Monthly Archives: November 2017

My Future with Fundamental Mathematics – Final Reflection on Discovering Mathematics

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Semester 1 of second year is nearing its end and we are all preparing for our assignments and exam. I am deliberately taking the time away from the heavy studying to reflect upon the learning I have gained in the Discovering Mathematics module as I think it will be beneficial to create a short final reflective post of all the things we have gained from the elective.

Firstly, I have to say that my entire perception of Mathematics has changed drastically because of the relaxed environment that we were within when experimenting with different mathematical concepts (something that was alien to me, as my main experience of maths was to study it in order to pass an exam to gain a qualification). Bello (2010), fittingly describes this re-awakened awareness one gains when relearning maths in adulthood:

“Entering the world of maths as an adult was very different from entering it as a child, when the requirement to pass exams means that often the really engrossing stuff is passed over. Now, I was free to wander down avenues just because they sounded curious and interesting.” (Bellos, 2010, pg. 10).

I felt that we were all in the same situation in terms of our perceptions in maths because, for the majority of us, our last experience of mathematics was within an exam hall. This module gave us the opportunity to step away from a regimented formulae-based learning to the subject and gave us various areas within wider society in which mathematics played an instrumental part to.

I had my own discoveries within mathematics this year:

  • Mathematics is literally everywhere – in the arts, in science, in architecture, in motorcycles, in shopping, and even in us (circadian rhythm)
  • The topics within mathematics overlap with one another (Ma’s concept of connectedness is a crucial point here)
  • Although logical, mathematics is far more creative than what people initially believe – as we explored the mathematics behind photography and the golden ratio
  • The word lunatic comes from the word lunar, which means moon, showing that a full moon has a greater affect on people’s actions due to its pull on the earth’s water (well, some people believe this, however, it has become somewhat of an urban legend with people disproving it’s scientific argument and basing the myth on a psychological illusory correlation) (Arkowitz & Lilienfeld, 2009)

Looking ahead, I am definitely going to view my future maths lessons with a finer eye for depth of what I can provide a class in terms of fundamental mathematical knowledge (as I have done with a reflection of previous math lessons I did during my first year placement). As practitioners, we have a huge responsibility in teaching mathematics because it is one of those touchy subjects where people can disconnect from it after one too many bad experiences with it during childhood.

Wider societal links, on top of a strong foundation of basic ideas within mathematics can set a student up for life in terms of their capabilities within the subject of mathematics, because context can widen one’s appreciation for a subject within the real world.

Furthermore, I didn’t come into the elective with any real irrational fear for mathematics; however, I did have an issue with doubting my calculations. Having openness about mathematics makes it far more easier to take a mistake as what it is: a simple error that can be corrected. I want to be able to establish a classroom that has this embracement of both our success in life and our shortcomings too.

Overall, Ma’s (2010) studies in mathematics has tied in really well with the premise of the Discovering Mathematics module, as we ourselves have expanded our mathematical horizons to see the subject in a new way just as her comparisons between China and the USA’s teachings had helped her come to a realisation of what makes a really worthwhile experience within mathematics.

To finish off my blogging for Discovering Mathematics, I collected all my blog posts and put them in Wordle to see what the most used words I have written during my ventures in maths. My favourite part of the wordle is probably the fact ‘mathematics’ and ‘life’ are almost in a pairing off at the side, which shows that human life and maths coexist to support one another. Without maths we would struggle with our day-to-day activities. We wouldn’t have any of the advanced technology we have today without someone being creative enough with numbers. ‘Students’, ‘teachers’, ‘numbers’, ‘subject’ and ‘systems’ are really at the forefront and are most prominent, however underlying them are the terms ‘art’, ‘beyond’, ‘shopping’, ‘important’ and ‘different’ to name a few, which I believe to be another fitting form of imagery. We might have the structuring of mathematics being the first thing we think of, but if we delve deeper we can see how far the roots of mathematics grow within various topics and how deep they can go. Finally, I like that students found its way neatly in the centre (almost like a nod to the child-centred approach), as it is the students that need to be thought of first and foremost.

The future looks far brighter for my practice than it did before starting this module, and for me, that is the best result I could have gained from any experience with mathematics.

