# Developing My Understanding of Mathemetics

Throughout my life I have always been quite anxious about maths. In primary school I was that child who would try and copy other people’s work and would be too scared to ask my teachers for help. Progressing onto secondary school, my confidence in maths was quite still quite low and, although I got the grades I needed, I would have a panic.

When I walked into the maths department on the day of results and told them I got an A in my second attempt at Intermediate 2 all the teachers cheered my favourite teacher, Miss McCutcheon, gave me a big hug. But I think this was because they’d finally be getting rid of me and my Negative Nelly approach to maths!

When starting my first year placement the thought of teaching maths to a P7 class petrified me. Of course I’d know how to do the maths that I’d be teaching them, but it was the panic of “how am I going to teach these children something that would have taken me so long to grasp when I was their age?!” But I learnt almost instantly that the more confident I was in front of them the more confident they were and with links to real life it made the lessons a lot more understandable.

I remember one lesson in particular and it was about nets of 3D shapes. I brought in tissue boxes and a Toblerone box and it was funny how engaged they were with such basic shapes. But being able to see real life examples proves the importance of maths and that we use it all the time.

This leads me on to say that the teaching of maths is more than just adding and knowing your times tables. It is about making it relatable to real life, engaging and fun and through studying and working through this elective I have seen such a huge variety of real life examples of maths.

Liping Ma (1999) also helps us understand the importance of having a Profound Understanding of Fundamental Mathematics (PUFM). She had four concepts that she believed that “elementary” teachers should have. She titled them basic ideas, longitudinal coherence, multiple perspectives and interconnectedness.  (Ma, 1999, pp.122)

She explains basic ideas as the idea of being able to appreciate the more simple, but impressive, forms of mathematics. This is vital in teaching maths to children as although it might be straightforward, it will be something which will help expand their understanding of other topics.

In my opinion, longitudinal coherence is the most important property Ma found. This is when a teacher knows that their pupils have a good understanding of a specific topic when they’re ready to rather than knowing it by a certain age. I feel that so many people are concerned about pupils passing and knowing how to do aspects of maths for statistics and league tables and not having pupils who are happy and confident in their understanding of maths.

Ma explains multiple perspectives as teachers and pupils being able to see that there can be a variety of ways of reaching a solution with advantages and disadvantages to them. This is something I noticed on placement as I needed to teach pupils how to understand different concepts in a different way than I’d previously explained and that there is usually more than one way to an answer.

Finally, she explains interconnectedness as topics within in maths and outwith maths being related. Through studying this elective I have seen this clearly with subjects like art, music and science and this allows me to develop lessons to allow them to relate to more than just one subject.

Studying Ma’s theory throughout this Discovering Mathematics elective has allowed me to see the importance of having a PUFM. As I previously stated, maths is more than adding and Ma’s theory clearly shows this. This is most definitely something I will apply to my professional development and I can safely say that my maths anxiety is something of the past.

# Time Goes By…

I’ve never really thought about time unless it’s how long my garlic bread is going to take in the oven or how much longer I need to wait on a train. But there is almost a kind of science behind time and it is actually something which influences the way the world works.

Something that we were asked was if animals could tell the time. Initially, I thought this was a ridiculous question. Of course they can’t because they can’t even read. But, animal can tell what time of year it is. Now is a perfect time to see this for ourselves as we can see birds in their arrow formation flying south for the winter to warmer climates to aid their survival.

We were also asked why it is important to humans and this list could be almost endless. We use time in everything we do. We need to know when to get up in the morning to stop us being late for lectures, for example. However, those who attend Christ Church College in Oxford can have that extra five minutes in bed we all want when that dreaded alarm goes off. The college stills keeps to tradition by tolling the Great Tom bell signalling 9am at five minutes past nine. This is similar to the clock at the Balmoral Hotel at Waverly station in Edinburgh. This is three minutes fast so if you look at it and you’re running late for your train you, hopefully, should not miss it!

Historically, the use of time was vital. Standardisation of time did not really come into play until the 17 and 1800s. This was when Britain first began trading with a variety of different continents and the time each ship would set sail was key in order for the goods from this country reaching a variety of places across the world. An important location for Dundee was India with the Jute industry with trading going in both directions.

We also change our clocks twice a year. The way I like to remember which direction we should do this in is spring forward, fall backwards”  as American refer to Autumn as fall. When we spring forward this is commonly known as “Day light saving time”. It was originally put in place in the United States during World War I and World War II in order to take advantage of longer hours of daylight and save energy for the war production. (Wonderopolis, no date)

When thinking about Ma’s idea of having a PUFM the concept she named Basic Ideas” comes to mind. A teacher who understands the Basic Ideas acknowledges the straightforward but impressive topics and principles of mathematics. The idea of time, in my opinion, is very basic. It’s something we are taught from a young age and is something that surrounds us and influences what we do on a day to day basis.

