Essay on Investigating Learning in Mathematics

Mathematics is undoubtedly a varied subject area, where many approaches to learning exist, with numerous theoretical underpinnings and rationale, all of which have some relevance to contemporary education. However, this assignment chooses to focus on two authoritative theories of pedagogy and learning, communicating the theory upon which they are based, simultaneously working out how they apply to the author’s own practice and in the school environment, particularly in accordance with mathematics. These two theories are constructivism and behaviourism, eminent approaches which have been applied to education for several decades. They can be linked with the subject of Mathematics, which has historically been a subject which is disliked by pupils, normally because it is quite hard for pupils to understand and it is seen as an activity which only a few students can do, such as those in the ‘top set’, who are normally perceived better by their teachers, although labelling children in such a way can make them more anxious and possibly derail their academic performance (Nardi and Stewart, 2003). Constructivism is different in that it is generally better received by students, and this is where students build their own set of problems and knowledge from what the instructor or teacher provides them with (Leonard, 2002).

Behaviourism and Constructivism are clearly different to each other, although they may not be complete opposites with some similarities between the two, which are expressed at a later stage of this assignment. Broadly, constructivism is a learning theory which argues that pupils ‘construct’ (which explains the origin of the word ‘constructivism’) understanding based on their interaction with the teacher which may be in a very individual manner. Jean Piaget is an eminent scholar in this field and he thinks that children make sense of the world around them by linking to prior knowledge which they have obtained and also assimilating this with their interpretation of environmental events based on the pre-existing cognitive structures which they have (Leonard, 2002). Capel, Leask and Turner (2002) talk of the beliefs that pupils may hold of things, which are mental structures abstracted from experience, which could explain why so many pupils seemingly dislike mathematics, as they have a negative view built up by the subject and have learned to dislike it. Constructivism also relates to the social environment (i.e. the classroom) which pupils exist in. Leonard (2002) references Bandura’s Social Learning Theory which suggests that children learn by copying the behaviour of others, particularly their fellow pupils and adults. In Piaget’s theory, this process is known as accommodation and assimilation, he also suggests that pupils go through stages of cognitive development (Jordan, Stack and Carlile, 2008). This point links slightly with constructivism, although this is more typical of cognitivism. Kozulin (2003) cites Vygostky’s work, arguing that the teacher has a role to play in the child’s cognitive and social development, through enhancing the child’s zone of proximal development (ZPD), which is difference between the child’s actual performance and their learning potential.

The process of expanding the child’s ZPD is learning:

Figure 1- Diagram of the intricacies of the Zone of Proximal Development

Behaviourism feels that pupils are influenced by other figures, but mainly they feel that it is the teacher who influences the pupils the most. Skinner is the main founder of this and he feels that instructors can provide learners with a set of learning outcomes and be able to fine tune and alter them as they wish (possibly through planning or other such mechanisms) to control both the behaviour (through punishments and rewards) and learning of the child (Leonard, 2002).

Taylor (2011), a widely-renowned government advisor of behaviour, articulates that rewards and punishments should be visibly displayed in each classroom, also advocating consistency in approach, with strong support from leadership and head teachers for practitioners imperative (Parliament, 2011). The implications of negative behaviour are outlined further by DfE (2010), in their authoritative White Paper, who recognise that behaviour management is one of the most important variables in the classroom and the perception that they will encounter challenging behaviour is something which deters prospective teachers from entering the profession, which could deprive education and the pupils within it from some talented and potentially inspirational teachers. Consequently, behaviourism may be imperative within the classroom, as if it is absent, then children may not learn the content as required. This does not take into account other variables which affect learning in the classroom such as motivation, mindset of pupils, personality and also how they interact with each other and the teacher, which constructivism considers more. Indeed, this is one of the main criticisms of behaviourism, that is focussed more on the learning output achieved by the child (what they learn and the numerical test scores they get), instead of their mental state and well-being. Despite this, it seems appropriate for me to be able to consult both behaviourism and constructivism to inform my teaching, so I can display behaviourism in regulating the behaviour of children and constructivism to get them interacting and cater for individuality.

