# Reflecting on Discovering Math

Reflecting on this course I believe it has given me a greater insight into the options I have when delivering maths to children within the primary setting. During placement, the teacher within my setting used a lot of textbook and worksheet maths however this module has shown me the different ways you can explore with the children that makes it enjoyable and provide the children with a context for their learning.
Throughout my learning career I think about all mathematics being from NHM textbooks and workbooks throughout primary school then moving to secondary where working from a text book or watching a teacher do equations on the board. When we asked in math class why we needed algebra or trigonometry the only reason we got was that it would be in the exam with no reason to why we would need it in later life.
This module has shown me how to make mathematics relevant to children through getting them to take part in real life activities. This can be both fun and interesting for the children bringing maths and other subjects together. This could be done through drama within a shop scenario or in art using measurement or pattern, even within literacy and trying to use mathematical language in a short story. It could also be done in a similar way to this module and allow the children to explore maths within an area of their choice.
This module has also shown me the importance of creating a math friendly classroom which is something I will take into my own practice to ensure that the children that I teach throughout my career have a positive attitude towards math and understand the importance of math within their life.

# Business and Math

During the input on demand planning I found it an interesting topic and especially though the activity I felt it would be an exciting lesson to do with children as it could be explored through a business angle, global warming, problem solving and design.
Before the activity we considered food miles, which is the way we measure the distance our food travels from farm to the supermarket shelves. I decided to consider something which was in the fridge at home and used an online calculator to calculate the average food miles travelled. The item I decided to calculate was the fresh coriander which was grown in Kenya. The Calculator then estimated that the coriander would have been sent from the Capital (Nairobi) and have been sent to London. From their calculations, the coriander will have travelled 4237 miles which equates to the production of approximately 1525kg of CO2. This was the calculation of CO2 without the calculations of distance from farm to airport then airport to the supermarket.
This could lead to a discussion with the children with the importance of buying locally sources food to support our local economy in addition to reducing the carbon footprint. This could lead to important discussions with the children within a setting about the statistics behind what would happen if we all swapped to locally sourced fruit and veg.
Another angle addressed was reducing the size of the item or the shape of the item to make it more shipping friendly meaning more can be fitted in a shipping container or fill the space fully so not to be transporting air that could be filled with produce. By being able to fit more produce into a container it would mean less trips would be needed to deliver the amount of product which would in turn make it more cost effective for the company and reduce the carbon footprint. Somewhere that has already attempted to change the shape of their produce is Japan where they decided to make watermelons into squares by allowing them to grow in glass boxes however this was not cost effective as the labour that was required in this meant that the watermelons cost around £85 to purchase. This could be explored with the children and get them to come up with new designs and problem solve how to effectively package different items.
Understanding demand is something that the supermarkets are required to do to ensure success within their business. Therefore, the supermarkets employ people to work with the statistics of what is in high demand at different times of year and calculate how much stock will be required for the store. During the activity during the lecture we had the chance to take part in this ourselves by purchasing goods over a period and charting how much profit was to be made. This was not done as successfully by myself and my partner however some groups stores had a massive amount of profit by the end of the task.
This input has inspired me for future lessons plans and ensuring that I can explore different areas of a topic and in more depth to facilitate the best possible learning experience for the children .

https://www.foodmiles.com/

http://news.bbc.co.uk/1/hi/world/asia-pacific/1390088.stm

# Counter intuitive maths

I enjoyed looking into counter intuitive maths which is being able to look into things that we believe to be true within mathematics yet is not necessarily the way we think.
First, we looked at the flipping of a coin and how as humans we would see this as being 50/50 as we as humans do not fully understand the meaning on Random. Our idea of random is not random at all as we don’t see random as having a possibility for a streak (landing on heads multiple times). As the probability with flipping a coin is 50/50 therefore the result of flipping a coin should be similar.
Problem solving and how many socks to make a pair activity. not until this activity did I realise my ability to over complicate a very simple problem from thinking the answer is ‘too obvious’. I recall back to school days and having to sit a standard grade general paper in mathematics and finding it more difficult than the general paper as I was trying to make it more difficult than it was. This is where I feel most of my maths anxiety comes from.
The final activity we looked at was the goat and car experiment which looks at a choice of three cups two of which have goat and one has a car underneath. Within this experiment it was seen that changing your answer once a cup was taken away increased your chances of getting the car. This was down to the probability of your original choice being 1/3 and if you change your decision after one being taken away it changes to 1/2. I found this interesting and tried it one people at home most of which said they would have originally stuck with their first choice like I has said before this experiment.
I found this input very interesting and would use these experiments with the children within my classroom in the future as I feel like it would be a good thing for them to explore.

