Category Archives: 2.2 Education Systems & Prof. Responsibilities

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.

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. 

Roman numerals, Ishango and all things prehistoric…

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Firstly, I’d like to you imagine something. Something unusual for many, but perhaps not so much for others. Imagine you had more than ten fingers. More than ten toes. More than two eyes, two ears, one nose and… Actually, imagining you had more than ten fingers will suffice.

We use methodology to take account of the value of things and this methodology is using number systems. Number systems provide symbols which represent digits which symbolise the value. This may seem confusing, but bare with me.

The terminology available to describe numbers, counting, digits or mathematical values, proves to be of a broad essence.

  • Cardinality – the number of elements in a set or other grouping, as a property of that grouping;
  • Ordinality  – ordinal number;
  • Integer – a number which is not a fraction – a whole number;
  • Numeral – a figure, symbol, or group of figures or symbols denoting a number/ a word expressing a number.
    – Oxford Dictionaries: Oxford University Press (2015)

So, it appears that numerals are symbols which represent the value, which fundamentally, is a number. Each symbol denotes a digit and looks something like: 0, 1, 2, 3, 4, 5, 6, 7, 9… (do bare in mind that number symbols do not look like this in particular countries… For example, have a look at the number symbols in the Chinese language).

How old are you, numeral?

Roman numerals is a number system going back to 3500bce, which makes it 5500 years old.
However, number sequences are suggestively not the same age. Take a look…

ishango_bone
 http://africanlegends.files.wordpress.com/2013/08/ishango_bone.jpg

This is the Ishango Bone is a mathematical resource, a ‘tool’ if you like, which was used to make tally marks to keep count. It is 22, 000 years old and in its original use, it did the job for people tracking environmental occurences, such as the factors indicating season changes, or light and dark weather. Today, clocks tell us the time and so if it is 11pm, we know, without looking outside, there will be a dark sky. Vice versa, if it is a dark sky outside, we will know, without checking the time, it will be somewhat late in the evening. Back when the Ishango Bone was used, there were no clocks to distinguish the time – the hours, days, weeks, months and seasons going by.

download
http://www.math.buffalo.edu/mad/Ancient-Africa/ishango_bone.jpg

 

Tally marks were notched into the bone to count or symbolise the number value. For example, one line = 1, two lines = 2, three lines = 3, and so on – each line is one  tally mark. Amazing, isn’t it?

Back to Roman Numerals

Perhaps Roman numerals is an easier way to educate learners about what a ‘numeral’ actually is, because Roman numerals, apart from the numerals we use today, are well known, so much so that they are still used on clock faces.

Here is an example of this! Roman numerals on the clock at The Steeple Church, Dundee. This picture was taken by myself. 
These are fundamental elements in maths – number recognition, counting, values and number
systems. Yet, it is so astonishing how number systems have evolved throughout the years of
maths going on, almost in a subconscious behaviour. I mean, when people used the Ishango

Bone, did they have the understanding they were using mathematics and, more amazingly, they

were practising and demonstrating an understanding of fundamental mathematics?

References
Mastin, L. (2010) Prehistoric Mathematics. Available at: http://www.storyofmathematics.com/prehistoric.html. Last Accessed: Dec 5 2015.
Ishango Bone – http://africanlegends.files.wordpress.com/2013/08/ishango_bone.jpg
Ishango Bone (2) – http://www.math.buffalo.edu/mad/Ancient-Africa/ishango_bone.jpg

Unimaginable Imagination

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Imagine an unimaginable imagination. One that no teacher, educator, facilitator, parent, caregiver or learner sets expectations for, standardises for, or… imagines.

Sarah Maxine Green, an American educational philosopher, author, social activist and teacher, stated,

“We want our classrooms to be just and caring, full of various conceptions of the good. We want them to be articulate, with the dialogue involving as many persons as possible, opening up to one another and to the world. We also want our students to be concerned for one another as we learn to be concerned for them. We want them to achieve friendships among one another as they move to a heightened sense of craft and wide-awakeness and a renewed consciousness of worth and possibility.”

