Category Archives: Discovering Maths

Is Curriculum for Excellence a ‘Sunk Cost Fallacy’?

For the past 8 years, Scotland has functioned on Curriculum for Excellence as a means for educating children. A curriculum that was created with a lot of hard work from government officials over a long period of time. However, many people would argue that the way the curriculum was formed has many flaws. One argument being, there was no communication with teachers until the curriculum had been made. Another argument is that education in Scotland is not as good as it could be and that, Curriculum for Excellence, is failing (BBC, 2017).

Which begs the question.. Could Curriculum for Excellence be a sunken cost?

A sunk cost is defined as something which has had a lot of work, time and money put into it by a company making them reluctant to stop or change it even if it starts to fail (Cambridge University, 2018). Priestley and Humes (2010) discuss Curriculum for Excellence in detail, more specifically the four capacities, what they mean and the general success of the curriculum. After being on placement, I witnessed a lack in cause for the four capacities meaning they were rarely mentioned and children were not aware of them. Priestley and Humes (2010) highlight that the four capacities have the potential to be highly beneficial, however, they currently are not used effectively in schools. Considering the four capacities role in the curriculum and what they stand for, children lacking awareness of them would suggest the way the curriculum is presented is failing as the foundations of the curriculum are not being discussed with pupils. Additionally, Priestley and Humes (2010) focus on the aims of the curriculum and put forward the argument that Curriculum for Excellence is not about the content but more about getting the right results. After experiencing placement, I can understand this argument as although what is learnt is essential. In maths, for example, we do not provide an education that allows children to have a deep understanding of maths, just an education which will enable them to pass a test.

A massive issue in education currently is Maths Anxiety in young children and can be caused by various things – failure, embarrassment or confusion. I believe that the way our curriculum is designed prohibits us from meeting every child’s needs. Realistically, with a curriculum so broad and full, we do not have time to spend ages on one topic to ensure a child understands – so most teachers move on. In terms of Liping Ma’s (2010) theory on the four concepts which make up a profound understanding of fundamental maths, pupils opportunities to experience these is limited due to the extent of the curriculum. Whilst having a wide variety of subject areas is beneficial, a curriculum which has too much crammed into it will result in students feeling stupid and anxious for not understanding a concept. If a child experiences this it can affect their confidence massively, and maths anxiety can start to form (University of Cambridge, 2017). If a child is experiencing Maths Anxiety the effects to their education can be significantly damaging as they will be reluctant to learn. The Scottish curriculum can be restrictive in terms of how we are supposed to teach things that it has the potential to damage the understanding of a child rather than enhance it. Going back to Priestley and Humes (2010) argument about the curriculum only providing what is needed for passing tests, could a lack of fundamental understanding be contributing to Maths Anxiety?. If we only teach children what they need to know to pass, how are they ever going to understand the real processes behind the maths they learn and how to link it to other areas?

The BBC (2017) put forward the idea that the current state of Curriculum for Excellence could potentially widen an already large attainment gap. Looking into this, it becomes clear that the reasoning for this is that due to the curriculum being so busy, we often turn to homework as a resource to try and consolidate any knowledge we couldn’t during class time. However, not every child is lucky enough to be born into a family which can viably support them through their education at home, therefore, causing children in lower classes to fall behind with homework as they do not have the same support system as someone coming from a numeracy rich environment. For these families, they rely solely on the education they receive at school – yet even then it has the potential to be better.

However, it would be unfair to look at this from a biased view and not consider the opposing argument as not all areas are failing. The Scottish Government (2017)  released statistics relating to the achievement of primary school kids in 2016/2017 in the areas of reading, writing and numeracy. These statistics highlight that pupils under Curriculum for Excellence are actually achieving in these areas with the lowest SIMD quantile at 66% for reading and 67% for numeracy highlighting that when we break down the success of the curriculum, whilst it has its flaws it is also providing young learners with an education. However, this could be improved significantly.

