Mathematics

Teaching School Subjects 11-19

Candia Morgan, Clare Tikly, Anne Watson

• Publisher: Routledge
• Year: 2012
• Print ISBN: 9780415321129
• eBook ISBN: 9781134359271
• Topic: Education
• Subtopic: Education General

This accessible and thought-provoking book considers what beginning teachers need to know about learning, teaching, assessment, curriculum and professional development, in the context of teaching mathematics to eleven to nineteen year olds. It is part of a new series of books that has as its starting point the fact that PGCE students are already subject specialists.

The authors show how mathematics teachers can communicate their own enthusiasm for the subject and inspire their pupils to learn and enjoy learning. They provide practical advice which will help teachers and student teachers to:

• plan, organize, manage and assess classroom work
• make decisions about the content, ordering and level of difficulty of lessons
• make sense of new subject material and how it contributes to wider educational aims
• develop professionally by developing the subject as a whole.

This is a comprehensive introduction to teaching mathematics in the secondary school which will be invaluable to teachers beginning their careers and those who are training to be mathematics teachers.

Chapter 1

Introduction

What is special about being a teacher of mathematics? How is it different from ‘just’ being a teacher or from being a teacher of another school subject? You may respond that you have your specialist knowledge of mathematics, your love of the subject, your desire to help young people share that knowledge and enthusiasm, and to help them to gain some of the skills and qualifications that will be of importance to them and to society in the future. ‘I know mathematics, and now I want to learn how to teach.’ These are some of the reasons commonly given by people applying for initial teacher training courses. Of course, these are all important prerequisites, but we will argue that there is more to becoming a mathematics teacher than learning how to teach. Crucially, we believe that continuing to learn and re-learn mathematics, and to learn about students learning mathematics are essential factors in your professional development at all stages in your career.

 

Example

One of the first topics Maria was expected to teach was quadratic equations. She prepared carefully for the lesson, refreshing her memory of solution methods, looking at the methods presented in the students’ textbook and mentioned in the school’s scheme of work, the National Curriculum and the examination syllabus. She also found out what sort of questions about quadratic equations her students were expected to be able to answer in the textbook and in examination questions. In the first lesson, she explained a method of solution by factorisation carefully and clearly, and was pleased to find that most of the class were able to answer correctly most of the questions in the textbook. A successful lesson, she concluded. After the lesson, her mentor, whose class she was teaching, congratulated her and then posed some questions that made her think:

• Some of the students got all the early questions in the exercise right but then gave the answers 8 and 12 to the equation x ² + 4x = 12; how might they be getting these answers? And how might you address this?
• How are you going to respond to the student who asked what the point of quadratic equations is?
• What ‘big mathematical ideas’ could you use this topic to develop?
• What sort of links could you help the students make with their learning in other areas of mathematics, including number and geometry?
• How are you going to respond to the student who looks up the quadratic formula and brings it to your next lesson?
• How are you going to respond to the student who looks up quadratic equations on the Internet and finds references to cubic, quartic and quintic equations, and the existence (and non-existence) of solution methods?

 

Maria started her teaching with a perspective (common to many beginning teachers) that took the mathematical content for granted, and assumed that the clarity and completeness of her explanations would be sufficient to communicate her knowledge to her students. Clearly, her personal mathematical knowledge, her research into the curriculum and her excellent communication skills were vitally important and provided a sound foundation for her development as a mathematics teacher. Yet her mentor’s questions highlighted other critical factors: the ways in which students may understand or misunderstand mathematical ideas; the ways in which students may or may not be motivated to learn; the image of mathematics as a whole that is communicated to students; the need not only to teach each topic separately but to make links between topics and to build strong foundations for future mathematical learning; variations between students and the need to be aware of these variations, and to value and support the learning of each individual. These are some of the issues that we will address in this book, perhaps not providing direct answers to the mentor’s questions, but suggesting ways of thinking about mathematics, about students, and about learning and teaching that will help you to find your own answers.

We imagine that most readers of this book are likely to be in the early years of their career. Yet we believe that more experienced teachers may also find some parts useful and that those working with beginning teachers may use it with those they are mentoring – and find that they are learning in the process. Certainly, in the process of writing the book we have found ourselves restructuring and developing our knowledge of mathematics, our ideas about learning and teaching mathematics, and our ways of working with our own student teachers.

