So, first Semester is coming to a close and Discovering Mathematics has ended. Overall, I have learnt that fundamental mathematics (connectedness, multiple perspectives, longitudinal coherence and basic ideas) is clearly essential in the wider society as it is present in almost every area of our personal lives and the world around us. I believe that my mathematics has developed throughout this module. At the beginning of this module I as nervous towards the subject of mathematics because it was also a weakness of mine, therefore I had anxiety towards teaching it. This module has however given me confidence in understanding mathematical topics and I now believe in myself to be able to teach it.

In conclusion, as a result of the new knowledge I have developed and the confidence I have gained, I feel I am now becoming far more of a “mathematical thinker” (Mason et al.,2010).

References

Ma, L., (2010) Knowing and teaching elementary mathematics (Anniversary Ed.) New York: Routledge.

Mason, J., Burton, L. and Stacey, K. (2010) Thinking Mathematically (2^{nd} ed.). Harlow: Pearson Education.

My family have always been into board games therefore I’ve grown up with the rituals of always playing them on boxing day etc. Thinking that there was maths behind these games and even simple mathematical methods used to win the game made it even more interesting and in intrigued me to investigate more.

In class we were told to investigate board games and analyse the fundamental maths utilised behind the game. On my table we had ‘The 5 Second Rule Mini Game’. This is a quick thinking and fast-talking game which puts the chosen individual under pressure. The question master will ask a question, for example, “Name 3 foods beginning with the letter ‘c’?”. The timer will be turned, and the chosen individual has five seconds to answer the question. This frantic game puts you under pressure and can make your mind go blank with the fast pace and the pressure!
After investigating this game we were able to analyse and find the mathematical connections. The five second timer is a mathematical link as there is a countdown from five to zero. There is also a mathematical link from the question being said aloud for the listener. The mathematical equation is how long the listener takes to process the question and understand what the question is asking them. The processing of this is different for everyone and some people are slower or faster than others.

We also discussed the technical game of the Rubik’s cube. The Rubik’s cube is a 3-D combination colour puzzle which was invented by Ernő Rubik, a Hungarian sculptor. The Rubik’s cube is also known as the magic cube and has six sides with six colours: white, red, blue, orange, green and yellow. The aim of the game is to make each face of the cube a full colour. After investigating this game, I realised that there could be cheats on how to solve the puzzle in the fastest time! Every square has eight corners, twelve edges, 6 faces. There are 40,320 combinations to solve a Rubik’s cube! The mathematical connection which I found interesting was that each equation/combination which solves the magic cube has a different amount of moves therefore some combinations are faster than others!

Here is a video of the world record of solving the Rubik’s cube. Patrick Ponce was able to solve it in 4.69 seconds! It’s interesting watching him look at the cube before he begins, analysing the combinations and sequences of the coloured squares starting positions. It’s weird to think that he solved this in 4 seconds and it’s averaged out that we blink nearly every 4 seconds. You could easily miss him solving the magic cube by blinking!

Being brought up in a very musical family meant that much to mum’s annoyance, there was never a quiet moment in the house. It also resulted in ‘happy birthday’ being sung as a riff off or in several harmonies! I had singing lessons up to the end of high school and thoroughly enjoyed being part of the school summer musical every year. I was also part of the school pipeband, playing both snare and tenor drum. This was a serious commitment as we won the World Championships several years in a row! I took music up to higher and it was a lesson I would look forward to going to every day. It’s clear that music is more than an enjoyable hobby to me and is something that I’ve grown up with and love.

“Rhythm depends on arithmetic, harmony draws from basic numerical relationships, and the development of musical themes reflects the world of symmetry and geometry. As Stravinsky once said: “The musician should find in mathematics a study as useful to him as the learning of another language is to a poet. Mathematics swims seductively just below the surface” (Marcus du Sautoy, 2011).

After having our music lesson, I went home and reread this quote repeatedly. The more I read it the more it made sense. I had never seen the relation of music and maths before and after realising all the connections it made a lot of concepts easier to understand.

