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.

reference

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.

Christmas in Mathematics – ideas for the classroom

Its that time of year again when the lights are up, songs are being sung, decorations are being hung and joy is absolutely everywhere. What is this time you ask? it is indeed Christmas. It is obvious that Christmas is a time for giving and thinking of loved ones, but after a recent blog post from Andy Hughes (titled ‘Christmaths’), I have been inspired to blog about how maths can be incorporated into Christmas in fun and inventive ways. In addition, it has made me want to uncover the fundamental principles of mathematics in Christmas.

Hughes (2015) in his blog post ‘Christmaths’ describes mathematics in Christmas. He states that  snowflake’s are a ‘fractal (a fractal is a shape that contains similar patterns which recur throughout the shape on a smaller scale)’. In addition he identifies that zooming in on a snowflake just produces more details and can splurge up fundamental processes such as scaling, shape and ratio – very fundamental processes. On a larger scale, lets take the fundamental process of shape. As I have mentioned numerous times before, maths is all around us. In a seasonal sense, we only need to look at our Christmas trees to see that shape is evident. For example, Christmas baubles. These are spherical in shape. The star on top of the tree also shines bright, but is still a shape. However, what is the maths behind a Christmas tree? It could be suggested that there is maths in assembling a Christmas tree (especially and artificial one. For example, you will need to have the right length of Christmas tinsel to pass around the tree – the mathematical concept of length and measurement. In addition, you will need to have enough Christmas baubles in order for the tree to be symmetrical. Talking about symmetry, this can be used widely as examples to help children understand this concept. For example, trees are symmetrical, as are snowmen, stars, snowflakes.. the list goes on. This can be used as a consolidation activity and it effectively ties Christmas into mathematics.

Moreover, Christmas time in the classroom should be an exciting and engaging time for children. So why can’t we use maths within Christmas? Even if it is not directly from a maths lesson. Board games are a fantastic way to do this and they challenge a range of mathematical concepts. Take monopoly, for example. Now, I am not going to lecture you on how to play monopoly as I am sure we all have a ‘profound understanding’ of how to play it. However, what does monopoly all boil down to? Money. In order to play this game, children must have a basic knowledge of money and will have to subtract or add together money when they land on someone’s property or even if they pass go! I find this to be pretty incredible – its fun and engaging, but you are learning at the same time… Do you see where I am going with this? Take snakes and ladders – counting is the fundamental principle there. Cluedo – problem solving to find out who really committed the crime. Even in dominos there is a basic element of counting. Probability is also another fundamental principle that can be seen in games that use dice – what is the probability of  landing on a six? Lets be honest, which games do not have dice in them?

It is obviously apparent that maths is incorporated into Christmas profoundly. If you even think about presents under your tree – some are cuboidal and some are spherical. Shape is such a common theme within Christmas. So when your opening your presents this year, have a second thought about how maths is incorporated into Christmas – it may even inspire you to write a blog post!!!!

reference:

Hughes, A. (2015) Christmaths. In a Class of my Own. Last accessed 06/12/15. Available at https://blogs.glowscotland.org.uk/glowblogs/ajhportfolio/2015/12/02/christmaths/

Mathematics in Demand Planning – Show me the Money!!!!

As mentioned in my previous blog posts, mathematics is incorporated in to absolutely everything in life. This idea was further embedded during a lecture on supply chain and logistics in maths. It was a three hour input, but It was honestly the best maths class to date. It was completely relevant, interactive and so much fun. This blog post aims to draw on points about fundamental mathematics on previous blog posts and how it is incorporated into demand planning.

Firstly, when we talk about supply chain and logistic planning, what springs to mind is food. Apart from the odd rumble of my stomach, there is actually some mathematical concept behind this all. lets take crisps for example. If we think about the bar code, it has a significant amount of numbers around it (see image below).

Within the red circle, is a best before date 06/11/10. The number to the side says 06:51 which would highlight the time where the big packet (if it is a multipack) was closed. The number code under 06:51 is 275 which means which day within the year it has been produced. These timings are absolutely crucial for the company to give an accurate sell by date which will prevent their customers from getting food poisoning. The fundamental mathematics behind this is predicting by using the numbers to provide an accurate time scale for consumption. To an extent, the fundamental mathematics behind it could be knowing the date and the days of the year. Furthermore, each packet of crisp has a certain weight which could identify that another aspect of fundamental mathematics is weight. Even when the small packets are being placed into the big multi-pack bag, there are still a specific number which is out in each bag. This takes the art of crisp making and manufacturing down to being able to count. Fundamentally, that is astounding. Come to think about it, think of all the activities that you could bring in to the classroom around crisps. Potentially you could go down the rout of how the crisps are made, what machinery is used to manufacture crisps and how they are programmed.. you could even take a stab at making your own crisps!! Of course, maths comes into every single one of those activities.

