Category Archives: Discovering Mathematics (ED21006)

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

99p – Maths Behind Consumerism

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During my first year placement, a key topic that I was regularly given the responsibility of planning lessons for within mathematics was budgeting. Now, within the Experiences and Outcomes documents there are various outcomes that cover this topic:

MNU 2-09a – “I can manage money, compare costs from different retailers, and determine what I can afford to buy.” (Scottish Government, 2009, pg.6)

MNU 2-09c – “I can use the terms profit and loss in buying and selling activities and can make simple calculations for this.” (Scottish Government, 2009, pg.6)

I write this blog post in the prospect of gaining a deeper reflection upon the experiences in relation to what we have been exploring during the Discovering Mathematics module.

Having an upper stages class allowed for more creative freedom in terms of setting up lessons that could be relevant to the learners within the class. More contextual and relevant aspects could be explored, with their existing knowledge of mathematics, in comparison to just establishing rules and procedures when calculating problems.

The community around the school had a large shopping centre where there were various shops that had large catalogues full of products for people to buy.

Using relevant resources, such as catalogues, allows for children to understand the connection between the ‘real world’ and the mathematical skills they learn

I used these catalogues in order to establish a lesson centred on the concept of working within a budget. I divided the class into varied ability groups, so that less confident students could be aided by those more confident in their calculations.

Their task was simple in its expectations: as a group, you have to decorate a living room whilst staying within budget. There was a list of required items they had to get and then there was space for free choice once they had got the basics (Sofas, TV, coffee table etc.) What the groups didn’t know was that I gave each group their own individual budget so that the types of furniture and the amount of furniture bought would be very different across the groups.

Once the groups had finished up with their purchases and calculation I brought them back as a whole class in order to gain some feedback on how successful they were with staying within budget. What I expected occurred: the students with the smaller budget struggled to stay in budget at first and had to adapt and change their expenditure. Also, the groups with larger budgets were able to buy more free choice products once they had worked out what money they had left over once getting the living room decorated.

“Why is it important to do calculations like this when buying things?” I asked the class.

The responses hit the nail on the head with the entire purpose of the lesson – so you know what money you actually have and so you know what you can afford. As much as the kids found it entertaining and different going catalogue shopping, it had a real underlying purpose that went beyond just reinforcing their mental math skills. The core purpose was to bring importance to skills they had learned, through the four operations, and bring a context that was familiar to them in order for them to see the relevance of learning mathematics in school. This lesson will no doubt occur for them once they reach adulthood and have to decorate their own homes.

Beyond this lesson, I also got the groups to use IT in order to explore other shopping websites to compare the prices of similar products (which taught them the importance of searching around when being restricted to a budget, as one price isn’t final) and I also wanted to delve into the marketing side of things when exploring the catalogues.

A key point made by a few of the students was that the majority of the products were not simply £15 or £50; they were £14.99 or £49.99. I knew that I couldn’t lose the opportunity to explore this topic further.

The whole consumerism psychology behind pricing of products has been thoroughly explored by these huge companies that we shop from. Psychological pricing is a phenomenon that is literally inescapable across the vast amounts of aisles within supermarkets and shopping centres. It is everywhere

ONLY 99P! SALE! ALBUMS UNDER £5! REDUCTIONS!

These bold, bright and in-your-face slogans are all there to get us to cave into buying something, to put it bluntly. These strategies are also there so that, when we buy something, we feel as though we have gained some form of saving in our spending. There are various theories and concepts of why .99 is so effectively used, however, a core reason that a price ending in .99 or .95 is chosen is because we read prices from left to right, so we associate the first number as being the overall price (Melina, 2011). Another example is that it is harder for us to calculate the total cost by the time we have amassed a large quantity of shopping in our carts by the time we arrive at the checkouts (in real life or online) and this can be another example of maths anxiety plaguing adults who fear working with numbers. We psychologically believe that £4.99 is cheaper than seeing £5 because our brains first see the 4. The ‘under £5’ slogan is one that is used regularly to heighten this idea of saving being gained, when in actual fact the product is probably £4.99 or £4.95. Factually it is under £5 but, is there really a massive saving here?

