Teaching Primary Mathematics

Sylvia Turner

• Publisher: SAGE Publications Ltd
• Year: 2012
• ISBN:
  9781446291399
• Edition: 1
• Topic: Education
• Subtopic: Teaching Mathematics

Teaching Primary Mathematics covers what student teachers really need to know and why, including approaches to teaching and learning, planning and assessment, and using resources in maths teaching. It also provides a brief historical overview of the teaching of mathematics and examines strategies to enhance learning and development as a confident mathematician in the primary classroom.

Informed by seminal and current research, and recent developments in education policy, the book also explores:

- the role of mathematics within the primary curriculum

- the development of mathematics as a subject of study

- the knowledge that can be gained from considering international approaches to mathematics.

This is essential reading for all students on primary initial teacher education courses including undergraduate (BEd, BA with QTS), postgraduate (PGCE, SCITT), and School Direct, and employment-based routes into teaching.

Sylvia Turner is Senior Lecturer in the Faculty of Education at the University of Winchester.

CHAPTER 1

TEACHERS AS MATHEMATICIANS

Aims

By the end of this chapter you should:

• be aware of factors that are specific to effective mathematics teaching;
• become aware of factors that may affect your teaching of mathematics;
• have identified your response to mathematics;
• have identified incidents and experiences in your autobiography that may have affected your response to mathematics.

Introduction

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In the United Kingdom state school teachers are expected to teach the full curriculum for children up to age 11. There are some exceptions such as in those areas that operate a ‘middle’ school system for children from the age of 8 or 9 years, where some specialist teaching takes place. Some schools have an element of specialist teaching but it tends to be for Foundation subjects such as music and Physical Education in which it is standard to acknowledge some teachers may lack sufficient subject knowledge. However, in core subjects such as mathematics such a view would appear not to be acceptable for someone with Qualified Teacher Status (QTS). Just as we would not assume that teachers are artists or have specialist skills to teach art, so we assume that those teaching in the primary age range do not require specialist skills to teach mathematics. The generalist nature of teaching in the first years of education has many benefits for children and teachers in terms of pastoral care of children and opportunities to make links between subjects. In recent years, however, these general assumptions have been questioned in terms of the challenges faced by the generalist in subject knowledge, curriculum organisation and delivery.

Particular concern has focused on mathematics and its falling ranking in relation to international comparisons. Data from the Trends in International Mathematics and Science Study (TIMSS) and, more recently, in the Programme for International Student Assessment PISA study by the Organisation for Economic Cooperation and Development (OECD) indicates that the UK does not compare favourably with other countries.

Beliefs and their effect on learning and teaching mathematics

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Where would you consider you are on the following scale in terms of your feelings about mathematics? Is it a subject you love or hate, or maybe your feelings are not so extreme and fall somewhere midway?

Mike Askew led a research project by King’s College, London that was sponsored by the Teacher Training Agency (TTA) (Askew et al., 1997). The aim was to investigate the distinctive characteristics of effective teachers of numeracy. Effectiveness was measured in a test administered to the pupils of the teachers in the sample at the beginning of the Autumn term and the end of the Spring term and the average gain calculated. This produced an indicator of teacher effectiveness.

Using data from case studies it emerged that teachers held certain individual sets of beliefs about mathematics. Such beliefs were found to be fundamental in influencing the way in which they taught. The beliefs were categorised in the following way.

• Connectionist – beliefs based around valuing pupils’ methods and teaching strategies with an emphasis on establishing connections within mathematics.
• Transmission – beliefs based around the primacy of teaching and a view of mathematics as a collection of separate routines and procedures.
• Discovery – beliefs clustered around the primacy of learning and a view of mathematics as being discovered by pupils.

(Askew et al., 1997)

One of the key findings was that effective teachers had connectionist beliefs. They made connections between different ideas and knew how to select and use effective and efficient strategies for calculation. They believed that their pupils could become numerate and that pupils develop strategies and connections between ideas or networks by being challenged to think by use of explanation, listening and problem-solving. There was little correlation between the effective teachers in the sample and their qualifications. However, there was evidence that they had undertaken mathematics professional development over an extended time.

Apart from the research led by Mike Askew, there has been considerable research carried out concerning beliefs about mathematics in both the adult population as a whole and more specifically in teachers of mathematics. In the UK alone there is a large body of evidence drawn from research over the last thirty years ranging from government reports, most significantly Cockcroft (1982) and Williams (2008), to research with teachers and those involved with education such as Buxton (1981), Haylock (2010), Jackson (2008), Boaler (2009) and Swan and Swain (2010). A brief overview of the research carried out may give you a sense of the significance of beliefs in the teaching of primary mathematics.

Over thirty years ago, Laurie Buxton, a mathematics adviser for the now defunct Inner London Education Authority, carried out research to investigate the aversion to mathematics that he was encountering among those involved with education. This group included a significant percentage of teachers who were otherwise high achievers and successful in their chosen career. The result was a series of detailed case studies based on individual interviews, group work and discussions that still have resonance today. Although the research was carried out with a small sample of nine adults, the fact that Buxton was seconded for a year enabled him to have time to work with the group in depth. He found that their emotional response was very important to learning mathematics and using it in an educational context.

