A Goodbye to ‘Discovering Mathematics’

So, the semester is ending which means that one of my absolutely favourite modules is too coming to an end. ‘Discovering Mathematics’ was my chosen elective for this semester, a module that I was right to assume would be extremely beneficial for my Education degree. I have to admit I am quite sad that we are now done with the module. It has been an interesting and enjoyable semester and I have been looking back at all of the things I have learned as a result.

The main purpose of the module was to become somewhat ‘at one’ with maths. It was heavily focused on developing an understanding of fundamental mathematics, something that Liping Ma has expressed to be a necessity for teachers. However, this module has taught me that having a PUFM (Profound Understanding of Fundamental Mathematics) does not mean that you have to magically become a maths genius. It is not about being able to fill a chalkboard with complex equations or suddenly being able to name every single digit of Pi. It is about understanding what lies at the very core in maths. It is about noticing the links that maths has within other learning aspects, being able to view maths problems from different perspectives. Having a PUFM is about understanding the basic principles of mathematics and the notion of maths continually progressing and developing. If we can understand these underlying principles of the subject then we deepen our PUFM, something I definitely think this module has helped me to achieve.

I feel that my understanding of Fundamental Mathematics has increasingly grown over the course of this module. Three months ago, if someone asked my opinions regarding longitudinal coherence in maths or about the ever present Fibonacci sequence my mind would have went completely blank and I would probably enter the panic zone. Now, I feel more calm in relation to maths. Mathematics and I have had a little bit of a rollercoaster relationship with plenty of ups and downs but I now feel that we are at peace with one another. I am aware of the maths around me, of it’s significance and I feel that I am working towards gaining that all powerful PUFM. I am not fully there yet. I feel that there is always room for progression and, as a teacher, I have plenty more to learn when it comes to the teaching of maths. However, I feel that on a personal level, my maths skills have significantly improved and my eyes have been opened to the range of possibilities there are in which to teach it. I can now see that maths is truly all around me and I often catch myself noticing the little things. Whether it be the detailed patterns hidden within nature in plain sight or the basic skills I use in work, I now realize that maths is ever present in society and always will be in more ways than we can possibly fathom.

So, the time has come to say goodbye to ‘Discovering Mathematics’. All I can say is thank you for teaching me so many valuable skills to take forward into my future teaching career. I have enjoyed the module immensely and have taken so much from it. For any future teachers that may be reading this, I highly recommend that you take this module, especially if you are feeling that maths anxiety niggling in your brain when it comes to the thought of teaching it in classes. This module will help you in so many ways and I am happy to have been a part of it.


Ma, L. (1999). Knowing and Teaching Elementary Mathematics: Teachers’ Understanding of Fundamental Mathematics in China and the United States. New Jersey: Lawrence Erlbaum Associates.


Discovering Mathematics- My helping hand in tutoring

This module has taught me so much valuable information regarding the subject of mathematics. From Artistic to logistical, my mathematics knowledge has expanded massively as a result of the module. However, for me there was an extra bonus side effect that has been completely fantastic. The skills I have developed as a result of participating in ‘Discovering mathematics’ have helped me become a better tutor, something I do after university twice a week.

I currently tutor two pupils, one a Primary 7 and the other a 1st year in high school. Both focus on mathematics in our sessions as it is a point of difficulty. As a teaching student, I have had placement in a primary 4 class meaning my focus regarding the subject so far has been rather simple i.e. looking at the basic principles of mathematics. This meant that for the first time since I was in their position, I would be practicing upper school mathematics. Originally the thought did not worry me, I mean I studied higher maths, I used to love the subject so shouldn’t be too difficult right? WRONG! teaching more difficult mathematical processes to those inexperienced with them is actually quite a challenge.

I then set out to find a more efficient way of teaching the subject so my tutoring would be successful. I want the best for those I tutor so I need to be committed 100%. That is where my ‘ Discovering Mathematics’ module came in handy. Through several engaging sessions, I realized that the key to teaching harder maths is to in fact break it down into the basic principles that lay at it’s foundations. Liping Ma (1999) highlighted this as being one of the four main factors within fundamental mathematics and I can honestly say that it makes a difference when it comes to teaching pupils. So, I began to start where anything really should, at the beginning. I slowly built up the knowledge of the subject, starting from simpler maths and by the time we reached the topics that initially were difficult, their understanding has solidified and they could complete the work to a higher standard.

