Category Archives: My educational philosophy

The end is near… My Profound Understanding of Fundamental Mathematics

As Semester one draws to its conclusion, I can’t help but feel a bit sentimental on my time in the ‘Discovering Mathematics’ elective. It has been a very enjoyable experience for me and I would encourage anyone who is going into second year to pick it as your elective. It is not about learning complex quadratic equations or even recapping on trigonometry, it actually is more useful that any of those concepts will ever be. I have mentioned in previous blogs that I was not the biggest fan of mathematics due to my stubbornness and having bad experiences at high school level. However, this elective has totally changed my ideology and it will definitely help me gain confidence as I go into my third year placement. I thought since the elective is over that I talk about my ideas and give my ideology and interpretation of what a profound understanding of mathematics is.

At the very beginning of this elective, the idea of fundamental mathematics was introduced. We were given some input on what it meant, but in all honesty I found it hard to understand. This was the case until I came across the work of Liping Ma. Ma conducted research in china and America to identify why the U.S was falling behind in terms of the world rankings and test results. She concluded that teachers in the U.S did not posses a deep understanding of elementary mathematics. Ma (2010) hypothesised that initial teacher training and that during this teachers should become familiar with basic mathematics (fundamental mathematics) much like her teachers in china. In hind sight, Ma (2010) came up with four characteristics which would allow a teacher to have a profound understanding of fundamental mathematics:

  1.  connectedness – this factor is about teachers emphasising how maths connects to other procedures that the pupils are learning.
  2. multiple perspectives – moving away from the idea that there is only one way to get an answer
  3. basic ideas (principles) – that teachers should bring ideas back to basics to encourage good attitudes in mathematics to promote understanding and a love for the subject
  4. longitudinal coherence – progression: the teacher needs to be able to see where the child is and how to further their progress in mathematics.

When you break these four principles of fundamental mathematics down, it is clear to see why maths is so important for everyone. When we think about connectedness, we need to look to our Curriculum for Excellence where cross curricular learning has to be incorporated. This is so important because maths is in everything that we see and do and it is also very important to see the links between maths and our world. Secondly, there is a huge maths myth that has been around for decades – ‘there is only one way to find an answer’… no there is not. There are multiple ways in which problems can be solved – its just about teaching maths in different ways. Thirdly, basic ideas is fundamental to fundamental mathematics because fundamental actually means ‘basic’. We as practitioners need to take maths back to its roots in order for children to progress and love maths at face value. In addition, basic does not mean that it can’t be challenging or fun (I will get to this). Finally, when we think about longitudinal coherence, we think of progression. Teachers need to know where each of their children are at in their learning and the teacher needs to take the steps necessary in order for our children to progress.

It is all very well talking about the theory of fundamental mathematics, but what have I learned about it? Well for starters, I can say that maths is absolutely everywhere… in the outdoors, in the weather – you name it and its there (links to connectedness). I believe that it is absolutely vital to make the connections that mathematics allows us to see. We need to make our children aware of this in order to heighten their love and interest for mathematics. I am aware that maths needs to be an active subject for children to really get their teeth into and enjoy. My blog post ‘active learning in mathematics’ covers this – it highlights that basic principles of mathematics can be taught in fun and interesting ways in which your children will understand. Furthermore, there is no better feeling in the world than seeing the children in your class engaged, learning and having fun. Maths allows us as practitioners to experiment and play with certain theories in order to evoke fun and enjoyment. For example, the lesson on demand planning was absolutely fantastic!! Moreover, enthusiasm counts for a lot in teaching. I mean why do you think I had such a negative view on mathematics? Its probably because my teachers in the past have never shown a passion for it. Our lecturers for this module have been the most enthusiastic people I have ever seen. It has made me become enthusiastic and, of course, if your enthusiastic as a teacher, your children will also become enthusiastic.

Therefore, I believe that fundamental mathematics for a practitioner to be confident and have a profound knowledge of basic maths in order for our children to understand and develop a love for mathematics. In addition, the practitioner needs to be able to encourage and motivate their children through meaningful and engaging activities that incorporates active learning. Furthermore, practitioners need to also make connections with mathematics in the real world and encourage their children  to make these connections as well. Finally, and I can’t emphasise this enough, we must be able to paint a picture of each child’s progress and be able to plan the steps in order for them to succeed and develop in mathematics.



Ma, L. (2010). Knowing and Teaching Elementary Mathematics: Teachers’ Understanding of Fundamental Mathematics in China and the United States. 2nd ed. New York: Routledge. P1-50.

