Category Archives: 3.1 Teaching & Learning

Profound Understanding of Fundamental Mathematics

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Four Properties – to ensure a profound understanding of fundamental mathematics

Liping Ma (2010) believes in four properties in teaching and learning which sum up the way in which a teacher’s profound understanding of fundamental mathematics can be represented in the classroom. By this, Ma means a teacher will demonstrate the four properties and if this is successful, he or she has a profound understanding of fundamental mathematics.
– If you are unsure about what this is – profound understanding of fundamental mathematics – scroll to the bottom to read quotes extracted from Liping Ma’s ‘Knowing and Teaching Elementary Mathematics’ (2010). This may aid your understanding of the Four Properties in this post. 
The first, connectedness, occurs when a learner has the intention of making connections between mathematical concepts and procedures. In pedagogical terms, the teacher will prevent the learning from being fragmented and instead, learners will develop the ability to make connections between underlying mathematical concepts that link.
The second property, multiple perspectives, is practised when a learner is able to take into account various perspectives when thinking in a mathematical way. This includes addressing pros and cons of all different viewpoints considered. In pedagogical terms, the teacher provides opportunity for their learners to have a flexible way of thinking and understanding concepts in maths.
Thirdly, basic ideas, is a way of thinking about maths in terms of equations. Ma refers to basic ideas as, “simple but powerful basic concepts and principles of mathematics” (2010, p. 122). Ma suggests, when practising the property ‘basic ideas’, learners are guided to conduct ‘real’ maths activity rather than just being encouraged to approach the problem. I suggest this means if a teacher is effectively implementing this property, he or she will be not just attempting to motivate the learners to approach the maths work, but instead, providing a solid and secure guide to the learners understanding the maths themselves.
The forth and final property, longitudinal coherence, is when a learner does not have a limit or boundary of knowledge. In other words, it is not possible to ‘categorise’ the learner or identify the learner as working at a specific level or stage in maths. Instead, he or she has achieved a holistic understanding of maths – a fundamental understanding. In pedagogical terms, Ma suggests a teacher who has achieved a profound understanding of mathematical understanding is one who is able to identify on demand the learning that has been previously obtained and will be learned later. Subsequently, the teacher will lay the fundamental maths as a foundation for later learning.

A few quotes extracted from Liping Ma’s Knowing and Teaching Elementary Mathematics (2010) that I feel define and summarise profound understanding of fundamental mathematics…

“The term ‘fundamental’ has three related meanings: foundational, primary, and elementary.”
– Ma (2010, page 120)

“By profound understanding I mean an understanding of the terrain of fundamental mathematics that is deep, broad and thorough. Although the term ‘profound’ is often considered to mean intellectual depth, it’s three connotations , deep, vast, and thorough, are interconnected.”
– Ma (2010, page 120)

“As a mathematics teacher one needs to know the location of each piece of knowledge in the whole mathematical system, its relation with previous knowledge.”
– Tr. Mao (2010, page 115)

“I have to know what knowledge will be built on what I am teaching today.”
– Tr. Mao (2010, page 115)

 

 

References

Ma, L. (2010) Knowing and Teaching Elementary Mathematics 2nd edn. New York: Routledge. Pages 115-122.

liping ma

‘The Secret Life of 4 Year Olds’

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slo4yo

I recently watched ‘The Secret Life of 4 Year Olds‘ broadcast by Channel 4 on Tuesday 3rd November (Watch here – http://www.channel4.com/programmes/the-secret-life-of-4-5-and-6-year-olds). As the episode progressed, I found myself realising I have definitely been underestimating children as young as four years old. The hour-long duration of the programme was enough to capture the essence of the life of a four-year-old and the day-to-day thoughts, activities and behaviour of the children – not to mention, the emotional roller-coaster they endure because their ‘best friend took their toy or decided to play with someone else.
Oh, to be four. 