For example, one of my tutor pupils struggled with chunking, a practice used to break down long division. Now, long division is a subject many dislike, myself included so i approached my tutor Richard to seek out some advice. He discussed some ideas with me but all the information led back to one concept- start from the basic principles and work up to the challenge. Following this conversation, I planned some activities that involved the breaking down of numbers. We used Lego blocks to visual represent factors within numbers i.e. Long blocks with 8 circles represent 8’s, 4 is 4 and so on. This was a very active and engaging activity that was massively enjoyed by both my student and I. We then progressively moved on and now their chunking practice has improved greatly.

I feel that my experience in tutoring paired with my learning in this module has shown me something hugely important. I feel that i have developed my own PUFM (Profound Understanding of Fundamental Mathematics) after putting it’s principles into my own personal and professional practice. Being a mathematics tutor has given me the opportunity to use the knowledge I have learned in this module and so far it has been a truly worthwhile experience.

Liping Ma’s Fundamental Mathematics

Throughout the ‘Discovering Mathematics’ course this semester there has been a focus on a key text – Knowing and Teaching Elementary Mathematics: Teachers’ Understanding of Fundamental Mathematics in China and the United States (1999 and 2010 edition) written by Liping Ma. Ma outlined what a teacher should know regarding maths in order to teach it successfully to young learners.

Ma describes the understanding that teachers in China have regarding Mathematics as a ‘Profound Understanding of Fundamental Mathematics’ or PUFM for short. She explains that while PUFM has been achieved in China, it is something that should be obtained by teachers worldwide to ensure that both they and their students can understand maths. It is important for teachers to be able to be confident in their own maths ability as this will be projected through how they teach in the classroom. As student teachers, it is easy to fall into the trap of dreading maths lessons or feeling out of practice in the subject. At the beginning of this module, we were asked to raise our hands is the idea of teaching maths is worrying to us. The majority of the class raised their hands with anxious looks on their faces. However, we were told not to fear as this module would help us become more confident and aware of mathematics, something I feel has in fact been achieved.

In order to understand and gain a PUFM, we have to strip mathematics down to it’s most basic properties. She wrote of four main factors that make up mathematics – Connectedness, Multiple Perspectives, Basic Principles and Longitudinal Coherence.  When we understand these four factors and can use them to teach maths confidently then a PUFM has been achieved.


Connectedness refers to how mathematics has links with several other topics. This module has proved this several times through inputs such as ‘Maths and Art’, ‘Maths and Music’, ‘Maths and Sports’ and many more. Through these inputs, it has become evident to me that maths has links and connections with several areas of education. This means that it can be taught in relation with a multitude of topics which well help  make them easier to understand for children in class.

Multiple Perspectives

Ma also emphasizes the importance of having multiple perspectives when understanding mathematics. In order to achieve a PUFM we must look at maths topics from several perspectives so it can be understood in different ways. When it comes to teaching, children may need to have topics explained to them in different ways as each has a different method of learning and understanding. This means that by having multiple perspectives regarding mathematics, we can actually aid all children in their understanding of it no matter what stye they prefer.

Basic Principles

In order to have a PUFM it is important to understand what lies at the core of mathematics, Mathematics is not solely complex equations and calculus that terrified us in school, it has simple principles at it’s core and when these are understood, so can all mathematics in time. Therefore, Ma highlights the importance of stripping maths back to these basic principles and using them as building blocks for a complete mathematical understanding.

Longitudinal Coherence

Maths is a subject that needs to continually move forward. It is not something that we study for a time in school and then push immediately out of our minds as soon as we leave (even though sometimes that is what it feels like!). The knowledge and skills we gain in mathematics are essential for our future. We need to be able to use these fundamental principles in a range of situations as highlighted to me through this module. Therefore, the final principle of fundamental mathematics is probably the most important in my opinion as it ensures that maths keeps expanding and growing both in society and our minds.