Beautiful Mathematics – an Outdoor Perspective

When we think of mathematics, we often think numbers, formulas, data handling and a whole host of other mathematical concepts. However, have you ever just looked around you whilst out in the outdoors and thought ‘maths is beautiful?” I guess its a thought that never springs to mind. However, mathematics is actually in everything we see in the outdoors (believe it or not) and this blog post aims to highlight how beautiful maths is within the context of the outdoors.

To see mathematics in the outdoors, we do not need to look far. It is in the very buildings that you see walking up and down the street. Here is a building that should all be too familiar:


For those of you who don’t know, this is the Dalhousie Building at Dundee University. I don’t know about you, but I feel believe that this is an architectural masterpiece. Firstly, if you look at the design, it is very visually appealing and it incorporates squares and curves to create a building that is grand in size. Where does the maths come into this? Well if we think back the original plans of the building, it had to be measured accurately in order for it to come together. Of course, there would have been slight room for error, but it had to be pretty accurate. If we think about the windows, the architects had to create enough space so they can tessellate perfectly. This absolutely astounds me. When looking back to a lecture on the golden ratio, you can almost see it happening here. To elaborate, the golden ratio was a ratio used since the 1500’s as it was perceived to be aesthetically pleasing. It uses the formula:

 \frac{a+b}{a} = \frac{a}{b} \ \stackrel{\text{def}}{=}\ \varphi,

This can be best described using this square:


Basically, A golden rectangle (in pink) with longer side a and shorter side b, when placed adjacent to a square with sides of length a, will produce a similar golden rectangle with longer side a + b and shorter side a. This illustrates the relationship  \frac{a+b}{a} = \frac{a}{b} \equiv \varphi.



This all together makes this pattern which can be recognised in Fibonacci’s sequence:

This is seen to make an aesthetically appealing design:

Picture Courtesy of apple.

So where does our Dalhousie building come into this? Well if we look at the elevation of the building (the front of building consisting with the front entrance and the classrooms in the second block, we get a (block two) and b the entrance which would create this perfect spiral. I find this absolutely intriguing. This is not just the case for Dalhousie, However, this is the case for most things in our world. If we look at this plant:

we can see the golden ratio coming into play along with Fibonacci’s sequences. This officially ties nature and mathematics together and the results are absolutely breath taking. In this image above, not one segment of the flower is out of place. They all spiral in the same direction towards the centre of flower which makes it symmetrical.

And here it is again (sorry, I couldn’t resist). Everything in this world is tied into mathematics and this is why maths is beautiful. Whether its looking at buildings or looking at flowers, the fundamental mathematics is there. With buildings, its all do with measuring and being precise and I guess with flowers you could say pattern. Whatever the outcome, just have a look at the world around you and it might amaze you like it has me. Maths is beautiful.
































Maths, Play and Stories

During the Discovering Mathematics elective, I have been amazed by some of the maths that I have encountered, particularly when linking maths to play and stories. I, beforehand, had a very narrow-minded view of what maths was as my school experience was not a good or successful one. I remember mostly being stuck and it made me think of myself as a failure. In addition, I vaguely remember my teacher(s) writing mathematical equations for on the board for us to copy and learn. Was it interesting? Most definitely not. I was disengaged most of the time which probably lead to me having bad memories about mathematics. However, this elective (along with my college experience) has opened my eyes to mathematics on a wide scale – especially through play and stories.

Lets go back in time to my second year of college. This is where my mind-set began to change on the whole concept of maths. As part of our early years placement, we had to make our own story sack which had to include many subjects from the CFE – one of them being maths. To say the least, I was pretty petrified. All that was going through my head was ‘how can I incorporate mathematics from a story?” And yes, the maths anxiety began to set in. However, after researching many different books to use for my story sack, I began to see the links between maths and stories. I chose the Wizard of Oz as my book and managed to include maths from ideologies like character order to the shape of the yellow brick road. Ultimately it opened my eyes and laid a basis for my love of maths to grow from.

Today in the primary school, children still have this maths anxiety. I thoroughly believe that it stems down from older generations. This is because learning and teaching has transformed in the modern day classroom from the mid 20th century. Older generations of people may see maths as copying down sums off the board in a kind of rote learning manner. This left these people with maths anxiety to have very bad memories of mathematics and generations of adults who don’t see the need to learn mathematical theories. Ultimately, our responsibility is to teach young learners that mathematics is relevant to every day life like money, time, fractions and so on. Ultimately, because our older generations do not have the greatest view of mathematics, it gets passed on to our children and they become “scared” of maths (Furner and Duffy, 2002). I believe this statement is true because our attitudes need to convey a love of maths in order for it to pass on to our children. Not only must we as teachers be enthusiastic and encouraging, we also need to introduce maths in fun and interactive ways so it will be memorable for our children (see my last blog post on active learning in mathematics for further information).

