Discovering Maths – Hannah Robertson

October 29, 2018November 26, 2018

Face, Maths and Evolution. How does it all mix?

Would you believe me if I said that mathematics can judge your facial proportion? Well unfortunately it can in a very accurate and understandable way. In our most recent input for our Discovering Mathematics module we unearthed how to draw a face accurately. For myself I have always had a creative ambition and enthusiasm, but I can say that I consciously – and even happily – avoided ever drawing an accurate face to gain my Higher Art and Design Qualification. It wasn’t until this input that I have ever successfully accomplished my avoided dream.

Introducing ‘The Golden Ratio’

Initially we were instructed to draw a face on one side of an A3 piece of paper. As you can imagine this triggered a few negative responses along with ironically a small part of art anxiety rather than maths anxiety. Like myself you will begin to wonder how on earth drawing a face has any connection to maths. It wasn’t until we were indulged on the beauty of drawing an accurate facial proportion. Everyday countless people have looked in the mirror and ridiculed there physical appearance. I would argue that at this moment in time there has never been a greater pressure worldwide to look, rather than be, your best. However, without knowing it you are judging the mathematical proportions of your facial features rather than your face itself.

To break it down simply, the shape and size of your face can be segmented into different groups. This discovery can unfold the ground breaking realisation that the human face is like a jigsaw. There are many different parts which aline and fit together to construct each of our individual faces. This is make clear by what is know as the Golden Ratio. This discovery by Dr. Stephen Marquardt identifies the proportions and measurements of each facial feature that would form the desirable face (Meisner, 2014) . In most cases this discovery has influenced reconstruction in cosmetic surgeries of people world wide. This is exactly what we considered when redrawing our original face. Ideally the stature of a perfectly sculpted face would include;

The inner and outer corners of the eyes to be in line with the center of the nose
The outer edge of each nostril to be in line with the inner corner of each eye
The center line of the chin to a line with center of the upper lip
The tops of the ears to be in line with the center line of the eyes
The bottom of the ears to be in line with the bottom of the nose

As seen in the picture above, when considering these proportions and measurements it made it dramatically easier to redraw our original portrait. The contrast when considering and understanding proportions is undeniable. Understanding the relationship between features meant that shape, scale and proportion became much more understandable. Mathematics has made this possible through comparing measurements of facial features, without this an ever-growing cosmetic would not be able to function. By segmenting features together and build considerable knowledge of these proportions, the end result has greater accuracy. It achieves realistic dimensions of the human face.

The golden ratio vs. primates

Studying this topic immediately fueled my interests to link the facial proportions of The Golden Ratio to the facial structure of primates. Science has generously allowed us to explore the connections and evolution of humans themselves. Without delving to much into scientific theory, the theory of evolution devised by Charles Darwin concludes that all species have devolved from a small string of lifeforms (BBC, 2014). This is where the link between humans and primates can be made. Over time genetic differences have reshaped our species both physically and mentally. The theory of evolution has brought to life scientific evidence that humans are derived from primates. Using my new profound knowledge of facial proportion I thought this may provide and enciteful comparison.

Neverse (no date)

Looking at the pictures above you can see how some proportions of The Golden Ratio can seen within the facial structure of these primates. Although there are some physical difference such are the ears, there is a clear connection between our faces and these primates. By showing that these primates do not have a the structure of The Golden Ratio this can in fact relate to the differences between physical appearance among humans. Our eyes don’t match. Our ears don’t match. Everything about our faces are different yet we all have small common forms of facial structure, just like primates. As explained by Burrow et al. (2014) humans and primates can be physically compared because of the way our muscles are positioned in our faces. It is this connection which makes it possible to see similarities between ourselves and primates using The Golden Ratio.

