Discovering mathematics reflection- Have I discovered it?

www.google.co.uk/mathslove

www.google.co.uk/mathslove

I’m not quite sure I have the same love for mathematics as the numbers above….however, I can say that my fear of maths has reduced dramatically over the course of this Discovering Mathematics elective!

I have just finished my assignment about having a ‘Profound Understanding of Fundamental Mathematics’ and I have Liping Ma to thank for making those four elements of PUFM torment me every night for the last 3 weeks whilst trying to get to sleep – CONNECTEDNESS, BASIC IDEAS,MULTIPLE PERSPECTIVES and LONGITUDINAL COHERENCE. It has taken me the length of the elective to get my head round these 4 elements and be able to apply them to my own mathematics experiences and to our lectures. I am grateful to Ma as It has been very beneficial being able to view maths as a coherent whole. I feel through my engagement with the lectures and wider reading, I have started to repair my fragmented view of mathematics as I begin to open up the underlying network between different mathematical concepts.

ENJOYMENT

Looking back over the discovering mathematics elective, I wanted to pick out a particular lecture that I enjoyed and show how it helped develop my understanding of how mathematics can be used in wider society. 

Maths – Data & Statistics (Dr Elanor Hothersall)

During this lecture we had a guest speaker, Dr Hotersall, who works for NHS Tayside as a consultant in public health. Dr Hotersall was quick to identify one of the main barriers for individuals working in the health profession – a lack of understanding of the basic mathematical concepts. This was in regard to nurses and doctors calculating drug dosages, fluid prescriptions, concentrations of medications, interpreting research and probabilities which leads to treatment decisions, biomechanics and in particular pharmacodynamics (what happens to your body when you swallow medicine). To be honest, before this talk, I never really considered the underpinning mathematical literacy that was required to work in the health field, yet I would want to be fully confident that my nurse/doctor was competent enough in mathematics to prescribe me the correct drug dose and be able to calculate the correct reduced dose if they were treating my son. Dr Hothersall also went on to discuss how in her job she regularly has to compare health information and statistics, identify patterns and anomalies in results and decide acceptable levels of variation. Overall it was a fascinating lecture and really made me think deeply about how basic mathematical ideas and concepts can be crucial when working in such an important profession.

Where do I go from here?

Although my maths anxiety has reduced from the start of this elective it is still clear in my mind that I need to participate in professional development by continuing to engage with mathematics topics, courses and training opportunities. I need to work hard to develop my confidence and understanding of basic mathematical concepts if I am going to be able to explain them well to my pupils and encourage them to explore them with confidence. It is really important to me that I do not allow my pupils to experience any negativity when it comes to maths. I want mistakes to be welcomed as long as we can work back from them, see why it went wrong and then discuss, debate and predict how we could get to the right answer. Collaboration and discussion is going to be vital to achieving success in my maths classroom! I like to talk things through and feel that children will benefit from dialogue when working through mathematical processes. I am happy with the wide variety or research I have done for this elective and I feel that my blogs show a high level of engagement with my professional development.

I want to thank Richard and Tara for all their lectures and hard work throughout the module – especially for putting up with my puzzled face and silly little questions. Also a big thanks to all the guest lecturers who provided us with a good insight into how fundamental mathematics can be applied in wider society and to other professions beyond educational applications.

 

 

Is Maths beautiful?

 The Golden Section and Mystery of Phi

When you look deeply at nature and our surroundings, you can begin to identify and analyse the role that mathematics plays in aspects of our daily life. The Golden Ratio is considered to be everywhere: nature, music, our fingerprints, logo’s, artwork and buildings. The Golden Ratio is aesthetically pleasing to our eyes. Look at the two pictures below and choose which one you like best.

