Monthly Archives: October 2018

Chance of Me Gaining an Appreciation for Maths – Extremely Likely

Probability – the next (and successful) leap in convincing me of how relevant mathematics actually is. This module has been effective in changing previously negative perceptions of maths, one of which in particular is that the terms ‘maths’ and ‘complex’ go together like bread and butter – this is not the case. Granted, parts of it can, as with anything. However I can vouch from my personal ongoing experience that if if we allow it to, it can be an enjoyable art.

Very briefly, probability is just what it sounds like – the likelihood of a certain event taking place or not. (Boaler, 2009). To use one of the most basic examples, the likelihood of rolling a ‘4’ on a dice. Well, there’s only 1 ‘4’ on a dice and this is known as the sample point. There are 6 possible outcomes overall (the values of 1-6), and this is known as our sample space. The probability of ending up with a ‘4’ on the roll of a dice is 1/6. Whilst the terminology was new to me, the concept is simple enough (Don’t Memorise, 2014).

Image result for probability

Schools Minister Nick Gibb identified 3 purposes to education in a 2015 speech, one of which was “preparation for adult life” – a quotation that is very fitting in regards to probability (The Purpose of Education, 2015). Like many other aspects of mathematics such as money and time, chance and probability is a relevant concept we use on a daily basis. The weather is an excellent example of this assertion. Before leaving for the bus in the morning, it’s very likely I’ll ask “Alexa, what is the weather in Dundee today?”. Now personification aside, I would be very surprised if she told me “It will definitely rain today for 3 hours straight”.

What she might tell me is “It probably won’t rain in Dundee today. There’s only a 25% chance“.

Since meteorologists cannot predict the exact weather conditions, they must make informed predictions (Tucker, 2018). The information we take from them may not emerge as being correct however – a 25% chance of rain does not eliminate the possibility. Regardless, it does help us make Image result for monopoly manjudgement calls – will I take a raincoat, or can I leave it in my back so there’s more space for collecting my library books? With only a 25% chance, its probably going to be the latter of the two. The decisions we make from probability do not just have to relate solely the weather, of course (Haylock, 2006). Informed decisions are a major part of any human’s life, and they can range from purchasing properties in Monopoly at a family game night, to choosing the best day to practice surfing based on the tidal/wave conditions, or indeed, making the right call in the world of gambling.

I myself work at an indoor amusements/arcade, however my responsibilities of managing children’s parties mean I have never dabbled with the ‘nudgers’ that to me are nothing more than complex machines with countless buttons and symbols on them. Unlike the simple example of rolling a singular die to get a number explained above, these machines have multiple possible combinations (i.e. the sample space).  Obviously, as with any business, the company intends to make a profit from the service it provides. As such, they are going to benefit in winnings a lot more than we are – even if we don’t witness that fact. The video below explains the basic functions of a slot machines, including what is meant by the terms theoretical payout and actual payout.

With these terms in mind, I spoke to one of our regular customers, who makes a guaranteed (okay, ‘extremely likely’) appearance at my work every weekend. I questioned him about his knowledge of gambling, and was consequently given a tour of the arcade, where he impressed me by pointing out which machines are best for winning (i.e. the best turnout rate, along with which ones were likely to give you an extra pound coin that may be stuck in the coin mechanism, or which machine is likely to have some loose change hidden at the back of the darkened retrieval pot. This man in particular has been coming to the establishment for years, so I was very interested to know how much profit his knowledge and skill set have rewarded him with – except he doesn’t. To lift his own words, “pure entertainment value”. He knows he is going to lose money regardless of what he plays, and this once again links back to how the actual payout of slot machines are ALWAYS greater than the money put in – you are technically losing more than getting back (tech4truth, 2010). This is not a new concept either. Charles Fey, who invented the first slot machine (3 reels, 5 symbols), had a 75% average pay-back! That may seem great at first glance, it only paid out 50% of the time (Valentine, 2018). There may be that faulty machine in the corner of the room, or perhaps you are being watched by a ‘lurker’ waiting to pounce at the right opportunity to steal your winnings. Even then though, they are using their knowledge of probability of knowing when to swoop in.

Image result for PHOEBE AND THE LURKER

To summarise all of this, probability and chance is something we use everyday in a variety of circumstances. It can become part of subconscious, much like the process of respiration and blinking, where we only realise we are doing it when we are told so. This area in particular is such a huge foundation in regards to our everyday choices. I’m truly beginning to realise that Maths isn’t just about sitting at a high school desk, attempting to find ‘x’ and losing your mind whilst you do it. There’s so much more to it, and just because it’s not an equation, that doesn’t make it any less maths-related.

Also, apologies – chances are you’re now aware you’re breathing and it won’t be a subconscious process for a little bit.

 

References

Boaler, J. (2009) The Elephant in the Classroom: Helping Children to Learn and Love Maths.  London: Souvenir Press Ltd.

Don’t Memorise (2014). Probability – Sample Space, Sample Points, Events! Available at: https://www.youtube.com/watch?v=5oI8-iQqPAI (Accessed: 18th October 2018)

Gambling Glossary: A Guide to Gambling Terms. (no date). Available at: https://www.gambling.net/glossary.php (Accessed: 18th October 2018)

Haylock, D. (2006) Mathematics explained for primary teachers. London: SAGE.

