Music and Maths

“Music is the pleasure the human mind experiences from counting without being aware that it is counting,” said seventeenth century philosopher Gottfried Leibniz. There are a number of connections between music and maths including beats in a bar, musical intervals and tempo. Without these, it would be extremely hard to compose and perform music as there would be no sequence, or rhythm. Therefore, maths is essential to the making of music!

Tempo

Tempo is the pace or speed at which a section of music is played. Timing is crucial to playing a piece of music. The speed at which music is played at can help create a desired atmosphere. For example the famous Jaws music starts slow and progressively gets faster. The timing makes the audience tense and aware that something is about to happen. Without this specific timing of music, the film just wouldn’t be what we all know it as today. This highlights the importance of maths in the making of music.

The Relationship between Music and Maths

When researching the relationship between music and maths further, I discovered the work of Beethoven. He is a famous composer, who in his career began to go death, however still managed to compose beautiful pieces of music. How did he do this? The answer is Maths!

This video explains the complexity of maths and music, however the main concepts are pretty simple. Beethoven used distance and patterns to help him compose music. This allowed him to see what notes and rhythms would sound nice together and which ones wouldn’t. The idea of this fascinates me, someone who is losing his hearing still capable of creating such famous pieces of music by the application of mathematics. This again just proves to me, how maths can be applied in numerous ways!

It’s Everywhere!

As I have said before, my views on maths weren’t exactly the best. I saw it as a hard subject that only ‘maths people’ were good at. When I think of maths, I instantly picture a desk, a textbook and puzzled faces. It has always been about answering set questions and getting them wrong or right. I have never shown an interest towards the subject, until this module, as it has opened maths up to me in a different way.

The endless variation in workshops, from counter intuitive maths to music and maths, has taught me how varied maths actually is and all the different aspects of the subject. Before I was very closed minded, however, now I realise how often we actually use maths subconsciously. It’s everywhere! Maths is so much more than equations, it is decision making, problem solving and so much more.

I use maths daily without even thinking about it, I will set an alarm for waking up, calculate how much I’m spending on my lunch, calculate how much I will get in my wage for the month and what I can and can’t spend, look at train timetables for making my way to uni and the list is endless. I can complete all these tasks successfully, therefore I’m good at maths? So it isn’t equations, but it still involves an element of maths and the ability to add up and calculate. I had never saw it in this way before. I was always the first to complain about maths and say how awful I am at it, but that was close minded and stereotyped maths. I can now recognise I am good at maths because I use it daily. This is something I want to portray as a teacher, and show my pupils that maths isn’t just about the equations, it is about all the daily interactions we make with it and how we use it to help us.

Netball with a Twist

I have never considered in depth the mathematical concepts that underpin sports. The scoring is about the only maths I can think of. However when given the task to create a new sport or update an existing one, I found that maths was key to making the game more challenging and exciting.

Our table looked at netball and how we could make some changes to make the game more intense and exciting to watch. The first observation we made was about the scoring, and how high the scoreboards can reach. It is a lot easier to score a goal in netball than it is in football, and the difference between the scores is never big. Compare this to a game of football were a score of 2-1 can be incredibly tense near the end of match, and in netball this difference can easily be equalled by the score of another goal within 2 minutes. Basically, scoring in netball is a lot easier to achieve compared to football and this is something we wanted to change.

Firstly, to make it harder to score points quickly, we made the length of the court bigger. This means that players will have to travel further in order to reach the nets to be able to score. Secondly we changed the layout of the hoops. We added in an extra two hoops and positioned them at different heights, to give players the opportunity to score more or less points when shooting through particular hoops. This adds more challenge into the game and will force players to make split second decisions on the court as to which hoop they are going to aim for. Lastly, we introduced a rule that meant you have to defend on the goal side. This means that players can’t immediately go to their goal, they have to take it up the court before coming back down and scoring. This will increase the challenge posed to the players and make scoring a lot more intense.

These applications of maths, length, height and position all helped us to improve the game and make it more intense to watch. This has made me realise that maths can be applied in any situation and can improve it on many occasions. Maths is everywhere!

Beans! Beans! Beans!

Today’s input was all about logistics and supply chain maths. We paired ourselves up and became the managers of our own stores. We were given a budget and a list of products, then told to choose 5 items we wanted to purchase and sell in our shop for the summer, autumn, winter and spring periods. We had to think about what would sell the best and how much of it to buy, we also had to take into consideration the sell by date of these items and how long they would last for. This was harder than you would think and we made a few mistakes buying certain items.

However, after the first round of purchasing, Beth and I discovered one simple trick that made us thousands.  The way to make money was to look at how much you were buying the product for and then how much you were selling it for. The bigger the increase the better. This was were our love for beans began. The beans were being sold for eight times higher the price than they were being bought for , which meant for a good profit. In addition, our beans would last and so any that weren’t sold could be taken over to our next season, therefore we weren’t losing money. By using this method we saw our profit doubling each time.

Not only did we have to calculate the basic sums of how much we were spending, how much was in the bank and how money we lost, we also had to use our initiative and look at what the best selling rate was. This input gave me an insight into how important the application of maths is in the retail industry, as small decisions can be costly for the business. In conclusion, just buy beans!

Can Animals Count?

There are similarities between humans and animals, yes, but do they have the ability to count? I would have originally said not a chance, however after today’s input I’m not so sure.

