Discovering Mathematics LAG – Just another blogs.glowscotland.org.uk

Mathematics is all around us- Play

To some the thought of mathematics can be tedious, so to imagine maths as play seems quite unusual.

However this is not the case as all games contain an element of mathematics.

In the input on play we were able to explore some of these areas:

Yahtzee: A game of chance and probability, made up of 5 regular dice and a score card the aim of the game is to fill the card with as many high scoring options as possible with the the total added up score of the highest number being the winner. Mathematical elements within this game are: addition, problem solving, probability/chance and number recognition.

Monopoly: Work around the board purchasing properties and avoiding paying competitors for landing on their purchase date squares while picking up chances and avoiding jail. The aim of the game is to bankrupt every other player or finish with the highest amount of money or property value. Mathematical elements within this game are: Money, addition, problem solving, subtraction, chance and negotiation.

Guess who: Using yes/no questions eliminate every character on your oppositions board apart from the one they have hidden on their chosen card, first person to guess the opponents character wins. This game is not evidently associated with mathematics however realising the number of characters with iconic similar features allows you to clear the board quicker also recognising patterns this way.

Scrabble: Each competitor gets 7 letters chosen at random each turn and can use as many of these to create words on the board, after the first word is palced the rest must always connect to another word already played. Letters are worth points with those that are more uncommon scoring higher, positions on the board also are worth more points than others. Mathematics is present in the addition of scores and multiplication when on appropriate board spaces.

Mathematics is all around us- Medicine

Medical personnel including doctors and nurses require a very solid understanding of mathematics in order to obtain exact figures.

Drug doses must be extremely accurate in order to avoid severe damage to patients. In Children the doses are measured in kg so the measurements must be correct. Doctors that have miss-calculated doses have been shamed by reporters and the outraged general public. The ability to identify how strong medication or aesthetics is requires an in depth understanding of the mathematics involved.

Another aspect of mathematics in medicine is the number of charts and graphs that are used and completed on a daily basis. Sews charts allows patients vital signs to be monitored to ensure their health is steady or improving.

Charts and graphs medical personnel to monitor statistics, patients progress and condition, notice patterns in diseases, and predicting results.
Charts and graphs provide a visual representation of the facts that allow doctors to make quick educated decisions about the patients treatment and care.

Am I understanding?

Having thought initially that this mathematics module would open my eyes to how to teach mathematics I would never have suspected how this would come about.

When asked what my thoughts on the fundamentals of mathematics are at the start of the semester I may have stared you blankly in the face or uttered the basics which I believed to be addition and subtraction.

However I now realize this is certainly not the case and there is far more to mathematics than + or -.

Mathematics is everywhere, from my previous blogs you may have realized that it is in fact all around us. All it takes is an understanding of connectedness, the ability to approach a problem from every angle, an understanding of the basic ideas, and to realize where problems are relevant and how to pass on tis information to others. Liping Ma defines these things the key elements. With an understanding of these 4 universal principles we can solve any math’s problem anywhere (Ma, 2010). How Exciting!

References

Ma, L. (2010) Knowing and Teaching Elementary Mathematics. New York: Routledge

Mathematics is all around us- Music

It might seem ridiculous already, maths in Art? Nature? and now Music! Alas it truly is all around us, put in your headphones and count out the beats to your favourite songs, recognise patterns and repetition in the rhythm, get louder and quieter throughout the tune. It is all mathematics.

Symmetry in music is a very regular occurrence, skip to 3.41 in the video below for an example of a very impressive yet simple example of not just symmetry, but repetition and pattern as well.

The tune in the video is made up of the same combination of notes just at varying speeds, or flipped by symmetry using several instruments. This piece was created by J.S.Bach and this style of composing was frequently adopted by him.

Another element of mathematics in music is pitch, pitch is measured in Hertz (Hz) and is the rate at which vibrations are produced. The mathematics in pitch is that it can be measured as a frequency (Wiggins, 2012).

The tones in music are all worth numerical values, it is these numbers that determine the sound of the tone. These are related to pitch also.

