Category Archives: Discovering Maths

The End of Discovering Maths

Before this module I saw maths as a boring subject which I wasn’t very good at and had no interest in. However now that the module has finished I see maths as a fascinating subject, which I use on a daily basis without even realising.

Three months ago I wrote my first blog post saying how I never had a good experience with maths over secondary school and how the thought of doing this module scared me. I can now happily sit here today, and say how I think this module has taken away my fears and anxieties I had relating to maths. Throughout my life I always heard people say ‘im bad at maths’ or ‘I’m not a maths person’, which I think made me believe I wasn’t either. Throughout this module I have learned that this isn’t the case as maths is involved in nearly all aspects of my life from baking, participating in sports and even planning what train I’m going to get, and considering I make my dinner and make uni on time (most times) I must not be too bad.

I have thoroughly enjoyed learning the four principles Liping Ma speaks of, ‘to fully promote mathematical learning’ (Ma, 2010, pp.210). Having a knowledge of the four principles connectedness, multiple perspective, basic ideas and longitudinal coherence will help me to ensure I teach my pupils about the links between mathematical topics and how each mathematical topic depends on one another. It will also remind me to promote the many different ways there are in order to reach an answer to a maths problem (Ma, 20100.

I am no longer scared at the thought of teaching my future class maths, as iv learned it doesn’t have to be all fractions and percentages from a textbook, instead I am exciting to make maths creative and show pupils the relevance of maths in their daily lives. One of my new aims I want to have as a teacher is to avoid maths anxiety in my classroom by incorporating the 6 ways I have learned maths anxiety can be avoided. These six things include: playing maths games, being aware of my own attitude towards maths, practising the subject with my class, getting rid of the idea that someone ‘cant be good at maths’, giving help from an early stage and getting children to shake off mistakes (Lee, 2016).

Lee, K. (2017) How You Can Help a Child Who Has Math Anxiety. Available at: https://www.verywell.com/how-you-can-help-a-child-who-has-math-anxiety-620833 (Accessed: December 2017)

Ma, L. (2010). Knowing and Teaching Elementary Mathematics: Teachers’ Understanding of Fundamental Mathematics in China and the United States. 2^nd ed. New York: Routledge.

The creativity of Maths

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Before this module I saw maths as a boring subject which involved sitting with a text book doing fractions and percentages but throughout this module I have realised there is more to maths than a boring subject which I needed to get into uni.

The first workshop which changed my opinion of maths and I particularly enjoyed was with Eddie. That was the input when I discovered maths can be used in many creative ways.

We began the workshop exploring tessellations, which is “a repeating pattern that fills a space without overlapping”. There are different types of tessellations including regular, which involves one regular shape being repeated and semi regular tessellations which involves more than one regular shape being repeated, both creating a pattern (Lee, 2011).

A ‘regular’ shape is one which has all its sides the same length and all its internal angles the same size. The regular shapes that tessellate are: squares, hexagons and equilateral triangles. ALL triangles and quadrilaterals also tile but they are not ‘regular’ shapes and you often have to rotate them to make them fit together.

The reason regular shapes tessellate is due to the angle they make where the vertices touch – 360⁰ (Harris, 2000).

Tessellations are common in Islamic art, with three fundamental shapes playing a large role in the design of the patterns, found in many Islamic buildings. The three shapes are:

Equilateral triangles- these represent harmony and human consciousness.
Squares – these represent the four corners of the earth.
Hexagons- these represent heaven (lee,2011).

Having a go at creating our own Tessellations

The second half of the input was used to let us create our own tessellations. We had the choice of shapes and colours we wanted to use, and it was a thoroughly enjoyable activity. This activity made me think of how other aspects of learning can be incorporated in this to further learning. For example if doing this with an early years class shapes and colour can be explored and with an upper years class angles, symmetry and size could be looked at.

As well as other principles of maths being incorporated, simple life skills could be taught too, along side this lesson. For example how to use scissors, how to manage time (if given a time limit) and how to work in groups (if made a group task).

Exploring maths and art together in the one input allowed me to see an example of how Lipings Ma’s “connectedness” can be brought into the classroom (Ma, 2010). I can now vision teaching my future pupils about tessellations in maths and then having a go at creating our own patterns in art, to highlight how some subjects across the curriculum have links with one another and also that maths can be creative !!!

