Reflection on Discovering Mathematics

Prior to starting the discovering mathematics module, I was rather anxious at the thought of reliving my fear of high school maths. However, to my delight, the module was far from my expectations. Throughout the module, we have examined how mathematics is relevant in all aspects of our lives. For example, we have looked into the importance of mathematics in sport, music, medicine, Fibonacci, games and gambling – highlighting to me that mathematics is in fact everywhere and also eliminating my initial idea that maths is just equations and tricky sums.

Liping Ma has also been a key focus throughout the module. The four principles, connectedness, multiple perspectives, basic ideas and longitudinal coherence, are all crucial for a teacher to acquire. Having a profound understanding of mathematics is much more than just having great knowledge of the subject – but it is having an understanding that topics within mathematics all link and rely on each other. Having a strong sense of PUFM also allows an individual to know that there are various ways to solve a problem, and then choose the appropriate one that is suited for the mathematical problem. The Discovering Mathematics module has made me realise that it is essential for a teacher to obtain a PUFM in order to teach maths effectively and to the highest of standards.

Overall, the Discovering Mathematics module has stretched my view of mathematics by conveying that maths is much more than what we all think – that it is in fact a part of our day to day lives. It has also made me realise the importance of PUFM from both a student and teacher’s perspective. The module has showed how mathematics can be much more than trigonometry and fractions, that maths can be made creative and also can be linked to different subjects within the curriculum.

The Next Apprentice Candidate?

After reflecting upon my time during the Discovering Mathematics, I have realised that my opinions on mathematics really have changed. The Discovering Mathematics module has really opened my eyes, to how much fun you can have with maths.

One example of this was when we looked at food supply and logistics in a lecture. We looked at food miles, which is the distance which food is transported from its place of production, to the consumer, e.g. a farm to a supermarket (Oxford Dictionary, 2017). Furthermore, we also looked into the factors which affect the transportation of food. For example; mass, the distribution of mass, the size of the product, its strength, temperature requirements and the products shelf life. We then went onto look at demand planning, and put what we have just learnt about food production and consumption, into practice to demonstrate how these areas of fundamental mathematics are used on a daily basis, all around the world.

Demand planning is;

“a multi-step operational supply chain management (SCM) process used to create reliable forecasts.  Effective demand planning can guide users to improve the accuracy of revenue forecasts, align inventory levels with peaks and troughs in demand, and enhance profitability for a given channel or product” (Rouse, 2010).

As also stated by Rouse (2010), successful demand planning includes various steps including; looking at historical sales data, creating statistical forecasts, looking at what the customers want and working with the customers, and many more.

The Game

Taking these rules into consideration, and applying what we had just learnt about logistics and food supply, we then split into pairs and worked on our business.

The rules of the game include;

As a team, we had 5000 euros to spendand had to look at what products would generate the most profit for each quarter. For example, between April and May, it would be stupid of us to put any money into the likes of Christmas selection boxes or luxury hampers as these would not generate any sort of profit. It would be clever however, to put your money into products such as milk, bread and beans, as these are staples of most people’s diet. Therefore, for each time period (April-June, July-August, October-December and January-March) we had to think about what products would sell the best and generate us the most profit.

1. Make sure we read the rules correctly and;
2. Check which product has the highest profit margin, and put all of our money into that.

Overall, the task was great fun and I believe that this would be a really fun and enjoyable task which children would love, and therefore, this is something which I hope to take forward into my future profession as a teacher.

References

Oxford Dictionaries | English. (2017). food mile | Definition of food mile in English by Oxford Dictionaries. [online] Available at: https://en.oxforddictionaries.com/definition/food_mile [Accessed 2 Dec. 2017].

Rouse, M. (2010). What is demand planning? – Definition from WhatIs.com. [online] SearchERP. Available at: http://searcherp.techtarget.com/definition/demand-planning [Accessed 2 Dec. 2017].