Musical Intervals correspond to precise and simple ratios, The harmonic series intervals with the simplest ratios are named “perfect” in music theory. Medieval composers favoured the P8, P5 and P4 as the most consonant intervals, partly because of their mathematical simplicity and elegance. (Rogers, 2004)

References

Rogers, G.L. (2004). Interdisciplinary Lessons in Musical Acoustics: The Science-Math-Music Connection Music Educators Journal . Available at: http://mej.sagepub.com/content/91/1/25.citation (Accessed: 01 December 2016)

Wiggins, G.A. (2012): Music, mind and mathematics: theory, reality and formality. Journal of Mathematics and Music: Mathematical and Computational Approaches to Music Theory, Analysis, Composition and Performance. Available at: http://dx.doi.org/10.1080/17459737.2012.694710 (Accessed: 01 December 2016)

Mathematics is all around us- Nature

Previously I discussed Fibonacci’s sequence and how it effects the golden ratio. Surprisingly art is not the only field that this effects as this pattern is common across nature.

A famous example of this is evident in the breeding patterns of rabbits…

Realistically the chances of every rabbit pairing consisting of one male and female is highly unlikely as well as other factors such as the amount of offspring, how many survive, the age of breeding etc., however the basic principle is sound and the application of Fibonacci’s numbers is very impressive.

Another impressive occurrence of mathematics in nature is the innate mathematic ability possessed by animals. Hans the horse was believed to be able to count until studies proved that the horses reaction was simply a response to his owners facial expressions.

However not all animals have shown mathematical ability with aide from humans. Chimpanzees being studied at Kyoto university have displayed extraordinary talents. Below is a video of Ayumu the chimp, he is able to recognize and retain numbers on a screen in a fraction of a second.

From this evidence it could be possible to argue that animals may even be better than humans at some mathematical problems.

dog-meme-25

Unfortunately not quite to this dogs ability just yet.

The Crutch

A wise secondary school maths teacher would often have us bring out our crutches. We were only allowed to use them however when the problem infringed upon us was to intricate to decipher at our own merits.

This crutch as Mr G would call it is in fact a calculator.

pink-calc

I’ve always appreciated this metaphor as I feel it is completely appropriate. Even when completing simple additions and subtractions we reach for the calculator app on our phones. “Nowadays… people plug numbers into a calculator without any intuitive sense of whether the answer is correct.” (Bellos, 2010).

The standard electronic calculators we have today were by no means the first calculating devices. Abacus’ are thought to have been used from as early as 2700BC, they are often constructed as a bamboo frame with beads sliding on wires, but originally they were beans or stones moved in grooves in sand or on tablets of wood, stone or metal. (‘Abacus’,2016)

Then came the Slide Rule “The Reverend William Oughtred and others developed the slide rule in the 17th century based on the emerging work on logarithms by John Napier. Before the advent of the electronic calculator, it was the most commonly used calculation tool in science and engineering. The use of slide rules continued to grow through the 1950s and 1960s even as digital computing devices were being gradually introduced; but around 1974 the handheld electronic scientific calculator made it largely obsolete and most suppliers left the business.” (‘Slide Rule’, 2016)

sliderule

There were a variety of other calculating creations through time but many came to a standstill after the creation of the HP-35. Hewlett-Packard launched his creation in 1972, at the time these devices of 35 buttons and a red LED display would cost £365. By 1978 the electronic calculator was growing and was eventually priced under £5 making it accessible to the general public (Bellos, 2010).

hp-35

References:

‘Abacus’ (2016) Wikipedia. Available at: https://en.wikipedia.org/wiki/Abacus (Accessed: 22 November 2016)

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

‘Slide Rule’ (2016) Wikipedia. Available at: https://en.wikipedia.org/wiki/Slide_rule (Accessed: 22 November 2016)

Mathematics is all around us- Art

The most common examples of maths in art are symmetry, tessellation, size and proportion. All of which are taught at primary school level mainly as an art subject than maths.

