# Maths is all around us!

What is Mathematics? It seems like a simple enough question, but is it?

On the surface, it’s obvious that maths is all to do with numbers. But delve a little deeper, and maths is revealed to be much more complex. In fact, it’s actually something that’s all around us and involved in almost everything we do. It’s in nature, art, technology, it’s the language of science, it’s in music and dance; the list is endless!

Ernest (2000) states that it is important to appreciate “the role of mathematics in life and work, the importance of mathematics in commerce, economics (such as the stock market), telecommunications, information and communication technology, and the role it plays in representing, coding and displaying information. Also how mathematics is forever becoming more central to, but also more deeply and invisibly embedded in, all aspects of our daily life and experience.”

I stumbled across this video and feel it sums up how integral mathematics is to our lives and within wider society.

References

Ernest, P. (2000) ‘Why teach mathematics?’ In Bramall, S. and White, J. (editors). Why Learn Maths? (pp.44-47). London: Bedford Way Papers.

The Beauty of Mathematics: available at: https://vimeo.com/77330591

# The Base 10 Place Value System

For as long as humans have been around, we have been counting things and looking for ways to keep track and represent the things that we count. The Ishango Bone is a great early example of this, but over time our method for counting and tracking numbers has evolved into a number system composed of ten numerals – 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 – which we know today as the base 10 system.

Understanding place value helps us determine the value of a numeral based on its position within a number, and since our number system is based on the idea of 10, to fully understand place value, children need to see how the pattern of ten repeats within numbers and how it is used to build numbers.

Children begin with rote counting, 1, 2, 3… From there they get to two digit numbers, 11, 12, 13 and to three digit numbers 100, 101, 102… And so to a child, the 1 in 1, 10 and 100 often means the same thing. However, in place value, a 1 is one, a 10 is 1 group of ten, 100, is ten tens or 1 group of 100.

This concept can be quite difficult to grasp, but base ten blocks can help children visualise place value a little better. The cubes represent one unit, strips represent ten units, flats represent 100 units and blocks represent 1,000 units. Children can easily see that 10 cubes fit into a 10 strip, 10 strips fit into the 100 flat and 10 100 flats fit into the 1000 block. Another way to understand the base 10 system is through the use of our fingers. We start with ones, being our fingers, so our counting units so to speak. Then we go to tens, being sets of fingers. Then we go to ten sets of sets of fingers, which is 10×10=100, so we go to hundreds. In other words, every time we go one place further to the left, that is, every time we go into a unit that is one times bigger than the previous place’s unit, we multiply by our base of ten.

Confused yet? Because I think I am! It’s easy to see why children struggle with this concept, but it’s a fundamental part of mathematics. Having a sound understanding of place value and our base 10 system are basic concepts that children need to be aware of, otherwise future learning will be compromised. That’s why it’s important to provide children with multiple perspectives, like the two methods mentioned above, to help highlight that there is more than one way to approach any given problem.

References

Bellos, A. and Riley, A. (2011) Alex’s Adventures in Numberland. London: Bloomsbury Publishing PLC.

‘Definitions of Base 10’ available at: http://math.about.com/od/glossaryofterms/g/Definition-Of-Base-10.htm

# The History of Mathematics

Mathematics stretches well into prehistoric history and knowledge of its origins and initial applications is mostly built on speculation. Because of this, it’s reasonable to assume that our prehistoric ancestors would have had a general understanding of amounts, and would have instinctively known the difference between the quantity of one thing and the quantity of two things. Basic counting like this would have likely been facilitated by the use of our hands, with each finger representing a numeral.

This would have worked well for small quantities, but how did our prehistoric ancestors count quantities bigger than ten? Evidence shows that they began using notches or marks to represent numerals. In doing so, they not only created one of the first mathematical records, but they also created the first known method of tracking larger numbers.

The Ishango Bone, discovered around 1960 on the border of Zaire and Uganda by the Belgian geologist Jean de Heinzelin, is considered by many to be the earliest mathematic artefact ever recovered. The origins of this artefact date to approximately 20,000 B.C.E. and consists of a series of tally marks carved in three rows running the length of the bone.

