Category Archives: 3.1 Teaching & Learning

Breaking down the idea of ‘Longitudinal coherence’ in mathematics

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Ma (2010) identified ‘Longitudinal Coherence’ as the final property of having a profound understanding of mathematics. If I am totally honest, this is the one that baffles me. I think this is because of my own fragmented experience of mathematics. When I was at primary school, I was never encouraged to link topics of learning, or reflect on more advanced learning, thinking about which concepts I had developed in order to get where I am now in my mathematics understanding.

I think I must have read the below definition of Longitudinal Coherence about 100 times:

‘Fundamental understanding of the whole mathematics curriculum and no limitation to the knowledge that should be taught in a certain grade. The ability to exploit an opportunity to review crucial concepts that students have studied previously and know what students are going to learn later and building the foundations for this future learning.’ (Ma, 2010, p.121)

After reading it 101 times and still feeling perplexed, I knew that I would have to do further reading to try and get different examples of what longitudinal coherence was in order to fully understand this property. Again, I found this difficult as every time I felt I was starting to get to grips with the concept, it began to feel like I was talking more about ‘connectedness’ than longitudinal coherence. I guess that it’s okay to have slightly different takes on the 4 crucial concepts of PUFM developed by Ma. I would say that connectedness and longitudinal coherence could have been combined as they do have very strong links with one another.

After a lot of research, I finally found some work which has helped me have a better understanding of what I believe to be longitudinal coherence from a teachers perspective:

“We produce many students who do not think globally – or to use a more common word these days, holistically- about mathematics. In the present context, teachers who come through such a training program may know the individual pieces of the school curriculum, but they are less adapt at seeing the interrelationships among topics of different grades. (Wu, 2002,p.19)

The above quote came with an example of helping students see the connections and coherent development of whole numbers all the way through to algebra:

Whole numbers ———> fractions —————> finite decimals, ratio, rates, percent, algebra (p.20)

Maybe Wu (2002) provided a simpler definition of longitudinal coherence than Ma (2010), or maybe because his description was accompanied by examples I was able to follow it better and have a clearer understanding. My role as a teacher is to continually encourage pupils to identify recurring themes and mathematical concepts when approaching new topics. Pupils should be able to see and draw on previous learning to help them develop new understanding. This should happen throughout the whole-school mathematics curriculum to enable students to see why previous learning was relevant and how it is supporting them in their current and future experiences.

Although this property initially baffled me, it is now the property which I connect with the most as I don’t feel I was given the opportunity to develop this at school. If anything, this places me in better stead for my future teaching pedagogies. I will always be able to look back on my own mathematics journey and ensure that I do the opposite to what I experienced at school.

Reflecting on my engagement with this module so far, I have found it extremely beneficial to breakdown the four properties of PUFM, (connectedness, multiple perspectives, basic ideas and longitudinal coherence), in order to develop my understanding of them. I feel that I can now engage with upcoming lectures with a different perspective and approach to mathematics. I want to be able to connect with the different topics we cover on a deeper level. I want to see how I can apply the 4 properties to help develop my own mathematics confidence and also my competence in developing positive teaching strategies.

Sources

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

Wu, H. (2002) “Longitudinal coherence of the curriculum” in What is so difficult about the preparation of mathematics teachers?. University of California: Berkley. Available at: https://math.berkeley.edu/~wu/pspd3d.pdf Accessed 31/10/15

Promoting multiple perspectives in mathematics!

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“The idea that learning mathematics requires little or no thought, as students are only required to reproduce procedures, suggests that students are engaging in ritualistic acts of knowledge production rather than thinking about the nature of the procedures and the reasons why and when they might be applied.” (Boaler, 2000, p.189)

This supports the argument for promoting ‘multiple perspectives’  in mathematics. If our students are taught one approach to solving a problem and are not encouraged to explore other ideas, to formulate their own strategies and discuss these with their peers, then essentially mathematics becomes a restricted subject.

“This idea follows a way of thinking that has been appearing in the last few decades, that doesn’t consider knowledge as given, established and transmissible, but where higher order and the critical thinking skills are privileged, where lectures are substituted by dialogue and discovery methods. Within this perspective problem solving tasks are powerful tools for teachers to use in their classroom. In particular patterns challenge students to use higher order thinking skills and emphasise exploration, investigation, conjecture and generalisation.”(Vale and Barbose, 2009, p.9)

I quote the above book extensively as I feel it contains a power message which encourages teachers to move away from single approach methods of learning mathematics towards finding a variety of solutions and being able to provide mathematical explanations for these different strategies. It provides opportunities to bring creativity and exploration into the classroom and when in my opinion, when children have the freedom to investigate and learn through trial and error, their motivation and enjoyment of the subject increases.