“[Consumers] have become conditioned to believe that they are getting a good deal when they buy something with a price ending in .99 even if the markdown is minimal” (Melina, 2011)

The children in my class were very aware of this aspect when we decided to explore the topic of shopping and budgeting further as a whole class. Links to buying their favourite sweets at the shop outside the school were made when exploring the fact that businesses are, economically, looking to make as much money from us as positively possible. Another important point that one of the kids brought up was that, when buying things, they mainly received back change after they had bought something.

Change is another tool utilised by businesses. When we purchase something, it is normally unlikely that we have the exact change outright, so we pay with something over the price and, in return, we receive the change in difference. Doesn’t seem complicated, does it? However, with fractional totals come more lucrative gains from vendors because studies have shown that we like receiving money back once we have spent, what was most likely a lot of money. It doesn’t make the blow of handing over cash so hard to take, continuing our spending because we aren’t going away completely empty-handed. (Bizer and Schindler, 2005)

Teaching children to be critical of pricing strategies used by big companies widens the importance of Mathematics

Overall, the various lessons that I planned on budgeting explored topics that go far beyond the realm of perceived primary school mathematics. Skills such as addition, subtraction, rounding, place value and more were utilised on top of a contextual learning space of consumerism, marketing awareness and psychological studies of how we shop! This ties in well with Ma’s theory on connectedness, which I wasn’t made aware of until studying this module.

Reflecting on placement now, in the midst of studying the Discovering Mathematics module, I can now see how my first experience with teaching mathematics was quite successful. Beforehand, I had to brush up on my mental arithmetic, explore the psychology of marketing and then construct lessons that fit towards the E’s and O’s. This shows that I was making myself aware of the ‘simple but powerful basic concepts of mathematics’ (Ma, 2010, pg. 122) in order to make my lessons more effective. This links well with Ma’s Basic Ideas in terms of the PUFM (profound understanding of fundamental mathematics) an educator must know in order to be successful in their teaching.

Progressing through the module, I am very glad that I chose it because it not only benefits my conceptualisation of mathematics for the future, but it is also reshaping my understanding of my previous experiences and sparking points of professional reflection (and reflection upon what money I’ve spent in the sales!).

Reference:

Bizer, George Y. and Schindler, Robert M. (2005) Direct evidence of ending-digit-drop-off in price information processing [Article] Available at: http://onlinelibrary.wiley.com/doi/10.1002/mar.20084/full (Accessed 25th of October 2017)

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

Melina, Remy (2011) Why Do Prices End in .99? [Article] Available at: https://www.livescience.com/33045-why-do-most-prices-end-in-99-cents-.html (Accessed 25th of October 2017)

Scottish Government (2009) Numeracy and mathematics: experiences and outcomes document [pdf] available at: https://www.education.gov.scot/Documents/numeracy-maths-eo.pdf 

Interesting Link:

http://news.bbc.co.uk/1/hi/magazine/7522426.stm Why is a 99p price tag so attractive?

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Maths Anxiety: What We Should All Fear…

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The subject of Maths is divisive, even beyond the system of education, and it has the potential to greatly impact people’s everyday life (both for good and for bad, depending on someone’s experience with it during their school years) (Bellos, 2010). It has been argued that it has the potential to separate humans into two distinctive categories; there are those who just “get Mathematics” and then there are people in society who think that it is an impossibility for them to ever understand the fundamental concepts of mathematics, so avoid maths for the rest of their lives (Foss, cited in Skemp, 1986). Today, we can understand this as a person being anxious about mathematics: Maths Anxiety.

Having a fear of anything related to mathematics has plagued society for generations and it continues to affect our young learners of today. An even scarier reality is that it even affects our educators.

Us.

It has been said that teachers that feel insecure within their knowledge of mathematics will pass on their worries to their students and they will instil negative connotations towards the subject because of the anxiety, resulting in their students not reaching their full potential (Haylock, 2014). Thus, resulting in a class-full worth of people being incapable or intolerant to working with maths (something that is essential to being successful in life i.e. being able to work with your finances). Therefore, it must be paramount that a teacher who feels jittery about mathematics seeks help for their fears. The only way to do that is through diving headfirst into the world of mathematical thinking.

