How Big is a Big Number?

Learning to teach mathematics in the primary school

Paul Killen, Sarah Hindhaugh, Author

• Publisher: Learning Matters
• Year: 2018
• Print ISBN: 9781526404145, 9781526404138
• eBook ISBN: 9781526447753
• Edition: 1
• Topic: Education
• Subtopic: Elementary Education

What, how and why?
If you don?t really understand the content of the primary mathematics curriculum, how can you teach it? This beautiful full colour book is here to help. It covers all you need to know to be an effective teacher of primary mathematics.

It shows you how to explore number, shape and pattern with the children you teach. It examines what we mean by ?mastery of mathematics? and reviews what we can learn from Asian maths teaching methods. It helps you to see how areas of mathematics fit together and how you can support children to build their own understanding of the subject.

This book goes beyond showing you how to teach. It shows you that process is as important as product. That getting it wrong can be as useful as getting it right and that children can?t really learn the what without understanding the why.

Masters of the classroom

• Defining a ‘master’

• Maths is a verb

• What does the National Curriculum say?

• Questions, questions

• Asian thinking

• Rounding up

• What maths can you see?

Masters of the classroom

Our book begins with consideration of what ‘mastery in mathematics’ actually is and what ‘mastery’ may look like in the classroom. We discuss key principles of mastery and how the teacher can develop mastery in their pupils.

Over the past few years, many educators, educational thinkers and politicians have promoted the use of the phrase ‘mastery’ in relation to mathematics. However, before talking about the value of any concept, we need to have at least some idea of what it is. Our, perhaps contentious, interpretation is that there are many commentators who use the term mastery, without knowing what it really is … but it does sound good!

Defining a ‘master’

It may surprise you that we choose to begin this book on mathematics with a little English grammar. However, in order for you to gain an understanding of the term ‘mastery’, it is useful to consider our everyday use of the term, as the word ‘master’ can be a noun, an adjective or a verb. Exploring these different uses may give us some insight into what the phrase means in terms of teaching mathematics.

As a noun, ‘master’ could be used to describe such diverse things as:

• a person who owns a dog or other animal

• a person who captains a merchant ship

• a male teacher

• a young man.

These different uses of the word ‘master’ lead us to a number of other descriptions including: mastery; mastermind; masterclass; master of ceremonies; grandmaster; chess master; masterpiece; masterstroke; master of disguise; old Master; headmaster; bandmaster; station master; postmaster.

Equally, ‘master’ can be used as an adjective to describe many roles or objects, such as master chef, master baker, master bedroom, master’s degree, master tape. When used as a verb, one may talk about ‘mastering a skill’ or even to ‘mastermind a crime’.

The Oxford English Dictionary describes the verb ‘to master’ as, gaining complete knowledge of, or skill in a subject, technique, and the word ‘mastery’ as, complete knowledge or command of a subject or skills.

These definitions may lead you to ask, What is a complete knowledge?

Perhaps the greatest of all mathematical thinkers was Albert Einstein, who once said:

Only one who devotes himself to a cause with his whole strength and soul can be a true master. For this reason mastery demands all of a person.

Wow! This sounds hard doesn’t it? The term ‘master’ seems to imply being some sort of expert or genius – or even a wizard!

Given these varied perspectives on what mastery may be, it is worth considering how this can possibly apply to primary mathematics. Can a child actually be a master of mathematics? And if so, what would this look like?

It is our view that a child can indeed be a master of mathematics; the art of the mathematics teacher is to ensure that children are presented with the opportunities to enable this to happen.

Maths is a verb

It is possible to find a book that will show you where you should place your fingers on a guitar to play a D or an A minor chord. Alternatively, someone may show you where to place your fingers on a piano keyboard to play a C chord. Does that mean you can learn to play the guitar or piano by just reading about it? Similarly, we have watched many cookery programmes that show us how to bake bread or make a soufflé. Does that mean therefore that after watching these programmes we can bake a tasty loaf or know that our soufflé will rise? Like many things, the knowledge of what to do is important, but to become proficient at any of these activities, practice is required. The only way to become proficient at a musical instrument or at cooking is to keep practising. The more we engage with the activity, the better at it we become. It is the same with mathematics.