Overall, I feel that the ‘Discovering Mathematics’ module has vastly improved my understanding of fundamental mathematics. The module has helped me experience all four of Ma’s important factors of it. I feel now that my knowledge on the topic has expanded and I can now carry this with me into the classroom to future classes. I know now that maths has to be broken down, explained in different ways and linked with other subjects in order to ensure it remains prominent in both my own mind and the minds of my pupils.


Ma, L. (1999). Knowing and Teaching Elementary Mathematics: Teachers’ Understanding of Fundamental Mathematics in China and the United States. New Jersey: Lawrence Erlbaum Associates.

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

Maths and Logistics

Let me start this post by saying, BUSINESS IS HARD. After a two hour activity in which me and Erynn were partners in business I was extremely stressed but did learn a lot about how maths affects it. It was an extremely fun activity but I have come to the conclusion that logistics in relation to business and food is not my best subject.

We began this input by discussing logistics in relation to food supply chains. It was really interesting to see how even something as basic as the food we eat can be linked to Ma’s Fundamental Mathematics (1999, 2010). We discussed food miles, meaning how far our food travels across the world to arrive on our dinner tables. It was fascinating, but what was more fascinating was that we were going to be getting the chance to open our own businesses and order in our own supplies. In order to do this effectively, we needed to understand certain patterns and trends in relation to supply chains so that we can order things that would be popular during certain times of the year. This is an example of how the mathematical topic of sequences and patterns is actually related to the operation of stores across the globe.

So, with this new logistical knowledge in mind, we set out to plan our business. Erynn and I decided to name our company E Squared as it seemed to suit both us and the module well. The first quarter was APRIL- JUNE so as these are warmer months we decided to buy in ice cream wafers, beer and cold drinks all of which are summer necessities in our heads. The resulting sales for the quarter were high meaning we gained profit. A bonus of the stock we produced is that it was long lasting, meaning that what remained would be carried over to our second quarter. By this stage we were getting into this activity and we put our business heads on for our next ordering strategy. JULY-SEPTEMBER was the next block and seeing as we hoped the sun would still be shining, we thought that cold refreshments were once again the way to go. This proved to be successful once more and we were proud of our growing earnings. We were halfway through our business year!

We only had one word for the next quarter (OCTOBER-DECEMBER)- CHRISTMAS. It was time for the most wonderful time of the year so our hypothetical shelves were stocked with everything one needs for the season. From Selection boxes to Champagne, we decided to invest a large percentage of our fund to our third quarter buys as we expected a huge increase in sales numbers. We were right. All of our bought stock sold at the rate of 100% meaning that the intake was rather spectacular. We took this moment to look around the room and we saw all the ‘Discovering Mathematics’ students were excitedly engaged in the activity. The Final quarter was the slow time of year being JANUARY-MARCH. This is when we decided to stock up on the basics as people would be struggling for money at this time of year. We purchased enough beans and bread to feed a university full of poor students and not so surprisingly, they were bestsellers! The profits kept rolling in and we had more than quadrupled our initial spending money. Unfortunately we noticed that we had actually made a few wrong calculations along the way so the exact profit we unknown but my partner and I enjoyed the session massively and had lots of laughs along the way. We discovered that the real key to this challenge was apparently beans. Yes, beans! The pair beside us at the table had changed their initial 5,000 to 184,000 by investing the majority of money on baked beans, they swear it was the key to success.

Overall, I found that our logistics input was extremely interesting and was definitely more fun than it originally sounded. I can see that fundamental mathematics is at the core of how stores and other business operate and while it was fun to get a glimpse into that world, I feel it is not where my mathematical strengths lie.

Maths and the Expressive Arts – Part 2

After the workshop about maths and creative art, one that I enjoyed massively, we were treated to another workshop based on maths and expressive art. Our next creative workshop was centered on maths in relation to music, something I was not aware had such a strong and vital link. I have to admit, music is not one of my strengths. It is something I am apprehensive about both teaching and participating in. However, I went into the workshop with an open mind and a sense of excitement at trying something new.

We began the workshop by stripping music down to the basics, quite like we have been doing in the ‘Discovering Mathematics’ workshop in relation to Liping Ma’s (1999) idea of having a ‘Profound Understanding of Fundamental Mathematics’ (PUFM). We began by looking at beats in a bar and how all music is composed of a number of sounds in an almost mathematical sequence. As a group we created different beat patterns in order to make a whole class rhythm. In all honesty I do feel that I was slightly off with my drum beats but it was all in good spirit. It was an activity I could picture doing with a class of my own one day and enjoying, something I never thought I would say in relation to teaching music.

