# Longitudinal Coherence – Two Definitions

During the process of writing my assignment, i came across two different meanings of longitudinal coherence which caused me to become slightly confused.

The mathematical term of longitudinal coherence, stated by Liping Ma, is described as the layering of the subject. This can be like the curriculum and how there are layers of various stages in each topic area and what each ability should be learning and implementing. Ma describes that the teachers must be aware of all levels and areas of the curriculum and not only the stages they are teaching or have taught. This, therefore, means that the teachers understand what their students have previously learnt and what they’ll be learning in the future in order to lay the right foundations for future lessons.

On the other hand, when I was searching for other definitions of longitudinal coherence, i found that this term also has a different scientific meaning. Within physics, this term is defined as the “distance over which two waves from the same source point with slightly different wavelengths will completely dephase”. As I have never understood physics, I don’t understand exactly what this describes. However, relating it to the assignment, I believed that it was explaining how two different topics may cross over or link as they contain basic areas which can be used within both or multiple topics. As this is very similar to interconnectedness, it caused confusion. However, i never realised that there are two completely different meanings for longitudinal coherence and the scientific definition, therefore, has no relation at all to the topic being discussed within my assignment.

Although the scientific meaning of longitudinal coherence relates in no form to the mathematical definition, i found it extremely intriguing. I believe the scientific meaning in a way, of what i understand, can in fact be an interesting description of how subjects or topic areas can be totally different, however, have similarities.

Segre, C. (2010) ‘Longitudinal Coherence’. PHYS 570: Physics. Available at: http://phys.iit.edu/~segre/phys570/10F/lecture_04.pdf (Accessed: 20 November 2016)

# Logistic and Supply Chain

The logistic and supply chain is describing the managing of planning, implementing and controlling the process of the shipment of goods. The best storage method during shipment must also be considered during this process in order to arrive safely to the supermarkets in order to meet the consumers’ requirements.

Food miles can also emphasise a way that mathematics is used in everyday life. These miles describe how many miles your food has travelled before arriving on your plate. This includes the distance from the producer to the supermarket and finally to the consumer. Food miles are a good example of mathematics. These miles can be broken down into kilometres, meters, centimetres and eventually millimetres. This can be broken down further by the description of number recognition and sequences. This emphasises the fundamental principles of mathematics as it can be broken down in to the most basic concept.  Food miles are important in everyday life and are used to encourage people to buy locally as the miles are often calculated through the pollution that was caused during the journey.

During the journey, a variety of other mathematical factors must be considered. When shipping the products, the correct method of packaging must be considered in order to carry the biggest amount of goods possible at once. This includes thinking about the weight, size and temperature requirements for each of the individual products. The shelf life must also be calculated through how long it took the food to travel to the supermarket and how long left it has left on it’s sell by date. All of these factors are influenced by mathematics whether it be amounts in a variety of forms or simple calculations.

Before any of the products even make their journey to the supermarkets, the businesses must demand plan. This is when the supermarket plans exactly what products they want and how much of each. This can be done by looking at the previous years data that was collected in order to judge whether they need more or less and what to get at certain times of the year. A good example of this is pumpkins. This is because buyers are usually only interested in them around the time of Halloween. During a clip from BBC1’s Supermarket Secrets (‘Autumns Supermarket Secrets’, 2015) it’s stated that “no one wants a pumpkin a day after Halloween. And the stores can’t run out too early.” This is a great example why supermarkets must look back at the data they collected on how much pumpkins were sold previously and on which particular dates as they don’t want to buy too many or too little.

In the future, this will help me develop my own health and well-being lesson on these areas. I believe it is important for children to understand how their food was produced and how far it’s travelled before they were able to buy it from the supermarket. Using food I bring into class, I will have the children calculate the food miles of each product through using the ‘Food Miles Calculator’ – accessible from https://www.foodmiles.com/. Although this information has gave me lesson ideas for when I’m teaching, I have learned how mathematics is needed throughout the logistic and supply chain which will be useful for me when ordering a number of resources for my classroom.

‘Autumn’s Supermarket Secrets’ (2015) Supermarket Secrets, BBC 1 London, 25 October. Available at: https://learningonscreen.ac.uk/ondemand/index.php/clip/27113 (Accessed: 20 November 2016)

# Data and Statistics

Data and statistics are an interesting area within mathematics which can be used widely within different subject areas. Data is defined as the collection of any facts and information, whereas statistics is specifically collecting and analysing numerical data.

It is known that the use of data and statistics reaches back to 35,000 years ago. At this time, the oldest mathematical tool, the Lebombo Bone, was used in order to collect and record data by bushmen in Namibia. This was similar to tally marks and was carved into the piece of bone – often the fibula of a baboon. This method of recording information was found to be used near the Border Cave in the Lebombo Mountains between South Africa and Swaziland. As it had 29 markings, it is believed that it could have been used to track the moon phases, menstrual cycles, or simply as a measuring stick.

This finding emphasises how long data and statistics have been around for and how it has developed over the years. Thousands of years on, we are now using software on computers to withhold information for us. We are able to create tables, whether on paper or within the computer, and input the data or statistics in order to collect all the information together.

