PGCE Secondary Mathematics: Assignment 1 – The Teaching and Learning of Mathematics
Part 1: The Nature and Place of Mathematics in the School Curriculum
When I studied maths at secondary school I had no real understanding of its importance in our society, I enjoyed the satisfaction of solving problems but, like many young people, believed that maths was going to be unnecessary in the real world. As an adult it became abundantly clear to me that mathematics was a vital component in not just personal success but in the success of society.
The notion of what mathematics should be taught and how it should be taught differs dramatically between different educational ideologies; Industrial Pragmatist, Mathematical Purists, Progressive Educator and Social Reformer. These ideologies are not real groups or organisations but represent the competing influences in mathematics education. “The industrial pragmatists see mathematics as an established collection of very useful techniques and skills that can be applied to a large range of technical and scientific contexts.” (Johnston-Wilder, Johnston-Wilder, Primm, & Lee, 2011)
Mathematics is needed in the school curriculum for the prosperity of our country; industry needs the future workforce to have the necessary skills to push the economy forward. For this reason it is seen as important for education to provide good standards of numeracy, data handling skills and use of ICT. Although I acknowledge this is an important requirement of mathematics’ place in the curriculum, it appears a very sterile reason for its inclusion. In a recent Y8 classroom observation the teacher told the class that “maths is just a toolbox for solving problems”. I was somewhat mortified by this definition and had to restrain myself from speaking out; why should a child be motivated to learn maths if that was all there was?
I had originally rejected the Industrial Pragmatist views but upon reflection I can see that there is a definite need for pupils to be ready for the working world and to have the skills to achieve their goals within it. The current curriculum seems to cover much of this useful mathematics, so this group’s view carries a lot of political weight. The Smith Report (2004), in its chapter “The Importance of Mathematics”, puts a very clear emphasis on maths usefulness to business and industry. It states that:
“Although the role of mathematics in underpinning science, technology and engineering is reasonably well recognized and acknowledged in the UK, the fundamental and all-pervasive role of mathematics throughout the rest of the economy is typically not well understood. To the layman it can appear that mathematics for the workplace has become less important because "everything is now done by computers". The clear message to the Inquiry from a wide range of leading industries and businesses is that this is absolutely not the case.” (Smith, 2004)
Maths can and will provide better prosperity both individually and nationally but I prefer to look to ways, other than financial, as to why maths deserves it place as a core subject in the curriculum. I have the view that it is the personal development of pupils that is the most important part of mathematics teaching and this is most closely associated with the thinking of the Progressive Educators and Social Reformers.
The Progressive Educators group “is primarily concerned with the personal development of pupils, with the individual child as the focus of attention. It rejects the adult-orientated nature of vocational development and supports social and academic development only to the extent that these encourage and serve the goal of personal development.” (Johnston-Wilder, Johnston-Wilder, Primm, & Lee, 2011)
Many of the ideas of Progressive Educators, such as self-discovery and positive feelings toward the subject, appeal to me; however the rejection of vocational development does now concern me.
I think it is maths ability of providing a person with a set of skills that allows them to develop personally that is its most important aspect. With the knowledge gained they can more readily pursue other subjects with confidence and have the ability to see previously hidden meaning. Science, design, geography etc. are brought to life with the understanding of maths, patterns and sequences are instinctively discovered which inspires.
Maths helps train your brain to respond logically, carefully and accurately to problems. This allows you to break down complicated situations into logical steps. Better life decisions can be made using mathematical knowledge, whether that is buying the best mortgage or weighing up the pros and cons of an emotional decision.
The Mathematical Purists views regarding maths aesthetic qualities and contributions to our cultural heritage inspire me, but their views on teaching methods seem somewhat dated and not in line with more current theories. It’s lecturing style with the use of standard texts and no use of resources other than traditional equipment will surely turn many children off the subject.
What the content of the curriculum will be is dependent on what society requires in 21st century Britain. The different ideologies will influence what is included but I believe that the maths content should be whatever is needed to inspire young people and encourage them to become lifelong learners, to give them the skills needed to improve their lives. It is important that students have historical/cultural understanding, know the context of what is being studied and appreciate that maths is creative. Statistics is one area that I think could be expanded, not only because of its everyday use in business, but because of the concepts of analysis it teaches and the personal benefits that can be drawn from them. Methods of prediction are important to personal success.