By providing multiple perspectives to a problem, teachers are also catering for the variety of learning styles within the classroom. Throughout my teacher training, one thing that has become clear is that it is my job to be able to discuss, explain and promote topics in different ways in order to provide equal learning opportunities for my students. If we do this with our teaching, surely we should encourage our children to do this with their learning. We want to encourage our pupils to think deeper about problems. We want them to have the confidence to analyse, predict, apply knowledge, reflect and evaluate. Even if their new strategy is unsuccessful, the learning gained from reflecting upon their work, thinking about what they could do differently next time and comparing strategies with peers is hugely beneficial.

The following video is an example of an alternative way of teaching students percentages. There is a strong focus on the use of reading skills and using the words of the question to break down the order of the calculation. For reasons stated above, it is important to present a variety of methods to solve a problem. Firstly, teachers must model solving the problem by using different approaches before children attempt them and consequently go on to formulate their own strategies. Teachers need to put in place the basic foundation of knowledge, as without this, children wouldn’t have the prior learning experiences and ‘basic ideas’ to assist them in trying to find alternative solutions.

My next blog will explore the notion of ‘basic ideas’ and what this means in terms of having a profound understanding of mathematics.

Sources

Boaler, J. (Ed) (2000) Multiple perspectives in mathematics teaching and learning. Greenwood: Praeger

Vale, I. and Barbosa, A. (2009) Multiple perspectives and contexts in mathematics education. Escola Politecnico de Viana do Castelo: Projecto Padroes. Available at: https://www.academia.edu/1485703/Patterns_multiple_perspectives_and_contexts_in_mathematics_education Accessed: 21.10.15

Extra reading and links

Lanarkshire resource – Problem solving and enquiry for secondary mathematics.

Promoting connectedness in mathematics!

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Liping Ma (2010) refers to ‘connectedness’ as being a key property of having a profound understanding of mathematics (p.122).

When I think back to my learning experiences at school, we were never made aware of, or allowed to explore, the interconnectedness of different mathematical concepts. Due to this, I now feel that I have a very fragmented view of the subject and I seem to approach different concepts as if they were isolated from all the others.

For my professional development, I think it is important for me to do some research around this area and to explore the importance of making connections; how does it help student’s learn? I hope this process will also allow me to have a better understanding of the underpinning network that runs through different concepts and begin to see mathematics as a coherent whole.

Education Scotland states “Within mathematics there are rich opportunities for links among different concepts.” (Mathematics Principles and Practice, p.4)

Before I did any research into the more complex connections between concepts, I asked myself, Where are the simple/basic connections?

I know that multiplication would be classed as a derived operation because it has links to the processes of addition. Then you have inverse operations; division is inverse (opposite) of multiplication, and addition is inverse of subtraction. These two processes, derived and inverse, are closely interwoven and tightly connect the four basic operations.

Internationally, the National Council of Teachers of Mathematics (2000) state that:

 “Mathematics is not a collection of separate strands or standards, even though it is often partitioned and presented in this manner. Rather, mathematics is an integrated field of study.” (p.4)

This notion of connectedness is promoted vastly throughout curriculums across the globe. Why? How do our students benefit from this approach?

If I had experienced this approach at school, I think I would have had a more coherent view of mathematics as a whole. I would have been able to connect my learning when dealing with different concepts and use these connections to investigate tasks and activities in other topics.

I believe that our students today are getting more than this. The are able to capitalise on the connections they build in mathematics. They use these connections to work through cross-curricular activities. For example, the use of data in tables to draw graphs and identify anomalies in science experiments. The opportunities to create mathematical connections in science is vast. These two subjects are tightly linked in school and further a field in employment and research-based jobs. There are links between shapes and symmetry in art activities, students may have to draw on previous mathematic concepts to make predictions or conjectures about the best way to draw a picture.