Following an input in Education Studies: Historical and Comparative Perspectives on Education module, I thought about this quote from Greene and I reflected on what it means to be in a classroom. What is our purpose of schooling? What is the need to teach such human traits as to be caring, respectful and open-minded? I argue that it is to broaden the mind of our learners. To allow them to visualise, to imagine their potential and teach them the skills they need to turn that imagination into a reality. Schooling is to support and encourage our learners in becoming well-rounded individuals.
I analysed this quote from Maxine Green and recorded a few key points I interpreted from it:

12248656_1059401164079490_206308192_n
Click on the picture for better quality.

Many of the traits Maxine Green talks about in the above statement, I interpret to making a number of links to the Curriculum for Excellence (2009), and in particular, the four capacities: successful learners, confident individuals, responsible citizens and effective contributors.

This just highlights to me that even though Maxine Green does not base her work in Scotland, around our Scottish Curriculum, based on our learners, our expectations and our way of working in schools, the idea and the imagination and thoughts are still on the same wavelength. We want our learners to achieve potential, bring their own views and to have the confidence to express them. Most importantly, for our learners to have an awareness and an imagination for what is possible.

In response to Greene’s statement, she is talking about creating an ethos and an influential environment in classrooms and amongst children as learners and individuals – one that demonstrates respect, appreciates friendships and understands voices and opinions. I strongly agree with what Greene pushes for by saying this, as she shows understanding of the profound importance of capturing the essence of the holistic child. 

References 

Greene, M. (2000). Releasing the Imagination. Essays on Education, The Arts and Social Change. San Francisco, CA: Jossey-Bass. Page 155.

Scottish Government (no date) Education Scotland: The purpose of the curriculum: The four capacities. Available at:
http://www.educationscotland.gov.uk/learningandteaching/thecurriculum/whatiscurriculumforexcellence/thepurposeofthecurriculum/. Last Accessed: Nov 17 2015.

There’s no avoiding it – Maths is everywhere!

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Throughout my inputs in Discovering Mathematics, I’ve been thinking a lot about the links between maths are other curricular areas. I’ve been making connections more and more between maths and other subjects and, I have discovered something about maths that perhaps I hadn’t thoroughly thought about before… It is everywhere. 

The idea to write a blog post centred around the fact that I am amazed elements of maths come in to everything, came from an idea I had to write an account reflecting on the connection between maths and one other particular topic which I knew linked and I had interest in. So, I started making a list and it went something like this…

  • Maths and food;
  • Maths and sport;
  • Maths and the environment;
  • Maths in literacy;
  • Maths and music;
  • Maths and dance;
  • As well as the above, the connections between maths and all curricular areas is something I would like to explore further.

I looked back at this list and as I read over it, I realised one thing – that list would most probably never end. Why? Well, because everything we do, somehow connects to a way of mathematical thinking – numbers, time, money, patterns, sequences, routines, timetables, schedules, musical beats, distance, speed, shape… I could go on. Whether it is basic, simple, easy-to-understand fundamental maths, or even the more abstract concepts that challenge us to think, reflect and solve… It is everywhere.

I decided to look at her writings in an extract called ‘Learning together series’ (no date) where Carol Skinner wrote about maths, titled, ‘Maths is everywhere’. Carole Skinner is a former teacher, Maths Specialist in Early Years and a writer with the belief that teaching maths, learning maths and experiencing maths holistically should be an exciting for all learners. At her time teaching at Brunel University, London, she was a Numeracy Strategy Consultant and an Early Years maths lecturer. Upon reading ‘Maths is everywhere’, my understanding of where maths links in to other areas of learning became more profound. Carole Skinner suggests maths can be acquired, learned and developed more effectively if children have a good start in being confident with counting. She goes on to explain maths has involvement in the simplest of tasks. For example, changing the channel for the television, pointing at numbers on the clock after recognising that they are in fact numbers, and even saying things like, ‘1, 2, 3, boo!’ to babies, toddlers and young children (no date). On reflection of reading this, I realise that maths is not solely about addition, subtraction, equations and problems. I would argue that if we think deeper about where the fundamental maths is that these more challenging aspects stem from, we can identify the fundamental maths more recognisably.