I think that maths is often best learnt by a process of self-discovery, there is nothing more satisfying that being pointed in the general direction of a solution but discovering it yourself. The connections are more firmly cemented in your mind if learning is done this way. I think that experimentation and creativity should be the backbone of learning. My vision for maths education is very much based on personal development and maths helping people achieve their goals. But there is also a big part of me that loves maths for maths sake and I believe that inspiring those few like me is also important.
I don’t believe that I fit into any one of the ideologies as I seem to see aspects of all of them that are important to teaching. I believe that this is a good approach because children have different ways of being motivated and if I were to restrict myself and become closed to new ideas then I will fail as a teacher. My approach to teaching will be one that is centred on personal development but I will draw in many other ideas to adapt to the needs of different pupils and society as a whole.
Part 2: Reflection on the teaching and learning of Mathematics
During my first day at placement school I heard two children say “I’m rubbish at maths”. In both cases they had been contributing good ideas in the small group discussion and when I complimented them on their efforts they immediately put themselves down. I was shocked by this reaction and concerned as to how I would be able to motivate such students. Motivation was an issue for me in many subjects when I was at school and I feel that I would have achieved a lot more during school if I had been motivated more in some way. I fell into the category of the student who believes they are reasonable enough to get through the work but won’t push their self any further, happy to sit back and not attract attention. I want to write this assignment about motivation so I am able to build the pupils self-belief as mathematicians and to be able to promote creativity. For this reason I will concentrate on the motivation associated with pupil identity and its links with creativity.
There are many definitions of motivation; the dictionary describes it as the reason or reasons for behaving in a particular way. This is not suitable for the context of this assignment as it links motivation solely to behaviour and the motivation required in the classroom is goal orientated. I prefer the definition:
“Motivation is the process whereby goal-directed activity is instigated and sustained.” (Pintrich & Schunk, 2002)
It is a process that requires effort and persistence to achieve the required goal. “Students who are motivated to learn about a topic are apt to engage in activities they believe will help them learn, such as attending carefully to instruction, mentally organising and rehearsing the material to be learned, taking notes to facilitate subsequent studying, checking their level of understanding and asking for help when they do not understand the material (Zimmerman & Martinez-Pons)” (Pintrich & Schunk, 2002). I have seen this behaviour, often in the top sets, in lessons I have observed; children who seem to be naturally motivated to learn. But what is motivating these children to push themselves towards their goals and can all children be motivated in a similar way? It is apparent when a pupil is motivated, however, what is motivating them is difficult to identify. As a teacher I need to be aware of as many attributing factors as possible.
“Pupils’ attitudes to school and motivation to learn are a result of a number of factors, including school ethos, class climate, past experiences, future expectations, peer group, teachers, gender, family background, culture, economic status and class. However, the link between motivation and educational performance and achievement is complex.” (Capel, Leask, & Turner, 2009)
Obviously, many of these factors cannot be affected by me as a teacher, but I still need to understand their contribution to the overall motivation of my pupils.
Motivation can be split in to two types; intrinsic and extrinsic. Intrinsic motivation is when a person engages in an activity for its own sake, people who are intrinsically motivated are persisting in a task because of personal enjoyment; it does not depend on rewards. Extrinsic motivation is motivation to engage in an activity as a means to an ends, persistence will result in a desirable reward or avoidance of punishment.
The four sources of intrinsic motivation for students which I am going to consider are; challenge, curiosity, control and fantasy (Lepper & Hodell, 1989) cited in (Pintrich & Schunk, 2002). This is by no means the definitive list, but it has similar characteristics to the requirements for creativity and for me this is useful. If I am going to improve the motivation of my pupils I need to understand these sources and be able to apply them in the classroom. The consensus among the literature is that the evidence points to the fact that intrinsic motivation can promote learning and achievement better than extrinsic. Also, despite some exceptions, extrinsic rewards can negatively affect intrinsic motivation.
“Intrinsic motivation fosters positive dispositions towards mathematics, which, in turn, encourage students to develop self-efficacy and mathematical autonomy as they discuss and share their understanding with their classmates.” (Mueller, Yankelewitz, & Maher, 2011)
Before I carry on I need to introduce the concept of creativity in education. I studied art at college after leaving school and have always considered myself a creative person. However, until studying with the Open University, I had not fully appreciated that mathematics was a creative discipline. It is not commonly referred to as creative subject and most people would struggle to find reasons to call it so.
