Tag Archives: Discovering Mathematics

Maths, this is not the end – I promise

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.

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…

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.

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

Beautiful Maths – Mathematics is beauty, literally

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During the course of the Discovering Mathematics module, I have grown to have a more active awareness and somewhat alertness about the Golden Ratio…

1.61803398875

It appears as a number we may dismiss, but its essence is something astonishing and that is, the Golden Ratio is a measurement buildings and other objects use to be proportioned and it is suggested and perhaps proven that if something is proportioned in accordance to the Golden Ratio, it appears more attractive.
The Golden Ratio primarily is the Fibonacci Sequence in practice, if you like. Fibonacci was an Italian Mathematician and the Fibonacci Sequence a theory of numbers, sequencing and ratio, used in architecture, buildings, as well as seen in nature (have look at pine cones). The sequence…

1, 1, 2, 3, 5, 8, 13, 21…

It is simple, arguably, to continue the sequence when you understand the theory behind it:
1+1 = 2
1+2 = 3
2+3 = 5
3+5 = 8
8+13 = 21…
Shown here is the sequence and continuation of calculations which construct the Fibonacci Sequence. Each number is put into the next calculation and so on.

This takes us onto the Fibonacci Spiral.

(Picture from: http://i.stack.imgur.com/eHWK9.png)

This is the Fibonacci Spiral drawn out and this is where the connections may make sense between maths ans beauty. Buildings, flowers, paintings and other art works are proportioned based on this mathematical concept and theory that using this ratio and scale will make things appear more attractive.

Here is a link if you would like to look further into the beauty and the art of maths and design:
Dynamical Systems by BBC News – http://news.bbc.co.uk/1/hi/sci/tech/7617191.stm

References

Bourne, M. (2015) The Math Behind the Beauty. Available at: http://www.intmath.com/numbers/math-of-beauty.php. Last Accessed: Dec 5 2015.

Image from – http://i.stack.imgur.com/eHWK9.png

Look around you!

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You’re probably thinking how monotonous it is that I continue to repeat this, but maths is everywhere! Again.

I will never lose the amazement or curiosity I have filled with, at the fact that maths is the fundamental principle behind the creation and design of many things – and, much to your shock, as you are about to discover, it’s even on your face! Keep updated on my blog and have a look at my next blog post if you want to know what I mean by this. But really, your face is maths in practice.

In my last maths blog post: There’s no avoiding it – Maths is everywhere! (you can find this at: https://blogs.glowscotland.org.uk/glowblogs/teachingjourney/2015/11/17/theres-no-avoi…-is-everywhere/)… I quite clearly conveyed my astonishment as I was discovering the honest truth that maths is everywhere. So, now it is my turn to shock you. Here are just a few places you’ll find maths…

Have a look at the tiles, perhaps in your kitchen or bathroom. This can be on the walls or the floor – if it’s the flooring, it may be wooden.
Like pineapples? If you do, you’re one step further. If you don’t like pineapples, look at a bar of chocolate.
If there happens to be a football kicking around – pardon the unintended pun – then you’ll find maths on that.
So that takes you outside – where you will see maths everywhere, but have a look specifically at cobbles, slabs or bricks on the pavements or roads – it’ll be there.

And you’ve got your answer…
Tessellation is a mathematical concept which the construction of a multiple number of identical copies of one shape. I exaggerate ‘identical’, as this is the reason tessellation occurs. For a shape to become a tessellation, they must be the same size and shape, to fit more than one copy together.

Oxford University Press (2015) defines ‘tessellation’ as:
“An arrangement of shapes closely fitted together, especially of polygons in a repeated pattern without gaps or overlapping.”

To demonstrate this and lay it out, I have drawn a picture showing tessellation:

In this picture, you can see the triangular shapes drawn are clearly equal in size and they touch with no gaps between each shape. I could have continued drawing triangles until the page was full, but I wanted to write about it instead! Tessellations also work with hexagons, squares and many more. Any comments with how many shapes you can think of which can tessellate, would be great!

To further explain how tessellations work, below is what a tessellation is not:

In the above drawing are four circles, equal in shape in size. So, they are equal in shape and size – shouldn’t they tessellate when they are drawn next to one another? Well, no – looking back at the first picture, there are no gaps between each shape. Now looking at this picture, you can see gaps between each shape = no tessellation.

