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**Investigating Learning in Mathematics**

**Introduction**

On Thursday 24^{th} November 2016 I delivered a lesson on adding and subtracting with decimal numbers to a year seven, set five (of seven) class. In the following session with the same class on Monday 28^{th} November 2016 I introduced multiplication and division with decimal numbers. Through this essay I will critically analyse my teaching methods and the effectiveness of the methodology on the learning of the students in this class.

The students of the class had already been introduced to the concept of decimal numbers before I took the aforementioned sessions. I had taught them in the two earlier sessions on how to put decimal numbers in order of size, and how to multiply and divide decimal numbers by powers of 10. studentsIn these two lessons I had picked up on the fact that one child in the class was struggling in particular despite appearing to make effort, and also having the support of a teaching assistant due to the fact that he is a SEND child. Also, some other studentsstudents had lacked engagement, which I believe was in part due to ineffective application of the behaviour management system that is being followed at the school.. It was after this instance that while planning for my lectures I had decided to make a particular effort to pull up these students up to the speed of the others so that those students not left behind again.

**The School as a Backdrop to Learning**

Claxton (2008, p.16) discusses the importance of schooling, or more accurately, it asks the very important question on the purpose of schooling. In chapter two, titled ‘*What is the point of school?*’, Claxton offers a damning assessment of schools. This chapter delves into every aspect of schooling; children leaving school lacking basic skills in Mathematics and English; declining attendance ratios; children being trained on how to pass tests rather than imbibing valuable skills, and even then many are failing in their exams; children simply aren’t finding their school to be an enjoyable place for learning; they aren’t feeling engaged; and teaching standards are letting children down.

After discussing the various concerns about the current situation of schooling, Claxton (2008, p.29) latches on to the idea that schooling is about preparing children for the future. Herein we see that how children learn in school is not necessarily limited to specific classroom activities, but also depends upon the environment of school, and its contribution to the learning process. Claxton (2008) in his work had listed a variety of potential future subjects such as ‘Human rights’, ‘How to think’, and ‘Body awareness’ and he had further added that none of these are what we would think of as conventional subjects. However, the school environment lends itself to discovery in these areas through the individual interactions within the school body. The relationships are developed along the lines that schools facilitate. So, it is fair to say that the school does play a role in the development and facilitation of the individual interactions.

Every school is different, and every pupil has an individual experience of his school. Capel *et al.* (2013, p.199) had noted the importance of taking the individual’s needs into account while teaching. A variety of reasons, both nurture and nature, impact the learning of the child. A variety of reasons impact what they will imbibe and what they will not, the speed of their development, how peer interaction affects their experience etc. Some things that also need a consideration are impact of the placement of certain children with other children on their development, impact of the external factors (before a child enters the classroom) on the state of mind the children, the quality, range, and availability of learning resources that the school can offer, and the interaction behaviour of the teacher with its students.

Oates (2011, p.122) argues that changes in the National Curriculum have not only created new problems, but old problems are still existing. The new curriculum introduced in 2014 saw a shift away from teaching methodology. Instead, it was focussed on the knowledge that students should develop through the course of their education. The underlying notion is to free up a teacher to adapt styles that suits the students’ needs as they see fit, while providing clear and concise documentation on the knowledge a pupil should have at each stage of their education, would be the best way to equip children for success in their adult life. The government came to this conclusion by employing a range of educational experts to merge successful styles of education from around the world with successful methods currently being employed within the English school system (BBC, 2014). The idea that teachers must be adaptive is reinforced in the chapter on Mathematics where it is stated that progress should only be made when students’ understanding has been secured. However, this does not square with the firm statements throughout the curriculum on what knowledge a pupil should have at each stage of their school careers.

In chapter 5, titled *Ways Students Learn* (Burton, 2013, p.307), the author begins with the identification of the theories of learning, and proceeds to give some examples, such as behaviourism, gestalt theory, and personality theories. Meanwhile, Jordan *et al.* (2008, p.55) groups theories of learning into three broad categories: behaviourism, constructivism, and cognitivism.

**Analysing the Lessons**

When planning a lesson, it is important to consider the age of the students in the class. Jean Piaget’s Cognitive Developmental Theory has been introduced (Jordan *et al.* 2008, p.55) as a stage-based understanding of how children learn. Piaget believed that the way that children learn is limited by their age, and their development can be cleanly broken up into progressive stages at which they become capable of successively higher levels of thinking. In the first stage, from 0-2 years old, children develop a sensory understanding of the world, which Piaget referred to as the sensory-motor stage. Then, from 2-6 years old, is the preoperational stage, where children begin to understand their place in the world, what other people and objects mean in relation to them. Following this is the concrete operational stage from 6-12 years old. At this stage logical thinking opens up within children, and they are able to grasp basic mathematical ideas. Finally, from age 12 onwards, is the formal operational stage. By this stage, children are able to contemplate abstractions and what ifs, and catalogue their knowledge.