Harris, A. (2000) The Mathematics of Tessellation. [Online]. Available at: http://ictedusrv.cumbria.ac.uk/maths/pgdl/unit9/Tessellation.pdf (Accessed: 1 December 2017).

Lee, A. (2011). [online] Tessellations in Islamic Art. Available at: https://classroom.synonym.com/tessellations-in-islamic-art-12085299.html [Accessed 1 Dec. 2017].

Ma, L. (2010) Knowing and Teaching Elementary Mathematics – Teachers’ Understanding of Fundamental Mathematics in China and The United States. London: Routledge

The Importance Of Maths…

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Before this module I never realised the extent to which maths is involved in every day life and how beneficial having a mathematical knowledge can really be.

In a recent module we had Dr Ellie (from the university medical school) come to talk to us about maths and statistics. She told us how important it was for professions such as doctors and nurses to have a profound understanding of fundamental maths. For example a nurse has to be able to measure correct amounts of medication in order to treat patients correctly (Education Scotland, 2008). This involves having an understanding of measurments and graphs.

Measurements are used by nurses in order to measure out the correct medicine dosage in relation to children’s weight. Nurses also have to have a knowledge of graphs and charts, in order to monitor progress and keep track of how much medication a patient has consumed (Hothersall, 2016), Thus showing how important a knowledge of fundamental maths can be in order to save lives.

Learning about graphs and data analysis is not only beneficial for employees to know but also children of a younger age. Teaching graphs from primary school will build on a child’s life skills by encouraging them to become confident individuals and function independently in society as they will have the ability to process information (Education Scotland, 2008). This may help with aspects of their future, from simple tasks like reading train times to understanding and adapting to the way in which the world is changing and be able to work out problems.

Yes, maths is essential in the workplace, but also during a persons regular day. From setting your alarm, planning how long a task will take, calculating money in a shop and even cooking dinner, many aspects of fundamental maths are involved. Such as simple estimating, problem solving and addition and subtraction. From now on instead of saying ‘im not a maths person’ I want to remind myself of the many aspects of maths I participate in pretty much every day.

Refrences

Education Scotland (2008). What Is Curriculum For Excellence? Available at: https://www.education.gov.scot/scottish-education-system/policy-for-scottish-education/policy-drivers/cfe-%28building-from-the-statement-appendix-incl-btc1-5%29/What%20is%20Curriculum%20for%20Excellence?(Accessed: 1 November 2017).

Hothersall, E. (2016) ‘Numeracy: Every contact counts (or something)’ [PowerPoint]. ED21006: Discovering Mathematics (17/18) Available at: https://my.dundee.ac.uk/webapps/blackboard/content/listContent.jsp?course_id=_56905_1&content_id=_4941433_1&mode=reset (Acessed: 1 November 2017).

Can Animals Count ?

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In a recent maths input I was asked the question, ‘Can animals count?’. Straight away I laughed and responded with not a chance, but after the input and doing some of my own research, I am now unsure of my answer.

The first piece of information which made me doubt myself was down to a horse called ‘Clever Hans’. In the 1900’s it was believed that Hans could count. Apparently when the horse was asked maths questions involving addition and division, it would answer by tapping out the answer using its hoof. It turned out that infact Hans couldn’t count and instead of understanding maths problems was understanding commands from his master ‘Van Osten’. Through experiments and observations it was discovered that instead of the animal quickly doing the sum in his head, he was responding to his owner’s movements to know how many times to tap his hoof (Jackson, 2005).

At this point in my research I was feeling pretty confident with my response of animals cannot count, however further into the input we looked at chimpanzees and ants understanding of maths which were the most convincing examples of all.

Over 30 years ago a professor created a study to demonstrate how Amuyu the chimp was a very intelligent animal and could respond to mathematical problems. The study showed Amuyu getting shown 9 numbers randomly placed across a screen for 60 milliseconds before they disappeared. He then was able to click the numbers in the correct order of what he had previously seen. After I had a shot and failing at this exact test I was amazed to see (in the video we were shown), the speed of which the chimp completed the task and also how when numbers were missing he was still able to remember the order they were in. This ultimately suggests that chimps can engage with mathematical skills, however I am still unsure if I’m convinced or believe this was down to Amuyu having remarkably good memory skills (BBC, 2017).

