One of the fundamental reasons as to why I chose the Discovering Mathematics module was to overcome my own problems in mathematics. Referring back to my first mathematics blog post, I discussed some of the issues I faced in mathematics and my hope of overcoming them throughout the module. One of the issues I aimed to overcome was the attitude of ‘why maths?’, why is mathematics so important to me?  Well, mathematics is important to me as it is part of absolutely everything I do in to my day-to-day life, and I am not over exaggerating when I say everything. The Discovering Mathematics module has opened up my eyes to a whole new perspective of mathematics and away from education.

I can’t say that immediately my view on mathematics changed it has been only very recently that everything started coming together. At the start of the module I found it difficult to grasp the concept of viewing mathematics away from educational applications and taking my teacher hat off. I realise now that as an education student, soon to be teacher, this is such an important trait to have and for that I am grateful to the module. Throughout my time on placement I can’t emphasize the amount of times I heard, “maths is boring”, “why do we have to do this” and to be honest I did not have an answer. I knew it was boring, and I wasn’t sure of the relevance of the topic, I just knew that I learned that, people before me learned that and that’s just what we had to do. Now, going in to next year’s placement I aim to challenge these views and now I can successfully give the children an answer.

Math is boring if you make it boring, the module has opened my eyes to a whole new way of teaching mathematics. Mathematics is a tricky topic and it is difficult to grasp however the current teaching of mathematics has no relevance to the children. The Discovering Mathematics showed me the mathematics that surrounds me and that is in my daily life. The buildings I enter are all built using mathematics; the golden ratio and measuring. The doctors who look after me use mathematics in prescribing my medicine. The music I listen to all includes mathematics such as counting beats, rhythm and patterns. Then of course the obvious mathematics I use such as telling the time, handling money and driving. There is never a truer statement when someone says mathematics is all around us. Nature represents mathematics such as the golden ratio which is represented in plants and the shape of countries. By making this clear with children and showing them mathematical links in the world away from “education” I aim to challenge the view of “why are we learning this?”

By looking at Liping Ma’s profound understanding of fundamental mathematics (PUFM) this has allowed me to understand there is more to mathematics than just knowing the methods. Through looking in great depth at PUFM and what makes a teacher have PUFM I feel this has again provided me with a new perspective of mathematics. Ma discusses that mathematics is about not just knowing how to do something but knowing why and knowing this with breadth, depth and thoroughness. Again, by applying this to the classroom will give children an understanding of why they are learning a particular topic instead of just learning the methods and memorising it. Another crucial part of PUFM is having the ability to make connections with mathematical concepts. This could also be beneficial within the classroom, if a child is not understanding a particular concept having the ability to show them similar techniques of a less complicated form could make mathematics less daunting for them and hopefully guidance they could begin to create their own links.

I am glad I picked the discovering mathematics module as although it was not designed for the purpose of education I feel it will benefit me greatly in teaching. It has challenged my own negative views of mathematics and that you don’t have to be a ‘whizz’ in order to be good at mathematics and has provided me with more confidence which as a result will ensure I am a more confident teacher. It also has allowed me to make mathematical connections with topic’s I would never had associated with mathematics.

Mathematics and Science

Mathematics has applications in many aspects of society and science is one of these. There are many mathematical elements within science. Arithmetic is used when working out values and to solve simple equations or formula. Mathematics is used in Astronomy for distances, sizes and masses. As these numbers are very large mathematics is used to portray the numbers in a smaller unit.  Algebra is used to show relationships before the measured numbers are used for calculations. Higher math is used for complex relationships between properties (Kurtus, 2013.)

During our lecture, we looked at scientific graphs and the mathematics behind them.

Some of the mathematics we used whilst studying the graphs:

• Algebra as we had to work out what each formula meant
• Understanding of coordinates
• Basic equations such as multiplication

We then had to make our own graphs. First, we had to complete an activity where we compared the distance and weight of two magnets using the device below.

We started with both magnets being 15cm apart and found the weight of the pressure that the magnets created which was around 280g. We decreased the distance by 1cm each time and recorded the weight. We found the weight never really changed until the distance was around 10cm apart. This was when we saw changes happening. By the time, we were around 2cm the weight was around 400g. We then created a graft using the weight and distance. We compared our graph to those we studied earlier to find which ours was most similar too.

Although this was a scientific experiment we used many aspects of mathematics such as measuring, weight and graph’s.

