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Mathematics and Motorcyclists

Today in Discovering Mathematics, we considered the mathematics behind motorcycles and racing. Motorcycles and motorcycle racing, is not something which I have ever had a particular interest in. However, it was interesting to see the mathematics behind them and understand how different mathematic concepts affect the way they drive, the speed which they drive at, and their turns in and on the road, which also influences the likes of cars and bicycles. Motorcyclists, as well as cyclists and drivers, are constantly making mathematical decisions in their head, and I have decided to consider this further.

First of all, we started looking at roads, and the different paths that you can take and how some paths taken on the same road at the same speed, can make you finish quicker than others. For example, the drawing attached represents a windy road and three different paths which you can take.

This allowed us to look at different routes and see what difference these made on distance, time and speed. For example, if you are taking the middle route you are cutting the corners and have more chance of finishing first.

Furthermore, we then went on to look at how the equations for speed, distance and time effect motorcyclists. These calculations can be worked out by;

– Distance = Speed X Time

– Time = Distance/Speed

– Speed = Distance/Time (BBC Bitesize, 2017).

This can be shown easily by the following triangles;

(Image taken from http://www.bbc.co.uk/bitesize/standard/maths_i/numbers/dst/revision/1/)

This is used particularly in racing, where motorcyclists can work out which route is best to take. For example, whether it is better to take a longer route to improve their current speed, or a shorter route, to improve their overall time. This also links to the racing line which is the line which most new racers decide to take to go around corners, because it allows them to keep up their speed whilst going around the corner straight. This video which I have attached explains the basics of the different racing lines which are used by competitors during races;

Furthermore, motorcyclists and normal cyclists, are constantly using mathematics and working out solutions in their heads to come over obstacles such as the weather, surfaces and other traffic or road users. For example, if there are pot holes on the road, the cyclist must work out how they are going to overcome this. At first this may not seem like maths, but because they are constantly making decisions, and working out solutions to problems, they are constantly problem solving which is a crucial skill involved with mathematics.

References

Bbc.co.uk. (2017). BBC – Standard Grade Bitesize Maths I – Distance, speed and time : Revision. [online] Available at: http://www.bbc.co.uk/bitesize/standard/maths_i/numbers/dst/revision/1/ [Accessed 16 Nov. 2017].

Yan, Tan, Tethera

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Number systems and place value was not something which I had thought about or realised to be so complex until now. When I was in school and learning how to count or about place value and systems, I understood it straight away and that was it. I never questioned the mathematics behind it or considered it further. It was never explained to me why we use the number system which we do or why there is units, tens and hundreds, but it was just that we do and that was it, case closed. Therefore, when I realised that this is different around the world, and that there was specific reasoning behind it, it intrigued me and I wanted to look into it further.

Why a base 10 number system?

A base 10 number system is the number system which we use in the UK every day. The base 10 number system is also known as the decimal system and has 10 digits to show all numbers: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, which uses place value and a decimal point to separate. The placing and positioning of the numbers in the system are based on powers of ten which is referring to tens, hundreds, thousands etc. Once you exceed the number 9, you then move onto the next highest position (Russell, 2017).

Before this input, I just presumed that everyone around the world used the same base 10 number system as us and that they would just translate it to fit their spoken language. To me it just makes sense, we have 10 fingers to help us count so we have a 10-base number system? No?

It appears however, that not all societies use a base 10 system. The Arara tribe in the Amazon for example use a two-base system:

anane (one)
adak (two)
adak anane (two, one)
adak adak (two, two)
adak adak anane (two, two, one)
adak adak adak (two, two, two) etc.

This confuses me as I do not understand how you are supposed to count by using the same two numbers which are simply repeated over and over again. How does that work when you are counting numbers as big as 100 or 1000? Then again, I suppose that if you are brought up using this, then it would seem normal, just like the base 10 number system seems normal to us. The Arara tribe may not need to count to excessive numbers and therefore, the base 2 system works best for them.

Another alien number system to me is the base 12 number system which is known as the dozenal base system. At first it felt familiar as they also use the numbers 0-9. However, the number 10 looks similar to an upside down 2 and is called ‘dek’, 11 looks similar to a reversed 3 and is called ‘el’ and then 12 looks identical to our number 10 and is called ‘do’ (Dvorsky, 2013). At first I was sceptical about a base 12 number system as a base 10 system seems so much easier to use. When I approached it with an open mind however, I realised that a base 12 number system makes sense for a multitude of reasons. Items such as eggs are measured in dozens, which is equal to 12, 12 has more factors than 10 and the clock dial is also numbered 1-12.