In terms of the bigger picture in depth exploring mathematical concepts within the primary school will open rather than close the mind of children. By weaving concepts together and making cross curricular links children will be able to access the world around them through a new lens. This will allow them to use imagination and fun within the world of maths rather than be tied to textbook procedural work. A key concept which drove the Profound Understanding of Fundamental Mathematics according to Ma (2010) is Basic Ideas. This allows teacher to craft they’re approach in a way the takes a component of maths (in this case proportion and measurement) and spark by guiding children through its exciting structure. Therefore, the maths curricula will not inhibit but expand children’s approaches and interests in the world we are a part of. Furthermore, this also signifies the importance of competence within teachers themselves. Unless and educators work individually, and together as a body, to acquire profound understanding of fundamental mathematics how can we expect pupils to feel the same? The level of understanding which educators are capable of mimic the level of understanding which their pupils can achieve (Bregner and Groth, 2006). Building on mathematical knowledge and relating it to other concepts throughout the education curriculum will only encourage children to achieve in maths rather than be limited by teachers’ deficiency’s. This attitude should be established both nationally and internationally in order to provide children with the opportunity achieve with flexible and stable knowledge and understanding.

References

BBC GCSE Bitesize (2014) Science: Evolution. Available at: http://www.bbc.co.uk/schools/gcsebitesize/science/edexcel/classification_inheritance/evolutionrev1.shtml (Accessed: 29 October 2018).

Bregner, R. Groth, J. (2006) Preservice Elementary Teachers’ Conceptual and Procedural Knowledge of Mean, Median and Mode. Available at: https://www-tandfonline-com.libezproxy.dundee.ac.uk/doi/abs/10.1207/s15327833mtl0801_3 (Accessed: 29 October 2018).

Burrows, A. et al. (2014) Humans Faces Are slower than Chimpanzees Faces. Available at: http://europepmc.org/backend/ptpmcrender.fcgi?accid=PMC4206419&blobtype=pdf (Accessed: 26 October 2018).

Neverse (no date) Available at: https://www.stockfreeimages.com/29164140/Pair-of-monkeys.html (Accessed: 29 October 2018).

Ma, L. (2010) Knowing and teaching elementary mathematics: teachers’ understanding of fundamental mathematics in China and the United States. Available at: https://ebookcentral.proquest.com/lib/dundee/detail.action?docID=481154. (Accessed: 25 October 2018).

Meisner, G. (2014) Available at: https://www.goldennumber.net/face/ (Accessed: 25 October 2018).

Meisner, G. (2014) Available at: https://www.goldennumber.net/beauty/ (Accessed: 25 October 2018).

October 29, 2018October 29, 2018

Can we touch the moon?

Education is one of the first places that allow children to build a realistic foundation of knowledge. It wasn’t until my first school placement that I began to understand the importance of putting fact into visual representation that all children understand. Therefore, if this is inaccurate it will result in a generation (or even generations) of flawed understanding. There is no better example of this than Space.

Without most teachers realising, there will be a time that a child in there class stares out of the window imagining what is beyond our drizzly, damp and occasionally blue sky. It is this unknown environment which beacons children to explore and build there experiences and knowledge of Space. I can happily say that I was most definitely one of those children. Primary 5 was the year of Space! Without a shadow of a doubt the experience that I had was incredible, I was well on my way to becoming a fully fledged astronaut. However, despite having amazing experiences, I cannot say that my knowledge of Space is at all realistic. Sadly this will be the case for many others who have or are in education at this time.

Until recently, I did not realise the enormity of Space. There is no word to describe how ginormous Space actually is and this is exactly where Space education must start. For me this is where my own knowledge is false because I was not aware of the scale of space.

To put this into perspective you can look at the distance between The Earth and The Moon. To the majority of people – including myself- believed that the The Moon was only a short distance away when in reality it is actually much greater than that. To scale you can approximately shrink The Earth down to the size of a basket ball and The Moon down to the size of a tennis ball. Using this scale this means that a basketball and The Earth along with a tennis ball and The Moon are at a ratio of 5, 280, 000: 1 (The Science Asylum, 2017).

Using this scale we can put the distance between The Earth and The Moon into a realistic visual perspective. The distance between the core of The Earth and The Moon is 238, 855 miles, when scale down using the ratio of the basket and tennis ball this distance is only 7.2 meters. In terms of what this looks like to scale it would look the picture below.