Reference:

Arkowitz, Hal and Lilienfeld, Scott O. (2009) Lunacy and the Full Moon [Article] Available at: https://www.scientificamerican.com/article/lunacy-and-the-full-moon/ (accessed 22nd of November 2017)

Bellos, Alex (2010) Alex’s Adventures in Numberland London: Bloomsbury

Ma, Liping (2010) Knowing and Teaching elementary mathematics: teachers’ understanding of fundamental mathematics in China and the United States New York: Routledge.

 

“Maths is truth, truth maths”: Romanticism, Poetry and Mathematics

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The Romantic Period, an intellectual, literary and artistic movement that swept across Europe, saw a change in the way society viewed the world during the late 1700s and early 1800s (Encyclopaedia Britannica, 2005). Nature was increasingly becoming more valuable to the Romantics during a time of industrial revolution, where trade and business was becoming king.

John Keats (1795 – 1821)

John Keats, one of the most famous Romantic poets, explored the natural world within his poetry and had a great fascination and desire to immerse his own being into nature itself – removing himself from the societal pressures life brought. Keats’ life was filled with much turmoil and his only escape was his own poetry and art of writing. Unfortunately, much of Keats’ work was not valued at when he was alive and he was heavily criticised by many. Thus, resulting in him believing he had failed in the art in which he blossomed…

You may question the mathematics behind a Romantic poet, whose main ideology was to distance oneself from life’s industrial pressures and structure, however, Keats’ art seeps with fundamental mathematics (Ma, 2010) underlying his literary prowess because he meticulously planned out his art to convey a particular theme or emotion and he understood the importance of selecting his words carefully in each of his poems. This blog post will explore this within his poetry and will also see the importance of mathematical thinking within creativity as a whole.

In “To Autumn” Keats breaks down the heavily structured way of writing by including an extra line in each stanza (a verse in poetry):

“For Summer has o’er-brimm’d their clammy cells” (Keats)

In his mind, autumn is a huge season filled with so much change and beauty. To convey this vast enormity, 11 lines are favoured over the traditional 10 to convey how excessive the nature of autumn is and how overwhelmingly beautiful it is to him. The extra line above even states that summer is overfilling (almost like a liquid going beyond the brim of a glass) into the orange richness of autumn, which Keats has shown through the lines literally overflowing beyond the constraints of typical poetry.

The Romantics had a key ideology of embracing self-awareness in people’s own emotions as a necessary way of improving society and bettering the human condition in a time of corruption and social class divides (Sallé, 1992). Keats effectively combines his art of poetry and his carefree beliefs with a structured and logical approach in formulation, similar to those who have the freedom to experiment and explore mathematics freely. Sadly, Keats’ work was not valued until after his death. I find this fitting very well with the mathematicians that believe that there is only one way to go about working through a problem. Multiple perspectives should be evident in both mathematics and the arts, because set rules only lead to confinement in gaining self-progress in both areas. There is more than one way to calculate a mathematical problem in the same way there is more than one way to write a poem. Keats could not reach his full potential as a writer due to the pressures placed upon him and this can be seen as an embodiment of a teacher or professional undermining the prospects of a student within mathematics – disaster will be the only outcome of negativity.

Poetry should also go beyond the words that are written on a page. During an input in Languages, about reading poetry, we were enthused to really appreciate the act of performing poetry aloud. This can be greatly identified in Keats’ poetry once more as he also saw the importance of rhythm in writing:

“Away! Away! For I will fly to thee,

Not charioted by Bacchus and his pards,

But on viewless wings of poesy” (Keats)

Within “Ode to a Nightingale”, Keats establishes an iambic pentameter as he picks each word systematically to follow a pattern, a key aspect within mathematical thinking (Bellos, 2010), of an unstressed syllable being preceded by a stressed syllable. Every line follows a da-dum da-dum rhythm so that the poem could be performed like a song or to a little tune. This iambic pentameter is used to symbolise the flight of a nightingale flying higher and lower, always changing and never following a set path. Keats explored the freedom of the bird and its stance in nature with its wings allowing it to go wherever its heart desire. This can also be connected with the mathematical structure of music, because songs are psychologically made to instil a mood, much like all aspects of the arts. For example, upbeat music is normally used to bring joy (Wall, 2013). This interconnects back to the Romanticism movement once more as all the arts saw a wave of change during this period, not just poetry and writing.

