# Profound Understanding of Fundamental Mathematics

If someone asked me two months ago what ‘profound understanding of fundamental mathematics’ was, all I could tell you it was something to do with maths. PUFM sounds very confusing and complicated, when in fact it is quite the opposite. Over the course of this module, my understanding of this statement has developed greatly, and I now feel much more confident in my abilities of teaching in a classroom setting.

Ma (2010) defines PUFM as “an understanding of the terrain of fundamental mathematics that Is deep, broad and thorough.” There are four principles in teaching and learning that represent a teachers understanding of fundamental maths in the classroom: interconnectedness, multiple perspectives, basic ideas and longitudinal coherence.

Interconnectedness –

Ma (2010) defines interconnectedness as when “a teacher with PUFM has a general intention to make connections among mathematical concepts and procedures.” Interconnectedness occurs when the learner is trying to make a connection between the mathematical concepts and procedures. Here, it is the teachers job to stop the learning from being fragmented, and help the students to develop the ability to make links between underlying mathematical concepts, and showing how maths topics rely on each other.

Multiple Perspectives –

“Those who have achieved PUFM appreciate different facets of an idea and various approaches to a solution, as well as their advantages and disadvantages” (Ma, 2010). Multiple perspectives is when a learner can approach mathematical problems in many ways, as they understand the various perspectives when taking on a maths questions, and are able to look at the pros and cons of the different viewpoints. It is the teachers job to provide opportunities for the learners to be able to think flexibly about the thinking and understanding of different concepts in maths.

Basic Ideas –

MA (2010) says that “teachers with PUFM display mathematical attitudes and are particularly aware of the ‘Simple but powerful basic concepts of mathematics’ (e.g. the idea of an equation.”  Basic ideas is a way of thinking about maths in terms of equations. Ma (2010) has suggested that learners should be guided to conduct ‘real’ maths activities rather than just approaching the problem when practising the property of basic ideas. If a teacher is implementing this principle effectively, then they will not only be motivating the learner to approach the maths problems, but will also be providing a guide to for the learners to help them understand the maths themselves.

Longitudinal Coherence –

“Teachers with PUFM are not limited to the knowledge that should be taught in a certain grade; rather they have achieved a fundamental understanding of the whole elementary mathematics curriculum” (Ma, 2010). Longitudinal coherence is when a learner recognises that each basic idea builds on each other. When a learner does not have a limit of knowledge, it is impossible to identify the level or stage that a learner is working at in maths. the learner has instead achieved a holistic understanding of maths (fundamental). A teacher who has a profound understanding of maths is one who can identify the learning that has previously been obtained. The teacher will then lay the fundamental maths as a foundation for learning later on (Ma, 2010).

The four principles are vital to having a deep understanding of maths, especially when teaching the subject. I will make sure I have PUFM before I begin teaching mathematics, as without it, I cannot teach the subject as in depth and as thoroughly as possible.

References:

Ma, L. (2010). Knowing and Teaching Elementary Mathematics. New York: Routledge.

# Maths, Play and Stories

How could maths, play and stories intertwine? How could a child possibly be learning maths whilst playing with their friends, or getting read their bed time story? Maths is all around us, even where we least expect it.

Parents as Teachers

Parents may think the only way they can engage their children at home with maths is by helping them with their homework. Some parents may not even be able to help their children with their homework due to the maths anxiety that many adults suffer. As I have spoken about in a previous blog post on maths anxiety, the negative connotations that parents have about maths, can easily be passed down to the child, and create them to have their own worries about maths. It is vital that parents provide a rich learning environment for mathematics at home, so that children can be learning at all times and reaching their full potential. Studies looking at how often numeracy activities occurred at home found that they often happened less than once a day. They also found that the activities rated by experts as having specific numeracy focus occurred more infrequently than the activities with ambiguous numeracy content (Skwarchuk, 2009). Much of a child’s early mathematical development is enhanced through communication. This is why parent should ask open-ended questions to support and challenge their child’s thinking. For example, using varied mathematical language like bigger, smaller, fewer etc.