12180149_10153047590482511_313530694_n Mondrain

In one of our recent lectures on Art and Maths, we were introduced to abstract art by artist Piet Mondrian. If you like the picture on the right, then I guess you are an exception to the rule. I drew that picture with no knowledge of how to create Mondrian abstract art and just experimented with lines and colour. If you like the picture on the left better, have a think about why? Is there anything in particular that jumps out at you? The picture on the left draws on the mathematical concepts and principles of the Fibonacci sequence. The measurements between the lines within the picture all fit in with the numbers on the sequence. To me, it is more aesthetically pleasing and it seems to have more structure to it.

http://www.tate.org.uk/art/artists/piet-mondrian-1651

http://www.tate.org.uk/art/artists/piet-mondrian-1651

It is worthwhile to bare in mind that because the golden ration is pleasing to the eye, we might show an unconscious preference to it. Artists may make choices in their work which result in approximations of the golden ration without drawing on the concept and principles directly. Could it be that the golden ration is not so mysterious and that it is somewhat learned rather than innate? Just some food for thought.

I absolutely loved the TV series 24. Kiefer Sutherland had me on the edge of my seat constantly. His new drama ‘Touch’ draws on mathematical concepts such as patterns relating to numbers and sequences. Take a look at the trailer below and see the golden ratio, and links between numbers, guide the drama and story line through exciting twists and turns.

Further reading and links

15 Uncanny examples of the Golden Ration in Nature

The Golden Ratio

The Golden Ratio and Aesthetics 

Bridget Riley 

Bridget Riley is a famous English artist who began her first Op Art paintings using only black and white colours and simple geometric shapes such as ovals, squares and lines. Throughout her career, Riley never admitted to using mathematics in her art work, but it is hard to dispute the mathematical features which her work comprises of.

Movement in Squares - http://www.op-art.co.uk/bridget-riley/

Movement in Squares – http://www.op-art.co.uk/bridget-riley/

Two-blues - http://www.op-art.co.uk/bridget-riley/

Two-blues – http://www.op-art.co.uk/bridget-riley/

Below is an interesting article written by Dodgson, who investigates whether mathematical measures can can characterise Bridget Riley’s stripe paintings.

Mathematical characterisation of Bridget Riley’s Stripe Paintings 

I hope my discussions above and the links to further reading has helped support some of the claims that maths can be beautiful and that it can be found in many aspects of our daily life.

Tennis and Mathematics

The obvious mathematical concept involved in tennis would be counting. The scoring system in tennis is unusual as it doesn’t go up in units of one, or even units of the same amount. the follow picture depicts the different scoring options that could take place during one tennis game. As you can see, the first two scores (if won by the same person) go up in units of 15 (15-0, 30-0). However if the same person won the next point, one would presume it would be 45-0, however, due to the unusual scoring system, the new score would be 40-0.

http://mycodehere.blogspot.co.uk/2011_02_01_archive.html

http://mycodehere.blogspot.co.uk/2011_02_01_archive.html

Are there any deeper mathematical concepts embedded in the game of tennis?

Analysing statistics is a vital aspect for anybody involved in professional sport. Statistics can help players identify individual strengths and weaknesses and also the weaknesses of their opponents, which might inform their strategy and tactics for future matches. Percentages are the usual way of comparing players. In tennis, the typical statistics you see after each set include: first-serve percentage, service games won, break points saved, second-serve return points won, break points converted, unforced errors, forehand winners and so on.

 

 

MURRAY

http://www.changeovertennis.com/wp-content/uploads/2013/01/matchstatsmurraydjok.png

 

Before Andy Murray takes to the court to play a big semi-final against Djockovic, members of his training team will have analysed Djockovic’s match statistics from that specific tournament so see if they can identify any trends, likely predictions of play and any areas which are particularly weak so that Murray is stepping onto the court with informed information. This may not alter how Murray chooses to play, however if the match is not going as he would like, it would make sense that he would draw on the information he had been given to try and change up his tactics and play more aggressively to his opponents current weaknesses.

What about geometry?

