Category Archives: 3.4 Prof. Reflection & Commitment

Maths, this is not the end – I promise

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Discovering Mathematics, I can honestly say, has been brilliant. Intriguing, eye-opening, shocking. It has left me sad it has reached an end, but I am 100% certain when I say that this is most definitely not where my new-found love for maths ends.

Where to start?
I selected ‘Discovering Mathematics’ as my elective module for MA2 and at the time, admittedly, I was not completely certain of what to expect to take from it. I knew it would entail learning about approaches towards learning mathematics and so this was an automatic go-to.
From the first introductory class we had, I thought to myself ‘yes, this is a good choice, I’m going to learn a lot’ and I was right!

Goodbye maths anxiety!
‘Maths anxiety’ is something I learned about in the module and, unbeknown to its title and ‘diagnosis’ if you like, experienced myself prior to the module. All thanks to the tutors who ran the inputs for the module, my lack of confidence in maths quickly banished and went to the back of my mind. Rather than dreading maths, I started to really look forward to the next maths input. I can confirm this was due to the tutors’ own love for learning maths which was an inspiring attitude that brushed off on me. Admittedly, there are still areas of maths I am not so confident in, as I am not qualified in the subject at a too-impressive level. However, the module has made me experience a sense of confidence in findings out things I did not know about in maths and question things upon professional and personal reflection.

Discovering Mathematics – Mathematics, you have been discovered
In summary, I mean what it says in the title. I would recommend this module highly as an elective choice for any MA Education students who have the option to look into choosing this. It’s a real eye-opener to ways of learning and approaches to methodology in teaching mathematics, which was not my strong point in professional practice. Now I am more keen than ever before to develop my maths practice in schools.

What’s next?
There are many thing I’ve learned about throughout this module which I would not have thought twice about in terms of making connections to mathematical concepts. For example, maths and dance, maths in nature and really, honestly, the fact maths is everywhere. It has changed my perspective of maths to one that it determined to explore maths further because now I know there is no escaping maths – it is in almost everything we do! Not only this, but I am driven to change people’s perceptions of maths – especially the way it is approached in schools. Let’s abolish the ‘rote-learning’ attitude and get adventurous!

 

Now Discovering Mathematics if finished, I think it’s time to go and eat some Christmas π.
… Sorry, I couldn’t resist.

Mundane Maths?

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I will not take this for an answer. Maths is not mundane. It is not tedious. Most of all, it is not boring! I have learned this only from teaching maths in my professional practice and still then I was not 100% confident maths could be full of fun and adventure. However, now, having completed the Discovering Mathematics module, my fun with maths is only just beginning.

Maths does not have to be a chore. It does not have to be the dreaded subject for a learner to approach. But this is up to you, the teach, the educator, the facilitator, the mathematician! I think it is important that in order for a learner to enjoy learning mathematics, the content delivery should be engaging, intriguing and taught with an approach oozing motivation and love for learning maths. Otherwise, what’s the point? You’ll be bored teaching it and you’ll be fed up learning it. Make it fun, it’s down to you to get involved.

Piaget (1953) expresses his view of learning maths:

“It is a great mistake that a child acquires the notion of number and other mathematical concepts just from teaching. On the contrary, to a remarkable degree he develops them himself independently and spontaneously. When adults try to impose mathematics concepts on a child prematurely, his learning is merely verbal; true understanding of them comes only with his mental growth.”
(p. 74)

My interpretation of this, is Piaget is saying maths does not solely have to be learned from the teacher teaching the subject content; honest understanding and real learning of maths happens through experience out-with the classroom, as well as in the class environment. I can vouch for this and state that I completely agree with this statement. I have learned through the course of Discovering Mathematics that maths can in fact be discovered and learned successfully by independent research and development. By development, I mean I have developed an appreciation for maths more than I had prior to this module. The way I have learned maths and been taught maths has made it intriguing maths rather than mundane, which is immensely important to note.

Bruner (1964) makes a valid point:

“Any idea or body of knowledge can be presented in a form simple enough so that any particular learner can understand it”.
(p. 44)

To me, this summarises and clarifies what fundamental mathematics is all about – why we talk about it, why we learn about it and why we use it to teach. It is so important to know about.
Prior to this module, I did not have a clue what ‘fundamental mathematics’ was. I mean, I could take somewhat-aimless guesses at what it meant, but I did not know how to approach understanding what it entailed and meant in the classroom context. Bruner (1964) suggests any content of learning has the ability to be translated in its notion, in order to allow learners to understand it at the appropriate level. In other words, abstract and ‘scary’ maths – which is commonly the root of maths anxiety (pardon the pun) – can be taught, delivered or learned in a different form, one that is more simple or fundamental, for the learner to have an easier understanding.