Horology – The History of Time

“Horology is quite literally the “study of time”, most often referred to as the art or science of time measurement.” (A World of Time, 2017)

Various methods and instruments were used to record time before the invention of the mechanical clock;

Sundials: Sundials are the oldest known method of telling time, dating back to 3500BC. The sundial relies on the rotation and movement of the sun. The shadows formed by the sun moving from east to west determine the time of the day. The time was measured based on the length of the shadow created. However, there was a downside to this method of time keeping. As the dial relied on the sunlight, this instrument would be useless if it was a cloudy day or if the sun was down. After discovering this issue, new methods of time keeping were developed that were far more reliable. (Marie, 2017)

Egyptian water clocks: The oldest known water clock dates back to 1500 BC. There are two versions of the water clock, outflow and inflow. The outflow clock was where a container (with markings down the side) was loaded with water, enabling the water to drip out an even pace. Individuals were able to tell the time by calculating the amount of water that had leaked from the container. The inflow clock followed the same method as the outflow clock, except instead of using one container, two containers were used. The water from the first container would drip into the second container at a steady pace. Observers were able to tell the time by the amount of water that had leaked from one container to the other. (Ancient Origins, 2014)

Sundials and Egyptian water clocks are just a few of many examples of time keeping. Others include, candle clocks, Astronomical clocks, incense clock are just a few methods that were used before the mechanical clock was created.

Dave Allen gives us an insight into the difficulties (and frustrations) of teaching children the analogue clock.

This video made me think of the importance of teaching children time. I think it is necessary that children recognise how far the various methods of telling time have developed over the years, as it will provide them with a better understanding of what time really is and will convey to the pupils how we came to using the analogue/digital clock. However, there has been speculation whether it is required for children to learn the analogue clock in our very digital world. I think, yes. Analogue clocks are everywhere, from the clock on our walls to famous landmarks, children need to be able to understand and interpret the time on the analogue clock. Furthermore, analog clocks help children understand the passage of time because they have hands that are consistently moving. Despite living in a high-tech 21st century, I don’t think it’s time to ditch the analogue clocks just yet.

Aworldoftime.co.uk. (2017). What is Horology – A WORLD OF TIME – HOROLOGIST. [online] Available at: http://www.aworldoftime.co.uk/what-is-horology—a-world-of-time—horologist.html [Accessed 27 Nov. 2017].

Ancient Origins. (2014). The Ancient Invention of the Water Clock. [online] Available at: http://www.ancient-origins.net/ancient-technology/ancient-invention-water-clock-001818 [Accessed 27 Nov. 2017].

Marie, N. (2017). When Time Began: The History and Science of Sundials. [online] Timecenter.com. Available at: https://www.timecenter.com/articles/when-time-began-the-history-and-science-of-sundials/ [Accessed 27 Nov. 2017].

Mathematics and Sport

Prior to our input with Richard, I was (yet again) confused to how sports and mathematics were linked. I was a keen netball player and coach when I was in school and my main aim when playing would be trying to win the match with my team by using various tactics; not by using mathematics to help achieve this. However, after our input I was eager to find out how maths and netball were connected.

Netball and Mathematics

• The court – the size of the court is 30.5 metres long and 15.25 metres wide and is divided into thirds. The centre circle has a diameter of 0.9 meters and the two shooting semi circles are me

Dsr.wa.gov.au. (2017). Netball. [online] Available at: https://www.dsr.wa.gov.au/support-and-advice/facility-management/developing-facilities/dimensions-guide/sport-specific-dimensions/netball [Accessed 9 Nov. 2017].

asured at 4.9 metres.
• Positioning – When a player attempts to defend or intercept the ball, they must be at least 0.9 meters away from the player with the ball.
• Speed and accuracy of the pass is key for moving forward and into the shooting semi-circles. The quicker and more accurate the pass is, the less likely the opponent team can intercept it.
• Shooting – A technique that many GS and GA use when shooting is positioning your elbow at a right angle when holding the ball, which improves stability.
• The timings – A netball match lasts an hour and is split into fifteen minute quarters.

