Throughout my teaching career, I have endeavoured to be a reflective practitioner who considers all attributes of my practice. Ellis (2012) emphasises the importance of the generality of teachers reflecting. Dadds (2009) extends this argument further, although the benefits of this seem to be constrained merely to theory and research. Weick and Sutcliffe (2001) highlight that the transition from theory to practice is not necessarily linear, something which will be disseminated throughout the course of this essay. Ofsted (2012) assert the further distinction of a mathematics teacher retrospectively considering their practice, due to the inherent complexity of the subject. Throughout my degree, I have continually tried to meet the standards stipulated by the Department of Education in meeting Qualified Teacher Status. I have made sufficient progress in meeting all standards, although there are two notable areas, which form the basis of this essay, that I am relatively deficient in. I have made reasonable progress in meeting standard 3 ‘Demonstrate good subject and curriculum knowledge’ (See highlighted GTA matrix in Appendix 1, p.15), but it is still something which I wish to improve in, specifically the sub-division of it which states that teachers should ‘promote high standards of literacy, articulacy and the correct use of standard English, whatever the teacher’s subject’ (DfE, 2014a). DfE (2012) emanate the significance of teachers being proficient in literacy and having the ability to incorporate it in their subject, particularly in stereotypically opposite subjects like English and Mathematics. This is stoked further by incumbent curriculum reforms with the new Mathematics programmes of study containing an increasing amount of worded questions, which require pupils to have the necessary literacy acumen to ‘decode’ them and plan a sequence of logical steps in order to answer it correctly (DfE, 2014b). This, coupled with my intrinsic interest in English, provides the rationale for my first topic of discussion being linking literacy with mathematics.
The second area concerns Standard 2: ‘Promote Good outcomes and progress by pupils’ (DfE, 2014a), something which I have made average progress in (See shaded GTA matrix in Appendix 1, p.15) and one which forms the basis of my targets at the end of my Year 2 placement (See the target entitled ‘promote good progress and outcomes- give feedback on pupil work’ in Appendix 2, p.16). Black and Wiliam (2009) make the valid point that differentiation and formative assessment are key components of this. Bartlett (2013) believes that targeted intervention with certain groups of pupils encompasses both of these elements, which seems a sound basis for selecting it as my second area of focus. Nevertheless, Faultley and Savage (2008) warn against the possible dangers of extrapolating pedagogical strategies gleaned from intervention to a class with a considerably larger amount of pupils, a variable which I will have to consider in my practice.
Linking Literacy with Mathematics
Sperry (1961) put forward a theory of hemispherical dominance, where people are predisposed to one side of the brain. Under the stipulations of his theory, people are ‘right-brained’ if they are inclined towards creative subjects like English and the Arts and ‘left-brained’ if they have a preference for more logical pursuits like Mathematics and Science. This seems to infer that English and Mathematics are at opposite ends of the cognitive spectrum, with a dearth of links and interrelating properties. However, this is contradicted by my experiences on placement. My subject mentor is an arch proponent of Literacy, making regular links to it within her Mathematics lessons. This correlates with my own educational philosophy and government reforms. This is illustrated perfectly by the lesson plan for a Year 11 class outlined in Appendix 3 (see highlighted segments in the lesson plan, pp.17-19). Although the topic taught in the lesson, Algebraic Division of Polynomials, on the surface seemingly had no literacy connections; I determined an abundance of these in my plan of and deliverance of the lesson.
Firstly, I constantly reminded and ingrained into the pupils the meanings of and the mathematical vocabulary involved with the topic (such as ‘Solve’, ‘Order’, ‘Quotient’ and ‘Polynomial’) and even dedicated a specific part of the lesson to allow the pupils to fully comprehend these topics. There seems to be a multitude of positive implications associated with this. Askew et al. (1997) purport that teaching in such a manner will imply pupils to cultivate a connections orientation and see the links and interweaving nature of mathematical topics, which could be advantageous in helping them to achieve a higher grade. This does seem to be reaffirmed in my lesson, as one explanation of Algebraic Division in the lesson involved a Formula Triangle (similar to the ones pupils regularly experience in GCSE Physics like the well-renowned Mass, Density and Volume) in that the Polynomial was a product of its constituent factors obtained from the Algebraic Division. This seemed to elicit good understanding from pupils (see lesson evaluation of Appendix 3) and they responded to the explanation well. Savage (2010) cites the multi-faceted nature of Mathematics and the abundant cross-curricular links which can be derived from it. However, this could also be construed as a deficiency of my Literacy-orientated approach. By concentrating on Literacy so much, this could distort the focus of my lesson and detract from the actual mathematics being taught. Johnston-Wilder et al. (2010) consider effective planning and a clear focus in the lesson as being the domains of a successful Mathematics teacher. Any implementation of such strategies in my future practice should include careful consideration of Literacy being secondary to, rather than superior, to the Mathematics being taught.
