Learning to Teach Mathematics in the Secondary School

A companion to school experience

Sue Johnston-Wilder, Clare Lee, David Pimm

• Publisher: Routledge
• Year: 2016
• ISBN:
  9781317372363
• Edition: 4
• Topic: Education
• Subtopic: Education General

Learning to Teach Mathematics in the Secondary School combines theory and practice to present a broad introduction to the opportunities and challenges of teaching mathematics in the secondary school classroom. This fourth edition has been fully updated to reflect the latest changes to the curriculum and research in the field, taking into account key developments in teacher training and education, including examinations and assessment.

Written specifically with the new and student teacher in mind, the book covers a wide range of issues related to the teaching of mathematics, such as:

• why we teach mathematics
• the place of mathematics in the National Curriculum
• planning, teaching and assessing for mathematics learning
• how to communicate mathematically
• using digital technology to advance mathematical learning
• working with students with special educational needs
• post-16 teaching
• the importance of professional development
• the affective dimension when learning mathematics, including motivation, confidence and resilience

Already a major text for many university teaching courses, this revised edition features a glossary of useful terms and carefully designed tasks to prompt critical reflection and support thinking and writing up to Masters Level. Issues of professional development are also examined, as well as a range of teaching approaches and styles from whole-class strategies to personalised learning, helping you to make the most of school experience, during your training and beyond.

Designed for use as a core textbook, Learning to Teach Mathematics in the Secondary School provides essential guidance and advice for all those who aspire to be effective mathematics teachers.

1 Why do we teach mathematics?

Robert Ward-Penny

Introduction

One of the most common questions that you are bound to hear as a secondary mathematics teacher is ‘Why do we need to do this?’ Although your students may sometimes ask you this just to distract you from your work, or to stop them from having to do theirs, they are touching on some important and far-reaching concerns. What makes it worthwhile learning a particular topic in mathematics? Why are students in secondary schools expected to spend so much time and effort on this one curriculum area? What is the point of teaching and learning mathematics?

Questions about the purpose of education rarely have simple answers; mathematics teaching is an endeavour with many aims and an enterprise with many stakeholders. However, these questions are important ones that will underpin all the work you will do as a secondary teacher. Reading through this introductory chapter, and working on the tasks, will help you explore some of the fundamental ideas that different people, including you, have about mathematics education. Your understanding of these ideas will help you answer the ‘Why do we need to do this?’ question, both for yourself and for your students, and also support you in making informed practical teaching choices.

Objectives

By the end of this chapter, you should:

• understand better how your own experiences have already shaped your views about mathematics education;

• be aware of some of the different reasons why mathematics is taught in schools, and why it is considered to be so important;

• be able to make connections among the different philosophical purposes of mathematics education and various practical aspects of teaching and learning mathematics;

• know some of the organisations that influence practice in mathematics education;

• be able to articulate more clearly your own current rationale for mathematics education.

Mathematics and you

You are about to begin exploring the field of mathematics education, but in reality you are not a beginner; you already have a wealth of experience upon which to draw. Whether you have studied mathematics at degree level or not, you will have spent many hours of your life engaged in learning and using mathematics – and in being taught it. Your views about what mathematics is and what mathematics teaching looks like, and your own philosophy of mathematics education, have therefore already been informed by all the memories, feelings and opinions you have gathered as a learner. It is therefore a valuable exercise to begin by looking back at and reflecting on where you have come from, what you are already aware of, and what opinions and biases you might bring to your training. Task 1.1 leads you through this process.

Task 1.1 Mathematics – your story so far

This task is structured to help you consider your experiences of mathematics and mathematics education to date. Give yourself some undisturbed time to read each of the bulleted points below slowly and reflect upon the thoughts stimulated by the questions. Note down any ideas or memories as they occur in a reflective journal.

• What is your earliest memory of learning mathematics? Call to mind where you are, what you are doing and how old you are. Is it a positive memory? What is your most recent memory of doing mathematics? Think about why you were doing it and how you felt at the time.

• How do you rate your intellectual capability in mathematics? Do you believe that your ability is something that you were born with or is it something that has developed over time? How does your mathematical capability compare with that of others? What basis are you using for your comparison?

• Who do you think was your best mathematics teacher? Consider what it was about this person that impressed you. Try to remember some particular moments that exemplify all that was good about their teaching. What aspects of this person’s teaching would you like to emulate?

• Who do you feel was your worst mathematics teacher? Think what it was about this person that led you to such a judgement. Try to remember some specific occasions that exemplify what was poor about their teaching. Which aspect of this person’s teaching would you most wish to avoid replicating your own practice? Is this likely to be an issue for you?