Angles, distance, time and speed are just a few of the basic concepts used in mathematics which links to what Liping Ma refers to (Ma, 2010).

This short video highlights how mathematics is used in sports as a whole. For example, discussing a players statistics which enables them to determine their strengths and weaknesses, deciding where a player is going to be positioned in the game/pitch or even the way in which the game is scored – mathematics is always involved. The clip then goes onto explore how mathematics plays a key role in cricket. Mathematics is used to calculate the total number of overs in a match (an over is a set of six balls bowled from one end of a cricket pitch) and calculate the average and strike of the batsmen and bowlers. The video also suggests that the measurement of the bat and ball can affect the player’s performance within a game. This is further supported by ‘The Physics of Cricket’ as they explain why lighter bats can be swung faster than heavier bats:

“Imagine hypothetically that the bat weighs 10 grams. If you swing it as fast as possible, you might get the tip to travel at 160km/hr. Now double the weight to 20 gm. This time the tip travels at about 159 km/hr. The problem here is that your arms weigh about 8kg all up, so the extra 0.01 kg is hardly noticeable. Most of the effort needed to swing a bat goes into the swinging the arms.” (Physics of Cricket, 2005).

Reflecting on this input, I have discovered that many fundamental concepts are used throughout sport; not only are these concepts reflected in the equipment used in a game e.g. a bat and ball, but are also used by sports players to improve their performance within a game.

Through my journey in discovering mathematics, it has become apparent to me that mathematics is everywhere. The idea of maths being all around us needs to be reflected in the classroom to minimise the idea that maths is not just about equations and fractions; but is in actual fact heavily reflected in our day to day lives e.g. in our favourite sports. Demonstrating how maths can link to pupils interests, makes the subject more enjoyable and relevant (The Scottish Government, 2008).

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

Physics of Cricket (2005) Available at: http://www.physics.usyd.edu.au/~cross/cricket.html (Accessed: 9 November 2017).

The Scottish Government (2008) curriculum for excellence building the curriculum 3 a framework for learning and teaching. Available at:  http://www.gov.scot/resource/doc/226155/0061245.pdf (Accessed: 9 November 2017).

Mathematics and Music

“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.”                                                                                                               Du Sautoy 2011

Prior to starting the discovering mathematics module, maths and music are very different subjects. I see mathematics as an academic subject whereas music is more creative; which highlights how closed-minded I was.

We were asked at the beginning of the input to think as many links that we could think of between music and maths. After struggling for a few minutes, I think in all we managed to come up with 3 links. We were then shown the list of connections and the list was endless –

• Note values/rhythms
• Beats in a bar
• Tuning/Pitch
• Chords
• Counting songs
• Fingering on music
• Time signature
• Figured bass
• Scales
• Musical Intervals
• Fibonacci sequence

This video explores how the famous composer Beethoven was able to create spectacular pieces of music despite spending most of his career being deaf. Beethoven discovered that he was able to see the patterns in the music, therefore was able to know what the music was going to sound like. The clip also explores how our ears are able to detect whether a piece of music is nice sounding (consonance) or not (dissonant) – and how maths is used to determine this.

Wave lengths also provide us with the answer to why it is impossible to tune a piano. The video explains that a when the string of the vibrates, it can only vibrate in certain waves (sin waves) due to the strings being fixed to the end of the instrument. The more bumps in the wave, the higher the pitch and the faster the string has to vibrate. The reason why it is impossible to tune a piano is due to the fact that the instrument has too many strings. The video then goes onto explain in depth why having too many strings is an issue and how they use mathematical equations to solve why it is impossible to tune a piano.

I have found it very fascinating discovering the links between mathematics and music. I will definitely consider combining both subjects together when teaching concepts such as sequence and patterns within my teaching practice.

