Category Archives: 3.4 Prof. Reflection & Commitment

LfL – Section 1

This post contains the information needed in section 1 of the portfolio.

This section of the Learning from Life portfolio should be completed prior to going on placement, which will allow a strong basis for continual reflection points during the placement.

Audit of Skills

Rating of skills (1=Not very developed; 3= very developed)

Skills and Abilities	1	2	3
Flexibility		*
Confidence		*
Self Discipline			*
Working Under Pressure			*
Setting Professional Goals			*
Taking Risks		*
Sharing Opinions Confidently	*
Teamwork		*
Acting as a Leader			*

Personal Attributes

Recognition	Reflection	Action
Skills already developed	How will I use these	How do I know (evidence)**
Teamwork	Working within ISS, I will be in contact with various staff members and will have to work alongside them in a collaborative manner.	Forming strong bonds with the staff at the International school of Stuttgart. Also, being approachable to the children that will be from vast spectrums of backgrounds due to it being an international school will ensure that I can work collaboratively with the children in a successful manner.
Sharing Opinions Confidently	In order to gain the most from the placement, I will need to be able to contribute my own opinion, which will need to be backed up with factual understanding in order for my opinions to be taken into real consideration.	When my opinion has been put forward and been fully considered by my peers.
Working Under Pressure	Not only will I be working in a different country that speaks a different main language from my own, I will also be working within a school that follows a different curriculum. Therefore, I will need to push myself beyond my limits in order to surpass the various pressures so that I can be successful.	I will have gained self-resilience alongside working under pressure and will be able to have sustained the expectations and workloads of the internship within the school.
Setting Professional Goals	Making goals prior to the placement and whilst working within the school in order to succeed in my VIVA at the end of the placement.	My continuing assessment of on going goals will ensure that I will have a successful VIVA.
Confidence	Being confident to express thoughts and opinions, which will be needed in both English and German	I will increase my confidence in both English and German.
Flexibility	Being open to change and being adaptable to the varying factors that affect the day-to-day workings of a school	When I can think on my feet when adjustment is needed to be made to practice whilst in the International school

Knowledge

I will be basing much of my knowledge on my first year placement and my previous experiences within working in primary schools prior to university, however, due to this being an international school in a different country, I will need to look at the placement through a different lens. Furthermore, I have learned the language of German up to an advanced level (advanced higher in school and an advanced level as an elective last year at the university) however, I will be experiencing the language in its natural environment. Also, due to the school being part of the International Baccalaureate, I will need to increase my knowledge in the differences in curriculum between that of the Scottish Curriculum for Excellence and that of ISS. Particularly as the International Baccalaureate strives for creating “internationally minded people” (International Baccalaureate, 2013, pg. 5)

Personal Attributes

I believe that I am a person that strives in challenging situations and I like to be pushed beyond my limits academically and within practice, and I believe that ISS is the perfect place for me to be able to, not only increase my professional development as a teacher, but also see beyond the teaching experience that I gained in Scotland. I want to also use my team working skills to form cohesive bonds with the staff and students at the school, as this will ensure that I will get the best results for my placement. Approachability, resilience and creativity are also attributes I hope to hone.

Professional Values and Personal Commitment (SPR 1) – Pre-Placement Reflections (General Teaching Council for Scotland, 2012)

1.1 Social Justice

I will need to embrace the different culture of the surrounding area of the school (Stuttgart, Germany) and take into consideration the various backgrounds of both the staff and the children that I will be working alongside in order to show a real dedication to aiming for social justice, particularly as the school will have people from across the globe under one cohort and student base.

1.2 Integrity

Criticality must be at the forefront of my practice, as this opportunity will allow me to see a completely different curriculum, which will allow for a real critical reflection upon my own educational philosophy.

1.3 Trust and Respect

Acting in a professional manner to be able to create an inclusive and accepting aura will ensure that both students and staff will feel respected and well encompassed from the get-go of my arrival. I want to be able to be seen in a similar fashion as the pre-existing staff members at ISS in that, I will be there to support all students and staff members.

1.4 Professional Commitment

Lifelong learning coinciding with continual collaborative practice will mean that I need to reach out as much as possible when opportunities arise for my services to be put to the test, for example, extra-curricular activities, staff meetings and CPD events, which will probably differ in comparison to those in Scotland, will be very beneficial for me so I should put myself forward for them when possible.