I have attached a video by Numberphile explaining the way the dozinal base system works and it’s benefits. One point which I found extremely interesting in the video however, was that some cultures still use their hands and fingers to count, a point which I raised earlier. Unlike counting their 10 fingers like we do, they count each individual segment on each of their fingers and their pinky which seems more confusing at first, but also makes sense.

One number system which I found confusing, but which I also found the most interesting is the number system which was most commonly used by farmers in North England to count sheep. This was a base 20 system, meaning that the farmers would count to 20 and then pick up a stone to represent the 20, and start again. I have attached a photo which shows the numbers which the farmers would use in Lincolnshire, Yorkshire, Derbyshire, County Durham and Lancashire. I found this interesting as it is a completely different way of counting, not using numbers as such but rather words. It has been argued that the number system used today in countries such as Spain and France, has been influenced by this system. This is because the word for 10 in North England is mostly either ‘Dix’ or ‘Dick’ which is like 10 in French which is ‘Dix’ and Spanish which is ‘Diez’.

After considering various different number systems, we then had the chance to create our own. However, we had to ensure that the numbers and symbols made sense when it comes to using decimal places and doing mathematical sums such as addition, multiplication, subtraction and division. My group decided to use the theme of flowers for our number system. I have attached a photo of my number sequence.

I feel that this activity really helped me put what we had been looking at in the lecture into context. I need to see things visually to help me fully understand it and this gave me the opportunity to do so. It also helped me to wrap my head around different number systems, a concept which I found something extremely confusing and challenging to begin with.

References

Bellos, A. (2010). Alex’s Adventures in Numberland. London: Bloomsbury.

Russell, D. (2017). What Base 10 Means in Mathematics. [online] ThoughtCo. Available at: https://www.thoughtco.com/definition-of-base-10-2312365 [Accessed 2 Nov. 2017].

Maths, Play and Stories

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“Play allows children to use their creativity while developing their imagination, dexterity, and physical, cognitive, and emotional strength” (Ginsburg, 2007).

Friedrich Froebel explains that a child does their best thinking and learning whilst they are playing. Play within education is fundamental to a child’s holistic development. It helps to make connections in their learning in a relaxed environment as it enables the children to experiment within their own learning and apply it to contexts which they are familiar with. Furthermore, Susan Isaacs also valued play based learning as she saw the value of play as a means to enable children the freedom to balance their ideas and feelings.

There are various types of play which can be involved within play based learning. These involve; symbolic, creative, discovery, physical, technological, games, environment and through books and language. During quality play within mathematics however, children are involved with making decisions, imagining, reasoning, predicting, planning, experimenting with strategies and recording (Lewis, cited in Pound, 1999). Playing around in maths allows the children to know and understand early maths language involved with basic mathematical skills such as measurement, time, shapes, spaces, positions, early numbers and order and patterns. One way to help children practice their new mathematical skills through play, is by learning number rhymes and songs such as; Five Currant Buns, Ten in the Bed and One, Two Buckle My Shoe. These are fun ways for children to connect with maths in a playful and exciting manner.

Play within learning encourages creative and flexible thinking, and is something which parents can get involved in. However, parents who are afraid of maths or dislike it, will pass this onto their children (Furner & Duffy, 2002). Therefore, it is important that when parents are encouraging maths in the home, that they promote positive mathematical experiences. Susan Isaacs, alongside Friedrich Froebel, has valued the importance of parents as educators. Parents of preschool children especially, are essential in their child’s early development (Pound, 2003). It is important for parents to encourage play based learning in their home as this will help to develop the learning of the child. Parents can be involved with helping children have fun with simple mathematical concepts such as numbers, shapes and measure. It is crucial however, to have a balance between child initiated play and adult initiated play. It is important to ensure that children are regularly at the forefront of their learning, as this is when the child will learn best. This can easily be done by allowing the children to create their own rules or their own games.

I have attached a picture which explains different ways in which parents can help children when learning maths.

