Category Archives: 2.1 Curriculum

Would You Chance it All?

How often do you take a chance? The answer would be every day. Every day we take a chance, whether it be crossing the road, going to the gym or simply choosing to watch the TV. We cannot predict the potential outcome of doing something, yet, without even thinking, we create the factors that are influenced by our decisions. Something of which has lead us to be where we are to this day.

Before this module began, I remembered learning about probability in high school. I was never intrigued by the prospect of learning about probability. However, after the lecture about this I have felt that I can relate chance and probability to my personal experiences and highlight to pupils the basic maths behind this, whilst making it enjoyable to learn about. Inter-connectedness is a concept of Liping MA’s that can be used between maths and our everyday lives (Ma 2010). By interconnecting our experiences with the subject of probability, basic mathematical skills can be integrated into our everyday experiences, something which I believe is important when obtain fundamental maths skills.

Probability is “the likelihood or chance that something may happen” (Turner, n.d. p5) and can be worked out by:

Probability of something happening = The number of ways it can happen – over – the total number of outcomes

For example, when rolling a dice there are 6 possible outcomes. However, if I were to only try and role a 5, the possibility of this would be 1 in 6. In many cases people try to predict the possible outcome, however as we have discovered in maths, it is not as simple as this.

Gambling has a profound and direct link to probability and chance. By taking this chance it can be the profit or the lost to some people’s bank balances. Those who are serious gamblers are costing the government £1.2 billion a year (IPPR, 2016). This is not only impacting the economy but can also cause extreme debt for some people, and a break down in relationships and mental health. This is why some people try to predict casino games and influence the outcome, in the hope they can solve the problem (Aasved, 2004).

There are aspects of gambling which are linked to multiple perspectives. This is about having a variety of different ways to reach an answer. For example, there are variety of meals you can have in a restaurant. Say for example there were 2 choices for a starter, 3 for main meal and 3 for dessert, you have to list of different meals that you can possibly have. Therefore, there are many different combinations and to figure this out multiple perspective is important.

Gambling is something which relies on randomness and probability (Turner, n.d). When discussing this in our lecture, we all thought that surely humans can create randomness effectively? Wrong! We wrote our predictions of either heads or tails, if we were to flip our coin 30 times. I thought I would try and mix it up a bit, put heads 4 times in a row, a couple tails etc, but the answers truly baffled me. It ended up being completely 50/50. Even when I was trying to be as random as possible. The results were similar on a larger scale of our class, with the majority of people being one or two off the other. When actually flipping a coin, my results were 21 heads and 9 tails, and this varied around the lecture room. Therefore, as humans, we THINK we are being random, however nothing is quite that simple and we actually try to create logic results rather than random. I remember when conducting my answers, I thought I had to put a head and not another tails because 4 tails in a row was being silly, yet the physical experiment proved this to happen with heads.

Using a coin is a simple way of introducing probability and chance to children, as there are only two outcomes. If they have the basic skills and understanding of what a half or 50% means then probability can quickly follow behind. This aids their longitudinal development, as they have the basic skills prior to probability and therefore they can build on this to have a deepened understanding of problem solving and the possibilities of finding the answer. It also helps them in more complex scenarios, for in class work and future work, if they were working in a restaurant for example. Therefore, multiple perspective, basic skills and longitudinal coherence, all key skills in profound fundamental maths (Ma, 2010) can be shadowed through everyday maths.

It is important that we recognise that gambling can possibly be predicted. However, slot machines and casinos profit more from us than we would ever earn back (unless you’re the 1 in 45057474 to win big on the lottery! (Lottoland, 2016)). Slot machines are to us, a quick and easy way to win money back. Charles Fey, developed the Liberty Bell machine in 1895, which has 3 reels and 5 symbols. This machine in particular pays out 50% of the time with an average of 75% pay-back. Therefore, although you seem to ‘win’ more, you are in fact, losing more!

Stefan Mandel, 1964, applied to buy every combination to the Romanian lottery. He followed this up by doing this with the Virginia state lottery. The video below highlights the result of this.

In conclusion, probability and chance is something that we use every day. Some people take this for granted and some take it to extremes. However, we have the ability to use multiple perspectives to figure out possible outcomes, which can be used in our daily lives and in maths in primary school. Yet, probability on a gambling scale, as seen in the video, can be on a completely different scale to our everyday probability. I believe that Liping MA’s principles are important here and they are concepts that I will look at deleloping in future placements and my teaching career.

References:

Aasved, M. (2004) The Biology of Gambling. Springfiled: Charles C Thomas.

LottoLand. (2018). The Probability of Winning the Lottery. Available: https://www.lottoland.co.uk/magazine/the-probability-of-winning-the-lottery-.html. (Accessed on: 17th October 2018)

Ma, L. (2010) Knowing and Teaching Elementary Mathematics (Anniversary Ed.) New York: Routledge.

