Lessons in Teaching Number and Place Value in Primary Schools

Kathleen Morgan, Stephanie Suter

• Publisher: Learning Matters
• Year: 2014
• ISBN:
  9781473908611
• Edition: 1
• Topic: Education
• Subtopic: Elementary Education

Lesson planning in line with the new Primary National Curriculum! Structured around the number and place value sections of the programmes of study within the National Curriculum (DfE, 2013), this book provides practical examples and lesson ideas on how number and place value can be taught within the primary classroom. Children need an in-depth understanding of our number system in order to access effectively other elements of the number curriculum; this is an area of maths that is a high focus throughout the primary curriculum. The accompanying commentary before and after each lesson plan, informs some of the decisions that you will make as part of the planning process. Potential challenges associated with the lesson, including common errors and misconceptions children encounter are also discussed. The authors recognise that each class is unique and so you will find suggestions after each lesson on how it can be adapted and developed to suit your teaching needs. Did you know that this book is part of the Lessons in Teaching series?
Table of Contents Teaching the national curriculum for mathematics / Teaching number and place value / Planning: telling the 'story' of the lesson / Year 1: One more than, one less than / year 2: Place value / Year 2: Using and = signs / Year 3: Patterns when counting in multiples / Year 4: Negative numbers / Year 4: Roman Numerals / Year 5: Big numbers / Year 6: Understanding decimals / Year 6: Reading scales / Moving on / Glossary of terms / Models, images and practical resources WHAT IS THE LESSONS IN TEACHING SERIES?
Suitable for any teacher at any stage of their career, the books in this series are packed with great ideas for teaching engaging, outstanding lessons in your primary classroom. The Companion Website accompanying the series includes extra resources including tips, lesson starters, videos and Pinterest boards.
Books in this series: Lessons in Teaching Grammar in Primary Schools, Lessons in Teaching Computing in Primary Schools, Lessons in Teaching Number and Place Value in Primary Schools, Lessons in Teaching Reading Comprehension in Primary Schools, Lesson in Teaching Phonics in Primary Schools

Chapter 1

Teaching the National Curriculum for mathematics

Learning outcomes

This chapter will help you to:

• engage with some of the aims that underpin the National Curriculum for mathematics;

• recognise the importance of developing both teacher and pupil confidence in numeracy and mathematical skills;

• consider the importance of providing opportunities for children to develop conceptual understanding, use mathematical language and solve problems in mathematics.

Maintained schools in England are legally required to follow the statutory National Curriculum and its programmes of study that outline the skills and processes that should be taught. However, although this core knowledge is outlined, schools and teachers need to make important decisions about how this content is taught. Indeed, in the overall aims of the National Curriculum (DfE, 2013, p6) the point is made that teachers can develop exciting and stimulating lessons to promote the development of pupils’ knowledge, understanding and skills.

Extensive literature has already been written about good mathematics teaching. It is therefore not the intention of this chapter to summarise the rich sources of research and literature that already exist. Instead this chapter will focus attention on some of the messages in the introductory sections of the National Curriculum. These sections can easily be missed as it is tempting to head straight for the programmes of study to look at the content that needs to be covered. However they contain some important messages that may influence your approach to teaching this content.

Confidence in numeracy and other mathematical skills

The National Curriculum, within the introductory paragraphs on ‘numeracy and mathematics’, states that Confidence in numeracy and other mathematical skills is a precondition of success across the National Curriculum (DfE, 2013, p9). This highlights one very important factor that teachers need to consider when teaching mathematics. Sir Peter Williams, in the Independent Review of Mathematics Teaching in Early Years Settings and Primary Schools, identifies how The United Kingdom is still one of the few advanced nations where it is socially acceptable – fashionable, even – to profess an inability to cope with the subject (2008, p4). You may very well have heard adults making statements such as, ‘I was never any good at mathematics’; however, it is interesting to consider whether you also hear adults making similar statements about other subjects, such as, ‘I was never any good at reading’. In order to break this cycle it is important that children gain confidence in the subject and as teachers we need to work hard to develop and instil this confidence. Haylock (2010) describes his experiences of working with trainee teachers who expressed anxiety about teaching the subject. He explored their attitudes and found that they can stem from feelings of helplessness, inadequacy and fear around the subject that often evolve from their own experiences of being a learner in the classroom. In order to instil confidence in the children, teachers therefore need to overcome their own anxieties and feelings so that these are not inadvertently passed on to the children they teach. Williams makes the important point that Confidence and dexterity in the classroom are essential prerequisites for the successful teacher of mathematics and children are perhaps the most acutely sensitive barometer of any uncertainty on their part (2008, p3).