Parents as Teachers
I found this concept to be rather interesting and intriguing because parent are children’s first point of contact in education. Parents are the child’s pioneer of education and it could be argued that parents are children’s first teachers. Within the context of maths, this should be no different. However, when it comes to children needing help with their maths homework, parents might try to avoid the matter. According to Pound (2003), parents of young children have a narrow minded view of mathematics and may not prioritise it within the home. I am not suggesting that this is the case for every child, but maths is a hard and abstract concept for children to understand and they probably will need help with their homework. This is why it is important for parents to get involved and convey a love for mathematics so their children can have a love for mathematics.

With regards to cognitive development, mathematical concepts allow children to think, reason, understand and learn. According to Piaget’s 4 staged theory, he believed that schemas are the way in which we organise information and for children to understand mathematics the should repeat their actions (sums and questions) in order to learn. In addition to this, Margaret Donaldson agreed with Piaget on some aspect but believed that if children think of abstract situations then they will fail. I do not believe that children will fail due to abstract concepts. I understand that maths is an abstract concept but with the right help and support, children will thrive. Maths is so beneficial for a child’s cognitive development due to the problem solving aspect of it and, without the children realising it, maths allows them to think articulately.

Fundamentally, we should see children as independent learners and should give them the space to work on an individual basis, This can be achieved through immersing the children in mathematical events before school (for example, ‘how many cows can you see in the field?’). As time progresses, children’s skills will develop, but it is important for parents to not give them direct instructions. Children naturally absorb the patterns and regularities that exist in the day to day natural and cultural world (Ginsburg, Cannon, Eisband, and Pappas, 2006). Effectively, routines are mathematical concepts and children pick up the patters very quickly. Of course, we as teachers need to be able to explain the maths correctly to the children in order for them to learn it (Henlock, 2003). In addition, this can be done through a number of strategies:

  • Open ended questions
  • Using mathematical language
  • Written communication through mark making

Effectively, these strategies should be used by teacher in order to develop their ideas and understanding of mathematical concepts. A good way to achieve this is through play. Play is important because it allows children to be a team players, communicate effectively, learn through interaction and ultimately allows children to express themselves in different ways. Furthermore, play is innate and allows children to make connections, be creative and use flexible thinking. Moreover, it refines and rehearses their skills and encourages perseverance. So where does this link into mathematics? Well mathematics allow children to make decisions through imagination, reasoning, predicting, planning, experimenting with strategies and learning through rhymes and songs.

In conclusion, I never realised that maths and play go hand in hand with one another. It is a fascinating thought that has allowed me to realise that there is more to maths than copying sums off the blackboard. Maths should be fun and inviting and I will definitely try to incorporate this into my lessons while on future placements and when I am a qualified teacher. However, parent play a huge role within mathematics. Parents should encourage a love for mathematics and view maths as a positive thing in order for our children to absorb a love for mathematics – lead our children forward!!


Co-operative Learning: why is it different to group work?

In this mornings lecture, a discussion about co-operative learning cropped up and it make me think about the importance of it in the primary classroom. Co-operative learning is a method of teaching and learning in which students team together to explore a significant question or create a meaningful project. It is a specific kind of collaborative learning in which students work together through discussions and pupils are individually accountable for their own work. However, is it just the same as group work? Or does revolutionise group work by allowing children to be more independent and appreciate working as part of a team? This blog post aims to delve into the concept of co-operative learning.

If we look back in time to 20th century education, co-operative learning would have never existed due to the way that pupils were taught by rote. Children were there to be seen and not heard and this would have made the classroom a very quite and eerie place to learn. When we think of classrooms in this day and age, they are not quiet places. However, this is not necessarily a bad thing. Learning is most effective when learners have the opportunity to think and talk together, to discuss ideas, question, analyse and solve problems, without the constant mediation of the teacher (Education Scotland, undated). This is why co-operative learning is fundamental within the context of the primary classroom. It stems from the work of Lev Vygosky and his view that learning is a social process. I do agree with this because in my experience you can learn more through interactions and discussions with your peers. Vygotsky’s ideas have been reflected in a number of teaching processes including critical skills and dialogic teaching. Both of these involve discussion and promote the idea that young peoples learning is best served when they have the opportunity to learn from each other.

