“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!

https://www.youtube.com/watch?v=zAxT0mRGuoY

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!

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.

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!

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.

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.

For me, even the word “maths” has connotations of dread, boredom and simply just not enjoyable. Traditionally, maths is taught on a blackboard, followed by textbook work to consolidate the learning. What if, however, you simply don’t understand the explanation, switch off halfway through because you aren’t engaged and then have no idea how to complete the textbook work? This cycle then repeats itself and you progressively become more and more lost, which kicks in feelings of panic and anxiety. I personally feel as though this was my main issue in school, resulting in my hatred towards the subject. How could this cycle be reversed? The key to this is making the subject engaging to pupils.

The Boring Subject

I think many of us would agree that maths is one of the more boring subjects out there. This was my view anyway, until after some of Eddies inputs. We were shown a few ways that maths can be taught in a more active and engaging way, which made me realise maths can be fun! I think it is extremely important that maths is active and engaging and had it been presented to me in this way, I would have a very different outlook on it. Making a maths lesson more memorable is more likely to have an impact on the pupil and make it easier for them to remember concepts and start to make links to other areas within the subject. In addition, if they are enjoying what they are doing they will associate that feeling with the subject. In turn this will decrease the feelings of maths anxiety.

Maths and Art

I was in Spain this summer and walked through many streets with pretty tiles, the floors in our hostel were even tiled. I didn’t look at them and think of maths, shape and symmetry, I just thought they were aesthetically pleasing. After Eddies workshop about creative maths, I realised that maths is all around us in a variety of creative ways. What I was looking at were different tessellations, that had all individually be planned out and put together using shapes that fit. We had the chance to create our own, and I found it very engaging.

We used three different shapes to create this, however, some that we were looking at had as many as 8 or more! This activity forced you to think about the different shapes and all the ways the shapes can fit, without realising you were doing maths. This was a stress free maths activity, one of the very few I have experienced. This is exactly the type of stress free environment I want to create in my classroom.

What is maths anxiety? According to Mark H Ashcraft, chair of the department of psychology at the University of Nevada, maths anxiety is “a feeling of tension, apprehension, or fear that interferes with math performance” (Guardian, 2015). These feelings of fear are something that I can relate to, as I have experienced them myself. This was one of the main reasons behind my decision to take this module, as I want to overcome this before going into the profession.

Where Does it Come From?

Maths anxiety typically stems from a previous bad experience relating to the subject. I can vividly remember sitting in my primary 5 class with tears rolling down my face, all because I couldn’t grasp the concept being taught. This then developed my hatred for maths and I began to believe I wasn’t a ‘maths person.’ Still to this day, the idea of completing a maths question makes my brain freeze.

Some pupils, however, have developed anxiety towards maths through their parents and this is alarming. Parents influence their child’s life heavily and therefore can pass down values and attitudes towards particular subjects in school. With maths being viewed, by many, as a subject you are either good or bad at, this makes it a popular subject to ‘hate’ and dismiss. The seeds of maths anxiety can be planted with a simple comment such as “I hated maths at school.” It is important that children don’t adopt these negative attitudes towards the subject as it can lead them into a negative mind set, which can often be difficult to reverse.

How it Effects The Profession

Maths anxiety is a ‘thing’ and needs to be recognised, as according to researchers it effects up to two million children! This is where the role of the teacher becomes important. We need to be able to identify maths anxiety in children and use a variety of methods to help the pupil to overcome and conquer this feeling. Some signs of maths anxiety include paranoia, the child believes they are the only one incapable of solving the question or passive behaviour, the child will do anything but solve the question they are faced with. Going slowly and using concepts children are already confident with to develop their skills further is one method that can help to ease maths anxiety in pupils.

Not only is it important for teachers to recognise maths anxiety in their pupils, but also in themselves. As stated previously maths anxiety can be passed down through parents and the same applies with teachers. We are also an influential figure to children and therefore it is crucial that we are confident with the subject ourselves. This will then enable us to provide thorough explanations of each concept and have a bank of solutions to help the pupils identify the easiest methods to suit their learning. This is especially important for me as I am aware of the feelings I have towards maths and I am conscious that they need to change, in order to provide the best education I can for my pupils. I also do not want the children I teach to experience maths in the same way that I did.

Moving Forward

Now that I am aware of the impact a teacher can have on how children view maths, I know that I need to change the way I feel about the subject. It will require self development and confidence in my own skills. I enjoy maths when I grasp the concept and see my answer match the answer section. I need to remember that feeling and apply it to all areas of maths.