Teaching for Mathematical Understanding

Practical ideas for outstanding primary lessons

Tony Cotton

• Publisher: Routledge
• Year: 2016
• eBook ISBN: 9781317443292
• Topic: Education
• Subtopic: Education General

Teaching for Mathematical Understanding develops the subject knowledge support and practical ideas from Tony Cotton's Understanding and Teaching Primary Mathematics into resources for full lessons. With an emphasis on developing outstanding lessons using a problem-solving approach, this highly practical guide is packed with activities that all trainee and practising teachers can use in the primary classroom.

Covering each area of mathematics, every activity offers helpful step-by-step guidance, including teaching and learning objectives; resources; lesson outlines; ideas for differentiation; assessment for learning and key probing questions. Also featured in this text are call-outs to the information contained in the book's companion website, a shared site with a range of relevant resources to support and consolidate your learning.

Teaching for Mathematical Understanding is an essential text for all trainee and practising teachers looking for inspiration and guidance towards outstanding mathematics teaching.

Companion website features include:

• Video clips in which primary school teachers demonstrate concepts covered in the book through teaching to a real class
• PowerPoint presentations which provide support for those using the book as part of a teacher training course
• updated weblinks to external sites with useful teaching information and resources.

CHAPTER 1
INTRODUCTION

Welcome to Teaching for Mathematical Understanding. I hope that the ideas in this book are exactly what I suggest in the title. I want you to find them practical – my aim is for you to think, ‘I could do that in my classroom’, with all of the activities. All of these activities are ones that I have used in many of the classrooms in which I have taught. I have also introduced them to trainees on teacher training courses and experienced teachers and know from their feedback that they have both found them useful and more importantly enjoyed using them. In that sense they have stood out for them from other activities – the genuine sense of ‘outstanding’.

Many of you will have bought this book as a result of reading Understanding and Teaching Primary Mathematics, also published by Routledge. I am fortunate that this book has become very well regarded by those training to teach and teacher trainers as a useful book to develop mathematical subject knowledge. People who use the book have told me its particular strengths are that it offers practical ideas for the classroom alongside subject knowledge content and that it sees problem solving as underpinning mathematical learning. The book you are now reading develops these practical ideas into resources for full lessons. If you are reading this alongside Understanding and Teaching Primary Mathematics this will allow you to try out the ideas in your classroom.

The book is also designed to stand alone however. The ideas will be useful to all of you who aspire to teach lessons which bring out the best in all of your learners through developing a problem-solving approach to learning and teaching mathematics.

What makes an ‘outstanding’ lesson?

In 2012 Ofsted outlined what they saw as the characteristics of outstanding mathematics learning and teaching. They expected to see lessons which:

• nurtured mathematical independence and allowed learners time to think and reflect on the learning that was taking place

• were planned to include problem solving, discussion and investigation and seen as central to learning mathematics rather than an ‘add on’

• saw misconceptions and errors as a step in the learning process which provided fruitful points for discussion

• allowed learners to make connections between topics and made links between mathematics, other subjects and with mathematics beyond the classroom.

You will see that the activities in this book meet these expectations. They are not designed as activities which you would use in ‘special’ problem-solving lessons or as one-off teaching sessions. I would hope that you would use these activities to support your day-to-day teaching of the whole mathematics curriculum. Learners who approach their mathematics as a problem-solving activity will be successful in tests as well as successful in using mathematics in their day-to-day life. This is because they see challenges and new problems as something that they can solve. They develop a resilience which means they will ‘have a go’. This is a useful mind set to have when facing questions that you do not immediately know how to answer or when you cannot immediately recall the method that you have previously employed.

Why a problem-solving approach?

The section above shows that teachers who use a problem-solving approach are meeting the requirements of the English inspection regime which sees lessons which ‘were planned to include problem solving, discussion and investigation’ as outstanding. This view is not limited to England however. International schools which follow the Primary Years Programme are encouraged to see mathematics as a ‘highly effective tool for solving problems’. The focus of the Primary Years Programme is to support learners in seeing themselves as mathematicians in the same way that we might see ourselves as ‘authors’ or ‘artists’. I would certainly hope that by working on the activities contained in this book both you and your pupils will see yourselves as mathematicians. Reallife mathematicians spend their time problem solving – they do not fill their days completing exercises that they already know the answer to.