Something I didn’t know before this module was how clever ants were. When ants leave their nest to go and get food they remember the distance they have travelled by apparently counting the number of steps they have taken in order to get back to the right place. This should then mean that ants have an understanding of numbers and counting…….

Researchers Martin Muller and Rudiger Wehner decided to investigate this theory and find out if that is the case. One of the tests they done was shortened the ants legs and the other was increasing the ants legs by adding match sticks to them, in order to see if the variables would affect the ants ability to calculate the distance they have travelled back to their nests. The researchers concluded that the ants with the longer legs continued to walk past their nests and the ants with shorter legs stopped before their nests. This was put down to the ants having internal pedometers, meaning that the ants do in fact count how many steps they have taken from leaving their nest and then take the exact amount of steps back. So by having longer or shorter legs, the ants still counted their steps back to their nests but just travelled a shorter or longer distance (Carey, 2006).

From the studies I have researched I still cannot pick a side as to whether animals can count or not. I do believe animals can understand the concept of counting (proven by the ant investigation), however I’m not sure if they can count as they cannot talk meaning they don’t have words for numbers.

In the future this would be a topic in which I would be interested in researching some more in order to have an answer. I believe this would be a great topic to look at with my future class to demonstrate how maths can be fun and associated with topics they are interested in (animals).

Can animals count ? The research continues…..

Refrences

BBC. (2017). Super Smart Animals – Ayumu – BBC One. [online] Available at: http://www.bbc.co.uk/programmes/profiles/31n3tMPHkZ8sQgxS5ZjdzN/ayumu [Accessed 6 Oct. 2017].

Carey, B. (2006) https://www.livescience.com/871-ants-marching-count-steps.html (accessed: 27 November)

Jackson, J. (2005) http://www.critical-thinking.org.uk/psychology/the-clever-hans-effect.php (accessed: 27 November)

Muldertn (2008) Chimps counting. Available at: https://www.youtube.com/watch?v=0cKg9D4QKCM (Accessed: 27 November 2017).

Maths in Business

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In a recent Maths workshop we looked at another way in which maths appears in our every day lives. Our focus of the workshop was supply chains and logistics.

We began by talking about food miles, which according to the oxford dictionary is, “a measure of the distance travelled by foods between the place where they are produced, and the place they are eaten”(Food Miles, 2017). So already before the consumer receives their food, maths plays an essential job in calculating the distance and time taken for the food to reach the consumer.

After this discussion we were put into pairs to play a business simulation game, as a demonstration of how fundamental maths is used in the real world on a daily basis.

The Rules of The Game

Each pair had 5,000 euros to start with.
Each pair had to pick five items only, to spend their money on (It was up to the pair how much of their budget they spent on each item).
These stages get repeated for each time period (April-June, July-August, October-December and January-March).

For each time period we had ten minutes to work out what products we wanted to buy, and how much of our budget we wanted to spend on each item. After each time period Richard revealed the percentage of the quantity sold of each item. From this percentage in our pairs we had to work out the amount of money we had made and the amount of leftover stock. Most leftover stock was able to be carried over to the next time period apart from milk and bread which would go out of date.

After each time period we also had to work out how much money we had spent, by adding the prices of what we had bought together, then subtracting the total away from the 5,000 euros we originally started with, or our new budget we had at the beginning of the time period, this left us with our new sum of money we were allowed to spend on our next 5 products.

Tactics

April-May

My partner and I began by spending most of our budget on crisps and beans, as both of these products were not only able to be carried on to future time periods (as they have a long shelf life), but also stable products which consumers buy all year round (because who doesn’t love some beans on toast). We were pretty happy with our decisions as after the first time period we had turned our 5,000 euro starting point into14,540 euros.

July-September

During this time period after looking at how much each product gets sold for (and looking at our neighbours), we discovered that buying champagne would give us a much higher profit than other items, meaning a large proportion of our budget was spent on champagne.

October-December

On the third time period our budget started at 18,380 euros (shout out to the consumers who bought champagne over Summer). Feeling extremely confident that champagne was going to do well again, especially as this time period was over Christmas. We thought tactically and spent all of our budget on seasonal goods (turkey, champagne and beer etc). We were then left with a whopping 35,920 euros, to spend on the last time period.

