# Maths in time or time in maths?

Throughout the discovering mathematics module I’ve learnt a lot about how maths is all around us disguised in our daily routines. Most recently, I was amazed just how much maths is in time but not in a way you would normally think. We are all fully aware that part of the curriculum for maths is teaching time. However, it is claimed that mathematics began through the study of time, particularly when it came to recording sequences of events. For example, understanding the seasons is essential if you are to successfully grow crops. You need to consider the right time to plant the crops, when the rain will come, when the rivers will flood, when you should harvest the crops. Knowing the length of the year was of vital importance, yet much less visible from the timekeepers in the sky (daily passage of the sun and monthly phases of the moon), leading to calculation where maths links in. It then became necessary to count days and months which lead to the creation of calendars. The earliest evidence of timekeeping dates right back to 20,000 years. The evidence of this comes from markings discovered on bones and sticks, for example, the Ishango bone.

Back in the day, it was extremely important for the Egyptians to know when the Nile would flood. Therefore, this played a huge role in how and why their calendar developed starting from the earliest version of roughly 4500 BC which was based on months. 4236 BC saw the beginning of the year as the helical rising of Sirius, the brightest star in the sky. This helical rising was seen to be the first appearance of the star after there being a period where it was too close to the sun for it to be seen. This occurred in what is now our July, which was said to be the start of the year. Not too long after this, the Nile flooded making it a very natural beginning to the year! Therefore, the helical rising of Sirius would be a signal for people to prepare for the flood. This year was calculated to be 365 days (as it is today) and by 2776 BC it was known to this degree of accuracy. Consequently, a calendar of 365 days was created in order to record dates. However, a later more accurate calculation of 365¼ days was worked out for the length of the year but the calendar was not changed. Interestingly, the 2 calendars ran in parallel, the one being used for the practical purposes (i.e. sowing crops, harvesting crops etc.) was based on the lunar month.

In the modern world, units of time require some way of measurement,one of the earliest devices used to measure time involved the sun. 3500 BC saw the gnomon being used which consisted of a vertical stick, using the shadow to indicate the time of day.

Further along the line in 1500 BC, the sundial was used. The problem with the sundial being that the sun took a different path through the sky throughout the year. To ensure the sundial would roughly produce the correct time all the year round it had to be set at exactly the correct angle.

Around 300 BC the hemispherical sundial was introduced to develop the sundial into a more accurate instrument. Just before the rise 27 BC,  Roman architect Vitruvius was able to describe 13 different designs of the sundial in his book.

But what happened at night? As the sun will have gone down! As the sun couldn’t be used to tell the time at night, water clocks were used in 1500 BC. Water would run out through a hole in the bottom of a vessel, the inside of this vessel had lines to indicate passage of time. Earlier versions didn’t consider the fact that water ran out more slowly when the pressure dropped. Additionally, sand was used and still is in the familiar hour glass in which sand trickles from a container, taking a certain length of time to run out.

So is it maths that is in time or time that is maths? Who knows! But one thing we know for sure is that time played a very important part in the beginnings of mathematical understanding as we know it. To refer back to Ma’s PUFM and the 4 interrelated properties, this particular post involves longitudinal coherence. When teaching time we tend to just jump straight into how to tell the time on a modern day clock, however, the history of time can be used to lay foundations for when they go on to learn history but also not limiting the knowledge learnt to just the curriculum and taking into account the subject as a whole. Additionally, connectedness can be found here as I made a connection to a previous post about prehistoric mathematics, linking the mathematical concepts. When teaching, it is important to make learners see the connections in their learning to ensure it doesn’t become fragmented and to reinforce previous learning. Overall, I have gained a lot from this module about just how vast maths is and it has opened my eyes to just how limited the maths taught in school is, leaving out all the fun and exciting parts. One of the things I believe to be very important when teaching maths is relevance. So often we see children disengaged in maths because it is not “relevant to their lives”, that is why it’s so important that we shed light on this issue and make sure we make connections to the real world whenever we teach maths. Taking this on board to the future with my teaching, I aim to ensure that all learners see the connections between mathematical concepts in the world, to show them just how fun maths can be and how it interlinks to just about every subject! By doing this, I hope to banish ‘maths anxiety’ and share my love for maths.