Friedrich, C. D. (1818) Der Wanderer über den Nebelmeer (Wanderer Above the Sea of Fog) – a Romantic painting that also explored man’s relationship with nature, showing the movement’s impact on the arts.

Delving deeper, the structuring of poetry and even language as a whole requires so many different parts (particularly within a persons time in education) to be taught and learned effectively in order for people to be able to communicate properly: spelling, grammar, sentence structure, punctuation, letters, symbols, words, paragraphs, essays… The list could go on and on. Mathematics breaths the same air in terms of its longitudinal coherence because we wouldn’t categorise the various aspects that make up language in the same way some people break mathematics down into specific topics. Teachers that make connections back to the fundamental skills of mathematics when exploring new areas with students provide the best learning experience, because students get to see the wider importance of maths (Ma, 2010). If we were to tackle language teaching in the same confining manner that maths is taught, then communication would be impossible because children wouldn’t see the importance without the contextualisation. In fictional writing, we normally get children to think outside the box and explore outlandish and creative environments, and yet, we then teach mathematics in a polar opposite manner of textbook work and worksheets (Haylock, 2014).

However, flipping the argument on its head, having too heavily a structured environment for writing could also hinder learners in the creative process (Perkins, 2012). Acrostic poems, rhyming schemes and other constraints being placed upon children when they first explore the art of poetry could paint the picture that, from the get-go, creativity and freedom to express one’s thoughts in writing has to conform to a set of rules and if it doesn’t, it isn’t valued. This can be interlinked with the emphasis on teaching through reciting formulae in order to deal with mathematical problems. Many children have experienced negative emotions with the subject when they see they have gotten a question incorrect when their mathematics might actually all be correct up until making a minor mistake.

Much like Keats’ poetry and the Romantics’ ideologies, we need to find a way of gaining a bounty of appreciation and understanding of the application of the fundamental principles of mathematics within life that go beyond the barriers that have been set by years of anxiety, years of dated practice and years of staying within the lines of convention.

The title of this blog comes from a very fitting last line of one of Keats’ poems, “Ode on a Grecian Urn”:

“Beauty is truth, truth beauty,” – that is all
Ye know on earth, and all ye need to know

The poet proclaims that the truth in our existence should be sourced through our own individual appreciation of life and shouldn’t be hindered by pure rationale. Once again, experimentation within mathematics should be heralded over utilising purely formulae in the subject because it has more substance for a learner. Life itself would be ultimately boring if we could answer everything with one answer; having a sense of discovery about existence is far more exciting and mathematics should be viewed in the same light. The art we create can transcend experiences, emotions and events in time and thats what Keats wanted to grasp within this last line. Utilising mathematics effectively, we could potentially do the same:

“Concepts such as active literacy and the natural learning environment have proved to be powerful tools in changing attitudes and practice in the field of language arts. Properly understood and adapted, the same concepts can work just as powerfully for us, and for our students, in mathematics.” (Monroe, 1996, pg. 369)

Overall, Keats and the other Romantics were controversial in their carefree beliefs during a time of structure and order; however, they themselves formulated structures within their creative art forms to emphasis that empathy and compassion were far more important to society than money and power. I think that we can take great points from the Romantics, poetry and writing as a whole when viewing mathematics as they have parts that overlap, just like the subject of mathematics. Education is, in itself, a wholesome topic and should be viewed in such a cross-curricular manner whether in language, mathematics or any subject we learn and teach.

All the extracts of poetry sourced from:

Keats, John (1994). The Complete Poems of John Keats (Wordsworth Poetry Library) Wordsworth Editions Limited: Hertfordshire.

Reference:

Bellos, Alex (2010) Alex’s Adventures in Numberland London: Bloomsbury

Sallé JC. (1992) Keats, John (1795–1821). In: Raimond J., Watson J.R. (eds) A Handbook to English Romanticism. Palgrave Macmillan: London

Encyclopaedia Britannica. (2005). Romanticism [Article] Available at: https://web.archive.org/web/20051013060413/http://www.britannica.com/eb/article-9083836 (Accessed 17th of November 2017)

Ma, Liping (2010) Knowing and Teaching elementary mathematics: teachers’ understanding of fundamental mathematics in China and the United States New York: Routledge.

