When taking a multiple-choice test, it often seems one answer is correct initially, however, on further reflection you think a different answer is correct. In this situation, is it better to switch or stick with your first answer?

Most people believe you should avoid changing your answer, but research has shown that most answers are changed from incorrect to correct, resulting in test scores improving when students change their answers (Kruger, Wirtz & Miller, 2005). But why do most of us still believe that we should stick with our first choice?

“The vast majority of over 70 years of research on answer changing, however, has questioned seriously the validity of this belief and the utility of the “always stick with your first instinct” test-taking strategy. The majority of answer changes are from incorrect to correct, and most people who change their answers usually improve their test scores.”(Kruger, Wirtz & Miller, 2005 p.725).

 Counterfactual thinking is the frustrating feeling that follows after the change of a right answer to a wrong answer (Kruger, Wirtz & Miller, 2005). As humans losing carries more impact than winning. This is because when we are disappointed by something it has a longer lasting impression. For example, most people will write a negative review on TripAdvisor over a positive review because the negative experience has made a greater impact. What you focus on is how you will feel, focus on the negative you will feel negative.

“changing an answer when one should have stuck with one’s original answer leads to more “if only…” (Kruger, Wirtz & Miller, 2005)

 By looking at the classic thought experiment in probability called ‘The Monty Hall problem’ we can understand the fundamental math’s used. The Monty Hall Problem is a counter-intuitive mathematical puzzle: There are 3 doors, behind which are two goats and a car. The player picks a door (call it door A). The player is hoping for the star prize, a car! The Game show host knows what is behind each door and will always pick the door with a goat behind it to show to the player.

Here’s the question: Do you stick with your original guess Door A or switch to the other unopened door? What are the consequences?

Surprisingly, the odds are not 50-50 which most people might believe, including myself when I was first introduced to the idea. However this is not correct, you actually have twice as much chance of winning if you switch your answer. Below is a diagram I have created to explain this baffling concept.

I tested the strategy by using the “pick and switch” approach. I repeated this 20 times and won 14 and lost 6. This is more than double!

Give it a go yourself:

https://betterexplained.com/articles/understanding-the-monty-hall-problem/

So the answer is you should always swap, as this gives twice the chance of winning the car. But why?

Probability = the likelihood of the event happening/ number of possible outcomes (wikihow.com)

If you chose ‘not to swap’ you have a probability of 1/3 or about 33% chance in not picking the car because there are three doors and only one car. Since there are three doors and 2 goats the probability of you picking a goat is 2/3 or about 66%. So by ‘swapping’ you have a 66% chance of winning the car and a 33% chance of winning a goat.

The trick of the game show is making it appear like the player has a 50-50 chance of winning the car just like the chances of landing on a tail when tossing a coin which sounds sensible but it not correct. So by using your mathematical knowledge of probability will you consider changing your answer the next time you are unsure? Because I certainly will!

References:

Kruger, J., Wirtz, D., & Miller, D. T. (2005). Counterfactual Thinking and the First Instinct Fallacy. Journal of Personality and Social Psychology, 88(5), 725-735. http://dx.doi.org.libezproxy.dundee.ac.uk/10.1037/0022-3514.88.5.725

Wikihow.com: https://www.wikihow.com/Calculate-Probability

Betterexplained.com:https://betterexplained.com/articles/understanding-the-monty-hall-problem/

The dreaded fear of a maths test resulting in; sweaty palms, increased heart rate and that tight knot feeling inside your stomach. This phenomenon is called ‘maths anxiety’. Researchers say around 20% of our nation suffer from ‘maths anxiety’ (Rubinsten, 2017).

What’s going on? And can it be fixed?

Maths anxiety is as simple as it sounds,  a fear of numbers. Robson (2015) states, maths anxiety is a well-studied psychological condition. For many, it can be a constant shadow of fear over their learning. People often assume they are anxious about maths because they are bad at it, however, it is the opposite. They are performing poorly because they are anxious. No wonder students are apprehensive about maths if their mathematical ability is associated with being smart.

People are struggling with basic mathematical skills like mental arithmetic that they had once mastered. Maths anxiety decreases our ‘Working Memory’ also known as Short Term memory. The pressure to solve questions quickly causes stress to build up resulting in it eating up our working memory, leaving less space available to tackle the maths itself (Rubinsten, 2017). Often in maths, the answer is either right or wrong which could also be a cause for you to worry about underperforming.