I myself can relate to the fact that teachers pass on their woes to their students as I have had many teachers tell me that mathematics is really tricky, which from the get-go, put boundaries between the subject of mathematics and I. However, to contrast this, I have had some amazing math teachers in high school when I was sitting my exams and their profound understanding of the subject allowed me to fully enjoy the subject and get the grade that I needed. The best teacher I had during my higher exams worked through topics with feedback from us, as students, to gauge what needed to be revised and revisited in the run up to the exam time.

However, once I did get the grade in higher Mathematics that was it for me with the subject. At least, that’s what I thought. Until it became clear that I myself was going to be teaching the subject.

I decided to choose the discovering mathematics module as an elective because I wanted to know the behind-the-scenes of what makes a successful teacher in mathematics and I felt that it would be in my best interest to study Mathematics in order to iron out any queries before teaching the subject myself. As I saw on placement, it isn’t enough just to know how to work out a problem. You also need to investigate the complexities of incorrect answers, alternative methods and the varying opinions and abilities of the subject within the classroom.

The main text of the module, Liping Ma’s “Knowing and Teaching Elementary Mathematics” is a great example of an academic text that picks apart the realities faced by teachers on practice. Not only that but, Ma (2010), contrasts and compares the teachings of practitioners from the United States and China, as it has been seen in the likes of the Programme for International Student Assessment (PISA tests) that the Chinese excel within mathematics and the sciences in terms of academic scores, whilst American students have stumbled (Serino, 2017). The investigations and research conducted by Ma found that, although the training wasn’t as extensive or as long as the USA, teachers in China were better equipped with a breadth of knowledge within the fundamental principles of elementary mathematics (Ma, 2010).

How could this be?

Before education is even taken into consideration, one aspect that came to my mind was the cultural differences between the countries. Firstly, it is regarded as being intellectual to understand mathematics within school within the United States (the same can also be said about societal beliefs here in the UK about those who can ‘get maths’) as students are increasingly only seeing it in isolation as a single subject (Green, 2014). So, many students feel that it is normal just to be bad at mathematics, as it has become the cultural norm. It is a bigger fear to fail at the subject than to just dismiss it completely. Those same students become the workforce that hold this opinion of the subject throughout their pathways through life; impacting their children, peers, students, colleagues, partners… you name it. This continues the cycle of fear.

Worldwide tests, such as PISA, have made education more competitive, which highlights what aspects of teaching mathematics needs to be taken into consideration when assessing the success of teaching the subject.

China, however, enthuses students and teachers alike to never give up and that anyone is possible of intellectual understanding through a hard work ethic. So much so, that “The Chinese teachers think that it is very important for a teacher to know the entire field of elementary mathematics as well as the whole process of learning it.” (Ma, 2010, pg.115) which highlights the severity the teachers in China place on their subject knowledge. They know how crucial they are to a child’s everlasting opinion on anything they come across when being taught.So, understanding this societal issue, we can then see how it translates in an educational setting when Chinese students are seeing a practitioner that knows the entire textbook by memory where as American (or in our case Scottish) students are taught topic-by-topic and their experience of mathematics is, traditionally, very linear.

Returning to the issue of Maths anxiety, I believe we need to change our societal opinions on education instead of just how we can tackle mathematics in isolation. In this way, we change the worries themselves. To do so, we need to encourage a you-can-do-it attitude, not only in school, but also for everyday life. Whilst on placement, my teacher was very adamant on being open with making errors within mathematics and heralded the students to call these ‘marvelous mistakes’. This worked effectively as it allowed for open dialogue, as a class, about how an error came about when working through problems. There was no shaming of who made the error because, in the end, we are all capable of failure. It was more about what we do with the failure that was important. I believe this scenario that I experienced is a fine example of a growth mindset approach (which the school utilised as a whole-school initiative). This is another aspect that needs to be at the forefront of any teaching: coherence. Green (2014), explains that many great ideas in teaching fail purely because teachers have not been sufficiently prepared collectively to tackle any given issue.

