Category Archives: Discovering Mathematics

Maths and astronomy

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for me, I have always been extremely interested in space, the universe and the stars and have always enquired about what more there is to find and what actually happens outside of the earths little bubble. Following the discovering mathematics module has enlightened me about many applications of maths that I had previously never thought about, and our discussion with Simon Reynolds (science learning manager, Dundee science center) was no exception.

Stars

I always knew that the observable universe was unfathomably big, and that the amount of stars that can be seen is around 10 to the power of 22. However seeing this number written out of the short form was truly amazing as it comes to 10,000,000,000,000,000,000,000, and that is only the amount of stars we can see! Using exponentiation (the power of) here is clearly extremely important as writing out a number with twenty two zeros can be a very strenuous task, so instead mathematicians looked to create exponents (the number to the right of the base) to create a shorthand for large numbers. In this case instead of Writing a 23 character number out, astronomers and mathematicians alike need only four, 10^22. This makes it easier for people to denote what they mean when writing and talking and is clearly a vital component when dealing with really really big numbers.

Distance

As most prominently portrayed in films, light-years are a  measure of distance, specifically, the distance light can travel in one year (6 trillion miles). The use of light-years here provide us with a relative understanding of how far that is and gives people a rough understanding of the sheer size of the universe. However 20th century astronomer  20th century astronomer Robert Burnham Jr created an ingenious way to portray the distance of one light-year and ultimately of expressing the distance scale of the universe, in understandable terms. He did this by relating the light-year to the astronomical unit (AU) – the Earth-sun distance. One Astronomical Unit, or AU, equals about 93 million miles (150 million km). Burnham noticed that, quite by coincidence, the number of astronomical units in one light-year and the number of inches in one mile are virtually the same. There are 63,000 astronomical units in one light-year, and 63,000 inches (160,000 cm) in one mile (1.6 km) (Mclure, 2016). This enables us to have a relative understanding of the distance between our planets in the Solar system and beyond.

Maths and astronomy are clearly closely linked together and the use of abbreviations or creating new number systems is vital in understanding and clearly expressing astronomical things. Reynolds lecture showed me that space isn’t anywhere near as confusing as first thought as he was able to give me examples that I could relate too, such as Burnhams earth-sun distance. this once again highlights the importance of having multiple perspectives and connectedness when teaching mathematics as he was able to explain to us an idea that we wouldn’t have thought of in a way that we were able to link back to what we previously knew.

McClure, B. (2016) How far is a light-year? Available at: http://earthsky.org/astronomy-essentials/how-far-is-a-light-year (Accessed: 1 December 2016).

Is Maths Fun?

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For me, I always enjoyed maths at school, I enjoyed having one answer to a question and being able to proof that answer yourself, and I was actually rather good at it. However coming into this module I knew this was not the case for many of my colleagues and that within schools ‘maths anxiety’ can be rife. I felt I was important for my own personal and professional development to try and find out why  this sense of ‘maths anxiety’ exists in a attempt to abolish it in my own future teaching.

I have found that a dislike for mathematics can be categorized into two groups; Environmental or individualized factors

Environmental

Instruction

Students who are taught in a way that relies too heavily on rote memorization isolated from meaning have difficulty recognizing and retaining math concepts and generalizations.

Curricular materials

Students who do not “get it” the first time are not likely to “get it” the next several times it is taught in the usual manner. Moreover, underachieving students are frequently assigned repetitious and uninteresting skill-and-drill work each year in order to teach them “the basics.” This type of work often represents a narrow view of mathematical foundations and a low level of expectation of students’ abilities. It limits opportunities to reason and problem solve.

The gap between he learner and subject matter

When the mathematics content being taught is unconnected to students’ ability level and/or experiences, serious achievement gaps result. This situation may occur if students are absent frequently or transfer to another school during the academic year.

Personal or individualized factors

Locus of control

Some students believe that their mathematical achievement is mainly attributable to factors beyond their control, such as luck. These students think that if they scored well on a mathematics assignment, they did so only because the content happened to be easy. They view their achievement as accidental and poor progress as inevitable. In doing so, they limit their capacity to study and move ahead (Beck, 2000; Phillips & Gully, 1997).

Memory ability

Some students lack well-developed mental strategies for remembering how to complete algorithmic procedures and combinations of basic facts.

