Essay on Mathematics Subject Knowledge Portfolio and Rationale

Teachers have a variety of roles in the classroom: role models, motivators, social workers, innovators and even substitute parents (in a case of loco parents). It could be argued however that the teacher’s most important job is to ensure that pupils learn what is being taught (Green and Leask 2009 pg 11). It is natural that teachers should have a deep and secure knowledge of the curriculum to ensure that pupils learn successfully. Despite unprecedented recent exam results which saw nearly a quarter of all GCSE papers awarded either an A or A* (Telegraph 2011) and an increase in the number of students taking core subjects as well as A-Level maths entries rising approximately 40 per cent in the last five years (Guardian 2011), there are still widespread concerns that students suffer from ‘test teaching’ and are not sufficiently mathematically prepared for future employment (Ofsted 2008). According to Ofsted (2008) subject content knowledge (deep knowledge of curriculum and mathematical methods) and subject pedagogical knowledge (a combination of : an awareness of pupils learn, how to teach a topic and being able to explain answers mathematically) are essential for effective mathematics teaching. Ofsted (2008) also highlights worries over the level of primary and secondary school teachers subject knowledge. Williams and Ryan (2007) again implies the importance of subject content knowledge as it suggests that some secondary school maths teachers have the same misconceptions as their pupils. It may also be beneficial that teachers are able to explain not only how to get an answer, but why the answer is that. This may enable students to develop a relational understanding (Skemp 1977) of a topic which may give them long-term security of that knowledge which would be useful to them in their chosen vocation. This may be difficult to do with certain topics. For example, when you subtract two negative numbers you obtain an answer which is greater than the original two negative integers. This is regarded as an arbitrary convention that a student is required to learn (Hewitt 1999) which may be difficult to explain mathematically. A possible solution for this is suggested in the first of my critical examples in my portfolio which illustrates how the numbers are represented in a real life context using temperatures of notable cities. The modest distinction between the terms position and operation is also emphasised. This may give pupils something to relate to and a deeper understanding of a topic and the fundamental mathematical properties behind it. To be able to achieve this, a teacher would have to have a large amount of subject knowledge, which may have to stretch beyond the mathematics curriculum.

I have created a portfolio which I hope will enable me to deepen and increase my subject knowledge so that it is at the level which a teacher needs. I am going to discuss how the portfolio has helped me to start to develop an appropriate level of subject content and subject pedagogical knowledge.

I audited my knowledge of the entire GCSE curriculum through a variety of sources including:

• Attempting GCSE Exam Questions
• Completing questions from a GCSE Textbook
• Reading and making notes from a GCSE textbook
• Reviewing my knowledge of topics using the NCETM portal
• Teaching the topic to a fellow student
• Creating a worksheet resource (that was originally intended for use as teaching practice) that I completed myself

This was done to try and ensure maximum coverage of the curriculum and that I had considerable experience of GCSE-standard questions. This enabled me to gain a perspective of the format of exam papers and possibly gain a greater understanding of the process that students go through in attempting to answer exam questions. I feel this enabled me to consolidate my understanding of the GCSE curriculum quite well although as I only conducted my self audit in accordance with one exam board (AQA) the scope of my audit may have been limited slightly. However, I feel this auditing my knowledge this way may have other benefits. By revising some topics with a friend, I have tried to appreciate different methods of revision (so I could perhaps see it from a learners point of view) and I also strived to gain the benefits of peer assessment as I have learnt different methods of answering a question. For example, when addressing the first issue in my audit, N6.4 Straight line graphs, I was attempting to work out the gradient of a line numerically but was struggling to do so. However, my partner (who I would later teach the topic to consolidate my understanding) showed me that by drawing the graph and gaining a visual representation of the problem made it easier to work out the answer. I also tried to develop an increased awareness of the different methods of explaining an answer by attempting question two of the subject knowledge audit discussion questions where I tried to prove that 10^0 = 1 by as many ways as possible. Hopefully this has allowed me to significantly increase my subject pedagogical knowledge as there will be inevitably many different learning styles within a class and a teacher may have to be prepared to explain things in a different way to certain groups of children. An awareness of the different ways to answer a question may allow the teacher to help students to learn maths through exploration and discovery learning (Bruner 1967) and develop mathematical understanding by being able to select and apply the correct method to answer an unfamiliar question (Ollerton 2009). Working collaboratively allowed me to try to gain the benefits of 2 learning styles suggested in Dfes (2007):

• Behaviourist: As my partner gave me feedback on both my subject content knowledge (by helping me understand the topic and working through some questions with me) and my subject pedagogical knowledge (by showing me alternative representations of questions and assessing my teaching style) which I tried to respond to by following their suggested improvements of my teaching style and subject knowledge.
• Social Constructivist: As I worked with a partner, I was able to learn from them as they addressed my misconceptions about how to find the gradient of a line.