“There are many misconceptions about creativity. Some people associate creative teaching with a lack of discipline in education. Others see creative ability as the preserve of a gifted few, rather than of the many; others associate it only with the arts.” (NACCCE, 1999)
Early research by Wallas (1926) splits creativity into four distinct parts; Preparation, Incubation, Illumination and Verification. It is quite easy to associate a maths problem with these headings; a problem is considered, it is thought about in detail, a solution is discovered and the solution is tested. These four parts could be useful in developing lesson plans for creative activities. The importance of mathematical creativity is stressed by its inclusion as one of the key concepts in the National Curriculum:
“Mathematics is a creative discipline. The language of mathematics is international. The subject transcends cultural boundaries and its importance is universally recognised. Mathematics has developed over time as a means of solving problems and also for its own sake.” (The National Curriculum 2007)
The Kyriacou & Goulding report places a lot of emphasis on pupil identity and its links with motivation. “Perhaps the most important area that has emerged from a consideration of 23 studies is that of pupil identity” (Kyriacou & Goulding, 2006). Pupil identity is regarded as the extent to which pupils identify themselves as mathematicians. Until very recently, when we were asked in a lecture, I had never considered myself a ‘mathematician’. I had never been asked before and, despite completing a maths degree, still imagined a mathematician as a professor in a white coat who devotes his life to the subject. I spoke of this to a teacher at my placement school and, to my surprise, he still didn’t consider himself a mathematician. It is common to call a child an artist or musician but a mathematician is seen by many as part of an elite group that few have the ability to enter. The studies in (Kyriacou & Goulding, 2006) indicate that to raise motivation through pupil identity the following strategies need to be implemented; fostering a caring and supportive classroom climate, challenging and enjoyable activities, enabling pupils to gain a deeper understanding of mathematics, group work and enabling the pupils to feel equally valued. I will look more closely at some of these in this assignment.
Achieving the correct classroom climate is a key aspect in improving pupil-identity and creativity. Without a positive climate there is little hope of improving motivation. My own experiences of what I consider a good classroom climate comes from my GCSE studies at school and more recent observations in my placement school.
Mrs Jarman was my maths teacher during my GCSEs; her rapport with the class was fantastic and empathy for the students was always present. It was an atmosphere that promoted learning and encouraged me to assist others who were less able; the best man at my wedding thanked me for getting him through his GCSE. It showed me that a caring environment has a very positive impact on students. She would make pupils feel valued when answering questions which meant they were not afraid to take the risk and speak in front of the class.
The most important aspect of classroom climate is the relationship between teacher and pupil (Muijs & Reynolds, 2011). Teachers should create an environment which if free from fear, where pupils opinions are valued and respected. Positive reactions should be given to wrong answers and they must be seen as part of the learning process; this is something that I have seen much of in my classroom observations, but I still do hear comments such as “are you with us?” and “you should know this”.
“Start by removing threats from the learning environment. No matter how excited you are about adding positives to the environment, first work to eliminate the negatives….There is no evidence that threats are an effective way to meet long-term academic goals” (Jensen, 1998, p. 30)
When a student is afraid, they will naturally resort to self-preservation and this will stop them from learning and developing new skills. This is connected with Maslow’s Hierarchy of needs theory, which states that after the physiological needs (e.g. food, warmth) comes the need for physical and psychological safety. Maslow’s theory is behind many of the requirements of the classroom climate; the need for love and belonging, the need to feel competent and gain recognition from others. Without fulfilling these needs, the level of self-actualisation, where creativity lies, cannot be achieved.
The enthusiasm shown by the teacher is another important part of classroom climate. The pupils of an unenthusiastic teacher, who shows little passion for their subject, will inevitably adopt the same attitude. A teacher who shows a love of their subject will motivate the class. In a recent lesson at my placement school, when delivering instructions for homework, the teacher said “I know homework is boring, but I have to give it to you”; this led to groans and much agreement among the class. Homework is obviously important, but this teacher lowered it to the level of a meaningless chore that even he is not interested in. In contrast to this, I witnessed another teacher saying “challenging homework this week, it’s important you complete it”; this led to a good deal of interest and very little negativity. I have been paying much attention in classroom observations to the amount of language used to increase pupil identity; so far I am seeing a distinct difference between the language in higher ability groups as opposed to lower ability groups. I have heard the term ‘mathematician’ used a number of times in the higher sets, but not in the lower sets.