Attention Span

Students may be mentally distracted and have difficulty focusing on multistep problems and procedures.

Understanding Mathematical language

Students are confused by words that also have special mathematical meaning, such as “volume,” “yard,” “power,” and “area.   (Sherman, Richardson, and Yard, 2014)

 

 

 

I feel as a future teacher it is vital to understand why children don’t enjoy/understand the maths that is being taught and I feel this has close links to Ma’s concept of teachers acquiring a profound understanding of mathematics in order to teach mathematics effectively. I feel a teachers job is to consider both environmental an personal motivators within peoples feelings towards maths and attempt to teach in a way that is universal, so that all children can understand to some degree. Having a PUFM will by and large take care of he environmental factors as a teacher would b teaching in a way that promotes the basic ideas, through multiple perspectives, connecting different mathematical ideas together, while understanding what each student has already done and where they are going (longitudinal coherence). If a teacher has a strong understanding of PUFM I feel the environmental factors that prohibit students during maths lessons are taken care of

However the personal and individual factors that can cause a dislike for maths can be far harder to counteract, and I feel it is the role of any good teacher of any subject to address these with students at a personal, individual level. giving students the tools to address their own issues with learning is a vital part of being a teacher; whether it be showing different revision methods, practicing older knowledge, extra help within lessons or simply sitting down and talking to children about their anxiety’s can be a huge help in dealing wit their personal motivators.

looking into what makes children dislike maths has informed me on a personal and professional level and I feel that it will help my understanding of the way children learn maths in the future

Sherman, H.J., Richardson, L.I. and Yard, G.J. (2014) Why do students struggle with mathematics. Available at: http://www.education.com/reference/article/why-students-struggle-mathematics/

 

Maths is art?

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Maths is everywhere, from structures and buildings to technology and science, mathematics can be seen by anyone at anytime. However, what some people fail to notice is the importance of maths within art, with sequences and ratios being used to create some of the most famous paintings and designs ever created. ad far back as Leonardo Da Vinci, art has been considered when creating beautiful masterpieces that survive generations.

Leonardo Da Vinci – The last supper

picture1

This famous depiction of Jesus’ last supper with his disciples is well regarded as one of the most famous paintings ever created, although many people do no know the importance of maths within the masterpiece. The pattern drawn over the painting is known by the term ‘Golden ratio’ and is thought to be aesthetically pleasing to the eye. The golden ratio is used to draw an onlookers eye to the focal point in the painting, in this case Jesus, then as the eyes look closer more details are noticed.  The golden ratio uses proportions, ratios, symmetry and balance to create an aesthetically pleasing look, and has been used in paintings and structure to create beautiful works of art.

Fractals – Fibonacci

picture2golden-rectangle

The Fibonacci sequence is a series of numbers where a number is found by adding up the two numbers before it, however, this sequence can also be used to create beautiful pieces of art containing geometric shapes. The sequence starts 1,1,2,3,5,8,13 and so on, and can be used to create the ‘perfect spiral’ in geometric form, this once again draws our eyes to the centre of the work and then presents the rest of the design.  

Maths and art lends its self to Liping Ma’s idea of ‘multiple perspectives’ (Ma, 2010) and how their are multiple ways of looking at and presenting one idea. The examples of art  I have presented show the importance of looking at maths in different lights. in the example of the Fibonacci, a simple sequence created by one man has been studied to the point that people noticed an artistic implementation that could be exploited. Therefore people having differing ideas on what certain math’s is trying to explain and show us Is vital in the human exploration of mathematics as there is always different ways to express the same subject matter.

 

Bookies or Bust

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One area the particularly interested me throughout my development within the module was the importance of math’s within gambling, something I myself occasionally partake in and something many of my friends do. Gambling consists of many forms including betting shops, casinos, and racing but for this I will focus on casinos in particular.

A study by the University of Las Vegas found that in the 2013 fiscal year, 23 casinos made over five billion dollars, averaging over $630,000 a day, per casino. This is clearly a big business and the reason for Las Vegas’ continuing success since 1941, but how do casinos continually make money? The answer is simple, probability and chance.

Casinos run on the idea of giving odds on a particular outcome to come to fruition, people then put money on the outcome that they feel is most likely and this results in either a person loosing their stake or winning money. A game I will focus on for this post is roulette, one of the more popular games and arguably the simplest.