So is CfE a sunk cost? Arguably yes. If we consider the practicality of changing a whole curricular area based on this, it just wouldn’t be realistic. The time and money that would be spent re-educating teachers to ensure there was a deep understanding of maths which they could pass on would be significant. Having spent so much on building the curriculum, educating people to teach it and presenting it to children, we are potentially in too deep to change – even when it is failing some children. No child should have to suffer Maths Anxiety as a result of the curriculum and the way it is taught. If we focus more on teaching children the basic ideas involved in the mathematical concepts they are taught, we could then introduce them to how it connects to everyday life and how they can use what they know to solve problems.

To finish, over the next 5 years, the Scottish government have committed to spending around 750 million pounds on closing the attainment gap (Scottish Government, 2016). The government are potentially putting money towards closing a gap in a curriculum which already has its own flaws. I cannot say for sure that it is a sunk cost, but after the lecture we had surrounding sunk costs, this sounds like it has the potential to be one.



BBC. (2017) New curriculum could be ‘disastrous’, says education expert. Available at: (Accessed: 24 October 2018).

Cambridge Dictionary. (2018) Sunk Cost Fallacy. Available at: (Accessed: 24 October 2018).

Ma, L. (2010) Knowing and teaching elementary mathematics. New York: Taylor and Francis.

Priestley, M. and Humes, W. (2010) The development of Scotland’s Curriculum for Excellence: amnesia and deja vu. Available at: (Accessed: 24 October 2018).

Scottish Government (2017) Achievement of Curriculum for Excellence Levels 2016/2017. Available at: (Accessed: 25 October 2018).

Scottish Government. (2016) Making Maths Count. Available at: (Accessed: 24 October 2018).

The University of Cambridge. (2017) The relationship between maths anxiety and maths performance. Available at: (Accessed: 25 October 2018).

Is there a place for old mathematical concepts in modern day maths?

This week we discovered Pascal’s Triangle – a number pattern in the form of a triangle where the numbers above added together give you a value for a new row. Within this number pattern, there are all different kinds of maths intertwined allowing us to explore it further in depth.

However, whilst learning about old mathematical theories and patterns is interesting, is there a place for them in the modern curriculum?

Although many of the theories were created decades ago they are still relevant and are the basis for the knowledge that is vital to understanding maths. Guiness (2003) highlights that whilst this information may have since been updated or expanded, the original idea that formed a mathematical concept could not have been created without this knowledge. Additionally, researchers across the world have concluded that children learn significantly better if they understand why they learn something and not just how (Alexander, 2017). A key CfE principle explored in schools is relevance (Scottish Government, 2010). It is KEY because by explaining and exploring where mathematical concepts come from and connecting it to aspects of the world, we make children’s learning relevant WHICH is more engaging than if this was left out and we begin to lose the ‘when will I ever use this’ line.

For that reason, I would argue that old theories do have a place in the modern curriculum through enhancing understanding.

So how does Pascal’s Triangle and the theory behind it fit into this?

Below is an example of Pascal’s Triangle made up of 12 rows (This can be endless). Each row is a result of the product above it. For example: the third row is made up of the numbers 1 3 3 1 and the fourth row is made up of the numbers 1 4 6 4 1. This is done by adding 1 + 3 to give us 4, 3 + 3 to give us 6 and again 1 + 3 to give us 4. What an interesting concept. However, this is not all the triangle does and this is where we can make an old number pattern and theory relevant to what the current curriculum teaches. After exploring Pascal’s Triangle, it became clear that when broken into smaller pieces it is actually made up of different number patterns.

Parts of Pascal’s triangle, when added together, make up square number patterns. As shown below, 1 + 3 is 4, 3 + 6 is 9, 6 + 10 is 16, 10 + 15 is 25 and so on. Whilst this is not a massive part of the primary curriculum, it shapes a section of secondary maths which is vital. By using basic addition, taught at school we start to create multiple patterns that we were previously oblivious to. Furthermore, a simpler way to argue its relevance in our curriculum is to go back to the basics of how we form the triangle. ADDITION. This is a basic concept taught from the age of 5 which allows pupils and teachers to connect concepts and work out solutions but also enables society to work (Haylock and Thangata, 2007). Teachers who will take on the challenge of teaching about Pascal’s Triangle are teachers who work through the stages of Liping Ma’s (2010) 4 concepts when they take a basic idea, explore it and then connect it to any relative areas in mathematics.

