Chance or luck? Which one is involved when guessing what side of a coin will land on or gambling at the roulette table? Turns out there is more maths involved than you would think. Probability is an aspect of maths that is used within our daily lives and influences the decisions we make. It also encompasses parts of Liping Ma’s fundamental principles.

Essentially, probability is “how likely something is to happen” (Maths is Fun, 2018), and this is the core idea of the concept. Though it is important to highlight the fact that we are surrounded by uncertainty and to predict everything that will happen is 100% impossible as well as it would be quite dull if we knew everything that was going to happen (Eastaway and Askew, 2010). Probability can give us a guess at the likelihood of the event happening on a scale based on the facts that are available to us (Haylock and Manning, 2014). There are various ways that we describe the probability of an event happening, highlighting during the lecture on probability;

• Percentages
• Decimals
• Likely/unlikely
• Good chance/no chance

Now think about how often you use these words during the day. It can be quite often when you think about it; there is 34% chance it will rain today, or it is likely that it will rain.

There are various methods to work out probability which links into Ma’s principle of “multiple perspectives” (Ma, 1999), as there are different processes that can be used and they are used depending on the situations/maths problem the children are confronted with. One technique that can be used is taking the “number of ways [an event] can happen” then divide it by the “total number of outcomes” (Maths is Fun, 2018). Use the example of a dice and the question ‘what is the probability that I roll a 6?’. There is only 1 way this event can happen and there are 6 sides to a dice so there are 6 possible outcomes – thus the probability of rolling a 6 is . This way means children are working with fractions and can even go on to converting the answer to a decimal or percentage answer – using other aspects of maths.

Another common way of working out probability is using a probability scale. This method of working out probability encourages children to use the terminology of probability that is part of the pupils “everyday language” (Haylock and Manning, 2014). Children can then progress to using a numbered probability scale to measure the events being questioned – for example, 0 stands for something that is completely impossible to happen and 100 as the opposite (Haylock and Manning, 2014). By using a numbered scale connects with the idea of cardinal and ordinal numbers – the scale has the numbers in a specific order.

Asides from being used to predict outcomes in our everyday lives, probability is applied to the industry of gambling. Personally, I had not realised the amount of maths that could be involved in gambling and thought it was primarily due to luck or less honest machines – linking into the idea of “gambler’s fallacy” (Bellos, 2010, p.322). Though it was the example of Stefan Mandel (a Romanian economist) that really made me think that maths was very much involved in gambling. In 1964, Mandel created an algorithm that generated all the possibilities that would get him five out of six numbers correct – which would win him second prize in the Romanian lottery (Bellos, 2010). He then progressed to assisting Australian businessmen to win the Virginia Lottery that was worth millions of dollars – using his algorithm to generate all the possible outcomes. This example completely blew my mind as using maths helped Mandel win the lottery which I had previously thought was completely random and draw of the luck.

Probability is an aspect of math that can be used by everyone and is ingrained in our daily lives. It influences the decisions we make – from whether to bring a coat in case it rains to winning at a casino. Probability assists us in gauging the likelihood of events happening in our lives – giving the unknown some structure and perspective.

REFERENCES:

Bellos, A. (2010) Alex’s Adventures in Numberland London: Bloomsbury

Haylock, D. and Manning, R. (2014) Mathematics Explained for Primary Teachers. 5^th edn. London: SAGE

Ma, L. (1999) Knowing and Teaching Elementary Mathematics – Teachers’ Understanding of Fundamental Mathematics in China and The United States. London: Routledge

Maths is fun [Online] Available at: http://www.mathsisfun.com/data/probability.html (Accessed 20th October 2018)

What is zero? A number or nothing at all? Before the lecture about place value, I had never considered the significance of zero or how anything interesting can be gleaned from it. Essentially, it was another number that was just there along with its friends 1,2,3 and so on. Though after some research, it becomes more apparent that zero has a lot more to say than I initially thought.

All numbers have a history – that had to begin somewhere and follow a path to become the number we use throughout maths. Zero is no different, slowly being developed by various civilisations. Its tale begins with the Babylonians, who helped developed the concept of place value using a variety of different symbols to represent their numbers (Haylock and Manning, 2014). Around 400BC it can be seen on clay slabs that the Babylonians began to include two wedge shaped marks between numbers where in modern times we would place a zero (Lamb, 2014). This was since a blank area to indicate a lack of value confused many, so the marks were used to avoid this (Bellos, 2010). Thus, begins the journey of zero.

