Category Archives: 3.3 Pupil Assessment

Longitudinal coherence

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WHAT?!

Well now I am aware it doesn’t seem so “profoundly” confusing. Ma (2010) outlines this concept as a fundamental principle in the learning and teaching of mathematics.

‘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, Pg. 121)

So with this in mind… Longitudinal coherence in more basic terms is actually the ability to build on and use previous learning to acquire more advanced understanding of progressive mathematical concepts. For example…

Subtraction -> division -> fractions = rates, percentages, algebra and decimals.

By building on previous learning we can cement and progress a child’s understanding in more complex areas. This progressive learning strategy is supported highly by myself, my previous post where I discussed my experiences at school… I asked those “why” questions and got a “because you do” answer. This would not be the case if I had those links and the explanations of WHY! Progression and depth are also two principles supported by CfE.

Depth

“There should be opportunities for children to develop their full capacity for different types of thinking and learning. As they progress, they should develop and apply increasing intellectual rigour, drawing different strands of learning together, and exploring and achieving more advanced levels of understanding” (Scottish Government, 2009).

Progression

“Each stage should build upon earlier knowledge and achievements. Children should be able to progress at a rate which meets their needs and aptitudes, and keep options open so that routes are not closed off too early” (Scottish Government, 2009).

I feel that these principles link closely with Ma’s concept of Longitudinal coherence and support the framework of CfE. This depth and progressive approach should in theory cement a life long understanding which can be used in other areas of the curriculum and in upper school, such as Science, Technology, Finance and Business.

 

 

Local opportunities to provide “Connectedness”

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One of Liping-Ma’s fundamental principles, Connectedness, is the opportunity to provide a linked and relative basis for learning mathematics, creating cross curricular learning in the classroom, benefitting a child’s progression and teachers ability to provide a sustained and relative learning experience.

“a general intention to make connections among mathematical concepts and procedures, from simple and superficial connections between individual pieces of knowledge to complicated underlying connections among different mathematical operations and subdomains” (Ma, 2010).

Creating these links provide children with the ability to “model real-life situations and make connections and informed predictions” as stated in the Principles and Practice document provided by the Scottish Government. This supports the belief of Ma that this intention should prevent students’ learning from being fragmented and instead of learning isolated topics, the knowledge will be unified. (Ma, 2010)

“Learning mathematics develops logical reasoning, analysis, problem-solving skills, creativity and the ability to think in abstract ways” (Scottish Government, 2009). Further to this the basic principles of CfE support Liping Ma’s concepts, the curriculum should be taught with breadth and progression in mind, in relation to Mathematics, science, medicine, technology and finance are, to name but a few, tightly related subjects which require the “basic” understanding of fundamental mathematical concepts in order to progress into such areas.

“To face the challenges of the 21st century, each young person needs to have confidence in using mathematical skills, and Scotland needs both specialist mathematicians and a highly numerate population.” Building the Curriculum 1

There is increasingly high demand for a high level of mathematical education in the UK, and to target this STEM was introduced and has become a high focus for the government to address a high demand in these areas. Science, Technology, Engineering and Mathematics are the areas identified. Relating this to my local area, Dundee and the place where I would hope to teach in the future I wanted to explore what opportunities there were available and how I would address my learners needs.

Science – Dundee Science Centre provides a great experience for children to involve themselves in an interactive, educational setting and also evolves to a deeper understanding that children can link their own experiences to. In settings like this, children are learning without realizing and this can be extremely successful, highlighting enjoyment and choice of learners.

dundeesci

Technology – Abertay University is regarded highly regarded in the Gaming industry and has homed some of the worlds most respected producers of games. Coding is an area which can be developed with children and I have experienced teaching this area. Using such a resource will mean that professionals can discuss with the children opportunities for development in technology and gaming.

abertay

Engineering – At first I found it difficult to think of an engineering learning opportunity in Dundee, however I then considered construction. The V&A museum is a great example to introduce a whole school to construction and architecture. As it is in the process of being built children can follow the progress over a period of time and explore the stages of development such a building requires.

v_a_at_dundee_building_u270112

Mathematics – A much wider topic which can be related to all of the above. I decided to think about cross curricular links that can be developed in the classroom alone. With so many opportunities I will explore this in a later blog post.

These opportunities provide learners with hands on experience, which they have a connection with. Connectedness is obtained by creating links and opportunities for learners to create their own understanding of basic ideas which will progress as learning manifests into other areas of the curriculum.

The beauty of Maths

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The Fibonacci sequence describes a concept constructed by Leonardo Pisano Bogollo (1170-1250), Italy.