A number of key points initiated…

  • Children, at 4, 5 and 6, are at a partial age;
  • Futures are formed from this young age;
  • These ages are a crucial stage for a child’s development – what they learn now is the ‘blueprint’ for adult life;
  • Moral argument can quickly become coercion;
  • Supportive friendships have the ability to rapidly change;
  • A history of friendships create expectations of behaviour;
  • Ambitions from the four-year-old children in the episode include, ‘save the planet’, doctor and hairdresser – at the same time, and ‘jelly maker’.

Dr. Sam Wass, Educational Psychologist – MRC Cambridge quotes,

“To establish and maintain relationships, one of the key tools that children need is language. And at four, the average girl tends to be five months ahead of the average boy, in terms of their language skills. This can put some boys at a disadvantage in their social interactions.”
On reflection, what is meant by ‘average’? Every individual child is different and unique in the way they learn. Therefore, arguably, we cannot generalise, label or categorise children’s abilities, to give us a specific indication of ability.

“They’re beginning to learn to regulate their emotions, to interact with each other and to understand that other people have feelings, too. These are lessons that will inform a lot of their future interactions.”
On reflection, children respond in a variety of ways in different situations and therefore express a range of emotions. For example, experiencing a tragic incident, being vulnerable to an unsafe environment, bullying, winning or losing, achievements and many more. I believe it is not possible to teach a child these emotions because to do that would mean telling  or showing a child which emotion ‘matches’, if you like, with which situation. Emotions are a natural human trait – they are intrinsic but often influenced by extrinsic factors. Therefore, we can only teach children how to cope with and respond to their emotions, by being a supportive role and most importantly, by understanding. This is a learning process which children are still going through at a young age.

He continues,
“You give a child a new abstract concept to play with such as the concept of a friendship, and the natural instinct of a child is to want to prod and explore what that idea means. They tug it around a bit, see if they can break it and by doing this, they learn more about what the concept of friendship means.”

Professor Paul Howard-Jones, Educational Neuroscientist – University of Bristol quotes,

“Competition is motivating, it’s exciting, but it’s also great learning experience.”
On reflection, competition is an issue that is widely debated: is competition a good thing? My viewpoint is that is can introduce diversity, which may be viewed as a positive. However, I think competition is an important thing to teach our children to deal with, by teaching coping strategies.

“Children at this age self-segregate on the basis of gender.”
On reflection, I remember at primary school having to choose partners, groups or team leaders and the majority of the time, boys would choose boys and girls would choose girls. It was rare that opposite genders would be paired together. What does this say about our society? If anything, what does it tell us about our teaching strategies? As teachers and educators, the fundamental basis of our teaching and learning is around equality of opportunity and inclusion. We teach children the morality that everyone is the same, despite gender, race or religion. So, why do children self-segregate on the basis of gender? I would be interested to see any comments on this post regarding this issue.

And one final thought I will leave with you – extracted from The Secret Life of 4 Year Olds:

“It’s really striking how much children have to achieve at four years old. It may look like play, but actually they’re working really, really hard and they’re having to learn an awful lot. The way that they’re communicating with each other, the way that they’re experimenting and finding things out is really, really important for them.”

– Professor Paul Howard-Jones, Educational Nueroscientist (2015)

Watch ‘The Secret Life of 4 Year Olds’ – http://www.channel4.com/programmes/the-secret-life-of-4-5-and-6-year-olds

Let me speak!

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Yesterday, in my place of work, I found myself astonished by the sight and sound of a mother silencing her child.

Allow me to contextualise the situation for you. I work in a restaurant located in our City Centre. It is always busy with guests coming in for the first time, the second time, or coming in for their weekly order. So yesterday, work was extremely busy and therefore the noise levels were expectedly high. However, I could not ignore what I heard one woman say.

I allowed my eyes to glance across tables and they stopped on one table in particular. At this table sat eight guests who had come in together and going by first judgement, they were a family – made up of what I can only presume was two brothers, around three and five years old, along with their parents, grandmother and perhaps other relatives or failing that, family friends.