January-March

Okay so this was it the final round. This was the moment we discovered beans were selling at 10 times the price we were buying them for, making us a 1.25 euro profit per pack of beans. After discovering this we went a bit mad and decided to buy 5,000 euros worth of beans and the rest of our budget on crisps, milk and bread. We were very proud to have come fourth place ending with a large sum of 98,194 euros, however the competitive side of me wanted to know where we had gone wrong.

After the Game

After asking the winners their tactics and reviewing our own work we came to the conclusion that we should have JUST BOUGHT BEANS. If I could give someone advice for doing this in the future it would be to look at what products make the highest profit. Basically try spend as little money as possible on products that will give a high profit. For example from the first round beans were giving us 8 times the amount of money we were buying them for.

I would also recommend to consider the products that go out of date, as on a number of occasions Rebecca and I lost some of our money by not all the products selling and us not being able to carry them on to future months.

The one thing we did do well was considered the time of year, as we spent most of our money on turkey and champagne during Christmas months which products had 100% selling quantity.

In the future this will be a great example to show my pupils or demonstrate to others how the fundamentals of maths are used in every day life even when we don’t know it. Also If I ever own a shop I now know what to sell….. BEANS.

Food Miles. (2017). In: Oxford Disctionary, 1^st ed. Oxford University Press.

Maths in Music

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Music has always been a subject which I love but have never been very good at, after TRYING to learn the violin (we wont talk about that today) I discovered I was never going to be a musical genius. I love nothing more than listening to a bit of Taylor Swift or watching a good musical. Sadly, that’s as far as my musical knowledge goes.

A recent maths input, which I very much enjoyed, was discovering the links between maths and music. At the beginning of the input, we were asked to think of as many links as possible, my list consisted of rhythm and beats in a bar. After being proud of coming up with 2 links you can imagine how amazed I was to discover the list is pretty much endless. Some links, which I wasn’t aware of included:

Note values
Tuning/pitch
Counting songs
Fingering on music
Time signature
Scales
Fibonacci sequence
Patterns and repetition

Patterns and Repetition

In any piece of music, pattern and repetition are usually involved in order to create a rhythm, which is a regular repeated pattern or sound (Rhythm, 2017). According to Professor Peter Schickele, the reason for this “is that regularity of pattern builds up expectations as to what is to come next”, thus making the piece more exciting and memorable. However when composing a piece of music it is essential to create the right balance of repetition as too much can make the piece boring and predictable (Schickele, 1980).

Fibonacci Sequence

The Fibonacci sequence, according to Elaine J. Home (2013) is “a series of numbers where a number is found by adding the two numbers before it”. For example, the sequence starts like 0,1,1,2,3,5,8,13,21. Lets take the number 8, this number belongs in the sequence as the two numbers found before it equals 8 (3+5+8).

In music, a chord is made up of the 1^st(the root), 3^rd and 5^th note on a scale. These are all Fibonacci numbers. Fibonacci numbers also re appear in the octave and multiple scales too (Meisner, 2012). The video below demonstrates how the Fibonacci sequence appears visually in a musical context.

The ‘music in maths’ input was one of my favourites as, even though music is something I listen to on a daily basis, I had never associated it with maths. This highlights the idea that maths is a common theme in many aspects of our lives. In the future, I would be interested in having a second go at trying to learn an instrument. After this input and with a basic fundamental mathematical knowledge, I might have more luck this time as I will be able to look out for the links!

References

Elaine J. (2013). ‘What is the Fibonacci Sequence?’ Available at: https://www.livescience.com/37470-fibonacci-sequence.html (Accessed 17 November 2017)

Meisner G. (2012). ‘Music and the Fibonacci Sequence and Phi’ Available at: https://www.goldennumber.net/music/ (Accessed 15 November 2017)

Rhythm. (2017). In: Oxford Disctionary, 1^st ed. Oxford University Press.

Schickele P. (1980). Symmetries. Symmetry in Music, p.238.

Maths In Sport

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When I first heard we were going to be having lecture on maths in sport, I was extremely confused as I couldn’t think of a link between the two subjects. However, after some research yesterday, and todays input, I have been made aware of the links between these subjects and discovered how the fundamentals of mathematics are used in the majority of sports.

