Category Archives: 3. Prof. Skills & Abilities

Planning – IB vs. CfE

Leave a reply

Within the “Making the PYP happen at ISS” document (2009), a key area that I found interesting to examine in a separate blog post was the difference in lesson planning for teachers in comparison to the traditional lesson planning format that we are used to at the University of Dundee.  I felt a separate blog post was needed to take into consideration its implications on the teachers at ISS in facilitating the learning for their clients (both the students and their teachers) was the PYP lesson planner.

Now, reflecting firstly upon the lesson plans that we were provided last year for 1PP1, I see that these were focused more on individual lessons that were centred around the planning of focused areas of learning objectives and success criteria, which does not fully represent the entirety of planning that a teacher must do in their practice, however, it served as an excellent basis for introductory planning for us as first-year practitioners.

Coming to ISS, I believe that I am at an excellent advantage to try and see the wider importance of teacher planning, as it will enable me to hone my skills within self-discipline, working under pressure and setting professional goals as I explore the different grades at ISS and see the planning that each teacher does to facilitate learning (and even just plan towards maintaining structure of their classroom). Teachers need to think beyond the days and weeks of their teaching time.

Cremin and Arthur (2014, p.148) believe that a practitioner that does not consider the greater importance in long-term planning will go into their in teaching that has been provided to them instead of engaging in a whole-staff base critically examining the practice that is being provided to their students, which in term formulates teachers that are not autonomous in their educational philosophy. If ISS wants to instil the values of autonomous learners within the classroom, they also need autonomous practitioners setting an example.

Whilst the grade 3s were with their specialist on Monday, I stayed with their homeroom teachers as they had a grade level meeting for planning. I gained a lot of insight in the depth of planning that the teachers go into. The consideration of even planning meetings for parents to get information of the class’ trips (that are months away still) showed me that looking ahead is just as important as planning for the learning now at ISS, particularly with the amount of time homeroom teachers get when their class is with specialist in PE, Art/Music and German/EAL.

This contrast with the landscape of teaching in Scotland as us primary teachers must also be the specialist teacher in today’s educational climate. The required balance between teaching and planning, I think, has a great say in the quality of teaching that the teachers provide. Teachers at ISS have adequate amounts of time to acquire the appropriate resources, plan out what learning needs to take place and consult their peers for feedback. A downside to this I heard from one of the teachers is that the amount of time spent with the children might not be enough to cover everything they hope to cover.

The teachers first spoke about the mathematics lessons they hoped to do that week, what that would lead to in the future and then they looked at their calendars to explore the events that were coming up in the future (such as the multi-cultural evening and the class trips). Also, the teachers talked in-depth about their Unit of Inquiry for their grade, which focuses on the categorisation of living things.

As these teachers are well aware of the layout of IB, they knew that their planning had to be in line with the International Baccalaureate curriculum, which also has its own PYP planner, which I received a copy of.

Examining the two frameworks for lesson planning, I can see there are areas of similarities and differences that I hope to deconstruct further below.

Firstly, The IB PYP Planner, which is used when planning towards learning in UOI (Unit of Inquiry), begins in a similar fashion as the lesson plans that we used. Information such as grade level, age group, title and proposed duration. Then, it leads a practitioner on to examine what planning must go into the inquiry they want their students to explore.

1. What is our purpose?

Teachers must choose a transdisciplinary theme and a central idea when planning towards a unit of inquiry and then they must also question how they will evidence this and assess the students, particularly when UOI is predominately student-led – “If there is no effective way that students can demonstrate their understanding of the central idea, the central idea will need to be revised so that students’ understanding of it can be shown. The articulation between the central idea and the summative assessment task(s) needs to be resolved before further planning takes place” (International Baccalaureate, 2009, p. 37). So, it is vital for teachers to not only know what skills they want to build upon or begin establishing through the outlet of a core idea, but also how to evidence this and see it within their students, otherwise they will need to re-assess their plans.

