Teaching Mathematics

A Handbook for Primary and Secondary School Teachers

Pamela Cowan

• Publisher: Routledge
• Year: 2006
• ISBN:
  9781134310838
• Edition: 1
• Topic: Education
• Subtopic: Education General

A practical introduction to Maths teaching designed specifically for beginning teachers in primary and secondary schools. It brings together the latest DfES and TTA guidelines and requirements with authoritative guidance, ensuring that readers feel confident about how to approach their role as a teacher. This book explores key issues in maths teaching today, including:

• planning and classroom management
• assessment, recording and reporting
• information and communication technology
• investigative mathematics
• equal opportunities, special needs and differentiation
• key skills and alternative mathematics qualifications
• being an effective maths teacher
• personal and professional development in the early stages of a teaching career.

1
Learning to teach

Introduction

This chapter aims to provide the reader with a broad overview of teaching in general to set the scene for the rest of the book, which deals with teaching mathematics. The process of teaching is highly complex and therefore to become an effective teacher is a great achievement. There are a number of ‘steps’ to help you reach this goal, so these steps will be the focus of this chapter. The best analogy to describe ‘becoming a teacher’ is that it is like a journey—one that lasts a lifetime, provides a wide variety of experiences and offers numerous ‘great wonders’ along the way. This book will focus on the early stages of your journey. It aims to guide you through Initial Teacher Education, securing your first post and introducing you to the requirements of Induction and Early Professional Development. From there, you should have the confidence to continue your journey of professional development with the support, advice and friendship of your colleagues.

Learning outcomes

At the end of this chapter you should be able to:

• describe the processes involved in teaching;
• list the qualities of a ‘modern and effective’ teacher;
• state the areas of competence to be developed as part of becoming a teacher.

What is teaching?

Teaching is more than just the delivery of strings of words to the listener: teaching must be meaningful to be remembered; it must be coherent to be understood; it must be planned to be continuous, and it must be enjoyed to be sustained for the rest of one’s life.

In its broadest sense, teaching occurs in a variety of places. Through watching other people’s actions, we are taught the basic skills needed to deal with everyday situations. For example, when shopping, we see others lifting their groceries and putting them in a trolley or basket; we notice people taking polythene bags and selecting their own fruit and vegetables; we observe queues forming at the checkout tills; we watch people paying by cash, writing cheques or using their credit cards, and we see people returning their trolleys to the trolley park to get their monetary deposit back. No one teaches us how to shop—we are taught by our peers and we learn by imitation. We are all teachers (and learners) in the supermarket. Similarly, by listening to other people, we are taught new vocabulary, new ideas, new information about the world around us and other countries. So does that mean newsreaders are teachers too?

We also teach ourselves about safety and dangers through personal experiences. How many times have you touched a shiny metal kettle when it has just boiled, or the hob when it has just been turned off, or a live wire? Probably just the once! Although we cannot always see the danger, we are aware that it exists and normally take the necessary precautions.

So teaching takes a variety of forms. Sometimes it is the demonstration of certain skills and physical activities (think of PE or ICT); often it is verbal communication in the form of exposition: learning facts and dates in history; frequently, it is through discovery (in science); and it may also be through problem-solving in mathematics. Sometimes, the teaching is planned and organised in advance (in school), while other times it occurs quite naturally (through playing, going for walks, participating in activities or hobbies).

In this book we are focusing on the planned and organised process of teaching. This is not to say that informal teaching should not occur in the classroom—a good teacher will be effective in both the formal and informal process of teaching, with the latter often going unnoticed by most pupils.

The processes of teaching

The National Curriculum is a statutory framework outlining the subjects to be addressed by all schools at Key Stages 1 to 4 (the compulsory years of schooling). It aims to provide depth and breadth in curriculum subjects, in addition to encouraging the development of the ‘whole child’ through the cross-curricular themes. Schools are therefore guided by the National Curriculum on the range of subjects to include at each Key Stage. Within each subject, a scheme of work is planned for each academic year to ensure continuity within and across Key Stages. The Scheme of Work provides a yearly plan of the topics to be covered that academic year. It is composed of Units of Work which offer a more detailed breakdown of the criteria to be addressed by the teachers. These Units of Work contain series of lessons, which are usually topic-based. For each series of lessons, the teacher develops his or her own individual lesson plans with the key lesson objectives or learning outcomes collectively meeting the criteria outlined in the corresponding Unit of Work.