# Maths and Music

Growing up maths and music were two of my most favourite subjects so I thought it was only right that I did a blog post about the links between them and the maths rooted in music. Having played the keyboard and piano since I was 8, I have a deep understanding of all things music, from the fractions involved in musical notes to counting semi-tones to find a chord. However, there is more maths in music than I ever realised which is a great discovery in terms of making maths enjoyable for learners and using different routes to teach it, for example through maths.

Firstly, in this weeks maths input I learnt about patterns in relation to music and maths which I found of particular interest after one of my previous blog posts on Islamic Tilings. Through my research, I discovered that music is an example of a hierarchical system as patterns are nested with larger patterns just like characters form words which form sentences, chapters and eventually a novel. Composers have been exploiting this concept of hierarchies for thousands of years, maybe even unconsciously, but surprisingly it’s only recently that these systems have been understood mathematically. This type of maths shows us that principles of musical composition are shared among diverse hierarchical systems (a form of patterns) which suggest there is still may more exciting avenues to explore.

Whole numbers and fractions are everywhere when it comes to music and are especially important as it enables musicians to read music and interpret the length of a beat easily. This is crucial when playing in an orchestra as everyone needs to stay in time with each other. So to give a bit of background for all the non-musicians out there, each note has a different shape which indicates its beat, length or time. Additionally, notes can be classified in terms of numbers. Whole notes consist of one note per measure, half notes – two notes per measure, quarter notes – four notes per measure, eighth notes – eight notes per measure and sixteenth notes – sixteen notes per measure. These particular numbers are used to signify how long the notes will last. For example, a whole note would last an entire measure whereas a quarter note would only last a quarter of a measure, therefore, there is time for four quarter notes in one measure. Which is expressed mathematical as 4 x 1/4 = 1. Furthermore, there is a dotted note which involves placing a dot after a note lengthening it by a half of that note. For example, a quarter note with a half would be held for 3/8 of a measure, expressed mathematical as: In conclusion, it is essential for musicians to understand the relationships and values of fractions in order to hold the note correctly, therefore, linking maths and music!

Finally we come to Fibonacci as believe it or not, it also links to music (and patterns!). As I’m sure you know, the Fibonacci sequence is a famous and well-known sequence that follows as: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89… and so on, where it uses the pattern of adding each term to the one before it to create the next term. For example, 5 + 8 = 13, 8 + 13 = 21, 13 + 21 = 34 and continuing to infinity. In music the Fibonacci can be seen in piano scales which is something I used to dread as a child (they’re no fun to learn). For example, the C scale consists of 13 keys from C to C (eight white keys and five black keys, with the black being arranged in groups of three and two. Additionally, the ratio between each term is very close to 0.618, which is known as the golden ratio.

I could continue on for pages about all the links that are present among maths and music. I find it particularly interesting as I had no clue just how many links can be made. In relation to Liping Ma’s 4 interrelated properties this particular post refers to longitudinal coherence as achieving a fundamental understanding of the whole curriculum you are prepared as a teacher to use any opportunity that may present during learning to review any critical concept where needed. For example, when teaching a music lesson if the opportunity where to arise, you could go over a mathematical concept to reinforce students previous learning. For future reference, what I have learnt through writing this blog post and my research will be taken into account when teaching as I think it will be of great benefit to the children to learn maths through music as for many it is their favourite subject whereas maths can be considered boring or difficult.

References

Buehler, M.J. (2014) The universal patterns of nature. Available at: http://www.slate.com/articles/health_and_science/new_scientist/2014/02/patterns_in_music_and_the_natural_world_creating_stronger_spider_silk_and.html (Accessed: 4 November 2016)

Glydon, N. (no date) Music, math, and patterns. Available at: http://mathcentral.uregina.ca/beyond/articles/Music/music1.html (Accessed: 4 November 2016).

# The Importance of Zero.

After yesterday’s lecture on place value, I was left feeling some what baffled by the binary system as even though we tried hard to crack it at our table, it wasn’t until I watched a video explaining it that it all made sense. Studying number systems closely over the past couple of weeks has sparked an interest in me for the particular digit that is zero. When making our own number system, zero was the one to cause a havoc for us as its importance is not always seen. As a matter of fact, the invention of zero was one of the most important breakthroughs in the history of civilisation.