In conclusion, having fear and anxieties about mathematics is very common and many of us suffer from it, however, we need to make it our mission to break away the years of instilled fear. To do so, we need to use the studies of scholars within our schools effectively and we also need to make sure we are open and honest about how we feel about the subject. Furthermore, we need ensure that we are consistently and constantly seeking various ways to tackle mathematical thinking through problems, which will enable our students to have a richer understanding in computing numbers and formulae.

Reference:

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

Green, Elizabeth (2014) Why do Americans Stink at Maths? [Article] Available at: https://www.nytimes.com/2014/07/27/magazine/why-do-americans-stink-at-math.html (Accessed 20th of October 2017)

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

Skemp, Richard R. (1986) The Psychology of Learning Mathematics, 2nd edn. London: Penguin Books

Serino, Louis (2017) What International Test Scores Reveal about American Education [Blog] Available at: https://www.brookings.edu/blog/brown-center-chalkboard/2017/04/07/what-international-test-scores-reveal-about-american-education (Accessed: 20th of October 2017)

Image sourced from – Flikr

Binary, Counting Horses, Indigenous Tribes… Oh my!

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Richard’s last two inputs about number systems and place value have left me perplexed to say the very least.

Binary, a counting horse and indigenous tribes…

All these aspects were covered in two inputs and they definitely broke down my structured beliefs on what mathematics really is. A key point that I took away from the lessons was to think beyond the confinements of what we know about the subject of mathematics and our 10-based numeral system.

It really is Discovering Mathematics all over again in a much deeper-rooted manner.

Rather than getting bogged down in the complexities of the possibilities of differing number systems and giving up, I embarked on reading Alex’s Adventures in Numberland in order to find an everyday answer:

“Without a sensible base, numbers are unmanageable” (Bellos, 2010, pg. 44).

Base systems of five, ten and twenty have been the most commonly used through the various cultures of mankind (Bellos, 2010) and it’s a pretty straightforward answer of why:

What is the most common tool a child (or anyone for that matter) would use in order to count? They use their fingers! In Early Years, “fingers are used in a range of ways and with varying levels of sophistication.” (Wright et al. 2006, pg. 13) Well, this instinctive notion towards mathematics has a rich meaning in terms of how we represent our numbers because, in reality, that is all a numeral system is: a way in which we express numbers and quantities of those numbers.

However, Richard introduced us to different variations on number systems that go beyond our commonly known systems. Not only that, but we were also shown the other number systems that were influenced by the culture that they were used within.

Number systems, in reality, are ways in which we give identity to a quantity. 1,2,3,4,5 are all just the symbols we have given to a quantity. Delving deeper into this concept of a numeral system, we need to first realise, how did we create such a vast amount of numbers?

Lets take an indigenous tribe like the Arara tribe in the Amazon for example; they only have base 2 number system, where they only have 2 words for 1 and 2, and anything after that is a combination of the two (anane =1, adake = 2, adake anan = 3, adake adake = 4 etc.) (Bellos, 2010).

Why? They have no real use for numbers beyond that. Their lives revolve around survival. A reserved community in the amazon are never going to need thousands or even hundreds of something, so they just don’t have it.

Farmers have also been shown to have their own number system where Base 20 is used. Farmers would count up (yan, tan, tethera) until they got up to 20 and then they would either pick up a stone or make a mark on the ground in order to indicate that he had got up to one set of 20 sheep and then he would begin again.

Yan. Tan. Tethera.

Could you imagine trying to quantify, say, a population of a whole country using these formats of number systems? The representations would be very time consuming! Once again, the tribes and farmers would not have a population that could equal the populations we have across the modern nations.

The fact that we have so many numbers is down to the fact that we have advanced to the point that we need a huge amount of numbers. We are beyond just surviving as a species, like the indigenous tribes or the independent farmers of the past. Similar to my post about the advancement in agricultural, we’ve adapted in order to advance and, in doing so, adopted a number system that allows us to easily distinguish between place value when putting a quantity on something (particularly large quantities). As we have multiplied, so have our quantities of population, food, cars, houses and so many more factors. An indigenous tribe does not need a number system that goes up to a million because that number has no right to exist. When are they ever going to need a million things of anything?