Identification of Skills and Knowledge

I have a strong understanding in terms of the environment of Scottish education due to placement and the multiple workshops during the 2 years of being in university, however, have limited knowledge in terms of the differing curriculum and philosophical approach towards education as a whole, thus making it an area that I will need to work on. This placement will play well into to the skills that I was able to develop during my first year placement in that I will be returning to the environment of a primary school. During the placement, i will be staying with a Germany family, which will require my skills in the German language being tested and explored within its naturally spoken environment. I will need to use my people skills in order to overcome the language barrier that will no doubt be a challenge at first, as it has been some time since I have utilised the language. Furthermore, I feel that much of my education in the language has been centred around the written format of language, which will be trickier to formulate into speech and to also be able to engage with spoken German that is being said to me, at first.

Reflection of Experience to Date

UK-German Connections Logo

Reflecting upon my experiences as a whole, I can utilise my pre-university experiences that I gained as a UK-German Youth Ambassador for the government-funded organisation that aimed to bring young people together to promote the learning of languages and my current university experiences to show my passion for both language and education. Dana and Yendol-Hoppey (2009) find that teachers need to be flexible to any given context they are put into, and that a welcoming attitude will serve any teacher well, thus meaning that I should be open to change, especially within a situation that is very abstract from my usual experiences.

What I hope to Achieve

I hope to improve both my understanding of the language of German and the culture of Germany, which will be best achieved through participation with my host family and any events at the school (Interlinks with Social Justice – SPR 1 as I will need to consider the local and global values that surround the school and my host family).
Gain a deeper understanding of the International Baccalaureate system, particularly the progression of the Primary Years Programme – I will be coming with a good knowledge in the Scottish education system, however, I feel it is necessary for me to learn as much as possible about the IB system, as it will give me more areas for reflection in terms of my educational philosophy when being faced with an entirely knew format to teach towards (Interlinks with Professional Commitment – SPR 1 as I need to be dedicated to continually learning new ways to hone my professional practice) – I also feel it would be beneficial to make contrasts and links between the two where possible.
Expand my professional knowledge in terms of strategies, teaching methods and approaches towards teaching through both observation and teaching of my own. I would also like to see more than one class levels to see a full scope in a child’s education within the International School of Stuttgart (and, in turn, understand the differing levels in education as a whole some more).

Reference:

Dana, N.F. and Yendol-Hoppey, D. (2009) The Reflective Educator’s guide to Classroom Research: learning to teach and teaching to learn through practitioner inquiry, California: Corwin Press.

General Teaching Council for Scotland (2012) The Standards for Registration: mandatory requirements for Registration with the General Teaching Council for Scotland [pdf] Available at: http://www.gtcs.org.uk/web/FILES/the-standards/standards-for-registration-1212.pdf (Accessed: 20 February 2018).

International Baccalaureate (2013) What is an IB Education? [pdf] Available at: http://www.ibo.org/globalassets/digital-tookit/brochures/what-is-an-ib-education-en.pdf (Accessed: 27 February 2018).

IB logo image sourced from wikimedia and UK-German Connection logo sourced from: http://www.ukgermanconnection.org/home

Learning from Life – A Month to Go…

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1 Month.

4 weeks.

28 days.

Time is passing quicker with each passing day. One specific date is highlighted in my calendar as having great significance and it is coming closer.

12th of March 2018.

Today, specifically, marks a month until I will be working in a school in Stuttgart, Germany.

Before this, however, we still have assignments that need to be completed for our semester 2 modules. Amongst the usual productive chaos of university we’ve had to organise our own professional placement in second year as part of a module called Learning from Life.

I write this post to reflect upon the process of sourcing my placement and to outline how I have developed even before going on the placement in terms of bringing to fruition a real finalised version of a planned 8-week placement in Deutschland! This is to coincide with section 1 of the Learning from Life portfolio, as we must record the progression of our planning and development of the placement. The rest of the content for section 1 can be found under the learning from life tag: https://blogs.glowscotland.org.uk/glowblogs/ajmeportfolio/category/lfl/.

From the offset I knew that I wanted to go to Germany when we were briefed about the possible prospects of students going abroad for their Learning from Life placements. This is due to a long-serving passion I have had for both the country and its language (along with the rich culture that intertwines the two and is further solidified by the diversity amongst the population) that stemmed from my work as a UK-German Youth Ambassador. Also, studying the language at Advanced Higher and as an elective at the university pushed me to new limits in terms of learning another language. However, I always knew that I could never truly comprehend the entire language and the culture of the country unless I was exposed to it within its natural environment.

Learning from Life’s core purpose within the course is for us, as future practitioners, to gain new insight and skills within an avenue out with the environment of a Scottish primary school. Doing so will broaden our experiences in life so that we can then utilise our new-found knowledge and skills when we return to a familiar classroom setting. I knew that I wanted to improve my German and wanted to go well beyond my limits in terms of my comfort zone.

