# Discovering Maths: Going Forward

After receiving our final input today for the Discovering Mathematics module, I realised how much my opinions and misconceptions of mathematics have changed. I wanted to write one final blog post to summarise some of the key discoveries I have made over the course of the semester and the ways in which I will use these discoveries in my teaching practice.

My opinion of mathematics

Before beginning this elective, I was extremely apprehensive about the thought of “doing mathematics”. What I mean by this is that I always viewed mathematics as complicated sums and equations which needed to be solved. I saw mathematics as a stand alone subject because my teachers never made explicit links to the mathematics I was learning and other curricular areas or the wider world. Now, after researching and questioning the mathematics in everyday life – such as in sport, music, art and games – I understand that maths really is everywhere! Even though the links to everyday life and maths may not be explicit, it is evident the importance that basic mathematical principles hold in so many aspects of life. My favourite example of this is the idea of thinking strategically, which is something I have always done when playing a board game for example, but I’ve never considered this to be a mathematical skill.

My favourite discoveries

One of my favourite discoveries I made during this module is the link between mathematics and sport. I particularly enjoyed thinking about the mathematics involved in the rules of sport and then reinventing these rules, using the mathematical principles I had identified (Coventry, 2017, a). This is an example of one of the moments when I realised I was really using mathematics rather than simply doing it to solve an equation. I also liked that I could relate mathematics to a dancing – a hobby I’ve enjoyed from an early age. Never did I think I’d be able to relate a passion of mine to a subject I never thought I’d use outside school (Coventry, 2017, b).

How this will influence my teaching practice

Going forward, I am excited to use some of the experiments and examples we have used in class in my own classroom with pupils. In particular I would love them to try and make links to mathematical principles and something they are passionate about such as a sport or a game. This will show the pupils that mathematics is relevant in their lives outside school. I will also be aware of the way I teach mathematics, making sure I issue work which is meaningful and enjoyable for the pupils. A good example I could look at with an upper years class is allowing them to practice their budgeting and money handling skills whilst looking at food chain supply (Sloan, 2017). This would allow pupils to work together in teams to decide what they want to spend their money on, whilst using basic arithmetic to calculate the profit they would make. This is something I feel as though children would become very invested in as they try and beat other groups to make the most money. This could also be a challenge which could be spread across an entire term, with prices for stock changing throughout the duration of the process.

This module has really opened my eyes to the ways in which maths can be explored in wide and meaningful contexts. It has also highlighted the issue that too often people presume they cannot do maths without thinking about what they do in everyday life which is underpinned by mathematical theory and practice. It is therefore vital for me, as a teacher, to make these links explicit to children so they can develop their interest in mathematics throughout their education and beyond.

References

Coventry, J. (2017, a) Discovering Maths in Sport. [Blog] Glow. Available at: https://blogs.glowscotland.org.uk/glowblogs/jceportfolio/2017/11/08/discovering-maths-in-sport/ [Accessed 28 Nov. 2017].

Coventry, J. (2017, b) Discovering Maths in Dance. [Blog] Glow. Available at: https://blogs.glowscotland.org.uk/glowblogs/jceportfolio/2017/09/14/discovering-maths-in-dance/ [Accessed 28 Nov. 2017]

Sloan, A. (2017) The Apprentice Activity. [Blog] Glow. Available at: https://blogs.glowscotland.org.uk/glowblogs/ahseportfolio/2017/11/16/the-apprentice-activity/ [Accessed 28 Nov. 2017]

# Mathematics in Computer Games 👾

As part of this module we have looked at many examples of where fundamental mathematics comes into different industries, one of which being the computer/video game industry.

There are many mathematical principles behind the creation of computer games including: geometry, vectors, transformations, matrices and physics (Goodman, 2011). For example, matrices relate to 3D graphics. Many games nowadays take place in a 3D virtual world. Objects and charactrs are created from a set of 3D points. As explained by (Wilkins, n.d., p.3) “these points are stored in a data structure as columns of coordinates relative to a convenient local coordinate system. These objects are manipulated (moved, rotated, scaled) to their desired shap and orientation then positioned in the world by a ‘change of coordinates’ to the world coordinate system”.