Monroe, E. E. (1996) Languages and Mathematics: a Natural Connection for Achieving Literacy Reading Horizons, 36 (5). Available from: http://scholarworks.wmich.edu/reading_horizons/vol36/iss5/1 (Accessed 17th of November 2017)

Perkins, Margaret (2012) Observing Primary Literacy London: SAGE Publications Inc.

Wall, Timothy (2013) Trying to be happier works when listening to upbeat music according to mu research [Article] Available at: http://munews.missouri.edu/news-releases/2013/0514-trying-to-be-happier-works-when-listening-to-upbeat-music-according-to-mu-research/ (Accessed 17th of November 2017)

Images sourced from: https://upload.wikimedia.org/

Two-kwondo: Breaking Down Maths in Taekwondo and Sports

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The sport industry is a huge sector in the market that people enjoy watching, participating in and involving themselves in someway. There is a sense of community with playing or being a fan of a particular sport which is crucial for people (children in particular) and their physical and mental wellbeing (Croddy, 2013). For our first task in exploring mathematics within sport, Richard wanted to show us the mathematics that goes into representing data of sport results.

We were given the first ever-recorded league table of football that was held in England in 1888 (Wright, 2016):

The first league table recorded for football in 1888. Can you see the differences between today’s recordings?

Now, we had to look at the current league tables of today and try and represent the original data as if it was held today. The clear differences were that the table of results was arranged in alphabetical order, in contrast to our current position-in-the-league formatting, which allows for easy distinguishing between who is on top and who is on bottom. This is what we created:

How the original league’s results would look like with today’s formatting

Another discovery that was made that, using today’s rules of scoring and goal differences, we placed Stoke 11th and Notts County 12th because they had a better goal difference of -31 to Notts’ -34. However, in 1888 it was Stoke that was at the bottom… How could this have been?

Well, the explanation showed us that rules within sports have progressed and developed as time has passed. Previously, the rule was that (prior to goal difference) the team that scored more goals against their opponents overall were higher placed in the league. With Notts having 39 goals over Stoke’s 26 it was clear why Stoke wasn’t so lucky back in 1888. Now, exploring mathematics further, we can see that representation of data is crucial within sports, because it can make the difference between rankings.

Our mathematics was correct when we changed the ranking between Notts and Stoke, because we calculated the difference in goals (number of goals scored – number of goals conceded = goal difference). This measurement and rule did not exist in the 1880s football therefore the overall result was totally different. It wasn’t until the 70s that the concept of goal difference was made official when ranking teams in a league.

So, the next task we were set was to do our own development of a particular sport and see what the mathematical consequences would be if we changed and adapted particular rules or layouts.

We chose Taekwondo.

The rules were quite easy to follow:

1 vs. 1 matches that are normally fought in a square ring.

You gain a point for a punch, two for a kick to body and three for a kick to the head. People are placed in divisions depending on weight.

Points can be lost with warnings for illegal activity.

The rules of Taekwondo

We decided to think about the size of the ring: giving competitors a smaller area to compete in means there is less space for them to avoid their opponents and try and waste time. We even thought about descaling the ring each round, in order to establish more pressure to the game, which could alter the performance of the competitors, thus possibly changing the outcome of the match with competitors breaking under the pressure due to the mathematics of the game becoming psychological (Bellos, 2010). We also thought of introducing different height divisions, as weight doesn’t have enough of a variable of competitors. Having competitors fight with equal stature could make the game fairer because they don’t have to fight someone much larger or much smaller than them in height. On top of the possibility of adjusting the size of the ring, we thought of decreasing the times per round, so that fighters have less amount of time to amass points against their opponent.

To change the game entirely, we even re-assessed the points system and allowed for more variables in the types of hits that can count towards points:

1 – punch to body

2 – punch to head

3 – kick to body

4 – kick to head

Another outlandish suggestion was making fights of doubles! 2 vs. 2 would add in 2 extra players into the match and would drastically alter how the sport was played. Other rules would also have to be enforced in order for fighters not to target just one person, for example. We even came up for a name for this new breed of taekwondo – two-kwondo. It evoked similar connotations to the matches held in the World Wrestling Entertainment (WWE) matches with tag teams.

Some of our outlandish changes of taekwondo. Some really got us examining the core mathematics within the martial art sport.