Thus, Germany was chosen.

I first contacted previous contacts I had in Germany, however, they were unable to source the type of placement I was looking for. Luckily, the main lecturer and head of the placement module had contacts with an international school in Stuttgart and was able to organise the placement. See my proposal, cover letter and more for further details surrounding the planning behind my Learning from Life placement.

Emails… Skype Interview… Paperwork… Applications… Accommodations… Before I knew it, the process was heading towards completion in terms of the planning that was necessary to secure my place at the school. A lot of hard work has went into being able to work within ISS.

Time itself is a complex thing. It feels as though it was yesterday I was uttering the words, “I hope to go to Germany next year for my placement.”

Now it’s almost here…

I wil be using my ePortfolio as my Learning from Life as my folder, as it will be easily accessible to be able to type up reflections of my work when I arrive.

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/

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 5^th 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 20^th 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

Reflecting on Semester One

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Being student teachers, we must constantly reflect upon the knowledge and skills we have gained in our everlasting learning process in becoming qualified teachers. As part of a TDT (tutor directed task) we are asked to reflect upon something that impacted us from our learning in semester 1 between the Values and Working Together modules. In particular, we must link this to the Standards established by the GTCS (General Teaching Council for Scotland). 3.4.2 tells us, as trainee teachers, we constantly need to “engage in reflective practice to develop and advance career-long professional learning and expertise” (GTCS, 2012, pg.12). This means that we can progress as life-long learners and gain further understanding through professional reflection.

A key aspect from last semester that has really stuck with me was Jill Shimi’s inputs for Values. As Jill was a primary school teacher herself, I could relate to her inputs on a personal level, as she brought her own experiences as a teacher and linked them with the social justice topics we were investigating.

One of her stories, which tied into the problem of social class structures within society and the rising awareness of the Getting It Right For Every Child approach, really impacted me.

When she was a teacher, she had a child in her class that would misbehave and lash out in an emotional way. However, they were not always an issue within her class and she knew that something must have happened in their life that had made them disconnect from their studies.

Jill decided that she needed to speak to the child on a one-to-one basis and discovered that something traumatic had happened at home. The child’s parent had been mistreating them and they were from an area that was deemed as being deprived. These two aspects put Jill’s student at a great disadvantage in life at such a young age and she knew that they would have a lot of problems that other more fortunate children would be less likely to have, which emphasised the point of the attainment gap hindering children due to their background.

“You just don’t know what issues each child faces once they go home. You really just don’t know.”

Jill’s words really resonated with me because it really hammers home that the school environment is never the same and it needs to adapt and change towards the needs of the children, which also vary from day-to-day.

What I loved about Jill’s ‘solution’ to the issue of the student being disruptive in the class was to have a genuine talk with the child. The GIRFEC approach did not exist when this case occurred and Jill’s hands were tied on how she could aid the child other than being open. She shared her own personal struggles with the student and she connected with them beyond just her duty of being an educator for them. This resulted in the behaviour improving.

Fortunately, there was a happy ending to the story as Jill saw the child a few years later doing well for themselves, going against society’s expectation of them.

Underpinning this personal story with reflection theory, Jill’s situation is a great example of a practitioner using Schön’s reflection-in-action concept as she had to use her own judgement, as a professional, in order to formulate a solution as the practice was unfolding in front of her eyes (Schön, 1987). She did not have any prior knowledge of the student facing these issues and she didn’t have any hindsight to work with. I, as a professional, will be thrown into similar situations where I will have to use my own judgement to tackle a problem within the classroom.

“The swampy lowlands, where situations are confusing messes incapable of technical solution and usually involve problems of greatest human concern” (Schön 1983, pg 42).

Schön explains, that real human problems cannot be fixed by legislation alone. He described professionals as being people in the ‘swampy lowlands’ meaning they are the people who are at the forefront of the problems faced in society.

I really commend Jill for her actions as a teacher and I am really glad she shared this story in the input because it allowed me to really delve into the Values and it emphasised their importance to me. She was a teacher who saw, firsthand, the injustices within society and that she had to find ways to tackle them.

Reference:

GTCS (2012) The General Teaching Council for Scotland – The Standards for Registration. Available at: http://www.gtcs.org.uk/web/Files/the-standards/standards-for-registration-1212.pdf (Accessed 20 January 2017)

Schön, D.A. (1983). The reflective practioner: How professionals think in action. New York: Basic Books.