Importance of Play

Lev Vygotsky says that the learning of a child takes place in the ‘zone of proximal development’ which represents the between what a child actually knows and what the child can learn with support from those who are more knowledgeable. He also believed that the teaching of maths can be influenced by relating the subject to the child’s own experiences. This helps us to understand why play is such an important aspect of learning maths, as it allows the child to personalise it and be able to relate it to themselves. Saracho (1986, cited in Saracho and Spodek, 2003, p.77) explains that when children play, they confront social circumstances and learn to collaborate,  help,  share,  and  resolve  social  difficulties. Play is extremely important to a child’s learning for some of the following reasons:

• Helps children to make connections in their learning
• Allows child to experiment
• Provides a meaningful context
• Promotes social learning
• Encourages perseverance

Friedrich Froebel was another theorist that also emphasised the importance of play in children. He viewed play as the ‘work of children’ and believed that children’s best thinking took place during play. During quality play, children are:

• making decisions
• imagining
• reasoning
• predicting
• planning
• experimenting with strategies
• recording

Susan Issacs had many of the same views as Froebel. She saw the value in play as a way to allow children to explore their ideas and feelings freely. Through play, children can move in and out of reality, and whilst doing this, she encouraged them to be curious and express their feelings. But how does this relate to maths? Nutbrown (1994) said that “mathematics is never far away from young children’s actions.”  All the different things that happen during quality play (as listed above) all link into mathematics. Predicting, experimenting, reasoning, making decisions are all needed in maths. Children using these strategies in play will help them to have a better understanding of the approaches they must use. It will also help them to understand why they are learning maths, as they can relate the problem solving and reasoning to their real-life experiences.

Maths in Stories

Another way that maths can be incorporated into a child’s everyday life is through stories.  Stories are something that are enjoyed by children, and can help eliminate that fear of maths that a lot of children have.  They can be used to introduce new mathematical concepts, or to build on ones that are already known. By showing children that maths can be fun and interactive, they will be much more willing to engage.

Here is an example of a math story book ‘A Place For Zero – A Math Adventure’. This story talks about the number zero and place value.

Stories like this one can help children to connect with, and understand the concept being portrayed in the book. Having a visual aid will allow the children engage in the book, and listen to the story line. At the time, they may not even realise that they are actually learning mathematical strategies.

Maths storybooks are not the only way to teach maths through stories. There are many popular children’s books that can easily be adapted to fit a mathematical story line. For example, ‘We’re Going On a Bear Hunt’ can easily be change to ‘We’re Going On a Square Hunt’. Using a familiar text will allow the child to acknowledge how maths can be related to them, and used in their day to day life. It may also make the experience more comforting, by having something the recognise, especially for those that suffer from maths anxiety. You should always remember to match the book and your discussions to the mathematical abilities and development of the children in your class.

From the workshop and my own research, I now understand the great importance of play in developing a child’s academic understanding, by allowing them to freely explore ideas and express their emotions. Not only in maths, but in every aspect of schooling, play can help children relate it to their own experiences. When working in classrooms in the future, I will make sure to incorporate all that I have learnt about play and stories to make the learning as fun and interesting as possible, as I now know the benefit of them.

References:

Ehrhart, M. (2014) A Place For Zero A Math Adventure [Online].  YouTube. Available at: https://www.youtube.com/watch?v=76–wKA1yYQ&t=487s [Accessed 1st November 2017]

Nutbrown, C. (1994) Threads of Thinking. London: Paul Chapman Publishing Ltd.

Saracho, O.N. and Spodek, B. (2003). Contemporary Perspectives on Play in Early Childhood Education. Conneticut: Information Age Publishing.

Skwarchuk, S. (2009). How Do Parents Support Preschooler’s Numeracy Learning Experiences at Home?. Early Childhood Education Journal, [Online] 37. Available at: https://link.springer.com/article/10.1007/s10643-009-0340-1 [Accessed 30th October 2017]

Valentine, E. (2017) “Maths, Play and Stories” [Powerpoint Presentation] ED21006: Discovering Mathematics [Accessed 31st October 2017]

# Is Maths a Language?

“Is maths a language?”  There are thousands of languages across the world – English, Spanish, German, French… all taught across schools and reinforced as being important to succeed in later life with jobs. So why is this attitude not portrayed with maths?