Du Sautoy, M. (2011). ‘Listen by numbers: music and maths’ Guardian. Available at: https://www.theguardian.com/music/2011/jun/27/music-mathematics-fibonacci (Accessed: 7 November 2017)

Fibonacci and Nature

“Is God a mathematician? Certainly the universe seems to be reliably understood using mathematics. Nature is mathematics.” (Pickover, 2009)

Mathsisfun.com. (2017). Fibonacci Sequence. [online] Available at: https://www.mathsisfun.com/numbers/fibonacci-sequence.html [Accessed 3 Nov. 2017

Throughout this module, there has been a reoccurring theme that maths is everywhere. In our input with Anna, we were introduced to the Italian mathematician Fibonacci. Fibonacci is known for the creation of many sequences, with the golden spiral being his most famous discovery. This particular sequence starts at 0 then 1 then you add the two numbers before you get the next number in the sequence; 0 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 and so on. Fibonacci discovered that if he used squares with this sequence, it would make the perfect spiral.

Fibonacci’s golden spiral can be seen throughout nature, sunflowers is a clear example of this theory. The spirals of the seed pattern of the sunflower contain Fibonacci’s sequence. But why do the sunflowers abide by Fibonacci’s theory? It has been suggested that the sunflowers can pack the maximum amount of seeds if it follows this particular sequence.

Spiral Galaxies is another example of where Fibonacci’s sequence is apparent. The milky way has several spiralled arms that follow in the Fibonacci sequence. After looking into this, I read about how the spiral galaxies does not fit the theories of astronomers. According to the astronomers, “the radial arms should become curved as the galaxies rotate. Subsequently, after a few rotations, spiral arms should start to wind around a galaxy. But they don’t.” (Planet Dolan, 2017).

Finally, the placement of a flowers petals also follows Fibonacci’s sequence. Examples of this is the lily (3 petals), buttercups (5 petals) and daisy’s which have 34 petals. It is said that the flowers follow the sequence of Fibonacci to maximise their exposure to sunlight, which is obviously beneficial for the flower.

These findings make it apparent that mathematics is purposeful. Whether it maximising sun exposure for a plant, or to maxmise the space within something; mathematics makes it beneficial.

Mathsisfun.com. (2017). Fibonacci Sequence. [online] Available at: https://www.mathsisfun.com/numbers/fibonacci-sequence.html [Accessed 3 Nov. 2017].

Pickover, C. A. (2009) The Math Book from Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics. London: Sterling.

Planet Dolan | Obscure Facts About Life. (2017). 15 Beautiful Examples of Mathematics in Nature. [online] Available at: http://www.planetdolan.com/15-beautiful-examples-of-mathematics-in-nature/ [Accessed 3 Nov. 2017].

Can Animals Count?

In this week’s discovering mathematics input, we looked into the origins of numbers and how number systems came about and also the big question; can animals count?

We first looked into Clever Hans, a horse that was known for his miraculous ability to ‘count’. When Hans’ owner asked him a basic sum, such as 3 plus 1, the horse would answer the question by tapping his hoof the correct number of times. The horse raised many questions to how he was able to acknowledge the question, then calculate the answer and then finally give his answer. It was then discovered after many scientific observations that the horse was in fact not able to calculate his owner’s questions, but was able to study his body language and cues. No one can dispute the horse’s intelligence; however, he could not count.

After learning about Clever Hans, I wanted to look into whether there have been other occasions where it has been speculated that animals can count. The question has been raised that if an animal is able to work out if a quantity of something e.g. how many animals are in a pack, is larger or smaller if that means they can count? An example of this is Lions. The concept of lions being able to count was tested by Karen McComb of the University of Sussex. Lions are highly territorial animals and will only attack another pack of lions if there are more in their own pack. So, the idea behind the experiment was to play the sound of a pack of lions and observe whether or not the lions would choose to run towards the sound or stay back. After playing the sound of a roar of 3 lions, the 5 lions that were a part of the experiment ran towards the noise. McComb stated “Lions were extremely good at weighing up their odds of success in terms of the number of themselves versus the number roaring from a loudspeaker”. However, it was discovered that after the roars on the loud speaker were upped to 6 lions, the lions in the experiment started to get confused whether or not they should attack or not (Silver, 2017). After looking into this experiment, there is no doubt that the lions were able to examine the situation and determine if it was safe for them to attack or not. However, whether they can count or not? I’m not convinced.

