Tag Archives: fundamental mathematics

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

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:

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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:

12312476_1068113746541565_938711779_n

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.

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