Category Archives: 3.4 Prof. Reflection & Commitment

Can Mathematics be Beautiful?

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Maths, beautiful? Before I would have never of put these two words in the same sentence. Now I will say maths can be beautiful. After allowing myself to really understand how this can be and how maths can make things more pleasing to the eye I can put maths and beautiful together.

The Rule of Thirds

The rule of thirds is the first thing that introduced my thinking of mathematics being beautiful. The rule of thirds is where an image is broken down into thirds- both horizontally and vertically- so you end up with 9 parts on the image (Rowse, 2006). Like this:

We then place the thing of interest (whether that’s a person, animal, object or even just a specific part of scenery/image) on one of the third lines or on one of the four intersections of the third line which will give a more aesthetically pleasing result rather than if we just centralised the thing of interest (Roswie, 2006). Roswie (2006) furthers this by highlighting that in 1797 John Thomas Smith explained that by using the rule of thirds makes our eyes naturally drawn to these intersection points and thereby makes the image more pleasing to the eye. Therefore, by framing an image in this way, using this rule, works with our natural direction rather than against it (Rowse, 2006).

After my lecturer in Discovering Maths explained this rule of thirds I wanted to explore this further to see if in fact this well-known rule of photography was being applied in the wider world and to see if by using this rule of thirds actually works.

Here is an example of an image using the rule of thirds where the wasps eye has become the point of focus.

Wasps, more often than not, are seen as not the most beautiful creatures however from using the rule of thirds within this image I think the image itself can be looked at as beautiful. I am aware however that this rule of thirds is not the only thing that impacts upon this image and I understand that there are other factors that contribute such like lighting and timing (Amirshahi et al, 2014). However, it is interesting to think that this “old” rule has been impacting on us when looking at images unconsciously, as before this Discovering Maths input, I had never heard of this rule before.

Another thing that can allow me to use ‘maths’ and ‘beautiful’ in the same sentence is when looking at intervening with our faces. It is thought that the more symmetrical our face is the more ‘beautiful’ our face is (Bader, 2014). One of the reasons for this is because it has been suggested by the Evolutionary Advantage Theory that the more symmetrical the face is the better a person’s health is (Bader, 2014). Dr. Stephen Marquardt (undated), along with many other facial surgeons and mathematicians, furthers this through his findings where he too found that people do find more symmetrical faces more attractive/beautiful. The Perceptual Bias Theory (Bader, 2014) agrees with this as it states that our brains work in a way that allow us to process symmetrical images easier than asymmetrical ones, thereby indicating that maths unconsciously effects our day to day lives in as much detail as what we find beautiful/attractive to look at (Perrett, 2001).

These two ideas – the rule of thirds and facial symmetry- are only just two examples of how maths is beautiful.

This then got me thinking even more. If we can use the rule of thirds to make an image more pleasing to the eye and by making a face more symmetrical we can make it more beautiful, can we make the same outcomes within the classroom?

Have a look at the layout of this classroom:

You can argue that this particular classroom, shown in the image, has slight symmetry and the use of the rule of thirds. For what I have previously researched and found out about the rule of thirds making things more pleasing to the eye (Roswie, 2006) and the that symmetry makes things more beautiful/attractive (Bader, 2014), can this be the same for this classroom? Would this make a difference to the learning and teaching which is created? This is what I now ask myself. It could be argued that because the rule of thirds makes thing more pleasing to the eye, if we were to layout our classroom using this rule of thirds, it could in fact make the room have a more pleasing feel to it. Dr. Sheryl Reinisch (2017) furthers this by saying that if then the classroom has a more pleasing feel to it this can impact greatly on the teaching and learning that goes on in this classroom. This is due to helping the children feel safe, secure and valued. Just by the way we layout our classroom can have a real impact on creating a more pleasing environment allowing children to feel more motivated and engaged (Reinsch, 2017).