The bone itself has been subject to a lot of interpretation.  At first, it was thought to be just a tally stick with a series of tally marks, but scientists have demonstrated that the groupings of tallies on the bone are indicative of a mathematical understanding which goes beyond simple counting.  In fact, many believe that the tallies follow a mathematical succession, with some going as far as to say it shows the earliest known demonstration of the prime number sequence.

The first row (a) can be divided into 4 groups, where each group has 9, 19, 21, and 11 tallies.  The sum of these 4 numbers is 60. The second row (b) can be divided in 4 groups, with each group possessing 19, 17, 13, and 11 tallies. These are the 4 successive prime numbers between 10 and 20.  This constitutes a quad of prime numbers. The sum of these is 60. The third row (c) can be divided into groups of 8 and features two sets of numbers (3 & 6 and 4 & 8) that may represent multiplication by 2, and one set of numbers (10 & 5) that relates to division by 2. Lastly, the first row (a) and second row (b) both add to 60, while the third row adds to 48. Both 48 and 60 are divisible by 12.

These mathematical trends suggest some basic understanding in terms of grouping numbers, but some people think they are not significant enough. The argument is strengthened by the absence of any other archaeological discoveries that indicate an understanding of the prime number sequence dating from that time period. However, despite the considerable debate surrounding the actual implications of the Ishango Bone and the fact we’ll never really know what these tallies represent, I’d like to believe that the tally marks are evidence of the first primitive counting tool.

The notion of humans understanding mathematics can also be illustrated in the Lascaux cave paintings in France.

Some of the markings indicate a basic human understanding of the 29-day lunar cycle. In one case, there are 14 dots followed by an empty square, which experts believe represent the 14 days that the moon is visible in the night sky and the day that it disappears. This suggests that part of our early mathematical understanding of counting derived in part from observational astronomy.

And so, whether it be counting or making sense of objects in the sky, it’s clear that early mathematics developed as a result of our ancestors’ need to understand and make sense of the world around them. And although the mathematics used today is vastly different, the same necessity still stands.

Drawing on history in this way can show children the need for mathematics and links in well with the four principles of PUFM. Being able to see the connections between the tally marks and their potential purpose helps us to understand and make sense of how our basic foundation for counting and recording has developed. Also realising that there’s no one right answer in terms of knowing what these tally marks or cave paintings mean highlights the need for multiple perspectives in terms of fostering a more flexible way of thinking.

References

‘A History of Mathematics’ available at: http://www.storyofmathematics.com/

# Profound Understanding of Fundamental Mathematics

During our first input, we were introduced to Liping Ma’s concept of profound understanding of fundamental mathematics (PUFM) and its importance in terms of enhancing teachers’ knowledge of, and ability to better teach, elementary mathematics.

Ma states that to “fully promote mathematics learning, teachers must first have a profound understanding of fundamental mathematics”. But having a PUFM is more than being able to understand elementary mathematics, it’s about being aware of the theoretical structure and basic attitudes of mathematics that are within elementary mathematics, and being able to use this as a foundation for which to instil the same awareness in the children that we teach. It’s important to have both a conceptual and procedural understanding – to know how and why we do something – that is deep, broad and thorough.

Ma identified four principles that are essential to gaining PUFM:

(Inter) connectedness: refers to being able to see connections between concepts and procedures. These connections allow for the use of prior learning and knowledge to be applied to new mathematical situations. This helps to ensure learning is not fragmented, but viewed instead as a unified body of knowledge.

Multiple perspectives: refers to seeing and appreciating different approaches to solving a problem. This encourages a much more flexible way of thinking as it is not restricted to any specific learning style.

Basic concepts: refers to being aware of the basic concepts within mathematics. It is important that these central ideas are revisited and reinforced as they provide the foundation upon which future concepts are learned.

Longitudinal coherence: refers to being aware of the entire mathematics curriculum and how this can be used to link previously learned knowledge with newly acquired knowledge (one basic idea/concept builds on another). This allows for there to be much more understanding and flexibility in terms of where learning is headed as lessons can be tailored with this in mind.