“Human beings are driven to be autonomous and self-determining. We will naturally resist when others attempts to exert undue control over us. By their very nature, however, schools are coercive organisations.” (Sullo, 2009)
There needs to be as much freedom and choice for pupils in lessons as possible to promote motivation and creativity. This initally sounds like an impossible task within the confines of school and the pressure of getting results, but there are simple methods that can be used. For example, pupils can be given the choice of two equally valid activities, they can be given the responsibility of how to display their project work and given the freedom to make up class rules. This is the ‘control’ source of intrinsic motivation mentioned earlier.
Borich (1996) identifies three types of classroom climate that can be used which offer varying degree of pupil-led activity; competitive, cooperitve and individualistic (Pintrich & Schunk, 2002). The competitive climate can motivate pupils, especially boys, but it can have a damaging effect to less able students because of the constant comparison. A cooperative climate is more pupil-led and allows for discussion and ideas to be shared, the teacher acts to stimulate and arbitrate the activity. This climate is good for social skills, helping develop thinking skills and, because pupils often enjoy working together, it can be highly motivating. Individualistic classrooms allow for pupils to work at their own level which encourages problem solving and independent learning skills, but lack of teacher guidance can mean low/average ability pupils do not make sufficient progress and they are unable to develop cooperative skill.
On their own these three styles of classroom climate are insufficient to motivate all pupils and will enevitably alienate some. If used in conjunction with one another and by mixing aspects together, a more rounded classroom climate can be achieved that will benefit all pupils much of the time. For example, I saw a game, called ‘castles’, being played in a Y7 class which used aspects of the competitive and cooperative climate. Pupils worked against each other in teams and scored points which were based on their cooperation skills. The game was universally liked and the pupils were very motivated.
“Activities that challenge students’ skills may be intrinsically motivating” (Deci, 1975; Harter, 1978; White, 1959) cited in (Pintrich & Schunk, 2002). Engaging in challenging activities is one of the sources of intrinsic motivation, it is also highlighted in the Kyriacou & Goulding report as a link to pupil-identity and promotes the kind of thinking that leads to creativity (Carr, 1996). Completion of challenging tasks leads to increased levels of competency which raises pupil-identity. However, I need to be careful not to set challenges too high or too low as this can have a detrimental affect on motivation. Csikszentmihalyi (1985) studied intrinsically motivating tasks and found a state of ‘flow’ which is defined as “the holistic sensation that people feel when they act with total involvement”. The flow state is the point where the amount of challenge and a persons’ capability meet. If a task is perceived as being too easy then a person will become bored; if a task is too difficult then they will become anxious. The ratio of challenge to skill can result in varying intensity of flow. In theory this sounds good but I see problems with applying this in lessons; in a class of thirty, how can you possibly get the level of task absolutely right for everyone. Extension work can be given to an able student after the initial task, but boredom might have already begun to affect motivation. Less able pupils with the same task might be suffering with anxiety which again affects motivation. Therefore a choice of tasks should be offered to the pupils or an activity which is more open-ended for which all abilities will get the most out of.
The appropriate level of challenge is associated with Vygotsky’s zone of proximal development; which is defined as the range in the level of difficulty between problems that a pupil can complete on their own and problems that can be completed only with the help of others. Teachers can improve motivation “by selecting tasks at the right level of difficulty and providing students with sufficient active instruction and follow-up monitoring of independent work activities to allow continuous progress through the curriculum with minimal frustration and failure.” (Brophy, 1985, p302) cited in (Carr, 1996).
The next source of intrinsic motivation is curiosity; the theory here is to present pupils with information or ideas that are discrepant from their present knowledge that will appear surprising. This increased level of interest will motivate pupils into finding a way to resolving the discrepancy. As with challenging activities, the level needs to carefully considered and should to lie within the zone of proximal development. Maths lends itself well to curiosity and there are many intriguing connections that can interest pupils; Fibonacci’s connections with nature and patterns in magic squares. Professor Ian Stewart’s books on mathematical treasures have scores of such problems and facts; for example, the number 153 is equal to the sum of its digits (1^3+5^3+3^3=1+125+27=153), can you find any other cases where this works?
