In a recent task assigned by Richard we were asked to explore the fundamental mathematics of a game. I have decided to investigate the mathematics involved in the seemingly simple game of ‘Connect 4’.

Connect 4 is a game with relatively simple rules (Victor Allis, 1988):

1. Each player in his turn drops one of his checkers down any of the slots in the top of the grid.
2. The play alternates until one of the players gets four checkers of his colour in a row. The four in a row can be horizontal, vertical, or diagonal.(See examples).
3. The first player to get four in a row wins.
4. If the board is filled with pieces and neither player has 4 in a row, then the game is a draw.

There is no set rule which states who takes the first turn, this is ultimately left to the players to decide. This decision may be crucial in deciding who wins the game…

There are several different ways in which this game can be approached. By creating a simple rule player 1 (black) can ensure they win or draw every game. The rule is: Suppose n is even. Group the columns in pairs, giving pairs of columns 1 & 2, 3 & 4, …, (n-1) & n. Each time White plays in one column of a pair of columns, Black plays in the other column of the pair. For n is odd, groups are made in the same way, leaving the n-th column as a single column. The same rules apply to this position. Only if White plays in the n-th column, Black plays in that column, too.

Fundamental mathematics can be found in Connect 4 as the principles of probability and prescription are apparent. A player must attempt to predict where their opponent will place their next move. They must estimate the probability of their opponent predicting their next move and try to prevent them from making a move which will trap them and make them lose.

In relation to Liping Ma (2010) a player must have multiple perspectives when playing connect 4. This means that a person must be able to see the game in various ways and must be able to approach it in multiple ways. They must be aware that their opponent may have various moves available and they must be able to prevent any moves which will result in them losing. A person must also be able to take into consideration not just their opponents next move, but the next few moves they could make in order to guarantee a win. This could be done by recognising patterns they make.

When I played this game when I was younger I would try my best to place my counters in such a way that I create multiple opportunities for winning. For example create two lines of three meaning my opponent can only block one of my potential wins and leaving me with a guaranteed win. I would do this by considering possible patterns of play, trying to predict where my opponent would place counters to block mine but in some ways it could be argued that the game is somewhat down to luck. For example your opponent may get distracted and miss an opportunity to win or fail to notice your plan.

Reference:

http://www.informatik.uni-trier.de/~fernau/DSL0607/Masterthesis-Viergewinnt.pdf  (Accessed on 27th November 2016)

Why choose maths?!

I guess I chose this elective as I enjoyed mathematics all through primary school and always felt particularly confident with the subject when it came to testing. My positive attitude towards maths remained with me until I decided to sit Mathematics at higher level. In one sense, my attitude changed simply because I did not like the teaching style my teacher used, I found she
took any enjoyment we had previously developed away from us. Another difficulty with Higher Maths was having two teachers who split our maths time between them. This resulted in us learning two completely separate topics at the same time which at such a high level was very confusing.

On reflection, I struggled to see the relevance of what we were learning. I constantly thought to myself when am I ever going to need this information again? In relation to Liping Ma (2010) I could consider myself as not having a profound understanding of mathematics as I struggled to see the connectedness between the two topics we were learning at the time. I also found it confusing when one teacher would explain something in one way and the other would
introduce a completely different strategy, I found that I liked having one way of approaching something and achieving a set answer. Reflecting back on this, I also lacked Liping Ma’s multiple perspectives as I was unable approach problems in multiple ways.

Although I put in a lot of hard work and still managed to achieve an A at higher, I still feel like the problems I faced in my last academic year of maths left me slightly anxious at the thought of teaching mathematics to the upper years. I know the importance of showing positive attitudes towards a subject as a teacher’s attitude highly effects how pupils view a subject’s importance. Liping Ma (2010) also discusses the theory that a teacher with a profound understanding of mathematics has a greater impact on students as to produce effective learning you yourself must have a specialist knowledge of the subject. This highlighted to me the great importance it is for me to deepen and broaden my own understanding of mathematics so that in future I can answer question and explain all curricular areas clearly and with confidence.

I chose Discovering Mathematics to re-discover my own enjoyment of maths and develop my understanding to improve my future teaching.

How did I find Discovering Maths?

I must admit I left multiple lectures feeling totally confused but in the best way possible. There were multiple lectures where I sat thinking ‘WOW, I did not know that’ and some where I had no idea what was going on. Through additional reading and blogging I feel like I have developed my understanding in the areas where I was slightly more puzzled. I honestly had no idea how much maths surrounded us. It was obvious to me that maths was all around us, but I had no idea of the degree to which we use maths in our everyday lives.

I particularly enjoyed how this module was organised. By bringing in guest speakers we were really getting a deep insight into how maths is used in different sectors. I found the lecture on data and statistics in health and medicine very eye-opening. One thing I took away from this lecture is to stop taking information as the truth, so many graphs have flaws that we seem to never question. Maths is key to improving the health of our nation and our ability to make measurements, predictions and calculations all contribute to our healthcare.