This is only one example of the relationships between mathematics and science. We looked at the scientific formula ‘E=MC2’ and the applications of this formula in to the wider world. This was a formula created by Albert Einstein. So, what does this formula actually mean? The ‘E’ represents the energy which is measured in Joules, the ‘M’ represents the mass of the object which is measured in kilograms and the ‘C’ represents the speed of light (Carroll, 2014.) Prior to the mathematics module I had never heard of this formula before let alone understand the significance that this plays in everyday life. This formula applies to medicine as it is used in pet scans as well as everyday items such as smoke detectors (Tyson, 2005.) Einstein’s formula is responsible for the creation of power and water stations and is one of the reasons we have electricity etc.   Einstein’s formula also had a negative impact on society. It was used in the creation of automatic bombs when used cause devastating effects.

Overall, we can see that mathematics plays a large part in science and is responsible for a lot appliances in our society today but how does this link to Liping Ma’s profound understanding of fundamental mathematics?

Basic Ideas: Throughout science, mathematical concepts are frequently revisited and reused for equations and formula.

Connectedness: Different mathematical concepts are used across the board within in science for example equations are used in physics, chemistry and biology. Although the equation may change according to the topic the mathematics remains the same, similar to the example below.

   S= D/T   T=S/D D=SxT

M=F/A  A=F/M F=MxA

Reference List

Carroll., J (2014) A fun way of understanding E=mc2. Available at: http://www.universetoday.com/114617/a-fun-way-of-understanding-emc2/ [Accessed on: 29 November 2016.]

Kurtus, R (2013) Using mathematics in physical science. Available at: http://www.school-for-champions.com/science/math.htm#.WD1OkfmLTIU [Accessed on 29 November 2016.]

Tyson., P (2005) The legacy of E=MC2. Available at: http://www.pbs.org/wgbh/nova/physics/legacy-of-e-equals-mc2.html [Accesed on 29 November 2016.]

In a recent lecture, we were looking at the mathematics in global food supply and demand planning. We started off by looking at the mathematics behind global food supply chain. It is a well-known fact that food is transported all around the world from country to country, but I had never stopped and thought about how much must be considered prior to transportation especially mathematically. There are many mathematical concepts to be considered in food transportation such as mass, distribution of mass, size, strength, temperature requirements and distance travelled i.e. for shelf life. Shape also plays a large part in food transportation for packing vans/ships etc. There are many things that must be considered such as the shape of each box and the best way to fit them all together. Also, the type of item that is being shipped i.e. eggs, heavy items could not be placed on top of these. In order to do this there must be some basic understanding of shape and what shapes fit best together. There also must be some understanding of weight to ensure no items become broken or squashed.

We also looked at demand planning. So, what is demand planning? It is used to predict/forecast businesses future sales (Demand Planning, 2016.) As a class, we were put to the test and asked to create our own business plan. We had to pick 5 items out of a list of 15 and had to decide what we thought would sell best. We were given £5000 to start off with. The business plan was from January to December. Once we had picked our items we were given further instructions which told us the success rate of those products and if we lost or gained money. This was the rule for throughout the game. It was a challenging task to say the least! It took a lot of getting in to and a bit of time to understand it. It was beneficial however as it allowed us to see for ourselves how exactly demanding planning works and everything that must be taken in to consideration. We had to consider how many items we had not sold, shelf-life, the money we would lose if we could not sell these items and at what time of year was best for products. Me and my partner started off well however we managed to lose a fair amount of money and although we finished with more than we started with it was far off some other teams. There are many aspects of mathematics involved in demand planning such as problem solving skills, data handling, money and basic equations.

There are elements of Liping Ma’s four characteristics both within global food supply and demand planning.

Basic Ideas; In order to understand demand planning we had to be able to work with basic equations such as addition, subtraction, multiplication and division. We also had to have basic problem solving skills and handling money.

Connectedness; Global food supply has connections with other mathematical topics such as tessellation. Tessellation is when many regular shapes fit together (Eastaway and Askew, 2013, p195.) This also applies to shipping food. Those who pack the ship must make sure all the shapes fit together to get all the items on board.

Multiple Perspectives: This applies to the demand planning. As the class was split in pairs, each pair had a different approach to the demand planning. Some tried to be tactical with their buying whereas others chose to buy large amounts of food. We each tackled it differently and all had different reasons as to why but were able to discuss which methods worked best in comparison to others.

