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.

References-

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

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

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.

References-

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

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.

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!

References-

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

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.

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.

Reference-

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