While I’m still getting to grips with what PUFM really means for me as a teacher, it’s amazing how being made aware of these four inter-related principles has really changed the way I look at mathematics, and will definitely influence how I teach mathematics in the future.

References

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

An interview with Liping Ma: https://www.eduplace.com/intervention/knowingmath/liping_ma/conversation.html

# Discovering Mathematics

Maths is a subject that I never really had much confidence in, and thinking back to when I was a child, I remember feeling such pressure to keep up with those who managed to grasp concepts that I found myself so often struggling to get my head around. I feared the thought of failure more than anything else and convinced myself that I was just one of those people who couldn’t do maths.

Now in my early twenties, I still feel the same way. During my placement in first year the thought of teaching maths filled me with such dread. I often found myself doubting my ability when explaining even the simplest of concepts. This made me realise that to teach maths successfully, I needed to have a deep, broad and thorough knowledge and understanding of maths, and by extension, the ability to explain concepts in a meaningful way. This meant going back to basics and trying to pinpoint the reasons for my lack of confidence in maths.

I chose this module as my elective for this exact reason; Discovering Mathematics is designed to help explore, reflect and critically examine various topics in fundamental mathematics and enhance subject knowledge.

I hope this module helps me to become a bit more confident in my ability to teach maths so I can be a better teacher, and one who has children that are more confident in their ability to learn maths.

# The Virtues of Teaching

In a recent input, we were asked what attributes we think teachers should possess in order to be regarded as both a professional and great practitioner. It’s difficult to pin-point just a handful of qualities, but here are a few that I think are important.

I think having integrity is something we as teachers definitely need. To me this means being honest and having strong moral principles. It’s important to be a good role model and someone who can show children what it means to have values. Being a person of integrity also means that my personal and professional relationships are genuine and I am a person who can be trusted, which is important considering children develop in the context of relationships.

I think having compassion is also important. Defined as “the sympathetic consciousness of others’ distress together with the desire to alleviate it”, we as teachers spend a great deal of time helping children understand and overcome issues large and small. Taking into consideration the thoughts and feelings of our children instead of solely responding to their words is great way to build strong relationships. For example, angry words may conceal fear, guilt may hide behind blame. Trying to understand and respond to the underlying emotion can result in the child being more likely to trust and open up a little more.

I think having fairness within the classroom is a definite must. Many define being ‘fair’ as treating everyone the same, but I would argue against this. Children are not the same. They have different motivations for their choices, different needs, different causes for misbehaviour and different goals. That being said, I do think having set boundaries and rules that apply to everyone gives each individual child a clear grasp of what’s expected (and not expected) and the consequences of their actions. But it’s vital to have an unbiased approach in teaching and I think making a real conscious effort not have ‘favourites’ is important.

I think having patience is a quality that we as teacher most definitely need in order to make this job possible! To deal with twenty odd children demanding your undivided attention can be a real task to juggle. Every child has different needs and varying ranges of ability and being able to take a step back and evaluate a situation is essential to staying cool, calm and collected.

And lastly, I think having respect is important in regards to all aspects of life, not only as a teacher. It shows that one values another as an individual and that they have regard for the feelings, wishes, or rights of others. Children judge the character of us as teachers based on how we treat them. Respect needs to be earned from both sides. When a child experiences respect, they know what it feels like and begin to understand how important it is. That’s why the best way to teach respect is to show respect.

# Plato’s Allegory of The Cave

What is reality? This is a question that Plato explored in his book, The Republic, by examining concepts such as truth and justice.

In his short story, he imagined a group of people born in the darkness of a cave, chained and facing away from the entrance with only reflections and noises made from people passing by outside the cave as their experience. They had no knowledge of the outside world. For them, the cave was their reality.

However, one day one of the people escape and set foot outside the cave. For the first time in his life he sees the outside world and is exposed to the true reality of life outside his own. It takes some time for him to adjust – to the revelation / enlightenment – because he can only understand what he has previously experienced. Driven by this new reality however, he tries to convince the rest of the chained people within the cave of what he has seen, but they don’t believe him. The people in the cave have had an intrinsic education which has formed the basis for their understanding of the world.

