Category Archives: Discovering Mathematics

The Art of Tessellations

One particular thing that interested me in the Discovering Mathematics elective was learning about tessellations. A tessellation is when shapes are fitted together and repeated over and over to cover a surface. There are no gaps between the shapes and the centre angles add up to 360 degrees. Shapes often used are equilateral triangles, squares, stars and hexagons. Tessellations originate from tiling found in the Alhambra in Spain in the 14th century. However, they were made popular by Dutch artist M. C. Escher in the 20th century. His work often featured animals unlike the typical geometric tessellations. I remember my teachers at primary school used to have a pad of colouring sheets that were filled with different intricate patterns and I loved getting one to colour in after finishing my work. I realise now that those patterns were tessellations. I was never explicitly taught about tessellations but I can see how they would be fun thing to do with the children in the future. Furthermore it links maths with art which would hopefully appeal to children and keep them engaged in the maths side of it.

Above is an example of making our own tessellations in class.

Above is an intricate example of tessellations in tiling.

Above is an example of where tessellations occur naturally within a bee hive.

Life long learning

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I feel like I have learnt a lot from the Discovering Mathematics elective. Even little random facts like what all the digits mean on a best before label on packets of food. It’s interesting because at one point or another I have wondered about that and now I know the answer. I have learnt practical skills like how to read a six grid reference off a map. I have learnt interesting things about maths and art such as tessellations and Islamic art. I’ve always had a passion for learning and this module has ignited a particular interest in maths. I never used to think maths was exciting, I just thought it was something you do.

The elective has opened my eyes to all the different fields maths features in and its fascinating. I have particularly enjoyed reading three of the books from the reading list: –

‘How many socks make a pair?’ by Rob Eastaway
‘Alex’s Adventures in Numberland’ by Alex Bellos
‘Maths for Mums and Dads’ by Rob Eastaway and Mike Askew

I think these books make maths fun and accessible for all abilities. Maths can be a scary thought for a lot of people, however, these books are quite light hearted about maths. When you read them, it doesn’t feel like maths at all, especially the Bellos book. That is written as if he is on an exciting round the world adventure. You forget that the book is actually about maths. The other two highlight some clever tricks and math problems that are quite intriguing and make you want to try yourself…and I did a few times.

A part of the assessment for this elective is to give evidence of how I will develop my understanding of mathematics further in the future. I don’t think I can provide hard evidence of how I will achieve this. However, what I will commit to doing is continue taking an interest in mathematics by reading books like the ones listed above. I enjoyed those therefore I wouldn’t see that kind of reading as a chore. I will strive to keep up to date with any developments in mathematics teaching and I will use what I have learned from fundamental mathematics and continue to apply my logical and creative thinking, promoted by multiple perspectives, to look for the maths in everyday contexts and in other subjects across the curriculum.

Fundamental Mathematics

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I discovered the real lesson learned from this elective is realising exactly what fundamental mathematics is. Previously I had thought that the fundamentals were just the four basic operations (addition. subtraction, multiplication and division) because they seem to underpin all other mathematical problems. I wasn’t completely wrong as basic ideas and principles is one element of fundamental mathematics as defined by Liping Ma (2010). She claims that the fundamentals are connectedness, multiple perspectives, basic ideas and longitudinal coherence. The more I have read about the four elements the more sense it makes. It seems obvious now that, that is what all our inputs were geared towards.

Connectedness (breadth and depth); not just making connections between the operations in order to understand one or the other better but making connections between the other three elements; connections across the curriculum e.g. maths and astronomy (science); connections to everyday life.

‘Mathematics does not consist of isolated ideas, but connected ideas.’ (Ma, 2010, p112).

Multiple perspectives; the ability to see different approaches to the same problem, knowing everyone learns differently so views things in a different light. That point is particularly important for teaching as teachers have to recognise different learning styles among the children in the class, therefore, adapting lessons to suit all. Also teaching children that there is more than one way to arrive at the right answer is a valuable lesson for problem solving later in life.

