The End Is Nigh!

The ‘Discovering Mathematics’ elective is finally drawing to a close and it is at this point I felt it appropriate to reflect on my development throughout this module.

Mathematics has always been an area in which I have found enjoyment, and in this I feel incredibly lucky.  Throughout the course of this module it has come to my attention how many people, and not just pupils but also teachers too, suffer from ‘mathematical anxiety’. This, of course, then becomes an area in which I must strive to work on throughout my practice so as to best address the issue and remove any mathematical anxiety from within my own classroom.

To do so, I turn to the work of Ma (2010) and the concept of a profound understanding of fundamental mathematics (which can be split into four characteristics; connectedness, multiple perspectives, basic principles and longitudinal coherence). This idea stems from the comparison between Chinese and American pupils that found those in China performing better in mathematics. Ma suggested that the observable difference in mathematical ability between students in America and those in China was due to a lack of fundamental understanding on the part of the teachers in America. From this it is clear that to teach maths in a way that is effective first the teacher must possess this understanding of fundamental mathematics.

But what have I really learned over this module? Of course it is obvious that I have clearly seen that to be able to effectively teach mathematics I need to posses a profound understanding of mathematics and that to develop this for myself I must continue to experience and explore mathematics in a variety of situations and contexts, and not to mention age ranges. I have also, as mentioned in frequent blog posts, learned that no matter where we look, we can find maths. Mathematics exists, in some form, within every possible environment and experience we can think of. This is something I must also consider in my future as a teacher, the interconnectedness between maths and other areas so as to be better able to engage and inspire learners.

It has also become clear to me that the use of active, rather than passive, maths within and without the classroom is vital in ensuring that every child is catered for and that nobody becomes disengaged.

The final area I have covered throughout the course of the ‘Discovering Mathematics’ module is enthusiasm from the educator. This is something myself, and many other ‘Discovering Mathematics’ blogs, have highlighted as lacking in the current school environment and Iddir (2015) states that the reason she began to feel excited about maths again was mainly due to the enthusiasm of the lecturer.

All these areas I must remember and continue to work on throughout my career to ensure that mathematics as a subject is taught in a way that is not only highly education but also engaging, active and inspiring.



It’s that time of year again. Everything seems to be Christmas themed, and not being one to rock the boat, I think it’s time to investigate the maths behind Christmas!

Snowflakes are something which from a young age have fascinated me. Although this may have been mostly due to their unparalleled ability to form snowballs which I could throw at people. None the less, as I have stated in previous blog posts; maths is in everything.  In particular I am going to discuss one snowflake. Koch’s snowflake. This snowflake is a fractal (a fractal is a shape that contains similar patterns which recur throughout the shape on a smaller scale) which is demonstrated in this video here.

As can be seen in the video, no matter how far we zoom into this snowflake there is always more detail to be revealed. An area like this, of course, opens up a variety of fundamental areas of mathematics which could be explored in the primary classroom (especially during the winter months) for example scaling, shape and ratio.

Kochs Snowflake

However taken a step further this simple snowflake becomes very interesting. Koch’s snowflake, as can be seen in the previously linked video is made up of triangles each a third size of the triangle it is attached to. Now, we know it to be true that no matter how far we zoom in, there will always be more detail revealed and as such we can state that to create this shape there are infinitely many triangles. Therefore we can assert that this shape has an infinite perimeter (this is because by adding up an infinite amount of triangles along its length we arrive at infinity). However Koch’s snowflake has a finite area, which can be clearly demonstrated by simply drawing a circle around the snowflake (we know the circle to have a finite area so therefore anything that can fit within the circle must have a finite area). We have now shown that the shape of a snowflake has an infinite perimeter but a finite area. Very strange!