Savage’s (2010) original point of the numerous cross-curricular connections in Mathematics could be another weakness of employing a plethora of Literacy links within a lesson. The topic being studied could be shrouded in ambiguity if the teacher tries to approach it from too many dimensions. However, this strategy could still be of a bipolar nature to the students in having positive and negative implications on their learning. Lave and Wenger (1990) argue that integrating Literacy effectively within a lesson can result in ‘situated learning’ where the pupils see the relevance of the topic in real-life. In this case, this lesson formed part of a Further Mathematics qualification (which bridges the gap between GCSE and post-16 study of Mathematics), so the ‘real-life’ applications may be confined to the pupils seeing a link between their GCSE course and A-Level Mathematics, instead of an actual real-life scenario. However, French (2004) makes the reasonable assumption that predominantly, Algebra topics in Mathematics have a paucity of links with the real world and richer Geometry and Data modules may be more successful in proliferating such links. This seems to be evidenced by the Algebra-heavy content of the Further Mathematics course. Nevertheless, Chambers (2008) hypothesises that this technique could motivate students more, particularly with this group of students who enjoyed mathematics and had an intrinsic interest in it (Coon and Mitterer, 2010). However, this raises another noteworthy point: pupils who do not have the same passion for mathematics may not be captivated by this strategy. This necessitates me thinking of alternative ways of linking literacy within my lessons for different classes.
A viable strategy of doing this is incorporating more discussion and collaborative activities within my lessons. This is something which I tried to promote on placement, with reasonable success. Principally, I achieved this through stimulating and challenging activities (see the extension of the Year 7 starter resource in Appendix 4, p.20). In theory, this should have allowed pupils to progress to highest level of Bloom’s (Krathwohl, 2002, p.212) Taxonomy (‘Evaluation’) where they were able to synthesise concepts and assimilate information effectively. This was partially replicated in practice; some pupils were able to reach this level of cognition, whilst others stagnated in the rudiments of the lesson. However, it could be assumed that pupils gained a least a modicum of the ‘relational’ understanding (long-term appreciation of concepts) prized by Skemp (1977, p.23) instead of the inferior instrumental understanding where pupils only have a superficial grasp of content. I found that the variation of ability in my classes was a significant barrier to the success of this strategy. For instance, my Year 7 bottom set encompassed a wide range of abilities, which made implementing such tasks problematic. Conversely, in my Year 11 Further Mathematics class, ability levels were much more uniform, which meant that most pupils were able to tackle the said tasks with ease.
To alleviate these problems, I followed the principles of co-operative learning, an authoritative learning strategy devised by Kagan (2009). This recommends that you place lower-ability pupils with students of a relatively higher capability within collaborative tasks to ensure that the optimum level of learning takes place. This is something which I instigated frequently within my lessons (See the Year 7 lesson plan in Appendix 5, p.21) as I believe in the ideals of cooperative learning. Vygotsky (1978) purports that this strategy will increase the ‘Zone of Proximal Development’ (the difference between what a child can do unaided and with the assistance of a More Knowledgeable Other (MKO) like a fellow peer or Adult) as the MKO (slightly more able pupil) helping the struggling pupil is not to dissimilar in terms of cognition. The benefits of this seems to be certified further by Dale’s (1970) Cone of Experience (See Appendix 6, p.22) which affirms that we learn ‘90% of what we teach others’. The model also asserts that we learn ‘50% of what we discuss’, which adds further justification for the inclusion of discursive activities in my Mathematics lessons. Theoretically, this should deepen and accentuate pupils’ knowledge of the content being caught whilst building rapports in the class. This was partially replicated in my classes as the pupils undoubtedly responded well to someone else explaining the content to them instead of a teacher. However, this seemed to slow down the progress of the brighter pupils, which is a common failure of peer-mediated instruction (Chan et al., 2009, p.877). Blatchford (2013) surmises that the effect of this strategy may be better in smaller class sizes, a logical, but impractical point, given the ever-increasing class sizes in contemporary education. Fundamentally, my strategies of incorporating Literacy into Mathematics have been fairly successful, but require further consideration to make full use of them.