Slot Machine History (2010). Who is Charles Fey? Available: http://slotmachineshistory.com/charles-fey.htm. (Assessed on: 15th October 2018)

The Progressive Policy Think Tank. (2016). Cards on the table: The Cost to Government Associated with People Who Are Problem Gamblers in Britain. Available: https://www.ippr.org/publications/cards-on-the-table. (Accessed on: 16th October 2018)

Tuner, N. (no date) Probability, Random Events, and the Mathematics of Gambling

Wherbert, P. (2010). Stefan Mandel (online video) Available: https://www.youtube.com/watch?v=4TqFp0efLK0 (Accessed on: 16^th October 2018)

What is a Number?

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We come into contact with numbers every day. Time, working out how many portions of dinner we’ll need to make or simply “hey what’s your number?”. But have we ever taken the time to think, what is a number and why do we even use them?

My first question is, why does the number three collectively represent other items in threes? Surely three dinosaurs would be more than three peas, even though they represent the same amount collectively. But it’s not due to weight, height or mass, its due to the value of the number three. Numbers are a language. For value, comparison and equivalence. They help us understand how many there is of something (value), help us compare that five dinosaurs are two more than three peas and that three peas and three dinosaurs are equal.

Numerals play a significant part in this. A numeral is the symbol or collection of symbols used to represent a number. A child’s first experience of a number is most likely going to be an adjective that would collectively describe a set of something e.g. five cats or two arms, this is more commonly known as the ‘cardinal aspect of a number’ (Haylock, 2014). This highlights that children can recognise an equivalence between two sets of objects and is why we understand that three dinosaurs are equal to three peas. At this stage numerals are the link to numbers. Therefore, children can look at numerals such as pictures of shapes or objects and count how many are there of one thing. This will therefore help them connect to real life situations and when shown beside numbers, they can begin to connect numerals and numbers together. By beginning to connect the two, children will have a deepened understanding of numbers and therefore their longitudinal coherence will develop. Starting their journey with numbers and maths simply and positively to prevent future maths anxiety (MA, 2010).

Numerals have been shown throughout our history and teach us a lot about the development of mathematics. One of the oldest is hieroglyphics, used by the Egyptians, dated as far back as 3000 BC. Their numerals were shown through drawings and symbols such as a bird and an Egyptian man (O’Connor and Robertson, 2000). Maths was commonly used by Egyptians if they were dividing food, solving problems for trade and market and most importantly for the pyramids (Mastin, 2010). This highlights the first use of maths for economy and trade. Wealth played a significant part in Egyptian life and social class was divided by money but more than anything, maths. Through looking at our history, we can see the similarities in the importance of economy then and now. We live in a world that’s economy changes daily and has the power to change and impact upon people’s lives. If we looked more in depth at historical economy such as Egyptian trade and social classes, we could learn and reflect on our economy nowadays and therefore maths is a historical aspect of economy that involves counting and numbers.

All factors of numbers have a place in fundamental maths, but most importantly teach us a lesson for now. We can apply what we have seen previously in history and the importance of numerals and numbers to aid our development with maths and see that it is essential for our wider society and not just basic classroom use. Relevance is a fundamental principle of Curriculum for Excellence (Scottish Government, 2010) and therefore it is important that teachers bring this forward from historical findings into our everyday maths lessons and make children think about their future with maths in society.

Number Patterns and Sequences

Another way of teaching maths can be through number patterns and sequences. This can be another way of making maths interesting for children whilst developing their relationship with numbers. Vale and Barbose (2009, p9), stated that the use of patterns in maths can challenge students to use “higher order thinking skills and emphasise exploration, investigation, conjecture and generalisation.” Therefore, basic skills in maths are used within problem solving to develop not only longitudinal coherence but also multiple perspectives as they can use different methods to find the solution to a problem.

An interesting number pattern is Pascal’s Triangle, named after Blaise Pascal, a French mathematician. This is a triangle that starts with the number ‘1’ and then below are the sum of the addition from the numbers above. For example, if 1 is beside 2 in the triangle the sum (number written below) would be 3. An example of this can be seen below.

Pascal’s Triangle makes maths fun for children whilst learning about addition and number patterns. Addition is a fundamental aspect of maths that it taught as early as ages 4-5 and can be continued to be taught throughout primary school through ways such as Pascal’s Triangle. The introduction of colours can also produce other findings within the triangle. For example, pupils can colour odd and even numbers or work out the horizontal sums of the triangle (Maths is Fun, 2017). This therefore keeps maths relevant and allows for differentiation in a classroom, as some can complete the main body of the triangle (addition) and some can move forward looking at other aspects of the triangle. Therefore old maths techniques and problem solving continue to find a relevance in our everyday maths and classroom.