Silver, K. (2017). The animals that have evolved the ability to count. [online] Bbc.co.uk. Available at: http://www.bbc.co.uk/earth/story/20150826-the-animals-that-can-count [Accessed 23 Oct. 2.

What is Mathematics? Why teach it?

Mathematics is “the language with which God has written the universe” (Galileo, 1564-1642). Mathematics is everywhere. It is something which we experience and practice every day. Whether it is to tell the time or using it to code computers, it is all around us.

Does this mean that everyone is a mathematician? Technically, yes. Mathematicians solve problems, investigate, explore, discover, collaborate and use symbols, tables and diagrams. These are skills which most people use in their everyday lives. It is therefore important that children understand the fundamental principles in mathematics, so that they can use these in their everyday lives.

However, some children and adults, suffer from maths anxiety, which is stopping them from engaging with maths as well as they could be.  Maths anxiety is “a general fear of contact with maths, including classes, homework and tests” (Hembree, 1990, p.45). It is medically recognised as stress and comes with multiple physical and emotional symptoms (Turner and Carroll, 1985). Maths anxiety causes headaches, muscle ache and shortness of breath, as well as confusion, intimidation, and concentration problems (Arem, 2010). Children then possess a negative attitude towards maths which leaves them disengaged with the subject and reluctant to use it.

It is therefore crucial, that not only teachers but parents to find a way of combatting this. Parents can do this by encouraging a positive attitude towards mathematics and allow the children to use their mathematical skills at home in various situations. These could include getting the children to tell the time or allowing them to help with cooking the tea and explaining to them how weight works. However, teachers can also help to address maths anxiety by making maths fun within the classroom. Instead of always doing textbook work, teachers could make the lessons more fun and enjoyable for the students. Teachers could also show the students how the mathematical skills and concepts which they are learning, fit into their everyday lives. By doing this, hopefully children will see the relevance and enjoyment of maths.

I personally hope to brush up on my mathematical skills and improve my confidence by engaging with the Discovering Mathematics module and by taking part in the Online Maths Assessment (OMA). The University have provided the OMA to help improve confidence within Education students and to also improve poor levels of mathematical competence (Henderson, 2010).

References:

Hembree, R. (1990) ‘The nature, effects and relief of mathematics anxiety’, Journal for Research in Mathematics Education, 21, pp.33-46.

Henderson, S. (2010) Mathematics Education: The Intertwining of Affect and Cognition. Unpublished doctoral thesis. D.Ed. University of Dundee.

Turner, J.R. & Carroll, D. (1985) ‘Heart rate and oxygen consumption during mental arithmetic, a video game, and graded exercise: further evidence of metabolically-exaggerated cardiac adjustments?’, Psychophysiology, 22(3), pp.261-267.

Why teaching?

From a young age, the idea of becoming a Primary Teacher has fascinated me, however it was volunteering at a local dance school that confirmed to me that I wanted to turn this idea into reality. For over a year and a half I devoted my Saturday mornings to working as a dance class assistant for young children from the ages of three to ten. It was here I realised how rewarding it was helping a child overcome their difficulties and seeing how proud they were after they had mastered a new skill. It also taught that with patience and extra help all children can learn no matter how challenging it may be to them. Watching each child develop their own personality as well as achieving their personal goals was equally rewarding.

I have a strong belief that Primary Teachers are the key to giving a child the best start to their life, which was another reason to why I was keen to be part of this valuable profession. It inspires me that every day will be a new challenge and I feel that with the skills I possess, and will enhance, that I will embrace it.