Schön, D. A. (1987). Educating the reflective practitioner: Toward a new design for teaching and learning in the professions. San Francisco: Jossey-Bass.

2017 – The Grand Scheme of Mankind in Agriculture

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Coming back to university after the Christmas break, I’ve made a few New Years Resolutions and two of them are really important to me: one is to cut back on the biscuits and the other is to keep up with the blog. Healthy body equates to a healthy mind and as they say, ‘new year, new me.’ Reflecting on the previous year, I think we can all agree that 2016 was nothing short of a phenomenon – both positively and negatively for mankind, as we know it.

However, in light of the many political controversies and celebrity deaths, I want to explore more on how we have survived as a species on this planet and how we have positioned ourselves as the powerhouse organisms of the Earth. Going into teaching, you’ve got to be open to constantly learning new aspects about the world in order to progress as active learners and professionals. We need to go beyond just taking things on face value and ensuring that we enthuse others to question what we are told in the media.

Christmas was full of indulgence for the majority of our population, myself included. The rush of shoppers to get presents, food, decorations, wrapping paper, refreshments and so on continues every year. It’s the tradition for many. One question that stuck in my mind, whilst being hurried through the aisles in the shops for sweets and shortbread, was how does it all come to be? We are so dependent on large supermarkets to provide us with fresh fruit, vegetables, meat and confectionaries everyday. Every. Single. Day. Christmas day is only one that has heightened importance in our consumerist eyes. 2016 was no different and 2017 will be just the same when it nears December 25^th.

We never really take a stand back to delve deeper on how these giant, industrial food chains exist and I think we need to question if they are sustainable for mankind. We are almost blind to the work that must go into getting fresh fruit and veg to the shelves.

An interesting documentary produced by the BBC saw Dallas Campbell explore some of man’s greatest creations in our modern world. “Supersized Earth” is the inspiration for this very post as it answered many of my questions about our understanding of agriculture and how science is being used to meet the demand of the ever-expanding populations across the nations.

Campbell travels the globe to find examples and sources of food, water and energy. They were nothing short of extraordinary: Cattle farms in Brazil filled with genetically enhanced ‘super cows’, the world’s largest 175-turbine offshore wind farm and the famous American landmark of Hoover Dam were just a few of the colossal constructions that we have set up as a species in order to survive and fight the complications of nature. Our expanding knowledge of technology means we can live in even the harshest of conditions. Like the rest of the animal kingdom, we are evolving and adapting.

One aspect that was really astonishing was the greenhouse farm of Costa del Sol. When we think of greenhouses, we imagine something along the lines of small, glass sheds hosting laborers who have great green fingers producing carrots, tomatoes or potatoes in their own garden or allotments. However, we couldn’t provide the 64 million people of the UK with enough crops from the simple gardens of suburban horticulturists.

Costa del Sol takes it to the extreme:

“A shimmering sea of polythene has consumed the small coastal plain of Campo de Dalías, some 30 km southwest of the city of Almería in southern Spain.” (Geography Field Work, 2016).

These polythene constructions allows for tomatoes to be grown beyond the constraints of Mother Nature herself. The tomatoes aren’t even grown in soil as it slows their development, which hinders the tight profit margins. It’s even been nicknamed ‘Costa del Polythene’ because of the huge concentration plastic_sea_almeria_spain
of greenhouses in one area. Around a ¼ of all the tomatoes produced end up shipped into the UK all year round. Yet, Costa del Sol is more famous as a holiday destination by many, proving our naivety of where our food really comes from. Tourists on the Spanish beaches have no idea of the real connection they have with the small region.

Farming was once filled with the challenges we had to accept with nature, however, we are now tackling it in order to compensate with the high demand of man’s need to eat. Growing population means more mouths to feed so more crops have to be produced. More crops mean more profits to be made for big business like the supermarkets that sell the crops. The cycle of business and agriculture is placed on a grand scale.

It would be great to try and incorporate these agriculture feats within classes in order for kids to learn about where their food comes from. We could show them how impactful the manmade constructions are to our planet and their lives. So much is going on with mankind and children within the classroom have the potential to have an impact on the agricultural industries across the world. Furthermore, the agricultural sector crosses over so many different subjects within the curriculum. Science, social studies, technologies and even mathematics are some of the subjects where the farming concept could be established in a classroom environment.

So, I’ve been able to come into 2017 a little bit more enlightened, a little bit healthier and, with a continued enthusiasm to learn.