Move forwards in time to the seventh century India where the Indians took the two wedge marks and developed it further, turning it into a real number. Indian mathematicians, like Brahmagupta, highlighted how zero effected other numbers and gave it the name ‘shunya’ (Bellos, 2010). Travelling to the Middle East, it was given an Arabic symbol – 0. Though it took longer for some countries to adopt the Indians method of using zero and the Arabic symbols for a variety of reasons – lack of trust of the Middle East or that many thought this new symbol could be easily manipulated to another number (Bellos, 2010). Eventually, everyone became more accustomed to the idea of zero and accepted this new number – obviously as we use it today during our own maths learning.

What is the importance of learning the history of zero? It covers a variety of civilisations around the globe which can be intriguing to pupils. Learning the history means incorporating another curricular area into maths, helping children to engage with the subject. Also, it shows how people used numbers before zero and provides an alternative way to using the place value system – linking into Ma’s fundamental principle of “multiple perspectives” as the way we use the place value system is not the only method. The place value connects with the children’s knowledge of numbers and working out which ones are larger than others. Take the number 109, the zero here represents the fact that there is no value in the tens column as well as highlight the fact that this number is ‘one hundred and nine’. Without the zero where it is, then it would be difficult to tell the difference between 109, 19, 190, 109 000 000 or 1090 apart (Haylock and Manning, 2014).

There is a quote from Alex Bellos’s book ‘Alex’s Adventures in Numberland ’ that made me stop to think about the number zero. “The present mathematical system considers zero as a non-existent entity’ (Bellos, 2010, p.138), from experience and initial outlook of zero I feel that this is a common misconception of zero. Many do not think zero is a significant number, I remember overhearing a child during a placement telling their friend that zero ‘was not a real number’. It is a real number and provides important functions in the maths universe. Take the example of positive and negative integers to emphasise the importance of zero. A positive integer is any number above zero and a negative integer are numbers lower than zero (Haylock and Manning, 2014). This idea would not be feasible if zero was not around – no zero means no numbers could come before. Linking this to the real world in terms of temperature, 0˚C can be felt be felt and when temperature sinks into the negatives this can be felt (memories of ‘Beast from the East’ anyone?) – and we need the idea of negative integers so this temperature can be measured. Additionally, there is a connection to another aspect of maths when it comes to counting numbers that are lesser than zero – cardinal (sets of things) and ordinal (the order things come in) numbers. We cannot count negative numbers in the cardinal way as it is impossible to have a negative quantity of numbers. So we count as ordinal numbers as this makes more sense (Haylock and Manning, 2014). This creates a “unified body of knowledge” (Ma, 1999, p.122) for the pupils.

Zero is a more interesting than I had anticipated – it has a colourful history spanning a variety of countries, had become deeply engrained into the maths we do and is nearly impossible to do mathematical concepts without. Zero is unique and links into a lot of Liping Ma’s fundamental principles.

REFERENCES

Bellos, A. (2010) Alex’s Adventures in Numberland London: Bloomsbury

Haylock, D. and Manning, R. (2014) Mathematics Explained for Primary Teachers. 5^th edn. London: SAGE

Ma, L. (1999) Knowing and Teaching Elementary Mathematics – Teachers’ Understanding of Fundamental Mathematics in China and The United States. London: Routledge

Maths Being Creative?

When I think of ways to describe Maths, creative definetly does not appear on the list or anywhere near it.  The common perception of Maths is long, difficult equations, hard to understand problems and a few questions involving graphs. Overall, it is something quite dreary and dull. However, Maths can be a creative subject that students can can have fun and be engaged with.

Every time I think of Maths lessons at school, the first memory is sitting at my desk and working through multiple problems trying to find the one solution – all so I could pass one final exam at the end of the year. The closest to doing something ‘creative’ on the course was drawing a graph – which for the most part, is not something fun to draw. As previously mentioned, most people will not consider Maths as being creative yet the Scottish Government expect this attribute to be developed within the classroom (Scottish Government, n.d.). It is important that children can see that this subject can be fun as this will hopefully make it appear less grey and boring.

During an input in Discovering Maths, we were discussing tessellations, “a pattern made by repeatedly fitting together without gas a collection of identical tiles” (Haylock and Manning, 2014), which is maths and is creative – combining the two unlikely things. The “basic idea” (Ma, 1999) that is at the centre of tessellations is 2D shapes. Children can learn about the types of triangles, quadrilaterals and building up to a variety of polygons  (e.g. hexagon, octagon and nonagon) – regular and irregular. It is from learning these shapes that children can learn which of these shapes tessellate and which do not. This gets them exploring 2D shapes and seeing if they can make a continuing pattern – this exploration stimulates their curiosity and appears less terrifying than other aspect of maths. Another benefit of tessellations is that they in the wider world – so children can begin looking for them, possibly with parents or guardians.