Zero has in fact baffled many who have studied and developed mathematical theories as how can nothing (i.e. zero) be something? First, lets start with where zero came from. It all started in 520 AD with the Indian Aryabhata who used a symbol he called “kha” as a place holder which has believed to have been the concept of zero. Brahmagupta, who lived back in the 5th century developed the Hindu-Arabic number system which interestingly included zero as an definite number in his system. Furthermore, other mathematicians such as al-Khwarizmi and Leonardo Fibonacci developed the concept of zero in the number system. This concept reached the Western society during the early 1200’s.

Now we’ve got some background on where zero actually originated we can look into how important zero is. As I’m sure you know zero is the number where negative numbers on the left stretch to infinity as do positive numbers on the right. Therefore, it is neither positive nor negative, hence why you see zero as the pivotal point on thermometers, the origin point for bathroom scales, coordinate axis, etc. Additionally, zero is extremely important as its value in place holding. For example, when writing five hundred and two, how do you do so that you understand that this number has no tens. Of course, you can’t just write it as 52 as this is in fact a completely different number, hence, proving the importance of zero.

Another key element when it comes to the importance of zero is the “additive identity element”. It may sound confusing but in actually fact it’s very simple. It simply means that when you add zero to any number, you get the number you started with. For example, 7 + 0 = 7. Now, I know this may seem obvious but it is very important to have a number like this. For example, if you’re manipulating some numerical quantity and you want to change its form but no its value you might add some fancy version of zero to it.

As we’ve found out in this blog post, zero is important and can sometimes be overlooked as without it what would we do? Not only does zero fulfil a central role in maths but as the additive identity of integers, real numbers and other algebraic structures as well as being used as a placeholder in place value systems. The whole concept of zero and number systems links in with my previous blog post on the Ishango bone as this is believed to have been the first piece of evidence we have where we started using numbers. All this links with connectedness as mentioned in a previous blog post as one of Liping Ma’s 4 properties of fundamental mathematics and as zero is one of the fundamental concepts behind maths, it all ties together nicely! The importance of zero is something I will keep in the back of my mind when teaching maths to my pupils as I believe it would be enjoyable for learners to try and use maths without zero.

The importance of the creation of the zero mark can never be exaggerated. This giving to airy nothing, not merely a local habitation and a name, a picture, a symbol, but helpful power, is the characteristic of the Hindu race from whence it sprang. It is like coining the Nirvana into dynamos. No single mathematical creation has been more potent for the general on-go of intelligence and power.” G.B Halsted

# Islamic Tilings

(Islamic Tiles Pattern)

Who knew maths could be so pretty and creative? Symmetry has to be the most significant and elegant connection between the boundaries of art, science and maths. These patterns create a visual language expressing order and generating appealing, fascinating compositions and it’s my favourite part of maths as it often surprises people! Islamic art is dictated by extravagant geometric decoration expressed by texture, colour, pattern and calligraphy. Remembering that the exquisite designs are not purely decorative are key to its understanding. They, in fact, represent a spiritual vision of the world- ‘Unity of God’. There is 3 fundamental shapes used in Islamic Art, the equilateral triangle, the square and the hexagon. In this post, I have created a short video showing you how to create a hexagon from a rectangular piece of paper. In Islamic Art, a hexagon represents heaven.

How to make a hexagon

Now that you’ve made a tile, you repeat that over till you have as many as you need and now it’s time for the tessellation process!

As seen in the photo below, you can fit the hexagons together in different ways till you get the tessellation you like. You can move the hexagons around at different angles to create different designs.

Tessellation is a great way to instil a love for maths into the children we teach as it combines maths along side many children’s favourite subject, art. I will definitely use what I have learnt about tessellation to show children that maths can be fun and creative. You can also show them how tessellation is used all over the world in buildings and even see if they can find tessellation in their world. This post refers to one of Liping Ma’s principle – connectedness as whilst learning how to make polygons and all about different shapes they also learn how they fit together to make tessellations. Therefore, students are learning a unified body of knowledge rather than just the topic of tessellations as it also connects to learning about shapes. This allows children to make connections to ensure their learning doesn’t become fragmented.