If I am completely honest, before learning and researching about this I would be very likely to layout my classroom the way it would look the prettiest to me and not really have a huge thought about it. However, now this has really made me think twice about this as it could have a real impact on the children’s learning before they even sit down.

Furthermore, this rule of thirds has also allowed me to begin to think on my wall displays in the future. Using this rule could make them much more pleasing to the eye and therefore more meaningful, as children would tend to refer to them more because they are pleasing on their eye. This would also allow for the discussion to take place on how, yes maths is used in everyday lives, but not just using the usual examples of counting money to buy sweets or working out bus timetables so you can get places. This would directly show the children that we use maths for almost everything, even when just take or displaying a simple photo.

 

 

References

Amirshahi,S et al. (2014) Evaluating the Rule of Thirds in Photographs and Paintings. Available at: http://booksandjournals.brillonline.com/docserver/journals/22134913/2/1-2/22134913_002_01-02_S11_text.pdf?expires=1540566335&id=id&accname=guest&checksum=3055FC5FFDC8C1AD326C6AB654328732 (Accessed: 6 November 2018).

Rowse, D. (2006) Rule of Thirds. Available at: https://digital-photography-school.com/rule-of-thirds/ (Accessed: 6 November 2018).

Bader, L. (2014) Facial Symmetry and Attractiveness. The Evolution of Human Sexuality. Available at:  https://sites.psu.edu/evolutionofhumansexuality/2014/03/24/facial-symmetry-and-attractiveness/ (Accessed: 6 November 2018)

Perrett, D. (1999) Symmetry and Human Facial Attractiveness.

Dr. Sheryl Reinisch (2017) https://education.cu-portland.edu/blog/classroom-resources/welcoming-classrooms-better-students/

https://www.google.co.uk/search?q=classroom+layout&tbm=isch&source=iu&ictx=1&fir=WPv9o2SVAH3AsM%253A%252CbTLiokDsO1T1KM%252C_&usg=AI4_-kRDyvtGzbDweqgy0q8h9lsLZI_84w&sa=X&ved=2ahUKEwidt6-Oz8LeAhVBKewKHcCsB8IQ9QEwBHoECAYQDA#imgrc=WPv9o2SVAH3AsM:

Thinking Fast & Thinking Slow

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According to Kahneman (2011), our thought process is determined by two different systems – impulsive/automatic and thoughtful/deliberate – also considered as system 1 and system 2. He makes apparent that for the majority of the time we should be using our system 2, however, this is not what happens. In fact, the reason we don’t use system 2 all of the time is because it is more exhausting, therefore, we resort to using system 1 (Kahneman, 2011).

To show how these systems work my lecturer used a very easy but clear example. He put this on the board:

We’ve used our system 1 process and immediately thought the answer is £10 – which is wrong. The answer in fact is £5- HOW?? Due to our system 1 thinking it automatically creates the calculation of £110 – £100 = £10, however, if and when we use our system 2 thinking –our thoughtful and deliberate thinking – we can see where we are going wrong. If we used system 2 thinking from the beginning, this is how we would work out the answer:

This isn’t “hard and complex” maths which makes us get the wrong answer. This has then made me wonder is it in fact Kahneman’s (2011) system 1 and system 2 approach to the way we think which affects the answer we first give?

So what? 

The ideas that Kahneman’s (2011) Thinking Fast & Thinking Slow give, really make up the whole spectrum of human thought, action and behaviour.

After getting my head around the “Thinking Fast & Thinking Slow” approach (Kahneman, 2011), it led me to further thinking on how, the way we think and process what we read, can affect the answers we give and its not nesessaraly because we do not know the answer. This got me thinking about exams within maths, in particular, and how the reason we give the answer may not be due to if the maths is ‘easy’ or ‘hard’. It could be caused by using only our system 1 (Kahneman, 2011) thinking.

Furthermore, still thinking about exams, this time multiple choice, our counter-intuitive thinking becomes apparent. This is when you put an answer but then go back and think it might be a different answer. Do you stick with the first answer you put or do you change to the other answer you think it might be?