This of course is a very abstract example of mathematics of which, unless it is actually snowing outside, the children are unlikely to find relevant and I’m sure you’re probably thinking that nothing about this idea is actually apparent in the real world. However, it is not merely just snowflakes which contain the property of infinite perimeter and finite area. Many other examples of this exists and is certainly an area that could be explored in the classroom to both find shapes that fit these rules but also why they fit these rules, which, of course, will require further exploration of fundamental mathematical ideas such as shape, ratio, division, and multiplication.

To get you started (in case it isn’t snowing when you next begin to think about fractals), despite how difficult it is for my mind to grasp the concept, Scotland is a perfect example of a shape which has an infinite perimeter but finite area.


I Need Some Space…

It hurts quite a bit when you stub your toe. Try being hit by a whole galaxy! Because that’s exactly what’s going to happen. The galaxy Andromeda (our closest spiral galaxy) is speeding towards us a massive 260,000 miles an hour…

Before you start planning for doomsday you might want to consider galactic distances. It is absolutely true that Andromeda is speeding towards us at an incredibly high speed, however, due to the gargantuan distance between us it will be more than 4 billion years before it gets anywhere near us. So you can stop breathing into a paper bag now.

This beautifully demonstrates the vastness of the universe in which we live. Something which we consider massive (take the earth for instance of which the diameter is 1.2756 × 10^7 or 12756000 meters in diameter) is tiny compared to the size of the universe (9.2 × 10^26 meters or 10 billion kilometres in diameter). This also illustrates the importance of fundamental areas of mathematics such as base systems. For example when using very large or very small numbers we use standard notation (used above) instead of attaching many zeros to a number to demonstrate its size. An example that shows why we switch when using extreme numbers is; imagine trying to give directions to Edinburgh using only millimetres. This would be very confusing, so instead we switch to using kilometres.


Pictured: Ngc 2207, Spiral Galaxy

However telling you that the universe is 10 billion kilometres wide does nothing really to explain just how massive it is. An area which I find fascinating is light years. This is the distance which light will have travelled within a year. As I am sure you are away, light travels unbelievably quickly (just think about how fast a room changes from light to dark when you switch a light on, it is near instantaneous). The actual distance light can travel in a year is 9.46 × 10^15 meters or 9.5 trillion miles. Now, the nearest star to our sun is Proxima Centauri, and I stress that this is the nearest to us. For light (the fastest moving thing we know of) to travel from Proxima Centauri to us would take 4 years (Yes, that does mean we see 4 years into the past when looking at Proxima Centauri). Now imagine how long it would take for light to travel across the universe! Or how long it would take for a human spacecraft to travel that far!

Before we were given an input on astronomy earlier this week, I knew very little of the content explored within this blog. This is something I think is a shame, as I’m sure this is an area I would have found hugely interesting as a child in primary, much more so than simply learning the names of the planets in our solar system and that they all go round the sun (though of course I concede this is vital knowledge to have).

As a future practitioner this is an area I would very much like to explore with my class. To learn about the solar system which we live in, but to also introduce the notion of fundamental mathematics within this subject area to further explore the world of astronomy and mathematics in an exciting and engaging way.

To Infinity…and Beyond!

Infinity. It’s about as big as it gets.

In our Discovering Mathematics input yesterday, upon discussing the vastness of the universe in which we live, the term infinity was mentioned. Simply put, infinity refers to something which never ends. A conceptual example of this would be natural numbers, of which, no matter how long you count for, there will always be more.

Greek mathematician Zeno of Erea is credited with being the first to investigate the infinity concept. Zeno’s exploration of infinity used an example of a journey to further understand the concept. He stated that to travel across a distance, first you must travel half the distance. So far not too difficult to understand. He then stated that from this halfway point you would then have to travel halfway again (this time half as far). Zeno claimed that even if this halfway process were completed an infinite amount of times, the destination could not be reached. Of course this exists as a paradox as we know that it is possibly to reach our destinations. A comparison could be drawn to the modern geometric progression that tends towards one (1/2 + 1/4 + 1/8 + …etc.) but never actually reaches the integer. Confusingly enough both of these processes involve an infinite number of steps to achieve a finite answer.