References

Demand Planning (2016) Available at: http://searchmanufacturingerp.techtarget.com/definition/demand-planning [Accessed on 29 November 2016]

Ma., L (2010) Knowing and teaching elementary mathematics. New York and London: Routledge.

What are the chances?

In a recent lecture, we were looking at probability and its application to real-life. So, first of all what is probability? Probability is a measurement that is applied to events. What is being measured is how much we believe that an event will or will not happen (Haylock, 2010.) Personally, I believe a lot of us use probability without consciously knowing we are doing so. I am a big football follower and I know I evaluate my chances constantly whilst watching it such as the chances of my team beating a team. Given the fact I am Dundee fan the chances are usually none or very unlikely! Prior to the discovering mathematics module, I never really thought about how much mathematics I do use every day and it has opened my eyes.

So, what is the mathematics behind probability? There are many aspects of mathematics which play a part in probability such as fractions, decimals, ratios and percentages. In order to understand probability, we must understand mathematical language such as impossible, evens and certain (Haylock, 2010.) There are various aspects of mathematics which you must fully understand such as percentages to  understand how probability works.

How do we apply probability to real life? Probability can occur in something as simple as playing a board game such as when rolling the dice what is the chance that you roll a number six. When flipping a coin, probability occurs as you have a 50/50 chance of it landing on either heads or tails. There is on aspect of the wider world which requires probability to function, gambling. The “fundamental principle at play in all casino games is the theory of probability. After all, gambling is all about what your chances are of winning or losing” (Mathematics of gambling, 2001.) Therefore, to succeed in gambling you must have some understanding of probability. In modern day gambling technology, does most of the mathematical thinking for you but we must have some mathematical concept such as the ‘odds’. Odds are “ratios of a player’s chances of losing or chances of winning, or the average frequency of a loss to the average frequency of a win. If a player owns 1 of 4 tickets, their probability is 1 in 4 but his/her odds are 3 to 1 (Tunrner, 2007.) Therefore, in order to have some understanding of how gambling works we must understand fractions and ratios.

This was only a brief example of how mathematics applies to the wider world and without mathematics some aspects of life would be virtually impossible i.e. gambling. I feel probability links to aspects of Liping Ma’s four properties.

Basic Ideas  – In order to understand probability we must have a profound understanding of basic equations, fractions, decimals, rations and percentages.

Connectedness –  Probability links with various aspects of mathematics i.e. fractions and decimals. Making these links allows us to develop a fuller understanding of how probability works.

Multiple Perspectives – Depending on the person people may have different perspectives of chance. For example how likely or unlikely a event is to happen.

References

Haylock., D (2010) Mathematics explained for primary teachers. Thousand Oaks CA: SAGE Publications. Edition 4.

Mathematics of gambling (no date) Available at: http://gambling-maths.co.uk/ [Accessed on 20 November 2016]

“Mathematics is all around us.”

This is a statement which I highly underestimated prior to the ‘Discovering Mathematics’ topic. I was more than aware of the amount of mathematics I used in my day to day life such as when I am driving, going in to a shop or cooking. However, I was very unaware of the mathematics surrounding me. From the buildings, I look at every single day to the tables and chairs I sit at. Mathematics has links in absolutely every part of life from art and music, science and technology to just general day to day activities.

Prior to the lecture I was aware of some of the mathematics involved in art such as symmetry, shape and measuring. However the mathematics we then went on to look at I had never heard of before. We looked at the Fibonacci sequence, the Golden Spiral and the Golden Ratio. To begin with each concept absolutely blew my mind however the further the lesson went on the more I began to grasp the idea.

So, what is the Fibonacci sequence? The Fibonacci Sequence is a series of numbers 0, 1, 1, 2, 3, 5, 8, 13, 21, 34… The next number is found by adding up the two numbers before it. The 2 is found by adding the two numbers before it (1+1) The 3 is found by adding the two numbers before it (1+2), and the 5 is (2+3), and so on!

The Golden Spiral is when we make squares using those widths. You then create a spiral from this. I feel this aspect of mathematics links to Liping Ma’s basic ideas. Without having a basic understanding of how to measure, count, use patterns and use mathematical tools such as a ruler and protractor this task would have been extremely difficult. Even with a basic understanding of each I still found the task quite difficult to understand at first. This shows links between mathematics and art as you are using a mathematical concept to create a piece of art.