Here is an interesting video by TED about the history of our numeral systems:

Binary, another spanner thrown into the math-works, was also something difficult to understand at first, due to it using the original place holder symbols of 1 and 0… and that’s it. Similar to the Arara’s, binary only uses two symbols to define various quantities. I vaguely remember aspects of binary being used way back in high school IT lessons; however, I didn’t really know the whole purpose behind it. Computers do not work the same way our brains do. Binary is used because a computer can only work through programming with a state of on or off. This is where the 2-based number system of binary comes into practice well:

The circuits in a computer’s processor consist of billions and billions of transistors. A transistor is basically a tiny switch that is initiated by signals of electricity passed through the computer. The digits 1 and 0 used in binary can reflect the on and off states of a transistor (BBC, 2017). So, computer-literate people can program commands into a computer using binary and the computer will be able to translate these codes (much quicker than the human brain could) into processes.

James May explains binary numbers within this video:

Now, if indigenous tribes, binary and abstract number systems weren’t enough to comprehend across two inputs, then this question that we were faced with will surely perplex you:

Can animals count?

Many opinions and theories circulated the room but the main thinking was… not really. An animal can maybe understand a form of quantity but they probably don’t know why they understand this.

An interesting video Richard showed us was about the enigmatic counting horse called Clever Hans. In the 1900s in Germany, Hans was taken around the country to demonstrate to people his great ability to work out arithmetic that his owner asked him to calculate… Could this possibly be true?!

Unfortunately, it was too good to be true. What Hans was actually doing was reacting to the positive praise through body language of his owner when given a sum. He would learn from cues when to facilitate an answer through tapping his hoof. Psychologist Oskar Pfungst investigated this and even discovered that the owner of the horse didn’t even know he was giving these positive cues, which revealed another theory years later known as observer-expectancy effect. This means that Han’s owner subconsciously gave the answer that he wanted through visual hints like a nod of the head.

Animal cognition is not the same as human cognition. Milius (2016) wrote an article about the topic of animals and mathematics and stated that “some nonhuman animals — a lot of them, actually — manage almost-math without a need for true numbers” and she explores how the argument has varying perspectives from psychologists and scientists alike. One theory is that animals just so happened to gain aspects of mathematical thinking through convergent evolution from similar ancestors as us. This evolution is similar to how bats and birds can fly however, are from completely different families and their wings derived in different pathways of evolution (Milius, 2016). It is also similar to sharks and dolphins both having to gain the best possible traits and abilities to survive in the ocean, yet neither are related in any format. Animals have gained the ability to understand some form of quantity in order to judge if there is 1 or many predators in front of them, however, they don’t have a numeral system to define this understanding.

In reality, much like the tribe, animals have no real use in knowing numbers because they do not think conceptually, like we do as a modern society.

Returning to the concept of place value within numeral systems, teachers need to be able to comprehend what the underlying meaning behind what place value really is. As Ma (2010) found in her studies, the students that excelled the most in mathematics in terms of comprehending number systems were the ones that were taught the appropriate measures when dealing with higher digit numbers when it comes to differing place value with subtraction and addition, for example.

Therefore, as educationalists, we need to know what the best methods for students to tackle number systems are. The answer? Preference is really down to the student. However, we need to be there to facilitate the various learning styles, challenges and boundaries that come our way in terms of learning mathematics – in a positive manner. This correlates well with Ma’s basis of multiple perspectives: teachers should be “…able to provide mathematical explanations of these various facets and approaches. In this way, teachers can lead their students to a flexible understanding of the discipline.” (Ma, 2010, pg.122). Giving children multiple avenues to explore problem solving, in terms of arithmetic, will only benefit their independent evaluation in terms of dealing with mathematical problems. It will benefit them far greater than giving them a formula.

In conclusion, Mathematics has various avenues when it comes the representing quantities and exploring huge amounts of quantities. Knowing the basics of 1,2,3 as teachers will only get us so far. It will also hinder our children greatly… Even discussing the great horse Clever Hans would be an interesting lesson to explore how different mathematics is between them and their pet peers. Being open to mathematics as a vast subject can only bring about great things within the classroom.