The cave leads to many fundamental questions such as: What is the origin of knowledge? What are the problems of representation? What is the nature of reality itself?

Plato believed that education was the answer to these questions and the key to living a true and just life. He further believed that because of this, education should be provided by the state, something which the Curriculum for Excellence is centred upon; this idea of citizenship and creating well rounded people who are just. Through education it can be ensured that children become conditioned to the values that are accepted, in Plato’s view.

And so the story provides a clear explanation of the difference between experience and reality. It shows that the experience of reality is conditioned on the experience of life. So in simple terms, the way you live your life is your reality.

This highlights the importance for teachers to allow children to explore and learn from experiences by exposing them to new ones, and pushing them out of their comfort zones and into the world of the unknown.

# We don’t learn, we remember…

I think it’s fair to say that the recent inputs from John on the philosophy of education have left me feeling a little bit confused. However, that being said, I feel I now have a very basic understanding of some of the key theories concerning the ways in which we learn and how these can be used to improve the ways in which we teach.

The Greek philosopher, Plato, believed that we don’t learn, we remember. This is because he believed that the inner part of us – known as the soul – doesn’t change and is immortal in the sense that it has been reborn, time and again. Because of this, it has seen things in both the ‘material’ world and the ‘real’ world.

Our world, the ‘material’ world, is constantly changing and we rely on our senses to understand what’s going on. The ‘real’ world is outside of this realm and is unchanging and eternal. We understand this world not with our senses but instead with our mind.

The soul is said however, to be captivated by the workings of the body (i.e. our senses) and so we struggle to see past the illusion of our current world. Plato argued that in order to realise the ultimate reality of nature, we need to think independently of our senses, as the ‘material’ world is simply a shadow of the ‘real’ world of forms.

Plato believed that all things have a true being, a concept he explained in his Theory of the Forms.

Take apples for example:

• What do they have in common?
• What makes them distinctly ‘apple’?
• What gives them their ‘appleness’?

Plato stated that there was only one form– or essence – of ‘appleness’ and that this forms the many. The form itself is unchanging and perfect (because it is from the ‘real’ world), but the apples (from the ‘material’ world) are simply an appearance of the form, which explains why they change and are imperfect.

We recognise these forms because we have a faint memory of them from our prior existence. It is this process of remembering the true reality of our world that allows us to learn, and therefore know that an apple is an apple, along with every other thing we interact with.

I think this is a really interesting concept and perhaps quite important in terms of our approach to teaching. This idea that we all have the prior knowledge there ready and waiting to be unlocked highlights the need for us as teachers to be willing to persevere with those kids who might be struggling. We need to help them rediscover what they already know.

# Gender Role Stereotyping

I never really experienced a great deal of gender role stereotyping when I was at school, or at least I like to think I didn’t. But a recent input from Jill has led me to think otherwise.

Looking back, I remember small things, like being told to line up separately or sit boy-girl. In the playground, most of the boys played football while the girls played hop-scotch or made up games of their own.  At the time, this didn’t feel wrong, but it’s only now that I realise the potentially negative implications of such behaviour in terms of gender stereotyping.

This led me to think about the way in which gender is constructed within the classroom and just how big of an impact it has on the learning environment. That’s why I think it’s so important to be aware of this as a teacher, because these stereotyped ideas about what’s suitable for boys and girls can limit children’s opportunities to learn and develop.

Take being lined up separately for example. It’s thought by some that boys get up to too much mischief if they’re left to their own devices and that the presence of girls have a positive, calming effect on their behaviour. While perhaps true on a certain level (although I’m not entirely sure I agree), it affirms this idea that boys and girls should be treated differently, which I think is wrong.

And I think this is one of the major challenges that teachers face – establishing environments both inside and outside of the classroom that don’t favour one group of students to the detriment of another group – and is definitely something that we need to tackle on a bigger scale.