I have displayed the two theories of my practice, which will become evident in the discussion which is outlined below. However, something which I also took into consideration was the spectrum of learning styles encompassed within my class, as it seems unwise to categorise pupils as having one learning style, instead cognition and learning is based on individuality and recognising that a child learn in different ways (Capel, Leask and Turner, 2013). Whilst recognising that children do indeed learn in different ways, it still seems relevant to consider the main theories of learning to assist my practice.

As previously mentioned, the constructivist approach emphasises the social element of children’s learning and how they interact with adults seems to have a significant impact on their learning. In Vygotsky’s theory in constructivism, this is known as ‘scaffolding’, where the teacher or adult works with the child, giving them the responsibility to complete a task which they may not have been able to do on their own (Verenikina, 2003). I try to do this within my own classroom, which is apparent through the many opportunities which I integrate into my lessons for discussion and interaction so that pupils can receive assistance from me and have access to the dialogue which will allow them to learn, although they may also be able to sustain this with other pupils. My practice is gradually changing. At the start of the placement, my lessons tended to be teacher-orientated, with more time dedicated to me talking and explaining concepts to the class as a whole, rather than individually and in groups which is the more student-centred style of pedagogy which I display now, which could allow my pupils to be successful learners who enjoy learning and achieve in the classroom (Johnston-Wilder et al., 2010). The effect of this could be increased if pupils are allowed responsibility more within a lesson, something which I have tried to incorporate into my lessons more. However, even though I displayed some of the ideals of the constructivist approach in designing the lesson and the learning activities which took place within it, I also had to adopt some of the principles of the behaviourist approach in keeping pupils on task and ensuring that there were no lapses in concentration which could occur in a lesson environment like this which is open and where discussion and dialogue activities were apparent. Prior to progressing to a more constructivist approach in the overall style of pedagogy I adopted, my approach was inclined more towards behaviourism as I changed the tactics I employed (punishments and rewards, along with the formality which I displayed) systematically to elicit the best possible response from learners (Leonard, 2002).

This mixed approach was evident in a lesson which I taught with pupils early on in my placement. The topic was a complex algebraic one (solving linear simultaneous equations), which the pupils (a Year 10 class) had little experience of in the past. Joos (1961) outlines 5 styles of spoken English: static, formal, consultative, casual and intimate. Perhaps mirroring the formal nature of the topic, I was quite formal in my approach and imposed strict regimes on the pupils, with clear adherence to the school’s behaviour policies and referencing them several times throughout the lesson. Here, I was sticking quite rigidly to the principles of behaviourism in believing that students could be regulated and conditioned by a strict set of rules and practices. However, this belief was mistaken as some students showed disinterest in the topic and were not appreciative of my teaching style. The learning style which I expected the pupils to display was a passive one, one more typical of higher education institutions rather than in a GCSE mathematics classroom. This learning style is typical of a behaviourist approach which places the teacher, rather than the student, at the forefront of the learning procedures which happen within a lesson. On reflection, I should have been more constructivist in nature and allowed pupils to learn through active or discovery learning, concentrating more on the intrinsic motivation of the child where they are actively involved in the formation of content (Leonard, 2002) so they could interpret the topic in a way which was mathematically correct, but still relevant to them. This may have been superior to the prescriptive method which I had articulated in my exposition (a step-by-step method involving making one set of coefficients equal to eliminate them, before subtracting them to form a linear equation where the value of one unknown could be found out and substituted into one of the original equations to find the value of the remaining unknown). It seems that there may be a tension between theory and practice in education, which could be particularly effected by the practical training of teachers (Oonk, 2009) and how they experience placement. This references the fact that how I predicted the pupils to react was not how they actually behaved. It is evident that a practitioner should be reflective in nature to pre-empt any possible difficulties which may have occurred in the lesson. I only adopted one theoretical stance (behaviourism) within my lesson which was insufficient and was not something which I applied in future lessons. Here, I went through a process of experiential learning where I experienced the event (misbehaviour among pupils), reflected upon my experiences (realising one school of thought was insufficient) and generalising them to a certain degree (as I am modifying my practice for all lessons in the future), before finally attempting to apply them to subsequent lessons (Kolb, 1984).

Although French (2002) feels that algebra is one of the more difficult topics in mathematics and contains many formal proofs, other mathematical topics may work better when used in conjunction with constructivist-based approaches.

