In class, we drew graphs to plot y=x, y=x^2, y=1/x and y=1/x^2. In order to draw these graphs, we first needed a fundamental understanding of basic concepts such as numbers, the sequence of numbers and being able to understand patterns in order to compare results and see relationships. Furthermore, we needed knowledge of the measurement of the squares on the page first, we needed to know measurements as one square is 1mm therefore, 10 of them make a centimeter so we can use 1mm to represent one of somethingg out of 10. Therefore, we demonstrated knowledge of fundamental understanding of mathematics as we used basic concepts and connecting them together in order to make relationships and associations between the graphs (Ma, 2010, p.104).
Science is defined by Oxford Living Dictionaries (2017) as “The intellectual and practical activity encompassing the systematic study of the structure and behaviour of the physical and natural world through observation and experiment”. So, science involves observation and experiment and these could not be carried out without maths because in order to observe for example the magnetic forces between magnets it involves recording observations on a graph which as I have stated above involves many basic mathematical concepts. In science, you are applying your use of mathematics to study the world. However, if science is the application of maths how is it that even Einstein (1943, quoted in Letters from and to children, 2004 ) one of the greatest scientists in history said that even he had great difficulty with maths?
Through our use of maths and science together to find the relationship between distance and forces with magnets. We found that distance and force has an inverse relationship which also links to electricity and gravity (Taylor, N. (2017). Therefore, once we understand these links we can explore what we can do more with our knowledge. For example, with electricity once this discovery was made it was used to make phones, dishwashers, washing machines, laptops, televisions and apple watches which are all everyday things that we may use in society but we don’t think about how the science and maths behind them all started out from basic concepts and relationships.
In order to develop children’s fundamental knowledge of mathematics in the future, I would need to start off teaching fundamental mathematical concepts such as measurement, how to draw a graph, the skills of observation used in science and how to plot a graph in order to show relationships between science and maths which links to connections between topics which Ma (2010, p.104) refers to as an element in developing your understanding of fundamental mathematics. Cross-curricular links could also be made to technology!
Taylor, N. (2017) Science and maths. [Lecture to Discovering Mathematics Year 2], ED21006: Discovering Mathematics (year 2) (17/18). University of Dundee. 20th November.
In a workshop, I took part in a business situation which demonstrated an application of Profound understanding of fundamental mathematics in the real world!
An overview of what was involved in this business simulation:
We had 5,000 Euros to spend for each quarter of the year
We had to choose a maximum of 5 items to sell for each quarter
We had to choose what quantity of items we wanted (quantity being a basic concept which is needed to have a profound understanding of fundamental mathematics according to Ma (2010, p.104))
We had to be careful and look at how much we were buying the item for to ensure we were not paying more for an item than it was being sold for such as the Christmas selection boxes in January.
We had to keep track of how many items were sold and whether they could be carried over into the next quota or whether it was a ‘write off’
We calculated how much we had spent by adding the prices together and subtracting what we had spent from the 5000 Euros or money we had to see what was left over. Therefore, through this process we were building upon each stage and adding to it over a period time which links to longitudinal coherence (Ma, 2010, p.104). Furthermore, we connected basic concepts together such as adding, subtracting, multiplying, percentages and decimals
We had to look at multiple perspectives to see what we were going to buy and sell however, we didn’t really look at multiple perspectives from a mathematical point as we continued to use the same methods of calculations throughout the session.
However, although we tried to be tackle by what we bought by guessing what items were more likely to be bought around a certain time of year, we could have looked ahead and seen that for January there was a sale price mark-up which mattered more. For example, beans only cost 25p to buy but they were being sold for 2.50 euro and 100% were sold. Therefore, there was a ten times mark up. Consequence, if I was to do this again i would pay more attention to the price per unit that we were buying it for and the seasonal price in order to make a bigger profit.
In the future, I would like to use this activity or at least one similar to further pupils understanding and application of fundamental mathematics in the real world or in the wider society. Therefore, they could connect and link various topics together for a meaningful purpose (Ma, 2010, p.104). It would be a fun and creative task or game for the children to do and could potentially due to the relevance of the task, it could give them an interest in business as a future career. However, in order to improve this activity I would like to further gain the children’s interest as well as emphasis the relevance by having the packaging of the actual items out on desks to further emphasise the relevance.
References:
Ma, L. (2010) Knowing and teaching elementary mathematics: teachers’ understanding of fundamental mathematics in China and United States. (Anniversary Ed.) New York: Routledge.
There’s a link between music, maths and our emotions. How can this be and why do major chords in songs make us happy? This is what I am going to explore in this blog post!
The links between maths and music:
beats (the value of each note)
timings
rhythm
string numbers
if you are changing key, you go up or down
scales: scales follow the same pattern, no matter what note
tempo: speed of the music (metronome)
Music makes us happy. I don’t know about you but I love listening to music and it makes me feel better no matter what mood I’m in. Kim (2015) states that the Feel Good Index (which is an equation itself) is the “… sum of all positive references in the lyrics, the song’s tempo in beats per minute and its key.” Having a fast tempo in a song makes us want to dance, we want to move and it makes us happy. Apparently this is because “Beats automatically activate motor areas of the brain.” Fernández-Sotos, Fernández-Caballero and Latorre (2016) also agree that the tempo impacts whether the music makes us happy or sad.