We also learned lots about maths in the sense of sequences in music. We played percussion instruments such as drums and xylophones to play sequences of notes and the underlying mathematical properties were evident. There were a certain number of notes to be played in a specific order to create music, something so clearly mathematical but with beautiful results. For me, the two hour session we took part in flew in as I was having such a good time. I laughed, I struggled but more importantly I learned. I learned that fundamental maths is at the root of music and that the two curricular subjects can be taught together to create a more enjoyable and worthwhile learning experience for children. Since the input, I have had so many ideas running through my mind of how to link the two subjects together and I have realized that it can be done in subtle and easy ways like shown in the following video-


Overall, this was another fantastic session of maths in relation to the expressive arts. I have said this before but once again I am fascinated by how the two can be taught together. As someone, with a slight aversion to certain artistic subjects and who struggles a little in this area, it is truly eye opening to realize that anyone can participate in these activities. Teaching the two subjects together has the benefit of both aiding children in mathematical and creative learning so a multitude of skills and useful experience can be gained through this form of teaching. I look forward to seeing what else this module has in store for us and to find out more new and exciting ways that I can teach maths to pupils in the future!


Ma, L. (1999). Knowing and Teaching Elementary Mathematics: Teachers’ Understanding of Fundamental Mathematics in China and the United States. New Jersey: Lawrence Erlbaum Associates.

Maths and the Expressive Arts

It is a popular belief that, when it comes to learning and skills, people tend to favour one aspect whether that be the logical, scientific subjects or the expressively creative ones. One person may be a wonderful painter who loves to play instruments and dance while the next loves to experiment with science and learn about astronomy. However, I feel that this is a myth. It is a myth I used to believe as I always felt I had a more mathematical mind than that of a creative one, but a myth nonetheless. Through my time on the ‘Discovering Mathematics’ module, my eyes have been opened to the fact that maths crosses over a vast amount of curricular content and is prominent pretty much everywhere. The lines connecting a subject like maths to that of the expressive arts were previously blurred but now I can see the strong direct links showing how they can be taught together.

There are many creative ways to promote Ma’s (1999) fundamental mathematical skills, one of which being through art. Last week, I attended the ‘Math and Art’ input of the module and had such an enjoyable time, despite my aversion of the arts so I was inspired to write this post about it. The lecture had us actively engaged in a range of art based mathematical activities that were easily accessible to and understood by all. An example that I found to be extremely interesting was the activity entitled ‘Who is the most beautiful?’. This activity highlighted the links between maths, art and history which established the principle of connectedness as a key concept on fundamental mathematics (Ma, 1999). Ma talks of connectedness as an awareness of the direct links that mathematics has with other curricular areas, something that was evident to me during this part of the input. In this activity, we related the golden ratio of Fibonacci to medieval stereotypes of beauty. Pairing this with mathematical measuring skills, we had an activity where we could actually mathematically calculate who at our table was the most beautiful.

After a thorough measurement and the careful calculations that followed, we discovered that each of us has a certain body area in which our ratios were mathematically perfect (mine was my waist to head ratio!). This activity  was exciting and we became active and energized as a result, This has made me realize that there are very creative ways to implement math learning in the classroom in a way that will actively engage students without them truly realizing that they are studying maths. It is almost the equivalent of slipping vegetables into a sauce so that kids don’t know they are eating them. It is a fantastic concept that I truly enjoyed, it may have even encouraged me to invent my own ways on incorporating maths into fun activities. I also found a video relating to art and mathematics found in nature which provides a range of ideas for lessons that may be interesting to watch! –


Overall, I think that this input has helped me overcome my art aversion which I hope to implement in my future lessons as a teacher. This will hopefully help me to prevent cases of possible maths anxiety and make mathematical learning more enjoyable for my future students.

Developing Maths in the Early Years

Maths anxiety, a fear and dread of anything related to mathematical processes, is prominent among adults in today’s society. However, this presents an interesting question- Is this seed of fear and doubt planted at an early age? The recent sessions of my ‘Discovering Mathematics’ course have shown me that a thorough and positive teaching of maths should begin at an early age but showed be implemented in creative ways.