Within the public health department, doctors must be numerate every day. This includes working out the doses of medication for individual patients or organising money for new drugs. Workers within this department also frequently use data and statistics to track the health of their patients. Graphs are used regularly in order to plot the rate a baby is growing within the womb, so they can track whether they’re growing at a healthy rate or not. They also use graphs to plot a patients height against their weight to be able to work out whether they’re under weight or obese for their height.

To conclude, data and statistics have been around for many years and is used within many subjects and vacations other than mathematics. Children learn how to collect data and statistics within maths, however, will use it within different areas such as topic work or ICT, etc. and will find it useful in the future. As a teacher i must make sure my lessons are effective and are stage appropriate in order for the children to learn successfully. I will also use data and statistics when teaching in order to note and calculate when assessing children. I will also use data with the class register as it will include the children’s full names, date of births and whether they have any additional support needs. Overall, data and statistics will be used regularly during my teaching career.

# Astronomy and Huge Numbers

Mathematics is widely used in astronomy. Astronomy, however, uses extremely large numbers since space is so huge. Maths can be used in astronomy to describe the distances between the earth and the moon, the earth and the sun, and between different planets. The distance from the earth to the moon is approximately 384,400km. This is a large distance considering there are things a lot further away form earth than what the moon is. The sun for example is 149,600,000 km away and that is still closer to the earth than what the dwarf planet, Pluto, is. Both of these large numbers don’t come close to how big our galaxy is alone never mind the distances between galaxies within the universe.

Light years is another example of the use of numbers to explain distance within space. Light year defined is how fast light can travel in 1 year and is used to describe the extremely large distances. In a vacuum, light can travel 300,000,000 metres per second. However, after using the speed = distance/time formula, you can calculate that in 1 year light can travel 9,460,800,000,000,000m. As this is such a huge distance, astronomers use this in order to describe the distance in space. They often use light years to emphasises how far different galaxies are away from the Milkyway.

Another example of using numbers in astronomy is to emphasise the amount of stars the universe contains. The universe approximately holds around 10,000,000,000,000,000,000,000 stars. This number is so big that people can’t get their brain to understand exactly what that amount of stars looks like. As it is such a long number, Astronomers use mathematics in order to simplify it’s written form. The amount of stars can also be described as 10²² (ten to the power of twenty-two). ‘To the power of twenty-two’ represents 22 zeros in the number and successfully breaks down the number to make it easier to read.

Overall, Maths is an important part in astronomy. I found this extremely interesting and literally couldn’t picture how big these distances are or how much stars that actually is. Even though i couldn’t picture exactly how far it was, the use of light years to describe a distance was useful as it puts it into better perspective than billions of miles away.

# Mathematics in Art

Who would have thought that maths is used within art. However, I have now learned that many artists have used a specific number sequence to create their artwork. These numbers are called the Fibonacci sequence.

The Fibonacci number sequence is made up of the fact that every number after the first two is the sum of the two preceding ones; 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144. This sequence was named after the mathematician, Leonard of Pisa, also known as Fibonacci. Details of the sequence were published within his book in 1202 which introduced it to into the mathematical world. He stated that he found a simple numerical series that is found commonly in nature.

This exact sequence has also been used by artists when creating their images. For example, Piet Mondrian has been known to have used it within his art work. During our workshop, we created an image in the style of Mondrian through using the Fibonacci sequence. We did this by the use of graph paper and a ruler. In order to create the different sized rectangles, we counted out the squares using the numbers from the sequence. I found this task to be great way in learning one way the sequence could be used within art and how simple it made creating the image be.

Within this work shop we also created a version of the Golden Spiral. Using the sequence, we were to create different sized rectangles in a certain order and then used a compass to create the spiral. Although I found this a little more confusing to create, it helped expand my knowledge on the topic.

Moving on to other forms of mathematics seen in art. In 1509, mathematician Luca Pacioli published an article on a number that is now broadly known as the ‘Golden Ratio.’ This is a special number approximately around 1.618. This ratio, symbolised by Phi (Φ) appears within mathematics, art, architecture and other areas. It indicates a special ratio of line sections when the line is divided through the use of an equation. In the workshop we tried to work out the ratio through inputting the short and long measurements of the lines into the equation (a+b)/b = b/a. We found that both sides calculated to be approx. 1.618.

This ratio was used by renaissance artists for beauty and balance in their artwork. Leonardo Da Vinci is once artists that has been associated with the golden ration and Fibonacci sequence. His painting “The Last Supper”, painted between 1494 and 1498, has clear examples of the design and architectural features to be using the golden ratio. Some also believe that Da Vinci even positioned the disciples around the table in proportion to Jesus using the ratio.

As the golden ratio has been used when creating artwork, it has also been used when creating buildings. It was in fact used in the designs and plans for creating one of the most famous buildings in the world, the Notre Dame in Paris.