‘Being able to calculate in multiple ways means that one has transcended the formality of an algorithm and reached the essence of the numerical operations – the underlying mathematical ideas and principles’ (Ma, 2010, p112).

Basic ideas; the foundations to advanced branches of mathematics. Often revisited in maths education as a sound understanding of them supports future learning.

Longitudinal coherence; thoroughness and progression. This element basically combines the other three to create a better understanding of mathematics as a whole. A sound knowledge of all the elements will lead to progression. Knowing the basics and building on them; the ability to see where and how these can be used and or developed later on. Gives a sense of purpose to what you are learning if you know it will be useful further down the line or in everyday life.

The Fibonacci Sequence

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The Fibonacci sequence is the series of numbers beginning 0,1,1,2,3,5,8,13,21…and so on. The next number in the sequence is found by adding the previous two together e.g. the five was made from adding the two and three together. The sequence is named after Fibonacci, real name Leonardo Pisano Bogollo, who lived between 1170 and 1250 in Italy. However, it is said the sequence was known hundreds of years before in India. He was also known for helping spread Hindu-Arabic numbers through Europe which replaced Roman numerals.

The sequence can be transformed into a spiral when squares are made with the widths of the numbers in the sequence.

It also relates to the Golden Ratio. The ratio between any two successive Fibonacci numbers is 1.6. This ratio is often used in architecture to design attractive buildings as 1.6 is the most aesthetically pleasing ratio. It is also frequently found in nature and art. During a Discovering Mathematics input we were learning about the golden ratios significance in art. We looked at Mondrian artwork. We were asked to first of all draw a free hand picture in the style of Mondrian. Then, we were asked to draw another one but this time draw it using the golden ratio of 1.6. As you can see from my art below, the ratio makes a massive difference in terms of which one is more pleasing to the eye.

Space…the final frontier!

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Another Discovering Mathematics input I enjoyed was Maths and Astronomy delivered by Dr. Simon Reynolds from Dundee Science Centre. Space has always been a particular favourite science topic of mine. It is often used as a topic in schools therefore it is perfect for highlighting cross-curricular links between maths and science.

Some of the maths involved in understanding space is scale (with regards to the size of the moon in relation to the earth) and distance (with regards to the distance of the moon from earth) which is 384,400 km. I think a lot of the time space in primary schools is taught with more emphasis of being able to name all the planets and put them in order from the sun, rather than some fundamental concepts like scale and distance.

I think children would be interested in learning about things such as lightyears. This is the measure of distance, not time. One lightyear is the distance light travels in one year. Light travels ,in a vacuum, at 300,000,000 metres per second.

During the input we also learned that there is no actual real photographs of the whole Milky Way as the galaxy is just far too big there is no way any human has ever been on the outside of it to be able to take a photo. This links to infinity (which is said to be the size of the universe). Infinity is an abstract concept used to describe something that knows no bounds; it is never ending. It is a concept widely used in mathematics and physics.

I want to be a demand planner

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Recently we had a Discovering mathematics input on supply chain and logistics and for a brief moment I actually wanted to a demand planner!

We were learning the importance of people in the position of supply chain demand planning and the numerous factors which need to be taken into account when deciding what and how many products are needed. Factors such as where are the products coming from and would that influence the cost. The size, shape and condition of the products (e.g. frozen) and how would that impact on the kind of transportation need to get it from manufacturer to shop and how to store it once it arrives. To be honest, I thought the lesson wasn’t going to particularly interest me. However, I couldn’t have been more wrong. It was favourite Discovering Mathematics input by far.

We were given a taste of what the job entails with a demand planning game. In pairs we were given order forms of a mixture of items typically bought in for supermarkets. We were allowed to spend up to £5000 in our first quarter and then any profits we made thereafter we were allowed to spend in the following quarters. I must admit I became totally engrossed in the task. There was something quite satisfying about just sitting working through the calculations and finding out what the final profit would be at the end of the year. Although I enjoyed it, I could see that in real life it would be a stressful job. Maybe as stressful as teaching.