With regards to my professional development I want to ensure that time is something that is taught throughout a child’s time at school and show the importance of it historically and how it influences them now.

References:

• Ma, L (1999) Knowing and Teaching Elementary Mathematics: Teachers’ Understanding of Fundamental Mathematics in China and the United States. 2nd New York: Taylor & Francis.
• Wonderopolis (no date), Why do we change the clocks twice a year?, Available at http://wonderopolis.org/wonder/why-do-we-change-the-clocks-twice-a-year (Accessed 14/11/16).

# 3..2..1.. Blastoff!

We were lucky enough to have Simon from Dundee Science centre come into our workshop and talk to us about Space. Simon’s passion and enthusiasm for our solar system and beyond was extremely motivational and has sparked an interest for me to research about how space can be taught in our schools.

Space is something which I have never really studied and this lecture inspired me to look into the subject in more depth for my own professional development. I believe that studying space in the primary curriculum is a fantastic opportunity to plan for cross curricular lessons. Space can be used in art and design, drama, history and, most importantly, maths!

Space is something which has been studied for thousands of years and will be for thousands more due to its sheer size. It is important to consider Ma’s concept of “multiple perspectives” when studying the solar systems and its relationship with maths. When Ma refers to multiple perspectives she wants people to understand that there are different routes to an answer. This links greatly into the teaching and understanding of space due to such a huge variety of people visiting the moon etc and finding out life changing facts and reaching their answers in different ways.

There is a huge link between maths and space and there is a  wide variety of things that pupils can be taught to link them together. When people associate space with mathematics I assume that they feel the same way as I do. Confused. I always just think that maths in this context is about extremely confusing equations and really long numbers.

This is not necessarily the case. When I teach space in the classroom I will remind myself that I am not an astronaut and neither are my pupils (yet!)  Pupils in the upper stages can be taught about speed, distance and time and this could be related to astronauts going out of the earth and to the moon/space. Younger years could possibly be taught about the names of the planets and ways to remember them.

The NASA website has a lot of good lesson plans which are extremely helpful and child friendly which can be found here: http://spaceplace.nasa.gov/math-activities/en/

Therefore, there are huge links between mathematics and space and this is something which can be made enjoyable in the primary classroom.

# Has Maths Ruined The Election?

On Tuesday Americans over the age of 18 had the opportunity to vote for who they would like to see as their next president. They had the choice between a variety of candidates but the real competition was between Republican candidate Donald Trump and Democratic candidate Hilary Clinton. This election has dominated the media over the past two years and many people are devastated by the result and I feel that this is due to the mathematics that was used.

This election had a terrible turnout. 230 million people in the US are entitled to vote but only 131 million actually did (Levine, 2016). This saddens me as so many people fought for the right to vote in the past and now people are not appreciating it.

The voting system used in America is called the Electoral College and is a system that could be made a lot simpler due to the confusion it causes for people like me who have never used this way of voting before. The system uses a lot of numbers and mathematics and shows which candidate will be in charge of the country for four or more years.

The Electoral College entitles each state to a certain number of votes dependent on its size. For example, California is entitled to 55 votes while North Dakota only has three. This is due to its population. California has a population of 38.8 million people and North Dakota has 739,482.

The Electoral College has 538 electors who choose the president and Americans go to polling stations and choose who they want their electoral points for their state to go to.  The winner needs to receive at least 270 of these electoral votes to win the majority.  The candidate who receives a majority of electoral votes (270) becomes president. The number 538 represents the country’s 435 Representatives, 100 Senators, and 3 electors given to the District of Columbia (The Huffington Post, 2012). Trump received 306 of the votes, so won by clear majority.

However, Trump did not actually receive the most votes from the American public, Clinton did. This is called the popular vote. 60.3 million voted for Clinton while only 59.9 million (Telegraph, 2016) and this is why I believe maths ruined the election.  The population of America chose Clinton. They didn’t choose Trump. But, due to their system with the number of Electoral Votes and the numbers for each state Trump won.

Therefore, this clearly highlights the importance of maths in every day life. It influences how a country will be run and could even possibly change the world over the next four years.