Where has this seed of fear come from?

Boaler (2016) talks about the myth ‘Maths Brain’. People believe you are either born with a ‘maths brain’ or not. But why have we got this perception towards Mathematics and not towards other subjects such as Geography? She stated, “until we change this single myth we will continue to have underachievement”. Maths anxiety is often a result of the way children are exposed to mathematics by their parents and teachers. Children can sense fear and internalise the way maths is spoken to them, for example: in a negative and unfamiliar way. Teachers with maths anxiety can spread their fear onto the next generations. Children are aware when a teacher is nervous, resulting in them being on the outlook for danger. The stereotypical myth that ‘girls are naturally not very good a maths’ is a cultural expectation which can also be a result of maths anxiety (Robson, 2015). The more anxious you feel about a situation the worse you perform.

As teachers it is very important we change the message children receive in classrooms about mathematics. We want to develop children to have a growth mindset so they believe they can grow and learn anything. We also want to highlight the importance of mistakes as they help your brain to grow by forming new synapsis (Boaler, 2016). Conceptual understanding is vital to mathematical confidence. Understanding the ‘why’ behind a concept is just if not more important than understanding the ‘how’. I believe that this is how we change pupils outlook of mathematics by making sense of the maths being taught. Creativity within maths allows children to explore maths in a unique and interesting way, helping them to approach a problem with more confidence. As professionals, we need to have positive attitudes and mathematical confidence to inspire confidence in all our pupils. To achieve this the children have to first understand the concept being taught, then we build confidence in their understanding before finally building competence through practice (Mason, Burton and Stacey, 2010).

Maybe it is comforting to put a name to the problem ‘Math Anxiety’ but I can assure you there are lots of people around you experiencing very similar feelings, including myself. This anxiety is not a reflection of your ability and can be overcome with time and consciousness. Let’s get rid of this lifelong fear of numbers!

References

Boaler, J. (2018). How you can be good at math, and other surprising facts about learning | Jo Boaler | TEDxStanford. [online] YouTube. Available at: https://www.youtube.com/watch?v=3icoSeGqQtY [Accessed 18 Oct. 2018].

Mason, J., Burton, L. and Stacey, K. (2010) Thinking Mathematically (2^nded.).  Harlow: Pearson Education Ltd.

Robson, D. (2018). Do you have ‘maths anxiety’?. [online] Bbc.com. Available at: http://www.bbc.com/future/story/20150619-do-you-have-maths-anxiety [Accessed 18 Oct. 2018].

Rubinsten, O. (2018). Why do people get so anxious about math? – Orly Rubinsten. [online] YouTube. Available at: https://www.youtube.com/watch?time_continue=257&v=7snnRaC4t5c [Accessed 18 Oct. 2018].

To most people the common perception of maths is it being never ended pages of addition, subtraction, multiplication and division sums, so imagine maths being taught creatively? Yes, I was surprised at this thought too. Personally, I am a very creative person and have a strong love for art, so I find the possibility of art being applied to maths very intriguing.

During one of the workshops in Discovering Maths, we were introduced to the concept of maths being taught through tessellations. Harris (2000) defines tessellation as “the covering of a surface with a repeating unit consisting of one or more shapes in such a way that there are no spaces between, and no overlapping of shapes.” Tessellation is often poorly taught in classrooms as its often used as a colouring in exercise. This leads to little progression as the children do not understand why some combinations of shapes tessellate and others do not (Harris, 2000). Haylock (2014) states “for children to enjoy learning mathematics it is essential that they should understand it; that they should make sense of what they are doing in the subject, and not just learn to reproduce learnt procedures and recipes that are low in meaningfulness and purposefulness.” This links to Ma (2010) Profound Understand of Fundamental Mathematics through the ‘Basic Concept’. The children must firstly have a strong understanding of their 2-Dimensional shapes e.g. Quadrilateral, Polygon, Regular, Irregular etc. It is by learning these shapes and their properties children can learn which shapes tessellate, which do not and why. This can be done through using templates. They should discover that the only three shapes that fit together without any gaps are a square, hexagon and an equilateral triangle. This is due to the angle they make where the vertices touch = 360 degrees, linking to Ma’s principle of ‘Longitudinal Coherence’. The children are using the basic concept of shape and are applying this knowledge to explain more complicated concepts, (Ma, 2010).  Showing the children how tessellation is applied in the real world, such as wall and floor tiling, demonstrates the value and purpose of understanding the mathematics they are being taught. Resulting in deeper engagement, according to Harris (2000).