In year 7, the students are aged between 11 &12 and they find themselves in between the last two stages of development according to Piaget. In my first lesson I introduced the concept of decimal addition with an example on the prices of some everyday items. I gave them the exercise studentsto add the prices of these items together. Jones and Edwards (2011, p.94) discussed the importance of considering relevant real world applications of mathematical theory. Taking this into consideration this seemed like an ideal way to introduce the topic as a confidence-building exercise since all students should have basic money management skills. Bartlett (2013, p.2) further talks about why mathematics must be made more relatable in order that students be engaged in the subject.

Once I was confident that the students were grounded and engaged with the subject matter, and they responded to this particular example as I expected they would, I saw it fit to broaden their horizons by introducing more abstract examples. I set a series of basic decimal addition and subtraction and differentiated them by level of difficulty. This was particularly based on the explanation by Jones and Edwards (2011, p.96) which tells the importance of differentiated tasks when in engaging a large class. Differentiation enables students of a wide range of abilities to embrace the subject matter, so struggling students can boost their confidence and get some practice with easier tasks while stronger students can push themselves and try to complete increasingly more challenging tasks. Basis my experience differentiation has enabled me to effectively circulate the classroom so that I can help those students who actually need it without being dragged into helping one pupil too much to the detriment of the other students in the class. I could also end the task feeling that most students have engaged with the subject matter to a sufficient degree.

Jordan *et al.* (2008, p.57) classifies Piaget as a constructivist, and goes into further depth, explaining that Piaget’s theories are based on the idea that children develop their thinking through their interaction with the world. Children learn by seeing things in action and understanding the cause and effect. He also stated that children learn when their beliefs are challenged by others and they are forced to reassess what they thought they knew. Piaget believed that children must be left to discover knowledge unimpeded and authority figures must be careful not to impose knowledge on them.

Another important thinker was Lev Vygotsky. Vygotsky came from a drastically different background to Piaget, having developed his ideas in the 1920s and 1930s in Soviet Russia (Burton, 2013, p.313). His research was censored by the authoritarian government; his work only came to prominence decades after his death. For all the differences between constructivism and cognitivism, it is important to note that both Piaget and Vygotsky place the most importance on the child in the learning process. However, Vygotsky recognised the importance of social interaction in the process of learning. Vygotsky believed language spawned from necessity as opposed to Piaget who merely saw it as another part of the learning apparatus.

One of the key concepts of Vygotsky’s theory is the Zone of Proximal Development (ZPD). This is the idea that if there’s something that a child struggles with today, then they can work to improve their understanding with support, and achieve it on their own tomorrow. The person playing the supporting role in ZPD can either be an adult teacher or a capable peer, the important aspect being that the person who supports them understands the concept themselves. Due to the period and place where Vygotsky developed these ideas he was in relative isolation, and his discoveries far predate those made by thinkers in more advanced countries. His work is still being translated into English to this day and is forming the basis for further research.

Anghileri (2006, p.33) claims that the social constructivist theory introduced by Vygotsky is what informs mathematics teaching. She comments on how conceptual learning in mathematics occurs by building upon previous learning and reconstructing previously held ideas with new learning. During my lesson on decimal multiplication and division, in order to introduce decimal multiplication, I introduced a basic decimal multiplication problem, 1.2 x 5.4. I then got the students discuss in pairs on how they thinkt they can solve this problem. After a few minutes of discussion, I asked for some students to explain how they solved this problem. I found that none of the students were capable of arriving at a correct solution. However, what I did perceive in the class was a heightened level of engagement. Students were keener to share their ideas first with their peers before the teacher and they also felt more confident to share their ideas with the teacher once they had split the responsibility for the idea with one of their peers. Once I modelled the solution with an effective method they appeared to be more receptive too.

In both of these lessons, and indeed most of my other lessons, I began with a starter task which would usually reference to the learning from the previous few lessons In the first class I set a starter task to order some sets of decimal numbers, referencing the previous lesson. In the second lesson I set some addition and subtraction problems, referencing the first lesson. As discussed earlier, I recognised the importance of building confidence in studentsstudents, and introducing a task that they should all be comfortably capable of completing allows all students to engage with the class. Morgan (2008, p.16) suggests splitting the lesson up into several small blocks in order to minimise behavioural issues. By beginning the lesson with a distinct starter task this also enables students to settle into the class without the need to distract or be distracted.

In stark contrast to both cognitivism and constructivism is behaviourist theory. Behaviourism is based on the idea that stimuli throughout the world generate a reaction, and certain stimuli can be placed in such a way to generate desired responses (Jordan *et al.* 2008, p.21). An early behaviourist thinker was Ivan Pavlov who experimented with dogs to show how certain triggers could cause a reaction in a dog, regardless of what the original reason for that reaction was. In order to do this, he first noted that the presence of food causes a dog to salivate. He then presented food to a dog, coinciding with the ringing of a bell, and repeated this process multiple times. Finally, he would simply ring the bell and not present any food to the dog. Upon doing this the dog would salivate, which is what he termed a conditional response. Following on from Pavlov’s ideas, Edward Thorndike experimented with reinforcement, in which he rewarded animals for ‘correct’ behaviours, while giving them nothing if they didn’t act in the desired way.