Not only is there fundamental mathematics behind the creation of the games but also for playing them. For example, one of the main mathematical skills required to succeed in playing most games is problem solving. In most popular and common games such as, FIFA, Call of Duty and Minecraft there are usually scenarios which require the player to overcome or solve (Tassi, 2016). For example, in FIFA players need to think strategically to work out the best way to tackle other players to get ball possession and the best angle for scoring goals (another mathematical concept!) The creators of these games need to look at the aspect of probability, to ensure players do not encounter the same obstacles all the time or so that they need to defeat these obstacles in different ways.

It is interesting to note that it is stereotypically boys who are deemed to be good at mathematics and science (Coughlan, 2015). Also, it is stereotypically boys who play more computer games (54%) (Takahashi, 2013). Could this therefore prove that video/computer games can enhance mathematical ability? Obviously, this is not a completely accurate argument considering that these are stereotypes, but it does account for the research you find on the two subjects.

As mentioned previously, I never realised the links between mathematics and computing and will therefore ensure that when I am teaching computing in the future I will make the links clear to pupils. Not only this, but hopefully encourage reluctant learners, who struggle with mathematics, that they are able to apply what they know to something fun and creative.

References

Coughlan, S. (2015). Clever girls lack confidence in science and maths. BBC News. [online] Available at: http://www.bbc.co.uk/news/education-31733742 [Accessed 14 Nov. 2017].

Goodman, D. (2011). The Use of Mathematics in Computer Games. NRICH. [online] Available at: https://nrich.maths.org/1374 [Accessed 14 Nov. 2017].

Ma, L. (1999). Knowing and teaching elementary mathematics : teachers’ understanding of fundamental mathematics in China and the United States. Mahwah, N.J.: Lawrence Erlbaum Associates.

Takahashi, D. (2013). More than 1.2 billion people are playing games. VentureBeat. [online] Available at: https://venturebeat.com/2013/11/25/more-than-1-2-billion-people-are-playing-games/ [Accessed 14 Nov. 2017].

Tassi, P. (2016). Here Are The Five Best-Selling Video Games Of All Time. Forbes. [online] Available at: https://www.forbes.com/sites/insertcoin/2016/07/08/here-are-the-five-best-selling-video-games-of-all-time/#7c3b3f025926 [Accessed 14 Nov. 2017].

Wilkins, K. (n.d.). MATHEMATICS FOR COMPUTER GAMES TECHNOLOGY. [ebook] Bathurst. Available at: http://users.math.uoc.gr/~ictm2/Proceedings/pap458.pdf [Accessed 14 Nov. 2017].

# Discovering Maths in Sport

Discovering Maths in Sport

Image credit: Thomas (2012) https://plus.maths.org/content/spinning-perfect-serve

As part of my journey in discovering mathematics, I have become aware of the fact that maths really is everywhere. So far, we have touched on maths in art, stories, music, computing, gambling and sport. In particular I’ve enjoyed looking at these aspects from a very open perspective, questioning the mathematical theory behind the subject. This was especially interesting to look at in relation to sport, where we took a step back from the rules of sport and completely reinvented them.

Firstly, it is important to understand how mathematics relates to sport. There are many specific examples, such as The Magnus effect in tennis which relates to how the ball spins as it moves through the air (Thomas, 2012). A similar example can be seen in badminton where the shuttlecock obeys Newton’s laws of motion. Its acceleration is controlled by the downward force of gravity and a “drag force” from the air that is proportional to the square of the shuttlecock’s speed through the air (The Plus Olympic calendar, 2012).

Although these are very specific examples, we can also see how particular aspects of mathematics relate to nearly all sports: the physics of sport, sporting strategy, architecture and infrastructure, predicting results and sporting statistics, scoring and ranking, and betting and odds (Teacher package: Mathematics in sport, 2010).