Another well-regarded international programme, Cambridge International Examinations, which is used in hundreds of schools around the world states that their ‘curriculum is dedicated to helping schools develop learners who are confident, responsible, reflective, innovative and engaged’. They will only endorse materials which have been ‘designed to engage learners in an active and creative learning journey’: another piece of evidence that if we follow a problem-solving approach we are following best practice.

There is more support from the United States. Professor Jo Boaler from Stanford University has set up the largest MOOC for parents and teachers who are interested in the best way to support learners in becoming confident mathematicians. Her work can be found at htt­p:/­/yo­ucu­bed­.st­anf­ord­.ed­u/o­urm­iss­ion­/ and this is another rich source of materials. Jo defines mathematics as

a performance, a living act, a way of interpreting the world. Imagine music lessons in which students worked through hundreds of hours of sheet music, adjusting the notes on the page, receiving ticks and crosses from the teachers but never playing the music. Students should not be just memorising past methods: they need to engage, do, act, perform, problem solve, for if they don’t use mathematics as they learn it they will find it very difficult to do so in other situations, including examinations.

(Jo Boaler, The Elephant in the Classroom, p30)

So – welcome to the international staff room which is the problem-solving approach to learning and teaching mathematics. It is very busy in here, but very welcoming and completely collaborative.

Which areas of mathematics do these activities cover?

As with Understanding and Teaching Primary Mathematics the curriculum that has been used is the most recent National Curriculum in England. However if you are working with other curriculum guidelines you will notice that there is a good match between this and other curricula and programmes of study such as the International Baccalaureate expectation for primary mathematics through its Primary Years Programme (which takes a cross-curricular approach to the curriculum) and the Cambridge Programme of Study used by many international schools around the world. This is explored in more detail in the next chapter which also unpicks the method of problem solving which underpins the approach taken throughout the book.

What do the lesson plans look like?

Each lesson – or activity (some may be more usefully explored over a series of lessons) begins with a stimulus. This may be a problem or an image which you can share with your pupils. (All these are available to download from the companion website.)

The lesson plan includes the key objectives that learners will meet as a result of working on the activity and the resources which teachers will need to have available. There are annotated lists of key vocabulary that teachers should focus on. The lesson is then outlined in detail including the way in which the teacher could introduce the session; ideas for differentiation; ways to encourage discussion whilst groups explore the activity including probing questions; and ways in which the learning can be brought together in a plenary session. Each plan also includes techniques for assessment for learning so that you and your learners know what has been learnt and the next steps in learning.

Outline of the book

The outline of the book mirrors Understanding and Teaching Primary Math ematics so that they are genuinely companion texts. Each ‘Teaching point’, ’Resource inspiration’, ‘Portfolio task’ and ‘In practice’ from Understanding and Teaching will be developed into a classroom activity. Each activity will take the form of a lesson outline. As mentioned above, Chapter 2 outlines the approach the book takes to learning and teaching mathematics. It offers a rationale for a problem-solving approach drawing on international comparisons and recent research into effective mathematics learning and teaching. I...

Table of contents

• Cover Page
• Half Title page
• Half Title
• Title Page
• Copyright Page
• Contents
• Half Title
• Chapter 1 Introduction
• Chapter 2 Learning Mathematics through Problem Solving
• Chapter 3 Problem Solving Using Mathematics
• Chapter 4 NUMBER Counting and place value, fractions, decimals, percentages and ratio and proportion
• Chapter 5 NUMBER Properties of numbers
• Chapter 6 NUMBER Calculating (addition, subtraction, multiplication, division)
• Chapter 7 Algebra
• Chapter 8 Geometry
• Chapter 9 Measurement
• Chapter 10 Statistics
• Glossary
• Index