Participants spoke of mathematics in terms of terror. When asked about their feelings towards mathematics phrases used ranged from worried to panic to the extent that one participant stated ‘“whenever I think back to it [mathematics], it’s always that dreadful numbing panic’ (Buxton, 1981). Buxton worked closely with Richard Skemp, a mathematician, educator and psychologist. He designed mathematics workshops to enable participants to develop confidence, which he believed to be fundamental to learning. Basic to the strategies Buxton used was the interaction of the individual’s emotions with mathematics. He found common feelings about mathematics among the group such as a belief that mathematics was a collection of incomprehensible rules and facts to be remembered that mainly involved computation. He acknowledged the validity of these perceptions as they came from the individuals’ experience. He believed that the experience of learning mathematics had to be changed if mathematics was to be seen as a comprehensible subject with interconnecting relationships.

The effect of emotion on mathematics was confirmed in the Cockcroft Report (1982) by surveys it commissioned. While acknowledging that many people coped with the demands of mathematics, it was found that among the adult population, mathematics induced feelings of anxiety, helplessness, fear and even guilt among interviewees and ‘No connection was found between the extent to which those interviewed used mathematics and the level of their educational qualifications “’ (Cockcroft, 1982: para. 2.20).

Despite the detailed recommendations of Cockcroft to address the problem in the intervening years, Ian Thompson found, when working with teachers on mathematical activity, they responded similarly to those cited in Cockcroft (1982) with the same feelings of panic, anxiety and guilt (Thompson, 2003). Similarly, Derek Haylock has considered the emotional response to mathematics with teaching students. In the fourth edition of his book Mathematics Explained for Primary Teachers (Haylock, 2010) he states ‘Even well-qualified graduates feel insecure and uncertain about much of the mathematics they have to teach’ (Haylock, 2010: xii).

Elizabeth Jackson carried out a literature review of mathematics anxiety and subsequently investigated the situation among primary teacher trainees (2008). She suggests that mathematical anxiety exists to the degree that the ability to do mathematics is strongly influenced by people’s attitudes rather than their cognitive skill.

Jo Boaler has carried out extensive research into mathematics teaching and learning in the USA and UK. Having completed longitudinal studies she has identified what she believes is wrong in classrooms today in terms of children’s mathematical experiences. She found that belief was significant in forming ideas about mathematics. Some teachers believed that success in mathematics was a sign of general intelligence and that it was a subject that some people can do and others cannot. Such ideas have the potential to be harmful to children as it makes them feel ‘helpless and stupid’ (Boaler, 2009: 1–2).

More recently, Malcolm Swan (Swan and Swain, 2010) has undertaken a number of research projects into the professional development of numeracy teachers with post-16 learners. Although relating to teaching those beyond statutory education the focus still relates to mathematics teachers. Teachers investigated eight research-based principles for teaching:

1. Build on the knowledge that learners bring to the session.
2. Expose and discuss common misconceptions.
3. Develop effective questioning.
4. Use cooperative small-group work.
5. Emphasise methods rather than answers.
6. Use rich mathematical tasks.
7. Create connections between mathematical topics.
8. Use technology in appropriate ways.

Existing beliefs and practices were recognised and alternative practices were offered and considered. The teachers were encouraged to adopt new practices and reflect on their experiences and beliefs. Results suggested that many of the teachers’ practices and beliefs were significantly affected. Their practice became less transmission-oriented and teachers began to create collaborative learning environments where students were challenged to confront difficulties and take on more active classroom roles.

Assessing your beliefs

Initially, the findings of the research cited above may seem somewhat negative in terms of teachers and the teaching of mathematics. However, it does give an insight into four key issues that can be identified from the research.

• The subject of mathematics can be seen to hold a level of mystique for a significant number of the adult population, including teachers.

Mystique is a characteristic that can mean charm and magic to some yet a level of impenetrability to others. It would seem that the teacher’s role is to enable children to engage in the subject and so find the magic it can yield.

• Adult mathematics qualifications and cognitive ability do not necessarily correlate with effective mathematics teaching.

Such findings may seem surprising but consideration of teaching you have received in the past may help clarify this point. You may remember being taught a subject or a skill by someone who was obviously an expert in their field but could not impart their knowledge in a way that made it accessible to you. Conversely, you may remember being in a similar situation but with someone who enabled you but was not an expert in their field. Awareness of how you were enabled can act as a starting point to making sense of generic teaching skills and lead on to relating and developing these skills in mathematics.

• Beliefs about mathematics are fundamental to engagement and enjoyment with the subject.

Feelings towards mathematics will come from a wide spectrum of possible responses from a complete lack of confidence in the subject in terms of personal understanding and teaching, to having strong confidence in the subject. How you feel about a subject can be affected by a range of factors. If you think back to your childhood, your feelings towards subjects may have been influenced by achievement, your teacher, fellow classmates, the setting or a combination of such facto...

Table of contents

• Cover Page
• Title
• Copyright
• Dedication
• Contents
• About the author
• Acknowledgements
• List of abbreviations
• Introduction
• The 2012 Teaching Standards
• 1 Teachers as mathematicians
• 2 The nature and language of mathematics
• 3 What do children really need to learn and why?
• 4 What should teachers know and why?
• 5 Teaching approaches
• 6 Assessment and planning in mathematics
• 7 Resources for mathematics
• 8 What can we learn from other countries?
• 9 Researching mathematics education – a case study
• 10 Conclusion
• 1 European Research Study
• 2 Reflection and ‘writing’ a reflective journal
• Index