In the past, maths was taught in a very linear way, a practice that is rather out of date meaning it fails to keep children interested, By incorporating play, art and other fun learning forms we can make the learning and teaching of maths an extremely enjoyable process.

First, lets take a step back and look at how children further their cognitive development. From an early age, children are very aware of their surroundings and can take in lots of valuable information through sensory exploration. The following video expands on this-

Children become slowly aware of concepts such as size, colour and language through observing and mimicking those around them. Therefore, an early exposure to mathematical concepts will be extremely valuable in helping children in later years. They can slowly pick things up if introduced to them at a young age.

Play is a fantastic way to do this. When children play they are able to explore and develop their knowledge.  Lev Vygotsky (1896-1934) emphasizes the fact that playing in a social environment develops learning. The following video shows how effective co-operative learning in relation to maths is an important tool in the classroom. I believe that if this was put into practice from the early stages of primary school, maths would be viewed in a more positive light, making it more understandable and enjoyable for children.

There are any ways to allow for maths based play such as the use of number or cubic blocks, sorting activities and linking animals for counting. All of these suggestions may not seem directly mathematical but they do enforce the fundamental mathematical skills described by Liping Ma (1999) which i find incredible!

Overall, I feel that it is vital that as a future teacher I promote and teach maths in a positive manner in the classroom. This module has made me realize that there are plenty of fun ways to implement maths teaching in the classroom but it is also important to recognize that this must be done from an early age. I want to introduce maths in a way that will engage young children and make them excited to learn more in the future. Play based activities are one effective way to do this and I hope to research more soon so that maths anxiety can be prevented before it begins to take shape.


Ma, L. (1999) ‘Knowing and Teaching Elementary Mathematics: Teachers’ understanding of Fundamental Mathematics in China and the United States’ , New Jersey: Lawrence Erlbaum Associates

Can Animals Count?

Can animals count? While it may be a rather strange question it is one that has kept me thinking. Can animals count? As a society we do recognize that animals share some of the same cognitive functions as humans such as our survival instincts but do they share some of the more developed traits that us humans possess? This was a topic discussed in one of this year’s ‘Discovering Mathematics’ inputs and it’s something I have been pondering ever since.

Those who study animal behavior have looked into this extensively however there is still debate over the validity of the research. Besides, some cases of counting animals can be discredited with ease, for example ‘Clever Hans’ the amazing counting horse. In the early 1900s, an Orlov Trotter horse named ‘Clever Hans’ became somewhat famous. This horse could apparently perform mathematical tasks which people paid to see. However, after some observation and research it was clear that the horse was not tapping out the answers to the questions asked, he was simply observing and mimicking the actions of his owner. This is an example which that does not prove that animals can count but is an interesting case that can be seen below –


A more convincing study in my opinion is one carried out by Karen McComb, an animal behavior specialist. She conducted a study on lions and there ability to count. Her evidence was that if a lion hears a set of roars in the distance, they can actually determine how many lions there are in that collective roar. This helps them to decide whether or not they will be outnumbered or in danger if they are to cross paths. I find this extremely interesting, that by sound alone a lion can know how many lions are approaching. By differentiating roar from roar they seem to possess the ability to count how many lions are near. While we cannot assume that they know exact numbers, surely this indicates an understanding of the concept of more and less? Does this indicate that lions are aware enough to be able to have this fundamental mathematical skill? I feel that there is a possibility in this case and I find it to be quite convincing. The case study can be read here and I highly recommend reading this interesting and quite amazing research-

Click to access Roaring_and_numerical_assessment_in_contests_between_groups_of_female_lions.pdf

Now these are just a few examples of situations in which animals have been believed to have counting abilities. I believe that animals do have some concept of mathematics. It may not be complex calculus but i feel that the underlying skills of fundamental mathematics are present somewhere in their minds. There are so many avenues that can be explored into this area of research, something I think I may look into in the future. But what do you think? Can your pet dog understand time? Does a sheep in a field know how many are in it’s flock? Can animals count?