Now, in my place of work is one huge stone oven which does not ignore the flames that provide it with a warm glow and extreme heat. The boy, of about five years of age, sat at the table and was staring at this oven in absolute amazement. Of course he would be – it’s an enormous oven and most definitely is not your standard oven in your kitchen at home. He was amazed. It was something new to him. Something wonderful and exciting.

He turned to his mother with absolute excitement lighting up his face, wide-eyed and open-mouthed and said,
“Look! Mum, look at that!!! Our pizza is in that oven, look!”.
At that moment lay an opportunity for the mother to endlessly discuss the most exciting thing this boy had discovered – the oven!

Instead, she turned to him, ignoring the subject that provided him with such amazement, and silenced him with,
“Sssh, be quiet.”

I was in shock. You may be wondering why I was left feeling shocked and quite simply empathetic towards this boy. You see, this child should be immersed in language. Engaged with language. Not silenced when something is open for discussing, explaining and being interested in. His mother could quite easily have turned to her son and described the oven, asked him questions about it, used language to indicate a sharing of excitement and amazement about what her son had sighted.

I am fully aware the oven is not the most exciting thing for an adult to lay eyes on. However, as teachers, parents, educators or caregivers, it is crucial that we recognise children’s learning is embedded from a young age, they are learning all the time; and that is what we need to get right – we need to identify the gaps for learning and fill those gaps with knowledge, vocabulary, insights and perspectives. With language, there are a mass amount of opportunities to do this.

It is moments like this when children are surprised, amazed and intrigued about something at which it is necessary to capture this interest and go with it. Silencing a child when they show interest in something can only teach them not to display signs of true hysteria.

Celebrate this, engage this, and most importantly ask questions. Be involved by talking, discussing and conversing using your language skills and understanding, in order to facilitate the child’s learning and awareness of language. Show emotions with language and use words the child will question the meaning of; use terminology to challenge the child appropriately and broaden the vocabulary of the child.

As cited in The Really Useful Literacy Book (3rd edn.), it is suggested that children learn by understanding and remembering, which is essentially achieved effectively by ensuring application and regular revision (Martin, T., Lovat, C., Purnell, G., 2012). I agree with this and I suggest that in order for children to learn, understand, remember and progress language skills, it is profound that they are immersed in a language-rich environment,

 

 

 The focus of this reflection is not about the oven. It is about spoken language.

 

Stress, stress, stress!

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I recently read the article on BBC News, “Stressed teachers being ‘reduced to tears'” by Hannah Richardson, BBC News Education Reporter, 22 October 2015 (see link below) and it really hit me. It made me think – teachers are crying out, literally, and what for? They are stressed.

This article is stating the lead up to the stress is due to the workload teachers are faced with. In the article, it is stated by Dr. Bousted, a writer for Times Educational Supplement:
“It seems that teacher stress is increasingly being regarded as par for the course and part of the job.”
I agree that the workload in teaching is part of the job, due to GTCS standards and requirements, paperwork must be done. However, that should not take away from the love, passion and fun that teaching should be for teachers undergoing current stress. Not only will the stress make you feel under pressure, it will have an impact on your learners as well as those around you – colleagues, friends and family.

Dr. Bousted continued,
“A newly qualified teacher, asking for help to deal with an impossible workload which took up every evening until 11pm and all of the weekend, was told by her line manager ‘that’s the way it is in teaching’.”
To say, “that’s the way it is in teaching”, is a harsh reality for some, however it does not have to be stressful, pressurised or looked upon negatively. As a current student teacher, I am still only partially aware of the workload required by qualified teachers. Of course, I have seen in practice the paperwork – planning, assessment and reports. My viewpoint is that if you are entering the teaching profession, it is profound you thoroughly understand what is expected of you – the teacher, the facilitator, the educator, the professional, the trusted and respected role model. In order to be these things, you have to do the work.

