Category Archives: Discovering Mathematics

Maths = Discovered?

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In all honestly maths has always been something that I have enjoyed as it is basically a universal language. Providing the use of the same number system many mathematical problems can be understood all over the world regardless of the language spoken and for some reason I feel comfort within this. Through the elective module I feel that I have truly discovered mathematics and it is something I will never stop discovering.

Through the module one of the many things I have been introduced to is the Fibonnaci sequence and the golden ratio; this is something I had no knowledge of before. Reflecting back I find it strange I haven’t come across this before considering how often it features in different aspects of life such as nature, buildings and even musical compositions! In order to develop my understanding of these concepts in the future I would like to investigate if they appear in any other areas of the curriculum and research ideas of how this can be implemented within the classroom. I really like the fact that these procedures are embedded into so many different areas and now I am aware of them I can begin to identify them.

I feel through engaging with the works of Liping Ma my understanding of what is needed when teaching mathematics has completely changed. Before I thought that a sound knowledge of the mathematics being taught would be suffice but now realise that to be a good teacher there is much more that is needed. The four principles connectedness, multiple perspectives, basic skills and longitudinal coherence will undoubtedly influence my future teaching practice. By thinking of my own learning experiences I can relate to the principles and realise that as a learner only knowing the mathematical procedures and not understanding the concept is not very beneficial as it makes it difficult to use the knowledge learned in other areas of mathematics. When I become a teacher I will make a conscious effort to gain a PUFM and teach mathematics in an engaging and efficient way.

Maths and Mozart

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In secondary school I was told by one of my teachers that before her higher maths exam she listened to Mozart because someone once told her that his music has hidden mathematical notation and apparently subconsciously unlocks mathematical parts of your brain. She ended up getting a great grade for maths and swears this was the reason why! I am unsure whether she was better at maths than she thought or if the music really did make a difference. However, when revising for my maths exam I of course had Mozart playing in the background. I am not sure whether Mozart is a mathematical genius or not but after sharing this story his track sales probably rocketed due to all the students giving it a try! The maths and music workshop we had reminded me of this and I decided to try and find out if there any evidence to support these claims and to my surprise there actually is!

http://www.bbc.co.uk/cambridgeshire/content/images/2006/10/17/mozart_maths_music_203_203x152.jpg

http://www.bbc.co.uk/cambridgeshire/content/images/2006/10/17/mozart_maths_music_203_203x152.jpg

There is a term known as the ‘Mozart Effect’ which suggests listening to Mozart could induce a short term improvement on the performance of certain kinds of mental tasks known as spatial-temporal reasoning. People who possess this type of reasoning can visualise how things fit together and how they can be manipulated into different patterns. They also have good problem solving skills. In early 1992, researchers from the University of California, Irvine discovered that college students  exhibited better temporal reasoning skills on Math tests while a Mozart Piano Sonata was being played. This was a significant discovery, but it only lasted 10 minutes.

According to Mozart’s sister,  he “talked of nothing, thought of nothing but figures”, it is also recognised that he noted mathematical equations in the margins of some of his compositions such as Fantasia and Fugue in C Major. These equations did not directly link to the music but it shows that he had an attraction to mathematics. Some believe he may have composed his music using mathematical equations and it is said that Mozart may have divided his piano sonnets using the golden ration. Livio says that there is mathematical symmetry to Mozart’s music. Mozart even had a musical composition dice game which had performers roll the dice to decide which bar would be played.

The Magic Flute is said to have the number three included in many ways such as the three-note rhythm sequence. It is written in E flat major which is a key with three flats, many of the characters come in threes and it has a three part harmony.

Overall, it has not been proved that Mozart used mathematics in his music or whether it has any lasting affect on a persons mathematical ability. However, it has not been proved that it does not. Did I mention I also did well in my maths exam, coincidence?

Refs.
Sautoy, M (2013) http://www.theguardian.com/music/2013/apr/05/mozart-bach-music-numbers-codes
Hunt, P (2013) http://www.electrummagazine.com/2013/06/mozart-and-mathematics/
http://musicmathsystem.com/mozart-effect/

Is Maths Natural?