Symmetry is almost like a reflection, commonly used to draw an exact image only flipped adjacent to the original.

symmetry

Image: (https://binged.it/2fxVXar)

Tessellation is a pattern of shapes that fits together perfectly.

Image: (https://binged.it/2eOCIPv)

Size and Proportion are simply how large or small, and how many times they need to be increased or decreased in size appropriate.

proportion

Image: (https://binged.it/2f9Zlvf)

Mathematics has been used in art for thousands of year, Ancient Greek architects and sculptors used the Golden Ratio to ensure buildings like the Parthenon in Athens was pleasing to the eye and the sculpture of David adheres to strict proportions.

Renaissance portrait painters ensured that proportions of their subjects’ facial features and the size of their subjects’ heads in proportion to the rest of their body followed mathematical ratios, e.g. the Mona Lisa.

Islamic art is heavily reliant on tessellating geometric shapes and is not purely decorative but represents a spiritual vision of the world.

Earlier I mentioned the Golden Ratio, this is a famous number sequence that appears frequently in our day to day lives. It is known by some other names, such as the Golden Spiral or The Fibonacci Sequence. The Fibonacci Sequence is characterized by the fact that every number after the first two is the sum of the two preceding ones. The Rule is x_n = x_n-1 + x_n-2.

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, … the list is infinite.

In art the Fibonacci Sequence can create the golden spiral.

spiral

Image: (https://binged.it/2eRQZoM)

An example of the golden ratio can be seen in a variety of famous paintings such as:

The girl with the Pear Earing

The Mona Lisa

and

The Last Supper.

goldan-spiral

The Exact Science

Mathematics is praised by many because of the fact it is an exact science. There commonly is always one answer and once you understand how to get to this answer you are sorted. But there’s your real problem. Getting to that answer.

There tend to be several steps in order to achieve a correct answer and if some where through the steps the tiniest mistake is made your answer could end up miles off correct, regardless of doing almost all your working out perfectly.

maths-no

Image: (https://uk.pinterest.com/pin/24347654215254549/)

This is where the fundamentals are important. By learning the most simple of the mathematical principals we are able to practice these, build on these, and then develop these. Even the most complex equations are made up of a variety of fundamental principles that support one another in an effort to discover the answer.

Complex mathematics should not be slated for its abundance in real life situations but praised as it allows us to work with our fundamentals, practice them and grow in our own understanding.

Maths, myself and I

Growing up in my household it was expected that you would either be a maths whizz or absolutely hopeless. My brother, father, gran, grandpa and great aunt all have the incredible ability to tackle most equations with ease. However half my family have never needed a calculator with more than 19 buttons.

Regardless of my background of secondary school maths teachers, a doctor and a lawyer I was not considered a maths whizz. BUT I wasn’t hopeless either. My relationship with maths is a fond one.

In primary school I was a square. This means I was in the middle group and as desperate I was to become a pentagon I never made it, but this didn’t dent my relationship with maths at all it merely pushed me to try harder. In secondary school my first maths teacher explained to my parents on a parents night that there was more than one way to cook an egg, me being the egg.

Beginning to understand more complex mathematics moving through high school I only gained confidence and regardless of small blips along the way I completed, Standard Grade, Intermediate 2, and Higher eventually. Although now I could probably not even hope to understand a higher maths paper I still consider myself highly confident in maths as a subject and how I use it in the real world.

That was until it came to teaching in my first placement. I struggled with creating maths lessons because I could not simplify in my head the topics so that I could convey it to the class in manageable chunks. With the help of my class teacher I was able to conjure up a few appropriate lessons for the kids but I was still so confused. How could I know exactly how to approach the maths and still not be able to simple it down and teach it? Why were all my ideas so complex? So now I’m stuck- I’m confident whilst being clueless. With this barrier I saw the “Discovering Mathematics” module as a perfect opportunity to understand why I struggled to teach my class, whilst cementing my own knowledge as well. I hope to use this module to get myself back to the fundamentals of mathematics and take it from there.

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