Dr. Bousted goes on to advise,
“Teachers, as professionals, expect to work hard but should not be expected to devote every minute of their lives to their work. Teachers need time to relax, to pursue hobbies, to talk to their families and friends. They need time to be human.”

Teaching is not all stress. The way I see it is that you will always have work to do. There will not be a day that comes when you will have completed everything on your ‘to-do’ list. But that is part of being a professional. It all comes down to commitment and dedication. 

 

References

Stressed teachers being ‘reduced to tears’ – http://www.bbc.co.uk/news/education-34602720

The educator’s conceptual view – know what you are teaching!

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“Limited subject matter knowledge restricts a teacher’s capacity to promote conceptual learning among students. Even a strong belief of “teaching mathematics for understanding” cannot remedy or supplement a teacher’s disadvantage in subject matter knowledge. A few beginning teachers in the procedurally directed group wanted to “teach for understanding.” They intended to involve students in the learning process, and to promote conceptual learning that explained the rationale underlying the procedure. However, because of their own deficiency in subject matter knowledge, their conception of teaching could not be realized. Mr. Felix, Ms. Fiona, Ms Francine, and Ms. Felice intended to promote conceptual learning. Ironically, with a limited knowledge of the topic, their perspectives in defining the students’ mistake and their approach to dealing with the problem were both procedurally focused. In describing his ideas about teaching, Mr. Felix said: “I want them to really think about it and really use manipulatives and things where they can see what they are doing here, why it makes sense to move it over one column. Why do we do that? I think that kids are capable of understanding a lot more rationale for behavior and actions and so on than we really give them credit for a lot of times. I think it is easier for anybody to do something and remember it once they understand why they are doing it that way“.”
– Liping Ma, Knowing and Teaching Elementary Mathematics (2010, page 36)

The most important thing to remember when teaching maths – when teaching anything – as the teacher, the educator and the facilitator, is that you must understand what you are teaching. This is what Ma (2010, p. 36) is talking about here.

As a teacher and a professional educator, you are responsible for providing knowledge to your learners, not just passing it to them as information in a book or in the form of confusing statistics and facts, but as an understood conceptual view of the content. If you do not understand what you are teaching, this may invite opportunity for confidence to fall in your learners – you are the trusted educator in the classroom, on which your learners depend on to provide subject matter with an understanding you have thoroughly revised, in order to adapt the content to best explain it to them.

Outsmarted?… Imagine this. You are planning a lesson – a maths lesson. You have a vague and somewhat passive understanding of the content you intend to teach. And so you think your learners will trust that you understand what input they are going to receive, because, after all, you are the teacher. Right? That passive understanding you have, is only going to brush off onto your learners. Children are observant and will easily pick up on your mistakes, your struggles and perhaps your lack of confidence when you are teaching them. So, you plan your lesson, still intact with your passive understanding of the content you intend to teach. Then it comes to your lesson and your learner outsmarts you. Perhaps in the form of a question, that you cannot answer. Is this due to your negligence?

Your learners depend on you to know what you are talking about, and here, Ma, explains the profound importance to approach your intended learning content with a conceptual view – if you understand, you have more chance of your learners understanding!

 

References

Ma, L. (2010) Knowing and Teaching Elementary Mathematics – Teachers’ Understanding of Fundamental Mathematics in China and The United States. London: Routledge.

A Helping Hand – Working Together

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An approach to learning. A learning style. Currently evolving. The sharing of knowledge. Expressing views and ideas. Assisting and helping. Working together. Collaborative Learning.

Collaborative Learning is an ongoing and widely debated topic in Scottish Education and a learning style encouraged and discouraged. It is an approach to learning and a way of working together and sharing ideas, building on knowledge and, most importantly, enhancing children’s confidence in being social. However, as a practitioner, I am also aware that collaborative learning does not always have a positive impact on children’s learning and development.