  • Understanding what numbers are – the fundamental maths in this is knowing that they are a way of keeping a count of the amount, quantity, value or measurement.
  • Recognising a number when you see it – and knowing in what context the number is used and it’s meaning.
  • Having ability to count – which, fundamentally, requires the learner to understand order, sequence and value; number patterns.
  • Pairing socks or matching socks in the home environment – the basic maths in this is knowing that in order to have one pair of something, you must have two single items of the same, put together.
  • Patterns and sequences – this can include in Art & Design; number patterns and sequences, which can entail having the ability to count and make calculations.
  • Routines, schedules and timetables – to understand a routine, you would have to understand the time involved in structuring the routine; also incorporated in making plans on timetables and schedules.
  • Keeping beats in music, songs, poems and dance – this entails having an awareness of timing, rhythm and regularity of beats in the tune, which most of the time – probably all of the time – entails the ability to count.
  • Maths in the home – in weighing and measuring; cooking and baking entails many different aspects of fundamental mathematics combined and connected to each other; setting out cutlery for meals and knowing how many plates, knives, forks, glasses etc. to set out in accordance to how many people need them; having a fundamental understanding of volume, in order to be able to run bath water correctly; setting an alarm clock which requires an understanding of the basic maths of knowing numbers and calculating time; planning routines such as breakfast, lunch, dinner, television programme schedules and hobbies etc.
  • Maths in the environment – environmental print can be largely beneficial to a learner’s developing understanding of fundamental mathematics and almost anything can be linked to maths.
  • Classroom – in the classroom, things like arrangement of tables includes maths and knowing numbers, which effectively entails the ability to count; organising children into groups or pairs requires the children to have this understanding, if they are to manage this themselves, of quantity, number value, division calculations, counting and addition; reward charts; homework charts; diaries, schedules and timetables in school and in the classroom; also all curricular areas children are learning, I would argue, relate to fundamental mathematical principles in some way.

Gathering these ideas, on reflection, has also made furthered my awareness of how maths is all connected. To justify this, in order to know how to do or understand one thing in maths, you must have grasped and understood the basic maths behind it – the fundamental principles.

 

Carole Skinner has written for a number of publishing companies, including BBC. Her latest book is titled ‘Maths Outdoors’ and is available to order – https://www.waterstones.com/book/maths-outdoors/carole-skinner/9780904187434

References

Skinner, C. (no date) Early Education: Learning together series. London: British Association for Early Childhood Education. Available at:
https://www.early-education.org.uk/sites/default/files/Maths%20is%20Everywhere.pdf
Last Accessed: Nov 17 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

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

The Meaning of Mathematics – Maths defined

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Numbers, sums, equations, patterns, sequences, problem-solving, formulas, confusion…

What comes into your head when you think about ‘maths’? Should we think more about ‘mathematical concepts’ and look deeper into what maths is all about, rather than accepting maths as being solely about numbers and work books?

‘Mathematics’ is defined as,
“The abstract science of number, quantity, and space, either as abstract concepts (pure mathematics) or as applied to other disciplines such as physics and engineering (applied mathematics)”.  (Oxford University Press, 2015)

So, what is maths? Maths can be confusing for many people. Many people believe you either have a ‘maths brain’ or you do not. I believe maths can be confusing, however I passionately argue that maths, the majority of the time is equations and formulas to follow. My view is maths can be straight-forward, if you allow it to be, or as the teacher, if you facilitate it right; it’s made up of steps and strategies. The difficult concept to grasp in maths is understanding those strategies, formulas and equations. Given you have that understanding, you are able to follow the steps and reach your solution – your answer.

However, saying that, is maths all about finding an answer? The problem-solving involved in mathematics is easier for me personally, because I enjoy being challenged to think and to think about problems from various perspectives. So, I enjoy the aspect of thinking about maths in contexts, as it has a purpose. I like to think of this as meaningful learning. In summary, I view problem-solving in mathematics as meaningful learning.

