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Paideias philosophy is based on the belief that schools can be informal and individualized, yet still educate well. (Paideia School)
How does this compare with my own philosophy about Mathematics? This has definitely changed and I believe for my own good. Previous education experiences left me very disillusioned, I wanted to change my opinions and beliefs, move forward, and enable others to change their philosophies about mathematics as well. Some people look upon mathematics as a taboo subject, and we, as teachers, need to ensure that everyone has opportunities opened to them to enhance and encourage their own academic learning.
We need to ensure that pupil's learning is individualised and that their experiences within the classroom are enjoyable, motivational yet challenging. Mike Ollerton (2009, page 4) suggests that
'The more confident any student becomes, the more competent they become. Achievement breeds confidence, confidence breeds competence and competence causes enjoyment'.
So what do I believe mathematics is? Mathematics for me is an international language, which describes, analyses and changes everything. It consists of a massive set of tools and we need to work out which tool to use when applying or solving mathematics. We learn new skills and must ensure we can use and apply them, by being able to do this makes us more competent to cope with daily life.
My own experiences of education connect very much to the information given in the Cockcroft Report (1982, chapter 9), which states explicit information on the education system of that particular era. However, the Smith Report (2004) (Chapter 3, pages 53-
55) identifies the current mathematics pathways for the National Curriculum (pre-16). This gives a more appropriate portrayal of the National Curriculum, which is broadly used, within schools in the United Kingdom.
Education throughout my own lifetime has greatly altered, so how has this affected my own learning, my pedagogy? Subject Knowledge is very important for a trainee teacher because you need to understand the subject and you need to be able to transmit it to pupils to ensure they can understand and interpret it fully to increase their learning capabilities. My own subject knowledge is continuously changing and advancing, teaching a topic allows oneself to improve and intensify the techniques for teaching a variety of topics and of the various ways, those pupils are able to access their learning.
In 1986, Shulman introduced the phrase pedagogical content knowledge. In his theoretical framework, he states:
'Teachers need to master two types of knowledge (a) content which is also known as "deep" knowledge of the subject itself, and (b) knowledge of the curricular development'.
Nevertheless, why is subject knowledge so important?
'Teachers who have effective subject expertise know how to structure learning in ways that allow pupils to connect apparently different topics, and build on their earlier learning'. (OFSTED, 2008)
To ensure my own subject knowledge continues to improve I utilize the National Centre for Excellence in the Teaching of Mathematics (NCETM). Using the NCETM enables me to continue improving my own pedagogy and my professional development, which then ensures that I can challenge and reassure pupils that they can reach their own academic potential as well. The Department for Education (DfE)] set up the National CentreÂ in response to Professor Adrian Smith's report Making Mathematics Count, (2004).
How can I ensure that pupils and I work collaboratively to enhance their academic learning potential? On my first placement I wanted the pupils to work together in pairs or groups; this is something that had not previously been evident and met with hostility from the pupils because it was something different.
'There is now general agreement in research that cooperative small group work has positive effects on both social skills and mathematics learning'. (Standard's Unit, 2005, by Malcolm Swan)
However, this approach needed more practise and I needed to ensure the mixture of the group would be able to work together. Once the pupils were more comfortable about working in pairs, I was then able to place them in specific groups and this encouraged them to work more collaboratively with different pupils and not feel uncomfortable or embarrassed about giving their views or answers. The groups became supportive and were able to develop their own knowledge and understanding even further. To encourage collaborative working amongst peers and other teaching staff ensures sharing of resources, lesson plans and different strategies, which work with different year or year groups. Being able to exchange ideas and different teaching styles improves learning expectations for both staff and students.
Within the approach for collaboration, I feel that the use of scaffolding is an area that worked not only between the pupils and me but also with the more able pupils and those with lower abilities. Originally introduced in the 1950's by Constructivist Jerome Bruner, however Vygotsky used a similar idea that learning depended on the zone of proximal development (ZPD), in which he defined:
"The distance between the actual developmental level as determined by independent problem-solving and the level of potential development as determined through problem-solving under adult guidance, or in collaboration with more capable peers."(Vygotsky, 1978)
In my first placement, I used scaffolding to support learning but I also found that by chunking certain bits of information the pupils were better able to retain the information.
Why is this? George A Miller (1956) produced landmark work on the short-term memory. He proposed that:
'Information is organized into units or "chunks", and the limitations on short term memory apply to the number of chunks an individual can hold in consciousness at one time. Chunks can have variable size, and the number of chunks humans can hold in short-term memory is 7 + 2'. (Miller, 1956)
Another learning theory that I connect with and use regularly is that of positive reinforcement (B. F. Skinner, 1971 - Operant conditioning), I find that using positive reinforcers for example stickers, raffle tickets improves not only the pupil's behaviour but also increases their motivational skills. Skinner suggests that:
'A reinforcer is anything that strengthens the desired response. It could be verbal praise, a good grade or a feeling of increased accomplishment or satisfaction'.