Digital Root Circles are another creative activity that can be done with pupils. A digital root is “the result of finding the digital sum of the digital sum of a natural number” (Haylock and Manning, 2014) until it has become a single number. For example;

6 x 4 = 24

2 + 4 = 6

So the number 6 is the digital root. This can be done with all the times tables which creates a pattern. The following circles are examples I have done from finding out the digital root:During this activity, it was interesting to see what patterns the digital root numbers produced and see what times tables produced the same pattern, for example the 5- and 11-times table produced the same version of the star. Children can explore this aspect of Maths in an enjoyable way without the negativity and a more interesting subject to look at.

Overall, Maths has the capability of being creative and can make children think it is intriguing to learn about. By doing more activities like this then can hopefully improve the outlook on Maths and encourage students to ‘give it a go’.

REFERENCES:

Haylock, D. and Manning, R. (2014) Mathematics Explained for Primary Teachers. 5^th edn. London: SAGE

Ma, L. (1999) Knowing and Teaching Elementary Mathematics – Teachers’ Understanding of Fundamental Mathematics in China and The United States. London: Routledge

Scottish Government (n.d) Curriculum for Excellence. Available at: https://education.gov.scot/Documents/All-experiencesoutcomes.pdf (Accessed: 25 September 2018)

Mathematics. It is a subject that fills most people with fear, dread and overwhelming anxiety. A subject that most will claim is too difficult to comprehend and only for those who are very smart. I have been in this mind-set for a long time, seeing a maths question and feeling that it was impossible to answer. That feeling has not left me though I do realise that there is maths involved in everyday activities – purchasing items, measuring ingredients and so forth. By taking Discovering Maths I can change my outlook on maths and find it less daunting.

During Primary and Secondary School, most of my maths lessons consisted of a short input onto what we would be looking at, followed by answering questions from a textbook. By the end of a lesson I would feel so overwhelmed by the number of steps and rules that it felt impossible to remember. Though I managed to get through my National 5 and Higher. It was not until the first week of this module that I realised that most of my learning had been focused on the ‘how’ instead of the ‘why’. We were asked what and angle was – answer, a measure of rotation. Over a decade in school and I had learned how to name the types of angles, use a protractor to measure angles, work out how to find the missing angle and other information about angles, but had not realised what it was. This example made me think about the amount of time teaching children the rules and how to get the answer – meaning children are missing out on a whole aspect of maths that could help their understanding of maths. Ford et al. (2005, cited in Haylock and Manning, 2014, p.6) highlights that those who are anxious/apprehensive about maths impacts how they handle maths in the classroom. This nervousness leads to the students need the support of rules to learn which adds to the anxiety – especially when confronted with something they have not seen. Haylock and Manning (2014) also go on to add that rote learning and anxiety both inhibit a child’s ability to work out a problem creatively. What does this actually mean? Exclusively focusing on the ‘how’ of maths means that the anxiety of children increases, and they become scarred by the subject. Children struggle in maths and another generation can go by with a negative outlook on maths.

It is becoming more apparent how much responsibility the teacher has when it comes to shaping a student’s view on maths and how they manage at the subject. It is down to us to think of creative ways to teach maths and to help support them discover the subject. During my placement with a Primary Seven class, I taught a series of maths lessons on 2D shapes and symmetry. It was not a topic that left me terrified but I did not foresee how nervous I would be teaching it to a real class. Researching common misconceptions surrounding the topic assisted to an extent as I could try covering these during the lesson. However, like any class there were unforeseen difficulties that arose which highlights every changing nature of a classroom. With the placement and the module, the importance of making maths interesting to the class has become more apparent and I also realise that I still have a lot more work to do to reach that point.

Overall, maths can be difficult, and it can take time to overcome and see it as something interesting and less challenging. Personally, I still remember the difficulty of maths throughout school, though with help and dedication of my maths teacher have realised that it is not something that is impossible to manage. Though this has definitely enabled me to empathise with students who struggle. By undergoing the Discovering Maths module, I will see maths as something interesting as opposed to a subject surrounded by anxiety.

References:

Haylock and Manning (2014) Mathematics Explained for Primary Teachers. London: SAGE