Something which we considered during our workshop was whether the popular sports we know well are structured in the best way possible. My group looked at the example of hockey. At first we struggled to come up with ways of improving the sport as we already thought that the rules and strategies of the sport were fair. Then we were posed with the question: what would make hockey more exciting watch? This started to make us think more creatively…

Firstly, we thought about the pitch layout – like most pitches the hockey field is flat. We considered increasing the difficulty of the sport by raising the surface of the area where the goal is (this area is known as the “D” due to its shape – another mathematical link!) If there was a platform of even just a few inches, this would require the players to be able to lift (or chip) the ball and retain control quickly after to avoid defenders “stealing” the ball. One of the key rules for hockey is that you are only allowed to score when you are in the “D”. We thought the game would be more exciting if players were able to score from anywhere on the pitch, meaning that defenders would always need to be alert. We thought we could also change the scoring system so that if players score outside the “D” they can win their team 3 points instead of 1.

Moreover, we thought of a way of motivating players to score more goals. We decided that for the last 10 minutes in each half (hockey is a game of two 35 minutes halves) the defenders would not be allowed to defend inside the “D”. This means that if the player manages to get past the defenders, it is 1v1 scenario with the goal keeper. This would allow the player to show off their scoring skills/abilities.

Below is an illustration of our reinvention of hockey where the crosses represent the players and dashed lines represent their formation:

The ideas that other groups developed also sparked a lot of imagination. I really liked the idea one group came up with in relation to Netball. They imagined what the sport would look like if there were 3 baskets of different heights (each worth different points) instead of just one. Like us, they also looked at the layout of the court but decided to make it larger, to encourage more changeover throughout the match.

This was an extremely engaging workshop which really allowed me to see sport from a different perspective than how I usually would (as very structured and fixed). As we have discovered, mathematics is not always fixed and there are often many ways of doing mathematics. I think this is a lesson I would like to look at in the future as a teacher – to allow the pupils to experiment with sports that they love (see previous blog post about mathematics in dance) (Coventry, 2017). It is also a great way for the children to understand the links and relevance of mathematics.

References

Coventry, J. (2017) Discovering Maths in Dance. [Blog] Glow. Available at: https://blogs.glowscotland.org.uk/glowblogs/jceportfolio/2017/09/14/discovering-maths-in-dance/ [Accessed 8 Nov. 2017].

Teacher package: Mathematics in sport. (2010). Plus. [online] Available at: https://plus.maths.org/content/teacher-package-mathematics-sport [Accessed 8 Nov. 2017].

The Plus Olympic calendar. (2012). Plus. [online] Available at: https://plus.maths.org/content/plus-olympic-calendar-monday-6th-august [Accessed 8 Nov. 2017].

Thomas, R. (2012). Spinning the perfect serve. Plus. [online] Available at: https://plus.maths.org/content/spinning-perfect-serve [Accessed 8 Nov. 2017].

# Analogue Clocks: Pointless and Confusing?

Living in a digital world, I ask the question: do we need to teach pupils how to read an analogue clock in schools?

Although most of us have converted to digital clocks in our online, digital world with our smartphones, computers and televisions; we cannot deny that our encounters with analogue clocks are not completely non-existent. There is an argument that in most places we visit – schools, work offices, supermarkets, restaurants and hotels – we most likely still encounter analogue clocks (Merz, 2014). Therefore, is teaching how to read analogue clocks not a necessary skill to teach pupils in school?

According to Merz (2014), many teachers are frustrated with the idea of this skill being disregarded, with the argument that analogue clocks can provide a vivid representation of time that digital clocks cannot – which can aide visual learners. Analogue clocks can also teach concepts including time management, the passage of time and how much time we have left to complete something (Merz, 2014).

However, with our fast-developing technological advances, it is difficult not to wonder if eventually analogue clocks will disappear in our society. Nowadays, we see plasma screen televisions or digital billboards nearly everywhere we go – displaying digital time. Although analogue clocks are often visually appealing and provide nice décor, they don’t really provide any use other than telling the time. It is therefore arguable that digital screens are much more valuable in society as they are multi-purposeful and allow for more creativity (The benefits of digital billboard advertising, 2015). For example, recently in a shopping centre in Edinburgh, I passed a large television screen which displayed the current top news stories, multiple adverts for new products which could be found in the centre, whilst also displaying the time.

