avoided if a child is given the MMR Vaccine.

Having an understanding of fair testing is pivotal to scientific literacy because without it a child will be unable to effectively draw evidence-based conclusions or fully understand and facilitate scientific knowledge- both equally important aspects to science literacy. Understanding what can be changed and what cannot be changed in an experiment to make it fair is massively important to ensuring the results are accurate. For example, if one is comparing and testing the fuel efficiency of cars and one is driven by an experienced professional driver and the other by a newly qualified driver, this would not be a fair test as there are too many variables between the drivers. In order for a test to be fair, there must only be one variable that is changed. For example, if one is testing the absorption rate of paper, then only the type of paper used should be changed but the liquid should remain the same. If both the variables were altered (the type of paper and the amount/density of liquid used) then the experiment would not produce accurate results and therefore would become pointless. Learning to become conscious of the importance of such variables is central to a child’s scientific literacy and therefore vitally important to teach. As scientific literacy is becoming the number one goal in primary schools for science, teaching the concept and rules of fair testing is consequently a vitally important aspect of what we want our pupils to know and understand by the time they leave primary school.

European Commission, (1995) White Paper on Education and Training

http://theconversation.com/muddied-waters-setting-the-record-straight-about-mmr-vaccinations-and-autism-3391

Millar. R, (2007) Scientific literacy; Can the School Science Curriculum Deliver? Communicating European Research 2005, Pages 145-150.

OECD [Organisation for Economic Co-Operation and Development](2003) The PISA 2003 Assessment Framework – Mathematics, Reading, Science and Problem Solving Knowledge and Skills. Paris: OECD

Oxford University Press, (2016) Literate – definition of literate in English from the Oxford dictionary [online]. Oxforddictionaries.com Available at: http://www.oxforddictionaries.com/definition/English/literate [assessed 9 Feb 2016]

Tim the Turtle Animation

The first ICT input we had in semester one was a much more hands on and exciting experience than I thought it would be.

For me at primary school, ICT was sitting in the computer suite making tedious and repetitive PowerPoint’s on Mary Queen of Scots or the Ancient Egyptians. That was not for me. However, working with Sharon Tonner, my eyes were opened to different the ways we can interact with ICT in a more interactive and stimulating way.

My favourite of the different methods Sharon showed us during the input was the ZU3D programme. Within this programme we used all the different features of ZU3D. From this we were able to then put together a short animation of which we included, title pages, music and the use of play dough in order to create characters and a setting.

Tim The Turtle