Delivering a lesson

In every lesson and for every series of lessons comprising a topic, a teacher goes through a sequence of steps to ensure that all the criteria outlined in the Unit of Work are met and that there is continuity and progression within and between lessons. These steps are:

• planning and preparing the lesson(s);
• presenting the lesson, taking account of classroom management issues;
• assessing the pupils to determine the effectiveness of the lesson;
• reviewing and evaluating the lesson(s).

This process is cyclical. After each lesson, the reviewing and evaluating step will inform the teacher’s approach to the next lesson, thereby feeding into the planning and preparation stage. At the end of a topic, the teacher reviews the series of lessons to ensure that all the criteria outlined in the Unit of Work have been addressed to the required level of proficiency. This review often takes the form of pupil assessment in order to ensure that learning has occurred.

Planning and preparation

It is a surprise to many people that planning and preparation is often considered to be the most important part of the lesson, particularly when there are no pupils involved in this step. A solid, detailed and thoughtful approach to lesson-planning is required to ensure that the teacher minimises the opportunities for disruption and confusion. By focusing on a limited number of criteria in each lesson, the teacher can develop the pupils’ understanding of the topic over a period of time. Normally, the basic or core skills within the topic are taught, practised and then developed further. Through the teacher’s own understanding and knowledge of the subject matter, this pedagogical approach to the material can be adopted to cover the content to the required level. Planning constitutes a sequencing of the order in which the content domain for each lesson will be addressed in the classroom.

Figure 1.1 Structure of the National Curriculum

It also includes the identification of the most appropriate teaching strategy for the delivery of the lesson and the range of resources needed to assist in the delivery.

Preparation is the gathering together of the resources needed in a lesson. Each lesson will have its own specific requirements, although it can be assumed that textbooks, whiteboard markers or chalk, and overhead projector (OHP) or data projector and laptop connected to an interactive whiteboard will be needed in every lesson. In the context of maths, for example, preparation may entail finding visual aids and graphics calculators, and creating worksheets for the introduction and/or presentation phase of each lesson. Additional preparation may include writing key formulae, criterial attributes or notes on an overhead transparency (OHT) or on the board prior to the pupils arriving in the room, bringing a class set of protractors or compasses to the room, or preparing a demonstration using a piece of mathematical software to facilitate an explanation or encourage discussion.

Classroom management and discipline

The level of planning and preparation has a strong impact on the success of the lesson in achieving its aims and consequently the level of classroom management needed within the lesson. An interesting, well-structured, organised and clearly explained lesson will offer little opportunity for pupils to become disengaged, confused or bored. As a result, classroom control will be maintained with relative ease. The majority of classroom management problems in a mathematics lesson result from:

• teachers ‘talking at’ the pupils, thereby curtailing any mathematical discussion, instead of encouraging them to participate in the lesson;
• pupils becoming confused due to an ill-prepared delivery of the content of the lesson, including skipping steps in a calculation;
• pupils becoming bored as the lesson has been pitched either too high or too low for their mathematical ability;
• pupils being unable to complete the textbook/worksheet exercises due to a mismatch between the material taught and that addressed in the exercise/worksheet;
• pupils lacking confidence in their mathematical ability.

In the chapters that follow, advice on dealing with pupil misbehaviour is offered in terms of the least-to-most intrusive strategies. In all cases, teachers need to be aware of the school’s discipline policy so that the correct sequence of events is adhered to during the assignment of sanctions.

Assessment

There are two main forms of assessment: formative and summative. Formative assessment provides information on a continuous basis throughout the duration of the course or module and is achieved through a combination of formal and informal strategies. The most common forms of formative assessment are:

• questioning the pupils in class;
• marking classwork exercises;
• marking homework exercises;
• grading project work, investigations, practical activities;
• end of topic tests or school examinations.

The results of formative assessment in mathematics lessons are used to inform the teacher of the pupils’ progress and to assist in the planning of subsequent lessons. Problems that arise in classwork or homework exercises must be addressed in the next lesson, otherwise further progression in the topic cannot be achieved.

Summative assessment is used to summarise the pupils’ achievements to date, normally against the National Curriculum levels. These are usually the formal examinations such as end-of-Key Stage tests, GCSEs and AS and A levels. The results also provide information about the pupils, although in this instance it is not used to inform subsequent teaching, but instead may be applied to subject or career choices for the pupil. Summative assessment is reported for each pupil, whereas formative assessment tends to ...

Table of contents

• Cover Page
• Title Page
• Copyright Page
• Acknowledgements
• 1. Learning to Teach
• Part I: Setting the Scene for Teaching Mathematics
• Part II: The ‘Bigger Picture’ In Teaching Mathematics
• Part III: In the Maths Classroom
• Part IV: Personal and Professional Development
• References