Reference:

BBC (2017) Bitesize: Binary [Website] Available at: https://www.bbc.co.uk/education/guides/z26rcdm/revision (Accessed 19th of October 2017)

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

Milius, Susan (2016) Animals can do ‘almost maths’ [Article] Available at: https://www.sciencenewsforstudents.org/article/animals-can-do-almost-math (Accessed 17th of October 2017)

Wright Martland Stafford Stranger (2006) Teaching Numbers: Advancing children’s skills and strategies 2nd edn. London: Sage Publishing Ltd.

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

Art in Maths and Maths in Art

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Art and artistic expression both have connotations of creativity, freedom and exploration for those that are deemed artistically imaginative to delve deep into their own vivid minds. One must be capable of thinking outside of the box of convention when viewing artwork, for example, to be appreciative of the emotions or message an artist is trying to convey.

Mathematics, on the other hand, follows formulae with the intentions of finding answers to problems… Doesn’t seem very creative, does it? Nor is it very thought provoking in terms of gaining a lasting emotion that viewing a controversial Banksy exhibit could produce, for example… I am digressing but one could argue that they’re more likely to be overwhelmed emotionally by maths than a painting…

Mathematics and Art don’t seem to go together quite so well do they?

A person that is seen as a creative being is less likely to be a ‘math person’ and be more likely to be a ‘free spirit’ who is in touch with the world. They do not feel the need to conform, right? Many of the great artists, such as El Greco, had this exact notion when being challenged by friends and family who thought of them as failures for taking up art as a profession. Free spirits break away from structure, order and routine that are upheld and followed rigorously by so many educators and scholars in the subject of Mathematics. Why?

Many people will agree that their math lessons at school did not involve any creative thinking, which, through recurring practices, disconnects the relationship between Art and Mathematics. This creates this societal view that a person can be naturally gifted in the arts or mathematics. Why not both?

Art and expression are the real world. Humans use art as a method to translate emotions (Mcniff, 2006). Creativity is not limited to artists, however. Across various industries and institutions people need to be innovative enough to conjure up ideas to tackle problems they are faced with in their profession and their everyday lives. So, mathematicians can be free spirits just like artists can be math people. A mathematician can be an artist just as an artist can do mathematics.

Any form of categorisation for determining the source of someone’s mathematical ability and their relationship with numbers is really a form of escapism from the real problem that people do not have profound enough knowledge of mathematics within the real world (Ma, 2010).

Da Vinci, Vitruvian Man (1490)

Leonardo da Vinci’s famous Vitruvian Man is a prime example of the everlasting marriage that Mathematics and Art have with each other. The piece depicts the dimensions of man in correlation to shape and symmetry. Both mathematically and artistically, the artwork shows that the ‘perfect human body’ is symmetrical in its measurements and dimensions, and da Vinci (and many others) argued that this was not by coincidence.

The Roman Architect and mathematician Vitruvius, who explored perfect proportions in building design and its connection with the human body, heavily inspired the Vitruvian man (hint is in the name, really) because his work led him to believe that we were the source of dimensions and that some higher power had granted us these tools for measurement. This was based upon the understanding that we were made up of symmetry.Two eyes that are near identical, two hand that have five fingers each and two feet that have five twos each being just a few examples.

Vitruvius used symmetry of the human form to aid his writing in various volumes of literature about architecture in Rome.

Another beauty of mathematics to behold is the golden ratio. The golden ratio was coined by the father of geometry, Euclid, and it is a number that is derived from taking a line and separating it into two in a manner that the ratio of the shortest segment to the longest is the same as the ratio of the longest to the original line (Bello, 2010).

Golden Ratio Diagram

You end up with a ratio of 1.6180339887, which cannot be represented as a whole fraction, deeming it an irrational number. The Greeks called this phi.

“The Greeks were fascinated by phi. They discovered it in the five pointed star, or pentagram, which was a revered symbol of the Pythagorean Brotherhood” (Bellos, 2010, pg. 284). Shape, a core visual element within art, is vital to mathematics, as geometry is a part of it.