# Prehistoric Mathematics and the Ishango bone

(One of the oldest surviving fragments of Euclid’s Elements, dated to circa AD 100)

Prehistoric Mathematics. Sounds boring, right? Wrong. Now I’m not one for history but after our workshop last week on the origin of number systems, I decided to look into the history of maths and number systems a bit more. If you (like me) think that prehistoric maths is something to that will put you to sleep, be prepared to be amazed as it really opened my eyes. So if you thought our current understanding maths was born overnight, you’re wrong as it actually toke several ages and to this day we are still discovering new mathematical concepts as maths research is very much ongoing and ever-growing. In fact, numerals are approximately 5,500 years old!

Back in the day, our prehistoric ancestors actually knew the difference between say one stick and two sticks, all through instinct as believe it or not school wasn’t around back then. However, the massive leap in intelligence from the idea of having 2 objects to the invention of a symbol (i.e. 2) or a word for this abstract idea (i.e. two) did take numerous centuries. Even in this day and age there are a few remote hunter-gatherer tribes in Amazonia whose number sequence simply consists of “one” “two” and “many”. Similarly, there are other tribes who only have words for up to five. How bizarre is that? Imagine trying to bake a cake using a recipe that just says ‘many’ eggs. However, without settled argiculture and trade there is, naturally, no need for a formal system of numbers like ours.

The earliest evidence of mankind thinking about numbers that we have today is the Ishango bone. The Ishango bone consists of notched bones from central Africa dating back to 35,000 to 20,000 years ago. However, as you can see from the image below, it was mere counting, what we would call tallying rather than mathematics.

(The Ishango bone)

What’s interesting is this bone was found in 1960 by Belgian Jean de Heinzelin de Braucourt among the remains of a small community in Africa that had been buried in a volcanic eruption. Whilst some interpret the bone to have that of tally marks, others suggest that the scratches may in fact be for getting a better grip on the handle or some other non-mathematical reason, it’s impossible to know and is down to individuals beliefs. Personally, I’d like to believe that the implement was used the construct a numeral system.

When analysed you can see that the bottom row (from right to left) begins with three notches and then doubles to 6 notches. The process is repeated for the number 4, which doubles to 8 notches, and then reversed for the number 10, which is halved to 5 notches. These numbers may not be purely random but could suggest some understanding of multiplication and division by two. Therefore, this bone may have been used as a counting tool for simple mathematical procedures.

In the book How Mathematics Happened: The First 50,000 Years, Peter Rudman disputes that the development of prime numbers could have only arouse after the concept of division, which he dates to after 10,000 BC, with prime numbers probably not being understood until about 500 BC. Additionally, he writes that “no attempt has been made to explain why a tally of something should exhibit multiples of two, prime numbers between 10 and 20, and some numbers that are almost multiples of 10.”

When Alexander Marshack examined the Ishango bone, he believed that it may represent a 6-month lunar calendar. However, Judy Robinson argued that he over interpreted the data and the evidence doesn’t suggest a lunar calendar. Furthermore,  Claudia Zaslavsky suggested it that this might indicate the bone was created by a woman tracking the lunar phase in relation to the menstrual cycle.

Moving on, mathematics properly developed as an acknowledgement to bureaucratic needs as a result of settlement and agriculture development (i.e. measurement of plots of land, tax, etc). The first time this occurred was in the Sumerian and Babylonian civilisations of Mesopotamia and in ancient Egypt.

To link back to more modern day mathematics, I started reading ‘Alex’s Adventures in Numberland’ which I highly recommend as it provides some eye opening moments and shows you just how maths isn’t boring and mind-boggling but in fact is fascinating and underpins our everyday lives. I can guarentee you will learn something new. For example, I didn’t know that every number can be winnowed down to a product of prime, try it for yourself! There is some more interesting theories as well that are still to be proven such as the Goldbach Conjecture (every even number bigger then 2 is the sum of 2 primes).