References

Bruner, J. S. (1964) Some theorems on instruction illustrated with reference to mathematics. In E.R. Hilgard (Eds.), Theories of Learning and Instruction: The sixty-third yearbook of the National Society for the Study of Education (NSSE) Chicago: The University of Chicago Press.

Mason, J., Burton, L. and Stacey, K. (2010) Thinking Mathematically (2nd ed.). Harlow: Pearson Education Ltd.

Piaget, J. (1953, reprinted 1997) The origins of intelligence in the child. Abingdon: Routledge.

First position – Dance and Maths

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First position. Second position, third, fourth, fifth position. Wall one, two, three, four. Corner five, six, seven, eight.

Maths, believe it or not, is a large fundamental element underlying choreography and dance.
From the age of two years and for the most-part of my teenage years, I was, unbeknown to my senses, experiencing mathematical thinking during rehearsals and practising dance. My passion for ballet, Scottish dance, jazz and tap meant I was dancing for years without consciously recognising the patterns of maths evolving. Now, since starting the Discovering Mathematics module, I have come to realise maths is monumental in its involvement in dance.

I first realised this whilst sitting in a Discovering Mathematics workshop, where my astonishment was rapidly growing about the connections maths has, to everything. I decided at that moment, to reflect on this. This led me to write about the links between maths and tasks we complete day-to-day. However, in hindsight, I realise I did not actually write about a specific link I’ve made to maths.

Generally, maths is in dance if you think about counting beats, speed, shapes made with the body, angles, position, timing in music and patterns in the choreography itself. For example, in the dance studio, you will, most of the time, be surrounded by four walls and four corners. You must learn the number each wall is labelled as and understand the directions in which you must face. To face whichever numbered wall or corner, you must understand the mathematical concept that is ‘rotation’ by understanding clockwise and anti-clockwise. Mathematical vocabulary is widely used in dance, as well as in drama and theatre performances:

  • CS – centre stage
  • CSL – centre stage left
  • CSR – centre stage right
  • USL – upper stage left
  • USR – upper stage right
  • USC – upper stage centre
  • DSL – down stage left
  • DSR – down stage right
  • DSC – down stage centre

These are named ‘stage directions’ and usually your choreographer or director will instruct you in accordance to the space. To be able to dance in accordance to this, you must understand the maths vocabulary used, which, in this example, is direction.

Dance does not require mathematical problem-solving or making calculations. Instead, it is simpler. More fundamental. It requires you to think mathematically. What I mean by this is you need to be able to have a sense of pace, time and speed in dance routines and therefore counting beats is a mathematical strategy in practice. It is debatable that counting beats is a musical skill, however I argue that this is mathematical, as well as musical.

Additionally, symmetry is largely used in dance. Numbers of routines and choreographed sequences are designed around the principle of symmetry – this requires the understanding of what symmetry is, what symmetry looks like and how symmetry is created. The fundamental understanding of symmetry is key in dance. Symmetry is also in occurrence when a dancer is balancing, because keeping the body symmetrical or in other words equal, aids balancing.

Dancers make shapes with their bodies in dance. Specifically, in ballet, dancers create triangular shapes and angles with their legs and arms. An understanding of straight, parallel, horizontal, and curvilinear needs to be understand, as this is important in ballet. Dancers should understand the fundamental learning of angles – specifically, understanding 40, 90, 180 and 360 degrees, in order to accurately use their bodies in pirouettes and developpes.

In summary, I have discussed the links between maths and dance, a real passion of mine and as dance as always been a strong commitment of mine I was enthusiastic about sharing its interconnected relationship with fundamental maths.
In terms of pedagogy, I aspire to learn about teaching maths through dance in my professional practice. Dance is an expressive art which is not implemented sufficiently in schools and I definitely intend on it having more consistent and regular involvement in aiding children’s learning.

Any fellow dancers, I would love to hear your thoughts on dance and maths. 

Look around you! (continuation – tessellation)

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Since my last blog post about tessellation – “Look around you!“, I have reflected deeper on what tessellation is and, more specifically, where the fundamental mathematics lies within.
You can read ‘Look around you!’ at:
https://blogs.glowscotland.org.uk/glowblogs/teachingjourney/2015/12/05/look-around-you/

Harris (2010) discusses the prior knowledge a learner must have acquired, in order to understand the mathematical concepts behind tessellation. The following is content the child should understand prior to learning about tessellations:

  • A whole turn around any point on a surface is 360°;
  • The sum of the angles of any triangle is 180°;
  • The sum of the angles of any quadrilateral is 360°;
  • How to calculate or measure the inner angles of polygons (a plane figure with at least three straight sides and angles.