# Numbers

Numbers are essential in the world we live in. Numbers are used within currency, in understanding time, distance and speed; without our number system, we would not have a way of trading, arriving on time to work or lectures, knowing how many miles of fuel we have left in or car or speed limits on motorways. Within societies where numbers are not necessary would be societies without these structures for example in an Amazonian tribe they only have the words for 1 and 2 and a word for anything more than that as they do not require the complex system which we use.
During an input, we explored creating our own number system. The number system we created was based upon our own base ten system. Something that came up during the input was that other groups had not created a symbol for zero therefore they would be presented with problems as they developed their number system. Binary was addressed which is a base 2 system which is what our technical devises use and looked at base 6 and 12 systems. We also considered the base twenty system of Yan Tan Tethera. This system I found interesting as it was one that I would easily understand unlike the other base systems that we looked into, I feel I found this easier to understand as it was most similar to our system.
Looking to our base ten number system I discovered it is derived from the Hindu- Arabic numerals which also used a set of ten symbols which originate from the 6th century and were brought to Europe from the publications of middle eastern mathematicians. From this I also found a video which explains why x is the symbol we use to represent the unknown. It goes back to the history of Algebra and its beginnings in Arabic mathematics and made its way to Europe (Spain) in the 11th century. In Arabic, the phrase (the unknown thing) they used could not be translated into Spanish due to the different phonic systems as Spanish has not sound like ‘SH’. The translators then had to borrow the sound from the Greek language ‘CK’ which is written as X.

I found this intriguing as we are never told the history behind our mathematic systems. We are taught the origin of our language but never of the mathematical language. Looking into the different number systems it has given me a great appreciation that we use a base 10 system as the other systems are so complex. It also allowed me to look at why we need number and how much we use it on a day to day basis even when we don’t realise.

# Art and Math

Within the six visual elements of Art and Design there are two mathematical terms within these. Shape and Pattern. Shape being an area of mathematics and patterns being what we look for within mathematics to gain greater understanding.
Photography
Looking into Mathematics within the arts the first thing that came to mind was the clear presence of mathematics in photography. Mathematics is within the technical aspects and within the artistry of photography. The math in the technical side of the camera includes a lot of math as you need to have the perfect balance of, the lenses with the IOS and aperture in addition to speed of the camera. All these elements need to be set to the correct balance for the perfect photo. Within the Art of photography there are certain rules which should be followed to create the ideal photo these are the Rule of thirds, Balancing elements, Leading Lines, Symmetry and patterns.
The rule of thirds is splitting a photograph into three equal sections whether that be vertically or horizontally. During school, I studies higher photography and within my final project I decided to focus on portrait photography. The rule of thirds was something which we addressed throughout the topic however I felt this worked best with landscape photography. Therefore, when it came to looking for inspiration for my project I looked into David Brandt who liked to break the rule of thirds and use half’s and quarters in his photography. I feel the rule of thirds in portrait and landscape photography gives you a focal point to focus in on whereas the rule which Brandt used allowed the eye to explore the image as a whole.
Balancing elements looks at ensuring the photograph is balanced. This is done by ensuring an image doesn’t have the focus at one side of the image and the rest of the image is left blank. This shows the use of symmetry behind the main focus within the background to balance out the image. Therefore, if there is something in the front left of the image there should be something in the back right of the image to balance it out.
Leading lines within an image are usually parallel lines which focus the views eye onto a specific area of the image. Pattern within an image is used to make it interesting.
Looking into photography and art it is vital to have some understanding of mathematics to be successful whether that is recognising parallel lines, and patterns, being able to use math to set up the camera correctly or knowing fractions and how to split an image.

# Teaching math

After the input about Intriguing maths I decided to consider the theory behind teaching of math. During the input, we explored Boalers research into the teaching of mathematics within two different settings.
Within the first setting the children were taught with the traditional set methods and procedures much like that of the 5-14 curriculum where the children were taught specific topics and were taught certain methods to solve problems. They found within this setting the children understood the rules and procedures for solving problems but Boaler found after time the children in this setting lost this information.
The second setting provided different math opportunities involving some degree of choice. The children were not provided with the answers or correct way of completing a calculation but were encouraged to find their own methods to finding answers and be able to give reason to their thinking. The setting did less textbook work which focused on specific topics. In this case Boaler found the children to have flexible thinking, can adapt their knowledge of one subject into another and adapt to new challenges.
Two psychologists that could give reasoning to the difference of these two settings and what the children got from their leaning. Piaget views cognitive development as a result of maturation and environmental experiences meaning the children’s original abilities in addition to the experiences they have been provided with have affected the way in which they process information provided. Therefore, the experience of the second setting would be an explanation for the different thinking processes when faced with a problem.
Vygotsky’s social development theory places more emphasis on cultural and social influences than Piaget. Vygotsky views social interaction as a vital part of development of thinking which can be seen within the second setting with teachers asking children to explain their thinking and discuss answers.
Therefore when teaching maths we should be looking into the best possible methods to ensure the children become effective learners and problem solvers.

