Mathematics For Primary Teachers

Valsa Koshy, Ron Casey, Paul Ernest, Valsa Koshy, Ron Casey, Paul Ernest

• Publisher: Routledge
• Year: 2014
• ISBN:
  9781134635917
• Edition: 1
• Topic: Didattica
• Subtopic: Didattica generale

This book combines accessible explanations of mathematical concepts with practical advice on effective ways of teaching the subject. Section A provides a framework of good practice. Section B aims to support and enhance teachers subject knowledge in mathematical topics beyond what is taught to primary children. Each chapter also highlights teaching issues and gives examples of tasks relevant to the classroom. Section C is a collection of papers from tutors from four universities centred around the theme of effective teaching and quality of learning during this crucial time for mathematics education.

Section B

The aim of this section is to support you to enhance your subject knowledge in mathematical topics. The topics included in this section are selected on the basis of what is considered to be necessary for a sound understanding of the contents of the National Curriculum at Key Stages 1, 2 and 3, the requirements of the Initial Training National Curriculum in mathematics set by the Teacher Training Agency and the Framework for teaching mathematics to implement the National Numeracy Strategy.

The following objectives guided the style and content of the chapters in this section. They are designed to:

• encourage you to think about the mathematics you already know;

• identify gaps in your knowledge and understanding of mathematical topics;

• consider mathematical topics at your own level through relevant contexts;

• make connections with various strands of mathematical topics;

• acquire or revise the correct terminology and language of mathematics;

• consider some key issues in the teaching of the topics to children.

Each chapter in this section begins with a list of the mathematics topics covered; sub-headings are used to guide the reader through the various topics included in the chapter. Mathematics is developed through examples. Emphasis is placed on addressing the underlying principles of a topic with the aim of facilitating greater understanding of the topic. Many of the principles are explained in order to facilitate thinking, in depth, about why many ‘procedures’ and ‘rules’ actually work. It is hoped that you will read the chapters in this section slowly and systematically; as the intention is to provide explanations of complex ideas rather than offer superficial discussions about mathematical topics. Within the text, key ideas about teaching the topics are briefly referred to, where appropriate, but it is assumed that you will refer to textbook schemes and sources for other practical ideas.

We recommend that you take time to read each section of the chapters. You may read a section about a topic that you are teaching to your class, or about a topic that is being covered on your course. Before reading a section it is a good idea to think about or write down the ideas you already know about the topic, also aspects of the topic you may feel anxious about or have difficulties with. While you read the section make notes about new ideas and vocabulary you come across. As you read through the text, it is also a good idea to give yourself some questions to tackle before you try the exercises at the end of the chapter. Teachers who trialled these sections found it useful to look at sections of children’s textbooks and teachers’ handbooks and relate the ideas to what is taught to children.

The chapters dealing with ‘number’ are longer than the rest. This is because the ‘number’ sections in both the National Curriculum and the TTA National Curriculum are substantially longer than the rest of the other sections. Also, in view of the emphasis placed by the National Numeracy Strategy on developing numerical skills and understanding, it was felt that you would appreciate opportunities to reflect on aspects of ‘number’ in greater detail than you have done in the past.

Finally, remember that learning and understanding mathematics takes time. As you read the chapters in this section, you should gain more insight into what each mathematics topic is about and develop your expertise and confidence to teach it. This, in turn, should enable you to teach it in such a way that the children you teach will both enjoy learning mathematics and understand what they are learning.

Auditing your subject knowledge

At the end of the Chapters 2 to 7 two types of tasks are provided. The first is a collection of tasks which enable you to think about the implications for teaching particular topics to children and the other is a set of tasks for you to try. Two sets of tests are included at the end of Chapter 8, which you may use for auditing your knowledge. The Record of Achievement and the audit grid in the appendices may also be of help.

Chapter 2

Whole numbers

This chapter focuses on:

2.1 Development of number concepts in the early years

2.2 The role of algorithms

2.3 Place value representation of numbers

2.4 Number operations

2.5 Factors and prime numbers

2.6 Negative numbers

Something to think about:

We have ten toes on our feet and ten fingers on our hands. It is natural for us to use a counting system based on ten. The sounds and sights we interpret to get information about our surroundings are received by two ears and two eyes. Is it therefore natural for us to use an information technology based on two?

2.1 Development of number concepts in the early years

Cardinal and ordinal numbers

Early experiences with counting make children deal with two aspects of number: the ordinality and the cardinality of number. Counting ‘one, two, three, four’ goldfish in a bowl or counting ‘one, two, three, four’ paws on a cat involves using 1, 2, 3 and 4 as ordinal numbers. Recognising that the goldfish and the paws of the cat have something in common -that they both consist of four things – involves insight into their common cardinality; the cardinal number 4 is used to describe the fourness of the goldfish and the paws. The same number symbol is used for both aspects – the ordinal and cardinal. The two aspects allow each number to have two roles. When 4 is being used in counting up to 4, it is playing its ordinal role, but when 4 is being used to indicate the size of a group of 4, it is playing its cardinal role.

Which aspect of number is involved when you see that a person or team is ranked fourth ?

Somehow, you need to count towards the person or team to see that the position is fourth; here, the ordinal aspect of the number four is involved.

Immediately spotting the cardinality of a group of things is possible, for most people, perhaps for small numbers.

The cardinality of a group of things is the number of things which are in the group.

Here is an activity for you to try.

Counting strategies

This activity is designed to help you to gain insight into the ability of adults and children to spot the cardinality of a small group of objects. Show a few adults and children a small collection of things, say 5, 6, 7 or 8. Find out how they determine the number in that set of things.

Do they count 1,2,3 and so on?

Do they use their fingers to count?

Do they count in their heads?

Do they ‘just know’ immediately by observing?

How large a group of objects can they spot immediately, without consciously doing anything?

Young children need to be provided with experiences to learn about the three aspects of number. Two have already been dealt with – the cardinal and the ordinal numbers. The third is the use of number symbols.

The cardinal aspect of a number is used to describe the number in a set: 10 beads in the set.

The ordinal aspect of a number refers to a number in relation to its position in the set: colour the fifth bead red.

A number symbol, say 9, is used both to express the cardinality of the number ‘nine’ and to show something in the 9th place. It is also sometimes used as a label: A...