# The System

Following a workshop on the idea of counter-intuitive mathematics, I was intrigued to find out more about how we can look past our intuition or gut feelings to understand mathematical concepts such as probability.

I am someone who struggles to grasp this mathematical concept, as I cannot see past the black and white picture: that everything that happens is completely random and cannot be controlled. To gain a more open mind about this topic I watched the documentary by Derren Brown – The System. This documentary focuses on trying to defy our understanding of probability and to see “the bigger picture”. Derren demonstrates this through inviting people to participate in horse race betting – telling them that he can guarantee they get a win every time.

We follow the journey of Kadesha, who has won every bet since being given anonymous tips from Derren Brown. When Kadesha eventually meets Derren, he explains to her that he has come up with the first “system” which will guarantee her a win for every single horse race she places a bet on. A mathematician worked out the probability of a system being possible to predict the horses accurately and came back with the result 1.48 billion:1 (which is not completely impossible but highly improbable.) So then how can Derren guarantee a win every time?

The truth is, he can’t. What we don’t know as we’re watching Kadesha’s winnings is that there have been 7,775 losses in the process. Altogether, Derren contacted 7,776 random people to take part in this experiment, knowing that someone was bound to win every race. This is because of the extremely high number of people taking part. For each race Derren randomly divided the participants into groups for each horse in the race (so inevitably, one group had to win each round). Kadesha was lucky enough to be part of the winning group every time until the 6th and final race when she was on her own and Derren could only choose one horse who he thought would win (unlike every other round when he could choose all the horses and at least one person would win their bet.)

This is something which didn’t even occur to me as I was trying to work out how this system was possible. I was honestly convinced Derren was a magician! But now, it makes perfect sense. The system works because Kadesha is not alone. This is exactly the way probability works. The more chances or opportunities you give yourself the more likely you are to be successful.

Derren used a similar example with flipping a coin. He explained that he would flip a coin 10 times and every time it would land on heads. What we don’t realise watching the show is that he had been flipping a coin all day until it came to the point when it landed on heads 10 times in a row – so inevitably it was going to happen eventually.

As stated by Derren: “to work out the system you need to understand that we can only know what comes from our own limited experience and our experience can be very far from the truth.” In my opinion this a great way of explaining counter-intuitive mathematics. If I was someone who had a greater understanding of probability, maybe I could have figured out the hidden secret behind Derren Brown’s “system”. This documentary has also highlighted the importance of looking at concepts from a very open mind (which is something I’ve always struggled with in mathematics and have mentioned in previous blog posts). These are both very important aspects of learning mathematics that I will take forward with me in the future: to look at mathematical concepts from a very open mind and to see how the concepts link with my own experiences or experiences of others.

I have linked below the documentary if you would like to find out if Kadesha won the final race or lost her £4,000 worth of savings… I have also linked an article which explains/criticizes the show.

http://www.independent.co.uk/news/media/sport-on-tv-hes-never-wrong-but-brown-fails-to-frighten-the-bookies-777421.html

References

Tong, A. (2008). Sport on TV: He’s never wrong but Brown fails to frighten the bookies. The Independent. [online] Available at: http://www.independent.co.uk/news/media/sport-on-tv-hes-never-wrong-but-brown-fails-to-frighten-the-bookies-777421.html [Accessed 30 Oct. 2017].

# Do Humans Really Understand Randomness?

After our input on chance and probability, I was intrigued to investigate the true meaning of randomness and how this relates to our everyday lives.

Randomness in our lives

Randomness is a concept I found very interesting when researching as I never realised how difficult an idea it is for humans to grasp. As humans, we intuitively use our memories to predict what will happen next, whereas true randomness has no memory of what came before (Bellos, 2010). Although we can make educated guesses as to what is to come based on experience and scientific explanation, we really cannot guarantee an outcome for anything.

A funny example of this, as explained by Bellos (2010), is when Apple CEO Steve Jobs had to change the programming behind the ‘shuffle’ feature on iPods. Customers complained that when they used this feature the songs that played were often from the same album or by the same artist. Yet this is extremely possible with randomness, as it does not consider what has already been played. Steve Jobs responded to this feedback by altering the shuffle feature to make it less random, defying the point of randomness altogether!

So why are humans so bad at understanding the concept of randomness?