Hole in one

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After an input on maths and the outdoors, I started thinking about where maths might feature in sport. As my knowledge of sport is pretty limited I decided to look at the one I know a little about…golf.

My dad has played golf for as long I can remember so I have grown up around it, especially watching it on it on TV. I started to wonder if there was any maths involved in a game of golf. The answer is yes. Maths features quite heavily in golf actually.

Golf consists of playing on a course that has 18 holes. Each hole has a par which is the number of shots it should take to get the ball from the tee to the hole. This is usually between 3 and 5 shots. Golfers are required to complete a score card on their round. This involves mental calculations of the score from every hole to be totalled at the end. Golfers need to know the yardage from every tee to the hole in order to plan their game and choose the right clubs. Golf clubs also involve maths as the face of the clubs are at different angles in order to send the ball further. Angles feature quite a lot as golfers have to calculate at what angle the ball is from the hole when putting so they can decide on the right direction.

Golfers also need to take into consideration the conditions of the golf course. The greens and fairways may be designed on an angle or an incline or decline. This has to be taken into account when choosing which club to use and how powerful the swing. They need to be aware of the weather as that will affect their game. They need to have a rough idea of wind speed and direction. There is even maths involved in the design of golf balls.

Golf…who knew it was so mathematical.

Maths at the movies

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Throughout this module we have been encouraged to think for ourselves regarding where maths might occur or be involved some way. As I enjoy watching films, and because we were set a TDT to watch a film about maths, I thought it would be interesting to combine the two and write a blog post on maths at the movies.

Some might argue that maths can be perceived as uncool or boring. However, make it the focus of a blockbuster movie starring famous actors such as Matt Damon and Ben Affleck and suddenly maths is cool and intriguing. Of course, I am talking about Good Will Hunting which featured a troubled young man, frequently in trouble with the law and working as a janitor. The connection to maths, however, is that he turns out to be a mathematical genius who just happens to be the janitor at one of the most prestigious universities in the United States. Basically, he secretly solves the complex maths problems put on a notice board intended for students to solve. Once discovered he is encouraged to put his talent to good use but he is very reluctant to comply. This film is actually one of my favourite films. I like how it highlights that absolutely anyone can have a talent for maths and that you shouldn’t judge a book by it’s cover.

Another famous film where the main character is gifted at maths is Rain Man starring Dustin Hoffman and Tom Cruise. Dustin Hoffman plays Rain Man, who is autistic but has excellent memory recall and possesses the ability to do very complex mathematical problems in his head. He ends up in Las Vegas with his onscreen brother, Tom Cruise, and they win a fortune on Blackjack through Hoffman card counting. The film won 4 Oscars.

Shawshank Redemption may not appear to be an obvious example of maths in movies but remember that Andy Dufresne was a banker before he was wrongly accused of his wife’s murder and imprisoned. While serving two life sentences he begins working for the Prison wardon as his personal accountant. Uncovering a lot of corruption, Dufresne manages to embezzle the money he laundered for the warden, which he collects after he escapes from prison.

Some films are written about real life mathematicians including ‘The Imitation Game’ which was about Alan Turing, and ‘The Theory of Everything’ which was about the life story of Stephen Hawking.

Why be a maths teacher when you can be a roller coaster designer!

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After having inputs from lecturers from the medical school such as Dr Ellie Hothersall, it was interesting to find out just how much mathematics is used in jobs in the medical field. Beforehand I would have thought it was just sciences like biology and chemistry that were needed to become successful doctors and other medical professionals. However, mathematics is quite essential. Its essential for calculating medicine dosages and recording and interpreting things that are measured or monitored such as blood pressure and heart rate. Its needed to make sense of patients medical charts and to figure out their next steps. Actually, it is quite obvious the more you think about it, just how relevant and important maths is in this field. Yet, as i said, it may not be the first skill set people would associate with doctors. Well, other than doctors are perceived to be very bright therefore their abilities in mathematics, perhaps, can be presumed to be very good.