Here is a mind map of the connections between maths and music:

The Fibonacci Sequence The Fibonacci sequence is something I had heard about before and after hearing it in the lecture I thought I’d do some more research on it. I never knew that this sequence was not only used in music but also in art and nature. From using the numbers from the Fibonacci sequence, it can lead to spiral patterns forming. Here are nature images which display these spiral patterns:

Scales Doing scales was something I always dreaded and personally, it was the least enjoyable part of the subject. However, after discussing Wiggins theory behind the relation of maths and scales in music it has made more sense and has helped me remember it more clearly.

We then learned about the 12 note row. This is something which anyone can do and is great fun to compose a piece like this! However, in my opinion it is note the most pleasant of music to listen to. Here is an example of what it sounds like:

Although music is a subject which I already knew a fair bit about, I have learnt a lot from the lecture and really enjoyed it!

Wiggins, G.A. (2012): Music, mind and mathematics: theory, reality and formality. Journal of Mathematics and Music: Mathematical and Computational Approaches to Music Theory, Analysis, Composition and Performance. Available http://dx.doi.org/10.1080/17459737.2012.694710 (Accessed: 6 November 2017)

Hockey is a sport which I have played since I’ve been in primary 4. It’s something I train for 3 times a week whilst playing 2 games throughout the week. However, even though I’ve played this sport for many years I had never thought about the connection and relation it had to maths. Therefore, after being told to think about this before class I was very surprised how many links I found there were.

The Rules for hockey are as follows:

1. It’s an eleven-sided game, meaning you are allowed to have a maximum of eleven players on each team playing on the pitch. Each team will have one goalkeeper and ten field players. Although only eleven players are allowed on the pitch, normally a hockey team will consist of fifteen players as subs will be needed.

In hockey the ball is not allowed to hit your feet unless you are the goalkeeper. If the ball is kicked then it’s a foul and the opposition gets the ball.

To score you must be in the attacking “d”. This is a semi-circle around the goal.

Raised balls are not allowed above knee height as this is classed as a ‘dangerous ball’ and will be a foul. However, a raised ball called an Ariel is allowed. This is a very high ball which is above head height.

Stick tackles are not allowed in the game of hockey. If there’s a slap to a stick and a loud noise is made when a tackle is made, then that is a stick tackle. This will be a foul and the ball will be given to the opposition to take from where the stick tackle occurred.

Strategies

I then used my own knowledge of the game in finding strategies for playing hockey effectively. In field hockey they are different strategies and tactics both defensively and attacking to result in success.
The main strategies for hockey are formations. With a goalie in the goal defending at the back, there are ten outfield players. This means you have ten players to create formations with. Below are two common different starting formations. Obviously, these will change depending on what team you’re playing and where their strong players are on the pitch. The way the opposition presses against your team’s hit out can also be dependent on your team’s formation. It also depends on where your area of strength is in the players you have on the pitch within your own team as well.

Hockey in relation to Maths

After doing research I realised the many connections field hockey has to maths. The design of the pitch is in the shape of rectangle which is separated into four sections. There are 3 lines marked across the pitch marking the 25m line, the 50m line (half pitch) and the 75m line. A hockey pitch also consists of two semi circles that make out 180 degrees which is called the “d”.
After our lecture on chance and probability, it made me think of this topic in relation to the game of hockey. It made me question what’s the probability of a successful attacking 3v2 situation? Or what’s the probability of 5 average attacking hockey players beating 3 very strong defenders? Or what are the chances for a goal in a 1v1 situation with a striker and a goal keeper?

After reading articles online of maths in hockey it made me realise how the smallest of things in the sport are related to maths! For example, the different shapes we form when passing, eg. triangle passing. As a defender it made me think how far away the defensive line should be from one another when we are talking a defensive hit out for it to be successful. For the hit out to be successful then it must eliminate some of the attacking line in the opposition. This topic has made me want to investigate this further and in more technical terms by measuring out the distance of how far away we should be standing from each other.

More math related aspects to field hockey

The make of the ball, for example whether it’s a kukri or Slazenger ball. Each make of ball feels different to play with.