Kutnick and Berdondini (2009) make the reasonable assumption that small-group teaching may be more effective than whole -class teaching due to pupils receiving more attention. A further advantage seems to arise from my own experience on placement, that in some ways, it is more effective than teaching a class as normal and more enjoyable for the pupils.
Cowley (2010) supposes that behaviour management in smaller groups like intervention is easier to manage in than in whole class situations. There seems to be an element of truth in this, particularly in what have observed in my Year 3 placement, where pupils seem to enjoy the closer interaction with the teacher compared to the often limited contact the student and teacher has in a normal class size. This could be due to the difference in formality between Intervention and Whole-Class teaching: teaching a whole class may require a formal tone rather than a more informal register than intervention is conducive to (Joos, 1972). Pupils may prefer seeing a different side to the teacher, which I regularly demonstrated in intervention and in extra-curricular activities. However, a practitioner to should take steps to ensure that the level of formality is not to low, as this may result in pupils becoming distracted and off-task and not fully engaged in the task. Ollerton and Sykes (2012) feel that this phenomenon is particularly pertinent in Mathematics lessons, a subject pupils stereotypically do not enjoy. Joos (1972) outlines a viable strategy to overcome this obstacle: that a teacher should use formal register to establish their authority at the start of a lesson before gradually progressing to displaying a consultative (duality of informal and formal lexis) tone once the class are on task. This seems to be a realistic method to employ which could be more simplistic to achieve in smaller groups of pupils (intervention) rather than in a larger class size. However, achieving the correct equilibrium of formality certainly seems to be an important variable. At my placement school, Year 11 students attend an optional after-school mathematics revision session to complete extra practice for their GCSE. However, due to the relaxed nature of the lesson, pupils often stray if task and only the most dedicated of pupils fulfil the true purpose of the intervention session. Haggarty (2002) suggests that in any intervention session, pupils need to have some level of autonomy and independence to fulfil the objectives of the scheme. This could be related to Goleman’s (1996) theory of Emotional Quotient: that pupils need to have the intelligence and maturity to show a commendable work ethic when they are not extrinsically motivated by a disciplinarian figure such as a teacher. However, it is questionable, as in the case of the scenario described above, whether pupils have the level of maturity commensurate with this level of responsibility. Biggs (1999) observes that the majority of independent learning undertaken by students occurs after secondary school, predominantly in their University degree. This seems to infer that expecting such diligence from pupils in intervention sessions may be unrealistic. However, this issue could at least be partially rectified by a strong teacher presence in an intervention session. In the compulsory intervention sessions I have observed and helped out in on placement, practitioners display a suitable measure of formality whilst simultaneously converging to the slightly informal register that a smaller class necessitates. Fundamentally, if a requiem of formality was applied in all intervention sessions, the success of them may be greater. However, ensuring the desired outcome of intervention may be due to an additional variable: the skills and expertise of the practitioner (s) overseeing it. There is a lack of an intervention specialist at my current placement school, a role which has had considerable success in other schemes like the Year 7 Numeracy and Literacy Catch-Up program (DfE, 2014c).
A common theme in my reflection is that I have developed in practice since my Year 2 placement and have achieved some successes, but further refinement of my practice is required. I have shown some competency in linking Literacy with Mathematics, primarily because of my proficiency in my subject, and this is something which will be improved with future practice. I have also gained more experience in Intervention strategies and have a more sophisticated grasp of the intricacies involved within such schemes. This is something else which I will enhance with more experience and knowledge. However, these 2 areas are just a portion of my practice as a teacher. To become an effective teacher, I will have to continuously improve on these variables in my future practice as well as a whole array of other variables such as subject knowledge, behaviour management and differentiation. Fundamentally, coupled with stringent reflection and deeply considering the totality of my practice, I feel that I have the attributes to become a successful teacher of Mathematics.
Askew, M., Brown, M., Rhodes, V., Wiliam, D. and Johnson, D. (1997) Effective Teachers of Numeracy: Report of a study carried out for the Teacher Training Agency. London: King’s College, University of London.