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When first introduced to the idea that maths is in nature it puzzled me as I had overlooked how maths is intertwined with non man-made items. I had only thought of maths being included in items such as buildings, roads and cars where humans have influence on the infrastructure and mathematics has been purposely included. Flowers, trees, animals and surprisingly people relate to mathematics! and once the seed had been planted (pardon the pun) I began to recognise mathematics everywhere!

Radial symmetry is a part of mathematics that can be identified in many natural items such as flowers, fruit, insects and starfish! Merriam-Webster defines this is as the condition of having similar parts regularly arranged around a central axis.

Helianthus_whorl

Sunflowers are appealing as they are bright and bold, and they are a prime example of radial symmetry although they relate to mathematics in many more ways. In the Fibonacci Sequence each number is determined by adding together the two numbers that preceded it.
1 2 3 5 8 13 21 34………. (and so on)
If all the seeds within a sunflower are added together they equal a number from this sequence. Actually many natural items make leaves, petals and seeds in quantities that equal numbers in the Fibonacci Sequence. This a why four-leaf clovers are so rare and are considered to be lucky!

Scientists believe the reason that plants follow mathematical rules simply due to efficiency. Sunflowers are able to have the maximum number of seeds if each is separated by an irrational-numbered angle. The most irrational number is known as the golden ratio which is as far away as any number can get from being a near any fraction.  It relates to the Fibonacci Sequence because if a number in the sequence is divided by the preceding number it produces a number exceptionally close to the golden ratio. We are apparently hardwired to be attracted to things in which the proportions are close to the golden ratio. I personally thought I found sunflowers aesthetically pleasing due to the bright colour but who knows it may be the golden ratio working its magic!

http://www.dangerdolan.tv/15-beautiful-examples-of-mathematics-in-nature/

Understanding Fundamental Mathematics

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In the Discovering Mathematics model I was introduced to the idea of fundamental mathematics. My understanding of this was that it was the skills that we use within maths such as multiplication, division, addition and subtraction. However, after reading Liping Ma I have realised it is so much more than this.

According to Ma (2010) there are four main principles that teachers need to know to have a reached a profound understanding of mathematics Connectedness,. Connectedness is one of these and teachers should be able to make connections among mathematical concepts and procedures. This was something I had not thought about before and led to me realise that there are some mathematical procedures I can carry put but have no idea of the reason why. An example would be when dividing fractions I have been taught to flip the second fraction and then multiply the top numbers and multiply the bottom numbers. I decided to try source the reasons for this and if I am honest some books and web pages confused me further and I could to relate why this is not taught as it may confuse the procedure further. I came across an explanation which instantly made sense to me by Raymond Johnson. He starts with the problem:

beg

and explains that it is easier to explain when it is written as a compound fraction:

compounf
He says that that dividing by a fraction is difficult but dividing by one is easy. To turn a fraction into 1 you multiply it by its recirprocal. Which would be the fraction divided by its inverse. 1 is the only number that can be multiplied with without changing the value and therefore the original fraction should be multiplied by:
invers
Therefore,

end

by doing this the bottom fractions become 1 which do not make an impact and therefore this is why the original factor becomes multiplies by the inverse.

Ma (2010) states that multiple perspectives is also a principle that is require within a profound understanding of fundamental mathematics. this requires teacher to be able to provide explanations of approaches as this leads to the students having a flexible understanding of mathematics. Now knowing the reasons for why you divide fractions in this way I can connect this idea to other areas of mathematics such as equivalent fractions.  Through investigating this I am begging to understand Ma’s four principles and their importance.

 

Johnson, R (2011) http://blog.mathed.net/2011/07/pretty-short-explanation-of-invert-and.html

Demand Planning

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Demand planning is not something I knew anything about before learning about it in the Discovering Mathematics elective module, and if I am truly honest the thought of a three hour workshop on this topic did not exactly excite me. However, I was completely proved wrong and in fact the demand planning workshop has been my favourite by far!

