# Can Mathematics be Beautiful?

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/

# Could the Curriculum for Excellence be a ‘Sunk Cost’?

Could the Curriculum for Excellence be a ‘Sunk Cost’’?

The aim of Curriculum for excellence and the reason it was put in place was to help children and young people, across Scotland, gain knowledge, skills and attributes needed for life as well as the skills needed for learning and work (Education Scotland, 2018). The Curriculum for Excellence has been the framework for Scottish education for the past 8 years, and was created by government officials over a long period of time to give teachers more flexibility over what and how they teach (BBC, 2017). Although the government would argue that the Curriculum for Excellence has a purpose to create successful learners, confident individuals, responsible citizens and effective contributors, formally known as the 4 capacities, and further framework to create the seven principles, to ensure children’s learning is specified to them, many people would argue that it has many flaws. One of the arguments that is made is that the Curriculum for Excellence could in fact be widening the attainment gap, not closing it, therefore it is failing (BBC, 2017).

What is a ‘sunk cost’?

A sunk cost is defined as something which has had a lot of time, money or effort put into it however if it starts to fail, those who have invested the time, money or effort, carry on with it as they don’t want it to appear wasteful (Arkes and Blumer, 1985).

Can this be what has happened with the Curriculum for Excellence?

For me what stands out the most from the arguments that have been highlighted about the flaws of Curriculum for Excellence is that it could in fact be widening the attainment gap and not closing it (BBC, 2017). Paterson (2018) makes this apparent when he highlights that Scotland use to be well ahead of the OECD (the programme for international student assessment) average however this has declined drastically. Paterson (2018) furthers this by summarising the results from the annual Scottish Survey of literacy and numeracy which show a fall in attainment in both of these curricular areas. This evidence then brings the realisation that the Curriculum for Excellence is not closing the attainment gap in this sense as attainment as a whole is falling from where it was before Curriculum for Excellence was introduced. Paterson (2018) then goes on to explain that on comparing the Curriculum for Excellence to other international educational structures, CfE can be seen to ‘neglect’ knowledge and focus more on skill. Hirsch (2016) furthers this and explains just how much knowledge matters. He does not dismiss the fact that skills do matter however shows the best way to gain skills is through gaining knowledge. Paterson (2018) goes on to highlight that schools provide the opportunity for children to gain knowledge who would not get it from home. Therefore, if schools stopped teaching structured knowledge, the inequality of knowledge will widen because the children of the well-educated and the wealthy will receive this structured knowledge in other ways. This then makes me indicates to me that curriculum for excellence can do better within this aspect and therefore makes me consider that is could in fact only be the start of its sunken cost.

On the other hand of this argument is that the Curriculum for Excellence is so deeply embedded, that removing it and creating something new would cause enormous upheaval. There would never be an easy way to change it and would mean that a whole generation would have been disadvantaged even more so within their education (Paterson, 2018). Therefore, meaning that the children of Scotland would be at an even bigger disadvantage within education if we were to move away from the Curriculum for Excellence compared to if we just stick to it and try and work on it.

Furthermore, looking at an opposing argument, the Scottish Government (2017) published information suggesting that those children because of Curriculum for Excellence are achieving within areas of reading, writing and numeracy. This then provides evidence that in fact the Curriculum for Excellence is providing the children of Scotland with a fairly good education however, it is the question of could it be better?

Overall, from my research I would say that arguably the Curriculum for Excellence is a sunk cost as it could be better. However, when looking at changing the whole education framework this is a very unrealistic approach as it would disrupt too much learn for very long period of time. The time, effort and money that would be spent creating a new framework, re-educating teachers would be significant. Having spent so much time, effort and money on the Curriculum for Excellence I feel we are in too deep now to change it. No child’s education should reduce due to a failing curriculum; therefore, I feel the best approach to this issue is that the Curriculum for Excellence is used as a very basic framework and we now begin to develop and work on the sections of it that are just not quite working.

References

BBC (2010) New curriculum could be ‘disastrous’, says education expert. Available at:  https://www.bbc.co.uk/news/uk-scotland-scotland-politics-41134835 (Accessed: 1 November 2018)

Education Scotland (2018) What is Curriculum for Excellence. Available at:  https://education.gov.scot/scottish-education-system/policy-for-scottish-education/policy-drivers/cfe-(building-from-the-statement-appendix-incl-btc1-5)/What%20is%20Curriculum%20for%20Excellence? (Accessed: 1 November 2018).

Arkes and Blumer (1985) The Psychology of sunk cost. Available at:https://www.sciencedirect.com/science/article/pii/0749597885900494 (Accessed: 2 November 2018).

Paterson, L. (2018) Scotland’s Curriculum for Excellence: the betrayal of a whole generation.

Hirsch, D.E. (2016) Why Knowledge Matters. Available at: https://eric.ed.gov/?id=ED568821 (Accessed: 2 November 2018).

Scottish Government (2017) Achievement of Curriculum for Excellence Levels 2016/2017. Available at: https://www.gov.scot/publications/achievement-curriculum-excellence-cfe-levels-2016-17/  (Accessed: 2 November 2018).