Much later in 1924, German mathematician David Hilbert designed a thought experiment to demonstrate the counter intuitive nature of infinite sets and further our understanding of the infinity concept. His example is know as the Hilbert Hotel paradox.

Consider for a moment a hotel which contains infinitely many rooms all of which were occupied by an infinite amount of guests. It would seem reasonable to assume that any newly arriving guests would have to be turned away. However this is where the idea of infinity becomes more complicated. Suppose a new guest were to arrive, we could (simultaneously) move each guest to the next room (Room 1 into room 2, room 2 into room 3 and so on in the fashion of moving n into n+1) therefore allowing room for the new guest. In this fashion it is always possible to accommodate a finite number of guests. But what if an infinite number of guests arrived? We know that there are an infinite amount of odd numbers, so the solution to accommodating an infinite number of new guests is to move our current guests into the infinitely even numbered rooms in the process of n into 2n, which will, in turn, free up an infinite number of odd rooms.


Hilbert’s explanation of infinity opens up the, frankly odd, concept that there are many different infinities, and some are bigger than others! This is most easily demonstrated by looking at the numbers between 1 and 5 and between 1 and 10. Due to the never ending nature of our decimal system there exists an infinite amount of numbers between 1 and 10, but there also exists an infinite amount of numbers between 1 and 5 and although the latter would appear to be half as big both sets have the same cardinality due to the presence of infinity. In fact, between every number you can think of there is an infinite amount of more precise decimals, which, in turn, means there are an infinite amount of infinities!

So, a vastly complicated area. However there are many applications of this concept, especially within the areas of physics and maths (although it could also be linked with philosophy, religion and language) and although infinity might be too complicated to be introduced lower down in a school it could certainly be broached with the older children, for example, as a way to better understand the universe in which they live.

It’s maths, Jim, but not as we know it…

When a string is plucked, it vibrates at a particular rate creating a note. A familiar concept to violinists, guitarists, cellists and…well anybody who plays a stringed instrument, and I’m sure anyone who has ever fired an elastic band across a room. The rate at which this string vibrates is known as the frequency (determined by string thickness, density, length and tension, f=(1/2L)√(T/μ)) which is measured in Hertz(Hz). As humans we perceive these frequencies within a given range as pitch. The range begins at about 20Hz and goes up to approximately 20,000Hz. For example concert A (the tuning pitch for orchestras) is 440Hz and middle C is 261.63Hz.


It’s probably at this point you’re beginning to wonder “In what way could this information possibly be useful?”.

Well, as many of you will know some notes sound better together than others (if you’re not aware of this, have a look at this video, about 3 minutes in). This has to do with how the two note’s frequencies are related. Notes whose frequencies are related by simple fractions tend to sound better together. Let’s take for example the note of A which we know to be 440Hz. If we half the frequency to 220Hz we arrive at another note of A only an octave down. Similarly if we double the frequency to 880Hz this is also the note of A, this time an octave up.

Taking this to the next level we can apply more complex fractions. If we take 3/2 of 440Hz we arrive at 660Hz or E and if we take 5/4 we arrive at 550Hz or C#. These are the fractional relationships between a major chord and by playing A, C#, and E we are actually playing the chord of A major. It is because of this fractional relationship we find harmonics (demonstrated rather nicely in this piece of music by Daniel Padim) situated exactly halfway, one third along, and one fifth along the string length of a guitar.

So, after all that, why would I write a blog about this? Apart from the fact that this is a very exciting application of maths. I suppose this was my rather long-winded way of demonstrating the often forgotten, cross-curricular nature of mathematics, and that perhaps (despite a very finely categorised curriculum) we should embrace the concept of learning in a way that accommodates individual difference and that allows for the blending of curriculum areas of which (Savage, 2010) describes as ‘sending a positive message to the children’ and as ‘rich pedagody’.