This was exemplified in another lesson which I taught, where the topic was a statistical investigation which delved into the relationship between someone’s height and their weight with a Year 10 class. Pupils were allowed to leave their seats and circulate the room to collect measurements from other members of the class which they recorded in a table, subsequently plotting the values on a scatter graph and drawing the line of best fit on to plot the trend of the data. Prior to this task, the pupils were given a series of steps to follow but these were not procedural or rigid: there was some flexibility within them for pupils to personalise their responses. Fundamentally, I was implementing a constructivist approach so that pupils could learn independently, with only prompts and gentle facilitation from me.

Jonassen (1994) values the importance of the learning environment and the effect which constructivism can have on it, theorising that it can ‘support the collaborative construction of knowledge through social negotiation’ (p.35). However, DCSF (2008) make the valid point that the environment which learning takes place in should be ‘enabling’ in that there should be a suitable bond and rapport between the teacher and students in order to foster students’ enquiry and independent thinking. Compared to the lesson which this piece mentioned at a previous stage, this lesson was much further into my placement and I had established a firm rapport with the students in the intervening period between the two lessons. The literature consulted seems to suggest that this bond allowed me to introduce an activity which was more constructivist in nature and enabled the pupils more independence in accessing and exploring the content as I had built a good working relationship with the pupils and trusted them to work relatively unsupervised. However, I did not use an exclusively constructivist approach, as my pedagogy did include some attributes of behaviourism. However, compared to the strong methods of discipline that I had employed in the first lesson mentioned in this assignment, in this lesson I was more of a facilitator, gently guiding students towards solutions and largely allowing them to work independently. I allowed pupils to make sense of the task themselves and build their own interpretation of it, an approach which Birenbaum (2002) conceptualises as self-directed active learning. Although Birenbaum’s research was predominantly conducted in primary schools, the principles of it can still be applied to secondary education. Such an approach could have a multitude of positive implications, with pupils becoming more intrinsically motivated as they are exploring the content in a way which is meaningful and relevant to them, obtain independent learning skills which could be useful for higher education, where teachers help students towards a solution, rather than do the work for them (Biggs, 1999) . The process of learning for pupils was helped further by the topic of the investigation which was relevant to real life and was something they could relate to. Lave and Wenger (1991) imply that ‘situated learning’, the context in which learning takes place, can be a highly productive and interactive learning process, something which I attempted to facilitate in my practice. The effect of this could have been intensified further by the abundant cross-curricular links within the lesson with pupils being called upon to explain their answers coherently (English) and suggest possible reasons for the relatively strong positive correlation between the two variables of height and weight (such as increased bone density and Body Mass Index and other scientific rationale). Savage (2010) articulates that context based learning actually facilitates cross-curricular pedagogies, so making the relationship between the two in my teaching seems to be a powerful tool. The pupils were not just learning in a constructivist manner, they were doing so at a deep level by evaluating the methods that they used (such as whether the scatter graph and line of best fit were fit for purpose) and the sample which they collected (questioning its advantages and limitations such as a small sample size and limited generalisability). They were accessing the upper levels of the hierarchy of Bloom’s Taxonomy (see revised version Krathwohl, 2002) in synthesising, analysing and evaluating the content.

The diagram below illustrates the interrelationship between Bloom’s Taxonomy and Surface/Deep Learning:

Figure 2- Diagram which illustrates the interaction of deep and surface learning and Bloom’s Revised Taxonomy

One of the main benefits of constructivism is that pupils gain ownership of what they learn, a sentiment which seems to have been validated in my practiceand I will utilise it in my future practice.

In essence, constructivism and behaviourism are essential components for any prospective or current teacher to study and use it in their approach to learning and also how they monitor and regulate behaviour. Fundamentally, they differ from each other, with constructivism being more concerned with social interaction and children making sense of the world around them, whereas behaviourism is more simplistic in nature, mainly regarding the conditioning of pupils and how they respond to certain rewards and punishments/sanctions (reinforcements). However, a way in which the theories align is that they both examine the relationship between a pupil and a teacher, whether it be the regime which a teacher follows in establishing their authority and keeping pupils on task (behaviourism) or how the teacher interacts with the pupils and builds their understanding and the collaboration between students and their peers (constructivism). Although the theories may oppose each other, they actually complement each other in practice and allow a teacher to make a lesson flexible in structure and adaptable to the pupils’ needs, whilst still keeping pupils on task. It seems a combination of the two approaches is most effective to producing optimal learning in the pupils, particularly when combined with other elements (such as a good rapport between the pupils and teacher).