So to link this all back to teaching as I’m sure you are wondering why on earth any of this will be relevant if I will never teach it in the classroom. Well, it all links to Liping Ma’s 4 interrelated properties. This post refers to ‘longitudinal coherence’ as by not limiting your knowledge to whatever is in the curriculum (i.e. prehistoric maths is not in the curriculum) you allow yourself to use every opportunity to learn about these crucial elements of maths and you never know you might even enjoy it! Additionally, there was a little bit of multiple perspectives in here as I considered all the various approaches to the origin of the Ishango bone which allowed me to have a flexible understanding of this concept. Surprisingly, I actually enjoyed discovering the understanding of maths long before my time and it’s encouraged me to open my eyes to a range of different mathematical topics that are out there rather than just the curriculum as I believe it will strongly benefit my teaching in later years as well as my overall understanding of mathematics.

References

Bellos, A. and Riley, A. (2011) Alex’s adventures in numberland. London: Bloomsbury Publishing PLC.

Mastin, L. (2010) Prehistoric mathematics – the story of mathematics. Available at: http://www.storyofmathematics.com/prehistoric.html (Accessed: 10 October 2016).

Rudman, P.S. (2006) How mathematics happened: The first 50, 000 years. United States: Prometheus Books.

# Profound Understanding of Fundamental Mathematics

To be honest, I hadn’t heard of PUFM (profound understanding of mathematics) until my first workshop for the math’s elective. I couldn’t even tell you one of the principles let alone four. However, now that I have done reading into Liping Ma’s research, it all makes sense. PUFM is essential if we as teachers are to enable learners to really get a grasp on mathematics.

So what is PUFM and why is it important? PUFM is more than just being able to understand elementary mathematics, it is being aware of the theoretical structure and attitudes of mathematics that are the foundations in elementary mathematics. Furthermore, being able to provide this foundation to instill those attitudes into learners. Having a PUFM is essential when teaching as in order for a learner to understand a concept, the teacher must fully understand it first both conceptually and procedurally.

Elementary mathematics is a range of depth, breadth and thoroughness. Teachers who are successful in achieving this deep, vast and thorough understanding will be able to reveal and represent the connections that lie between and among mathematical ideas in their teaching. These 4 interrelated properties are the principles that lead to these different aspects of meaningful understanding in mathematics:

Connectedness refers to connections made among mathematical concepts and procedures, anything from basic connections between individual strands of knowledge to more complex, underlying connections that link mathematical operations. When put into practice, this should ensure the learning that takes place doesn’t become fragmented. For example, instead of learning isolated topics, learners will engage with a unified body of knowledge.

Multiple Perspectives refers to when a teacher has achieved the PUFM and can appreciate different facets to an idea or various approaches to a solution including the advantages and disadvantage. Additionally, they can provide mathematical explanations in relation to these facets and approaches. In teaching, this can lead to learners having a flexible understanding of a concept.

Basic Ideas refers to displaying mathematical attitudes and being aware of the “simple but powerful basic concepts and principles of mathematics”. Teachers with PUFM will tend to revisit and reinforce these basic ideas. This leads students to be guided to conduct real mathematical activity.

Longitudinal Coherence refers to not limiting the knowledge that is being taught as you will have achieved a fundamental understanding of the whole curriculum. Therefore, you are ready to exploit any opportunity in order to review crucial concepts. Additionally, they have an understanding of what students will go on to learn later and take these opportunities to lay a real foundation.

All four of these principles are equally as important when gaining a sound understanding of elementary mathematics. This is extremely important when teaching mathematics as how can you teach and inspire young minds without having a deep understanding yourself? Not only is this relevant to maths but it is something I will keep in mind when teaching any subject as each of these principles can be applied to different topics. I found Liping MA’s research extremely eye opening and I will continue to use these principles and keep them in mind when bringing new mathematical concepts into my future classroom.

# Why does everyone hate mathematics?

“If I had 50p for every time I failed a maths exam, I’d have £6.30” (Twitter)

It has recently occurred to me that maths gets such a bad reputation amongst learners of all ages and as someone who enjoyed maths at school it’s hard to pin point why. After I stumbled upon this tweet it really got me thinking about why it is that children dislike maths and consider themselves ‘maths dyslexic’. I consider it an important issue in our education system and has been for a while as maths being hard or boring isn’t a new opinion. This is why one of my goals as a teacher is to instil the love of maths I have into the pupils I will teach. I am strong believer in ‘teachers who love teaching, teach children to love learning’ which applies to every subject whether the teacher in question likes it or not.  A pupils’ love for a subject tends to stem from the way the teacher taught the subject. Therefore, it’s important to never forget the influence you have over a learners enjoyment for subjects.