At present, mathematics teaching in England and Wales is regulated by the National Curriculum (DfEE, 1999) and, at Key Stage 3, is guided by the National Strategy and the National Framework for Mathematics (DfES, 2001). Teacher education is regulated by standards that must be met by all teachers in order to achieve Qualified Teacher Status (QTS) (DfES, 2002) and to have this status confirmed at the end of an induction period. The details of these regulations and guidance are available in the documents cited, but these are subject to revision as policies change. Rather than attempt to refer in detail to each curriculum objective or teaching standard, our aim in this book is to provide a more holistic approach to mathematics and to teaching mathematics. We will address all aspects of the standards for QTS and for induction, and will work with examples of mathematical content relevant to the National Curriculum and curricula for post-16 students, but we will not generally give specific references to the official documents. This approach reflects our belief that mathematics is more than a list of discrete topics and that becoming a teacher of mathematics involves more than acquiring a defined set of knowledge and skills.

Similarly, you will find no separate section devoted to the use of information and communication technology (ICT). This is not because we do not think that ICT has an important role to play in the learning and teaching of mathematics (indeed, some types of ICT use have great potential to transform mathematical activity) but because we see it as one set of tools among many others – and one that is changing rapidly with new developments in technology. Reference to the use of various forms of ICT occurs throughout the book, integrated into the context of discussions of mathematics, learning and teaching.

We have included a number of references to the work of researchers and other authors in the field of mathematics education and to research that we have carried out ourselves. This is not just a claim to academic respectability! We believe that research and the accumulated knowledge of other professionals can and should play an important part in informing our thinking about teaching and our practice. For example, in Chapter 8 we cite some results of research about children’s understanding of specific mathematical concepts and show how these might affect your design of questions for your own students to assess their prior knowledge before starting to teach a topic. Indeed, in later chapters we suggest that undertaking research into your own teaching and the learning of your students is a powerful way of enhancing your professional development and contributing to the development of the profession itself. At some points we have tried to draw your attention to books and other sources that can be particularly useful to you, providing information and evidence to inform your planning and teaching, or offering further suggestions of lesson ideas.

The chapters in the rest of the book are arranged in three parts (corresponding approximately, but not exclusively, to the main sections of the current Professional Standards for Qualified Teacher Status (DfES, 2002): Knowledge and Understanding; Teaching; Professional Values and Practice). The theme of the first part, Framing the subject, is mathematics itself: its place in schools and in society; the relationship between the mathematics taught in school and that encountered in the university and in the ‘real world’; alternative ways of thinking about the mathematics in the school curriculum. The subject mathematics is not the same in different contexts and, as you prepare to teach it in school, you need to consider its complexity and connectedness, its fundamental ideas, its applications and how these may relate to the beliefs, understandings and motivations of your students. We believe that teachers’ knowledge and understanding of mathematics continue to develop throughout their professional career as they prepare and teach their lessons, and as they observe and reflect on the ways in which their students learn. This part does not, therefore, seek to give a complete picture of mathematical subject knowledge for teaching. Rather, it provides examples and ways of thinking about restructuring and recontextualising your own knowledge to enhance this developmental process. The issues, examples and ‘big mathematical ideas’ introduced here form a basis for thinking about learning and teaching the subject in later parts.

The four chapters in the second part, Learning and teaching mathematics, focus on students and the mathematics classroom. We start by thinking about students and the ways in which they may learn and be motivated to learn mathematics. This understanding of learners and learning, together with the understanding of mathematics, underpins the planning process. In Chapters 6 and 7, we address planning for teaching, not by providing a comprehensive guide or framework for lesson planning (these are generally provided by initial training courses and may be found published elsewhere, for example, Johnston-Wilder et al., 1999) but by offering some example lessons and discussing the principles and thought processes behind their design. While the examples may provide some immediately practical ideas for lessons, we hope that the discussion will provide more generally applicable approaches to planning as you develop your own lessons. In the final chapter of this part, we turn to assessment, focusing primarily on the role that assessment plays in the learning and teaching process. We consider what information about students’ mathematical understanding may be useful (or, indeed, essential) for teachers and suggest approaches to gathering that information and using it to inform and enhance learning and teaching.