Grade 3s central idea was around living things and Grade 4s was centred around the impact of humans being both suppliers and consumers in a media-rich world (which then branched into the themes of internet safety, stereotypes being emphasised through advertisement and the problems of photoshop on body image).

2. What do we want to learn? 

This concerns the key ideas (form, function, causation, change, connection, perspective, responsibility, reflection) that will be focused upon in the inquiry. the guidance is that no more than three of the key concepts should be selected to focus on.

Now, the UOI may be student-led, however, teachers have the responsibility to frame the areas of inquiry in the beginning of the unit through provocations they provide. They also need to provide open-ended questions that are interlinked with the central idea that needs to be explored so that students will be prompted to investigate further.

This relates to our format of planning, however, we must relate our areas of inquiry with the experiences and outcomes. However, the philosophy towards education is greatly different between Curriculum for Excellence and the International Baccalaureate, as there is more focus on the students going out to source their own learning with teachers being their to guide them; CfE instead is more linear in the sense that teachers to make sure that students have met the outcomes and success criteria at the end of how many lessons they believe are necessary for any given topic.

3. How might we know what we have learned? 

This section deals with the prior knowledge that students will bring to the table (something we also need to consider with CfE). IB emphasises that regular feedback needs to be given to students in order to identify progress and areas for growth that they can explore though self and peer assessment. Teachers need to make sure they have proof of each student’s learning in a format that the student can then reflect on their work. “a well-designed assessment task becomes, in and of itself, a learning experience because it provides opportunities to reinforce or extend the learning” (International Baccalaureate, 2009, p. 38)

4. How best might we learn? 

This section must be referred to in unison with question 3 as the two are interrelated. One cannot really know how they can best learn without knowing what they already know. Both teachers and students alike should be able to propose ways that people can go about learning their unit of inquiry. This will allow students to then make connections between the key concepts proposed.

5. What resources need to be gathered? 

Another key area that relates with the planning that I am familiar with. Teachers need to know prior to the learning occurring, what materials they and their students will need to ensure that the best quality of learning occurs. Furthermore, additional technological resources need to be organised effectively in a school where many other staff members will also want to use similar resources for their learning.

6. To what extent did we achieve our purpose? 

Reflection is at the core of all formats of teaching, as a post-reflection is also necessary when planning in Scotland so that teachers can see where they have areas that were strong and where they have areas that need refinement in the future. IB states that examples of work should be attached to this also. It is not enough either just to question the success of the UOI as a whole, specific areas such as the assessment tasks and the connections between the central idea and the transdisciplinary theme must also be evident.

7. To what extent did we include the element of the PYP? 

This section is to explore the learning experiences that really enabled students to be able to fully comprehend what they were learning in the unit of inquiry and how the UOI developed particular attributes of the learner profile (the core concept that is evident throughout IB in shaping internationally minded students).

8. What student-initiated inquiries arose from the learning? 

I feel this is an area that the Scottish format of planning towards learning could take note from. This is where a record of the areas that the students really picked apart or brought up themselves during the inquiry process. Some student-initiated inquiries will be particularly influential in determining the nature of the inquiry and therefore should be evidenced by the teacher, as they can have a great influence informing planning when the inquiry is next visited.

I feel that experiencing a different curriculum from the one I am used to in Scotland has really extended my view upon planning towards learning. It is crucial that we see beyond the boxes that are put in front of us as students in terms of “we must complete this, this and this, and that is it”. Planning should be viewed in a less clinical format, which I think IB has focused on this well with their inquiry-based format of planning. The learning is thought about before, during and well-after a unit of inquiry has been explored. This also shows me an example of what sort of questions are needed in a longer term planner, as we have only used the daily lesson planner.

Reference:

Cremin, T. and Arthur, J. (2014) Learning to Teach in the Primary School. New York: Routledge.

International Baccalaureate (2009) Making the PYP happen at ISS: A curriculum framework for international primary education. International Baccalaureate: Cardiff.