Phi within a pentagram

The golden ratio can even be extended to Fibonacci sequences (1,1, 2, 3, 5, 8…) as adding two previous terms to get the next equates to the golden ratio of 1.618033… as Bello (2010) states:

“adding two consecutive terms in a sequence to make the next one is so powerful that whatever two numbers you start with, the ratio of consecutive terms always converge to phi.” (pg. 291)

The Golden Ratio seen within the make up of a sunflower

This emphasises that phi is so crucial to the natural world and it’s symmetrical properties and that it not just a random number chosen by Euclid. So much so, that we, as a species, have utilised it when exploring painting, architecture and nature. These are all areas stemming back to creativity and art and can be seen being explored by great mathematicians and artists alike. A crucial part of Fibonacci sequence theory is that it is periodic, which means that every new term can only be created by the value of the previous terms. This stems well with a theory that links Fibonacci sequences and phi to nature because plant life forms expand through recurrence and can be followed through a being’s lineage (Atanassov, 2002).

Da Vinci used the golden ratio in many of his artworks and numerous historians argue that this is why we find them so aesthetically pleasing. It’s natural, based on the concept of phi.

What I found really important about this discovery was the connectedness of Mathematics through the history of art, and it’s prominent role on expression from the greats like da Vinci. This brought me back to Ma’s belief that the core aspects of mathematics coexist and are forever dependent on one another:

“A teacher with profound understanding of fundamental mathematics has a general intention to make connections among mathematical concepts and procedures…” (Ma, 2010, pg. 122).

In order for students to really grasp what they are learning, they need to see how their learning is important in a wider context. They need to be made aware of the journey that they are venturing on in their academia. Gone are the days of learning formulas for the sake of passing an exam.

I have found this blog post to be very therapeutic because I have had to learn so much in order to formulate the complexities of mathematics within art in my own words. I feel that this has benefitted my overall understanding of the link between mathematics and art. On practice, Fibonacci sequences could be interlinked with a topic on plants and could allow children to see the cross-curricular benefits mathematics has, not only in school but also in the real world. Not only that, a class could create their own symmetrical art using shapes that follow the golden ratio… The possibilities are endless!

Overall, art and mathematics have long been connected throughout time. It has only been through the dated teaching methods of rote learning and regurgitation of formulae that has hindered the broader prospects mathematics has on the wider world, in particular with creativity and artistic freedom.

Reference:

Atanassov, Krassimir T. (2002) New Visual Perspectives on Fibonacci Numbers New Jersey: River Edge

Bello, 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.

McNiff, Shaun (2006) Art-based research 5th edn. Jessica Kingsley Publishers: London

Time – The Underlying Mathematics and Science

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As a species, we form our daily lives around clocks, calendars and alarms. It would be extremely difficult for us to cope if we didn’t know what time it was or what day it might be because the concept of time is at the core of our society and our civilisation. From what I gathered before investigating into the concept further, it is the manmade vehicle that traverses us through our entire existence on our planet and beyond. I always believed that time was something that we just made up ourselves… My discoveries proved me wrong.

Firstly, let’s take an example: what does a normal morning begin with for many?

An alarm blares at 8:00am to sound that it is time for us to get up and start the day. However, the snooze button delays the awakening to 8:12am (12 more minutes still leaves us a sufficient amount of time). Washed, dressed and ready; our phone reads 8:54am. We’ve wasted too much time because we need to be at our destination by 9:00am and we know 6 minutes is not long enough for a journey that takes 10 minutes. We’re going to be late. We need to be more organised next time.

You may or may not know it but this little scenario – that may be all too familiar – is oozing with mathematics.

It may seem like common sense to the average person, but planning towards time is all linked with having skill and knowledge within the fundamental principles of mathematics: estimation, planning, problem solving, sequencing events, organisation and so much more. They’re all how
we go about our days. Without being competent in these various fundamental skills, we’d be at a huge loss. Ma (2010) categorised 4 aspects of mathematics that teachers need to tap into in order for their students to have a rich understanding in their learning in maths during her investigations in teaching in China and the United States. They are: interconnectedness, multiple perspectives, basic ideas and longitudinal coherence.