# Perspectives

Liping Ma defines the Key principles of mathematics as:
• Interconnectedness
• Multiple perspectives
• Basic ideas
• Longitudinal coherence

The principle I will be focusing on in this post is the idea of multiple perspectives in mathematics. Looking into this topic I found a talk by Roger Antonsen who believes mathematics is made up of patterns which he addresses as being connections which you need to find in order to understand. He explains this idea of finding patterns everywhere by considering the different methods of tying a tie and shoe laces.
He begins looking into different perspectives by looking into a simple equation x+x= 2x. The equals sign in this equation shows the break of the two different perspectives. This can be seen through all mathematical equations as something equals something else meaning you are looking at the same thing from two different perspectives.
To strengthen his idea of perspectives and understanding of mathematics giving a greater understanding of the world by looking into the number 4/3. He began looking at the number from the perspective of a decimal then changes the base system to show different perspectives. He then uses lines going around circles at 4/3 and uses then to draw the ‘image of 3/4’.
He then looked at the number from a musical perspective creating the sound of 4/3 then the beat of 4/3. He then looks at the shape created by 4/3 which is an octahedron. He shows how changing perspectives of a shape and taking it apart and reconstructing you learn more about the object.
This talk helped me to understand the importance of perspectives in mathematics as you can never fully understand until you are willing to explore different avenues. If someone is telling a story from one perspective you aren’t getting the full story with all the information which is why it is necessary to have different angles from different people. Therefore, I have grown to understand the importance of perspectives not only in mathematics but in the world around us.

# Reflection

I feel the module I gained a vast amount of knowledge from last semester was the working together module. This is particularly due to the focus that was paced upon working within a partnership and building relationships with those people you need to work with. This was a useful insight for me as it allowed me to experience what it would be like within a working environment and the difficulties that you can face as a professional. Before entering onto this course I had some experience of reflection in practice from college where I had to reflect on lessons and activities that I had planned. I learned the importance of reflective practice though my time doing the HNC as it was vital for responsive planning and understanding the children’s needs. I am excited to begin placement and further my knowledge of reflective practice.

# Relationships

The main issue in both the Suzanne Zeedyke and John Carnochan videos is that the early stages of brain development within babies is crucial and the main thing that can promote or hinder the development is the child’s environment and their interactions with other humans. The connected baby was very focused on the biological effects of different environments for children and how they create different chemical pathways within the brain from pre-birth to three. She explained how a child from a domestic violence background could have a high level of stress hormone within the brain which could restrict the level of concentration they have and limit their development. This is important to understand within our profession as we are teachers we would need to understand that not all children will come into the classroom with the same levels of ability. In addition, they will all have very different life experiences which may mean that some children react differently in situations than others and as teachers we would need to find ways to benefit each child’s capacity to focus and learn. There was also mention that the brain continues to develop until the age of 20, therefore as teachers we have the ability to still make a change within an individual’s life by being a positive role model within their life. The John Carnochan video explained the need for focus to switch to early years as changes in early years could be early intervention for some children which he believes could change their pathway in life. He believed that interactions as a baby could mean the difference between important social and physical skills being developed within a child. This is important to think about in practice as it shows the interactions with the children within any particular setting and forming positive relationships with these children could make a change within their lives.

# Dance workshop

My own experience of dance is very limited. A years’ experience of dance class, Scottish country dancing throughout high school and dancing to the radio in the kitchen is as far as it goes. Before the dance input I was apprehensive as I only ever dance in front of family and close friends purely to make them giggle. However, the thought of dancing and making a fool out of myself in front of people I barely knew was not something I welcomed with open arms. Leaving the dance session, I still felt a little embarrassed but a lot less apprehensive over teaching dance within the primary setting as I learned that to teach dance you didn’t need to be a professional but instead have the ability to facilitate the children to be able to enjoy dance. The challenges I feel I may face when teaching dance would be to be able to push myself out of my comfort zone of my kitchen to actually joining in with the children and experiencing the lesson with them. This I feel would display to the children that there is nothing to be embarrassed about during a dance lesson. In addition, I feel getting the older children within a setting to get involved with dance would be difficult as from my own experience of being apprehensive within those lessons that it could be difficult to get children who feel the same as I do/did to be able to give as much as they can to the class to get out a lot from the sessions. I feel a way to work round this would be having quick fire warm ups to get the children into the swing of things and to feel less self-conscious before beginning the main bulk of the lesson. My professional goal for the next four years in particular is to be less self-aware when participating in these kinds of workshops or even when delivering them to a class.