International School of Stuttgart – Structure of Placement Provider (Section 2)

Leave a reply

The International School of Stuttgart is an educational institute that aims to create and shape globally-minded students that strive to see their prospects in life with a wide scope. The school follows the international baccalaureate diploma and provides education from Kindergarten through to grade 12:

“ISS is a vibrant, student-centered learning community where both subject rigor and character development are fostered, where tradition is valued and innovation embraced and where partnership with parents is a fundamental part of our philosophy.” (ISS, no date)

Particularly, within the “Making the Primary Year Program Happen” Document (2009), it finds that education that is international needs to be both inclusive and challenging to be able to shape a globally-minded being. A key area that I find relevant that interlinks with CfE is that International Schools aim to utilise the transdisciplinary model “whereby themes of global significance that transcend the confines of the traditional subject areas frame the learning throughout the primary years, including the early years.” (ISS, 2009, pg. 6).

International Baccalaureate, in a similar fashion to Curriculum for Excellence, strives IB learners  to be:

Inquirers – Ensuring students are independent in their studies and are studying for the sake of learning and to enjoy the process of learning itself

Thinkers – Critical and creative thinking are needed in order to both understand problems that one faces on an academic level, but also beyond in real-life situations.

Principled – Thinking and acting with respect and taking into consideration of fairness and justice (which resembles the General Teaching Council for Scotland’s SPR 1)

Knowledgeable – Exploring knowledge in an interdisciplinary manner that allows

Communicators – Being able to communicate and express oneself in more than one language

Open-Minded – Having integrity to understand and value ones’s own historical heritage whilst also seeking the stories and beliefs of others to mould our perceptions and thoughts on the world around us.

Caring – Having compassion towards others and striving towards doing the most that we can to support one another.

Risk-Takers – learners must see challenges within their lives as areas for great development and critical reflection to occur when learning takes place.

Balanced – understanding the importance of balancing and juggling the aspects of our lives – physical,

Reflective – Being mindful of our strengths and weaknesses in life and being able to see where we need to go in life in order to achieve our goals.

All of these attributes culminate towards creating an IB learning profile that is fit for being a responsible citizen that can contribute both locally and globally with compassion and wisdom (International Baccalaureate, 2013, pg. 5).A key area that I find relevant that interlinks with CfE is that International Schools aim to utilise the transdisciplinary model “whereby themes of global significance that transcend the confines of the traditional subject areas frame the learning throughout the primary years, including the early years.” (ISS, 2009, pg. 6).

Furthermore, the transdisciplinary model, particularly within the development of skills, is seeped in pedagogical findings, particularly within the development of gross and fine motor skills (ISS, 2009, pg. 23), which Boyd & Bee (2014) emphasises that is core in the overall physical, mental and perceptual development of a child, ensuring that they are adequately equipped for their futures ahead. The international baccalaureate emphasises that the utilisation of large muscles alongside delicate muscles for precision need to be development (this is within the branch of Self-management skills.

Now, looking at ISS in a deeper lens, we can see that there are various stakeholders that impact the school but are also who are impacted by the institution itself. I will explore these within this post below.

Stakeholders:

“Our parents, as key stakeholders in the school, are an essential part in the education of children and an active Parent Teacher Association also helps relocating families to settle in, make new friends, find their way around and to feel part of the school and the local community.” (How to Germany, 2018)

Looking through the website and speaking to staff members, I have found that the school has a deep connection between staff members, students and their parents, as they are all stakeholders within the school. Particularly, the Board of Trustees. The Board of Trustees comprise the ISS Supervisory Board and members are elected to serve a four-year term of office and devote their time and energies on a voluntary basis. Those elected serve as being advisors for the School Director. Due to the school being a private non-profit organisation, the stakeholders, whom invest money into the school in order to see that students are adequately educated, have the right to have their voices heard in terms of the practices used within the school. Also, a predominate feature of the board of trustees is that they have children at the school of ISS, which also factors in the great deal of importance they place on their hopes for education. They will consider necessary funding and initiatives through a parental lens but also within a business lens – how will this improve/benefits the students/teachers at ISS?