It seems that wherever I look there is always some aspect of maths to be found. Perhaps it could be combined with more curricular areas than just music…

Base Jumping

Imagine we had been born with 12 fingers. An odd thought I admit but allow me to explain. We work in a base 10 number system in mathematics which uses ten digits (0,1,2,3,4,5,6,7,8,9), but it wasn’t always this way and nor is it this way everywhere in the world. Binary, for example, is a base 2 number system and in the north of England a system known as ‘Yan tan’ exists for counting sheep which contains 20 individual digits and hence is a base 20 number system. We even switch to a different base when telling the time! (Base 60, 24, or 12).

So the question I am really asking is ‘Why base 10?’ Certainly it has been suggested that by switching to a base 12 number system containing 12 digits (0,1,2,3,4,5,6,7,8,9,Χ,ε) we could simplify the learning and teaching of maths in many areas such as fractions. For instance the fraction 1/3 in our current system is written 0.333333….(a recurring number). 1/3 written in base 12 would be written 0.4 and besides being far more aesthetically pleasing this decimal is much easier to use in calculations. It is because of this easier division of fractions we work in dozens when buying and selling food. The implications of this are that the conversion of fractions to decimal, a subject once feared by teacher and student alike, becomes not only easier to teach but easier to learn. The full extent of the advantages of a base 12 number system can be found here.

So if simpler number systems exist, again I ask ‘Why base 10?’ It comes down to 10 simple things. Our fingers. We have 10 of them. Our love of counting on our fingers has dictated the use of a second rate number system. I do concede that during the initial learning of mathematics being able to count on our hands is incredibly helpful. However in retaliation to this children could just as easily be taught to count on the joints of their fingers on each hand, which, luckily enough, there happen to be 12 of…


So, it would seem that a shift in our number system is in order. But is it really that much better? I have already, paraphrasing the Dozenal Society PDF (linked earlier in this post), stated that fractions would become easier to convert to decimal values however this is not entirely true. 1/5 in our current number system coverts to 0.2. 1/5 in the dozenal system becomes 0.41666667. Not exactly easy to work with.

In conclusion, is the slightly easier fractional system (and only in some cases) really worth completly upending our entire concept of number for? …Well, in my opinion, no. The transition for children already versed in our current system would be confusing and messy and I suspect we won’t be switching any time soon.


I Hate Maths

“I hate maths.”

I’ve heard it a hundred times before. In fact if you’re a parent or teacher or have anything to do with children, you’ve probably heard it too. But why? What is it about mathematics we find so dislikeable? To me, the answer is simple. We are not teaching maths.

Of course, there is a slot on our timetable blocked out for ‘Mathematics’ which we slog our way through at 9.15 each morning. Where our class learn the ins and outs of addition, multiplication, percentages and the like. But is that all there really is to mathematics?

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Frankly no. These areas are merely the foundations upon which the world of mathematics is built. The ‘grammar’ of the maths world, comparable to a score which when brought to life creates a beautiful piece of music, or a script waiting for the actors to transform into something awe inspiring, and although an essential part of mathematics there is so much more to maths than just this.

Unfortunately this ‘grammar’ of maths is all that children know and, as far as many are concerned, all that children need to know about mathematics. There is no room to allow for the creative side of maths. In fact, many are unaware that there even is a creative side to maths, and it’s no wonder!  Where is the creativity in learning our tables by rote? What innovation can possibly be found in mapping a small square on the left hand side of our page into a bigger square on the right hand side of our page?

Why not teach mathematics through its applications? Why not teach the exciting side of maths?

Could we not teach ratio by creating a scale drawing of a T-Rex on the playground in chalk? Why not teach multiplication by calculating the age of a tree (Diameter × Growth Factor for anyone interested)? Why don’t we introduce children to the Fibonacci sequence through spiral art and the composition of natural objects?

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“I hate maths.”

It was in fact this exact phrase which turned me on to teaching and is why I decided to take the mathematics elective. It is my goal to share my love and enthusiasm of mathematics to inspire those who I teach, so I never again have to hear the words “I hate maths.”