Math is a language that has been used thousands of years. It is spoken universally, and can be understood by all no matter the age, religion or culture. Sure, different countries may have different symbols or words for aspects of it, but the profound fundamental understanding of maths is the same no matter where you go. Paying for your shopping in a supermarket uses the same knowledge of maths whether you’re paying in pounds, rupees, yen or euros.

Some anthropologists suggest that the global language of maths was needed in order to trade. Many different countries were trading, and were not able to communicate with each other as there was such a wide variety of languages, so a universal language that could be understood by all needed to be implemented. Roman numerals were the most dominant number system used in trade. It was created on the base 10 system but was not directly position and did not include a value for zero (Mastin, 2010). The base 10 system is a system used today in every country, and our understanding of place value is based on this. It is thought that this system was introduced at least as early at 2700 BCE by the Egyptians (Mastin, 2010). This system is used widely and is an understood language across the world, even though it appears to have begun in Egypt.

Europeans were still using Roman numerals in the 13th Century, but found that they were difficult to work with when trying to divide or multiply. This is when Italian mathematician Fibonacci introduced Arabic numerals into Europe. These are the numerals that we know and use today to represent values of numbers. The difficulty of the Roman numerals led to merchants and bankers embracing the simpler Arabic system (Maths Careers, n.d.). This number system eventually spread across the globe, as the inclusion of zero meant that so much more could be done.

Here is a great video from Dr. Randy Palisoc, talking about maths as a language. This video also touches on maths anxiety, and how looking at maths as a language can help to eradicate the anxiety and fear around maths.

References:

Mastin, L. (2010). Egyptian Mathematics. [Online]. Available at: http://www.storyofmathematics.com/egyptian.html [Accessed 23rd October 2017]

Mastin, L. (2010). Roman Mathematics. [Online]. Available at: http://www.storyofmathematics.com/roman.html [Accessed 23rd October 2017]

Maths Careers. (No Date). A Universal Language. [Online] Available at: http://www.mathscareers.org.uk/article/universal-language/ [Accessed 23rd October 2017]

TEDx Talks (2014) Math isn’t hard, it’s a language | Randy Palisoc | TEDxManhattanBeach [Online].  YouTube. Available at:  https://www.youtube.com/watch?v=V6yixyiJcos [Accessed 23rd October 2017]

# Maths Anxiety

I always enjoyed mathematics all the way through Primary, and never once felt anxious about participating in it. Even during the early years of High School, I still had no worries about maths and liked going to classes. This all began to change around about 3rd/4th year, when the maths became harder and more stressful. Trying to memorise formulas just to pass my exams without fully understanding how or why I was learning this is where my maths anxiety stemmed from. This anxiety is still something that I carry with me now.

I decided to choose ‘Discovering Mathematics’ as my second year elective, as I wanted to try and further my understanding of maths and try to get over my maths anxiety. In one of our first few workshops, maths anxiety was one of the topics that was discussed. Hembree (1990, p.45) describes maths anxiety as “a general fear of contact mathematics, including classes, homework and tests.”  This anxiety of maths can have both physical and psychological effects on students. These include headaches, muscle spasms, shortness of breath, dizziness, confusion, mind blanks, incoherent thinking and many more (Arem, 2010, p.30).This anxiety of maths can cause the pupil to become disengaged in their learning, as they lose some self-esteem, and in turn, the anxiety increases. I found it really interesting that maths anxiety can be considered a diagnosable condition, as I always felt that it was just my own fault for not being great at maths.

For most children, this anxiety continues into adulthood and can affect their confidence in tasks such as paying bills and handling their finances. If as kids they did not learn the basic mathematics as a result of maths anxiety, this could potentially affect them for the rest of their lives. This anxiety can also be transferred into their own children, giving them a negative impression of maths. Many parents will be unwilling or unable to help children with their homework, which will also greatly affect the child.

Maths anxiety in teachers also greatly affects the student’s performance. The teachers on the Maths and Science Survey (TIMSS) it is shown that Scottish P5 pupils are scoring below international average, and S2 pupils are scoring well below the international average. Even the highest achieving pupils in Scotland scoring well below the international average, which is very worrying figures for the country (IEA, 2008). As a teacher, I want to try and get over my maths anxiety, after seeing how greatly it can impact on your class’ performance, and help the children that do suffer it to see maths in a different light.

This video by TED-Ed does a great job of explaining what maths anxiety is and ways in which it can be helped.