At the beginning of the semester, we had an input from Susan on the sociology of education. Its aim was to aid our understanding of education systems and create awareness of social context and diversity within primary schools.

The video below is a recording of a lecture given by Sir Ken Robinson, a speaker and international advisor of education, on the subject of education paradigms. Robinson talks about how the current educational systems of the world are flawed and need to go through some major changes.

One of the main problems is that the current systems of education were designed during the industrial revolution years, which focused primarily on academic performance. We no longer live in such a society, so why then, in the words of Robinson, are we “trying to meet the future by doing what [we] did in the past?” It seems nonsensical considering our technologically advancements and the various new and exciting outlets for learning. It makes me question why our education systems have failed to follow the trend.

Which leads onto another issue affecting our current education system; the fact that it generally only benefits those children who fall under the category of academically intelligent. Those who aren’t as academically intelligent are forgotten about in a sense, which results in them losing interest and falling away from their studies.

This taps into an interesting point Robinson makes about Divergent Thinking (a thought process or method used to generate creative ideas by exploring many possible solutions) and how young children have an almost natural ability to think about various different concepts, but as they grow older and become ‘educated’, it begins to fade.

So how then are we as teachers meant to raise attainment if the systems put in place fail our children? The answer is, we can’t. Not unless we take a step back and rethink these traditional approaches to learning to ensure no child is left behind.

That way, we will be able to ensure we are flexible to the needs of our children, instead of stifling them with outdated systems of education.

# Smarter Study Skills

Trying to understand what university involves is a daunting process for most first time students. In those first few weeks, a lot is happening and it can be a somewhat overwhelming experience. ‘The Study Skills Book’ is a great reference manual to help with this process; it provides a simple yet comprehensive insight into all of the different aspects of student life, along with various hints and tips on how to make the most out of university.

Before the book begins to delve into the details of what to expect from university, it asks the reader to think about where they see themselves in five and then ten years’ time. This made me realise that, in the space of a short (and what already feels like will be a quick) four year degree, I’ll be starting out on my first year as a fully qualified teacher.

One of the main differences between secondary school and university is this concept of students being fully responsible for their own learning, and this book provides many useful hints and tips on how to cope with the transition, such as:

• Goal setting – making short term goals will make the long term goals much more achievable
• Communication with tutors / staff members – If I have an question or issue, speaking to the relevant people will ensure there is a quick resolution
• Organisation – essential for keeping on top of course work and meeting deadlines, ensures effective time management, and having a balance between work and play
• Learning new skills – being prepared to improve on existing skills, as well as those that are unfamiliar, and being open to the new ideas and concepts of learning
• Looking after yourself – going to university is a big step and sometimes all you need is a little advice with how to cope with certain difficulties, but there are many services available ranging from academic skills advice to a counselling service

The book also details differences across other areas such as attendance, teaching strategies, learning requirements, and written work requirements. These are set out in a table while allows the reader to see the main differences at a quick glance.

These differences run parallel to the differences I’ll probably experience once I graduate from university and begin to venture out into the world of employment. The skills I develop now will stay with me throughout my life and the book highlights their importance to employers.

Working on these skills will make it easier to see the areas where I feel most competent, the areas I feel need more work, and the areas where I feel I need to dedicate time to the development of new skills.

# Why teaching?

I can’t really remember ever wanting to pursue anything other than teaching, and I’m sure if you were to go back in time and ask five year old me, I would say the same! I loved learning as a child, especially in primary school. My teacher(s) were a big reason for this, as was my mum, and as cliché as it sounds, I wouldn’t be where I am today if it wasn’t for them. So in a way, I guess part of the reason I’m so interested in teaching stems from wanting to give children a similar experience to the one I had as a child.

Teaching is such an incredibly rewarding profession, and I think children actually teach us as much as we teach them. Every day is different, and although challenging, packed full of fun. I can’t wait for that moment of realisation on a child’s face when they finally ‘get it’.

I hope to become a teacher who has children excited for what their school day has to offer, to have children inspired and eager to learn. I want to ensure they feel supported and challenged. But most of all, I hope to have children who have as fond memories as I do of my time at school.