Another way in which children can learn maths is through stories. Stories allow children to make sense of both the real world and the imaginary world. A mathematically themed story can be shared either on a 1-to-1 basis or also within a group or classroom environment. Stories could be read aloud at home by parents to their children, or teachers could use stories to support their pupils’ mathematical learning and understanding. Furthermore, the pictures within a story book are also a good stimulus for the development of mathematical discussion. However, it is important to ensure that the questions asked and discussions had, are relevant to the children’s stage of mathematical development. If this is not the case, the children will not benefit from the story and instead this may confuse them. Using story books within mathematics can also support other areas of the curriculum too. For example, the children could act out the stories, linking to drama and performing arts, and place emphasis on the mathematical language or concepts involved in the stories, and how this links to what they are learning at the moment in class. Furthermore, reading story books to help support maths, will also improve the children’s language skills and influence their love of reading and language.

I have attached a video of a mathematical story book being read aloud, which is a perfect example of the kind of story books which could be used to assist mathematics for children.

References

Ginsburg, K. (2017). The Importance of Play in Promoting Healthy Child Development and Maintaining Strong Parent-Child Bonds.

Pound, L. (2003) Supporting Mathematical Development in the Early Years. Buckingham: Open University Press.

Pound, L. (2008) Thinking and Learning about Mathematics in the Early Years. Oxon: Routledge.

Skwarchuk, S. (2009) How do parents support preschoolers’ numeracy learning experiences at home? Early Childhood Education Journal, 37(3), pp.189-197. Doi:10.1007/s10643-009-0340-1.

Maths is everywhere!

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Maths has been described as “the language with which God has written the universe” (Galileo, 1564-1642).

During my time at school, I probably would have turned my nose up at this quote and disagreed with it. Although, this was probably because I hated maths at school, but didn’t everyone? After taking the Discovering Mathematics module this year however, I have noticed that mathematics truly is at the heart of everything that we do. Whether it is as simple as telling the time or as complex as working out the sale price of a pair of jeans in Topshop, we constantly use maths in our everyday lives.

The Scottish Curriculum for Excellence (2016) states that “mathematics is important in our everyday life, allowing us to make sense of the world around us and to manage our lives”. Furthermore, the Curriculum’s principles and practice documents also explain, that mathematics is crucial because it is implemented and used throughout the whole curriculum. For example, in art and music you use patterns, sequencing and symmetry and in science you are often using graphs and charts when it comes to experiments. Moreover, maths is also used in health and wellbeing, modern languages, design and technology, and ICT (CfE, 2016). In addition to this, Haylock (2014, p286) states that “mathematics provides us with knowledge and skills that are valuable not just for their own sake but because we can apply them to situations in real life and across the curriculum”. It is therefore crucial that we as teachers encourage children to engage with maths in all its purest forms. It is important that children not only learn maths within the classroom, but also outside the classroom.

Education through play is a great way for children to take the mathematical skills which they have learnt in the classroom, and use them in their everyday lives. Edwards (1998) states that both children and adults learn better when they are interested and motivated to learn, as this makes learning a purposeful and pleasurable experience. She goes on to state, that the presentation of mathematics through play is a great way to sustain the motivation and interest of the children. In my MA1 placement class, they had active maths once a week, where the children used the mathematical skills which they had learnt recently, for example percentages and decimals, and applied them to game like situations. I was in charge of organising and creating the games during my placement. I would create my own board games and crosswords etc. for the children. Doing this highlighted to me the importance of play involved with learning, especially within mathematics as it helps the children to put the skills which they have learnt into real life situations and contexts.

The following TED Talk by Jim Patrick, a budding mathematician at the young age of 6, explains the importance of maths and how we use maths in our everyday lives.

References

Curriculum for Excellence: Mathematics Principles and Practices. (2016). [ebook] Education Scotland, pp.1-4. Available at: https://education.gov.scot/Documents/mathematics-pp.pdf [Accessed 13 Nov. 2017].

Edwards, S. (1998). Managing effective teaching of mathematics 3-8. London: Paul Chapman, pp.2-7.

Haylock, D. and Cockburn, A. (2014). Understanding mathematics for young children. 4th ed. Los Angeles [u.a.]: Sage, pp.286-291.