I have also been told that the board meet on a regular basis (a minimum of once a year) with staff members in order to discuss the school.

Furthermore, the core stakeholders that are evident with every school are obviously the parents, the teachers, the staff members and, of course, the children. These are then interlinked with those affiliated with the school itself – finance providers, large resource companies (with the example of Michael Schmitt Gastro, who supplies the food in the canteen), and the International Baccalaureate as a whole. Not only does the school need to adhere to the laws of education for Germany, they must also cover the guidelines established by the IB pathway.

Leadership Team (Chief Roles): 

There are many people involved within the International School of Stuttgart that go beyond just educating the children. Teachers are supported by both a leadership team and a human resources branch within the school, with various receptionists, support staff and involved parents participating in implementing the International Baccalaureate education.

The main responsibility within ISS lies with the School Director, who shares strategic and pastoral initiatives, as well as operational review, amongst the leadership teams, which “consists of the Business Manager, the Division Principals, the Head of Campus in Sindelfingen and the Director of Admissions and Enrolment.” (ISS – Leadership Team)

Beyond this, there are two principals for the lower and upper stages of the school (this is also due to the buildings being separate for the lower grades and upper grades). There is also a Business Manager who’s main role is focusing on the finances of the school. Human Resources Manager, Directors of Administrations and Enrolment, and even a head of another campus are some of the leaders that command the ongoings of the school. Also, working within the school I have seen that there are various other staff members that work with the children in differing formats. There are specialist teachers whose role it is to facilitate the education of a particular subject area (for example, students will go to art lessons in the art room that has a specialist, the same can be said with P.E and Music). This also allows teachers to have non-contact time to plan lessons and seek the required resources that are necessary to meet the learning intentions of their students.

My main points of contact are with the Lower School Principal, who needs to ensure that the school is running smoothly and accordingly as possible but also ensure the safety of the children within the school (a key example I saw on placement was a child was unable to eat and the head teacher had to make the executive decision to send them home because it impacted their capabilities to learn) and the Lower school assistant principal, who orchestrates much of the meetings for teacher planning. The assistant principal will meet with the teachers from each grade level to plan out the learning they hope to achieve, highlight any concerns or questions they have and make goals towards what they hope to achieve in their practice. I can see this being a massive job to undertake, but one of massive importance as it allows management staff and teachers to be on the same page in terms of the progression of learning.

Furthermore, the class teacher I was placed with first indicated to me that as an international school teacher she was able to source continual professional development workshops or courses, bring them up to management and possibly have the course paid for. So long as it was meaningful for the practitioner and the school could gain benefits from it. This can also be found on the school’s website amongst its Erasmus+ information, as a teacher can take part in training in any European country.ISS state that between the years of 2014 and 2017, 65 teachers and staff members benefitted from professional development that was funded through Erasmus+. Also, from discussions in the school, I have found that the school is in partnership with other schools across the EU (Poland, Italy and the UK being three examples of links that the staff have mentioned to me) to host students to come and learn about the school and also share their background, showing that the stakeholders are global.

Reference:

Boyd, D.R. & Bee, H.L. (2014) The Developing Child 13th edn. Essex: Pearson Education

How to Germany (2018) International School of Stuttgart Storefront [Website] Available at: https://www.howtogermany.com/storefronts/iss.html (Accessed 9th of March 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 10th of March 2018)

International Baccalaureate (2012) The IB Primary Years Programme: Education for a better world Cardiff: International Baccalaureate.

ISS (no date) About ISS [Website] Available: http://www.issev.de/about-iss (Accessed 10th of March 2018)

ISS (2009) Making the PYP happen at ISS: A curriculum framework for international primary education Cardiff: International Baccalaureate.

LfL – Section 1

Leave a reply

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 Connection

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…

Leave a reply

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

1 Reply

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

Leave a reply

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/

99p – Maths Behind Consumerism

Leave a reply

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?

Image 

 

Maths Anxiety: What We Should All Fear…

Leave a reply

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!

Leave a reply

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

Leave a reply

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