We looked at various examples of animals “counting” the first being Clever Hans. Clever Hans was around in the 1900’s, he was a horse that could supposedly do simple mathematics, like counting and square roots. His owner would ask him a question for example 1 + 1, and Clever Hans would answer the question by in this case tapping his hoof twice. The initial idea of this is incredible, an animal working out basic maths! When you dig deeper however things are what they seem. The owner of the horse, Wihelm von Osten, would perform these tricks with his horse in front of crowds in public and amaze them. It was after a formal investigation by psychologist Oskar Pfungst that revealed the horse was not actually performing these tricks, instead he was reacting to the observers around him. This showed a fault in this experiment, as there were no variables to prove the horses tricks. Although on the outside it looks like this horse can count, with a little deeper dig it shows that he can’t technically count. I do wonder if the horse does have a sense of maths, by being able to tap his foot twice?

Another example we looked at was chimps counting and remembering sequences of numbers. The video showed chimps counting up to nine, touching the right order of numbers on screen. It then showed the chimps being shown numbers on a screen and then disappearing. They then had to remember where the numbers were and count in order. Humans then did the same test and actually did considerably worse than the chimps! The video was pretty impressive and after having a go as a class, it revealed how hard the test is, so for chimps to be able to complete it was fascinating! One explanation, however, could be that the reward of peanuts means more to chimps than humans. The chimp still had a concept of numbers and the sequence, so therefore it can count?

Finally, another example of animals counting is the chicks. The scientists conducting the experiment had two screens, and some plastic balls attached to string. They would move the balls behind the screens as the chickens watched from inside a clear box. Once there was 3 balls behind one screen and 2 behind the other, the chicks were released and each time the experiment was conducted, the chicks would go to the screen that had the most balls behind it. This showed that the chicks were able to count how many balls went behind each screen and then remember which one had the most. This is related to instinct, as chicks will also follow the larger group when they are first born. However this experiment shows that these untrained, young chicks had an awareness of numbers and counting.

So, even though not all the experiments were valid as a result of some influencing factors, I believe that overall animals have an awareness of numbers and to a certain extent, the ability to count. I knew animals were clever but I wouldn’t of agreed with anyone that told me they could count until now.

Counter Intuitive Maths – The Monty Hall Problem

I found the concept of counter intuitive maths very interesting as it applies to many situations I have faced, like sitting a multiple choice test. Counterintuitive is something that goes against what you believe to be right based on common sense or logic. Learning about the psychology and thinking process behind it has been interesting as I have found myself thinking the same when carrying out a quiz or making a decision. An example of counter intuitive maths would be taking a multiple choice test. You decide that the answer is A but then you have a moment of doubt and second guess your answer, as you think it might be B. The majority of people won’t change their answer. This is because if you change your answer and then get it incorrect it has more of a negative impact on you than if you stick with your answer and get it incorrect. There is too much of a risk and so people stick with the safe option. I am in the same boat and tend not to change my answer if I doubt myself, however, is this the mathematically correct thing to do?

We looked at The Monty Hall Problem to test this and the results are pretty fascinating. If I think about this too much my brain hurts, but when I look at the basic principle of the concept I understand the Monty hall problem. I am applying my knowledge of fundamental maths to understand this.

Pretend you are on a game show, there are three doors, behind once is a brand new car, behind the other two are goats. You pick a door, and then the host opens one door revealing a goat. He then gives you the chance to change to the other door. If you stick with your original door there is a 1/3 chance you will pick the car, however, if you change to the other door, you are giving yourself a 2/3 chance of winning the car. This picture explains It more clearly.

You are not guaranteed to win the car by changing your answer however you are increasing your chances from around 33% to around 66%. Therefore, you have a better chance of winning the car by changing your answer. This has made me re-evaluate my thought process and I will now be making sure I go with my second answer to increase my chances of success.

Maths and Play

Play is an important part of learning and is vital to a child’s development. Play is used across the curriculum to aid learning. There is a focus on active learning in Curriculum for Excellence, it aims to provide opportunities for investigation and exploring. Through play, children can discover and ask questions they may not of thought of, solely working from a textbook. It is a popular method of teaching among other subject areas, so why not maths?

Friedrich Frobel believed that children’s best thinking is done when they are playing. Some may question how play can be achieved during maths, even I wouldn’t have immediately paired the two together. Maths, however, is connected to maths in numerous ways. During play, children are making decisions, predicting, experimenting with strategies and more! Developing these basic skills will flow into their overall learning, and therefore they are more likely to remember the lesson if they enjoy it. I know that the lesson Eddie showed us has stuck with me.

We looked at measurement and we did this by using small linking elephants and a beaker of water. We were able to see how much the water rose and work out its weight. I was able to see in front of me how this was maths, and putting this in perspective made my own thinking clearer. By making the teaching active and playful, you are making it easier for the child to explore and clarify their thinking.

This input inspired me to look at ways I can make maths fun and create a relaxed, comfortable environment whilst learning.

The energy and enthusiasm that these pupils show towards learning maths is something I never experienced, however believe that this is the best way to learn. One of the teachers pointed out that she aims to have the children learn without realising they’re learning. One way to achieve this, is taking the pressure of children. When they don’t feel as if they have to get everything right and learn statically, they are more open to taking risks in their learning and are then able to learn from their mistakes without feeling judged. Math’s is stereotyped as a subject that needs to be learned at desks from a series of textbooks, this module however is helping me to break my own stereotypes and focus on how to change this view. Encouraging children’s enthusiasm towards maths right from the get go sets the right attitude towards the subject and will help to prevent maths anxiety from developing. Finally, if children see that the teacher is having fun they will see maths as a life long subject.