Before todays input, I researched the ways in which maths is used in basketball, and was amazed by the many ways in which maths can influence a player’s performance. I discovered that the force, the length of the players arms and their speed are only some factors which influence a players shot. Angles are another mathematical concept which regularly appear in basketball. For example, if shooting behind the free through line, a smaller angle is necessary as the player is closer to the hoop, and when a defender is trying to block a player a higher shot is necessary. To achieve this, elbows should be as close to the face as possible and the shot should be taken from a 45 degree angle (Mathsblaster.com, 2017).

At the beginning of todays input, we studied an old football league table in our groups. We then rearranged it to look like a modern league table, which we would see today. My group began by noting down the differences between the old and new league tables, and then started making changes to the old table. First we rearranged the teams in order of who came first, depending on the amount of games won. We also used simple algebra to work out the amount of points each team had, as we discovered that, on the old league table, if you multiplied the team’s amount of games won by two and added the number of games they drew, you would receive the final answer of how many points they had (winnings x2 + number of games drawn = total number of points).

After this activity, my group decided to create a new set of rules for netball. We considered the length of the court, the distance in which each player can move and the likelihood of teams scoring, while constructing the new set of rules.

By extending the courts length, more opportunities would be available for changeovers, as more space provided a longer distance for the ball to travel. Providing three hoops instead of one creates a higher chance for points to be made, as players have a choice to decide which hoop they are more likely to score depending on the angle in which they are positioned.

We also made a rule where the closer the player shoots from the hoop the more points their team will receive. This makes the game more exciting and competitive.

Another factor we could have considered is the surface area of the court. Having the court raised in the middle would make it difficult for players to get the ball to the other side of the pitch, as gravity would be forcing the ball back down the hill (Gravity, 2017).

In this new version of netball with new rules, players would have to consider maths constantly throughout the game. They would have to visualise the angles they should use to position themselves in order to have a higher chance of shooting. They would have to calculate the distance in which they have to pass the ball as a longer court means a greater distance for the ball to travel. Players would also have to use basic addition and subtraction skills to work out which hoop would give their team the most points and how many points are needed to win the game.

From this input, I have discovered how many fundamental mathematical concepts are used by, not only league table creators, but also by sports players to improve their performance e.g. distance, speed and time.

In the future when I have my own class, I would like to highlight to pupils how basic mathematical concepts are used in every day activites such as sport. This will provide a good example to show how maths is relevant in other subjects they may be interested in (The Scottish Government, 2008).

References

Gravity. (2017). In: Oxford Dictionary, 1^st ed. Oxford: Oxford University Press.

Mathsblaster.com. (2017) Maths in Basketball-How Maths is used in sport. (Online) Available at: http://www.mathblaster.com/coolmath/articles/basketball-and-math (Accessed 6 Nov. 2016).

The Scottish Government (2008) curriculum for excellence building the curriculum. Available at: http://www.gov.scot/resource/doc/226155/0061245.pdf (Accessed: 7 November 2017).

Profound Understanding Of Fundamental Mathematics

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Before starting the “discovering maths” module, I had never heard of PUFM (profound understanding of fundamental maths). To be honest I thought it sounded quite daunting, however, after doing some research and breaking it down I discovered that PUFM is not quite as intimidating as I had thought. It consists of 4 principles and according to Lipping MA (2010), “to fully promote mathematical learning, teachers must have a profound understanding of fundamental mathematics”. In other words teachers must have a deep, broad and thorough understanding of the mathematical topics in which they are and will be teaching (Ma, 2010).

If you are still as confused as i was initially, think of a bus driver. Bus drivers know the roads well and know how to make short cuts and reroute if needs be. Teachers with PUFM are like bus drivers, only relating to maths. They know the topics they are teaching very well and can take students from their understanding of maths to future learning, and in the process of this they know many ways in which the journey can do it (National Research Council, 2001).