By attempting to increase my subject content knowledge in this manner, I have tried to increase my understanding of maths as much as possible. This would hopefully have a direct benefit in lessons on the children. Vygotsky (1978) suggests that learning occurs in the ‘zone of proximal development’ (the difference between what a pupil learns working independently or working in collaboration with a ‘more knowledgeable other’ such as a teacher). This would seem to suggest that the greater a teacher’s subject knowledge is, the more the zone of proximal development increases and hence greater learning takes place for the pupil. With continually trying to extend my subject knowledge as much as possible in the future, this may allow me to help as many children as possible reach their full potential.

For the topic of angles I:

• Constructed a spider diagram
• Created a sequential flow diagram
• Tracked the topic from KS1 to KS5

It may have been advantageous to draw the spider diagram as it reminded me of all the relevant curriculum content associated with the topic as it tracking the topic from KS1 to KS5. Although these two activities may have had useful applications in increasing my subject content knowledge (and possibly producing schemes of work) they did not really develop my subject pedagogical knowledge. On the other hand, creating a sequential flow diagram may have been beneficial in that it allowed me to see the progression of ideas in a topic and the connections that could be made between them such as circle theorems and geometric proof. As well as having the possibility of altering my teaching style to have more of a connectionist (Askew 1997) orientation and promote links between topics, it may allow the pupils to learn more effectively: if teaching is structured by the conceptual structure of a topic and ideas are presented in a logical sequence, this may allow pupil’s brains to link new ideas to old ones and thus retrieve and recall information better (Ausubel 1968, Gagne 1977 and Stones 1992 as cited in Burton 2009).

Overall it would seem that teachers need an extensive amount of subject content knowledge and subject pedagogical knowledge to be able to teach effectively and this knowledge would probably have to go beyond the level that pupils need to pass their exams. I have attempted to start to develop my subject content and pedagogical knowledge by creating a detailed portfolio which has required me to complete tasks such as rigorously auditing my knowledge of the GCSE curriculum and looking at topics such as angles, straight line graphs and directed numbers in a vast amount of pedagogical detail. In the near future and throughout my career as a teacher, I will carry on developing my subject content and pedagogical knowledge so that it is at the required standard to help students learn as much as possible.

References

Books

• Ollerton, M. (2009) The Mathematics Teacher’s Handbook. London: Continium
• Ryan, J. and Williams, J. (2007) Children’s Mathematics 4-15: Learning from errors and misconceptions. Maidenhead: Open University Press
• Bruner, J.S. (1967). On knowing: Essays for the left hand. Cambridge, Mass: Harvard University Press.
• Vygotsky, L.S. (1978) Mind and society: The development of higher mental processes. Cambridge: Harvard University Press
• Green, A. and Leask, M. (2009) ‘What do teachers do?’ in Capel, S., Leask, M. and Turner, T. (eds.) Learning to teach in the secondary school: A companion to school experience. Oxon: Routledge, pp. 9-21
• Burton, D. (2009) ‘Ways pupils learn’ in Capel, S., Leask, M. and Turner,T. (eds.) Learning to teach in the secondary school: A companion to school experience. Oxon: Routledge, pp. 251-266

Websites

• Shepherd, J. (2011) The Guardian. Available at: http://www.guardian.co.uk/education/2011/aug/18/a-levels-boys-close-maths-sciences-gap/ (Accessed: 24 December 2011).
• Paton,G. (2011) The Telegraph. Available at: http://www.telegraph.co.uk/education/educationnews/8720988/GCSE-results-2011-quarter-of-exams-graded-an-A.html/ (Accessed: 26 December 2011).

Articles

• Hewitt, D. (1999) ‘Arbitrary and Necessary Part 1: A way of viewing the Mathematics Curriculum’ For the learning of mathematics, 19 (3), pp. 2-9, [Online]. Available at: http://www.eric.ed.gov/ERICWebPortal/search/detailmini.jsp?_nfpb=true&_&ERICExtSearch_SearchValue_0=EJ607163&ERICExtSearch_SearchType_0=no&accno=EJ607163/ (Accessed: 28 December 2011).
• Askew, M., Brown, M., Rhodes, V., Wiliam, D. and Johnson, D. (1997) Effective Teachers of Numeracy in Primary Schools: Teachers’ Beliefs, Practices and Pupils’ Learning. Paper presented at the British Educational Research Association Annual Conference. (September 11-14 1997: University of York)
• Skemp, R. (1977) Relational Understanding and Instrumental Understanding. Mathematics Teaching 77: 20-26.

Government publications

• United Kingdom. Ofsted (2008) Mathematics: Understanding the Score [Online]. Available at: http://www.ofsted.gov.uk/resources/mathematics-understanding-score (Accessed: 30 December 2011).
• United Kingdom. Department for education and skills (2007) Pedagogy and Personalisation [Online]. Available at: http:// exeter.ac.uk/download.php?id=14921 (Accessed: 31 December 2011).