He continues to explain children are required to know about the angle properties of all polygons – regular and irregular – in order to understand the maths in tessellation (2010, p. 4).

So, having read this report by Harris: “The Mathematics of Tessellation” (2010), I now know there is more fundamental elements than I previously assumed. Prior to reading Harris’ work, I thought the only fundamental maths in tessellation was knowing the shapes in use. I did have an awareness of the angles having an importance, but as I knew the shapes I demonstrated worked in tessellation anyway, I did not think twice about needing to know the angles of the shapes.

If you would like to find out more about the mathematics in tessellation, follow the link below!
https://my.dundee.ac.uk/bbcswebdav/pid-4544087-dt-content-rid-2917269_2/courses/ED21006_SEM0000_1516/Tessellation.pdf

References 

Dickson, R. (2015) Look around you! Available at: https://blogs.glowscotland.org.uk/glowblogs/teachingjourney/2015/12/05/look-around-you/

Harris, A. (2010) The Mathematics of Tessellation. Available at: https://my.dundee.ac.uk/bbcswebdav/pid-4544087-dt-content-rid-2917269_2/courses/ED21006_SEM0000_1516/Tessellation.pdf. Last Accessed: Dec 5 2015.

Profound Understanding of Fundamental Mathematics

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Four Properties – to ensure a profound understanding of fundamental mathematics

Liping Ma (2010) believes in four properties in teaching and learning which sum up the way in which a teacher’s profound understanding of fundamental mathematics can be represented in the classroom. By this, Ma means a teacher will demonstrate the four properties and if this is successful, he or she has a profound understanding of fundamental mathematics.
– If you are unsure about what this is – profound understanding of fundamental mathematics – scroll to the bottom to read quotes extracted from Liping Ma’s ‘Knowing and Teaching Elementary Mathematics’ (2010). This may aid your understanding of the Four Properties in this post. 
The first, connectedness, occurs when a learner has the intention of making connections between mathematical concepts and procedures. In pedagogical terms, the teacher will prevent the learning from being fragmented and instead, learners will develop the ability to make connections between underlying mathematical concepts that link.
The second property, multiple perspectives, is practised when a learner is able to take into account various perspectives when thinking in a mathematical way. This includes addressing pros and cons of all different viewpoints considered. In pedagogical terms, the teacher provides opportunity for their learners to have a flexible way of thinking and understanding concepts in maths.
Thirdly, basic ideas, is a way of thinking about maths in terms of equations. Ma refers to basic ideas as, “simple but powerful basic concepts and principles of mathematics” (2010, p. 122). Ma suggests, when practising the property ‘basic ideas’, learners are guided to conduct ‘real’ maths activity rather than just being encouraged to approach the problem. I suggest this means if a teacher is effectively implementing this property, he or she will be not just attempting to motivate the learners to approach the maths work, but instead, providing a solid and secure guide to the learners understanding the maths themselves.
The forth and final property, longitudinal coherence, is when a learner does not have a limit or boundary of knowledge. In other words, it is not possible to ‘categorise’ the learner or identify the learner as working at a specific level or stage in maths. Instead, he or she has achieved a holistic understanding of maths – a fundamental understanding. In pedagogical terms, Ma suggests a teacher who has achieved a profound understanding of mathematical understanding is one who is able to identify on demand the learning that has been previously obtained and will be learned later. Subsequently, the teacher will lay the fundamental maths as a foundation for later learning.

A few quotes extracted from Liping Ma’s Knowing and Teaching Elementary Mathematics (2010) that I feel define and summarise profound understanding of fundamental mathematics…

“The term ‘fundamental’ has three related meanings: foundational, primary, and elementary.”
– Ma (2010, page 120)

“By profound understanding I mean an understanding of the terrain of fundamental mathematics that is deep, broad and thorough. Although the term ‘profound’ is often considered to mean intellectual depth, it’s three connotations , deep, vast, and thorough, are interconnected.”
– Ma (2010, page 120)

“As a mathematics teacher one needs to know the location of each piece of knowledge in the whole mathematical system, its relation with previous knowledge.”
– Tr. Mao (2010, page 115)

“I have to know what knowledge will be built on what I am teaching today.”
– Tr. Mao (2010, page 115)

 

 

References

Ma, L. (2010) Knowing and Teaching Elementary Mathematics 2nd edn. New York: Routledge. Pages 115-122.