It is in fact because we have no control over it! As humans, our natural instinct is to give explanations for what happens in our lives – by projecting patterns – and having control in our situations (Bellos, 2010).

I think this relates to the problem humans have when they blame their situations to bad or good luck. Some of us are so invested into the ideas of karma or superstition that we start to lose sight of all the various factors which contribute to our fortunes/misfortunes. Although we might be reluctant to admit it, the random occurrences we encounter come down to chance (Lane, 2011).

An interesting discovery based on luck however, is that “people who believe they are more lucky, are actually likely to be more lucky, because they are more willing to take advantage of opportunities” (Lane, 2011). Furthermore, after carrying out research over several years, Wiseman, (2003) suggests that good luck and bad luck come down to our behaviour and attitudes more than anything else. He explains that lucky people generate good fortune through four basic principles: creating and noticing chance opportunities, listening to their intuitions, having positive expectations, and adopting a resilient attitude (Wiseman, 2003).

It is important to note here that Lane (2011) and Wiseman (2003) suggest that we have control over how we experience a situation rather than have control over the situation itself. For example, someone who considers themselves as lucky may feel lucky about breaking their leg after falling because at least it wasn’t their neck. I think this is where the confusion comes in for humans. We often put a lot of faith in luck, instead of accepting the randomness of events which occur in our lives and facing these events with a positive attitude.

In my opinion, it is not so much the issue that humans do not understand randomness – it’s that we need to accept it! If we accept randomness it is said that we can live a much more carefree and optimistic life (Lane, 2011). It is vital that we teach the concept of probability and chance in school from an early age – not only because it enhances prediction and problem solving skills but so that children can get to grips with this concept and can explore how it relates to their everyday lives (Taylor, n.d.).

Image credit: EX UNO PLURA (2015) (https://www.exunoplura.com/tag/randomness/)

References

Bellos, A. (2010). And now for something completely random. The Daily Mail. [online] Available at: http://www.dailymail.co.uk/home/moslive/article-1334712/Humans-concept-randomness-hard-understand.html [Accessed 16 Oct. 2017].

Lane, M. (2011). Why do we believe in luck?. BBC News: Magazine. [online] Available at: http://www.bbc.co.uk/news/magazine-12934253 [Accessed 16 Oct. 2017].

Taylor, F. (n.d.). Why Teach Probability in the Elementary Classroom?. lamath.org. [online] Available at: http://www.lamath.org/journal/Vol2/taylor.pdf [Accessed 16 Oct. 2017].

Wiseman, R. (2003). Be lucky – it’s an easy skill to learn. The Telegraph. [online] Available at: http://www.telegraph.co.uk/technology/3304496/Be-lucky-its-an-easy-skill-to-learn.html [Accessed 16 Oct. 2017].

# Place Value, Number Systems and Their Complexities

Following a workshop this week on the concept of place value, I was inspired to write a blog post on the difficulties and complexities found in these seemingly simple, basic concepts of maths.

As someone who learned mathematics in a procedural fashion, it is very difficult for me to wrap my head around the concepts and understanding behind the maths itself. I therefore found this workshop very challenging when I had to look past the mathematics I have learned and replace it with concepts either made up or used in minority areas across the globe. A good example of this was when we had a look at our number base system…

Understanding place value

Universally the most common number base system is 10 i.e. 1, 2, 3, 4, 5, 6, 7, 8, 9; then using the concept of exchanging or “borrowing” (which is the common term used in schools) to make the number 10 – where the number 1 represents 10 units (Russell, 2017).

I feel like in my experience at school, exchanging (or borrowing as it was called) was never really explained to me. I understood the concept of hundreds, tens and units – but I could never picture that the number 654 could be the same as saying 65 tens and 4 units or that the number 6717 could be the same as saying 67 hundreds and 17 units. To be able to see this and understand it, a key understanding of place value is required.

This was something which I saw as a maths stater activity when I was out on placement. The teacher would ask the children to break down a number and to think of as many ways as possible to express it. This then helped the children as they moved onto their multiplication and division work (where the concept of exchanging is extremely important). I therefore think it is vital that when I go out to teach, I also touch on the theory and concepts behind the topic the children are working on.