A day would not be a day without a reference to what the digits on a digital clock read or where the hands were pointing on the analogue equivalent. But what really is a ‘day’? How have we measured 24 hours as a full day? I asked this question to the Internet and even myself multiple times. This led me to the discovery of the Circadian Rhythm:

The number 24 was not chosen out of sheer randomness, it is a crucial number that correlates to various living beings on the planet.

(Latin) Circa – about

(Latin) Diem – day

The phrase Circadian rhythm, broken down, literally means about-a-day rhythm.

In short, the circadian rhythm, a phrase coined by scientist Franz Halberg (2003), is an organisms’ body clock that indicates what they need to be doing at any given time across a 24-hour cycle. Sleeping, waking up and eating are examples of where the circadian rhythm is at work. It is heavily influenced by environmental factors. The sun and the moon indicate to our bodies when to rise and when to sleep (phone and computer screens being great deceivers to our body clock’s perception of night and day). Similarly, plants’ leaves adapt to the environment by moving in order to attract pollinators depending on the time of day.

Maths is natural to us.

Plants have a body clock too

Discovering the underlying biology to how we’ve conjured up time has led me to really appreciate why we need the manmade structure of clocks to keep us on track through our natural daily lives. This has shown me the real importance of mathematics having a relationship with the earth and it’s creatures. Its context is so core to every little thing we do, that we don’t even realise the underlying principles behind it. The mathematical ideas we are using to problem solve, estimate, decide and sequence events are intertwined with our bodies.

The clocks, calendars, phones and timers are all mathematical tools made from our innate ability and urge to define time and to quantify our instinctive movements. Furthermore, this further exemplifies Liping Ma’s theory of [inter]connectedness, as the various tools and formulae of mathematics are linked with, not only with each other but also with the real world (Ma, 2010). Tapping into this, as professionals, will be the difference between a student who can answer questions and a student who fully comprehends the work that they are doing. Knowledge in time is a topic that is heavily linked with the real world and children need to be competent with working with numbers. “Understanding relationships between numbers, and progressively developing methods of computation, has become the focus for learning, replacing the traditional ‘four rules of arithmetic’” (Skemp, 1986, Pg. 7).

Relating this further towards education, children, even from a very early age, have a great understanding of the concept of time. Toddlers “become familiar with the routine of their day” (Early Years, no date, pg. 2) and know, logically, what they’re doing and when they’re doing it. They may not know how to read what time it is when they have a snack or go for a nap, but they know instinctively when they are actually going through with consistent tasks (their circadian rhythm are already keeping them on track from the get-go). This, although it may seem minimal, is a child’s early access to problem solving mathematics.

Overall, my investigations into the concept of time have only scratched the surface of what is to come within the Discovering Mathematics module, and in my professional development as a student teacher.

Circadian Rhythm

Looking ahead, I know now why we must teach time to children, as it is part of their being. Furthermore, having the underlying knowledge of the basic ideas, coined by Ma (2010), will improve how deep a teacher’s teaching roots can grow in a child’s ability to truly grasp mathematics and go beyond just the academic mathematics that we throw onto a child.

I finish this post with a pop song that explores our fascination with what is possible in 24 hours:

“I wish these 24 hours

would never end,

oh in these 24 hours,

 wish the clock had no hands”

(Ferreira, 2013)

Reference:

Early Years (no date) Maths through Play [brochure] Available at: http://www.early-years.org/parents/docs/maths-through-play.pdf (accessed 22nd of September 2017)

Ferreira, Sky (2013) 24 Hours In: Night Time, My Time [CD] 0602537712793 Capitol Records.

Halberg, Franz. (2003) Journal of Circadian Rhythm: Transdisciplinary unifying implications of circadian findings in the 1950s [article] Available at: https://www.jcircadianrhythms.com/articles/10.1186/1740-3391-1-2/ (Accessed 20th of September 2017)

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

Skemp, Richard R. (1986) The Psychology of Learning Mathematics, Second Edition, London: Penguin Books.

Useful Link:

https://sleepfoundation.org/sleep-topics/what-circadian-rhythm