The Golden Ratio is when we take any two successive Fibonacci Numbers, and divide the larger number by the smaller. The answer is very close to the Golden Ratio which is approximately 1.618034… We tried this in our lecture using the numbers from our Golden Spiral. I chose the numbers 21/13, the result of this calculation was 1. 6153.. The Golden Ratio! The Golden Ratio is used for building and art work such as the Parthenon in Greece, but it is not known if it was designed that way. It was also used to design the Notre Dame in Paris. The ratio features also in the United Nations building and the pyramids in Egypt (Boaler, 2016.)

Not only does the ratio exist in buildings it exists in nature.
Flower seeds that grow in  spira they grow in a ratio of 1.618:1… the golden ratio.

This is only one example of how mathematics play
s a part in art. When using pattern in art there are aspects of mathematics. Pattern is a combination of  elements or shapes repeated in a recurring and regular arrangement. Repetition also occurs in patterns and this refers to one object or shape repeated. Pattern and repetition is used in Buddhist mandala. Buddhist mandala is a spiritual and ritual symbol in Indian religions, representing the universe. This is an example of mathematics being used to create religious art.

As we can see mathematics plays a huge part in art and day to day life, whether its creating a picture or designing a building, there always aspects of mathematics being used. Prior to the Discovering Mathematics I was unaware of how much of an influence mathematics has in everyday life. It is not just the obvious mathematics such as dealing with money or driving a car but the design of the building we walk in. Mathematics is all around us and as the module progresses this is becoming more and more evident.

Profound Understanding of Fundamental Mathematics 

Basic Ideas – In order to understand Fibonacci sequence I first of all had to understand basic equations. Without having a secure knowledge of addition it would be difficult to understand how the Fibonacci sequence works. As well as understanding of measuring etc previously mentioned when discussing the golden spiral.

Multiple Perspectives – When carrying out the Golden Spiral we each had a different approach which we found the simplest for us. The approach I was taken was over complicated and one of my fellow peers showed me her approach which I found much easier.

References
Ma, L. (2010) Knowing and teaching mathematics: Teacher’s understanding of fundamental mathematics in China and the United States. 2nd edn. New York: Taylor & Francis

Fibonacci Sequence (2016) Available at: https://www.mathsisfun.com/numbers/fibonacci-sequence.html [Accessed on 29 October 2016]

Design in art: Repetition, pattern and rhythm (2006) Available at: https://www.sophia.org/tutorials/design-in-art-repetition-pattern-and-rhythm [Accessed on 29 October 2016]

In a recent lecture, we were looking at mathematics and play. I feel this is a very divided topic. Some people are in favour of introducing more play into mathematics, others however would rather keep mathematics traditional. I can understand why, as many people view mathematics as traditional subject which is very much textbook based. I enjoyed sitting down with a text book and working my way through problems, there was no greater feeling than understanding a complicated mathematical concept and firing your way through a series of questions (and that is from someone who was never fond of mathematics.) However, education changes frequently and we must welcome these suggestions with a more open mind. I feel play is important especially in early years for allowing children to put their knowledge to use and to engage them.

My own personal of mathematics was never a positive one. I found mathematics difficult for as long as I can remember. I was never in the top math setting nor was I ever super speedy with my equations. It also took me a vast amount of time to understand mathematical concepts especially throughout my high school years. Would I have felt different about mathematics if it was more active? Yes, I think I would.

I feel at this point I must define what is meant by ‘play’ in mathematics. For me personally I don’t define play as children ‘playing’, I feel it is more about children being active, engaged and driving their own learning. Play can be physical such as children using cubes to create shapes or for working on their symmetry. They can be using sand or water for making shapes and patterns and filling boxes and materials of different shapes and sizes to compare weight and quantity. Play can be imaginative such as having a class café, they can be looking at recipes, measuring, dealing with money and time management. It can be something as basic as a jigsaw puzzle. Play can promote more mathematical discussion amongst the children and provide more opportunities for the use of mathematical language such as first, second, third, heavy and full. Now I understand that this does seem more directed towards younger children. However, there can be opportunities for older children to be more active in mathematics. They can be measuring the playground, running a shop or even developing mathematical skills on the computer.

I am also not stating that mathematics should be all play and absolutely no written work. I feel that there should be a balance between both. When introducing mathematical concepts to children it would be difficult to do this through play at first. They should be shown how to do the methods, provided with examples and provided with opportunities to do the questions themselves. Incorporating play after this point may be beneficial as it allows the children to reinforce their knowledge.