We also explored how children’s knowledge of 2D shapes can be developed through paper folding. It is important for the adult to be reinforcing the mathematical language associated with shapes, for example, sides, angles and symmetry.

This is an example of paper folded shapes.

Digital root circles are another creative activity that can be used in the primary school. Digital roots provide good opportunities for children to find and create visual patterns. They also strengthen the children’s knowledge of their multiplication tables. The activity starts with the children working out the digital roots of the numbers in each of the multiplication tables, starting with the one times table, and progressing to the twelve times table. This creates a more enjoyable way of learning the dreaded time’s table.

Example

5 x 3 = 15

1 + 5 = 6

So the digital root is 6. Here are some examples, I have done from finding out the digital root:

By doing more creative activities like these maths has the potential to be an intriguing and exciting subject to learn. We as teachers can hopefully change children’s outlook on maths through more creative and stimulating lessons. It is vital as teachers to have a clear understanding of the underlying structure of the mathematics being learnt to provide the best learning opportunities for our pupils, through promoting understanding and confidence in mathematics. We must also develop our pupils understanding through exploration, problems solving, discussion and practical experience according to Haylock (2014).

So to answer the question, yes maths can be taught creatively!.

References  

Ma, L. (2010) Knowing and Teaching Elementary Mathematics – Teachers’ Understanding of Fundamental Mathematics in China and The United States.London: Routledge

Harris, A. (2000) The Mathematics of Tessellation. [Online]. Available at:https://my.dundee.ac.uk/bbcswebdav/pid-5217937-dt-content-rid-3675302_2/courses/ED21006_SEM0000_1819/Tessellation.pdf

Haylock, D. (2014) Mathematics explained for primary teachers, 5th edn.,: SAGE.

When I was first asked to describe what profound understanding of fundamental mathematics (PUFM) meant, I immediately thought it meant being a maths genius. It sounded very complicated. However it soon become clear that this was not true.

Liping Ma created four universal principles that would enable a teacher to have a profound understanding of fundamental mathematics. These four principles consisted of inter-connectedness, multiple perspectives, basic ideas and longitudinal coherence. Ma (2010) believes if a teacher can demonstrate these four principles within a classroom then they have a profound understanding of fundamental mathematics.

 Inter Connectedness can be describedas the ability to make connections between different concepts. When teachers apply this principle within the classroom pupils learning will become one body of knowledge instead of being disjointed.

Multiple perspectivesis the idea of seeing and appreciating different approaches to solving a problem. This stops your pupils being restricted to a specific learning cycle as there are multiple ways to reach the same answer. Demonstrating that the myth “there is only one process of finding an answer” is incorrect. It allows a more flexible way of thinking within mathematics.

 Basic concepts within mathematics refers to the basic ideas such as ordering. We must be aware of the basic ideas and apricate their importance. Successful teacher will revisit and reinforce these basic ideas throughout their pupils learning, as they provide the foundation upon which future concepts are learned.

Longitudinal coherence is an understanding of the whole curriculum and how one idea can build on another idea. A teacher with PUFM can identify where a pupil is with their learning and support them with the correct learning to build their mathematical knowledge and skills. They are also aware of the different mathematical concepts their class is going to cover in the future, allowing them to effectively lay solid foundations ahead of time.

After looking into the four inter-related principles which make up mathematics I can see the importance of learning mathematics and exploring the different aspects of mathematics around us. When I become a qualified teacher I will highlight all four principles within my classroom to allow the best learning and teaching for my pupils.

Moving forward in this module I would like to study the wider context of mathematics and how mathematics is used in everyday life through play, music and art. I also want to explore the different ways of making mathematics meaningful and exciting for the pupils as I understand the importance of inspiring young children to want to learn mathematics for themselves and not because they have to.

References:

Ma, L. (2010). Knowing and Teaching Elementary Mathematics: Teachers’ Understanding of Fundamental Mathematics in China and the United States. 2^nd ed. New York: Routledge.