Could tennis just be a problem of geometry? When playing a game of tennis, your aim is to win by the point by hitting the ball hard over the net, with spin, to a position where you opponent cannot return the ball. Is geometry involved in this process? In the path of the ball? The position of the player? The spin on the ball? It has been said by many commentators and some of the greats of the game that tennis is about controlling the middle of the court, cutting of angles and geometrical concepts support players in achieving this.

www.google.co.uk/tennisangles

www.google.co.uk/tennisangles

An attacking shot in tennis why relies on the play having a knowledge of angles is the angled approach shot. Hitting a short angled ball over the net forces your opponent forward, scooping up their return which should theoretically leave you with an easy volley to put away. You create a greater number of steeper angles when you step into the court and this forces your opponent to move around playing less confident and challenging returns. The following article discusses some of the geometric concepts in tennis further and provides some interesting food for thought for the next time you step out onto the tennis court.

Geometry and the Art of Tennis

What about the impact of top spin?

www.google.com/tennisbounce

www.google.com/tennisbounce

The ball bounce can greatly affect how an opponent returns your shot. Two properties which are involved in this process are Coefficient of restitution and Coefficient of kinetic friction. These two elements involve mathematical concepts such as ration, speed, distance and angles. The above diagram shows the bounce of a flat shot with no spin. The angle of incidence is approximately equal to the angle of reflection. This means the angle at which the balls impacts the ground and which the ball leaves the ground are almost the same. The ball slows down after it bounces due to the impact when hitting the surface.

www.google.co.uk/tennisbouncespin

www.google.co.uk/tennisbouncespin

The diagram above shows that when a ball is hit with top spin the angle of reflection is lower than the angle of incidence. The ball is not affected by the friction when impacting the surface of the court and the forward spin pushes the ball forward with greater speed after it bounces.

The following link is great for describing the effect different spins have on the ball bounce and how the different spins change the velocity, speed and direction of the ball.

The Physics of tennis 

 

The things I love and their affair with mathematics!

The ‘Discovering Mathematics’ module has encouraged me to identify and explore mathematical concepts that occur regularly throughout our lives. I have found it very interesting to do this with things I encounter everyday and wouldn’t normally associate with mathematics.

PARENTING

I have a two year old son who is now at the stage where he is exploring his surroundings, asking questions, solving basic problems and interacting with others around him. When I was sitting with him the other day, playing with his toys, I decided to look at how many of his toys incorporated some sort of mathematical concepts. The books that I would read to him encouraged him to identify shapes, count objects, try help the character solve problems like getting through a maze. One of his favourites is playing with shape toys and fitting them into a container. He enjoys sorting objects into shape, size or colour and I often find him counting his way up the stairs to his bedroom. All of these are very basic mathematical concepts: shape, organising and sorting, counting, simple problem solving etc. When reflecting on this, what hit home was the notion that my son enjoys doing all of these activities, he doesn’t shy away from it. The activities excite him and give him a sense of achievement and I wonder if these feelings and motivation leave children when they experience the mathematics curriculum when they start school.

 

HAIR & MAKEUP

Anyone that knows me well knows that I am obsessed with hair and makeup!! I am always waiting for payday to come so that I can visit the local MAC counter and purchase the latest lipstick or foundation. I decided to explore makeup and hair further to see if I could identify any mathematical concepts that are involved in these processes.

Hairdressers need to have a sound level of confidence and competence when dealing with basic mathematical principles. In their profession, they regularly work with; angles when cutting hair, ratios and percentages when mixing colours together, and symmetry when finishing off styles.

The following link shows a short video clip of how the mathematical concepts discussed above can be applied in hairdressing.

In makeup, concepts such as angles, symmetry, geometry and shape are regularly used to achieve the ‘perfect look’. The following video shows how makeup artists used measurements, and a compass-like tool, to apply makeup in the 1930’s for women with different shaped faces. The artist would measure the distance between the forehead and bridge of the nose and then from the nose to the chin. These measurements would then act as a geometrical guideline for outlining where to apply makeup to the cheeks, eyes and lips.