References:

Hembree, R. (1990) ‘The nature, effects and relief of mathematics anxiety’, Journal for Research in Mathematics Education, 21.

IEA (2008) Trends in Mathematics and Science Survey 2007. Lynch School of Education, Boston College: International Association for the Evaluation of Educational Achievement.

TED-Ed (2017) Why do people get so anxious about math? [online].  YouTube. Available at: https://www.youtube.com/watch?v=7snnRaC4t5c [Accessed 4th October 2017]

# Health and Wellbeing – Relationships

Yesterday, we had a health and wellbeing lecture about relationships. It was very interesting to learn about the kind of relationships children form at each age group and how they change as they grow older. To help further our understanding of the importance of relationships in the early years of a child’s life, we were asked to watch two videos,  one from Suzanne Zeedyk and  one from John Carnochan.

In the video, Zeedyk explains that human babies are born prematurely in comparison to other mammal species. This results in the human babies being born with an undeveloped brain. The brain is left to develop outside of the womb and the environment that the child is in can have a significant impact, positive or negative,on the development of the child’s brain.  The relationship’s that the baby forms in the first few years of its life are vital. The first four years of a child’s life are the most important years and can impact them for the rest of their life.

If a child is living in a household with domestic abuse, then their brain has to develop to cope with the threatening nature of this environment. As a result of this, they are using so much energy looking for their next threat that they can’t concentrate or learn. Carnochan goes onto mention that children need consistency in their lives. They may not be getting this at home in their threatening environment, so going to school can be their one happy place away from their troubles. As a teacher, it is valuable to recognise the importance of making your classroom a fun, safe and welcoming place for every child, especially if it is going to be an escape from their home life. Even children that are not facing difficulties at home need this environment at school.

After watching these two videos, it has made me more aware of the importance of relationships in a young child’s life and the valuable role of teachers if these relationships cannot be formed at home. Seeing how this affects a child’s learning and capabilities, it will allow me to have a wider understanding of every child and why they are acting the way they are. In turn I can accommodate my teaching methods to fit to every child and make them feel happy and safe inside my classroom.

# Online Unit 1 – Identifying Skills and Abilities

Improvement is impossible without reflection. Reflection allows you to look back at yourself as a person and see where you are personally and professionally. Reflecting on and identifying your skills and abilities is important as it lets you recognise your strengths and weaknesses. Without knowing these, you wouldn’t be able to advance as a person.

For Activity 1 of the Online Units, I rated my skills and abilities on a scale of 1 (not very well developed) to 3 (very well developed).

Throughout the year, I hope to advance and improve many of my skills, and hopefully gain more confidence within myself and my abilities.

 Skills and Abilities 1 2 3 Flexibility * Self Confidence * Self Discipline * Work Under Pressure * Set Personal Goals * Take Risks * Share Opinions Confidently * Team work * Take Responsibility * Build Social Networks * Manage Time * Act as a Leader * Negotiation * Make presentations * Listen to Others * Debate Formally and Informally * Contribute to Discussions * Converse Confidently * Take Notes * Write for Academic Purposes * Computing Skills * Be Creative * Use Technology * Problem Solve * Generate New Ideas * Work on Own Initiative * Organise and Plan * Think Critically * Evaluate Information *

# My First Attempt At The OLA and NOMA

When I first found out that there was a maths and literacy assessment, my mind began to panic. It soon was put to rest though, after I discovered that it was only for your own benefit and to help you improve your basic maths and literacy skills. It has been at least 6 years since I’ve been at primary school, so how was I ever meant to remember all the things I had learnt there.

On Tuesday, I had a few hours to spare in between lectures, so I decided to head to the library and do my first attempt at the OLA (Online Literacy Assessment) and the NOMA (National Online Maths Assessment). I decided to start off with the OLA, as I thought it would be a bit ‘easier’ than the NOMA. It did not start off smoothly though. The audio did not seem to work properly, even though I was doing it in the library, so I had to guess the first few answers. The rest of the test was not as difficult as I had initially expected, and my final score was 27 out of 35 (77%). I was quite happy with this score, as I knew there was some room for improvement. Hopefully at my next attempt, I can achieve at least 85%.