Mathematics has a body of knowledge that requires acquisition. However, that knowledge alone is insufficient for us to become proficient in it. We need to practise using this new knowledge. First, perhaps by relating it to existing knowledge. Then, this knowledge needs to be applied, in a variety of situations, moving from the familiar to the unfamiliar. In this sense, we see mathematics as a verb – a ‘doing’ word. Deep learning comes from doing mathematics. The Chinese philosopher Confucius famously said:

I hear and I forget. I see and I remember. I do and I understand.

This sums up the way we would like children to learn mathematics.

Just do it

It may have been easy for Confucius to say we should get children to ‘do’ mathematics. But what does that mean? For some teachers, in days gone by, that may have meant finding a worksheet with enough questions to reduce the chance that a child could actually complete their work within the lesson. In such lessons, differentiation of work can become using the same questions with harder numbers.

Many American school textbooks have perhaps over a hundred questions on each topic. The evenly numbered questions have the answers at the back and are completed in the lesson. Only the Teacher’s Guide has the answers to the odd numbers, so they become the homework.

Mastery moves completely away from this idea. Maths mastery is not about children completing the most questions, nor children completing similar questions with bigger numbers.

Reflection

Suppose you are planning to teach long multiplication. How would you know if each child in your class has learned long multiplication?

How many long multiplication questions would you expect children to do by themselves in order for you to be assured that they have understood the concept?

What does the National Curriculum say?

You may think that a good place to start considering any questions about maths mastery is the mathematics National Curriculum. Does it surprise you therefore that the words ‘master’, ‘mastery’ and ‘mastering’ do not appear anywhere in the mathematics content of the Primary National Curriculum? The words you do see in the mathematics National Curriculum include: fluency; conceptual understanding; mathematical reasoning; problem solving; enjoyment; curiosity and application. These are what we would like to define as some of the principles of mastery. It is perhaps by thinking of a child who can evidence some of these principles that we can move towards an image of mastery.

How to master maths

The primary mathematics curriculum is clear that children need to challenged by being offered

rich and sophisticated problems before any acceleration through new content.

You may be able to recall lessons from your own mathematics education where the teacher finished a topic before you had really understood it and then quickly moved on to something different. This is what we call aquaplanic learning as only the surface of each topic is considered. Children can ‘aquaplane’ from one topic to another, making no connections whatsoever between any of them. Such approaches result in learning not being secure and may lead to a child learning only what they cannot do. A mastery approach seeks to eliminate aquaplanic learning.

We see ‘mastery’ as a pedagogical approach to teaching and learning within mathematics. Any aspect of the mathematics curriculum can be developed to mastery level: by this, we mean that new learning is assimilated into each child’s existing knowledge and understanding. Children need to be able to apply new knowledge in a wide variety of contexts to fully demonstrate deep understanding of each new topic. By this token, a child ...

Table of contents

• Cover Page
• Halftitle
• Advertisement
• Title
• Copyright
• Contents
• Acknowledgements
• About the Authors
• Introduction
• 1 Masters of the Classroom
• 2 Not as Easy as 1, 2, 3
• 3 A Ray of Sunshine
• 4 Maths is Mental
• 5 Making Sum Sense
• 6 Recurring Problems
• 7 The Joy of X
• 8 The Shape of Things to Come
• 9 Losing the Plot
• 10 Made to Measure
• 11 Statistically Speaking
• 12 Moving on
• Appendix 1 What Maths can you see?
• Appendix 2 Task Answers
• Appendix 3 Mathematics National Curriculum by Topic
• Appendix 4 Roman Numerals 1–1,000
• Index