One way for primary teachers to gain confidence and overcome any feelings of helplessness and fear is to develop their own understanding and knowledge of the subject. However it takes a particular kind of ‘subject knowledge’ to become a confident teacher of mathematics. It might be assumed that those with higher formal mathematics qualifications are best placed to teach mathematics. Askew et al. (1997), however, found that this was not necessarily the case and that it was not the teachers with higher formal mathematics qualifications who necessarily helped children to make the highest gains. Instead this research found that the most effective teachers were those who had three types of knowledge: a deep knowledge and understanding of the mathematics they were teaching, a good knowledge of the children they were teaching and a good knowledge of relevant teaching approaches they could employ. It can therefore be argued that good knowledge of the subject involves a detailed understanding of the ideas and concepts that are being taught. This involves skills such as the ability to break mathematics down into steps and stages the children can understand and relate to and to be aware of the errors and misconceptions that children can make. Good knowledge also requires a detailed knowledge of what the children already know, understand and can do and their next steps in learning. Finally good knowledge also requires good pedagogical knowledge so that a variety of teaching approaches can be employed to help the children to learn effectively and to become confident mathematicians so that they do not go on to form the next generation of adults who profess an inability to cope with the subject.

The next sections in this chapter will consider some aspects of pedagogical knowledge. The introductory paragraphs of the mathematics National Curriculum (DfE, 2013, p99) identify the importance of children gaining an enjoyment and curiosity about the subject and also state that:

The National Curriculum for mathematics aims to ensure that all pupils:

• become fluent in the fundamentals of mathematics, including through varied and frequent practice with increasingly complex problems over time, so that pupils develop conceptual understanding and the ability to recall and apply knowledge rapidly and accurately;

• reason mathematically by following a line of enquiry, conjecturing relationships and generalisations, and developing an argument, justification or proof using mathematical language;

• can solve problems by applying their mathematics to a variety of routine and non-routine problems with increasing sophistication, including breaking down problems into a series of simpler steps and persevering in seeking solutions.

There is not scope in this chapter to discuss all of these ideas in detail. Therefore just one aspect from each aim will be discussed: conceptual understanding, using mathematical language and solving problems.

Conceptual understanding

Haylock (2010) suggests that, in order to promote positive attitudes to mathematics, the subject needs to be explained – an approach that requires deep understanding of mathematics, good knowledge of the children and the use of effective teaching approaches. He outlines the importance of shifting perceptions away from the idea that the subject can be taught by following rules and recipes towards an approach where understanding is developed. Skemp (1976) explored these two different approaches in depth and named them instrumental and relational understanding. He identified ‘instrumental understanding’ as learning mathematics by memorising and following sets of rules but without having an understanding of the reasons why these rules work. For example, when you learnt mathematics at school you may have been given the rule for multiplying numbers by ten as ‘add a zero’ and may have then been given the opportunity to use this rule by completing some examples. However you might not have been given the opportunity to explore this rule, to understand why this rule might work and to find out whether it does work in every circumstance. In contrast, Skemp therefore also identifies ‘relational understanding’. This differs in that it involves not only knowing what to do to solve some mathematics but also understanding why and how it works. When using this type of teaching approach a teacher might instead model to children multiplying by ten by demonstrating how all of the digits move one place to the left and then a zero is added to the empty column as a place holder. While both approaches can in some cases lead to the correct answer, the second approach helps to avoid part-learnt or half-forgotten rules and the creation of easily avoided misconceptions when the rule does not apply to all numbers (for example 2.5 × 10 ≠ 2.50). It is therefore relational understanding that supports the development of conceptual understanding and as teachers we need to help children to understand how and why the mathematics they are exploring works. This is also recognised in Understanding the Score (2008), an Ofsted report that provides an analysis of inspections of mathematics teaching. This report identifies the dangers of fragmenting mathematics and discourages teachers from presenting children with sets of rules to memorise without them understanding the contexts in which these rules work.