My favourite lecture was the input on logistics and supply chains. This was a great example of how to make maths fun and active, although it did bring out a competitive side in some of us! The aim of this lecture was to make predictions and calculate the probability or risk of an item on our list being sold. Using the probability that the item would sell we had to decide how much of the product to buy, taking into account the time of year, and whether we had perishable items etc. This highlighted to me that maths is apparent even in supermarkets and there are people dedicated to predicting how much of an item will sell and when these items will sell best. This is done by using previous data, forming graphs and using this to make accurate predications. (The lady in the video clip shown takes into account a bank holiday which she believes will influence sales.)

This module has also opened my eyes to the history of maths. I have never really thought about where maths evolved from or how we got to the stage where we are. I now know that our base system is based upon the fact we have 10 fingers (therefore it makes perfect sense to use them to help us count). There is a huge amount of history in maths that I was blind to before, I have learnt about the Babylonian base systems and the Ishango bone which shows that maths is over 20,000 years old! I have also discovered that there is maths in both music and art which I had never ever considered.

I can truly say that this module has brought back my enjoyment in maths. The enthusiasm and knowledge our lecturers (particularly Richard) showed throughout the module has modeled to us the attitudes we should have in the classroom. The positivity from lecturers definitely affected my feelings towards the module in the best way.

How will this impact me in the future?

Although this course has not made me a mathematician, it has made maths and the thought of teaching maths a lot less daunting for me. It has definitely shown me multiple fun ways to introduce and develop maths in the classroom and the importance of relating the maths we learn to real life situations. I hope to continue to improve my understanding of maths in future and foster the enthusiasm that our lecturers displayed to create a positive learning environment in my own classroom!

After reading Alex’s Adventures in Numberland I was made more aware of the different types of risks we take in everyday life. This chapter explores the risks people take when gambling, or playing games involving money. It is clear that there is a huge money making opportunity bas

Slot machines are a huge profit maker in Casinos

-ed around the risks people are willing to take to make easy money. Bellos (2010, p) states that:

“slot machines make $25 billion dollars per year (USA) after paying out prizes.”

Clearly, there is a huge market for exploiting people’s willingness to take risk. But what does this mean?

The Oxford Dictionary (no date) defines risk as: “A person or thing regarded as likely to turn out well or badly in a particular context or respect.” Taking a risk means we are taking a chance on an unpredictable outcome. The outcome may result in something bad happening but there is a chance that something good may arise from the risk we have taken. In order to understand risk

many people dream of winning big money, but at what risk?

properly we must be able to calculate the probability of an outcome. Probability is the study of chance and with a mathematical approach unpredictability becomes very predictable. (Bellos, 2010) Casino games and bets require the player to have an expected value, this is their expectation of what they can get out of a bet. For example a £10 bet could be calculated as: (chances of winning £10) x £10 + (chances of losing £10) x-£10. In casinos players should expect to lose money, but the chance of winning big means there is a great desire to take risk for an unpredictable outcome.

Probability can be used to calculate the probability of throwing a certain score on dice. When you throw a die there a 6 possibilities (1,2,3,4,5 or 6). BBC bitesize explores this further and states that there are three ways of getting an odd number (1, 3 or 5).

So the probability of getting an odd number = the number of ways of getting an odd number / total number of possible outcomes. = 3/6 = 1/3 or 0.5 or 50%.

Here there is a connectedness (linking back to Liping Ma’s (2010) definition of PUFM explored in the essay and other blog posts) as to understand what these numbers mean, a person must know fractions, decimals and percentages. They must also have the understanding of equivalent values. By calculating the outcome of probability when throwing dice, a person is able to calculate the risk that the die may/may not land on a number they have betted on.

Why is probability and risk important in our everyday lives?

In our everyday lives we take many risks. Risks may arise on simple tasks like crossing the road, being able to calculate the speed of the car across the distance you need to travel. You may decide that it is highly probable that you will get across the road safely but there is always an element of risk as you cannot control the drivers speed, the driver may not stop (unlikely but possible). Risk in our everyday lives may be about calculating danger but can be as simple as sending a text that may have a negative response. In my job as a lifeguard I am constantly aware of risks in the workplace and thinking about the likelihood of an accident arising. Being able to identify risks in the workplace means you need to be able to identify what may cause harm to people and deciding if action is necessary to prevent incidents from happening.

The law states that businesses should carry out risk assessments. A risk assessment means a worker is taking measures to control or limit the risks in the workplace. This means that workers must have a basic understanding of mathematics in order to calculate the risk and work out the probability of an accident happening. Another example of risks in our everyday lives could be revising for examinations. There is an element of risk in revising that you may miss out key information, or you may look at past papers and work out the probability of certain questions coming up in the exam, meaning you only study for these questions. Again, maths would need to be understood in order to calculate the likelihood of a question being in an exam paper. The risk in this would be that exam questions are often random, and there are simply somethings in live that we cannot predict. By limiting revision to questions that you have predicted will come up, you have taken a risk. The risk may be that although highly probable, the questions you predicted were not in the exam and everything you revised is of little use. This could also relate to Liping Ma’s (2010) PUFM as she explores a person having the ability to look at things with multiple perspectives. In risk and probability you must be able to approach a problem with multiple views as there is often more than one outcome. A worker needs to be equipped with the mathematical knowledge to allow them to evaluate and calculate risk in the workplace in order for a business to thrive.