I still had some time left so I decided to attempt the NOMA, although I did have to rush the last few questions to make it to my lecture. Maths has never been my strong point, so I was a bit apprehensive. I was pleasantly surprised though, as the actual questions  were not difficult, it was trying to remember the formulas that was a bit of a struggle. I scored 41 out of 54 (76%) which I was pleased with. I knew that I needed to go and re-learn many of the formulas such as volume of a pyramid, area of a trapezium and so on.

I really like how both the OLA and NOMA give you feedback and show you where you need to improve. I shall use the feedback given and re-attempt both assessments in the near future, hopefully improving my score greatly.

# Why Me?

Following our very first ‘values module’ lecture on Tuesday, we then had our first values workshop later in the day. When I arrived, I sat down at table, not knowing that my choice had a significant effect  on my experience of the workshop. After everyone had arrived, large brown envelopes were placed onto the centre of the table and the task was explained. Using the resources that were in the envelope, each group (we were group three) had to create something that would be beneficial to a student just starting at the University of Dundee, just as we are.

We opened the envelope and all that it contained was: 3 sheets of paper (2 white and 1 blue); 3 rubber bands; 3 paper clips; a small white envelope; 2 post-it notes; a small lump of blue-tack; a pen and a pencil. We sat for a while, thinking of what we could possibly do with the lack of resources. We finally came up with a survival guide that contained a timetable, a map, top tips and much more. We saw Derek walking around giving praise to the first 2 groups, and barely glancing in our direction. The time came to present our idea back to the other groups. Alan was chosen to talk about what we were planning to create and he explained our idea very well. However, Derek looked less than impressed with it.

After presenting our ideas back the other groups, the time came to actually create the survival guide. We used all of the resources given, and did a pretty good job to create what we did, with as little as we had. We even added some hashtags to the front page (#uodedu) in an effort to impress Derek. The final product was then presented back to the other groups, and Derek was to rate them out of 10. It then became clear that as the groups progressed from 1-4, the less resources that they had received, with group 4 only have a pencil, a post-it note and a few paper clips.  Group one received 9, group two received 7, group three (us) received 4 and group four received 2!

We were shocked when we only got a score of 4/10; even so much that the girl sitting next to me, Kirsten, turned to me and said “Why doesn’t he like us?”. We were sure that Derek held a grudge against us as our idea was not bad enough to only be scored a 4.  I was left thinking ‘why me?’. I felt disheartened and also slight anger at the favouritism that had clearly been shown towards group one and two.  It wasn’t fair that we had been given less materials than the other groups and then scored without that being taken into consideration. It all became clear soon enough though that it was, in fact, all a wind up. I have to admit that Derek’s acting was impressive as I had really believed that our work was disappointing and he truly didn’t like us.

Being one of the groups that had the negative experience, it really opened my eyes to a few things. The same results cannot be expected from everyone if they do not have the same resources, and this needs to be taken into consideration. Encouragement is also a major factor in this. I realised that being put down for my work had such a backwards effect, and made me not want to continue and improve. Someone should not be put down just because they are not performing to the standards of everyone else. Every pupil needs to be looked at as an individual, as everyone is going through different things in life and may not always have what they need, whether that be resources or even support.

This is an experience I will certainly never forget and will carry the experience of it with me into the classroom.

# Why Teaching?

Whenever the question ‘What do you want to be when you grow up?’ arose, all I ever thought was a teacher. This has been the case for as long as I can remember. I have always had a passion for helping, encouraging and supporting people, especially with my peers and younger sibling.  I loved to help around the classroom and was especially inspired by a Primary Teacher of my own. She had a great enthusiasm for teaching and always made the lessons fun and interesting. She showed me what learning should be all about.

My real love for teaching, however, didn’t develop until I began work experience in my 3rd year of High School. My first placement was at the Primary School that I had went to as a child. Seeing how the children learnt and developed in such a short space of time was amazing to me. There was such a joy and pleasure spread by the children when they achieved something or overcame an obstacle. The children were also incredibly sweet, caring and easy to develop a bond with; something that can be hard to find in another occupation.

I also volunteered at a holiday club at a school during the Easter Holidays. We worked with the kids to decide the activities that would be taking place that day. I loved working with the kids to make decisions rather than just telling them as it made them feel more independent. It was such a fun experience and made my love for teaching grow even more.