I first experienced co-operative learning during my first year placement. It was an abstract concept to me as I had never observed it being put into practice. To begin, the teacher would organise the children into their home groups (every group had a specific name and emblem) and the teacher instructed the children to perform their team greet. I was specifically interested in this part because each team had a different greeting and I think it allowed each child to feel part of the team – a lovely touch!! The children were left to number themselves 1-4 and each number had a specific task. I cant remember off the top of my head, but I think this was the structure:

1-resource manager
2- time keeper
3- motivator
4- collector of resources

What fascinated me the ‘motivator’ position because this person had be enthusiastic and push his or hers team on. It was refreshing to see that the children were motivated by this and praise from the motivator allowed the children to feel good about their work. The task itself was to make a badge that displayed their hobbies and interests. Each hobby or interest were to be drawn at the corners of the badge (the badge was a small A5 rectangle) with their name in the middle. In addition, the teacher would ask the children 4 questions like, “which is your favourite cartoon?” after each question, the teacher would put a digital clock on the whiteboard in so the time keepers can say ‘we have 10 seconds left’ and this allowed the team to know when their time was up. It was really refreshing and exciting to see the children work together in such a way. They were all actively involved in their own learning as each badge was personal to each individual. The children took responsibility for their own learning and it was enjoyable to observe (I even made my own badge and joined one of the groups).

This was a very good experience for me, but I began to think that it is a bit like group work. After discussing this point with my teacher after the lesson, she identified a number of key points that makes it a whole lot different from group work. She suggested that group work is more focussed with learning partner work, in which you can complete your work with your learning, but no meaningful discussion is being produced. She suggested that an activity allows children to take control of their own learning through discussion and learning off one another. She identified that each individual part of the home teams contribute effectively to each others learning through interactions and ‘motivating’ one another. Fundamentally, these skills need to be taught to the children before co-operative learning can take place. The children need to learn to organise themselves and once this happens, they are effectively taking part in their own learning.

Effectively, co-operative learning is a fantastic way of allowing children to take part in their education through means of discussion and learning from each other. It is a tried and tested method of collaborative learning and it is different from just normal group work in the sense that the children have more freedom and control over their learning. I will definitely be promoting this in future placements and I am eager to see it in practice again.


The Curriculum for Excellence: A comparison between our own curriculum and the Scandinavian Approach

The Curriculum for Excellence was introduced by the Scottish Government in 2010 – 2011 and was meant to revolutionise education in Scotland by providing children and young people between the ages of 3-18 a coherent, more flexible and enriched curriculum (Education Scotland, undated). In addition, the curriculum includes the totality of experiences which are planned for children and young people through their education, wherever they are being educated. The CFE is meant to be a holistic and child centred curriculum very much like the Scandinavian education systems. However, why might this not be the case? This blog aims to investigate the CFE in a critical manner and this will be used as a comparison to Scandinavian approaches in countries such as Sweden.

Lets travel back to the 1990’s where the 5-14 curriculum was implemented in Scottish schools. I very much remember being in lessons such as maths and English and completing my end of unit level C test. This was a very anxious time as testing was a fairly new concept to me but it now apparent to me that testing was done in order to raise attainment in schools. I distinctly remember some of my friends and classmates would deliberately take  a sick day in order not to do the tests which, to me, felt wrong and, in the end, they would need to sit the tests when they came back. It made us feel anxious. As practitioners, it is important that our children are comfortable and feel safe in the school environment. Testing did not make me feel safe and comfortable because I did not want to fail. So why did it change? why was there a big push for a more holistic and child centred curriculum? The Scottish Government wanted to make guidelines more fit for their purpose by providing the right level of detail for teachers so they could maintain the current level of specificity where that makes sense. In addition, the CFE proposed to remove unnecessary detail from existing 5-14 guidelines in curriculum areas such as Expressive Arts and Environmental Studies to allow teachers more flexibility and scope to provide rich and varied experiences, and reduce the time spent on assessment. In addition, the Scottish Government wanted to bring the 3 to 5 and 5-14 curriculum guidelines together to ensure a smooth transition in what children have learned and also in how they learn. This will mean extending the approaches which are used in pre-school into the early years of primary, emphasising the importance of opportunities for children to learn through purposeful, well-planned play. This to me sounds fantastic as children can broaden their horizons through meaningful activities that incorporates play into their every day learning. However, how did the Scottish government come up with the idea to introduce this into the CFE? We need to look over to Scandinavian countries such as Sweden.

