Monthly Archives: October 2018

Space is big, really REALLY big

One of the ways that teachers across Scotland are trying to encourage positive affective domains within children in their maths is by making it more relevant.

By applying mathematical concepts to everyday real life concepts, we can encourage children to appreciate the application of maths in society. For example, as teachers we cover a range of inter-disciplinary topics as part of the Curriculum for Excellence (Scottish Government, 2008), which often allows them to apply mathematical concepts that they have been taught to real life scenarios. However, do we allow our children to recognise the importance of these topics, rather than them seeing it as just another area which they need to learn?

One of these topics – a topic familiar with every child – is Space. My understanding of space at Primary School was that we had a sun and a moon and 7 other planets surrounding us in an area known as our solar system. However, is this where our knowledge of space should end? At the edge of our solar system? Could we not explore our universe with them even further by delving deeper into the whole new worlds which surround our solar system, allowing children to recognise the incredibly vast space that we live in. Thus, I ask myself…

Could maths give us a better understanding of the depth of our universe?

Early thoughts…

Greek astronomers such as Ptolemy believed that our solar system was geocentric – that the earth was in the centre of our solar system and the other planets orbited around it. He also believed that the planets were an equal distance away from one another. As he was unable to use experimentation to prove his ideas, he tailored his maths to fit them and went on to make everyone believe his predictions through his use of mathematics (Falkner, 2011).

Image 1 – Geocentric model

Ptolemy used maths to gain a better understanding of our universe, but yet his knowledge of it was not correct. He showed how maths can be adapted to fit our beliefs, but it may not give us an understanding of the underlying reality.

However, his maths would have given him a greater understanding of the universe if he had conducted experiments in regards to the distances between the planets. He then could have went on to use maths to confirm his experiment results.

Another Greek astronomer Copernicus did exactly this. He used constant and more in depth observations than Ptolemy to figure out that we have a ‘heliocentric system’ (where the sun is at the centre of the solar system) and that the planets are unequal distances apart. Through this, he was then able to use maths to prove his theory to the public (Falkner, 2011).

So how can we use the maths and the theories that Copernicus created to give us a better understanding of our universe?

I was challenged to recreate our current solar system as we know it using the distances each planet is from the sun, and here is how it turned out (Apologies for the poor quality)…

Image 2 – Planets in our solar system distance from the sun

*Please note that the planets are not to scale with the toilet roll, although they are to scale with each other.

It took me approximately 30 minutes to devise a scale for my planets to sit on due to the HUGE differences in distances that the planets are from the sun. The first four planets were all between 57-230 million km away from the sun whereas the furthest away planet, Neptune, was over 78 times further away from the sun than the nearest planet Mercury! This made it incredibly difficult to devise a scale which would clearly show the difference in distances between the planets and the sun.

Upon finally deciding that I would use 500 million km per toilet ply as a scale for my solar system, I was finally able to align my planets! Mercury, venus, mars and earth all appear incredibly close to one another, when in actual fact they are at least 50 million km away from each other. If it wasn’t for the maths underlying my model I would have assumed that the planets are actually pretty close together. The maths involved in this tells me that actually the pictures we were shown in primary of planets being perfectly aligned away from the sun were in fact, incorrect. (See an example of these types of images below.)

Image 3

If you then consider other entities within our milky way you can see how small we really are in this vast world. We are just one of hundreds of thousands of solar systems that make up the milky way (Reynolds, 2018).

If the diameter of our solar system is around 250 billion km, then can you imagine the enormity of our milky way. But it doesn’t just stop there. There are 100-1000 billion galaxies in our universe. IC 1101 is the biggest, at 11,504,106,950,000,000,000 km (Creighton, 2016).

Implementing this in the classroom

It is these numbers that help us to gain a better understanding of the vast universe that we live in. Basic ideas can be explored using the maths that Copernicus created, such as scale, shape and area. These basic ideas are what Ma (2010) describes as the fundamental principles of mathematics. This suggests that by developing children’s spatial awareness further when exploring the topic of space, we are also helping them to become more competent in their maths skills. It is these skills that would allow them to recognise a wrongly scaled solar system if they ever saw one!

They could then go on to implement these skills in other areas of the curriculum such as art and design technology. For example, using shapes and scale to draw a correctly proportioned human face (see next post). These interconnections between their areas of learning is something that Ma (2010) suggests is key is allowing children to see the relevance and importance in what they are being taught.