Our beliefs about mathematics can clearly influence the way in which we teach the subject. Askew et al. (1997) identified three different sets of beliefs about the way mathematics is taught and their research found that, although teachers might at times hold a range of different beliefs, they generally have a disposition towards one particular approach. In their research they identified ‘discovery teachers’ as those who place more emphasis on learning than teaching and view mathematics as something that should be discovered by children. The ‘transmission teachers’ were those who placed more emphasis on teaching than learning and view mathematics as a set of rules and procedures children should learn and use. Finally the ‘connectionist teachers’ were those who had a clear focus on helping children to establish connections between different aspects of mathematics and different representations of mathematics. The research found that teachers who had a strong connectionist orientation were more likely to have classes that made greater gains than those with a strong transmission or discovery orientation. It is therefore useful to consider ways in which mathematics can be represented.

Haylock and Cockburn (2013) provide a useful model that can be used when teaching children to develop understanding and make connections between new and previous learning and ways of representing mathematics. Figure 1.1 shows the four types of experiences that can be integrated into number-based mathematics lessons. Real objects can be provided for the children to handle and manipulate in order to give them physical experience of the mathematics and pictures can be used to show children images of the mathematics and how it might be drawn (the models, images and practical resources section at the back of this book provides examples of real objects and images that can be used when teaching number and place value). Alongside this, mathematical language can be used to describe the concrete experiences and images and mathematical symbols can be used to show how we might record them. This model links back to the introductory section of the mathematics National Curriculum, where the interconnected nature of the subject is recognised and therefore children need to be able to move fluently between representations of mathematical ideas (DfE, 2013, p99).

Figure 1.1 Important connections

Using mathematical language

Talk is a vital element of mathematics lessons and we need to encourage children from a young age to explain their thinking and justify their reasoning. This can be very difficult if they don’t have the language in which to express themselves.

The National Curriculum acknowledges how The quality and variety of language that pupils hear and speak are key factors in developing their mathematical vocabulary and presenting a mathematical justification, argument or proof (DfE, 2013, p99). Therefore teachers need to actively support children in developing their use of increasingly precise mathematical language. Askew (2012) makes the useful distinction between talking about mathematics and talking mathematics. When children are learning a language such as Italian, we want them to talk Italian and not just talk about Italian. In the same way we need to engage children in the language of mathematics by talking mathematics rather than just talking about it. Enabling children to talk mathematics takes careful planning and consideration, particularly when children need to become familiar with a wide range of technical vocabulary that they are unlikely to use regularly and hear outside of the classroom in everyday conversation.

The Mathematical Vocabulary book (DfEE, 1999) provides guidance on ways in which you can plan opportunities to develop children’s use of mathematical vocabulary. This includes providing the opportunity for children to listen to adults using words precisely and ensuring that children regularly have the opportunity to describe and compare the mathematics they are engaged with, to discuss how they are approaching and tackling the problems they have been set and to justify their methods, solutions and reasoning to others. Being able to express your thoughts in a logical and coherent way takes practice and an audience. The suggested lesson plans within this book suggest ways in which regular opportunities can be provided for children to engage in discussion in pairs, with the whole class and with the teacher. This provides children with opportunities to practise their explanations and to refine them from feedback and, importantly, helps teachers to gain an understanding of children’s thinking and how this can be further developed. In order for meaningful discussion to take place children need something meaningful to talk about. To give children opportunities to develop an argument and have opportunities to explain and justify their thinking children need to be presented with problems to solve.

Solving problems

Problem solving should be a regular and integrated part of mathematics teaching and learning so that children have the opportunity to apply and reinforce the knowledge and skills they are learning. To identify when something actually poses a problem, Haylock (2010) identifies three Gs of problem solving: the given, the gap and the goal. A problem is only really a...

Table of contents

• Cover Page
• Halftitle
• Title
• Copyright
• Contents
• The Authors
• Acknowledgements
• Introduction
• 1 Teaching the National Curriculum for mathematics
• 2 Teaching number and place value
• 3 Planning: telling the ‘story’ of the lesson
• 4 Year 1: One more than, one less than
• 5 Year 2: Place value
• 6 Year 2: Using < > and = signs
• 7 Year 3: Patterns when counting in multiples
• 8 Year 4: Negative numbers
• 9 Year 4: Roman numerals
• 10 Year 5: Big numbers
• 11 Year 6: Understanding decimals
• 12 Year 6: Reading scales
• 13 Moving on
• Glossary of terms
• Models, images and practical resources
• Index