Understanding and Teaching Primary Mathematics

Tony Cotton

• Publisher: Routledge
• Year: 2020
• ISBN:
  9781000173826
• Edition: 4
• Topic: Education
• Subtopic: Education General

Written by an experienced teacher and teacher educator with widespread experience of teaching mathematics in the UK and internationally, Understanding and Teaching Primary Mathematics combines pedagogy and subject knowledge to build confidence and equip you with all the skills and know-how you need to successfully teach mathematics to children of any age.

This fourth edition has been fully updated to reflect the latest research developments and initiatives in the field, including a brand-new chapter on 'Mastery and mathematics' and 'The Singapore approach' which reflects the current international interest in these approaches to learning and teaching mathematics. Extra features also include helpful callouts to the book's revised and updated companion website, which offers a shared site with a range of resources relevant to both this book and its companion volume, Teaching for Mathematical Understanding.

Stimulating, accessible and well-illustrated, with comprehensive coverage of subject knowledge and pedagogy, Understanding and Teaching Primary Mathematics is an essential purchase for trainee and practising teachers alike.

CHAPTER 1

TEACHING AND LEARNING PRIMARY MATHEMATICS

Introduction

Learning maths was like a dagger going through my skull – no matter how hard I tried, it wasn’t good enough. This was really frustrating ’cause I knew there were really important things to discover, and when I’m teaching and the kids really get it it’s like they’ve found diamonds. I don’t know whether it was my fault or the teacher’s.

This is the way a student of mine shared the experience of being both a learner and a teacher of mathematics. I had asked a group to describe to me their memories of learning mathematics at primary school. It details vividly the concerns that some of you may have that you do not have sufficient mathematical understanding to be as effective a teacher as you want to be. It also describes the joys that come with seeing that you have been able to explain a complex mathematical concept well. During the same activity another student remembered how a feeling of panic can set in if you feel as though you have not got a good grasp of the area that you are teaching.

My problem is being put on the spot – I can’t think then, lots of little warnings run through my head the moment I’m asked a question.

Some of you will have had very positive experiences of learning mathematics and some of you may have felt as though you could not understand what was being asked of you. Whichever is true for you, working at your own subject knowledge will support you in feeling much more able to find those diamonds that are lurking and to feel confident that you will be able to answer whatever question a pupil throws at you.

Through working on research projects such as the one I have described here, I have become convinced that the best way to develop mathematical subject knowledge is by engaging teachers in actively questioning and reflecting on their learning. Some beginning teachers do not see themselves as ‘good’ at mathematics because of the way in which they have experienced mathematics as a learner. Through exploring mathematics in a new way, they gradually change their view of what being ‘good at mathematics’ is and can begin to see themselves as mathematicians. Introducing their pupils to a form of mathematics that they have recently experienced themselves allows them to be both tentative in terms of the possible outcomes whilst being secure in the process. They know, or they trust, that interesting things will happen in terms of children learning mathematics. And they notice that through exploring mathematics in an open and investigatory way, their learners are themselves developing as young mathematicians. These beginning teachers are moving towards a belief that mathematics is about questioning, exploring and justification.

The beginning of this process is to understand what we might mean by subject knowledge. This chapter offers a definition of subject knowledge for teaching mathematics, as well as setting the scene for the rest of the book. It describes how working with the book will support you in developing your own mathematical subject knowledge. Armed with this understanding, you can plan how best you might use the book.

Starting point

Whenever I am asked to describe a teacher with good mathematical subject knowledge, I use the following example from my observation of a trainee teacher. The trainee had placed a multiplication grid (you fill in the blanks by multiplying together the numbers at the end of the row and column) on the whiteboard for the pupils in the class to complete as the mental/oral starter to a mathematics lesson.

As the multiplication grid was revealed, there were sounds of complaint from the pupils. One of the pupils said to the trainee, ‘You don’t put fractions in a number grid and we only go up to 10!’ The trainee, patiently, persuaded the pupils to accept this ‘new’ version of what had become an everyday activity for the pupils and then asked them to complete the grid. After 10 minutes the trainee stopped the class and asked, ‘Which column did you fill in first?’ One of the pupils put their hand up and said ‘48’. The trainee paused and said, ‘I don’t want you to tell me any answers. I want you to tell me which column you filled in first.’ At first the pupils did not understand why the answer for the top-left square wasn’t seen as important. Then realisation struck. One pupil said, ‘You could do the twos first. That would be really easy.’

×	8	2	$\frac{1}{2}$	5	$\frac{1}{4}$	10	4
6
3
7
9
4
5
12

Another pupil suggested starting with the tens for the same reason. By the end of the discussion, the class had come to understand that mathematical thinking was about looking carefully at a problem and finding the most effective, or efficient, way of solving it rather than simply following a process which had worked previously. So, for this grid, if you complete the twos column you can then complete the fours and eights columns, by doubling. Similarly, filling in the tens column allows you to complete the fives column by halving. The same process will sort out the half and quarter columns.

I often use this activity with teachers and trainee teachers. Most people will start at the top-left corner and work systematically through the grid. This shows how deeply engrained the feeling is that there is a ‘right way’ to complete the activity. When I suggest the alternative, several people have described how this ‘feels like cheating’. It is hard to break away from the image of learning mathematics that we carry with us from our own experiences at school. However, if you do not feel confident in your own mathematics knowledge, this is, in part, due to your experiences as a learner of mathematics. I hope this book will allow you to make such a break.

Why is mathematical subject knowledge important?

Whether you are training to become a teacher, or are an experienced teacher who wants to improve their mathematical subject knowledge, the process is the same. Someone with good mathematical subject knowledge is able to be confident in what they are teaching and, more importantly, will be seen as confident by their pupils. We all learn better from someone who we believe is both confident in their own knowledge and who is passionate about sharing that knowledge with us.

That is why the focus of this book is on developing your mathematical subject knowledge by sharing with you activities you can use in the classroom. I hope that by engaging you in mathematics that you see as relevant to your teaching, you can become as passionate as me about mathematics itself. This will ensure that the pupils you teach will learn well. I will have been successful as a writer if you, as a teacher, and your pupils as learners enjoy your mathematics and if you become more confident learners of mathematics together.

I would argue that there is a direct link between good mathematical subject knowledge and effective teaching and learning of mathematics. I would also suggest that there is a direct connection between having good subject knowledge and being able to make appropriate choices about the way in which to teach particular mathematical ideas. Before we move on, I should define exactly what we mean by ‘good mathematical subject knowledge’.

What is good mathematical subject knowledge?

Research on teaching suggests that teachers draw on three forms of knowledge in order to teach effectively. The first is knowledge of the subject itself – you need to feel confident in your own mathematical subject knowledge in order to be able to teach effectively. Teachers also need an understanding of the curriculum they are expected to teach, so you need to be clear which mathematical ideas and concepts are appropriate to the age range of the pupils you are teaching. Finally, you need to understand which are the most appropriate strategies and activities to engage your pupils in learning a particular mathematical idea. In order to develop your skills in teaching and learning primary mathematics, this book must focus on these three areas. The book...