Learning mathematics can occur in a variety of different situations. What is most important is that the pupil uses a method that works for them and teachers are aware of a variety of different methods to ensure students are achieving their full academic potential.
Currently, I am experimenting with different starter activities and plenaries. At placement one, I experimented with rich activities that I produced, tarzia puzzles; loop cards and mini-whiteboards because the students were not familiar with them, by introducing these different aspects slowly allowed me to let the pupils extend/further their own learning comfortably without feeling over challenged. I feel that the starter gets the pupils in the mood for learning - by giving them a variety of challenging activities gets them thinking about the mathematics involved in the tasks. My plenaries combine to show how the pupils have understood the work covered in class and how it links to the learning objectives/outcomes for the lesson.
'Effective teachers make good use of starters and plenaries in the context of interactive whole-class teaching to engage all pupils in constructive deep learning'. (Department for Education and Skills, Sept 2004)
Another area that I have been experimenting with is my questioning techniques. I have also found Bloom's taxonomy useful in planning objectives for lessons in my second placement and for increasingly challenging questions using the cognitive objectives. Why is this?
'You can use the steps in the taxonomy to help plan objectives for lessons over a period of time to ensure that lessons are making increasingly challenging cognitive demands on pupils. You can also use them to plan sequences of questions in a lesson. By sequencing questions in this way, you can help pupils to deepen their understanding, to develop their thinking skills and for them to become more effective learners'.
(Department for Education and Skills, 2004)
Another area for experimentation has been the explanation of principles. By observing a variety of different teaching staff, I can identify and learn how to approach different topics especially the higher-level work. Within this section, I feel that I am way out of my comfort zone but I need to challenge this and ensure that I improve my subject knowledge. I do not think that I will ever stop learning different aspects of mathematics however, by taking part in extra observations; I am able to move away from my comfort zone and extend my own learning and understanding. I am able to observe how to explain key concepts and the changes needed in explanations for the pupils of different abilities.
What or who is informing my own practice? I have found that the people who have influenced my teaching practice are primarily the mentors both at University and at the schools. The constructive feedback from lesson observations is invaluable and enables oneself to adjust their own practice to incorporate new ideas and strategies. By being able to observe a variety of different teaching staff, I am able to gain greater skills and attitudes on how to approach new topics, behaviour management and strategies to cope with 'the bad lesson experience'.
Another good source towards informing my practice has been the Schemes of Work related to different examination boards. My first placement covered the AQA syllabus, the system was well prepared and set up. However, my second placement uses the
Kangaroo Maths syllabus for KS3 pupils and a variety of OCR syllabuses covering the linear and modular exam boards for the KS4 pupils.
My uses of different teaching styles are also developing within my teaching repertoire. I like to ensure that my lessons incorporate different aspects of the Visual, Auditory, Kinaesthetic and Tactile styles to ensure that all pupils are able to access relevant information in their own preferred learning style ensuring that Every Child does Matter. Psychologists and teaching specialists first developed the VAK concept, theory and methods in the 1920's. Howard Gardner (1983) suggests that:
'The VAK model provides a different perspective for understanding and explaining a person's preferred or dominant thinking and learning style, and strengths'.
So what about the future of mathematics and ICT?
British Educational and Communications Technology Association (BECTA), now axed under the new government, was one of the main drives into getting computers into school.
BECTA ran Home Access, the former Labour government's scheme to bridge the "digital divide" by ensuring all children have a computer at home especially amongst the poorest families in the United Kingdom. (BECTA, 2010)
A SecEd symposium held in March 2010 stated that:
'Students today are growing up in a digital age, communicating and learning via technology now more than ever before. To prepare students for future success, schools are embracing technology to enhance the teaching and learning experience and transform
educationâ€¦to truly transform learning schools must integrate technology tools that can broaden the education experience and connect the classroom with the 21st century'.
Within my first placement, the school used 'Mathswatch' as a resource to use in lessons and at home by the pupils. I was also able to use the graphics calculators to help pupils
recognize and identify different aspects of graphical work. This was very beneficial for the pupils and they enjoyed using them. My second placement, is a Technology College, where ICT is far more extensive. Interactive whiteboards are in all classrooms, ICT is a separate department as recommended in the Smith Report 2004. The use of Mymaths enables homework to be utilized as a resource that gives instant assessment. This is another area of experimentation for me, being able to set homework using ICT and getting instant accessible results, this could be useful when writing reports and compiling spreadsheets of the pupil's assessment results.
The Smith Report (2004) recognises the relevance of mathematics to everyone and I feel stresses the importance of why mathematics still deserves its place in the school curriculum:
"It is importance for its own sake, as an intellectual discipline; for the knowledge economy; for science, technology and engineering; for the workplace; and for the individual citizen. (Chapter 1, page 3, point 0.12)
This point emphasises the value of mathematics within various different aspects of everyday life. We need to ensure that pupils leave school with the appropriate mathematical skills that will enable them to cope in their future careers.
My final point comes from a rabbinical saying:
'Don't limit a child to your own learning, for that child was born in another time.'
Word count (2155)
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