References:

• The Huffington Post (2016) What is the Electoral College and why it matters. Available at: http://www.huffingtonpost.com/2012/11/06/what-is-the-electoral-college_n_2078970.html (Accessed 11/11/16)
• Levine, D. (2016) Over 90 Million Eligible Voters Didn’t Vote in the 2016 Presidential Election. Available at: http://heavy.com/news/2016/11/eligible-voter-turnout-for-2016-data-hillary-clinton-donald-trump-republican-democrat-popular-vote-registered-results/ (Accessed: 11/11/16)
• The Telegraph (2016)  US election results: The maps and analysis that explain Donald Trump’s shock victory to become President. Available at: http://www.telegraph.co.uk/news/0/us-election-results-and-state-by-state-maps/ (Accessed: 11/11/16)

# The Sound of Maths

Our most recent workshop with Anna was all about music. Music is a subject which I did not really enjoy in school but I would love in my free time. I have a burning passion for musical theatre and the art of performance is something that I love. The picture above shows me playing Velma Von Tussle in Hairspray in 2012 which I enjoyed so much.

We learnt a lot in Anna’s workshop which I did not know about as my knowledge of music is fairly limited. However, what we were shown was that music and maths walk hand in hand to make the music enjoyable.

The link between maths and music is so important and in this workshop we were shown many different ways where we use maths in this creative subject. One activity we did was clapping to a beat. One person in the group clapped a basic beat while the rest of the group were split into four groups. We were to listen to the beat that the student was clapping while looking at a line on the board and clapping when it was yellow. For this we really had to concentrate and listen to the beat to ensure we did not ruin the sound. But, most importantly, we had to count up to 8 which was not difficult for us but this could be a tricky challenge for young children.

I feel like this is a very good activity to do with school pupils and I will definitely use it when teaching music to my classes. It will be good for pupils of any age in primary school as it allows them to understand the importance of keeping a rhythm. In the early stages of school this will help pupils with their counting as it can be done out loud and also allow them to understand basic rhythm, while the older pupils can use this is a warm up activity to remind them of the importance of keeping a beat and rhythm.

We also had fun in this workshop by getting to use the xylophones and glockenspiels and I feel that having a practical element to the lesson was essential as some of the things we were taught were confusing but being able to apply what we were taught to a fun activity made it a lot more understandable and fun.

Music is a good point to look at to see Ma’s idea of longitudinal coherence in practice. This is when the teacher knows that their pupils have a good understanding of the current subject that they’re learning about. I think this links in well with applying mathematics to music lessons as the teacher can ask the children to count or use their timetables when making a beat.

Therefore, I think that music and maths work perfectly together and the use of maths in this context allows us to have a much sounder of music and the theory behind it.

# The Only One Who Wins at the Bookies is the Bookmaker

Our most recent workshop was about probability and chance. This is most definitely something which we see every single day and affects people in both positive and negative ways.

So, what is probability? Probability is defined as “the study of chance” in one of our key texts: “Alex’s adventure in Numberland.”

Gambling is strongly connected to maths and is all about chance and luck and helps us see that Ma’s idea of “interconnectedness” is always apparent and important. There are so many aspects of maths in gambling such as money, probability and percentages (in particular; the amount actually paid out in comparison to what was paid in).

Here, in gambling and chance, all three topics that I previously stated need each other to allow people to win (or more than likely lose!) Think of it this way:

“How much cash should I put in to this game? How likely am I to actually win with this amount of money? What percentage of that have I won?”

There are also aspects in gambling which are linked to “multiple perspectives”. This is about having different ways to reach an answer. In the workshop we were shown an example of a made up restaurant. It had 2 choices of starter, 3 choices of main, 2 choices for dessert and we had to figure out how many different possible combinations we were. I managed to see multiple perspectives put into life here. The way I would have done it was long and confusing: I just wrote out a letter for each option and kept writing and writing until I had run out. But, another student used a “branch” type system and reached the answer a lot quicker than I did. Therefore, showed me that even as student teachers we are on our way to achieving a PUFM.

Unfortunately (in my opinion), gambling is something which affects many people’s lives on a daily basis and it, more often than not, involves money. Let’s look at fruit machines. These are something which are seen in almost every pub and casino that people go into and the aesthetic appeal of most of these machines encourage people to waste their money and use them. The first fruit machine was invented by Charles Fey from California in 1887. (Slot Machine History, 2010) and since then their popularity has continued to grow.

However, it’s not all fun and games. If you were to win on a slot machine there is only a 70% pay-out rate. (If you could win £10 you would actually only win £7).

This leads me on to talking about probability. For example, imagine we had a slot machine with three reels (the “screens” to see the symbols) and 15 symbols. To find the number of combinations we have to multiply the symbols by the number of symbols on the remaining reels. So, for a three reel machine that has 15 symbols per reel we have to do 15x15x15 which equals 3375 combinations of slot symbols.