McComb. K, Packer. C, Pusey. A, (1994), Roaring and Numerical Assessment in contests between groups of female lions, Panthera leo, available at:

https://www.cbs.umn.edu/sites/cbs.umn.edu/files/public/downloads/Roaring_and_numerical_assessment_in_contests_between_groups_of_female_lions.pdf : Accessed (16/10/17)

Why Discovering Mathematics?

For my second year elective in my MA Primary Education course I have chosen Discovering Mathematics, a module highly recommended to me be older MA students. However this was not my sole reason for selecting the module.

During high school, my favourite subject was maths. While my friends were all creative individuals, I seemed to have a more mathematical mind. I would happily sit through my algebra homework as they compiled art folios with ease. However, in schools I witnessed a staggering number of cases in which children were claiming to not understand maths, therefore closing their minds completely to the idea of them ever being able to in the future. Arem (2009) suggests that it is easy for pupils to get overwhelmed when working in maths causing them and others to believe they are in the early stages of developing maths anxiety, a term coined to describe the fear many possess when it comes to dealing with mathematics.

I do believe that this is something that can be overcome as I am a prime example. I did not enjoy or feel very capable when doing maths in primary school. I hated certain topics with a passion and upon starting secondary school was placed in a medium to low level ability class. However, when my high school teacher began our maths course something clicked. I was able to understand things due to the way she explained them and progressed to the top class by the end of the year. It made me aware that a different style of teaching was necessary to get my out of the seemingly endless cycle of ‘I just can’t do this’.

This is why I have chosen this module. I want to learn different methods of teaching maths, different ways of explaining topics as that is what each individual child is- different. In order to appeal to each and every learner when teaching maths, I need to be able to teach in various different methods and be able to explain things in a variety of ways to aid all learners in mathematical thinking. I want to help guide those who believe they hate and are incapable of doing maths into seeing that everyone can do it and that it can be done in fun and active ways. Over the course of the module I hope to gain as much experience in active maths teaching as I can so that for my future classes, maths anxiety is a thing of the past.


Arem, C, (2009), Conquering Maths Anxiety, 3rd Edition, Canada: Cenage Learning Inc.

Moment of Reflection- Semester 1

In semester one,  ‘Working Together’ was one of the studied modules. This module highlighted the importance of collaboration within the workplace and taught of the professions that need to work together within a school environment to benefit those within. As part of the course we were put into set groups ( mix of Education, Social work and CLD students) with which we had to work weekly, go on a visit and produce a group presentation.

The day of our assessed group presentation stood out for me in terms of the module as a whole. As a team, we met early to go over our talk and notes. We encouraged and supported each other throughout the morning turning a somewhat nerve-wracking experience into a rather enjoyable one. There was a prominent enthusiastic atmosphere surrounding the group helping put us at ease.  It was this day that I realised how much our group had progressed.

In the beginning, our group started off quietly. We were all first year students in a new module with new people so were feeling slightly nervous. As the weeks went by, we continued to work together, learn from each other and bond as a working group. The progress we made from the start was astounding and I realised that we all felt so much more engaged in our work when we had a team atmosphere in which to complete it.

From this experience, I have learned about the importance of having confidence to talk and engage with new people. Perhaps if we had been more social earlier in the project we may have produced better work due to team ethic. However, by the end we were a well integrated group who worked together extremely well, gaining an A5 for our group presentation. I feel our collaborative practice was furthered thanks to reading such as  Huxham and Siv Vangen 2005 whose work emphasises the importance and necessity of it.

I also feel that I have made progress when it comes to delivering a talk. The idea of standing up in front of people and presenting something is somewhat daunting making many people ( yes, even teaching students) anxious and uncomfortable. The support from my group gave me the confidence to be vocal and an active participant throughout the assessment as we completed it as a unit.

While I feel this group presentation did help a great deal with my confidence, there is a part of my that is still reluctant to comfortably speak out in front of people. I should start to resolve this issue slowly by offering more answers during lectures and workshops. I have started to speak out more in this way especially in group 1 workshops as I am starting to feel more comfortable with the group as we spend more time together. My next step will be to offer more suggestions in lectures to ensure that I often speak in front of large amounts of people until it becomes a normal occurrence for me. This will benefit my greatly in the long run especially concerning my upcoming placement.

Reference to-

Managing to collaborate: the theory and practice of collaborative advantageChris Huxham, Siv Vangen 2005