In conclusion, Copernicus’ theories surrounding the scale of our solar system allows us to gain a better understanding of the magnitude of our universe. The magnitude of the numbers he devised to prove his theory gives us an indication of just how small we actually are in this world. Using the numbers he came up with, we can create a correctly scaled replica of our solar system and other bodies in the universe to gain further knowledge and understanding of the world we live in. Children can use the math surrounding his theories to develop the basic ideas of shape and scale, whilst being able to acknowledge that their is more to the universe than just our solar system!!

References

Creighton, J. (2016) The Largest Galaxy In the Known Universe: IC 1101. Available at: https://futurism.com/ic-1101-the-largest-galaxy-ever-found [Accessed 25 October 2018.

Falkner, D.E. (2011) The Mythology of the Night Sky: An Amateur Astronomer’s Guide to the Ancient Greek and Roman Legends. Springer Science+Business Media: London.

Ma, L (2010) Knowing and Teaching Elementary Mathematics. Oxon: Routledge.

Reynolds. S. (2018) Maths in Astronomy [PowerPoint Presentation]. ED21006: Discovering Maths. University of Dundee.

The Scottish Government (2008) The Curriculum for Excellence: Building the Curriculum 3. Edinburgh. Available at: https://www.gov.scot/resource/doc/226155/0061245.pdf [Accessed 23 October 2018].

Image 1 – Bfritz (2013) Available at: https://www.proprofs.com/quiz-school/story.php?title=history-astronomy_1 [Accessed 25 October 2018].

Image 3 – Pixabay (2017) Available at: https://pixabay.com/en/solar-system-space-planets-order-2453896/ [Accessed 25 October 2018].

What do you mean numbers don’t exist?!

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Yes, you did see it right. Apparently, numbers do not exist. My mind is still trying to process how such a concept could even be possible after believing all my life that numbers are the fundamental basis of mathematics. After much time pondering and challenging this, it appears to be true…

We use numbers everyday, from the numerals on our watches, the numbers on our calendars to the sizing on our clothing. Each of these numbers we use represent something, but what if we substituted our common base-ten numbers for something different? Would we still be able to count, tell the time, use money etc?

Collins (1998) suggests that by taking a nominal view of numbers, we can. He suggests that we can use any number system in the world (or a completely new one if we wished) to represent a size and it would still make sense, as long as we knew what each symbol represented. Society has constructed these notions that 1 is less than 4, and 4 is 3 more than 1 etc. However, Collins (1998) argues that we could equally use the symbol ‘4’ to represent one item and the symbol ‘1’ to represent four items, thus showing the flexibility of numbers as we know it. This is something I would never have considered as I thought the number system was a rigid structure, something which could never be changed. Nominalists see numbers as ‘fictions’ – symbols which aid and support our understanding of mathematics and everyday life, meaning we could use other symbols or entities to do the exact same job. The fact that we use different number systems around the world, such as octal, mayan and Arabic, supports this theory, and suggests that numbers are something which we have been predisposed to use through the work of many mathematicians which have came before us.

On the other hand, Collins (1998) explores the view that many realists take. We use numbers everyday in order to make sense of, and complete a wide variety of tasks everyday and thus, numbers must be real if we use them. Fine (2009) suggests the links that we have formed between numbers and the properties we have given them suggests that their existence must be true. For example, we have developed the concept of ‘prime’ numbers within our number system. Although, is this not another made up concept used to help us understand and make links within our made up number system? Confusing, isn’t it!

We could have used the word prime to represent another property of numbers, such as those divisible by 2, or every number ending in 1. Thus, this could be applied to other items we represent using language in our everyday lives. We could use the word ‘table’ to represent a ‘chair’ or vice versa, as long as we had an agreed understanding as a society that a ‘table’ actually represented what we know today to be chair.

Randy Palisoc develops this idea is his ted talk (see video below) by suggesting that numbers are a language we use to make sense of the world around us (TEDx Talks, 2014), just like we have constructed the letter system (the alphabet) and thus words to be able to communicate with one another.

Overall, I believe that the debate on the existence of numbers is an incredibly interesting one. I acknowledge that we use numbers everyday to perform basic functions, and thus they must be real as such. I also share Collin’s (1998) nominalist view that we could substitute other symbols for our base-ten number system and would still be able to perform basic arithmetic and daily tasks involving what our numbers represent, whether that be time, quantity of money or clothes sizes – the list goes on. The language we use for numbers, whatever this may be, is one which we will most likely always require to be a fully functioning member of society. Through this, is it clear to see that is imperative that we have a number system in place which allows us to represent different quantities and sizes etc, even if it is made up!