If a jackpot offered on this machine pays on 7 7 7 and only one 7 symbol is on each reel, then the probability of hitting this jackpot is 1/15 x 1/15 x 1/15 or one in 3375.

When it comes to teaching probability there is much simpler way to do this: flip a coin. There is 50% or one in two chance of guessing heads of tails.

With regards to my professional development probability and chance will be taught. But, this workshop has also made me realise the importance and seriousness of gambling and that it is fun but can lead to debts and problems in later life.

References

• Bellos, A. (2010). Alex’s Adventures in Numberland (Chapter nine). London: Bloomsbury.
• Slot Machine History. 2010. Available at http://slotmachineshistory.com/charles-fey.htm(Accessed on: 28th October 2016)

# Maths and Play

We recently had a workshop with Wendee which was all about learning about mathematics through play. It was great as it let us pretend that we were children again!

Play has an extremely important part in a child’s development. It allows them to learn new skills, such as socialising with their peers. But, it also allows them to revise and revisit the abilities they have already learnt through previous education and interaction.

Thinking back to my own primary school experience with maths and play I always remember the little plastic blocks like the ones shown. It makes me feel quite nostalgic and no one can deny that they all smelt the same!

These blocks would be used to help us to learn how to count one by one, then move onto addition and subtraction and in later stages they could be used to understand volume and cubic centimetres.

I feel that using these blocks for play links in with Liping Ma’s theory of basic ideas. Counting is simple, to us, but this understanding of basic ideas will allow children to learn to count then move onto addition and continue to work up to more complex mathematical situations confidently.

The importance of these blocks was that children could use their senses. They could physically see addition and subtraction taking place and this would allow them to apply these skills to mathematics in the future.

But, in my opinion, play now is very different to what it was when I was at primary school. Children are so advanced with technology so using things such as iPads and computers can benefit them with their mathematical knowledge through the use of  apps and websites.

I think that technology is a fantastic way for children to learn as the majority of the time they do not even realise they are applying their knowledge because they are having fun.

Here is a website with a variety of apps that children can use to apply their mathematical knowledge to games: http://www.pcadvisor.co.uk/feature/software/best-maths-apps-for-children-3380559/

A study from Davies (1995) found a huge variety if reasons why games are beneficial in a child’s learning and understanding of maths. The one which stood out most for me here was motivation. From my experience on placement trying to motivate pupils to learn and engage with my maths lessons proved to be quite difficult. But using games in maths allowed children to totally engage because they wanted to win. I feel this was due to the very competitive nature of the pupils in my class.

Additionally, he stated that games were beneficial because it allowed children to have a bit of independence from the teacher. I agree that this is beneficial because children do like to occasionally work on something themselves or in a group in order to meet a goal or, in this case, to win a game.

Therefore, I believe that play is vital for pupils. It is important for their social development but it also allows them to realise that maths is fun.

# The Art of Mathematics

So far in this elective I have seen that maths is used in everything we do every single day. From looking at the clock when we wake up and how much flour we should put in that cake we really shouldn’t be having. But, in a recent workshop with Wendee we have seen that maths is extremely important in art. Throughout this post I will prove that maths is used in different types of art and is in a lot of things we see everyday. Art is a subject which is about creativity and expression and, possibly, has no correct “answer” as such. So how can a subject like this be related to maths?

Throughout this elective we have been told to read the work of Liping Ma. She explains there are four fundamental properties within mathematics. She refers to one of these properties as “interconnectedness” which is how topics within maths depend on each other and work together.

Islamic art is a great aspect to look at when studying the relationship between maths and art, but also exploring the property of interconnectedness. Islamic art uses 4 main geometric shapes: circles, squares, equilateral triangles and hexagons. Richard (2015) tells us that each shape represents something which is important in Islamic religion, for example, the circle represents unity and something which is never ending. Additionally Hussain (2009) states that these complex and beautiful geometric designs create the idea of continuous repetition, and this allows a person to understand the idea of the infinite nature of Allah.

During our lecture the “Golden Ratio” was mentioned and I think this is a vital concept which is key in understanding the strong bond between mathematics and art and also maths in everyday life. To give a bit of background; the Golden Ratio is a number which is approximately equal to 1.618 and also known as “Phi” in the Greek alphabet. This is how the ratio looks visually:

The Golden Ratio is used in geometry in mathematics and it is how symmetry is used to make a balance look in pictures. You start with a basic rectangle which is drawn using the ratio of 1.618…. If you were to draw a line in your rectangle to make a perfect square the rest of the rectangle will have the ratio of 1.618: the same as the original rectangle. You can carry on doing this on the rectangle and that is why it is so special. (Sincere apologies if that has bored you!)