That’s not to abolish that other elements of mathematics are not intended purposeful learning. As stated by Scottish Government,
“Mathematics is important in our everyday life. It equips us with the skills we need to interpret and analyse information, simplify and solve problems, assess risk and make informed decisions.” (Scottish Government, Education Scotland, 2015)

I agree with this – maths is important. Maths can be used everywhere in situations, without us recognising that we are using our mathematical understanding. How would we be able to tell the time? How would we be able to implement time management skills? How would we know to recognise significant dates? Would you know when your own birthday is approaching? How would we manage finances and handle money? I could not think of one occupation or career that does not involve mathematical elements in some way. Could you?

Maths is fundamentally important in every day life, I agree.

References

Oxford University Press (2015) Oxford Dictionaries: Language Matters. Available at: http://www.oxforddictionaries.com/definition/english/mathematics. Last Accessed: Nov 5 2015.

Scottish Government (2015) Education Scotland: Mathematics. Available at: http://www.educationscotland.gov.uk/learningandteaching/curriculumareas/mathematics/. Last Accessed: Nov 5 2015.

The educator’s conceptual view – know what you are teaching!

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“Limited subject matter knowledge restricts a teacher’s capacity to promote conceptual learning among students. Even a strong belief of “teaching mathematics for understanding” cannot remedy or supplement a teacher’s disadvantage in subject matter knowledge. A few beginning teachers in the procedurally directed group wanted to “teach for understanding.” They intended to involve students in the learning process, and to promote conceptual learning that explained the rationale underlying the procedure. However, because of their own deficiency in subject matter knowledge, their conception of teaching could not be realized. Mr. Felix, Ms. Fiona, Ms Francine, and Ms. Felice intended to promote conceptual learning. Ironically, with a limited knowledge of the topic, their perspectives in defining the students’ mistake and their approach to dealing with the problem were both procedurally focused. In describing his ideas about teaching, Mr. Felix said: “I want them to really think about it and really use manipulatives and things where they can see what they are doing here, why it makes sense to move it over one column. Why do we do that? I think that kids are capable of understanding a lot more rationale for behavior and actions and so on than we really give them credit for a lot of times. I think it is easier for anybody to do something and remember it once they understand why they are doing it that way“.”
– Liping Ma, Knowing and Teaching Elementary Mathematics (2010, page 36)

The most important thing to remember when teaching maths – when teaching anything – as the teacher, the educator and the facilitator, is that you must understand what you are teaching. This is what Ma (2010, p. 36) is talking about here.

As a teacher and a professional educator, you are responsible for providing knowledge to your learners, not just passing it to them as information in a book or in the form of confusing statistics and facts, but as an understood conceptual view of the content. If you do not understand what you are teaching, this may invite opportunity for confidence to fall in your learners – you are the trusted educator in the classroom, on which your learners depend on to provide subject matter with an understanding you have thoroughly revised, in order to adapt the content to best explain it to them.

Outsmarted?… Imagine this. You are planning a lesson – a maths lesson. You have a vague and somewhat passive understanding of the content you intend to teach. And so you think your learners will trust that you understand what input they are going to receive, because, after all, you are the teacher. Right? That passive understanding you have, is only going to brush off onto your learners. Children are observant and will easily pick up on your mistakes, your struggles and perhaps your lack of confidence when you are teaching them. So, you plan your lesson, still intact with your passive understanding of the content you intend to teach. Then it comes to your lesson and your learner outsmarts you. Perhaps in the form of a question, that you cannot answer. Is this due to your negligence?

Your learners depend on you to know what you are talking about, and here, Ma, explains the profound importance to approach your intended learning content with a conceptual view – if you understand, you have more chance of your learners understanding!

 

References

Ma, L. (2010) Knowing and Teaching Elementary Mathematics – Teachers’ Understanding of Fundamental Mathematics in China and The United States. London: Routledge.