Reference:

Geography Field Work (2016) Costa del Polythene: a sea of plastic Available at: http://geographyfieldwork.com/CostadelPolythene.htm (accessed 12th of January 2017)

BBC (2012) Supersized Earth Available at: http://www.bbc.co.uk/iplayer/episode/b01p9f4n/supersized-earth-3-food-fire-and-water (accessed 12th of January 2017)

Identifying Skills and Abilities – Activity 1

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Being a teacher involves more than just learning a curriculum.

Standing up day in and day out teaching a course to a class is no good for the students nor is it any good for the teacher themselves. It is an occupation that requires self-evaluation on a constant basis. Reflecting on your own skills and abilities allows for you to not only understand where you are as a professional, but also where the people you interact with are. Knowing your strengths and your weaknesses is the first step of being able to go further as a person both personally and professionally.

Our first activity on the online unit was to identify skills and abilities and pin point how confident we are in employing them. We also had to give a rating (1 being not very developed and 3 being very well developed) on how strongly we feel about each skill and ability currently.

As we progress through the year, we are to update this table and expand on our strategy to develop the skills we may not feel very confident in using. We also have to comment how we hope to maintain the skills and abilities we may already have. Furthermore, activity 2 delves deeper into our own reflection of our skills and how we take/will take action on improving our skills.

Moving forward, I hope to work on the skills that I lack confidence in and to hopefully bring them up to a very well development on the table.

Skills and Abilities	1	2	3
Flexibility		*
Self Confidence	*
Self Discipline		*
Working Under Pressure			*
Setting Personal Goals			*
Taking Risks	*
Sharing Opinions Confidently	*
Teamwork		*
Taking Responsibility			*
Building Social Networks		*
Managing Time		*
Acting as a Leader		*
Negotiation		*
Making Presentations			*
Listening to Others			*
Debating Formally/Informally		*
Contributing to Discussions	*
Conversing		*
Taking Notes			*
Writing for academic purposes			*
Computing Skills		*
Being Creative			*
Using Technology		*
Problem Solving		*
Generating New Ideas		*
Working on Initiative			*
Organisation and Planning			*
Critical Thinking			*
Evaluating Information		*

Why Teaching?

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Why Teaching? I ask myself… Why Teaching?

This question of ‘Why Teaching?’ repeats over and over within my mind as I scramble to look for the answer. Why have I chosen this path in life? What do I aim to achieve by becoming a teacher?

Firstly, I feel it’s necessary to start with a brief introduction to myself as an undergraduate student on the Education course at the University of Dundee. My name is Alan Macdonald and I am hoping to become a primary school teacher.

Much of my own time in education was, in my eyes, very successful. This was due partly to the fact it shaped me from being the shy and timid only child that I was in nursery and primary, into a confident and independent citizen by the time I left high school. I felt ready to face the world.

This growth process flourished mainly because of the great teachers I had to support me in my studies. Whether that be because of a teacher’s teaching methods, their enthusiasm in their own profession, or their approach to learning as a whole. Great teachers really make the difference; a difference that I’d love to make myself someday.

My German teacher, in particular, is the person that really sourced my likening towards the idea of going into teaching. Back in 2014 she told our German class about a language scheme that believed in promoting language learning, particularly German, in young people. I was the only male pupil in an already small class of 15 people that decided to take German further. So, I really wanted to get involved in a scheme that could lead to future generations continuing language learning as I felt it was a huge issue, particularly in Scotland.

The scheme was set up by the government-funded organisation UK-German Connection who aim to bring the young people of Germany and the UK together.

UK-German Connections

UK-German Connection is dedicated to increasing contacts and understanding between young people in the UK and Germany – UK-German Connections

Being an Ambassador during 2014-15, I represented the organisation through the projects I planned within a primary school. I set up weekly classes that involved the kids at Murroes primary school learning German on a wider and more dynamic scale, with emphasise on interactive activities that challenged the norm of language learning at a primary level. I covered topics such as History, Geography and even Music within the grand scheme of teaching the kids German. Every lesson incorporated new vocabulary that related to the task of the lesson.

Having the freedom to able to create my own class plans allowed me to see my potential in going into a career in primary teaching. What made it even better was that, it was not only the kids who enjoyed my classes; I enjoyed planning, creating and presenting the classes to a group of enthusiastic children who were eager to learn more.

An example of one of my classes

To wrap up the year, the ambassadors and I attended an evaluation seminar in Berlin where all the different projects were summarised and discussed as a group and we all talked about our plans for the future. My plan was made clear by the ambassador scheme and I knew that the University of Dundee was right for me.

Looking ahead, I really can’t wait to delve deeper into my studies at the University of Dundee and really learn what it takes to become a successful educator that can shape future generations just as my teachers impacted my life successes.

Alan Macdonald