• Try to remember some occasions when learning mathematics was difficult and some when it was easy. Why was there a difference? Consider other people with whom you have learnt mathematics. Did they find it easier or harder than you? Why do you think that was? Do you think you have a preferred way of learning mathematics?

• Think about your friends or peers from secondary school who were taught in different groups or sets for mathematics. In what ways do you think their experiences of learning mathematics would have been qualitatively different from your own?

• Consider each of the different stages of your own mathematics education. At each stage, what did you think was the point of learning mathematics? Did you ever discuss this with a teacher, advisor or lecturer? When and why did you decide to become a mathematics teacher?

Having looked back over some of your past encounters with mathematics, you are now in a better position to look forward. In order to grow and develop as a mathematics teacher, you will need to go on reflecting about mathematics, about education and about your place in relation to both. You may like to return to your notes during or after your teaching practice, to help place your own experience in a wider context.

Key purposes of mathematics education

It might seem peculiar to begin this book with a chapter that focuses on abstract issues and the purposes of mathematics education; you may be keen to get on with the likely somewhat daunting matter of surviving in a classroom full of students. However, it is important to understand at this early stage how widely such issues can impinge upon practical matters. You, your students, your school and your government will all have different aspirations and ambitions related to what happens in your classroom. (And remember your classroom is not solely a mathematics one: see Capel, Leask and Turner, 2013, Unit 7.2.) It is important that you are aware of these from the very outset, so that you can better balance the various needs, outcomes and pressures. This section will therefore outline six of the purposes of mathematics education and discuss how each might be reflected in your practice.

Task 1.2 Painting by numbers

To begin with, consider this somewhat mundane mathematics question:

Caroline has used three cans of paint to cover one room. How many cans of paint will she need to cover four rooms? Write down a formula to describe this situation.

Imagine that one of your students has asked you ‘Why do we need to do this?’ Before reading on, consider how you might answer their question. How many different responses could you offer this student?

Everyday mathematics and the development of numeracy

Perhaps the most obvious answer to this student’s question is ‘In order to check you can multiply three by four, and in case you ever need to find out how many cans you should buy to paint four rooms’. Although this scenario may be unlikely in isolation, there is a place in the secondary classroom for the practice of simple problems with immediate applications. One of the most fundamental purposes of mathematics education is to ensure that all learners can apply basic techniques of number and measure in commonplace situations. This area of mathematics is sometimes called numeracy or functional mathematics.

Part of this involves an awareness of certain basic aspects of modelling and some specific assumptions that are tacitly involved in this mathematical question to render it answerable, including whether the other rooms are all the same size as the first. Even if the dimensions are comparable, it is not a question of volume but whether there is the same surface area to be painted in each room (see Keitel, 1989, for an account of a related painting task in the classroom, the discussion of which does not go at all as the author of the article expected).

As a successful learner of mathematics, you might be surprised at some of the basic techniques that secondary mathematics teachers are often required to teach. The remit of the mathematics department not only includes teaching all students to use units of measurement and perform simple arithmetic, but also checking that each learner can read a clock and use money confidently. You might also be shocked to find out that many who leave school still struggle with applying mathematics in everyday contexts: one meta-analysis of 13–19-year-olds conducted by Rashid and Brooks (2010) estimated that 22% of young people in England are not able ‘to deal confidently with many of the mathematical challenges of contemporary life’ (p. 71). Many learners subsequently choose to enrol on adult education numeracy courses in an attempt to gain such basic competence that they missed out on at secondary school, and in addition to overcome the difficulties of getting a job, which can be exacerbated or even caused by poor numeracy. There is certainly still a need for mathematics teachers to ensure that all their students can use basic mathematical ideas and techniques confidently and effectively.

Straightforward arithmetic word problems such as Caroline’s paint problem place a mathematical operation in a context and may start to help students make connections between mathematics and the outside world. However, there are many other tasks that foreground everyday abilities much more effectively and start to bring reality (in all its various forms and guises) into the classroom. For instance, you might ask your students to budget for a family holiday using brochures; to plan a trip to the seaside using timetables; to compare different mobile phone tariffs in a catalogue; to read and interpret some utility bills. These tasks may not always be at an appropriate level of difficulty though – and as they become more involved and realistic, they can also become more time-consuming, lasting for entire lessons or even longer. One of your responsibilities as a mathematics teacher will be to determine if, when and to what extent such tasks and explorations are appropriate, as well as to balance the time you spend working towards each of your goals.