# Thinking Fast & Thinking Slow

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.

# Can a raise in attainment within mathematics be created and maths anxiety destroyed?

Discovering maths has really opened my mind and has got me thinking about how I want to teach maths in my future classroom and what kind of teacher in maths I want to be.

After one of the discovering maths lectures it allowed me to begin to ask myself a series of questions around the way maths is perceived within schools and the way in which maths is taught. I now ask myself, can we raise attainment within maths, not by doing more maths but by doing different maths?

When I think back to my experience of maths in both primary and secondary school they are very similar – sitting at a desk, the teacher talking a new concept to you, you then completed some examples and then you were assessed on your understanding … or more like memory. Nothing ‘fun’, no enthusiasm or creativity presented. It was JUST maths.

Jo Boaler’s (2009) study has really begun to form a foundation to my thinking of my future teaching of mathematics. The study carried out was on 2 schools, Amber Hill which was a traditional school which followed set methods and procedures, distinct topics and closed mathematical problems and Phoenix Park which was a progressive school where there was a degree of choice, openness and mathematically rich experiences. From a series of assessments Amber Hill pupils performed worse than Phoenix Park. Amber Hill pupils had a broad understanding of facts, rules and procedures however they found them difficult to remember over time. Whereas Phoenix Park pupils were flexible and adaptable and they were able to see their knowledge in different situations.

Going back to my question of can we raise attainment within maths, not by doing more but by doing different maths, Amber Hill pupils had not learned any less maths than those at Phoenix Park, but different maths. Jo Boaler concluded from her study that transmitting mathematics is less helpful, which those at Amber Hill experienced, than classrooms where pupils are “apprenticed into a system of knowing, thinking and doing”(Boaler, 2009), which those at Phoenix Park experienced. Therefore, this has allowed me to reflect on the teaching of maths I have experienced and has really opened my eyes to some of the ‘could be’ causes of maths anxiety amongst both pupils and teachers.

Through Jo Boaler’s findings it has become apparent that Amber Hill pupils demonstrated instrumental understanding and Phoenix Park pupils demonstrated relational understanding(Skemp,1989). The best way to think about these concepts is that those pupils who demonstrated relational understandings are like a chef, if they are missing an ingredient to their recipe they know what they can use as a substitute to make what their making taste the same. Whereas those pupils who demonstrated an instrumental understanding are like recipe followers, meaning that if they are missing an ingredient they would have no idea what they could use as a substitute, therefore would be unable to complete the recipe. In a more mathematical example, mathematics is hidden in our everyday lives which some people are unaware of (Haylock, 2010, p13) such like telling the time, counting money, reading bus timetables as they forget that this is ‘maths’, therefore unconsciously allow themselves to overcome their confusion/fear of maths with no realisation that they are doing it, because it is relevant at that time. However, as soon as maths is disclosed, it causes fear and uncertainty across many (Haylock, 2014, p4). Therefore, again implying that when people have that basic understanding, know where maths can be used in their own lives and are able to use their knowledge in different situations, it begins to erase this maths anxiety (Haylock, 2014).

From this it has highlighted to me that we need to create ‘chefs’ within maths in schools, where pupils are enthused, engaged and enjoy maths because it should not just be about learning a concept, having to memorise it, just to get the answer correct when completing a test because asHaylock (2010, p5) goes on to describe, a lack of confidence within maths is formed from thinking you have to get the answer correct. This is where from the discovering maths lectures, already have got me thinking about how I can have an influence in reducing the fear and anxiety of maths amongst pupil. I am now aware that in the classrooms today we need to create an understanding, which allows for wrong answers and allows children to focus on the methods and thinking behind the answer (Hansen et al, 2017, p3) and move away from this ‘traditional’ maths and begin to teach ‘different’ maths.

Overall, I think maths anxiety can be destroyed within the younger generation and I believe that a raise in attainment within maths can be created by doing ‘different’ maths, other than just the traditional set methods and procedures, distinct topics and closed mathematical problems. I think that if we teach maths to create basic understandings where choice, openness and experiences are present and we create ‘chefs’ within the classroom, children will have a much better chance of having flexibility and adaptability within their learning, being able to transfer their knowledge to different situations, therefore meaning that when they do come across concepts/problems that are more difficult and that they haven’t seen before, the pupils can begin to look at them with a much more open mind set.

References

Boaler, J. (2009) The Elephant in the Classroom: Helping Children to Learn and Love Maths.  London: Souvenir Press Ltd.

Skemp, R. R. (1989) Mathematics in the Primary School. London: Routledge.

Hansen, A., Drews, D. and Dudgeon, J. (2017) Children’s Errors in Mathematics. 4thedn. London: Learning Matters

Haylock, D. (2010) Mathematics Explained for Primary Teachers. 4thedn. London: SAGE Publications Ltd.

Haylock, D. (2014) Mathematics Explained for Primary Teachers. 5thedn. London: SAGE Publications Ltd.