“Intrinsic motivation can be promoted with activities that involve learners in fantasy and make-believe through simulations and games that present them with situations that are not actually present” (Lepper & Hodell, 1989) cited in (Pintrich & Schunk, 2002). Maths classrooms have notoriety amongst children as being boring, adding fantasy elements to lesson will engage pupils and aid motivation. I observed a low ability year seven class being taught inequality with the use of ‘Fred the hungry crocodile’. The teacher drew a crocodile with its mouth as the greater than sign, the class were then told that Fred was greedy and would only eat the biggest number. Fred was then placed between two different numbers and the class were asked which one he would eat. This fantasy element engaged the pupils far more than could have been achieved using a standard approach. Converting a bland list of questions into a team game will improve motivation to complete the task; the ‘castles’ game that I mentioned earlier was steeped in fantasy, with the pupils building castles out of tables and firing cannons at each other. This kind of activity is enjoyable not just for the pupils, the adults in the class can be equally enthused by the fantasy element; and enthusiasm breeds enthusiasm.
The use of extrinsic motivation in the classroom is not straight forward, as it can negatively affect motivation and creativity if not used correctly. “Intrinsic motivation is supportive of creativity. Some types of extrinsic motivation are harmful to creativity, some are not. Of course, our task is to determine which is which and how best to use them in the classroom. (Starko, 2001, pp325-6)” cited in (Fautley & Savage, 2007).
Extrinsic motivation often comes in the form of physical rewards; this can unintentionally imply to pupils that the hard work and learning is not valuable in itself (Sullo, 2009). I have seen rewards such as ‘board games at the end of lesson’, ‘less homework if you finish work in class’ and chocolate bars offered to pupils; but these tactics showed no sign of improving enjoyment of the subject, instead pupils trudged through the work with the reward the only motivating factor.
Praise is a form of extrinsic motivation that is widespread in schools; this is a much researched area and is anything but simple. O’Leary and O’Leary (1977) found that for praise to be effective, it must be based on performance of the behaviour, be specific and be viewed by the pupil as sincere and credible (Pintrich & Schunk, 2002). Teachers who praise students for success at easy tasks are suggesting to them that they believe they are of low ability. Praise used too often is seen by students as unimportant and has little effect (Pintrich & Schunk, 2002). In my recent observations at school I have seen a great deal of praise given, both good and bad. One teacher in a low ability group would issue praise continuously for the most trivial of things, from an outside perspective this looked patronising and I could see that the pupils took no pleasure in receiving the praise. Another teacher used it sparingly and only when a difficult task had been achieved, this teacher was highly regarded and praise was held in high esteem by the pupils. Most other teachers delivery of praise seemed to lie somewhere in the middle and could be somewhat inconsistent.

Part 3: Synthesis and implications for practice
My own views on the nature of mathematics will impact on my teaching style and the methods I use in class, but I have to be aware of the requirements of all my pupils. “In order for students to be motivated to learn mathematics, the teacher must strive to match his or her views of the nature of mathematics and mathematics learning with those of the student, or to somehow alter the views of the student to correspond with his or her own convictions” (Carr, 1996). If a class is used to a certain approach to mathematics learning and they are then expected to suddenly change this approach, motivation can be negatively affected. I need to be aware of a class’s previous method of being taught and think carefully about the introduction of new ones.
At the beginning of the course I had no idea that motivation was such a complex area, I thought that it was mainly an inherent quality. Because of my own experiences of education I was more inclined to think that extrinsic methods of motivation were the most useful and most commonly used. Through reading relevant literature, attending lectures and classroom observations I have come to understand that intrinsic motivation has a much greater impact on learning. Motivation is very personal and discovering the sources of motivation for every pupil is my classes will be an uphill struggle but by putting challenge, curiosity, control and fantasy into activities I should be able to promote intrinsic motivation that leads to an increased level of pupil-identity and ultimately creativity. I appreciate that extrinsic motivation is going to be used in all schools and that is has good points if used to acknowledge learning behaviour, but I now see that it sometimes has negative effects if used as a means to an end. I will not be offering to reduce homework as a reward for completing an assignment.
I am passionate about creating the caring and supportive classroom climate as I see this as a cornerstone to everything that goes on in a class. My enduring memory from my favourite teacher was the feeling of belonging; this is what I want to offer my students.
If I am to improve the motivation of my students I need to put all the ideas from this assignment into practice in my classroom and be able to reflect on the success or lack of it. I aim to be able to make pupils understand that they are ‘mathematicians’ and mathematics is creative.
Bibliography
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