References:

BBC, Bitesize Revision, Probability (no date) Available at: http://www.bbc.co.uk/education/guides/zxfj6sg/revision/6 (Accessed on 8th November 2016)

Bellos, Alex (2010), Alex’s Adventures in Numberland

The Oxford Dictionary, Definition of Risk, no date, Available at: https://en.oxforddictionaries.com/definition/risk (Accessed on 8^th November 2016)

Ma, Liping (2010) Knowing and Teaching Elementary Mathematics: teachers’ understanding of fundamental mathematics in China and the United States

It seems that there is virtually no escaping maths. In a recent lecture with Anna Robb we
explored the idea of maths in music. Although I have played violin and piano for a large part of my life, I have never really appreciated the large amount of mathematics found in music.

When I first began playing violin I was taught about time signatures and the values of different

Learning note values and equivalent values is important in music.

notes. For example a crotchet is one beat, a minum two and a quaver is half. To understand this basic concept I must’ve had an understanding of fundamental mathematics as I could understand the value of each note and how to count to that value. Even being able to associate notes to which finger to put down on the fingerboard required some aspect of maths I was unaware of at the time. I knew that on each string the higher the note required to play meant the higher number of finger to use. (e.g. playing a G on the D string uses the 3rd finger which is a note higher than F which uses the 2nd finger) It all seems more complex trying to explain in words.

Different music also required different speeds. The speed was dictated at the top of each piece of music by stating a note (mainly crotchet) and a number which represented how many beats were to be played per minute. ( crotchet = 128 bpm, or quaver = 140 bpm, the higher the number generally the quicker the pace of the music. This can also be put into a metronome which produces a sound at the speed requested allowing you to play music to the correct speed whilst keeping to the beat.

Time signatures also dictate how many beats are in a bar and the value of the beat. More maths! 4/4 is the most basic and most used timesignature in music and consists of 4 crotchet beats per bar. This tells the musician that there are four beats in the bar and also that when to play the downbeat. 3/4 is commonly used in Waltz as it is a simple 1,2,3 1,2,3, 1,2,3. In music exams you are also required to identify how many beats in a bar are in certain pieces of music played by the examiner, this means you must have a basic knowledge of maths as you need to know the value of the beat and count.

Patterns and repetition are also apparent in nearly all music. A composer often repeats the same pattern, or creates a pattern to invert, mirror or play at a different pitch. Patterns are often recurring. We have already discovered the mathematics behind forming patterns so patterns in music is another example of maths. Music can take different form, ABA, ABCD, ABACAD. Each letter has a different theme or pattern. In ABA, the A is the repeated pattern.

Today’s lecture also showed me that there is a concept in maths, firstly which I had never heard of, but secondly that I would have never associated with music. This is the Fibonacci sequence. The Fibonacci sequence is a series of numbers where a number is found by adding up the two numbers before it.(Hom, 2014) Starting with 0 and 1, the sequence goes 0, 1, 1, 2, 3, 5, 8, 13, 21, 34 (the rule for this can be expressed as x_n = x_n–1 + x_n-2 ). It is interesting to note that in music a chord is made up of the root (1st note of a scale), the 3rd and the 5th. These are all Fibonacci numbers, infact Fibonacci numbers are apparent in

An example of Debussy’s work where is a clear pattern of Fibonacci numbers.

the octave and multiple scales too! In a previous post I discussed the use of the Golden Ratio in Art. I have now discovered that the Golden Ratio is also closely linked to music and is infact used when designing instruments like the violin. (Something I will be investigating when I am reunited with my own violin). The Fibonacci sequence is used in many melodies and harmonies as the pattern of notes is most pleasing to the ear and can be heard in many of Claude Debussy’s compositions (Shah, 2010). Although when listening to music it may be hard to pinpoint exactly where the sequence is present, when reading the music itself this becomes slightly clearer. (I have uploaded a picture of an extract from Debussy’s work where the notes, 1,2,3,5,8, and 13 form the main melody).

Music in maths is something I found very interesting as even though music is something I play or listen to everyday I don’t tend to associate with maths. It reinforces the idea that maths is crucial to our understanding of multiple aspects of our lives. Music provided me with so many different opportunities and experiences, my favourite being touring Italy with Perth Youth Orchestra.  I can’t imagine a life without music and it seems absurd to me that if I didn’t have a basic knowledge of fundamental mathematics I wouldn’t have been able to progress this far in music.

http://www.perth-youth-orchestra.org.uk/lord-of-the-dance/ – Link to one of my favourite performances from my time at PYO (skip to 3:48)

References:

Elaine J. Hom, 2014, What is the Fibonacci Sequence? Available at: http://bit.ly/2dtNlVm (accessed on 9th November 2016)