Rose (undated) describes ‘Demand planning is a multi-step operational supply chain management (SCM) process used to create reliable forecasts.  Effective demand planning can guide users to improve the accuracy of revenue forecasts, align inventory levels with peaks and troughs in demand, and enhance profitability for a given channel or product’.

With this in mind we participated in a game which involved us ordering products whilst considering the demand for this due to the time or year or the selling price of the item.  For example it is less likely you are going to buy a Christmas selection box in April – unless you’re highly organised! Not only did the game involve us making simple calculations such us multiplying the number ordered by the price we then had to work out how many were sold using percentages. The profit margin on items also had to be considered as we were trying to make as much money as possible. Whilst working through the activity all purchases, stock, profit and loss was recorded on a printed spreadsheet. Working in pairs made this activity really exciting and I think for a split second I’d spent my share of £69,00! Throughout this activity I kept thinking if adults could be so absorbed by this activity imagine how engaged children would be.

In my future teaching I will definitely incorporate this task as it can be adapted to different ages. It provides opportunity to introduce business like language such as profit and loss, starting budget and mark up on products. I like the fact the activity included thinking of a company name as I feel children like to personalise and have ownership of their work. The activity can include other aspects of the curriculum as the children could produce formulas to include in the spreadsheets to work out calculations. Overall the activity is a way of showing how mathematics is used in the wider world whilst being a highly engaging and enjoyable experience.

 

 

Rose, M (undated) http://searchmanufacturingerp.techtarget.com/definition/demand-planning

Op Art – A Mathematical Art Lesson

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When I was on my first year placement I was faced with the challenge of choosing which curricular area I would like to teach for my first lesson. After a discussion with my class teacher we decided that I would deliver an art lesson as this is subject I have a personal interest in. However, I decided I would like to incorporate another curricular area to the lesson to cover interdisciplinary learning. This lead me to the work of Victor Vasarely and Op Art.

Op Art is an art form which is a mix between abstract art and optical illusions.

Vega – Victor Vasarely 1957 Acrylic on canvas 195x130cm http://www.op-art.co.uk/victor-vasar

 

 

 

 

 

 

Cassiopee II – Victor Vasarely 1958 Acrylic on canvas 195×130 http://www.op-art.co.uk/victor-vasarely/

Vega (1957) was the inspiration behind my lesson because it is striking and interesting in my opinion . I also felt it had the potential to allow me to include mathematics into my lesson due to the various shapes and sequential pattern. I took myself to Pinterest to research Vasarely further and discovered art lessons which were based upon recreating his work. I took some ideas from this and adapted this to meet the needs of my class. Whilst researching the lesson I came across a blog www.artfulartsyamy.com which was very helpful on explaining how to recreate the spheres and this set the foundation for my lesson.

I began the lesson with a Powerpoint about Vasarely and discussed the images with the class and asked them what they found interesting relating to shape, colour and composition. The art work was completed on squared paper to ensure all the background squares were of equal size. Various sized circles from the flat shapes in the classroom were made available and the children chose how many they would like to use and which sizes. The children were asked to place the circles on the paper, draw around them and then divide them into quarters. Next they were asked to draw the lines that would help transform them into sphere like shapes.

http://www.artfulartsyamy.com/2012/01/lesson-plan-op-art-spheres-easy-way.html

Once completed the children were asked to choose one or two colours and alternate these with each square and repeat this over the whole piece. These were the results…

 

IMG_3374 IMG_3377 IMG_3375

 

Overall I feel that this lesson was very successful and the mathematical elements made it more accessible for the children who did not feel that art was their strongest subject. Not only do the finished pieces look great the children were very proud of them as it took patience and effort to ensure the finished product had the magnified effect!

Maths TDT

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Beyond Legend: Stand and Deliver as a study in School Organizational Culture – Roger C. House
Stand and Deliver is a movie about a teacher who is on a journey to turn a school around, the teacher says “if life is unfair you must work harder!”. The film is has received some harsh critisms stating that the film is one-dimensional, socially disturbing and racially oppressive. Also it has been said that the clichéd plot distracts from the deeper meaning of the film. However, the film does successfully illustrate the way in which schools function as social systems in deprived areas.