Here are the chances of each outcome in the game roulette:

Image result for roulette chances table

(No Date) Available at: http://www.roulette.co.uk/wp-content/uploads/roulette-odds1.png (Accessed: 2 December 2016).

As we can see the odds of winning are better within Europe which means that American casinos will inevitably earn more money than European ones. Furthermore it is clear to see that the chance of winning is always slightly more than the pay-out, this is for two reasons, firstly to save the casinos as much money as possible when they do lose, and secondly to provide an easy pay-out, using whole numbers, thus removing any need for small money when paying out.

Casinos rely on an elementary concept taught in all primary schools, probability and chance. Whether it be tree diagrams, percentages, decimals or fractions, children will learn how to calculate the probability of one thing happening over another. As a teacher it is important to show real world examples of this however a casino and the idea of gambling probably isn’t the most appropriate way of showing a real world application. A more appropriate way may be to calculate the probability of fruit growing on a tree etc.

Ma’s principle of multiple perspectives links perfectly to teaching probability and chance as their are multiple methods of expressing and showing probability and it is important that children understand that there is more than one way to work out and express a chance of an outcome happening. It is for this reason a teacher must have  Profound understanding of mathematics, in order to communicate effectively the many ways of expression and calculation when teaching this concept. It is vital that a teacher understands that all children understand things differently an that although 10% may be the easiest way for one child to express probability, 0.1 or 1 tenth may be easier for other children.

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Making Mathematical Connections

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During my engagement with the discovering mathematics module, and my research for my essay, I came across an enlightening post from Linda M. Gojak, the president of the National Council of Teachers of Mathematics (NCTM). Within the post she states the importance of making connections within mathematics, and how one experience of connections being made within topics gave her one of her most memorable experiences whilst teaching.

“One day, a student commented that this was just like what they had studied at the beginning of the year. When I gave a puzzled look, the class pointed to the posters still on the wall from our first unit of study and said, “You know, that factor and multiple stuff.” I had a new appreciation for the power of providing experiences that enable students to make connections among mathematical ideas.” http://www.nctm.org/News-and-Calendar/Messages-from-the-President/Archive/Linda-M_-Gojak/Making-Mathematical-Connections/

Linda’s experience of a class recognising connections between two otherwise unrelated topics shows how learning topics in isolation within mathematics is far less efficient than taking a mathematical concept and considering how it originates, extends, and connects with other concepts. Teaching maths as a unified body will inevitably help teachers develop a profound understanding of mathematics (Ma, 2010) and in turn will give students the tools and instruction that students need to make their own connections and create their own perception of the mathematical topics and processes.

 

 

 

Base 10 or not?

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A numeral system is a writing system for expressing numbers, using digits or other symbols in a consistent manner. Here in the United Kingdom we use a simple base 10 system or decimal number system. In base 10, each digit in a position of a number can have an integer value ranging from 0 to 9 (10 possibilities). The places or positions of the numbers are based on powers of ten (e.g., hundredths, tenths, tens, hundreds, thousands). Exceeding the number 9 in a position starts counting in the next highest position. This seems simple to those who have been brought up using the base 10 system, but is it the only system we use?

Lets start with the most relatable, a base 60 system that is used all around the world every single second of every day. TIME! We use this base 60 system everyday, with both seconds and minutes using the numeral system. The base 60 system counts up from zero all the way to 59 before changing into another set, for example; if we had 57 seconds on a timer, but there was 6 seconds to be added, we would say that it took 1 minute 3 seconds, and not 63 seconds. The use of a base 60 system is actually older than that of the base 10 system. The ancient Sumerians, the oldest known civilization on Earth (c4500–c1700 B.C.) used a sexagesimal system too that can still be seen today, in angles, bearing and most importantly time. But why are base systems so important?

The use of a base system makes it easier to count large numbers, using the example of the seconds can show this perfectly. There are 86400 seconds in one day, but this is a mouthful and would be hard to keep track of throughout the day, using the base 60 system of seconds we can reduce this down to 1440 minutes in a day, making it much easier to work with. But wait, Minutes too use a base 60 system so we can reduce the 1440 minutes even further to 24 hours, thus creating time as we know it today in the 24 hour clock. Other number systems can include a dozenal system (base 12), Binary (base 2) and many many more. Base systems, put simply, are just ways of communicating numbers in the simplest form necessary for what you are trying to count.