Overall, before the workshop, I would have never believed that art can hold mathematics in such depth as the Fibonacci sequence and the golden ratio. I never thought that portraits would have hidden mathematics within them. I think that the use of mathematics is a successful and effective way to create images and has been proven by various artists. In the future, I will now consider mathematical concepts to have been used within the creation of artwork and will even try these methods with my own class to see what they can create.

# What is Mathematics?

To a lot of people, mathematics is only about numbers and calculations. They also believe that, apart from the basics, a lot of the more difficult areas taught are pointless and will never be used outwith maths class. It’s emphasised that mathematics is all about finding the one correct answer and has been stated to be all about “remembering and applying the correct rule when the teacher asks a question; and mathematical truth is determined when the answer is ratified by the teacher” (Lampert, 1990, p.30). However, maths does not only consist of numbers but is made up of rhythms, sequences, patterns, time, etc. Marcus du Sautoy considers maths to be in the world around us – nature and man made.

Mathematics can be found in various areas throughout the man made world. It can be seen in shops and supermarkets through the use of prices, amounts or weights, VAT, or even calculating the total and change due. It can be found in buildings or building plans through the number of windows and doors the building consists of, if the building is symmetric, or even the use of the golden ratio in order to create sizes for rooms – without mathematics, architectures would not be able to create successful structures. Artists have also used the golden ratio in order to create proportionate artwork or interesting designs and spirals.

The golden ratio can also be found in nature within the spiral of sea shells and plants. Other ways mathematics can be seen in the world around us is through patterns created within plants and flowers or even the hexagonal shapes within the honeycomb of a bees nest.

Overall, mathematics is not only about numbers but holds many other components which can be seen and used within everyday life. Mathematics can’t fully be used everyday, however, basic aspects can such as number, sequence, pattern, etc.

Lampert, M. (1990) ‘When the problem is not the problem and the solution is not the answer: Mathematical knowing and teaching’, American Research Journal, 27(1), pp.29-63.

# Can Animals Count?

Many people, pet owners or not, have the belief that animals in fact have the ability to be able to count. This has caused a large debate, however, many scientists became doubtful ever since the case of ‘Clever Hans’ about 100 years ago.  Hans was a horse that had his owner believing that he had a mathematical ability. Hans’ owner said to have taught the horse to add, subtract, multiply, divide, tell the time, work with fractions, and keep track of the calendar. It was said that the horse would answer both oral and written mathematical question by tapping his hoof the correct amount of times. Hans would perform his talent in front of crowds all over Germany, free of charge, and amazed many people including his owner.

As Clever Hans became well known throughout the country, Scientists became known of his abilities and were interested to investigate. Carl Stumpf, a psychologist, gathered a group in order to study the horse. It was observed through experiments that 89% of the time, Hans gave the correct answer when able to see the person who asked the question. However, when the person was out of the horse’s sight, the answers were only accurate 6% of the time. It was also found that Hans couldn’t answer correctly if the person themselves didn’t know the answer.

The overall conclusion the scientists found was disappointing for the owner of the ‘clever’ horse. It was explained that when answering a question, Hans could sense when to stop tapping due to the person’s reaction. He would begin to tap more slowly when he got nearer to the answer and eventually knew when to stop through sensing the expectation from the person. The owner did not realise that, by looking at Hans, he was giving him unspoken signals.

Through the tests conducted by the scientists, it was proven that Hans in fact didn’t understand concepts of mathematics, however, was clever in order to understand what was expected of him. This became known as the “Clever Hans effect” which is used in psychology to describe when an animal or person can sense what someone wants them to do without using deliberate signals. It’s now important to take this into consideration when testing an animal’s or a human’s intelligence. However, the question still remains. Can animals count? or are they in fact sensing what us humans want them to answer.

# Why I Chose to Discover Mathematics.

During  my first year placement within a primary 6/7 class, i found myself finding the maths lessons very difficult to teach. The highest maths group were learning about problem solving using decimals. For some reason i was finding this extremely hard to break down and explain. I then realised that i was finding some simple things difficult in order to understand, let alone how to explain it effectively to the children.

I couldn’t get my head round why i was finding simple maths questions so difficult and i was too embarrassed to admit it and ask for help. I then realised through a conversation with the class teacher that I wasn’t taught the basics effectively during primary school due to asking for help and getting told to “sit down and work it out myself”. This wasn’t helpful. This advice caused me to often sit for the rest of the lesson, stuck on the same question. I’d then get into trouble for not finishing the page. How was this method of teaching fair? Due to a teacher being bored of me asking for help too often, my missing knowledge is now effecting me in later life.

This is why i have chosen this elective. I don’t want to be that teacher who sat behind her desk and didn’t see a child was struggling and needed extra help. I want to be able to be confident whilst teaching maths and be able to assist the children effectively. I don’t want my pupils being effected in later life due to my non-existing support as i myself didn’t have a full understanding on the topic.

What’s the point in becoming a primary teacher when you can’t be bothered with your pupils asking questions or get annoyed when they approach your desk more than once. Teachers are in place to support children throughout their time at primary school and make sure they have the key skills and knowledge to prepare them for later education. Hence why I want to make sure I enhance my knowledge within maths in order to ensure my pupils get the correct information and support.