This theory was also tested during the course of my sessions. The moment I used to write a problem sum on the board, the students would make groups immediately to solve the problem. Also it was seen that the students of comparable calibre used to group together. When I started giving chocolates for every correct answer, the weaker group of students also came into action and started giving a tough competition to the brilliant ones. This was pretty much in line with the behaviourist theory wherein the behaviour of the students had changed with the associated rewards.

All of the theories on learning have a definite place & role in a child’s education. In Mathematics, it’s fundamental that any practitioner understands the order of operations. The order of operations is entirely arbitrary, but they are a universal standard and they are vital in order to communicate mathematically. It is possible that you could make the case that a child would eventually discover the order of operations given enough examples, but in this instance it would certainly be more expedient to simply tell a child what the order of operations are, enabling them to discover higher level mathematics in shorter course. This is one instance where a case can be made for rote memorisation, utilising behaviourist ideals.

Further to this, behaviourism is often applied in mathematics in order to instil a particular formula or a procedure. Whilst I allowed my students to discuss how they might derive their own methods for solving problems, such as with decimal multiplication as discussed earlier, I eventually modelled a straightforward procedure, and then set some problems. Behaviourist theory suggests that by repeating this operation many times it will eventually become ingrained.

Cognitivist and constructivist ideas are key in developing mathematical understanding. For example, a teacher may demonstrate algebraic principles by showing children some apples and oranges, and prompting them to discover that the two distinct types of fruit can’t be grouped together. It’s possible to have children memorise the rules of algebra, but conceptual understanding won’t necessarily develop as a result. Children will see meaning in an example involving fruit and consequently be capable of applying the ideas to other scenarios and transferring the skills they have developed.

**Conclusion**

Claxton (2008, p.88) suggests that children are born with an intrinsic desire to learn. People derive great joy from gaining new knowledge and abilities. The problems occur in school when children have demands placed on them to learn in a specific way and setting. What appears on the face of it to be a lack of motivation in many cases is simply the result of their school experience not being tailored to their needs. Claxton returns to the idea that school is about preparing children for the wider world, and if we are to do that effectively then their in-school learning needs to have some context in order to relate it to the real world. The challenge in doing this is that the school is an artificial environment where problems are pre-packaged whereas this is rarely the case in the real world.

In an attempt to break this jinx, I devised my teaching style on the basis of the cognitivist, behaviourist and constructivist theories. I used multiple techniques that had involved more involvement, peer group learning opportunity as well as direct comparison based on the items of interest. The results were very encouraging and it so appeared that the prolonged and persistent weak links as well as bottlenecks in the current education methodology can be overcome with the use of these theories. It is quite evident from the outcome of my teaching sessions that many theories of learning can be applied in conjunction in order to create a desirable learning environment. While some theories may be more applicable in varying situations, there is not a single theory that is universally applicable, and each theory has its own strengths and weaknesses. Based on my research I would tend to favour Piaget’s cognitivist ideas where possible. If children are allowed to take ownership of their education, then they will feel empowered to learn and one would hope that this will have positive lifelong consequences.

At this early stage in my career it appears to be more beneficial to focus on issues such as behaviour management than how effective varying styles of learning can be. As long as I control the behaviour of the class then I should be able to have some effect on the learning of my classes, but if I lose control of the class then there is very little chance of learning taking place. As I mature as a teacher I will find ways to apply the theories that I have learnt about in a more natural style.

**References**

Anghileri, J. (2006) ‘Scaffolding Practices that Enhance Mathematics Learning’, Journal of Mathematics Teacher Education, vol.9(1), p. 33-52

Bartlett, J. (2013) Becoming an Outstanding Maths Teacher, Abingdon, Oxon: Routledge

BBC (2014) How is the national curriculum changing. Available at: http://www.bbc.co.uk/news/education-28989714 (Accessed: 27th September 2016)

Burton, D. (2013) ‘Ways Students Learn’ in Capel, S., Leask, M. and Turner, T. Learning to Teach in the Secondary School: A Companion to School Experience. 6th edn. Hoboken: Taylor and Francis pp.307-314

Claxton, G. (2008) *What’s the Point of School? Rediscovering the Heart of Education*. New York: Oneworld Publications

Capel, S., Leask, M. and Turner, T. (2013) *Learning to Teach in the Secondary School: A Companion to School Experience*. 6^{th} edn. Hoboken: Taylor and Francis

Jones, K and Edwards, J. (2011) ‘Planning for Mathematics Learning’, in Johnstone-Wilder, S. et al. Learning to Teach Mathematics in the Secondary School. 3rd edn. Hoboken: Taylor and Francis, pp. 79-100

Jordan, A., Carlile, O. and Stack, A. (2008) Approaches to Learning: A Guide for Teachers. Maidenhead: McGraw-Hill Education

Morgan, N. (2008) Quick, Easy and Effective Behaviour Management Ideas for the Classroom. London: Jessica Kingsley Publishers

Oates, T (2011) Could do better: using international comparisons to refine the National Curriculum in England. *The Curriculum Journal* Vol. 22, No. 2 pp 121-150