References:

Haylock, D. (2014) Mathematics Explained for Primary Teachers. 5th edition: SAGE.

L, Mastin. (2010) Egyptian Mathematics https://www.storyofmathematics.com/egyptian.html (Accessed: 5th October)

J, O’Connor and E, Robertson. (2000) Egyptian Numerals. http://www-history.mcs.st-andrews.ac.uk/HistTopics/Egyptian_numerals.html (Accessed: 5th October)

Ma, L. (2010) Knowing and Teaching Elementary Mathematics. (Anniversary Ed.) New York: Routledge.

N/A (2017) Pascal’s Triangle. https://www.mathsisfun.com/pascals-triangle.html (Accessed: 6th October 2018).

Scottish Government. (2010) Curriculum for Excellence Building the Curriculum 3 A Framework for Learning and Teaching: Key ideas and Priorities. Available at: http://dera.ioe.ac.uk/1240/7/0099598_Redacted.pdf (Accessed: 5th October 2018).

Vale, I. and Barbosa, A. (2009) Multiple Perspectives and Contexts in Mathematics Education. Available at: https://www.academia.edu/1485703/Patterns_multiple_perspectives_and_contexts_in_mathematics_education (Accessed: 5th October 2018).

The Art of Tessellation

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When Jonathan first said the word “tessellation”, I immediately thought what on earth is he on about?! Yet, following this lecture I now understand that tessellation is something that surrounds us.

Tessellation is the arrangement of identical shapes that fit together perfectly to create a pattern. These shapes have to fit precisely beside one another, meaning they’ll leave no gaps. If we look closer at items that we come into contact with on a regular basis, such as chocolate, footballs and kitchen tiles, we can see that there are shapes such as hexagons, squares and triangles that are joined together to form tessellation.

But how does tessellation link into a classroom setting?

Tessellation can occur through two different types of shapes. The first are regular. These include squares, hexagons and equilateral triangles and therefore form a more simplistic tessellation, for example, in the form of chocolate squares. Regular shapes, unlike irregular, have the ability to interconnect as all the vertices meet one another and therefore create the sum of 360 degrees. The second are irregular shapes, these are shapes such as pentagons, octagons and isosceles and scalene triangles (Maths is Fun, 2018). These work similarly to regular shapes, however, the shapes must be cut and pasted to a different part of the shape to be able to interlock with the other identical shapes. An example of this is shown below in the creation of a horse.

A form of tessellation can particularly be seen in Islamic religion through mosaics and geometric patterns (Hames, 2017). Islamic art focus on the creation of stars through tessellation. For example, they particularly use equilateral triangles to create 6 to 12 points of stars. These represent and symbolise harmony and hum consciousness. These features can be introduced into a classroom. Using maths (tessellation) and interconnecting it with art is a great way of introducing a calm and settled environment to the classroom. Boaler (2009), states that completing tasks in different ways therefore allows children to see that there are different methods to learning maths and therefore maths can be enjoyable for everyone.

Liping MA’s idea of inter-connectedness is highlighted through the use of maths and art. By using a mixture of the two, children who feel anxious about maths will therefore find a task such as creating Islamic art, as a more relaxed approach to maths. For example, if they enjoy art they believe it is more about art than the maths. Furthermore, this will lead to their longitudinal coherence. This is because they will have the basic understandings of shapes and therefore children have a sound enough understanding to bring this information forward to more complex areas such as tesselation.

An example of a lesson that could be used for tessellation is multiplication to create stars. By finding the different digital roots e.g. 4 times 6 = 24 which therefore this simplifies to 2 + 4 = 6, you can start at a point in the circle and continue to connect to the following dots (answer). When completed the pupils can colour these in and therefore maths and art have been interconnected in a lesson, helping those who have a passion in art have a profound understanding of maths.

Example of tesselaltion star from the digital root of the 4 times tables.

Overall, tessellation is a great lesson to introduce differentiation within a classroom. It allows for both art and maths to be taught at the same time, making maths fun and achieveable for those suffering from maths anxiety. Tesslation links into our classroom setting through a number of different lessons and has a major link to pupils’ understanding of shapes. A basic concept of maths that is learnt thoroughly to bring forward. This lecture in particular is one that I will continue to revisit when teaching, as I have not only learnt how maths can be fun, but have learnt about a different culture in the process. Therefore, I think this topic could be integrated into the classroom in a number of ways such as a class topic or investigation task.

References:

Boaler, J. (2010). The Elephant in the Classroom: Helping Children Learn and Love Maths. London: Souvenir Press.

Giganti, P. (2010) Anatomy of an Escher Flying Horse. Available at: https://www.youtube.com/watch?v=NYGIhZ_HWfg (Accessed on: 25th September 2018).