During our lecture, we were provided with an opportunity to implement play. We were provided with cubes which we had to pretend were a chocolate bar. We placed them together and count how many snaps it would take to break the bar into singular pieces and provide a formula. If this task had to be carried out individually and without physically being able to break the pieces, it would have proved difficult. However, as we worked in a group it was an enjoyable experience.

I feel play in mathematics particularly the activity we carried out in our lecture links to aspects of Liping Ma’s fundamental mathematics.

Basic Ideas – In order for us to create our own formula we had to have a basic understanding of how to do this i.e. basic calculation.

Multiple Perspectives – As we were working in a group each person had a different way of carrying out the calculation for the formula. We developed our mathematical language whilst discussing what would be the best way to do the formula.

Longitudinal Coherence – I feel this would be a good starting point for moving on to more complicated mathematics such as algebra.

By examining mathematics and play I feel it has developed my mathematic understandings as it has encouraged me to look at this topic from a new perspective. I am a firm believer of introducing mathematics in play, especially due to my terrible experience of mathematics. However, through discussion with peers and further reading I now appreciate that ensuring there is a balance of both play and textbook work is key. Mathematics is an ever evolving subject and it is important as professionals to embrace change. However, I feel there must still be traditional aspects involved in mathematics such as repetition and text book to ensure methods are embedded in children’s knowledge.

References

Ma, L. (2010) Knowing and teaching mathematics: Teacher’s understanding of fundamental mathematics in China and the United States. 2nd edn. New York: Taylor & Francis

Maths. That word has filled me with fear and dread for my whole life. Maths is a subject which I find extremely difficult. The different concepts, the rules, there being only one correct answer is something I have had difficulty understanding for years. I’m not sure if my difficulties in maths developed from the idea that “you can only be good at either English or maths; you cannot be good at both.” From a young age this is the statement you hear from parents, friends and teachers. There may have been a point in my life where I just accepted my strength was English and that if you could only be good at one, why try? After school when again will I need maths? How wrong was I… 

Firstly, the idea of you can only be good at maths or English is just ludicrous. There are many people who have a natural ability to be good at both, I know as I have witnessed it. From colleagues, to friends and the pupils I taught on placement they have opened my eyes to the fact that yes, you can be good at both. Do not get me wrong you may be stronger at one or enjoy one more but what is stopping a person being good at both. This is backed up by Eastaway and Askew who state that there is “no such thing as a maths gene.” (Eastaway and Askew, 2013, p14.) They discuss that mathematics has been around for hundreds of years and that today’s society are much more mathematically sophisticated compared to those in medieval times, therefore how could there be a gene that programmed us to “be good at maths.” (Eastaway and Askey, 2013, p15.) I understand people can argue that maths has developed heavily over the last few centuries however my argument would be if we were all genetically programmed with a “maths genes” then why were people in the medieval times not as developed as us?

Secondly, the concept of you do not need maths after you have left school does make me laugh, now. For years I was a firm believer of “when will I need maths again?” Fast forward a few years and here I am standing in front of a class teaching them angles, fractions and decimals. All the things “I would never need to use again”. Maths plays such a heavy influence in day to day life. We need it for such basic things such as setting an alarm clock or calculating change. As we grow older we need maths for knowing how to drive or buying a house. Without mathematics these things we do each day without thinking would prove extremely difficult. In modern day society maths is an essential qualification to have when applying for most jobs. Whether we like it or not we do need maths and it is deemed a requirement in society. (Eastaway and Askey, 2013, p21.)

Predominately, the reason why I chose the ‘Discovering Mathematics’ elective was due to my anxiety about maths. People’s anxiety from mathematics can stem from many different things. Eastaway and Askey state that people’s mathematics anxiety can develop from a parent or teacher but mainly it is not the fear of maths itself but the fear of being shamed. (Eastaway and Askey, 2013, p15.) Personally, I feel one of the reason as to why I feel anxious mathematics was due to the way I was taught it. We were given brief explanations about the methods and explanations and then set on to a task from a text-book. This method never worked for me. As a learner I need to be able to relate it too real-life. When I become a teacher I strive to ensure that when teaching mathematics, I will relate it too real-life as much as I possibly can. This will give the children an insight as to why they need maths and I hope that they will never be asking themselves “when will I use this again?”

Eastaway, R., Askew, M. (2013) Maths for Mums and Dads. Square Peg. London.