References

Collins, A.W. (1998) On the Question ‘Do Numbers Exist?’, The Philosophical Quarterly, 48 (190), pp. 23-36. Available at: https://www.jstor.org/stable/2660370?seq=1#page_scan_tab_contents [Accessed 8 October 2018].

Fine, K (2009) The question of ontology, Metametaphysics: New Essays on the Foundations of Ontology, pp. 157-177. Available at: https://philpapers.org/rec/FINTQO-2 [Accessed 8 October 2018].

Image – Wikipedia (2018) Numeral Systems of the World. Available at: https://commons.wikimedia.org/wiki/File:Numeral_Systems_of_the_World.svg [Accessed 8 October 2018].

Video – TEDx Talks (2014) Math isn’t hard, it’s a language | Randy Palisoc | TEDxManhattanBeach. Available at: https://www.youtube.com/watch?v=V6yixyiJcos [Accessed 8 October 2018].

Maths IS important!

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“Is maths important?” my lecturer asked during a recent Discovering Mathematics workshop. My exam driven former self would have explicitly answered no to this question, as I had all sorts of weird and wonderful equations and rules drilled into my head at this point – none of which I have actually implemented within my everyday life, but hey! At least I passed my exam…

As I am no longer in this environment, and have begun my journey back into the
Primary School setting, I have began to re-establish the view that maths IS important. But what caused my shift in opinion on the importance of maths when I reached High School?

I believe that in Primary School we are taught in such a way that allows us to have a conceptual understanding (the HOW and WHY) of the area we are learning, formally known as relative understanding (Skemp, 1989). Whereas in High School, my experience was being taught the ‘HOW’ of a branch of mathematics – known as a procedural or instrumental understanding(Skemp, 1989). Upon asking my Higher maths teacher why we needed to know the likes of the quadratic formula, the cosine rules and how to differentiate an equation, he replied “you don’t – you just need to know HOW to do it”. The fact that my own maths teacher was teaching us based on instrumental understanding may suggest that either; he underestimated the power of understanding the WHY, or he was not required to teach it, so he simply didn’t. As long as we managed to get the answers right, even if we did not understand why it was right, he would say we were doing a good job and were heading on route to pass the exam.

Not knowing the WHY behind the mathematics I was being taught made it seem irrelevant and useless to me, hence why I refused to see its importance at the time. Ma (2010) suggests that having a profound (or relative) understanding of mathematics enables learners to see the ‘interconnectedness’ between concepts, thus they are able to apply their knowledge to a wider range of scenarios than having an instrumental understanding alone. His theory dismisses teaching towards an procedural understanding (like my teacher did), as it restricts the wider connections an individual can make between what they are learning and other areas of the curriculum, and also in their everyday life (depending on the nature of the topic being explored).

My belief that High Schools are teaching towards the objective of passing exams is strengthened by the data released by the Scottish Government (2012), as it shows the percentage of pupils performing very well at the level they are working at in maths decreases by 18% from Primary 6 to S2 (see graph below). This statistic, although not surprising to me, is a shocking reminder that the strategies for teaching maths in High Schools is not as effective as those demonstrated within Primary School.

Overall, I believe that mathematics is an important aspect of every child’s education. Although, I believe that its importance is lost throughout High School due to a lack of relative understanding of the concepts that students are being taught, meaning they are less likely to recognise links between concepts and overcome difficulties they encounter within maths problems (Skemp, 1989). This means that they are more likely to adopt a negative attitude to learning maths in the classroom. By giving our children space to be creative (see previous post titled ‘Maths can be… creative?’) and delve deeper into their understanding of different mathematical concepts, we are more likely to develop their awareness of the transferable nature of the skills that can be applied to different areas of maths and other wider connections. This leads to the development of ‘interconnectedness’ that Ma (2010) suggests is critical to having a profound understanding of mathematics, and thus recognising its importance in everyday life.

References

Ma, L (2010) Knowing and Teaching Elementary Mathematics. Oxon: Routledge.

Scottish Government (2012) Scottish Survey of Literacy and Numeracy 2011 (Numeracy). Available online at: http://www.scotland.gov.uk/Publications/2012/03/5285/downloads#res390565 [Accessed 3 October 2018].

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