We recently had a lecture that focused on the process of tessellation. If I am totally honest, I had never heard that term used in my life and I considered myself a beginner learner in that class. (That being said, I am now learning to see this as advantage because, if I haven’t dealt with a topic before, I have no reason to feel anxious about it or let any previous experiences obstruct my confidence and motivation in the topic.)

Tessellation is the fitting together of shapes without having any overlaps or gaps between them.  We spent time discussing in groups and predicting which shapes we thought would tessellate (we had a selection of shaped on our table to use for our discussion). I never thought to think deeper into the activity and use my knowledge of regular and irregular shapes, polygons and congruent shapes to help me make an informed prediction about tessellation. I was pretty much taking a gamble for my predictions. As teachers we always encourage our students to use prior knowledge and experiences to help make predictions so that they have some meaning and thought behind them. So, in this situation I was contradicting myself and just jumping in without linking or drawing on prior learning.

Throughout the activity, it became clear that shapes would only tessellate if the angle they made when the vertices touched added up to 360 degrees. Therefore, I could have used this information to make a prediction about other regular shapes. (Regular shape has all its sides the same length and the internal angles are all the same size.) This approach would have linked to the process of connectedness and the underlying links between different mathematical concepts.

 

http://mathforum.org/sanders/geometry/GP07Tessellations.html

http://mathforum.org/sanders/geometry/GP07Tessellations.html

 

 

 

 

 

 

 

 

 

In conclusion, ‘connectedness’ is a crucial aspect of having a profound understanding of fundamental mathematics. It allows teachers, students and other professionals to see mathematics as a coherent whole and develop skills in connecting and linking different mathematical concepts together. These connections allow you to use prior learning and knowledge and apply it to new situations and contexts.

I end with the following clip which I was fascinated by. This teachers has a brilliant way of encouraging and supporting children to use and connect different concepts and apply them in new situations. There is a big emphasis on the vocabulary used, which is vital in making clear links in mathematics and the praise and feedback given by the teacher encourages motivation and problem solving within the activities.

Further reading and links:

Tessellation – Easy to understand!

Promoting connectedness in early mathematics education – This paper, written by Abigail Sawyer, describes the perspectives of two early years teachers involved in a new approach which encourages teachers to support students in making connections between mathematics and real-life experience. It is a very interesting read.

Sources:

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

Scottish Government (2009) Curriculum for excellence Mathematics principles and practice. Available at: https://www.educationscotland.gov.uk/Images/mathematics_principles_practice_tcm4-540176.pdf Accessed: 20/10/15

The National Council of Teachers of Mathematics (2000) Executive Summary: Principles and Standards for school mathematics. Available at: https://www.nctm.org/uploadedFiles/Standards_and_Positions/PSSM_ExecutiveSummary.pdf Accessed:20/10/15

Modern Languages – Isolated or integrated approach?

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If I am honest, I have always felt more comfortable with the thought of a specialist language teacher delivering the modern language curriculum to my class. My justification for this was that these teachers have specialist knowledge and understanding of the language which would allow them to teach the language more effectively than I could. The children would then get an informed and specialist educated professional delivering the lessons and their learning experiences would be greater than what I could offer. I think this train of thought is encouraged by my lack of confidence in modern languages and the fear that I would teach something incorrectly.

After leaving my first Modern Languages tutorial, I was surprised at how much French I could actually remember from school and I was happy that I could follow most of the discussions. This tutorial gave us some brilliant teaching strategies for a beginners language class and the emphasis on ‘why these strategies were effective’ was really valuable. Some of these strategies were: repetition, voice modulation, clear and precise dictation (providing a good speech model), mime, gestures, facial expressions, eye contact and games. It’s all good and well choosing a teaching strategy, but if you don’t know ‘why’ and ‘how’ it is going to help pupils learn and develop, then you will not be able to deliver your lesson effectively.

After some reading and research into the strategies used to develop a modern language (ML), I have been able to identify some criticisms of my initial ‘Isolated-approach’ method delivered by a specialist. As a result of a ML being taught by a language specialist, there is very little room for embedding modern languages into the curriculum as teachers wouldn’t have the knowledge or skills to provide cross-curricular learning experiences. There is also limited opportunity for children to learn the target language in meaningful contexts as all language learning would be restricted to small slot in the timetable.