Probability. (no date) Available at: https://www.mathsisfun.com/data/probability.html (Accessed: 18th October 2018)

tech4truth (2010) Slot Machine Paybacks and Slot Odds Explained (Tech4Truth Episode 3) Available at: https://www.youtube.com/watch?v=4wzg-8QKC5s (Accessed: 18th October 2018)

The purpose of education. (2015) Available at: https://www.gov.uk/government/speeches/the-purpose-of-education. (Accessed: 18th October 2018)

Tucker, K. (2018) Examples of Real Life Probability. https://sciencing.com/examples-of-real-life-probability-12746354.html (Accessed: 18th October 2018)

Valentine, E. (2018). ‘Chance and probability’ [PowerPoint presentation]. ED21006: Discovering Mathematics. Available at: https://my.dundee.ac.uk/webapps/blackboard/execute/displayLearningUnit?course_id=_58988_1&content_id=_5217952_1. (Accessed: 18th October 2018)

Images Used

https://www.wikihow.com/Calculate-Probability

https://www.pinterest.co.uk/pin/330873903853107243/?lp=true

https://www.pinterest.co.uk/pin/451134087656430346/?lp=true

Fibonacci Sequence – Significant Coincidence?

The Fibonacci Sequence is a fairly new concept to me, having only seen a flash of the term in a textbook during my MA1 school placement. The Discovering Maths module is responsible for properly introducing me to this concept, however it was a shock to learn that in reality, this peculiar set of numerals have been very much right in front of me all my life.

In a TED talk, Arthur Benjamin emphasises my shared belief that the vast majority of mathematics taught in education is primarily surrounds solving calculations with countless rules, often labelled as having one correct answer – especially in comparison to other fields such as English, where more flexible and diverse answers can be accepted (2013). Benjamin accentuates that more time should be spent on application of maths, and that it is both more interesting and enjoyable if we allowed ourselves to go beyond the HOW, and contemplate the WHY. The Fibonacci Sequence is a perfect example for this.

The idea of the Fibonacci Sequence is fairly simple. You take the first two numbers (0 and 1) and add them together to calculate the next digit – in this case, 1. This is followed by adding the new value with the previous – in this case, 1 and 1 to make 2, and so on. The image below lists the first set of numbers in the sequence.

Image result for fibonacci sequence

There we have our calculation – and a pretty basic one at that. However, to perpetuate Benjamin’s statements, mathematics does not have to be a stale pattern of addition – it can be astonishingly magnificent, relevant and beautiful (Erikson, 2011).

To discover this, we link this set of numbers to the concept of ‘Ratio’. A ratio can be described as how much of one thing there is compared to something else. (Haylock, 2006). In regards to having a ‘perfect ratio’ where two things are perfectly proportioned, this value would emerge as 1.618034.  The image below explains the method of obtaining this value. 

Related image

When we put these two mechanisms together, we are able to uncover a startling connection. It has been proven that the further along to Fibonacci Sequence we travel, the closer the resulting ratio between two consecutive numbers is to the Golden Ratio (Boaler, 2009).

For example, 13/8 = 1.625

Much further along, we can divide 4181/2584 to obtain 1.618034!

The link between the Fibonacci Sequence and the Golden Ratio is abundant – however humans have yet to figure out what it is. Whilst I personally have no hope or intention of uncovering the meaning of this relationship, I am able to come to a personal conclusion as to whether or not this is all just a coincidence or if there are greater forces at play.

Many famous landmarks such as the Taj Mahal and Notre Dame have been labelled as having the ‘Golden Ratio’. Unfortunately, the age of these constructions mean we aren’t even sure if this was an intentional decision (Strange Mysteries, 2017). Could the Fibonacci sequence just be something we are naturally attracted to, and is it only limited to human beings?

Image result for taj mahal golden ratio

Thanks to the nature, we can safely say the answer is no… at least to the latter question. Perhaps we are just naturally attracted by these numbers, yet man-made structures are not the only things that have the Golden Ratio present (Boaler, 2009). The image of the shell below conveys the lines clearly as the sequence increases into an elegant curve.

Much like the supposed mathematical construction decisions surrounding the Taj Mahal, but far more philosophical in depth, we currently don’t know and probably will never have concrete evidence on how our universe was created – just a possible evolved understanding of it (Francis, 2013). But the abundance of the Fibonacci sequence makes it very difficult to accept that this is all a coincidence. If the buildings and shells do not provide enough confirmation, we need only look at other natural examples that humans could not possibly interfere with – whether that be hurricanes/storms, flower petals, or indeed our own body.

References

Boaler, J. (2009) The Elephant in the Classroom: Helping Children to Learn and Love Maths.  London: Souvenir Press Ltd.

Erickson, M. (2011) Beautiful mathematics, Washington, D.C.: Mathematical Association of America.

Francis, M (2013) Will we ever… know what happened before the big bang? Available at: http://www.bbc.com/future/story/20130502-what-came-before-the-big-bang (Accessed: 9th October 2018)

Haylock, D. (2006) Mathematics explained for primary teachers. London: SAGE.

Strange Mysteries (2017) Why is 1.618034 So Important? Available at: https://www.youtube.com/watch?v=keLN89CWZ-A (Accessed: 9th October 2018)

TED (2013) The magic of Fibonacci numbers | Arthur Benjamin. Available at: https://www.youtube.com/watch?v=SjSHVDfXHQ4&t=287s (Accessed: 9th October 2018)

Images Used

https://www.pinterest.co.uk/pin/500181102340885176/?lp=true

https://www.quora.com/What-is-so-special-about-the-golden-ratio

https://www.phimatrix.com/architectural-design-golden-ratio/

https://www.theodysseyonline.com/7-beautiful-examples-fibonacci-sequence-nature