This way of teaching maths, for a child, is so much more intriguing than textbook work. Yet, Robert Floden – dean of the College of Education at Michigan State University (Hartnett, 2016) puts forward an interesting argument about how we are trying to teach maths. He argues that often, when we try to teach maths actively and through games, we actually reduce a child’s learning as they end up carrying on and not actually learning. He believes that by trying to make maths active, children are missing out on fundamental understanding needed to be efficient in areas of maths. Whilst in some cases this may be true, I believe this has more to do with class management and not how we teach maths and for that reason, we should not stop engaging activities. If we set rules around activities, we can reduce the likelihood of bad behaviour when it comes to actually using them.

Boaler (2009) explored two schools who taught maths in very different ways. One school (Amber Hill) taught in a very traditional way. They had goals in the curriculum to meet and stuck to the path to get there. When explored, this type of learning showed that children found it difficult to remember how to do things after a certain amount of time. The other school (Phoenix Park), took a more relaxed approach whereby each concept was explored in a way that could be connected to another making it relevant and enjoyable. The outcome of this was that pupils who had been taught at Phoenix Park had a better understanding of their learning and because of this, they were able to apply it in different situations. This study highlights the point that if we are creative in our ways of teaching maths we open up more opportunities for understanding. By taking basic concepts such as addition and multiplication, we are able to connect these to other aspects of maths such as Pascal’s Triangle and expand our knowledge through our learning. If we were to be influenced by Robert Floden, we would essentially be following the teaching shown at Amber Hill, using a more traditional approach. However, as shown the more traditional approach is not always the most effective approach.

Overall I believe that old mathematical theories and patterns do have a place in our modern day curriculum, however, only when used a specific way. I believe we should use these theories and patterns to enhance maths learning by making it engaging and fun. If we make learning intriguing for children they are less likely to feel negatively towards it.


Alexander, P. (2017) ‘The Relevance of Relevance for Learning and Performance’, Taylor and Francis Online. Available at: (Accessed: 4 October 2018).

Boaler, J. (2009) The Elephant in the Classroom: Helping Children to Learn and Love Maths.  London: Souvenir Press Ltd.

Guiness, I. (2003) The mathematics of the past: distinguishing its history from our heritage. England: Middlesex University.

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

Hartnett, K. (2016) ‘Meet the new math, unlike the old math’, Wired, 10 August 2016. Available at: (Accessed: 4 October 2018).

Scottish Government. (2010) Curriculum for excellence building the curriculum 3 a framework for learning and teaching: key ideas and priorities. Available at: (Accessed: 3 October 2018).

Discovering Creativity in Maths

If you had said to me this time last year that today I would be sat in lectures genuinely interested in the small parts that make up mathematical concepts, I would probably have laughed. When I was taught maths, it was just maths. You learnt a concept, did some examples and then were assessed on your understanding.. or memory.

When we were introduced to tessellation, I was so intrigued, yet shocked, at my obliviousness to this concept and the amazing things that can be done with it.This week’s lectures in particular have opened my eyes to the idea that maths is everywhere. In music, in art, in biology and so on.

Tessellation is based around shapes which all join together to create a bigger shape or picture with no gaps. For example, regular shapes such as squares, equilateral triangles and hexagons are perfect for tessellation as all the corners in these shapes have a point which links them together. These are known as regular shapes. On the other hand, irregular shapes such as normal triangles, octagons and circles can still be used in tessellation by changing the properties of the shape. This is done in a cut and paste structure and if done correctly can produce some great pictures and great examples for pupils. Escher was a graphic artist who used shapes, adjusted them and then created new shapes that were in the form of animals or humans. However, in order to understand tessellation children must first be taught the properties of these basic shapes and how to identify irregular or regular shapes. In all areas of maths children must have a profound understanding of the chosen area in order for them to maximise their success.