Overall, what we were doing was examining the rules and variables that make up a sport and how altering even just one of the variables has a drastic change on the outcome. This could be correlated to conducting science experiments because we were hypothesising what the changes would do to the results and thinking about how drastic a change could be on a sport. This further emphasises the points made by Ma (2010) where she believes that mathematics cannot be viewed in a singular way, as it impacts various areas of life (in our case, sport). This can also represent the avenue of multiple perspectives as we had to examine the variables we had changed and be critical on whether the changes would be beneficial to the sport, or whether they’d hinder the game just as you would in examining the different procedures in calculating a mathematics problem.

This creative way of examining sports could be an interesting way for children to pick apart the rules of their favourite sports and see the various mathematical concepts that are used to determine the rules of any game. I think this would be far more interesting and context-based than using textbooks to reinforce formulas and skills.

Ending this post, I feel a lot more enlightened in terms of mathematics in sport. We can go beyond just thinking about how points are earned and actually think about the parameters of specific rules and the layout of the environment a sport is held in.

Reference:

Bellos, Alex (2010) Alex’s Adventures in Numberland London: Bloomsbury

Croddy, Kristi (2013) Importance of Sports and Games in Schools [article] Available at: https://www.livestrong.com/article/367838-importance-of-sports-games-in-school/ (Accessed 11th of November 2017)

Ma, Liping (2010) Knowing and Teaching elementary mathematics: teachers’ understanding of fundamental mathematics in China and the United States New York: Routledge.

Wright, Chris (2016) On this day in 1888, Preston the big winners as first ever football league results recorded [Article] Available at: http://www.whoateallthepies.tv/retro/243184/on-this-day-in-1888-preston-the-big-winners-as-first-ever-football-league-results-recorded.html (Accessed 11th of November 2017)

The Truth is; The Game Was Rigged From The Very Start

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“The good news is that in every deck of fifty-two cards there are 2 598 960 possible hands. The bad news is that you are only going to be dealt one of them” Anthony Holden (no date)

A famous saying goes, that one must “play the hand they are dealt” with in life. This is of course a reference to the more infamous than famous game of poker… or maybe it was blackjack? The game doesn’t matter because the truth is; the game of gambling was rigged from the start.

If we were to look at it more in depth, the phrase itself has more to do with the ordinary pack of cards used to play the games of chance. Eastaway (2010) believes that it is crazy that we, as a species, have conjured up vast amounts of trivial games that can be played across the globe with just simple mathematical rules and a deck of playing cards. Whole empires of gambling have been created around it in Las Vegas with casinos making huge amount of money off of revelers who come in hopes of making it big. Now, this is where the concept of gambling comes onto the table…

The concept of gambling was explored during an input in the Discovering Mathematics module where the principles of probability and chance were examined further. Gambling itself is all a game of luck, isn’t it really? Not necessarily as the odds are further explored in many of the popular games that are legal in casinos.

Games that are present in casinos across the globe are set up against you from the very beginning: “All casino games involve negative-expectation bets; in other words, in these games gamblers should expect to lose money” (Bellos, 2010, pg. 314). Psychologically, we can see that people that win when gambling get a massive sensation of glee when they unexpectedly win because it doesn’t happen very often. However, the people in charge of these games know all this and they purposefully set up the mathematical odds just right in order to keep people spending cash. It has been shown that if a casino was to make the odds harder for people to win, they lose money because people no longer want to keep losing at a game. Moreover, it has also been shown that if people win too much then they do not have an incentive to continue playing; the thrill is no longer there in the game (which loses the casino more money once again.) (Bellos, 2010). So, gambler tycoons need to make sure that revelers are feeling optimistic about playing a game, however, they also need to make sure they aren’t too optimistic or they will see no risk when parting with their pennies.

Returning to the Richard’s input, we first talked about the chances of someone winning the lottery. The game of the National Lottery in the UK revolves around choosing a set of numbers in hopes that they will appear in a draw (like any other lottery, really). You pick 6 numbers between 1 and 59 (formerly, it was 1 and 49, however, they have added more numbers making the odds of someone winning even harder) and in order to reap the luxuries of the millions you need to be able to get all 6 numbers. Now, mathematically, we can calculate the probability of someone being able to accomplish such a feat, by wagering up the variables.