Brownstein et al (2000) make apparent that those who go ahead and change their answer actually change it to the wrong answer. However, Kruger et al (2005) creates an alternative argument to Brownstein et al (2000) and highlights that over 70 years of research which has been focused on changing answers and this belief that if you change your answer your more likely to change to the wrong answer, that this is not the case. In fact, Kruger et al (2005) make clear that people who have this counter-intuitive thinking, and who then go on to change their answer, majority of which end up changing their answer from the wrong one to the right one. This then improves their overall result/mark.

So why is this the case? Well my lecturer presented a problem to prove how this can mathematically work. He asked us this:

 

Straight away, using our system 1 approach (Kahneman, 2011), we think the answer is 50/50. He then showed us the mathematical proof that there is more chance of winning the car if you switch doors.

After my lecturer explain why this works, making me use my system 2 approach, it got me thinking even more about Kahneman’s (2011) Thinking Fast and Thinking Slow approach in that is our counter-intuitive thinking really just questions tricking our system 1 thinking so we then use our system 2 thinking. This meaning that when it comes to changing your answer, really we should change it because as Kruger et al (2005) state, there is more a chance of us changing it to the correct answer and there is more of a chance this is us then using our system 2 thinking which Kahneman (2011) would argue is the system we should be using all of the time. Finally, this has highlighted to me that throughout teaching mathematics we should really consider this Thinking Fast & Thinking Slow (Kahneman, 2011) approach as this could be a major factor which affects a child’s answer being wrong and not actually how difficult the mathematics is.

 

References

Kahneman, D. (2011) Thinking Fast & Slow. New York :Farrar, Straus and Giroux
Brownstein, Wolf, and Green, Barron’s ‘How to Prepare for the GRE: Graduate Record Examination’, 2000, p. 6 as cited in  Kruger, Wirtz & Miller, 2005, p. 725
Kruger, J., Wirtz, D. & Miller, D. (2005) ‘Counter factual thinking and the first instance fallacy’, Journal of Personality and Social Psychology, 88(5), pp. 725-735.

Reflection of semester 1

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Semester one was not what I was initially expecting. I thought it would of been about getting to grips with Curriculum for Excellence and Pedagogical studies, not a direct focus on working collaboratively and values, as I thought these things were “common sense”. However, the fact that we did these modules and worked with social work and CLD students, I feel was very beneficial and a real eye opener as these areas are much more than what I assumed to be “common sense”. Being honest before starting University I had no idea what CLD even consisted of.

I feel as though the most important part of semester one for me and my professional development was the Working Together module because it has given me a real in depth understanding of what social work and CLD actually consists of, therefore erasing any stereotype definitions I had assumed previous. This module really helped me to understand just how important and crucial it is for a child’s welfare that all three of these disciplines (teachers, social worker and CLD) work collaboratively to ensure the best outcome for a child, giving me indication that for some children the GIRFEC approach can only be fully fulfilled if all three disciplines have this close working partnership. Therefore, indicating this module impacted on my professional development significantly as before I never thought that these three disciplines would be so closely connected in my teacher career.

I much preferred the assessment structure for this module as it allowed me to really think and reflect on my own thoughts and opinions however, it also allowed me to work and listen to others whose opinions may have differed from my own but also allowed others to question my opinions which encouraged for me to further reflect my own thinking. I also benefited greatly from working with others in the different disciplines due to them having slightly different angled views, therefore allowing me to explore different angles of a situation presented with an open mind. When it came to the presentation I did feel very nervous as I did not want to let my group down, however I feel this was an appropriate and more meaningful way to present all our work from our agency visit and long discussions.

Moving forward, I feel as though that working with the different disciplines in expressing my views and the presentation has given me a boost of confidence to begin to speak out in lectures and tutorials, which I feel will really benefit my professional development as I can fully engage and benefit from all materials presented in future.