I will say this again, ‘MATHS YOU HAVE AMAZED ME!!!’  in addition, you actually do this within the context of the class because I feel that this is so beneficial for children to understand and so they can continue there love for mathematics.

Life in Poetry

Poetry is a word that can bring someone a sense of fear depending on the poem. I, for example, are one of those many people that remember dissecting and analysing poems in high school for months on end to then find out that that specific poem didn’t appear in the final exam. Pointless and time consuming? Perhaps. However, after thinking about poetry on a vast scale, outwith the environment of the high school classroom, it is apparent that poetry is a fun and memorable thing to do within the context of a language lesson. So why poetry and why incorporate it into a language lesson. It could be suggested that poetry breaths life into language, and I hope to convey this within the context of this blog post.

Firstly, it could be argued that people’s love for poetry dies when they reach high school level. This is down to the fact that poems are over analysed and deciphered to a point that it becomes a tedious task. This feeling was always at the forefront of my mind in high school and it completely sucked the life out of the poetry that I was trying to learn. However, I loved poetry in primary school, especially Robert Burns. Every January my school had a recital competition which required everyone to learn a Robert Burns poem. The person who won received a certificate and an immense sense of pride in memorising Burn’s tricky poems in Scots tongue. Competitions like this have stood the test of time and are still apparent in schools all over Scotland. This, along with this afternoons poetry lecture, allowed me to see a different side to the ‘frightening’ concept of poetry that I learned to loath.

So what is poetry? Carter (2012) describes his interpretation of what poetry is:

• it is a box of spoken words
• a spoken song
• a fresh way of looking at something
• language at its most playful
• music and meaning
• repetition
• the most fun you can have with language

I found Carter’s interpretation of Poetry to be quite intriguing. I never knew that poetry could be related to music. However, after thinking about it, poetry could actually be made into a song, or it is music without the melody or instruments. It does have rhythm, however, and you could almost imagine the beat and the rhythm in your head. This is good for children because it can help them grasp the concept of rhythm and beat and this could be an activity to do with the children (clapping and singing to the poem). In addition, poetry is much more achievable to perform and write than writing an actual story. The children can play with the language and delve into it and this, to me, seems much more enjoyable than writing a descriptive and detailed story. The fact that poems come in many shapes and forms allows the children to be more creative and structure their own poems in their own ways. The only thing, however, is that the children need to have a good idea of language to convey emotions and meanings because the space in a poem is very short. However, poems do not have to be complicated pieces of text. For children, poems need to be short, chunks of text in order for the children to understand and comprehend the meaning of the poem. However, some style of poems like sonnets may require a bit more work (especially Shakespeare). This is because the language is very complex and this could be difficult for children.

Poetry should display a number of key characteristics (Cremin, 2009):

• Sound Effects – including repetition, alliteration, onomatopoeia, rhythm and rhyme
• Visual Effects – including similes, personification and metaphors
• Powerful vocabulary in order to convey emotion and passion
• surprising word combinations in order to encourage discussion
• repetition

This is very beneficial for the language development of children because they are learning new skills and this will convey their understanding of the poem and make it exciting and fun. Effectively, poetry should demonstrate the playfulness and musicality of language so children can enjoy and find poetry fun. In addition, poems encourage children to explore feelings which allow them to develop empathy and self-awareness. Ultimately, children can explore the emotions within the context of a certain poem and they can feel attached and sad towards a specific character in the poem. In addition, the children are picking up key terminology like stanza and metre which is good for their understanding of poetry because they become more aware of structure and can contribute well to discussions about certain poems.

In conclusion, poetry needs to be playful. In early to middle years, poetry must be enjoyable and the children should have the opportunity to play with the language and the nonsense of poems. In the upper years, the children can delve into why specific language has been used in certain poems and they should experiment using their own language in their own poems. Secondary school pupils should be able to deconstruct and analyse poems. However, I have rekindled my love for poetry within the primary school context. It all links back to active learning and the children having fun and enjoying learning about poems.

here is my favourite poem from my childhood – enjoy!!!!!

Crocodile

by J.K. Annand

When doukin in the River Nile
I met a muckle crocodile. He flicked his tail, he blinked his ee,
Syne bared his ugsome teeth at me.

Says I, “I never saw the like.
Cleaning your teeth maun be a fyke!
What sort of besom do ye hae
To brush a set o teeth like thae?”

The crocodile said, “Nane ava.
I never brush my teeth at aa!
A wee bird redds them up, ye see,
And saves me monie a dentist’s fee.”

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

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.