Encouraging ourselves as Educators to understand scale is extremely important, not only to physically represent Space in our own mind but in the minds of the those within the classroom. As fun as it may sound, taking children to Space is not a class trip that can be offered within my lifetime. This is why it is so important to allow children to experience accurate scale. This can be linked to the ideas of a logarithmic scale present in the minds of those inadequate and unrealistic experiences. Bellos (2010) maintains that children believe that with unresolved understanding will be unable to fully interpret the realistic size of maximal numbers. From experience children believer that space looks simplistic; with planets knitted closely together on a perfectly circular orbit surrounding the sun.

Generic image of Space.

In reality it is believed that space actually looks like this!

This is only a small part of Space. Space is actual made up of millions of solar systems that most of which can be easily understood. In other words the universe is made us of billions of stars, these stars form galaxies and galaxies form the universe. Overall, in order to promote a Profound Understanding of Fundamental Mathematics educators must invite children to build knowledge into compound understanding. This can relate to Ma’s (2010) key concept of connectedness which focuses on connecting mathematical procedures to wider concepts, thus enabling a greater understanding. In terms of Space, knowing the size of an the environment highlights intellectual solidity (Frobisher, 2007). Only when this happens children will be able to be able to explore the immensity of Space purposefully.

References

Bellos, A. Riley, A. (2010) Alex’s adventures in numberland. London: Bloomsbury.

Frobisher, L.J. (2007) Learning to teach shape and space: a handbook for students and teachers in the primary school. Cheltenham: Nelson Thornes.

The Science Asylum (2017) How far is the Moon? Available at: https://www.youtube.com/channel/UCXgNowiGxwwnLeQ7DXTwXPg. (Accessed: 25 October 2018).

September 29, 2018September 29, 2018

the maths and creativity sandwich.

Never would I have imagined that myself or anyone could sandwich together maths and creativity. Yet what a wonderful sandwich it is! Realistically, the majority of people would strongly argue against this opening statement, my self being one of them, however let me tell you that it is more than possible.

Like most sandwiches it all begins with the bread and in this case it begins with MATHS and Art. If you wanted to find these breads on a supermarket shelf you would instinctively look at opposite ends. However, this is not true because they could actually be found right next door to each other. My own experience of maths and art at school was not in anyway this experience. I would have confidently argued during my time at school that art was the elegant French baguette – thick, crunchy and popular- and on the other hand maths was the sourdough of all breads – bland, odd tasting and for the select few. Although reflecting on this now I think differently. This week in discovering maths we were exposed the creative aspects of this once bland subject.

This adventure was sparked by looking in detail at shapes. We discussed the names, number of side and angles of a variety of 2D shapes such as triangles, squares and hexagons.

You are now wondering how does this relate to maths? And it begins by introducing the idea of tessellation. ‘Tessellation (or tiling) is a repeating pattern of shapes that fit perfectly together without any overlaps or gaps.’ Brown (2018). Simple shapes such as triangle and squares can tesselate because their angles can make a full rotation. But how do you do make it personal?….

Take an original shape, such as a square, and cut segments out of it.
You then take your segments and add them back onto a different side of the square.
You can then repeatedly join this new shape together by repeating, rotating or mirroring it.
Repeat it all over the page, your final result should be a wonderful tessellated pattern.

When this is practiced you can make magnificent patterns and works of art. Traditionally this commonly used within Islamic art and patterns.

(Please watch this short clip to see many different types of visual tessellations)

Watson, C (2015)

As I discussed this shows that maths can be used in an engaging and exciting way and this is what is extremely important when introducing maths into any classroom. I believe that when you begin a maths lesson you have only a few moments to make it interesting otherwise children will switch off. This what brings me back to the sandwich. Do NOT present maths as the sourdough bread! Within tessellation alone there are hundreds of opportunities for children to put there own creative stamp on their maths sandwich. They can experiment with fillings, experiment with topping, experiment with size and most importantly of all they will understand how the sandwich is made.

This reiterates the concepts of Profound Understanding of Fundamental Mathematics (PUFM). For myself, by investigating this topic of tessellation alone my view of PUFM has evolved because I can see it represented in Maths! The root of tessellation is shape. Children’s basic understanding of shape will be to name the shape they see. However, if pupils have PUFM they can understand that if you alter the shape it will still have the same area. In other words pupils will not only be able to name the types of bread, they will understand how the bread is actually made.