Probability – a calculation of how plausible a particular event may occur (Holme, 2017)

“Probability is the study of chance.” (Bellos, 2010, pg. 303)

“Mathematically speaking, lotteries are by far the worst type of legal bet.” (Bello, 2010, pg. 329). One can calculate their chances of winning the lottery by using fractions and probability. A person would need to assess all the variables (6 numbers between an integer of 1 and 59) and then they would be able to see the staggering statistic: 1 in 45 million (this was originally 1 in 14 million before the introduction of the extra numbers).

The odds are really against a person participating in the national lottery when you look at the maths behind it

You are more likely to be crushed by a meteor than win the lottery (1 in 700000 chance of that happening) and being struck by lightning is almost 4 times more likely to happen to you than being able to get the jackpot (Khan, 2016)

A bigger theme that came to fruition during this input was the societal impact of teaching probability and chance in relation to gambling. Gambling can be extremely addictive and has ruined many peoples lives, due to the everlasting hope of being able to feel that elation at winning a jackpot. Could these aspects be taught in a primary school setting? Personally, I think it is a topic that should be explored because it is a prime example of mathematics being evident in the real world, and many people probably don’t realise the true mathematical thinking that goes into a game of poker or blackjack. Furthermore, bringing awareness to the odds being against the participants engaging in gambling from the get-go means that children can establish their own perceptions on gambling before being consumed by it naively once they are old enough to gamble. Also, in a less serious note, simple card games they play themselves could be a basis of exploring probability as the kids can work out within a pack of cards what are the chances of them getting a specific card.

A clip that we watched during the input, that I think would be beneficial for children to be made aware of, demonstrated that a lottery could be rigged using mathematics. A man named Stefan Mandel and his investors were responsible for being able to rig the Virginia State Lottery in the 90s:

Mathematics was the very thing that created this cycle of gambling but it can also be the thing that breaks down the unfair games in order to allow people to beat them. The basis behind the rigging of the Virginia state lottery was quite simple: the jackpot exceeded the investment needed to buy every combination of numbers, so if someone were to buy every ticket they would win. They used a buy-every-combination approach in order to remove the chances of anyone else being able to get the winning combination.

“Lottery officials speculate that the investors may have chosen Virginia for two reasons. The state had the biggest jackpot in the country that weekend. And the seven million entries required to cover all the combinations in a 44-number lottery is just half the number needed in a 49-number lottery, like Florida’s. California has 51 numbers and New York has 54. Improving the Odds” (New York Times, 1992).

Overall, probability was not a huge subject that I remember being explored a great deal back at school. All that I remember was that one mathematics teacher told us, in the run up to our exam, that there was only going to be one question in the actual examination that dealt with probability and that it wouldn’t be worth that many marks… A negative view on the whole concept prevented our class from being able to see the full potential that probability can provide for one’s societal understanding of things like the lottery, casinos and card games. Luckily, we are exploring these avenues within this module, for I would have never have fully understood the impact probability has on our lives.

Reference:

Bello, Alex (2010) Alex’s Adventures in Numberland London: Bloomsbury

Eastaway, Rob. (2010) How Many Socks Make a Pair? Surprisingly Interesting Everyday Maths, London: JR Books

Holden, Anthony (no date) Quotation on playing cards [Quote] Available at: https://quotefancy.com/quote/1729230/Anthony-Holden-The-good-news-is-that-in-every-deck-of-fifty-two-cards-there-are-2-598-960 (Accessed 3rd of November 2017)

Holme, Richard (2017) Chance and Probability [PowerPoint Presentation] Available at: https://my.dundee.ac.uk/webapps/blackboard/execute/displayLearningUnit?course_id=_56905_1&content_id=_4941456_1 (Accessed 3rd of November 2017)

Khan, Shehab (2016) 11 things that are more likely than winning the lottery [Article] Available at: http://www.independent.co.uk/news/uk/home-news/11-things-that-are-more-likely-than-winning-the-lottery-a6798856.html (Accessed 3rd of November 2017)

New York Times (1992) Group Invests 5 Million to hedge bets in lottery [Article] Available at: http://www.nytimes.com/1992/02/25/us/group-invests-5-million-to-hedge-bets-in-lottery.html?pagewanted=all (Accessed 3rd of November 2017)

Table accessed from: https://www.lottery.co.uk/lotto/odds