However, sad music can make us happy. Sometimes I’ll be in the mood to listen to sad and slow songs but they make me happy however, this contradicts what is said above? The lyrics are sad, not pleasant. This is because whilst the emotion of sadness is seen as negative, in artistic form, sadness can be felt, sensed or understood differently. Therefore, songs that are in a minor key which are identified as being sad songs do not always cause a negative emotion (Kawakami et al. (2013, pp. 1-2). Personally I think the reason for this is more than just the links to the maths behind the music such as a slower tempo or a minor key. It’s also because we connect with the feeling, the emotion through the song or lyrics and this connection is pleasant to us (Nield, 2016). The emotions are caused as the music brings back previous experiences that can make us happy or sad. (Konečni, 2008, cited in Hunter, Schellenberg, and Schimmack, 2010, p.54).
Additionally, why do certain pop songs in the charts make us feel happy because their tempos are fast, around 116 beats per minutes and have a major third musical key (Kim, 2015)? You might be thinking, what does music have to do with maths? Well, music is made up of “…pleasurable patterns of rhythm, beat, harmony and melody” (Gupta, 2009). If you are still asking what do tempo and beats have to with music well, according to Fernández-Sotos, Fernández-Caballero and Latorre (2016),
Tempo is “…the speed of a composition’s rhythm, and it is measured according to beats per minute.”
“Beat is the regular pulse of music which may be dictated by the rise or fall of the hand or baton of the conductor, by a metronome, or by the accents in music.”
So, what makes a pop song catchy? Why do we enjoy hearing the next new hit song on the radio and want to sing and dance along? The University of Bristol asked the same questions. They formed a mathematical equation to work out what makes a popular song popular. They created a diagram to compare pop songs patterns. The coloured parts of it represented beats and the connections seen in the diagram were sections of music that join. What they found was that many pop songs had a very similar pattern Seeker (2014).
A drawback to the equation that The University of Bristol developed (which they recognise themselves), is that it will need adapted as what becomes popular changes since, over time the songs that have become popular are ones that are getting louder (University of Bristol, 2017). Why is it that what is popular changes over time?
Why do these pop songs make us feel good? In one of my previous posts (Noble, 2017), I talked about how according to Burkeman (2011) gambling is satisfying as we are addicted to the potential of getting a reward and the satisfaction is due to the chemical dopamine being released (How the Brain Gets Addicted to Gambling, 2017). This same chemical is released when we listen to these pop songs therefore, making us feel satisfied, please, happy and feel good. However, why do we not get bored of these songs? In the video linked below it demonstrates how a lot of the famous pop songs are repetitive, they use the same four chords (random804, 2009).
So, we seem to be satisfied by the same types of songs. These songs that are popular, are popular because they fit “[I]mplicitly learned patterns…” or their patterns only differ by a slight bit (Wheatley, no date, cited in Hughes, 2013). This slight difference must be what keeps us satisfied as we would be bored if it was the exact same every time.
Critically, maths can be non-existent in music according to Sangster (2017). For example, every pitch has a different frequency but a piano note can’t be tuned to the exact frequency that it mathematically should be or it doesn’t sound right musically. This is where maths cannot be applied to music. This video below demonstrates why it’s impossible to tune the piano notes to the exact mathematical frequency although it demonstrates why this is so using maths (Minutephysics, 2015).
In the future, I would like to apply my knowledge and understanding of how maths underpins music to teach pupils about maths. Pupils could make their own short songs, using their maths skills such as counting, timings, rhythms and beats! This would be a fun activity showing the relevance that maths has in the wider environment and in everyday society.
In conclusion, maths is behind music. Music is composed by using basic concepts in maths such as the speed of the music, timings and counting the beats per minute. In order to count beats, people need to know how to count in a sequence, and how long a minute is. All these basic concepts are put together and built upon to make music over a long period of time. All of which relate to what Ma (2010, p. 104) says are the principles of fundamental mathematics. Furthermore, the type of music that is produced from using and building upon all these basic mathematical concepts can have an affect on peoples emotions, making them either happy or sad and formulas can even be created to determine or predict what songs will be big hits!
Hunter, P., Schellenberg, G., and Schimmack, U. (2010) ‘Feelings and Perceptions of Happiness and Sadness Induced by Music: Similarities, Differences, and Mixed Emotions’, Psychology of Aesthetics, Creativity, and the Arts, 4(1), pp. 47-56. Available at: http://www.erin.utoronto.ca/~w3psygs/HunterEtAl2010.pdf (Accessed: 9 November 2017).
Sangster, P. (2017) Discovering Maths: Music and Mathematics. [Lecture to Discovering Mathematics Year 2], ED21006: Discovering Mathematics (year 2) (17/18). University of Dundee. Day and month
g out of 10. Therefore, we demonstrated knowledge of fundamental understanding of mathematics as we used basic concepts and connecting them together in order to make relationships and associations between the graphs (Ma, 2010, p.104).