It is a sequence of numbers starting at 0, 1 and the next number is found by adding up the two numbers before it. E.g. 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584. The combination of numbers formulates a diagram which is used progressively and consistently in daily life. For example the construction of buildings, architectural design, and natural growth i.e. flowers and plants. It can also be used in areas such as logistics, planning and analysis in business.

This spiral design can be seen in the following natural formations for plants, flowers and shells.

fibo1 fibo2 fibo3 fibo4

The Fibonacci sequence is also used to calculate the “Golden Ratio,” the most aesthetically pleasing equation “φ” which is approximately 1.618034. Using any number from the sequence and dividing it from another should result in approximately the same answer.

A B   B / A
2 3 1.5
3 5 1.666666666…
5 8 1.6
8 13 1.625

 

In analyzing the role of Fibonacci sequence I investigated highly regarded artists to consider their approach to creating some of the most inspiring and popular artwork. In a recent lecture we discovered the designs of Mondrian a French artist who used the sequence to divide his designs. We attempted to recreate his art without the knowledge of his association and then with the knowledge recreated our initial attempts using the sequence.

mond1 mond2

 

 

 

 

 

From creating this design I can now appreciate the visual aspect which the concept can interpret. I much prefer the second design, more interesting and eye catching than my first attempt which displays to me the aesthetic nature in which mathematical concepts can be used. Linking this to pedagogy I now have a greater understanding of why links are so important to draw learners into cross curricular learning opportunities, in this case Art. This further links to Ma’s description that teaching Maths should be interlinked and meaningful when discussing fundamental mathematics. The fundamental mathematics which can be explored in my own example could be fractions, adding and subtraction, division, symmetry and area and perimeter. I also appreciate the concept of differentiation which can be applied when discussing this concept and using a multitude of learning opportunities targets all learners and their cognitive and physical needs.

Through this varied and interactive approach children can investigate and engage with mathematics freely without fear of error, with creativity and choice. Making mathematics fun and accessible is a key platform to building a child’s confidence and willingness to engage with a negatively perceived subject by many young children.

ren2 ren1

I recently came across this design which reminded me of the Scottish artist Charles Rennie MacIntosh but is actually an example of the Fibonacci sequence, displaying the “golden ratio.” I am a huge fan of Charles Rennie MacIntosh, finding his art work simplistically stunning. I am really excited by the prospect that he has used the Fibonacci sequence in creating his designs, and I can now appreciate how it can be interpreted in different ways by different people in this case an abstract form. My preference toward Charles Rennie MacIntosh’s designs also highlighted my lecturer’s description that this is often an unconscious preference to the aesthetic aspect of the golden ratio.

Combining Mathematics and Art

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In a recent lecture we explored the concepts of Mathematics in relation to aspects of ancient art and how these can be taught with cross curricular intent. Symmetry and pattern are an example of key mathematical concepts which can be explored using art from around the world.

is art 2

Islamic art uses fantastic pattern and repetition, creating fascinating and eye catching designs used in architecture, religion and clothing to name but a few. Using shapes and folding techniques we explored the possibilities for classroom learning. Shape, position and movement are concepts which can be addressed as early as nursery learning.

folding

tiling

Repeating patterns can create dramatic displays for the children to appreciate and encourage their learning, and demonstrate appreciation for their involvement. It is important to be aware of the difficulties some children may face when creating pattern and using tools such as scissors or coloured pattern. Some examples of this can be physical and visual impairments, concentration and proportional and spatial awareness. Creating achievable, differentiated outcomes for children is key to their attainment and this can supported by allowing the children to create their own success criteria.

Constructivist Learning

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Recently, I have watched a video clip where a secondary school teacher is teaching Pythagoras. He leads the classroom with a concept unrelated to maths, in this case the game of golf. He creates opportunities for discussion and exploration for pupils by introducing them to a relatable context, this gains thundering discussion in the classroom and all learners become instantly engaged, possibly because they do not see the relation to mathematical concepts. As the lesson continues the teacher offers the children the chance to explore two methods by which to solve the initial problem he has constructed. Once again the children respond in an engaged manner and they discuss with their peers to investigate.

The outcome of this strategy is that all learners are then given a similar concept to solve, and due to their new experience of calculations adopt a new strategy one that involves calculations and a deeper understanding on the mathematical concept.
Relative context in this lesson has been key to the learner’s understanding, which supports my previous engagement in mathematical pedagogy.
By creating a variety of opportunities for engagement the learners are empowered by their previous knowledge to explore and create deeper understanding of what can be a confusing concept… in my own opinion!

Offering learners the opportunity to discuss and share findings is an excellent method of assessment for the teacher. Often learners can explain things to other learners in a way which is more accessible to a teacher’s method.