Hames, S. (2017). Tessellations in Islamic Art. Available at: https://classroom.synonym.com/tessellations-in-islamic-art-12085299.html. (Accessed on: 12th November 2018).

Ma, L. (2010) Knowing and Teaching Elementary Mathematics. (Anniversary Ed.) New York: Routledge.

Maths is Fun. (2018). Tesselation. Available at: https://www.mathsisfun.com/geometry/tessellation.html (Accessed on: 24th September).

Warner, M. (no date) Digital Root Patterns Available at: https://www.teachingideas.co.uk/number-patterns/digital-root-patterns (Accessed on: 26th September 2018).

Scientific Literacy TDT

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An understanding of science and the skills developed through the subject are becoming increasingly more acknowledged within Primary schools. Rutledge (2010) states; “children will not be able to connect up with their own ideas without skills such as predicting, and they will not be able to challenge their ideas without skills such as looking for patterns in results”. This links well into the importance of scientific literacy as it exhibits that children can develop skills that are learned within science, which can be transferable across the curriculum such as measuring, predicting and evaluating.

Scientific Literacy falls into four stages; Nominal, Functional, Conceptual and Multidimensional. This starts with recognition of a scientific term such as mass, following up to a full understanding of the term and applying it to their own lives and linking it into areas, such as history or a wider setting.

If there is a misinterpretation within scientific literacy and a person does not have the knowledge and skills to understand the science within conducting experiments; this could lead to false scientific conclusions being made. An example of when a lack of scientific literacy has led to false media representation is the MMR vaccine scare. The MMR vaccine protects against three illness, those of which are; measles, mumps and rubella. Dr Andrew Wakefield publicised that he believed this vaccine has links to autism and bowel disorders which resulted in vaccine rates of the MMR to decrease dramatically due to a fear within the public. It took over five years for this scare to finally be ruled out due to the high amount of allocations made by witnesses. It was said by the General Medical Council that Dr Andrew Wakefield “abused his position of trust” by conducting tests in which were unnecessary without the appropriate qualifications or ethical consent. Due to a failure of disclosure of questions that conflicted his argument the original paper that published this scare removed the article and his allegations were deemed as false (NHS Choices, 2010).

One of the key issues which must be enforced throughout science education in primary school is the importance of scientific literacy. As described by the OECD (2013, p9), academic literacy is the ability to use scientific fact and draw an accurate conclusion based on such knowledge. The importance of encouraging academic literacy from the earliest years of education cannot be underestimated. When teaching science from the earliest stage, academic literacy must become part of a routine. This will therefore allow accurate findings to be made which can support an individual’s research. As future educators, we must allow children to explore the ways of carrying out research responsibly throughout their time at school. Children must be motivated to work from a Nominal level (where by pupil recognises scientific vocabulary and concepts but their understanding is vague) – up to a Multidimensional Level of Scientific literacy (whereby the pupils can use scientific concepts and vocabulary in relation to other curriculum areas) (Holbrook & Rannikmae, p279). In order to understand how easy, it is to alter a substantial amount of results from the click of a button, children must see the damage inaccuracy can have. This allows them to understand that the implications are not just on an individual level but effect greater society. From a societal point of view, inaccuracy can cause extreme conflict between mind and matter. As a result, people most commonly suffer from severe mental of physical implications, which can be life changing and threating. Therefore, the importance of enforcing a secure structure of research, from a young age, will mean that in future the integrity of science can be protected.

To conclude, there must be more focus upon scientific literacy within primary schools. This will allow children to explore science and its connections to wider society, making it more relevant to them and their future learning. It will also provide them with the fundamental skills to; predict and challenge their learning and numerous transferable skills, all of which will accommodate their skills within school.

Jess Millar, Kimberly Waddell, Hannah Robertson, Rebecca Potter

References –

Holbrook, J. and Rannikmae, M. (2009). The Nature of Science Education for Enhancing Scientific Literacy. International Journal of Science Education, 4.

Nhs.uk. (2010). Ruling on doctor in MMR scare. https://www.nhs.uk/news/medical-practice/ruling-on-doctor-in-mmr-scare/ (Accessed on: 10^th February 2018)

Nhs.uk. (2018). MMR vaccine ‘does not cause autism’. https://www.nhs.uk/news/pregnancy-and-child/mmr-vaccine-does-not-cause-autism/ (Accessed on: 10^th February 2018)

Oecd.org. (2015). PISA 2015. https://www.oecd.org/pisa/pisaproducts/Draft%20PISA%202015%20Science%20Framework%20.pdf (Accessed on: 8^th February 2018)

Rutledge, N. (2010) Primary Science: Teaching the Tricky Bits. Maidenhead: McGraw-Hill Education.

The Science Behind the Experiment