Bartlett, J. (2013) Becoming an Outstanding Mathematics Teacher. London: Routledge.
Biggs, J. (1999) Teaching for quality learning at university. Buckingham: Society for Research into Higher Education and Open University Press.
Black, P. and Wiliam, D. (2009) Developing the Theory of Formative Assessment. London: Springer.
Blatchford, P. (2013) Is it true that class size does not matter? A critical review of research on class size effects. Public Seminar at Hong Kong Institute of Education, March 2013.
Chambers, P. (2008) Teaching Mathematics: Developing as a Reflective Secondary Teacher. London: Sage.
Chan, J. M., Lang, R., Rispoli, M., O’Reilly, M., Sigafoos, J. and Cole, H. (2009) ‘Use of peer-mediated interventions in the treatment of autism spectrum disorders: A systematic review’, Research in Autism Spectrum Disorders, 3, pp. 876-889.
Coon, D. and Mitterer, J. O. (2010) Introduction to Psychology: A Gateway to Mind and Behaviour. New York: Cengage Learning.
Cowley, S. (2010) Getting the Buggers to Behave. London: Continuum.
Dadds, M. (2009) From Passionate Enquiry to Loving Detachment. The Sage Handbook of Educational Action Research. Los Angeles: Sage.
Dale, E. (1970) ‘Cone of Learning’, Theory into Practice, 9 (2), pp. 89-95.
Ellis, V. (2011) Learning and Teaching in Secondary Schools. 4th edn. Learning Matters: Exeter.
Faultley, M. and Savage, J. (2008) Assessment for Learning and Teaching in Secondary Schools. Learning Matters: Exeter.
French, D. (2004) Teaching and Learning Algebra. London: Continuum.
Goleman, D. (1996) Emotional Intelligence: Why it can matter more than IQ. London: Bloomsbury.
Great Britain. Department for Education (2012) Schools’ White Paper: The Importance of Teaching. London: DfE.
Great Britain. Department for Education (2014a) Standards for Meeting Qualified Teacher Status. London: DfE.
Great Britain. Department for Education (2014b) New GCSE Mathematics Programme of Study. London: DfE.
Great Britain. Department for Education (2014c) Year 7 literacy and numeracy catch-up premium: guide for schools. London: DfE.
Great Britain. Ofsted (2012) Mathematics: Made to Measure. London: Ofsted.
Haggarty, L. (2002) Teaching Mathematics in Secondary Schools: A reader. Oxon: Routledge Falmer.
Johnston-Wilder, S., Johnston-Wilder, P., Lee, C. and Pimm, D. (2010) Learning to teach Mathematics in the Secondary School: A companion to School Experience. London: Routledge.
Joos, M. (1972) Language and cultural diversity in American education. Englewood Cliffs, NJ: Prentice-Hall.
Kagan, S. (2009) Kagan Cooperative Learning. San Clemente, CA: Kagan Publishing.
Krathwohl, D. R. (2002) ‘A revision of Bloom’s taxonomy: An overview’, Theory Into Practice, 41 (4), pp. 212–18.
Kutnick, P. and Berdondini, L. (2009) ‘Can the enhancement of group working in classrooms provide a basis for effective communication’, Cambridge Journal in Education, 39, 1, pp. 71-94.
Lave, J. and Wenger, E. (1991) Situated Learning: Legitimate Peripheral Participation. Cambridge: Cambridge University Press.
Ollerton, M. and Sykes, P. (2012) Getting the Buggers to Add Up. London: Bloomsburg.
Savage, J. (2010) Cross Curricular Learning and Teaching in the Secondary School. London: Routledge.
Schön, D. A. (1991) The Reflective Turn: Case Studies In and On Educational Practice. New York: Teachers Press, Columbia University.
Sperry, R. W. (1961) ‘Cerebral Organization and Behavior: The split brain behaves in many respects like two separate brains, providing new research possibilities.’ Science, 133 (3466), pp. 1749–1757.
Skemp, R. (1977) ‘Relational Understanding and Instrumental Understanding’, Mathematics Teaching, 77, pp. 20-26.
Vygotsky, L. S. (1978) Mind in society: The development of higher psychological processes. Cambridge, MA: Harvard University Press.
Weick, K. and Sutcliffe, M. (2001) Managing the unexpected: assuring high performance in an age of complexity. San Francisco: Jossey-Bass.