Sometimes a learners believes that mathematical achievement is due to factors beyond their control such as luck rather than their own hard work (locus of control). These learners may think (for example) that if they scored well on a maths test it is because the content happened to be easy. Additionally, they believe that if they fail it is due to either the lack of innate mathematical inability or level of intelligence. By viewing their achievement as ‘an accident’ or lack of progress inevitable leads to them limiting their capacity to do well and study productively.

Attention span is another factor that can contribute to a dislike for maths. Students can be mentally distracted and can have difficulty focusing on problems especially when they involve multiple steps. For example, dealing with long-term projects or more than one variables can interfere with their achievement. An effective teacher should use attention grabbers such as visual aids. Additionally, students who can work together can help each other stay on task, however, this can also have the opposite effect!

The last factor I am going to talk about is understand the language of maths. It is often true that students can become confused by words that have some sort of special mathematical meaning. For example, “volume”, “power”, “area”, “divisor”, “factor” or “denominator”. Terms such as these can seriously hinder student’s abilities to focus and understand these terms for mathematical operations. Memorising such terms without context that they can relate to is not productive. That’s why it is important when teaching a new topic in maths that involves new terms that the meaning is clear and something they can comprehend. For example, when introducing estimation many learners won’t understand it until you relate it to a situation they have experienced such as when the opening of a local shop is delayed (the estimation was a bit off).

These are just a few of the many factors that can effect how a learner perceives maths. Therefore, it’s our jobs as teachers to push past these factors and adjust their learning environments to instil a love for learning maths in the classroom for all the learners, whoever they are. As part of my journey to becoming an effective educator, I would like to do further research into why maths causes such anxiety amongst learners.

# Why are people afraid of Maths?

Following the 2 inputs we’ve had so far on Maths, I’ve decided to write this post as part of the TDT. Personally, I have no anxiety at all regarding Maths, it’s a subject I’m totally comfortable with and would choose to teach it instead of any other curricular area any day of the week. I’m not quite sure where my love for maths came from as neither of my parents really share my enjoyment for it and even my brother wasn’t all that keen on maths at school. All through Primary School I was okay at Maths, never good nor bad just doing fine. Then all of a sudden in third year at High School I undertook Standard Grade and all of sudden I realised how much I liked it and that I was actually really good at it. So much so that I even took Advanced Higher which many of my friends couldn’t understand as that would be their worst nightmare!

There’s just something about doing calculations and solving problems that I really enjoy. There’s always an answer. I don’t feel the need to guess on a whim because I can always find the answer even if it takes me 50 attempts and when you do, there’s nothing more satisfying than having got to the right answer eventually. Maths is everywhere, it’s in everything you do such as counting the change you’ve got or telling the time. It’s definitely an important part of the curriculum and always has been, I have strong memories of reciting my times tables to my Mum as part of our homework.

Something I found really interesting in the inputs was how being illiterate is seen as being unacceptable (which is completely understandable) and yet it’s acceptable to be innumerate? Its definitely not. Everyday life would be extremely difficult if you were innumerate, I don’t think people realise quite how important Maths is. This is why it’s extremely important to inflict an enjoyment for Maths on children from the moment they walk into a classroom for the first time. I can see why people are afraid of Maths because it can be very daunting especially if all your friends can do it and yet you just can’t seem to get a grasp on it. This could be partly down to influences from home as it seems to be very common nationwide to have a fear of Maths. As I have a passion for the subject, I feel like it is my job to make sure the pupils I teach aren’t afraid of Maths and I can make it as enjoyable as possible.

However, teaching Maths is a whole other ball game. To be perfectly honest, I would have no idea where to start on how to make Maths enjoyable, especially as it’s compulsory which can make it all the more scary. When teaching Maths I will try to demonstrate how fun Maths can be and hopefully the pupils will pick up on my enthusiasm! My goal for placement regarding Maths would be to try and find innovative ways to teach it that will spark an enjoyment for the children. How I will do this, is something I will have to figure out in the mean time!