The spiral is the main part to focus on here:

It starts at the bottom left then hits the opposite corner of each square within the rectangle. This spiral is extremely pleasing to the eye and is found in a huge variety of things we look at every day such as plants and even the human face.

It has been found that the Golden Ratio is found in art work such as the “Mona Lisa” and “The Last Supper” by Leonardo DaVinci. By following the spiral round our eyes are drawn into the main focus of this picture.

I really enjoyed this lecture. Throughout my time at school I always loved art and I think it is fascinating how so many different aspects of maths can be used in a variety of different pieces and seen in objects we use every day.

References:

• Henry, R. (2015) Geometry- The Language of Symmetry in Islamic Art. Available at: http://artofislamicpattern.com/resources/educational-posters/ (Accessed 4th October 2016.)
• Hussain, Z. (2009) Introduction to Islamic Art. Available at: http://www.bbc.co.uk/religion/religions/islam/art/art_1.shtml (Accessed 4th October 2016.)

# Discovering the ‘M’ word…

On Monday I started my second year at the University of Dundee (time is going too quickly!) which means I am now one step closer to achieving my dream and goal of becoming a primary school teacher.

This year I have chosen the elective titled “Discovering Mathematics”. I have taken a piece of my previous blog post about maths and will write about why I think I will enjoy this elective so much.

“Yes, the M word… Maths. Maths is a subject which I have never had huge confidence in so knowing that on Wednesday I had to sit in a workshop all about my most dreaded subject was something that I was really not looking forward to. I was actually very nervous. I had this image in my head of being sat in groups and working out maths problems in a scary kind of silence. But, that was not the case at all.

Throughout my time at primary school I hugely lacked in confidence when it came to maths. Ialways remember being that child who was sat beside a pupil, who I always thought, was smarter than me and felt I needed to look at their answers just so I didn’t get in trouble for not completing the work or having too many wrong answers. Of course this was not the case. The teacher I had would always give me the support I needed- it was just a fear of being embarrassed that I always seemed to have.

Moving on to secondary school- maths was still the subject I dreaded. And this time, it was worse in my mind because ‘high school maths is much worse and harder than primary school maths.’ Even just the little thing of having to buy a scientific calculator rather than just a normal one seemed daunting enough. However, in first year I was put into the middle set which was almost a confidence boost. Although I wasn’t in the top group, I wasn’t in the bottom one either.

I had a teacher called Miss McCutcheon and she managed to entirely change my view on maths. She made me, and my whole class, understand that maths is not something that you should be scared of and always made lessons enjoyable. She was a teacher who believed in every single one of her pupils and knew that they would succeed. In third and fourth year I was put into the middle set again aiming for an Intermediate 2 in maths. I feel this is when my maths anxiety returned because I knew I was going to be officially assessed on my mathematical knowledge. I had a different teacher whose teaching style was different from Miss McCutcheon’s, but she took a maths revision class after school on a Thursday. This was great as I could still rely on her for support and that positive push I needed. When teaching maths, I want to be as motivational and encouraging as she was for me throughout my whole time at high school. In fourth year I obtained a C in maths. The fact I’d passed was an absolute miracle in my eyes, but I knew I needed at least a B to get into university. So, I did Intermediate 2 maths again in 5th year, but this time I managed to get an A.”

This was definitely one of my greatest achievements during school. I never thought I could say that I had an A in maths and I couldn’t have done that without the support of my teacher and when I am a teacher I want to be as understanding as her when it comes to dealing with pupils’ anxieties in maths and all other aspects of the curriculum.

From my previous experiences with maths I think that choosing the Discovering Mathematics elective will greatly benefit me. The course will continually show us maths in aspects of everyday life and we will discover how to include maths in daily activities.

The elective aims to make us, enquiring practitioners and student teachers, reflect on and be able to examine a variety of topics within mathematics. And, maybe even make maths enjoyable?!

Therefore, I am excited to start this elective and I hope you will enjoy reading my blog posts and watch me progress with my understanding and hopefully see me say that I love maths!

Boyle, C. (2016) ‘The ‘M’ Word’, My ePortfolio- Carys Boyle, 17 January 2016. Available at: https://blogs.glowscotland.org.uk/glowblogs/myeportfoliocb/2016/01/17/the-m-word// (Accessed: 13 September 2016)