The four principles in which PUFM is made up of are:

Interconnectedness: A teacher with PUFM is able to identify and make connections between the mathematical topics they are teaching (Ma, 2010). They are able to make links with the pupils current knowledge and their future learning, expressing how these depend on each other.
Multiple Perspective: This refers to a teacher being able to approach solutions and mathematical problems in a number of ways (Cuarezma, 2013). This can ensure flexible learning within the classroom as teacher are able to adapt their teaching and give mathematical explanation in regards to their approach (National Research Council, 2001).
Basic Ideas: This involves teachers knowing the, ‘simple but powerful basic concepts and principles of mathematics’. This allows teachers to be able to reinforce basic maths ideas within students, as some of the most basic principles in maths recur through most learning (Cuarezma, 2013).
Longitudinal Coherence: As Teachers, we have to understand that one mathematical topic builds on another, and what we teach students today is a base for future learning (Ma, 2010). A teacher must be able to revisit topics to sole date learning but also move on with their teaching in order to cover the whole curriculum while ensuring the needs of the learners are being met (Curezma, 2013).

Having PUFM as a teacher is essential as teachers must have a deep and thorough understanding of the concepts they are teaching in order to teach them to others. I will be remembering the 4 principles to PUFM when I teach maths in the future to identify how they can be applied when teaching each topic.

Cuarezma, A. (2013). Q & A with Liping Ma: The New York Times. (online). Available at: html http://www.nytimes.com/2013/12/18/opinion/q-a-with-liping-ma.html (Accessed 1 Nov. 2017).

Ma, L. (2010) Knowing and Teaching Elementary Maths. 2^nd ed. New York: Routledge Publications. pp, 12, 14-19, 34-37.

National Research Council. (2001) Knowing and Learning Mathematics for Teaching. 1^st ed. Washington. The National Academies Press. pp, 11-20.

Maths Anxiety

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Looking back to seven years ago, I was in primary seven getting ready to start secondary school. Maths was something I loved and a subject I felt confident in.

Six years ago I remember sitting through a maths class in S1 looking at the teacher who was desperately trying to teach me how to do ‘The St. Andrews flag method’. I had no idea what she was talking about and remember feeling like I wanted to cry. How and when did I get so bad at maths?

From S1 to S4 maths was a struggle I remember fearing every lesson and trying to think of any excuse to avoid class whether it was ‘feeling sick’ or having ‘dentist appointments’. This of course made things worse as I began to make myself believe maths was more scary than it actually was. I began to give up listening in class or paying attention as I believed I would never understand and paying attention was in vain.

This evidently affected me as I sat crying the night before my national 5 maths exam. Frantically looking through every textbook and jotter praying for a miracle that would somehow turn me into a math genius. This of course didn’t happen. I ended up spending my whole night watching ‘Mr maths genius’ on Youtube explaning ‘sin, cos and tan’ which, to me, were just random words. To cut a long story short, I failed the exam.

Despite getting good grades in my other subjects I remember feeling disappointed in myself. Looking through every university’s entry requirements, I began to realise if I was serious about becoming a primary teacher I was going to have to try again.

Round 2….

Despite hating the walk to maths and dreading every lesson, I made sure I concentrated in class. After a year of sticking it out, I achieved an A at nat 5 maths. Out of all my grades I had achieved at Higher level, I felt most proud of this as it meant my dream of becoming a teacher could happen and maths was no longer holding me back.

A month ago I found myself sitting at a desk looking at a power point which read ‘Discovering Maths’ and thought “what have I done…?” The sheer mention of the word maths made me want be sick.

The first topic we looked at was maths anxiety. Reading through the symptoms and causes, I realised this was something I suffer from.

Maths anxiety is something I had never heard of until recently, and according to Hill, ‘is a debilitating negative emotional reaction towards mathematics’ (Hill, 2016). After researching I have discovered I am not alone in feeling anxious towards maths in fact it is believed that a quarter of the worlds population share this feeing (Brain, 2012).

When I become a teacher, maths anxiety is something I cannot bring into the classroom. I will refuse to pass my fear of maths onto any future generations. It is, therefore, my aim to become more confident in my maths abilities and defeat the monster that maths is.

Brian, F. (2012). Maths Anxiety: The numbers are mounting. The Guardian. (online) Available at: https://www.theguardian.com/education/2012/apr/30/maths-anxiety-school-support (Accessed 16 October. 2017).

Hill, F. (2016). ‘Maths anxiety in primary and secondary school students’, Learning and Individual Differences, pp. 63-68.