liping ma

‘The Secret Life of 4 Year Olds’

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slo4yo

I recently watched ‘The Secret Life of 4 Year Olds‘ broadcast by Channel 4 on Tuesday 3rd November (Watch here – http://www.channel4.com/programmes/the-secret-life-of-4-5-and-6-year-olds). As the episode progressed, I found myself realising I have definitely been underestimating children as young as four years old. The hour-long duration of the programme was enough to capture the essence of the life of a four-year-old and the day-to-day thoughts, activities and behaviour of the children – not to mention, the emotional roller-coaster they endure because their ‘best friend took their toy or decided to play with someone else.
Oh, to be four. 

A number of key points initiated…

  • Children, at 4, 5 and 6, are at a partial age;
  • Futures are formed from this young age;
  • These ages are a crucial stage for a child’s development – what they learn now is the ‘blueprint’ for adult life;
  • Moral argument can quickly become coercion;
  • Supportive friendships have the ability to rapidly change;
  • A history of friendships create expectations of behaviour;
  • Ambitions from the four-year-old children in the episode include, ‘save the planet’, doctor and hairdresser – at the same time, and ‘jelly maker’.

Dr. Sam Wass, Educational Psychologist – MRC Cambridge quotes,

“To establish and maintain relationships, one of the key tools that children need is language. And at four, the average girl tends to be five months ahead of the average boy, in terms of their language skills. This can put some boys at a disadvantage in their social interactions.”
On reflection, what is meant by ‘average’? Every individual child is different and unique in the way they learn. Therefore, arguably, we cannot generalise, label or categorise children’s abilities, to give us a specific indication of ability.

“They’re beginning to learn to regulate their emotions, to interact with each other and to understand that other people have feelings, too. These are lessons that will inform a lot of their future interactions.”
On reflection, children respond in a variety of ways in different situations and therefore express a range of emotions. For example, experiencing a tragic incident, being vulnerable to an unsafe environment, bullying, winning or losing, achievements and many more. I believe it is not possible to teach a child these emotions because to do that would mean telling  or showing a child which emotion ‘matches’, if you like, with which situation. Emotions are a natural human trait – they are intrinsic but often influenced by extrinsic factors. Therefore, we can only teach children how to cope with and respond to their emotions, by being a supportive role and most importantly, by understanding. This is a learning process which children are still going through at a young age.

He continues,
“You give a child a new abstract concept to play with such as the concept of a friendship, and the natural instinct of a child is to want to prod and explore what that idea means. They tug it around a bit, see if they can break it and by doing this, they learn more about what the concept of friendship means.”

Professor Paul Howard-Jones, Educational Neuroscientist – University of Bristol quotes,

“Competition is motivating, it’s exciting, but it’s also great learning experience.”
On reflection, competition is an issue that is widely debated: is competition a good thing? My viewpoint is that is can introduce diversity, which may be viewed as a positive. However, I think competition is an important thing to teach our children to deal with, by teaching coping strategies.

“Children at this age self-segregate on the basis of gender.”
On reflection, I remember at primary school having to choose partners, groups or team leaders and the majority of the time, boys would choose boys and girls would choose girls. It was rare that opposite genders would be paired together. What does this say about our society? If anything, what does it tell us about our teaching strategies? As teachers and educators, the fundamental basis of our teaching and learning is around equality of opportunity and inclusion. We teach children the morality that everyone is the same, despite gender, race or religion. So, why do children self-segregate on the basis of gender? I would be interested to see any comments on this post regarding this issue.

And one final thought I will leave with you – extracted from The Secret Life of 4 Year Olds:

“It’s really striking how much children have to achieve at four years old. It may look like play, but actually they’re working really, really hard and they’re having to learn an awful lot. The way that they’re communicating with each other, the way that they’re experimenting and finding things out is really, really important for them.”

– Professor Paul Howard-Jones, Educational Nueroscientist (2015)

Watch ‘The Secret Life of 4 Year Olds’ – http://www.channel4.com/programmes/the-secret-life-of-4-5-and-6-year-olds

Let me speak!

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Yesterday, in my place of work, I found myself astonished by the sight and sound of a mother silencing her child.

Allow me to contextualise the situation for you. I work in a restaurant located in our City Centre. It is always busy with guests coming in for the first time, the second time, or coming in for their weekly order. So yesterday, work was extremely busy and therefore the noise levels were expectedly high. However, I could not ignore what I heard one woman say.