“Pupils need to encounter, process and use new language in different, meaningful contexts in order for it to be embedded in their minds.” (McLachlan and Jones 2009)

As my understanding in this field has grown, I now consider a whole-school approach as being vital to the success and effective delivery of ML.  By this, I mean that class teachers have the support of their management team and other members of staff in promoting Modern Languages around the school and embedding it within daily routines. For example, this could be ML display boards in the school reception promoting welcoming phrases and increasing the status of languages for parents and visitors. Simple classroom routines such as the register, instructions, behaviour management and timetabling could be introduced in the target language. Welcome and introductions in assemblies could be done in the target language which would create a positive whole-school attitude towards ML.

“A clear and shared vision for the whole school’s present and future language provision will considerably enhance the initial classroom teaching.” (Hood and Tobutt, 2009)

This approach could be hindered by teacher’s lack of confidence in Modern Languages. Being able to provide cross-curricular learning experiences in the target language can be challenging and a daunting experience for members of staff and this could work against the integrated approach which schools are aiming for. A way in which schools could tackle this issue would be promoting professional development opportunities in Modern Languages. Schools need to give teachers the opportunity to develop their own knowledge and understanding of the languages being taught which would increase their confidence and would take a step in the right direction for embedding language in the whole curriculum. Another option would be to work with a Language Specialist, or another teacher who is particular competent in languages, to plan cross-curicular learning experiences and work through some lesson plans to ensure the class teacher felt secure in delivering them. Collaborating with other professionals is a valuable experience and one that allows you to share good practice, come up with new ideas and benefit from each other’s strengths.

An important point to consider when teaching a new language is, should my lessons be delivered only in the target language? Or, should I be using the english language to teach the new language? Teachers have differing views on this issue. Some say that the english helps to establish meaning in the target language and will prevent students from getting lost and de-motivated during lessons. Others say it doesn’t make sense to learn a new language by speaking your native tongue language. I fall in-between these two statements with a slight tendency to lean towards the latter. I do believe that children should be immersed in the target language and be able to develop the four language skills through this immersion. However, I agree that in the early stages of learning language, it is appropriate and often necessary to return to english to establish meaning, to cover tricky grammar rules and instructions and to check for your pupils’ understanding.

The video below is an interview with an experienced language teachers and she discusses her views on using the target language when teaching ML.

 

For my 2CM6 assignment, I am going to do further research into the use of singing, rhymes, games and story-telling to develop the target language. I want to develop my understanding of why and how teachers are slowing moving away from rote learning in ML acquisition to strategies which introduce Knowledge about language, language exploration and language comparisons.

References

Hood,P. and Tobutt, K. (2009) Modern languages in the primary school. London: SAGE Publications

McLachlan, A. and Jones, J. (2009) Primary languages in practice: a guide to teaching and learning. Maidenhead: Open University Press

Can we teach ‘Reading for pleasure’?

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Reflecting upon today’s lecture, it has become clear that ‘Reading for pleasure’ cannot be taught. It is my role as a teacher to create a culture in my classroom that introduces, encourages and supports learners in embarking upon their reading journey. Personal preference plays a big role in this and it is vital that I take this into consideration when trying to encourage reading for pleasure. I can model my love of reading and make sure that I immerse myself in children’s literature to ensure that I have sound knowledge of the books my pupils might be interested in. I need to have an open attitude to the types of literature and reading that my pupils might want to read. I can support them in their choices and try and advise appropriate reading options but it is ultimately their decision and it is important to not take away their right to choose.

‘The Rights of the Reader’ poster by Daniel Pennac clearly outlines the important factors of reading for pleasure. Looking at the factors from a critical point of view, I am slightly concerned about the implications some of these may cause for the classroom. For example: the right to read anything. There are some genres of books that I feel would be totally inappropriate for primary school aged children. Although these couldn’t be accessed in the school environment, teachers do not have supervision of the types of books children have access to at home. The right to not finish a book is an interesting factor. I believe that this is correct and that children should not be forced to finish a book of their choice if they are no longer enjoying it. However, I would be concerned that this attitude could begin to penetrate other areas of their learning and children may start to decide not to finish classwork because it is ‘their right to choose not to finish.’

When I am a teacher, I need to ensure that I create a reading culture within my classroom that encourages children to have free choice over books. I would like to have a section of time during the week where children could share what they are reading with the class and make recommendations to each other about books that they like. I feel that this would encourage reading for pleasure as children would be given the time to voice their opinions and be listened to. It would also be nice for children in class to know what each other are reading and it could create good discussion opportunities.