The surface of the ball – some balls have a smooth surface all around the ball, whereas some have little circular dips around it.

The speed of the ball when it is passed. The speed changes depending on if the ball was sent as a slap, a push pass, a sweep, a hit, a small raised ball or an aerial.

Position of hands on the stick, finger positions.

Angle and position of the stick when the ball is received and passed.

Angle and position of body and stick in a tackle.

Eye level behind the ball when passed/hit.

Body weight distributed between both feet whilst hitting the ball.

Motion of swing when hitting the ball.

The strength behind each pass.

The pressure of how tight you hold the hockey stick.

Today in class I was proposed to the question, “Can animals count?”. If I’m honest, this is something I have never really thought about. I’ve never had a pet before so have never been heavily interested in animals, so this is something which had never crossed my mind. At first, I was 100% sure that animals were not able to count, then we were introduced to the story of the horse, Clever Hans. This made me think carefully about In the 1900s, it was announced that this special horse was able to add, multiply, subtract and divide, ‘Berlin’s wonderful horse; He can do almost everything but talk – how he was taught’ (Heyn, E. T., 1904). Hans would tap his hoof the amount of times he thought the answer was. However, while it might’ve seemed to the public that he was able to count I wasn’t convinced and as video continued it explained that after investigation, it was proven that Hans wasn’t performing these intelligent tasks, but was watching his owners’ movement to tell him how many times to tap his hoof.

AYUMU

We were then introduced to the case study of the chimpanzee, Ayumu. I found this very interesting at how fast Ayumu could process and remember the correct number order and where each number was placed on the screen. As a class we then tried to see if we were able to beat Ayumu. Personally, after many attempts I wasn’t successful at beating him! I found this very fascinating at how fast the chimpanzee and was extremely impressed how he was able to process what was going on at such an extreme speed. It makes me wonder what will be happening in several years to come, will animals be interacting more with humans? Will animals be used more with technology?

According to Galileo (1564-1642), “the language with which God has written the universe”. A fact that I loved learning in today’s class is that Mathematics is a universal language. Knowing that it’s the same in every country across the world fascinates me. Something which I didn’t realise was that maths isn’t just in the classroom, it is all around us! For example, mathematics relates to everyday situations, such as, bills, cooking, weather, stock markets, travelling and baking.

An important factor in teaching maths is the attitude of the students learning it. How people feel about maths has an impact on their ability to do and interest in learning maths. Something which needs to be addressed is maths anxiety.

Here are examples of both physical and psychological anxiety symptoms related to mathematics:

As a teacher, my aim is to to spark the student’s enthusiasm towards the subject of maths by teaching the subject in a motivating and interesting way.

Today’s class was all about mathematical shape involvement in visual illusions, symmetry and the importance of active learning. I learnt how to make and augment regular polygons in order to create an Escher-inspired tessellation or Islamic art-inspired tiling. We explored the relationship between art and mathematics and how throughout history, artists have used the properties of symmetry, tessellation and proportion to create artistic masterpieces.

Maths is a subject feared by majority of children and this class has made me determined to try and change the way children view maths. I learnt the importance in involving students in the learning process of maths. By making children do fun activities related to the maths topic it keeps them involved and being productive in the topic. It helps children get excited over maths, keep interest, and enjoy what they’re learning.

Today we learnt how to make our own tessellations. A tessellation (or tiling) is a repeating pattern of shapes that fit perfectly together without any gaps or overlaps. Regular tessellations are made up of only one regular shape repeated, whilst semi – regular tessellations are made up of two or more regular shapes tiled to create a repeating pattern.

Here is the beginning of my own tessellation. I have used three different shapes. However, as you can see there are gaps so it isn’t finished yet!

Personally, maths was never one of my strongest subjects at school as I found it very challenging and difficult. After dropping it as a subject in school I rarely ever needed to use the mathematic skills I had learnt, therefore, leading to me forgetting what I had learnt. As I’ve gotten older my anxiety towards maths has grown and from placement last year I noticed that it was a subject I felt very nervous about teaching. I’m hoping this module will help build my confidence in mathematics and I’m excited to discover more about the subject.

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