This led me to think about other possible jobs that require or involve mathematics that is not necessarily obvious. I did a little research to learn more and found some fun and interesting jobs that involve mathematics as well as discovering ones I didn’t even know existed, for example, an actuary. Apparently an actuary is responsible for interpreting statistics to determine probabilities of accidents, injury, sickness, death, disability, unemployment and retirement as well as loss of property from theft and natural disasters. In addition, they are also tasked with the job of designing insurance and pension plans. The maths being used here, then, include probability, data analysis and statistics.

Another career involving maths , which is also quite cool, is a roller coaster designer. Again, I think this is one of those jobs you didn’t know was an actual thing but clearly someone has to design them. I learned that these designers use maths to calculate velocity and to help them understand the mathematical properties of curves on a roller coaster and material strength. There were actually a lot of careers in the design and creative/artistic fields involving maths. Some of those include an animator, professional photographer, fashion designer and car designer. For an animator, trigonometry is very useful for rotating and moving characters. Professional photographers need maths to calculate depth of field and to determine the correct film speed, shutter speed and exposure. Fashion designers work with area, perimeter and diameter as well as calculating the amount and cost of fabric needed. Car designers find geometry essential to designing cars that look good and perform well. They also work closely with engineers who use calculus to design powerful and economical engines.

The inputs from the Discovering Mathematics module that were delivered by people outside of the school of education have made me think about how maths can be applied elsewhere in everyday life and jobs. Maths is not only useful for those pursuing a career as a maths teacher. It goes wider than that. I have a work colleague who is studying maths here at Dundee University and I have to admit that when she told me what shes studies, I replied with ‘do you want to be a maths teacher then?’ I assumed that, that is the only possible route for her when in fact its not. I know that because I have found a huge list of jobs and careers involving maths but has nothing to do with teaching it. Some are obvious, some not so, some sound exciting and interesting, and some I knew nothing about.

Maths is all around me

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Although I have always felt relatively comfortable and confident in my maths ability, I chose this elective because I felt the area I struggled with was knowing how to break key concepts down and teach them to children. However, so far, this module has taught me more about fundamental mathematics and how maths is actually everywhere!

It actually is everywhere! This module has made me think about where maths occurs in everyday life. There are the obvious ones like handling money while shopping or weighing out ingredients to bake a cake or being able to tell the time in order to show up to lectures on time. Yet, whenever I see flowers in a vase, pine cones on the ground or even cutting up a pineapple to use in a fruit salad, I never realised maths was involved. Not in the cutting itself, although there might be and I just haven’t twigged yet, but in the shape and make up of the pineapple. There is something called the Golden Ratio at work here. I will go into more detail about this in a blog about the Fibonacci Sequence. Ultimately, the lesson I learned was that maths is evident in nature.

This elective is also teaching me that maths is evident in art. I’ve always loved art and found it very expressive and free with no set rules or structures to it therefore at first I found it hard to grasp the fact that maths would be involved. I could understand its function in architecture but not pieces of artwork. However, through exploring tessallations and Islamic art, it was beginning to make sense and I was able to identify familiar mathematical shapes such as hexagons and pentagons. I also found these types of artwork were quite pleasing to the eye. This, again, relates to the golden ratio.

To be honest this module, at first, probably confused me more than it enlightened me. I thought I had a good understanding of maths but soon realised I only had good knowledge of formulas and how to use them to work out equations and problems. I can tell time, add, subtract, multiply and divide but never really knew what the fundamentals of maths were. This module is making me think of when and where maths is or can be used. Thinking about the clothes I put on in the morning; designers would have worked with measurements. Thinking about the car I drove to Uni; engineering is very closely linked with maths. Wondering if the architect who designed Dalhousie used the Golden ratio?