Haylock (2007) discusses the importance of cross curricular activity throughout school and highlights how vital it is for pupils to be informed of the links between subjects. A geometric multiplication circle is something which we studied this week in Discovering Maths and a tool which would be great to use with pupils in a class who struggle to enjoy maths.

A theorist came up with the idea that by using geometry and times tables, art could be formed in the shape of stars. The process works by having a circle with numbers one through to nine surrounding the circle. By picking a times table such as 6×7 we get 42. This simplified gives us 6 and from there on we can join the dots creating Islamic Art, or as kids would say ‘stars’. This shows the importance of primary school kids being able to take a subject like maths and use their knowledge to enhance their creativity in art. By giving the pupils these opportunities we are taking MA (2010) ideas of basic ideas and inter-connectedness and linking them to other aspects of the curriculum allowing pupils to use their multiplication skills and logic.

By carrying out geometric multiplication circles, pupils are participating in maths without realising they are doing a lot of it. A teacher with a profound understanding for fundamental mathematics would take a topic such as shape, apply the basic ideas of the topic such as the sides and angles, connect it to other areas in maths and as a result expand the child’s overall understanding of maths whilst making it significantly more interesting. These children who have these experiences also have the ability to transfer skills from one area to another enabling them to think and connect all subjects to their mathematical understanding.

Here are some examples of Geometric Multiplication Circles:


Haylock, D. (2007) Key Concepts in teaching primary mathematics. Los Angeles: SAGE.

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

Warner, M. (n/a) Digital Root Patterns. Available at:  (Accessed 27 September 2018).


A Fear of Mathematics

When it came to picking my elective for second year I was extremely hesitant to choose Maths. I thought it would be so interesting, however a part of me could not get over my original fear of maths itself – a fear most people share and have had from a very early age.

As a primary school kid, I went through some states trying to complete my maths homework and failing to understand what it was that I was doing wrong. It would constantly cause arguments with my dad, who was trying to help, because his way of solving a maths problem was different to the way I had been taught. I imagine kids today would look at my way of learning and agree that their way of learning is significantly different too. This is what makes the approach teachers take so important as it can significantly affect the child’s confidence and ability in maths.

Despite this, every time I think of maths I am reminded of those nights spent in tears over my maths homework fearing I had done it wrong. As a training teacher, this is a feeling I hope my pupils never go through. Every day children learn new things and so do adults.

It would be hard to find someone who could honestly say maths hasn’t scared them once in their life because the reality is that everyone has a small fear of it. I believe this can stem from lacking basic knowledge of concepts which therefore inhibits a pupils’ ability to understand how to complete a task. Often, if the topic is broken down into small bits a child will know how to do most of it but because they can’t yet grasp one aspect of it, they say they can’t do maths. This is where it is vital that we provide the support necessary for pupils to develop confidence and understanding in maths. Maths is essential for everyday life and because of this it is vital that we find a way to avoid children fearing it.

The University of Cambridge (2017a) discusses maths anxiety and how it can affect a child. Maths anxiety can affect pupils ability to develop skills in maths as it causes significant fear when the subject is mentioned. There are numerous things that could contribute to math anxiety however a large factor that could cause maths anxiety is that the child has encountered embarrassment and failure within maths so now associates it with fear (Joseph, 2017). The University of Cambridge (2017b) released data showing that if a child has poor maths skills they are more likely to suffer from maths anxiety and if pupils have maths anxiety, their ability to complete maths problems will decrease significantly. I believe that maths anxiety can be avoided or overcome by taking the right approach to teaching maths. Maths can, and should be fun. Instead of associating it with textbooks and texts we should associate it with fun and activity.

As I said at the start of this blog, I was so scared to take this module however after one lecture I soon realised that I was not the only person who had this fear of maths, we learn something new every day and that with the right approach and knowledge, maths doesn’t need to be scary…

Oh, and I learnt what an angle was… at the age of 18.


Joseph, A. (2017) Sciencing. Available at: (Accessed: 14 September 2018).

The University of Cambridge. (2017a) Maths Anxiety. Available at: (Accessed: 14 September 2018).

The University of Cambridge. (2017b) The relationship between maths anxiety and maths performance. Available at: (Accessed: 14 September 2018).