Preparation for work and vocational development

A second answer to your student’s question is ‘Because resolving this problem practises certain mathematical ideas and techniques that you might need to use in your job in the future’. For example, in this instance, the relationship between the number of rooms R and the number of paint cans C can be written as C = 3R. While this particular formula is of limited use, many professions involve formulae of some sort: nurses use them to calculate safe dosages; account managers use them when setting up spreadsheets; special effects organisers can use them to calculate safe distances when working with pyrotechnics. Therefore it is valuable for students to practise coming up with, writing and reading formulae, since the ability to express relationships symbolically and to work with algebraic expressions is essential in so many careers involving science, technology or engineering. Similar arguments can be made for much of the mathematics curriculum; for example, probability is used by insurance companies, weather forecasters and call centres, whereas statistics is deployed in fields ranging from art history to zoology.

While some aspect of mathematics can be used valuably in almost any career, there is also at present a focus on jobs that require a significant amount of mathematical competence. Official reports – such as Roberts (2002) – recognise that there is a shortfall of individuals in STEM (science, technology, engineering and mathematics) careers; although such jobs are essential in a modern economy, there are not enough graduates in many STEM fields, and there is a deficit in some significant areas such as finance. Consequently, there are many interest groups who work to promote STEM subjects and careers, and you may come across some of these either indirectly or directly in your work as a mathematics teacher.

It can be argued that repetition and practice are effective in preparing students for employment, as they work towards ensuring that learners can perform a set range of relevant tasks correctly and reliably. There are many other approaches to mathematics teaching which foreground the goal of preparing students for future occupations, combining the ‘functional mathematics’ described above with concepts and methods borrowed from outside mathematics. Taking examples from within STEM fields, you might: show your students how to work with numbers in standard form by borrowing figures and contexts from astronomy; illustrate sample spaces in probability by talking about dominant and recessive genes; practise algebraic substitution with authentic formulae taken from a field such as engineering. Such exercises can be very valuable, as they can demonstrate to your students how mathematics is used via genuine contexts, and that it is not just limited to contrived and simplistic scenarios (such as Caroline’s painting problem). You may even choose as a mathematics teacher to develop this approach further, using fuller and longer cross-curricular projects in your classroom, perhaps bringing in any expertise that you have from a previous career in industry or another discipline, or working with members of staff from another department.

The aim of preparing students to use mathematics in occupations outside of school can also influence how you use technology in the classroom. Adults in employment have access to a range of technologies, including calculators, spreadsheets, dynamic geometry software (such as Cabri or The Geometer’s Sketchpad) and even computer algebra systems. You will therefore need to consider when, and in what ways, you allow your students to use these tools in their work, and how often you insist that they work unaided instead. Similarly, workplaces often involve people working in teams: is this something that you feel should be replicated more in the classroom?

Thinking techniques – habits of mind and personal development

Thinking mathematically is not an end in itself. It is a process by which we increase our understanding of the world and extend our choices. Because it is a way of proceeding, it has widespread application, not only to attacking problems which are mathematical or scientific, but more generally.

(Mason, Burton and Stacey, 1982, p. 178)

A third answer for any genuinely interested student is that learning mathematics can lead to good thinking habits and develop the brain in a specific and valuable way. Adults regularly use aspects of logical thinking in their organisation and decision-making processes, and these have been developed over time through training ...

Table of contents

• Cover Page
• Learning to Teach Mathematics in the Secondary School
• series
• Title Page
• Copyright
• Dedication
• Contents
• List of Illustrations
• List of Tasks
• Notes on Contributors
• Foreword
• Series Editors’ Preface
• Preface to the Fourth Edition
• Introduction
• Acknowledgements
• 1 Why do we Teach Mathematics?
• 2 Mathematics in the National Curriculum
• 3 Learning Mathematics: A Cognitive Focus
• 4 Learning Mathematics: An Affective focus
• 5 Planning for Mathematics Learning
• 6 Teaching for Mathematics Learning
• 7 Assessing Mathematics Learning
• 8 Communicating Mathematically
• 9 Learning Mathematics Through Digital Technology
• 10 Perspectives and Practices in the Mathematics Classroom
• 11 Working Mathematically with Students with Special Educational Needs
• 12 Teaching Mathematics Post-16
• 13 Getting the Whole Picture
• 14 Professional Development
• Appendix 1 Glossary of Terms
• Appendix 2 Sources and resources
• Appendix 3 Making Closed Tasks Open
• Appendix 4 Practical task – rolling balls
• Appendix 5 Group task
• References
• Name and Author Index
• Subject Index