Due to my ongoing collaborative practice in university classes, as well as having observed collaboration of between children in my placement, I find that working collaboratively can have the positive impact of enhancing confidence and building on knowledge. Working with others allows opportunity to gain an insight to other people’s ideas and ways of thinking, which in itself puts into practice communication skills we teach our learners…
– Listening
– Talking
– Turn-taking

“Jigsaw Learning”
– http://www.schoolsworld.tv/node/1247

I was unaware of just how interesting and helpful collaborative learning can be – the outcomes. However, after watching this video, my insight has been enhanced and I can relate it to what I have observed in practice.

“Jigsaw Learning”, as seen in this video, is an approach to learning which is active-based. Active Learning is an important approach the Curriculum for Excellence promotes and jigsaw learning encourages this. This is a learning style in which children participate in game-­like activities, working together. In Park Lane Primary School, seen in the video, the children demonstrate collaborative learning by researching and creating presentations on provided poems. This is an example of Jigsaw Learning, which places responsibility on the children and encourages equality of opportunity and teamwork as the research-based task is controlled by the group of children. As the groups alternate, the children develop socially, as well as sharing their knowledge.

Collaborative Learning allows the teacher and the children to be flexible and adaptable to every child’s stage of development. In this video, the children peer assess their work and if any corrections are required to be made, this follow ­up is done collaboratively. To encourage Collaborative Learning, in Park Lane Primary School, the children are seated at tables of mixed ability groups, which are changed every two weeks. This forces children to work together to overcome struggles or difficulties, which often encourages scaffolding to take place – children of higher ability can assist and support children of lower ability (Vygotsky, 1978). However, having seen this arrangement in my placement setting, I am aware that this can also have a negative effect on children’s confidence in using their voice with their peers, as, in some cases, their classmates are not the friends they are most comfortable in using their social skills with.

Not only has my awareness of Collaborative Learning been enhanced by this video, my understanding has developed of how Collaborative Learning works in a beneficial way for each individual child involved in the learning experience. For example, the children are assessing each others’ work, explaining their feedback. The outcome of this is that children understand areas to improve on, as well as recognising their next steps. Peer support is an important trait in working together which proves effective for many children, as it allows children to assist their peers in overcoming difficulties they may have reached themselves

– children being the teachers at this time. 

Collaborative Learning works within the classroom between the teacher and the pupils, as well as pupil-to-pupil. In this video, the class teacher provides the children with an insight to their aims and targets for their stages of all curricular subject areas. The teacher shows the targets to the children in the class and following this, the children think of activities and tasks to carry out, to allow them to meet their targets. This allows the children to have an understanding of the content they are learning and, subsequently, having pupil-directed input of how they want to learn it. The children then peer-assess, making links to the success criteria and this allows each individual child to have an awareness of their progress and an understanding of areas of improvement.

One pupil in Park Lane Primary School, as seen in the video, explains,
“some people might not understand the work, but someone else might understand the work really well. I think it works really well because they can help each other”.

Working together, in pairs, small groups or larger groups, is a memorable experience for myself. I remember reading books in groups, throughout Primary School. I remember, paired-­reading, in which I worked with a younger pupil within the school, assisting her in reading. As well as the pupil developing her Literacy, it benefited myself and my social development.

I would best summarise my understanding of Collaborative Learning as the approach the Curriculum for Excellence takes of theorist Lev Vygotsky, that learning is a “social process” and learning is most effective when people are learning from each other. It is the group tasks which are the most memorable. Collaborative Learning is most important.

Is teaching a profession? Are teachers professionals?

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Professionalism in the teaching discipline is diversely considered. Many people have the viewpoint that professionals are of high status in society, an example being lawyers. We must acknowledge the responsibility teachers hold in the profession and identify their roles as professionals, deservingly.