It is also vital to use the correct terminology when teaching mathematical concepts like place value. Liping Ma (2010, p.37) explains that 86 percent of Chinese teachers made the change of saying “composing” or “decomposing” a number, rather than using the term “borrowing”. This mathematically makes more sense because when we do a sum like 32 – 15 we need to break the number 32 down to see how we can possibly take the number 5 away from 2. We are not simply “borrowing” but taking 10 units from the tens column. This is something I will take into consideration in my own teaching practice in the future.

Particularly for place value, I have found some great videos online which show how you can teach the concept to children. This is a great example of teaching place value and also using a child’s mistake to create a new learning opportunity:

(Singapore Maths Place Value Lesson, 2015)

The binary system – a great way to visualise place value!

As I feel I was not given the opportunity when I was younger to fully explore the idea of place value, it is difficult for me to see past a 10 base number system when using the concept of place value. In our workshop we were introduced to the most basic system – the binary system. The binary system is a system mainly used for computer systems and technology. A binary number is made up of only 0s and 1s (Mathsisfun.com, 2016). So, the way I explained it to myself, if the highest unit you can have is 1 then the number 2 must be 10, the number three 11, and the number 4 100. This took me a while to grasp as I just couldn’t imagine the hundreds, tens and units columns when working it out in my head. Luckily, we got given a spreadsheet that made the concept much easier to visualise, which was then when I managed to wrap my head around it. I have inserted a similar picture which helps to demonstrate this concept:

Image credit: J.D. Casnig, Knowgramming.com (http://knowgramming.com/nanosemaphore/a_bit_about_binary.htm)

As you can see, the two base system (binary system) is at the bottom. This picture also gives you examples of what it would look like for other number base systems.

10 base system – the best system?

The discussion of a binary system also led to the discussion of a dozenal system. It is argued by many that this makes more mathematical sense than our 10 base number system. Some good examples can be found on the dozenal society website, including: packing/packaging (e.g. we get a dozen eggs), a clock dial is numbered 1-12, and the factors of 12 being more useful than the factors of 10 (Dozenalsociety.org.uk, 2017). This is something which really should get us thinking. I assumed that everything we learn in maths is what makes the most sense but this has highlighted to me the importance of questioning the concepts we learn as there could always be better ways…

So what would a 12 base system look like?

Numbers 1-9 stays the same. The number 10 looks like a rotated 2 and is called dek. The number 11 looks like a rotated 3 and is called el. The number 12 looks like the number 10 and is called doh  (Dvorsky, 2013).

An interesting argument for the 10 base system is that it is easier for learners to use their hands to help them count. However, as argued by Dvorsky (2013),  we have three separations on each finger, which if used separately would be extremely useful for counting using the dozenal system. If we use our thumb as a pointer, and start with the index finger we can work our way from the bottom of our index finger until we reach the number 12 (or 10) at the top of our pinky. This is difficult to explain without a visualisation so I have attached an illustration below. Using this system, gives us a total of 24 numbers to work with using just our hands!

As noted, there are many complications and arguments for and against a dozenal system that I could write a whole blog post about it. However, I wanted to touch upon it here as something which has inspired me to open my eyes in terms of not seeing mathematics as a set and fixed structure.

Place value and number base systems are both concepts which I have never considered to be so complex until now. Even after doing my research for this blog post and revisiting my notes from the workshop, I am still struggling to pick these concepts apart and look at them from a very open mind. As I develop my thinking throughout this module, I am hoping it will become slightly easier to be able to analyse and question the mathematical concepts I have learned.

References

Casnig, J.D. (2013). A Bit About Binary. [online] Knowgramming.com. Available at: http://knowgramming.com/nanosemaphore/a_bit_about_binary.htm [Accessed 4 Oct. 2017].

Dozenalsociety.org.uk. (2017). DSGB. [online] Available at: http://www.dozenalsociety.org.uk/ [Accessed 6 Oct. 2017].

Ma, L., (2010) Knowing and teaching elementary mathematics (Anniversary Ed.)New York: Routledge.

Mathsisfun.com. (2016). Binary Number System. [online] Available at: https://www.mathsisfun.com/binary-number-system.html [Accessed 4 Oct. 2017].

Russell, D. (2017). Basic Math Concepts: What Is Place Value?. [online] ThoughtCo. Available at: https://www.thoughtco.com/understanding-place-value-2312089 [Accessed 4 Oct. 2017].

Singapore Maths Place Value Lesson. (2015) (video) YouTube: Singapore Maths Academy (UK).