In preparation for the CFE for to be implemented into Scottish schools, momentous research had to be carried out and the Scottish government looked towards Scandinavian countries to gain some influence because these countries top the leader boards in an educational sense. With regards to Early years education in these countries, there is a strong financial support for families with young children, and this is fully funded by the state so all young children can have a formal early education. In Sweden, children are guaranteed a place in pre-school for working or studying parents within a few months of applying. There is an emphasis for outdoor learning all year round and the children in these countries are generally more happier and confident from and early age. With regards to Primary Education, the children start at around age 6 or 7 as this is the perfect age to start. In Scotland, there could be children as young as 4 in a Primary 1 class. This is because in Scotland there is a work force interest which is aimed at parents getting back into work sooner in order to contribute back to the economy. As a result, more funding is needed (especially in early years). If a small independent countries can do this, then why can’t we?

In Sweden, children address teachers by their first names unlike here where we say “Mr”, “MRS” or “Miss”.  I think it allows the teacher to be seen as a real person instead of a hierarchical figure and this would ultimately allow the children to feel more comfortable. Teachers are trusted and highly respected and in Scotland it could be argued that this is not the case. This point is also emphasised because no inspectors come into Swedish schools. In addition to this, the Swedish school resembles a family home in the sense that the staff and children take their shoes off and the fact that there is a kitchen in every room. This, again, allows the children to ‘feel at home’ and  allows the children to feel more comfortable in their education. In pre-school education, the children have meals around the table with the teachers and again emphasises the notion that their schools are like family homes.

In Sweden, there is no testing and the children have unfinished work trays where they can pick and choose to finish work when they want. The work is very much set at the children’s pace and the children can, as a result, take control of their own education. In Scotland, there is a massive incentive for children to finish work as quickly as they can which, I feel, is necessary in some cases for short tasks, but we have to remember that children move at different paces to one another and we need to take differentiation into account. In comparison, Scotland does not have a form of national testing. However, the Scottish Government are trying to re-introduce standardised testing into Scottish Schools in order to raise attainment. However, I think this poses a very big problem. As we know, in every subject area of the CFE there are different levels. For example, children within the first level can range from Primary 1 all the way to Primary 4 – my point being is that how can you give a primary 1 pupil the same test as a standardised test as a primary 4? To me it seems completely unrealistic and I doubt that attainment will raise with in Scottish schools. I believe that the Scottish government is too focussed on statistics and aren’t focussed enough on the children. It seems extremely ironic because the CFE is considered as a ‘child centred’ curriculum and I do not believe that the children are being into account.

Testing out the way, lets get down to the facts. Swedish children feel secure within their education. The less informal approach incorporated by the teachers allows the children to feel relaxed and, as mentioned before, it almost feels like a family home. The children have large amounts of unrestricted floor space to play, build and be creative. This encourages the children to plan their own play and this is not the case in Scottish nurseries where play could be very much structured and the children might not be able to explore freely. The Swedish curriculum is grounded and play and is influenced by the theory of Froebel (the founder of kindergarten). The teachers, who are experts in their field, are there to communicate and understand and respond to the children’s needs. I am convinced this happens in Scottish schools as inclusion is an important matter within the Scottish school setting along with GIRFEC and, naturally, teachers should be empathetic and understanding. However, I feel that there is a hierarchy between teachers and pupil and this could make children feel uncomfortable at times.

According to the Swedish Ministry of Education (1998), ” learning should be based not only on interactions between adults and children, but also what they learn from each other”. This comes back to the idea that communication is key and that collaborative learning is a very useful tool for children and teachers alike. Teachers should constantly be looking for opportunities to learn with the children because knowledge is not found in the child or the adult, its found within the world around them and the beauty of education is finding things out together. Collaborative learning is a huge aspect of the CFE and this is something that works. Its all about experiencing new things everyday and this is why outdoor learning is extremely important in any class in any country. The significant thing about the Swedish curriculum is that it is based on a division of responsibilities where each individual state determines the overall goals and guidelines for the schools within each state. What is alarming is that there is no national curriculum and they are still above Scotland in the ranks.

Therefore, why are Scandinavian countries above us in the rankings and miles ahead of us? Well for starters, the CFE has cherry picked certain aspects from these countries to create the ‘perfect curriculum’ but this cannot be the case unless every aspect has been taken from these Scandinavian approaches. That is why the CFE has been criticised on a national scale for being hollow and unrealistic. Our teachers are anxious about how ‘vague’ the CFE is where as Scandinavian teachers are happy in their work. Fundamentally, it is the children that we should all be thinking about and maybe one day the CFE will be completely child centred and holistic. Well, that is goal and the end of the day.

If you would like to find out more about Swedish education, refer to the video below.