These are basic examples. More abstract designs using two or more different shapes can still tessellate, because they can be in order and start to design a pattern.

The fundamental mathematics behind tessellations is the shapes, sizes, scaling and quantity. The most basic idea is shape. In order to begin to tessellate a shape, you need to know the number of sides the shape has. For example, if I, at random, chose the circle to tessellate then began drawing it, I would soon discover it does not work – this is because it has one edge which is rounded. Therefore, clarifying my point that the fundamentality behind tessellations is shape.

UPDATE
A great discovery I have made… Have a look:

Harris, A. (2000) The Mathematics of Tessellation. [Online]. Available at: http://ictedusrv.cumbria.ac.uk/maths/pgdl/unit9/Tessellation.pdf Last Accessed: Dec 5 2015.

References

Dickson, R. (2015) There’s no avoiding it – Maths is everywhere! Available at: https://blogs.glowscotland.org.uk/glowblogs/teachingjourney/2015/11/17/theres-no-avoi…-is-everywhere/

Oxford University Press (2015). Available at: http://www.oxforddictionaries.com/definition/english/tessellation?q=tessellations. Last Accessed: Dec 5 2015.

Nothing but anxiety: Maths anxiety explained

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Yes, ‘maths anxiety’ is real. Not only is it real, it is more common than you may think.
Many people I know go by saying, ‘I hate maths’ or when doing maths work, will say, ‘I don’t get it’ or ‘I’m confused’. I would say, I hear this more during maths than I do in any other subject. So I’m here to get to the root of the problem.

What is maths anxiety?

Cognitive Psychologist Mark H. Ashcraft provides a definition,
“Math anxiety is a phenomenon that is often considered, when examining students’ problems in mathematics. Mark H. Ashcraft defines math anxiety as “a feeling of tension, apprehension, or fear that interferes with math performance.”

I would best describe ‘maths anxiety’ as feeling fear, pressure, worry and lack of confidence when thinking about or doing mathematics.

Why do we experience anxiety about maths?

I think this is built up worry and anxiety due to past failure and mistake-making, or simply lack of confidence in ability – which again, could be caused by past failure or mistake-making. Making comparisons of yourself to others can also influence your attitude towards maths, just like it would in any curricular area. I think the only way to get to the bottom of maths anxiety is to change your outlook on it and realise that mistake-making, if that is the root of your maths anxiety, is permitted and it is expected. In fact, mistake-making can be the only indication of your learning and areas of progression. No learner should feel anxious because of this!

As the teacher, the educator, the supported and the motivating role, we must first battle our fear of maths. If we are confident in learning and teaching mathematics, I strongly believe our learners will too! Would you be comfortable with an anxious driver teaching you to drive?

I believe there are a number of factors behind this, but they go under two headings – the learner and the adult:

LEARNER – ethos, motivation, confidence and perceptions of maths comes from the educator’s
approach to maths
– negative attitude
– disengaged, unmotivated, uninterested in the learning
– insecure understanding
– fear or worry about mistake-making
– lack of confidence and little self-esteem regarding ability

ADULT – passing on of negative views and beliefs onto children (parents, teachers etc.)
– parents or others believing maths is ‘irrelevant’ or a ‘waste of time’; often the
questions are asked, ‘why are we doing this?’ or ‘when am I ever going to need
to know this?’ – passed onto the children
– lack of confidence regarding dealing with finance etc.

How do we overcome maths anxiety?

Maths anxiety, I believe, may always occur in some people – children and adults. I say this because, in maths, you can be confident in one area and not in another. It is the responsibility of us – the teachers, the educators, the supporters – to provide encouragement, motivation, support and guidance to our learners. Most importantly, it is crucial that we educate our learners about the positive of making mistakes or errors – this gives the teacher and the learner a good indication of where to take the next steps. Then it is up to the adult to give quality guidance to ensure progression in every learner.

References

Ashcraft, M. H. and Kirk, E. P. (1999) The Relationships Among Working Memory, Math Anxiety, and Performance. Available at: http://www.apa.org/news/press/releases/xge1302224.pdf. Last Accessed: Dec 6 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.

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