However because of constraints children will not have time to explore this and there for be unable to sandwich maths with creativity. So how do educators step of our this narrow box. Haylock and Thangata (2007) argue that drill like teaching methods which are reused over decades betray creativity. Thus how maths is taught in the classroom can either uplift or damage creative the link between maths and creativity. Similarly Maths needs to be understood by the educator before it can be understood by pupils (Setati, 2011). As a future teacher I will continue to encourage creative thinking and tasks classroom maths topics. If this is done by all it can transform Maths from a bland sourdough into a baguette.

References:

Brown, J. (2018) ‘Maths, creative? No way!’ ED21006: Discovering Maths. Available at: https://my.dundee.ac.uk/webapps/blackboard/execute/displayLearningUnit?course_id=_58988_1&content_id=_5217933_1 (Accessed: 29 September 2018).

Haylock, D. and Thangata, F. (2007) Key concepts in teaching primary mathematics. London: SAGE.

Setati, M. (2011) Mathematics in Multilingual Classrooms in South Africa: From Understanding the Problem to Exploring Possible Solutions. Dordrecht: Springer Netherlands 2012.

Scottish Government (no date) curriculum for excellence: mathematics principles and practice. Available: https://education.gov.scot/Documents/mathematics-pp.pdf (Accessed: 29 September 2018).

Watson, C (2015) What is Tessellation? Available at: https://www.youtube.com/watch?v=7GiKeeWSf4s (Accessed: 29 September 2018).

September 19, 2018September 24, 2018

Do you know what an angle really is?

Most recently I have began the studies of my elective module Discovering Maths at University. Although we are only breaking into the second week of this module, I have immediately found it abundantly clear that this module will serve more than just knowledge of the Primary School mathematics curriculum; it will indefinitely open my eyes to the cracks of this subject.

On our very first input the class was asked how well we believe we know mathematical topics. Quickly I began to think that, like all others in the room, we would at least have a National 5 qualification in mathematics, therefore our knowledge of maths would be quite solid. Yet, is it actually? This was quickly answered when my lecturer was discussing angles he asked ‘What is an angle? It is the measurement of a rotation.’

Image result for boom

In this single moment I realised that my knowledge of angles was molded into a way that I could only answer textbook questions. In my thirteen years of schooling I had never once understood what an angle was. My head was filled with knowledge about seeing right angles in every stair, corner and cupboard at my home, knowing how to measure them with a protractor and being able to name the different types of angles at the drop of a hat. Looking back on my experiences at school now, I know that I do have valuable knowledge about angles but none of it made sense until that moment. This is because I understood what an angle is.

It is moments like this that every child must have within their learning. As a student teacher there is an expectation that we must equip children with the knowledge to meet curriculum requirements. In many lessons this is the case, however knowledge should never be put in front of understanding. We can teach children a million different facts about the world around us, but if they do not understand these facts how will they be of any value to them? As a future teacher I now find that it is crucial that this should be a part of all learning because it will equip children with the ability to see and make links within their learning. This matters seems controversial throughout schools across the world as many have differing opinions about what the purpose of mathematics is.

Understanding mathematics is key aspect of specialist knowledge of fundamental mathematics. My early understanding of this phrase so far is that it is understanding the thing itself and all of its properties. Enthusiasts of maths in education such as Liping Ma highlight that understanding of mathematics in crucial in making sure that students have the greatest success (2010). Therefore, if children can understand the roots of mathematical topics, not just what they look like, this will allow them to have a profound understanding needed to progress learning. Similarly Haylock et. all (2007) found that mathematics promotes profound learning that allows children to understand the world around them. Thus, mathematics in school should not just require children to solve problems; they must create links with how these issues relate to everyday life. Looking forward I am excited to find out how my experience in this module will allow my conception of understand in maths to flourish and develop.

References:

Haylock, D. and Thangata, F. (2007) Key concepts i teaching primary mathematics. London: SAGE.

Ma, L. (2010) Knowing and teaching mathematics: teachers’ understanding of fundamental mathematics in China and the United States. New York: Routledge.