 

All girls know about the ‘Big C’ when talking makeup. It usually is accompanied by a picture of Kim Kardashian. Yes, I am talking about contouring. If you are a man reading this, then you will probably want to switch off now. Contouring is a work of art. It is a mission and can sometimes feel like mission impossible. Contouring works to give shape to an area of your face and then enhance your facial structure through makeup. It can give the illusion of higher cheekbones, slimmer nose and chin, and it can be used to help alter your face shape. Similarly to how makeup was done in the 1930’s, contouring draws on concepts such as shape, symmetry and measurement.

12233353_10153072637347511_98309362_n                             12272733_10153072637422511_540116491_n

http://4.bp.blogspot.com/-xz83kd-zKqc/U_XppFboYWI/AAAAAAAAAg4/HLJa-xfg6o4/s1600/b1d33a589ca631d11d054eaf25b42893.jpg

http://4.bp.blogspot.com/-xz83kd-zKqc/U_XppFboYWI/AAAAAAAAAg4/HLJa-xfg6o4/s1600/b1d33a589ca631d11d054eaf25b42893.jpg

Links

Cut it out and curl it up – Maths Careers 

Geometric beauty

 

Breaking down the idea of ‘Longitudinal coherence’ in mathematics

Ma (2010) identified ‘Longitudinal Coherence’ as the final property of having a profound understanding of mathematics. If I am totally honest, this is the one that baffles me. I think this is because of my own fragmented experience of mathematics. When I was at primary school, I was never encouraged to link topics of learning, or reflect on more advanced learning, thinking about which concepts I had developed in order to get where I am now in my mathematics understanding.

I think I must have read the below definition of Longitudinal Coherence about 100 times:

‘Fundamental understanding of the whole mathematics curriculum and no limitation to the knowledge that should be taught in a certain grade. The ability to exploit an opportunity to review crucial concepts that students have studied previously and know what students are going to learn later and building the foundations for this future learning.’ (Ma, 2010, p.121)

After reading it 101 times and still feeling perplexed, I knew that I would have to do further reading to try and get different examples of what longitudinal coherence was in order to fully understand this property. Again, I found this difficult as every time I felt I was starting to get to grips with the concept, it began to feel like I was talking more about ‘connectedness’ than longitudinal coherence. I guess that it’s okay to have slightly different takes on the 4 crucial concepts of PUFM developed by Ma. I would say that connectedness and longitudinal coherence could have been combined as they do have very strong links with one another.

After a lot of research, I finally found some work which has helped me have a better understanding of what I believe to be longitudinal coherence from a teachers perspective:

“We produce many students who do not think globally – or to use a more common word these days, holistically- about mathematics. In the present context, teachers who come through such a training program may know the individual pieces of the school curriculum, but they are less adapt at seeing the interrelationships among topics of different grades. (Wu, 2002,p.19)

The above quote came with an example of helping students see the connections and coherent development of whole numbers all the way through to algebra:

Whole numbers ———> fractions —————> finite decimals, ratio, rates, percent, algebra (p.20)

Maybe Wu (2002) provided a simpler definition of longitudinal coherence than Ma (2010), or maybe because his description was accompanied by examples I was able to follow it better and have a clearer understanding. My role as a teacher is to continually encourage pupils to identify recurring themes and mathematical concepts when approaching new topics. Pupils should be able to see and draw on previous learning to help them develop new understanding. This should happen throughout the whole-school mathematics curriculum to enable students to see why previous learning was relevant and how it is supporting them in their current and future experiences.

Although this property initially baffled me, it is now the property which I connect with the most as I don’t feel I was given the opportunity to develop this at school. If anything, this places me in better stead for my future teaching pedagogies. I will always be able to look back on my own mathematics journey and ensure that I do the opposite to what I experienced at school.

Reflecting on my engagement with this module so far, I have found it extremely beneficial to breakdown the four properties of PUFM, (connectedness, multiple perspectives, basic ideas and longitudinal coherence), in order to develop my understanding of them. I feel that I can now engage with upcoming lectures with a different perspective and approach to mathematics. I want to be able to connect with the different topics we cover on a deeper level. I want to see how I can apply the 4 properties to help develop my own mathematics confidence and also my competence in developing positive teaching strategies.