Teaching is a paid occupation, therefore I believe should be defined as a profession. Oxford English Dictionary (2012) supports this by stating that a profession is a paid occupation following extended training and formal qualifications. This definition serves the suggestion that teaching is in fact a profession, because to graduate with a teaching degree, you are required to be relevantly and formally qualified and to have completed the appropriate teacher training. Saying this, many teachers are being given jobs in schools in England without having achieved the formal qualifications. So, does this mean they are not part of this profession and subsequent to this, not professionals, because they do not have the formal qualifications? Profession is defined as having gone through appropriate training in the specified field of study, so if an individual has relevant work experience, should formal qualifications be a necessity? An article on The General Teaching Council for Wales brought to my attention the belief that teaching is not an isolated profession. Instead, it is much more: it is working with social workers, therapists, specialists and many more who bring support to teachers and children. This raises the question: are teachers professionals in more jobs and roles than teaching solely?

Oxford English Dictionary defines ‘professional’ as ‘relating to or belonging to a profession’. Although this definition is understood as meaning, for example, professionals in healthcare potentially belong to a Healthcare profession, there are many attributes of an individual which make them a professional. Being a professional means meeting standards and expectations of society, policy and those set yourself with consideration of ethics, values and beliefs. An important element of being a successful teacher is being a professional; the way you conduct yourself, behave and how your professional role is portrayed. To be a professional in teaching, it is important to ensure you are a good example for young children to follow. It is also important to consider the way you conduct yourself to your peers and to wider society, not just to your colleagues, because teachers being known as professionals can depend highly on the way they are viewed by society, which relies on the way the individual is portrayed to others.

I strongly think teachers’ roles being viewed as professionals is hugely dependent on the close community and wider society. The General Teaching Council for Scotland (GTCS) sets out a Code of Professionalism and Conduct (COPAC) in order to put in place key principles and standards expected of registered teachers, to ensure members of society have trust in teachers. The Code of Professionalism and Conduct is not only for registered teachers, but is also available to members of the public, therefore supporting the suggestion that teachers being viewed as professionals or unprofessionals is dependent on society. Saying this, it may also influence members of public to have a deep understanding of what is expected by teachers and therefore raise issues where teachers are identified as not meeting the standards. Therefore, this resulting in teachers not being appreciated as professionals. Although every teacher is different and will carry differing points of view on issues, varying values and personal beliefs, The General Teaching Council for Scotland strives to improve professional standards for teachers.

Yes, teachers should be identified as professionals, however it is dependent on how individual teachers conduct themselves, behave and portray their personality that defines whether they are being a professional.

professional

Let’s talk philosophy… MA1

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It is fundamental for any teacher to have their own philosophy of education, which is made up of their values and beliefs about learning, teaching and education, because this is an influential element of the learning, planning and reflecting process in the classroom, of which the teacher is responsible for.

Education is a never-ending, lifelong learning experience throughout which an individual to learn information, process knowledge and develop skills and abilities through experience and practice, which will develop the individual in becoming well-rounded. The value education holds is powerful and significant to many aspects of life including society, relationships and personal development; we develop by learning and being educated. Education and the value found in education is somewhat dependent around the individual’s learning experience.

To me, education is learning, knowledge, understanding, developing, experience and adaptation. I believe it is important for the teacher and the children to engage to their full capacity, always; be interested; and for the teacher to have and promote a ‘thirst for knowledge’. To experience valuable education, approaches and attitudes to acknowledge and portray are: mutual respect, equality, consideration, honesty, loyalty, integrity, justice, trust and fairness.

The teacher is the role model to the children he or she is responsible for; if the teacher portrays a thoughtful, considerate and interested attitude, the children will follow this. The responsibility the teacher holds for the learning and development of each child is highly important; the teacher must always ask, ‘what is the impact on learning?’, when he or she goes through the planning and assessing cycle. It is the teacher’s responsibility for the learning and education within the classroom – this is why it is of paramount importance the teacher has his or her own Philosophy of Education.

 

TEACH