Returning to this blog post I have reflected on this teachers approach. I can now connect this method of teaching to Ma’s description of multiple perspectives. This teacher displayed an appreciation for “difference facets of an idea and various approaches to a solution, as well as their advantages and disadvantages”  (Ma, 2010, Pg. 122). Offering multiple solution processes for learners allows children to adopt a more flexible and insightful approach to a discipline. I can also relate this to differentiation, with many learners differing in their learning strategies, this approach encourages learners to explore opportunities for problem solving.

 

The Story of Maths, BBC Four

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As discussed in my previous entry, on a recent placement in a school, I was asked to create a maths lesson linked to my learners topic work on Ancient Egypt. Watching this documentary has given me a wealth of inspiration regarding the links I could, and should have created when producing my lesson plans.


The paths I could have explored with the learners were endless, and I can guarantee that most learners would have been fascinated by the first 15 minutes of this documentary.
Introducing media in the classroom to build a foundation of understanding is something I had not considered and coincidently this documentary would have been a fantastic aid for my learners, and myself.

 

The secret life of numbers engagement

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On reading this article, I related closely my experience in school of mathematics and the pedagogy it was determined by. We learned the basic concepts… enough to pass exams, and even then it was not a guarantee as much of the time we were unable to identify the relation to everyday life. I recall asking a “why” question to my standard grade maths teacher which was responded to with a “because you do” answer. This was a regular occurrence throughout my school existence, the explanations and relevance were brief and uninsightful.
The introduction of Curriculum for Excellence threatened to challenge such isolated pedagogy, but my experience in the classroom suggests otherwise. Mathematics is being loosely taught following the new guidelines while old techniques of teaching the subject are clearly prevalent. Cross curricular links are being used but not with the depth and breadth they could be.
During my time in one school I found myself with the concept of teaching a lesson on area and perimeter. Coincidently, as described in this article, the classrooms topic work was focusing on Ancient Egypt. I’d like to say that as stated by Marcus du Sautoy I related the history of why mathematics was so relevant in Ancient Egyptian times, unfortunately I did not and created a worksheet for the class as suggested by my classroom teacher. *Sigh*. The children in the classroom enjoyed completing the worksheet, but on reading this I am disappointed in my own pedagogy. I realise now, that a basic understanding of mathematics will not be enough for all learners. Many, like myself, will need a much broader understanding of the relevance and history behind such concepts to make coherent sense of their learning. http://www.theguardian.com/education/2009/jun/23/maths-marcus-du-sautoy

 

It’s my philosophy

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I believe that the educational system today should be used to encourage, motivate, inspire and allow children to follow the path they wish to. Primarily, I believe, there are social aspects of education that must be addressed and a society’s values and norms should be displayed so that children are able to function effectively in society. However I don’t believe that this is for the benefit of society; it is for the benefit of the individual. This will allow them to form relationships, whether it be family, friends, colleagues or clients. It is important to gain these understandings at a young age as these values and norms that are expected will allow a child to form their own understanding with time and life experience. Instilling this cultural awareness is crucial.

Education should not just be for learning knowledge. In my view, some of the most intelligent people I know have little social skills and ability to adapt in challenging situations such as interviews for jobs. These are the sort of skills children should be encouraged to adapt in the early years through communication and understanding of different opinions, perspectives and cultures. I believe that these skills are equally if not more important than vast knowledge in particular subjects.

Children should have free choice of the subjects they wish to explore, and although the curriculum is key in allowing children to receive a fair and equal education, it is fundamental that children are heard and feel valued. Children should be allowed to explore their learning in a controlled way in which they can be kept on task, while using a wide variety of facilities, environments and tools. This independence will encourage a child’s imagination to grow and they will gradually form their own opinions of the world. I do not want to contribute to a society full of single minded individuals. I want to be inspired by the children in the same way that I inspire them! It is a key aspect of the educational system that the teachers are continuously developing and trying new approaches to learning.

With regard to discipline, I hope to improve behaviour, not necessarily discipline. I want children to recognise the rights and wrongs, but also to understand why they are expected to adopt particular manners. This would not only allow the children to develop their social skills but to also to make the choice of how they wish to act. I believe that if children feel like they have some control of the situations in class then they are more likely to cooperate. In my experience, explanations and understanding are key to the improvement of behaviour in a child. It is not only important for the child to know what’s gone wrong, but also to recognise myself. Has it potentially been something that I could have avoided myself? I aim to reflect daily of the situations which arise in the classroom and this should allow me to evaluate the strengths and weaknesses in order to adapt my approach towards the individuals of my concerns.