Maths Anxiety

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Mark H. Ashcraft describes maths anxiety as “a feeling of tension, apprehension, or fear that interferes with math performance” (2002, p. 181). Math anxiety is a recognized stress disorder which, according to research shown in the attached video, 20% of our population suffer from (Turner & Carroll, 1985). It ranges from feelings of mild tension to strong fear when involved with any kind of mathematics. Physical symptoms include headaches, muscles spasms and aches, shortness of breath and increased heart rate. Furthermore, math anxiety also causes psychological problems such as confusion, the inability to concentrate and incoherent thinking (Arem, 2010, p.30).

Math anxiety does not only affect children in school, but adults too. For example, suffering from math anxiety may stop adults applying for certain jobs as they have a fear of failure and lack in confidence. Additionally, adults may struggle to deal with the mathematical challenges involved with dealing with personal finances, bills and mortgages.

The shorter video attached explains what math anxiety is, but it also highlights a problem which I myself face. During school I was never bad at math, in fact I was always in the higher classes and did manage to achieve a B in Higher Maths. However, this was not an easy process for myself. Maths was always something which took up a lot of my time, as I was always working to better myself. I knew that if I wanted to do well, I would have to work hard at it. However, there is still many areas of mathematics which I struggle with, which worries me for when it comes to teaching my own class. The attached video explains how pupils can easily sense if teachers suffer from math anxiety themselves and I am worried that this will be conveyed across to the children during my math lessons. If I myself suffer from math anxiety, how am I supposed to feel confident and competent in my own mathematical skills, when it comes to teaching the children? By taking the Discovering Mathematics module, I hope to improve my confidence in mathematics, so that when it comes to teaching the children, I can give them the best mathematical experience possible.

Furthermore, the University of Dundee have also researched into ways of targeting maths anxiety. They have introduced the Online Maths Assessment (OMA) to improve poor levels of mathematics and confidence within education students (Henderson, 2010). By doing this, the University of Dundee hope to target maths anxiety in student teachers so that they do not implement negative views towards mathematics in their future classes.

References

Ashcraft, M. (2002). Math Anxiety: Personal, Educational, and Cognitive Consequences. [ebook] Cleveland, Ohio: Blackwell Publishing Inc., pp.181-185. Available at: http://www.thinkingahead.com.au/Documents/math_anxiety-consequences.pdf [Accessed 30 Sep. 2017].

Henderson, S. (2010). Mathematics Education: The Intertwining of Affect and Cognition. Unpublished doctoral thesis. D.Ed. University of Dundee.

Nuffieldfoundation.org. (2017). Understanding mathematics anxiety | Nuffield Foundation. [online] Available at: http://www.nuffieldfoundation.org/understanding-mathematics-anxiety [Accessed 30 Sep. 2017].

Turner, J.R. & Carroll, D. (1985) ‘Heart rate and oxygen consumption during mental arithmetic, a video game, and graded exercise: further evidence of metabolically-exaggerated cardiac adjustments?’, Psychophysiology , 22(3), pp.261-267.

Why teaching?

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When asked why I want to be a teacher, there is simply only one answer.

As cliché as it may sound, I have always wanted to be a teacher. Ever since starting school at the age of 5, I have always looked up to teachers and admired what they do and the role they play in helping and assisting children in their education and personal development. It was my primary three teacher Miss Graham however, who inspired me and made me want to become a primary school teacher. She was not only amazing at her job but was also kind, caring, and funny and made me feel more excited and enthusiastic to learn. She changed the way I thought about school and learning and really made an impact on my school life, even from such a young age. Ever since then I have always wanted to become a primary school teacher and be exactly like her.

Having known for a long time that I wanted to become a primary teacher, I have taken every opportunity that I can to work with children both within and out with schools. For four and a half years I volunteered at my local Rainbows Unit, 1st Annan Rainbows, in which I planned activities, organised events and worked closely with young children between the ages of 5 and 7 on a weekly basis. This allowed me to develop my communication and leadership skills and enhance my patience and understanding when working with young children. Furthermore, working with these children also gave me the ability to relate to how a child’s mind works and seeing the girls develop and build in character within the years spent with then was such an exciting and rewarding experience and I know that is exactly how I would feel everyday as a primary teacher.

After so many years of wanting to be a primary teacher I am now more excited than ever to actually be at the University of Dundee studying Primary Education. I am looking forward to gaining more knowledge and experience over the next four years that will help and prepare me for my future as a primary school teacher. I am most of all looking forward to the placements that I will take part in where I will be able to put everything which I have learned over the many years of volunteering and work experience into practice in the classroom.