The final part of the book, Professional values and practice, takes a broader view of what it means to be a mathematics teacher, considering how teachers can take responsibility for their own practices and for the development of better mathematics education for all. We start by examining some of the assumptions that are commonly made about what kinds of mathematics curriculum and teaching may be offered to various groups of students. Having high expectations of all results in teachers constantly challenging these assumptions, while having the realistic and practical means of providing access to the curriculum for all students.

Entering teaching and developing as a teacher are not just about gaining skills and experience. Becoming a mathematics teacher can be seen as learning to participate in a community of practice that continually evolves as it responds to changes in wider society and as new participants bring with them new perspectives, energy and initiative. We believe that it is important that teachers as professionals neither adopt traditional practices uncritically nor develop their practices only in response to external pressures. Chapter 10 looks at the ways in which teachers may be involved in the processes of curriculum development and in the development of approaches to teaching. This chapter reviews teacher involvement in developments at local and national level, and suggests ways in which individual teachers can play an active role in developing their own practice and contributing to the development of the wider profession.

Continuing professional development is the theme of the final chapter. We see professional development as something that continues throughout a teacher’s career, involving a deepening understanding of the processes of learning and teaching mathematics and of mathematics itself, and a developing ambition and vision to improve the learning experience for students of mathematics. For many, it will also lead to career enhancement, taking opportunities to lead a mathematics department, to support the development of new teachers, to develop some aspect of the mathematics curriculum within a school or as part of a local or national project. We discuss ways in which mathematics teachers can actively promote their own professional development. A key to this is working with fellow teachers and other professionals both within the walls of the classroom and the school, and outside the school as part of the broader community of mathematics teachers.

Part I

Framing the subject

What do teachers need to know about mathematics in order to teach it? It seems natural to assume that well-qualified graduates of mathematics or of other highly quantitative disciplines are sure to know enough. Moreover, most programmes of initial teacher education for graduates are too short and crowded to allow much time for developing thinking about the subject matter itself. Yet our experience, and that of many other teachers we have known, is that the knowledge needed to be a successful learner and to pass examinations is only a part of the mathematical knowledge that is needed to be a successful teacher, to communicate effectively with a wide range of learners, to structure their learning experiences, to address their difficulties constructively, to inspire students with enthusiasm for mathematics and with an understanding of its potential use. We hope to prompt you to reflect on mathematics and on the ways in which you and others may know and understand it.

The chapters in this part address the nature of mathematics, its privileged place in the school curriculum and its role in the ‘real world’. The mathematics studied at university often seems to be a completely different subject from that found in school, but we would argue that insight into advanced mathematics and its application is an essential basis for teaching. We explore some of the ‘big ideas’ and connections between mathematical topics and consider how these can be used to inform teaching at secondary school level. The theme of connections within mathematics and to other domains of human experience is important, not only in this part, where we look at mathematics itself, but throughout the book as we try to find ways of helping learners to make sense of mathematics.

Chapter 2

Why teach mathematics? Why learn mathematics?

KEY QUESTIONS

• What are the aims of mathematics education?
• What reasons may students have for wanting to learn mathematics?
• Why does mathematics hold such a central position in the school curriculum?

Why teach mathematics indeed! With mathematics as a high proportion of your degree studies you have very marketable skills. You may already have been working in industry, commerce or the public services. You may be a recent graduate who has focused for some time on preparing to become a mathematics teacher or you may be ‘testing the water’ in the classroom, trying to decide if a career in education suits you. Your own education may have been entirely in England or you may have firsthand experience of one or more other national education systems. Mathematics teachers as a group have a diverse range of knowledge, attributes and experiences to contribute to our schools but generally share some common aims. Significantly, you are likely on the one hand to enjoy mathematical activity yourself and on the other hand to enjoy working with children and young people, and to have a desire to contribute to their lives so that they might derive benefit from their mathematics, as you have done. In making a decision to enter mathematics teaching, you have probably already reflected on your personal reasons for wanting to teach and for choosing mathematics as your subject. But why should mathematics be taught? In this chapter we discuss reasons for the place of mathematics in the school curriculum and the ways in which studying mathematics may impact on learners’ lives and on society more broadly.

The most recent version of the National Curriculum for England and Wales states some of the ways in which mathematics is important: in everyday life and employment; as a powerful set of reasoning and problem-...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright
• Contents
• List of illustrations
• Preface
• List of abbreviations
• 1. Introduction
• Part I: Framing the subject
• Part II: Learning and teaching mathematics
• Part III: Professional values and practice
• Notes
• References
• Index