Constructivism and behaviourism are theories which contribute partially to my emerging philosophy of teaching: I believe that the teacher should have the required authority within the classroom, possibly through behaviourist approaches at a formative stage and through building and sustaining a rapport through constructivist approaches of pedagogy later on, specifically through the interactions and dialogues which I share with my pupils. However, these are not the only things which inform and affect my personal philosophy as a teacher: how I believe mathematics should be taught, observing and adapting parts of other practitioners’ practice and tailoring my approach to suit the needs of my pupils are other variables which will impact on my subsequent practice as a teacher. Furthermore, with regards to Mathematics, the topic which is being taught may also affect the approach that I adopt. As evidenced throughout this essay, certain modules may be suited better to a constructivist approach than others such as those which allow scope for discussion and are less formal (geometry and statistical investigations for example).

Fundamentally, the mathematical topics which have sound links to English and other subjects in the curriculum may be work well with a constructivist approach whereas those which are more formal with no distinct curriculum links may require a different type of educational theory, although it is arguable that constructivist elements could be integrated within it, given the prevalence of this branch of philosophy in the field of education. Ultimately, my personal teacher identity and philosophy is beginning to form, which will become more clear and definitive as I gain more experience and establish a firm set of beliefs and values which inform my practice.

References

Biggs, J. (1999) Teaching for Quality Learning at University. SHRE: Open University Press.

Birenbaum, M. (2002) ‘Assessing self-directed active learning in primary schools.’ Assessment in Education, 9(1): 119-38.

Capel, S., Leask, M. and Turner, T. (2005) Learning to Teach in The Secondary School. London: Routledge Falmer.

Capel, S., Leask, M. and Turner, T. (2013) Learning to Teach in The Secondary School.  6^th edn. London: Routledge.

Department for Children, Schools and Families (2008) Independent Learning: Literature Review. [Online]. Available at: https://www.gov.uk/government/uploads/system/uploads/attachment_data/file/222277/DCSF-RR051.pdf (Accessed: 29 November 2014).

Department for Education (2010) The importance of Teaching: The Schools’ White Paper. [Online]. Available at: https://www.gov.uk/government/uploads/system/uploads/attachment_data/file/175429/CM-7980.pdf (Accessed: 28 November 2014).

French, D. (2002) Teaching and Learning Algebra. London: Continuum.

Johnston-Wilder, S., Johnston-Wilder, P., Pimm, D. and Lee, C. (2010) Learning to Teach Mathematics in the Secondary School: A Companion to School Experience. London: Routledge.

Jordan, A., Stack, H. and Carlile, O. (2008) Approaches to Learning: A Guide for Teachers. Maidenhead: Open University Press.

Jonassen, D. H. (1994) ‘Thinking Technology: Toward a Constructivist Design Model.’ Educational Technology, 34(4): 34-37.

Krathwohl, D. (2002) ‘A revision of Bloom’s taxonomy: An overview.’ Theory Into Practice, 41(4), 212-218.

Kozulin, A. (2003) Vygotsky’s Educational Theory in A Cultural Context. Cambridge: Cambridge University Press.

Lave, E. and Wenger, E. (1991) Legitimate Peripheral Participation. Cambridge: Cambridge University Press.

Leonard, D. C. (2002) Learning Theories: A- Z. London: Greenwood Press.

Nardi, E. and Stewart, S. (2003) ‘Is Mathematics becoming T.I.R.E.D? A profile of quiet disaffection in the secondary mathematics classroom’, British Education Research Journal, 29, 3: 345-367.

Oonk, W. (2009) Theory and Practice in Teacher Education.

Parliament (2011) Behaviour and Discipline in Schools. [Online]. Available at: http://www.publications.parliament.uk/pa/cm201011/cmselect/cmeduc/516/516i.pdf (Accessed: 28 November 2014).

Savage, J. (2010) Cross Curricular Learning and Teaching in the Secondary School. London: Routledge.

Taylor, C. (2011) Getting the simple things right: Charlie Taylor’s behaviour checklists. London: DfE.

Verenikina, I. (2003) Understanding Scaffolding and the ZPD inEducational Research. Faculty of Education, University of Wollongong, NSW, Australia.