Moreover, one of the key issues with teaching pupils about the analogue clock in schools, is how complex it is for pupils to grasp and understand. This light-hearted, comical video highlights the difficulties for young learners learning how to read time:

(Dave Allen – “Teaching Your Kid Time” – ’93 – stereo HQ., 2009)

I partly decided to write this blog post as I was one of the learners in primary school who had difficulties learning about time. I could not wrap my head around the idea of ‘quarter past’, ‘half past’ and ‘quarter to’ (considering we represented every other number on the clock as a number). I also struggled with the concept that there were different ways of reading the clock (e.g. saying 35 past 7 or 25 to 8) which would both be correct. This raises key issues of problem solving and looking at a mathematical concept from multiple perspectives (key skills which are transferrable across all mathematical topics.)

It is important to note that these are aspects of telling the time which apply to both reading the analogue AND digital clock. It is therefore my opinion that the real issue with teaching time to pupils is the concept itself, rather than teaching pupils how to read a particular type of clock. The video above does highlight the difficulties of learning to read an analogue clock – however with the fundamental understanding of the concept of telling the time, I believe that most pupils would welcome the challenge of applying their knowledge to reading an analogue clock. For example, it is vital that children have a strong understanding that 6 is half of 12 to be able to understand why we use the term half past. Another skill which would benefit children before reading an analogue clock is knowing the 5 times table. According to Drabble (2013), without knowing the 5 times table, “anything beyond the o’clocks becomes almost unotainable.” This relates to the idea of longitudinal coherence, introduced by Ma (1999) who states that teachers should use children’s prior knowledge to enhance learning in the topic at hand. It also links with what she writes about basic ideas, meaning that children should revisit the basic concepts they have learned (i.e. fractions and times tables) to understand that they are required for other areas of mathematics (Ma, 1999).

In conclusion, after doing research online and through my own experiences, I believe there is a necessity for teaching pupils about digital and analogue clocks. I believe that we currently live in a world where analogue and digital clocks are both relevant and should therefore both be exposed to pupils. I have realised since studying this issue, that it is important to ensure that pupils understand the principles behind telling the time before introducing them to an analogue OR a digital clock. Furthermore, learning how to read two types of clocks reinforces pupils’ understanding about the concept of time and allows them to practice telling the time from different contexts. This reflects the work of Ma (1999), who highlights the importance of connectedness – meaning that children can link what they have learned to different contexts.

This picture reflects what I saw on my first year placement and shows how to make reading the time on an analogue clock more visually appealing for pupils, whilst also acting as a visual aide (however it is important that pupils realise that they cannot rely on this, as every other analogue clock they see will not be represented in this way!):

Image credit: Teacher’s Pet (2014) www.tpet.co.uk (http://displays.tpet.co.uk/?resource=1507#/ViewResource/id1507)

References

Dave Allen – “Teaching Your Kid Time” – ’93 – stereo HQ. (2009). (Video) YouTube: davidwrightatloppers.

Drabble, E. (2013). How to teach … telling the time. The Guardian. [online] Available at: https://www.theguardian.com/education/teacher-blog/2013/aug/05/telling-the-time-teaching-resources [Accessed 4 Nov. 2017].

Ma, L. (1999). Knowing and teaching elementary mathematics : teachers’ understanding of fundamental mathematics in China and the United States. Mahwah, N.J.: Lawrence Erlbaum Associates.

Merz, S. (2014). Should We Still Teach Analog Clocks?. [Blog] Stories From Schoolaz. Available at: http://www.storiesfromschoolaz.org/still-teach-analog-clocks/ [Accessed 4 Nov. 2017].

The benefits of digital billboard advertising. (2015). [Blog] Signkick. Available at: http://www.signkick.co.uk/blog/the-benefits-of-digital-billboard-advertising/ [Accessed 4 Nov. 2017].