Sources

Ma, L. (2010) Knowing and Teaching Elementary Mathematics – Teachers’ Understanding of Fundamental Mathematics in China and The United States. London: Routledge

Wu, H. (2002) “Longitudinal coherence of the curriculum” in What is so difficult about the preparation of mathematics teachers?. University of California: Berkley. Available at: https://math.berkeley.edu/~wu/pspd3d.pdf Accessed 31/10/15

Basic Ideas – THANK YOU!

I am over the moon to be writing a blog post about the basic ideas of mathematics! During this module we have been encouraged to look past the simple ideas and working on developing a relational understanding between different mathematical concepts. SO, to be able to take a step back and explore the advantages of basic ideas is music to my ears! Basic maths is definitely my type of maths!

Ma (2010) describes the property of basic ideas as being ‘ideas that recur throughout mathematics learning creating a solid foundation for future learning’ (p.121). As teachers, this is an essential element which we need to encourage pupils to identify when approaching different mathematical situations. The basic/simple concepts link together individual topics and allow students to see patterns between the different strands, exploring the interconnectedness of mathematics. Rutherford (1990) states ” A central line of investigation in theoretical mathematics is identifying in each field of study a small set of basic ideas and rules from which all other interesting ideas and rules in that field can be logically deduced.” Once these basic ideas have been learned, pupils can use this existing knowledge to help them solve more complex calculations.

I have chosen the topic of fractions to further explore the notion of basic ideas in mathematics.

What are the basic ideas of fractions? I perceive the basic ideas to be:

  • Fractions represent a number of equal parts of a unit/whole number
  • Fractions can represent a point on a line
  • Numerator (top number) represents how many sections of the whole part you have
  • Denominator (bottom number) represents how many sections are in the whole.

What are the more complex ideas of fractions? I see the more complex ideas to be:

  • They can represented as a proportion of a set ( leaners need to build on the basic idea of fractions representing a number of equal parts of a unit to extend the meaning to include a number of equal parts of a set.) (Haylock, 2014)
  • They can be used to model division – 3/8 can also be read as 3 divided by 8.
  • Knowledge of fractions can be extended to knowledge of ratio – 4/7 can be see as 4:7 E.g. For every 4 females, there are 7 males.
  • Improper fractions (top-heavy fractions) -leaners need to understand the basic idea of numerator and denominator to be able to solve problems involving improper fractions. For example; knowing that 6/6 is equal to 1 whole, 12/6 is equal to two, 18/6 is equal to three.
  • Mixed fractions – using the knowledge of numerators and denominators to know that 3 1/3 represents 3 wholes and then 1 section of 3 out of a whole.
  • Addition and subtraction with mixed fractions – need knowledge of common factors to be able to convert both denominators into the same times table. (links to the idea of connectedness)

 “Knowing about a key idea in mathematics, such as fractions, involves knowing how fractions relate to whole numbers, where they belong on a number line, how they link to ideas of ration and proportion, the connection between fractions and the division operation, the links between a range of modes of representing fractions, and a host of other points” (Haylock, 2014,p.9).

Understanding the basic ideas and concepts of fractions creates a web from which we can build on to develop our knowledge and relational understanding of more complex fractional calculations.

The following video is a good resource for introducing the bass ideas and concepts of fractions.

 

Links and extra reading

Fractions – The basics  – Good resource starting with the basic ideas and concepts involved in fractions.

Sources 

Haylock, D. (2014) Mathematics Explained for Primary Teachers (5th edn.). London: SAGE

Ma, L. (2010) Knowing and Teaching Elementary Mathematics – Teachers’ Understanding of Fundamental Mathematics in China and The United States. London: Routledge

Rutherford, J.F (1990) “The Nature of Mathematics” in Science for all americans. Oxford: Oxford University Press. Available at: http://www.project2061.org/publications/sfaa/online/chap2.htm Accessed: 31/10/15