The Guardian Article

Who I am and why I’m here!

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I am Sara Chalmers in the University of Dundee studying for a MA in Education.

I left school in 2010 having secured the conditions required to start my BA in Sport and Active Lifestyles Promotion at Glasgow Caledonian University. Shortly after starting my course, I realised that I wasn’t yet ready for the ‘university experience’ and felt that I was struggling to settle in. Luckily, I was able to transfer to my local college back home, and there I completed my HNC in Fitness, Health and Exercise.

Teaching had always been in the back of my mind as a career choice but it was extremely important to me that I gained some practical experience within a school setting before embarking on a degree at university. I thought this would be a good opportunity to combine working with traveling and gain further life experience. I managed to secure a job as a Teaching Assistant (TA) at the British School of Bahrain. It was my two years working there that confirmed that teaching the only career I wanted to pursue. As a TA, I was able to observe and support extremely talented individuals within the profession, and experience different teaching styles and the effects these had on learners.I was able to get involved in coaching extra-curricular activities and really get to know the students and how best they learned as individuals. During my second year, I decided it was the right time to apply to the University of Dundee to study education. I remember being so motivated and inspired by the course lecturers on the day of my interview and I was absolutely thrilled when I received an unconditional offer for the following academic year.

My primary goal is to become a successful student teacher at the University of Dundee. Over the next four years I want to take full advantage of the breadth of knowledge and experience that my tutors, lecturers and fellow students have to offer. I am excited to work alongside those doing Social Work and Community Learning and Development, and building a good inter-agency partnership with them as this will be vital in the future. I have identified that to become a ‘successful student’, I need to maintain my good organisational skills and a high level of self-discipline with regard to time management and punctuality. An area of my learning that I need to develop is my ability to be perceptive and not be afraid to challenge and explore in-depth the reasoning behind teaching techniques, theories and what evidence there is to support them. In order to achieve this, I will need to familiarise myself with the university library and do extensive reading and research. This in turn, will increase my subject knowledge and may even highlight an area within education that I may like to specialise in further down the line.

Looking ahead to my career after my time at Dundee University, my goal is to become a teacher who regularly sets targets. I want to constantly reflect on my teaching and identify areas that require further development so that I am continually striving to support my learners in the best possible way. I believe that I will be able to do this by attending regular professional development training courses, increasing my subject knowledge through research and reading and maintaining good professional relationships with all those involved in a child’s learning journey. I support the progressions currently happening within education and welcome the changes still to come. Children and young people are being encouraged to take ownership of their own learning. They are no longer consumers of education, they are co-producers, and it wouldn’t surprise me if children are the directors of their own education by the time I retire.

For me, there is no end goal for my teaching as it is a life-long learning experience. There will always be an area that I can develop, another book I can read or the latest tablet device to get to grips with. Most importantly, every year there will be a fresh set of faces bustling through my classroom door, all desperate to teach me something new, and I cannot wait.

My Educational Philosophy

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I value education as it has the opportunity to support and guide children through their learning and development with the aim of producing confident, independent individuals who are able to contribute effectively to society and the economy.

Many people have differing views on the purpose of education. My belief is that its primary role is to provide an equal and active learning experience for all children and young people regardless of gender, race, religion or social classification. It is important that schools adopt an ethos which encourages pupils to have a sense of belonging to a community; with children feeling united and part of something great. Through achieving this, schools and education will be able to combine academic learning experiences and the transmission of morals and values with the intention of creating a fair and more egalitarian society.

It is absolutely critical that children are at the heart of the education process. They deserve to have a voice and play an active role in their learning. This differs from the idea that education is consumed and that it involves the creation of accepting minds rather than creative individuals. Having looked at practices such as Montessori schooling, I can see the benefits of children having the freedom to engage in activities which interest them and how this can support their development. I also like how the focus is on the process of learning and not the end result. Children are rewarded on effort and not by test results or grades. I believe that this approach is a crucial aspect of intrinsic motivation and one that will result in confident, independent thinkers who are not afraid of facing challenges.

In summary, education plays a vital role in creating  ethical, independent  and  successful individuals. It is a teachers role to provide an engaging and stimulating learning environment which allows children to grow and explore through social and environmental interactions. It is absolutely critical that this opportunity is provided to all children and young people.