I allowed my eyes to glance across tables and they stopped on one table in particular. At this table sat eight guests who had come in together and going by first judgement, they were a family – made up of what I can only presume was two brothers, around three and five years old, along with their parents, grandmother and perhaps other relatives or failing that, family friends.

Now, in my place of work is one huge stone oven which does not ignore the flames that provide it with a warm glow and extreme heat. The boy, of about five years of age, sat at the table and was staring at this oven in absolute amazement. Of course he would be – it’s an enormous oven and most definitely is not your standard oven in your kitchen at home. He was amazed. It was something new to him. Something wonderful and exciting.

He turned to his mother with absolute excitement lighting up his face, wide-eyed and open-mouthed and said,
“Look! Mum, look at that!!! Our pizza is in that oven, look!”.
At that moment lay an opportunity for the mother to endlessly discuss the most exciting thing this boy had discovered – the oven!

Instead, she turned to him, ignoring the subject that provided him with such amazement, and silenced him with,
“Sssh, be quiet.”

I was in shock. You may be wondering why I was left feeling shocked and quite simply empathetic towards this boy. You see, this child should be immersed in language. Engaged with language. Not silenced when something is open for discussing, explaining and being interested in. His mother could quite easily have turned to her son and described the oven, asked him questions about it, used language to indicate a sharing of excitement and amazement about what her son had sighted.

I am fully aware the oven is not the most exciting thing for an adult to lay eyes on. However, as teachers, parents, educators or caregivers, it is crucial that we recognise children’s learning is embedded from a young age, they are learning all the time; and that is what we need to get right – we need to identify the gaps for learning and fill those gaps with knowledge, vocabulary, insights and perspectives. With language, there are a mass amount of opportunities to do this.

It is moments like this when children are surprised, amazed and intrigued about something at which it is necessary to capture this interest and go with it. Silencing a child when they show interest in something can only teach them not to display signs of true hysteria.

Celebrate this, engage this, and most importantly ask questions. Be involved by talking, discussing and conversing using your language skills and understanding, in order to facilitate the child’s learning and awareness of language. Show emotions with language and use words the child will question the meaning of; use terminology to challenge the child appropriately and broaden the vocabulary of the child.

As cited in The Really Useful Literacy Book (3rd edn.), it is suggested that children learn by understanding and remembering, which is essentially achieved effectively by ensuring application and regular revision (Martin, T., Lovat, C., Purnell, G., 2012). I agree with this and I suggest that in order for children to learn, understand, remember and progress language skills, it is profound that they are immersed in a language-rich environment,

 

 

 The focus of this reflection is not about the oven. It is about spoken language.

 

Stress, stress, stress!

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I recently read the article on BBC News, “Stressed teachers being ‘reduced to tears'” by Hannah Richardson, BBC News Education Reporter, 22 October 2015 (see link below) and it really hit me. It made me think – teachers are crying out, literally, and what for? They are stressed.

This article is stating the lead up to the stress is due to the workload teachers are faced with. In the article, it is stated by Dr. Bousted, a writer for Times Educational Supplement:
“It seems that teacher stress is increasingly being regarded as par for the course and part of the job.”
I agree that the workload in teaching is part of the job, due to GTCS standards and requirements, paperwork must be done. However, that should not take away from the love, passion and fun that teaching should be for teachers undergoing current stress. Not only will the stress make you feel under pressure, it will have an impact on your learners as well as those around you – colleagues, friends and family.

Dr. Bousted continued,
“A newly qualified teacher, asking for help to deal with an impossible workload which took up every evening until 11pm and all of the weekend, was told by her line manager ‘that’s the way it is in teaching’.”
To say, “that’s the way it is in teaching”, is a harsh reality for some, however it does not have to be stressful, pressurised or looked upon negatively. As a current student teacher, I am still only partially aware of the workload required by qualified teachers. Of course, I have seen in practice the paperwork – planning, assessment and reports. My viewpoint is that if you are entering the teaching profession, it is profound you thoroughly understand what is expected of you – the teacher, the facilitator, the educator, the professional, the trusted and respected role model. In order to be these things, you have to do the work.

Dr. Bousted goes on to advise,
“Teachers, as professionals, expect to work hard but should not be expected to devote every minute of their lives to their work. Teachers need time to relax, to pursue hobbies, to talk to their families and friends. They need time to be human.”

Teaching is not all stress. The way I see it is that you will always have work to do. There will not be a day that comes when you will have completed everything on your ‘to-do’ list. But that is part of being a professional. It all comes down to commitment and dedication. 

 

References

Stressed teachers being ‘reduced to tears’ – http://www.bbc.co.uk/news/education-34602720