Myths and Legends of Mastery in the Mathematics Curriculum

Enhancing the breadth and depth of mathematics learning in primary schools

Pinky Jain, Rosalyn Hyde

• Publisher: Learning Matters
• Year: 2020
• ISBN:
  9781526482266
• Edition: 1
• Topic: Education
• Subtopic: Elementary Education

This book helps you understand what ?mastery? is and how to effectively integrate it into classroom teaching. It explores how ?mastery? is viewed and supported in other countries and encourages a critical examination of this topical theme.
The book includes practical advice and examples of learning activities for primary teaching. It also outlines how to support children who might be weaker in their mathematical abilities and still ensure that all children master mathematics.
The text also supports those who are developing whole school mastery approaches and looks at how we can assess ?mastery? as well as how we can be confident that it is supporting good progress.

1 Defining mastery

Anne Mulligan

Diane Cochrane

Keywords: mastery; deep learning; conceptual understanding; variation; fluency; reasoning

Chapter objectives

This chapter will allow you to achieve the following outcomes:

• understand what is meant by ‘mastery’ in relation to mathematical learning;
• know some of the historical and theoretical background to ‘mastery teaching and learning’;
• consider some of the implications for schools of developing a mastery approach to teaching and learning mathematics.

Introduction

If we are able to develop mastery learning in students, we must be able to recognize when students have achieved it. We must be able to define what we mean by mastery and we must be able to collect the necessary evidence to establish whether or not a student has achieved it.

(Bloom, 1968, p8)

The term ‘mastery’ and its use in relation to mathematics teaching and learning in schools is becoming familiar, with teachers adopting a ‘mastery approach’ and schools using a ’mastery curriculum’ and assessing learners as being at ‘mastery level’. Before using the term ‘mastery’ in relation to mathematics, it is important to understand first what is meant by mastery, and second how the term applies to mathematics. It is also worth considering whether schools have a shared understanding of mastery in mathematics and how it is enacted in different school settings.

The National Curriculum makes clear its expectations for all learners by the end of their Key Stage by stating that they will ‘know, apply and understand the matters, skills and processes specified in the relevant programme of study’ (DfE, 2014, p4), and although it does not explicitly mention mastering the subject, the implication is there. The task for schools then becomes integrating a mastery approach to mathematics with the expectations of the National Curriculum; this, in theory, should be straightforward. The difficulty arises when schools and teachers do not have a clear understanding of what mastery involves and focus more on the term than on its meaning.

It is important to understand why there has come to be such a strong focus on the term ‘mastery’ in relation to current mathematics teaching. Mastery is not a new idea, but has had a resurgence of interest due to concerns over the performance of English learners in international tests (OECD, 2016; TIMSS, 2015) compared to those in countries such as Singapore and China, who are taught through what we term a ‘mastery’ approach. Comparisons between the education systems in England and Singapore and Shanghai in China have resulted in some East Asian approaches to teaching being adopted in English schools. Specific aspects include what has become known as ‘teaching for mastery’ and the use of textbooks. Concrete and manipulative resources are being used more widely as mathematical representations to support learners’ conceptual understanding and there is less emphasis on moving through topics quickly.

This chapter aims to define mastery with respect to mathematics teaching and learning. The historical development of mastery from its origins to its current use in schools will be outlined, as well as how it is being interpreted in some settings. Some of the key theoretical perspectives underpinning mastery teaching and learning will be highlighted and some of the implications for schools will be considered.

Defining mastery

What is ‘mastery’ and what does it mean in terms of mathematics? Various dictionaries define mastery as being comprehensive knowledge or having learnt and understood something to the extent that it can be used without any difficulty. A master is seen as someone who is dominant in a particular field with ‘exceptional skills’ or knowledge and who is ‘thoroughly proficient in their use’ (Collins English Dictionary, 2006, p528). The implication for mathematics is that learners who have mastered a concept are proficient in the application of the mathematical skills and knowledge associated with that concept. Nunes and Bryant (1998) use the terms ‘mastery’ or ‘mastered’ throughout their discussion of children’s mathematical understanding. They do not explicitly define the terms, but the implication from their discussion is that when learners have mastered a concept, they are deemed to have a good understanding of that concept, can make connections between it and other concepts, can reason with it, and can apply it in different contexts.

The teaching of mathematics in Shanghai is different to that in England and is considered to take a mastery approach. According to Boylan et al. (2019), it emphasises:

Whole-class interactive teaching to develop conceptual understanding and procedural fluency, using carefully designed tasks and skilful questioning. To ensure pupils progress together, tasks are designed to allow for extension by deepening understanding of concepts and procedures, and daily intervention is used to support those needing extra tuition.

(p15)

Recent approaches to mastery in mathematics are based on East Asian pedagogy, one example being the Mathematics Mastery (2019) programme. This was set up by the Ark network of schools in order to improve attainment of all learners by enabling them to develop deeper conceptual understanding and promote the use of a ‘mastery curriculum’ based on those of Shanghai and Singapore. It places emphasis on developing depth of understanding rather than breadth of content, and also focuses on whole-class teaching where most learners progress at the same pace. Studying topics in greater depth is intended to reduce the need to revisit topics as often as would normally be the case. The Mathematics Mastery programme also emphasises the importance of language, mathematical representations and having high expectations of learners. The aim of this approach is that learners become fluent in their use of mathematical concepts while developing understanding of them so that deeper conceptual understanding is developed (Vignoles et al., 2015).

While the current National Curriculum does not make specific mention of ‘mastery’, it is clear that the aims set out in the programmes of study for Key Stages 1–4 require learners to be able to ‘become fluent in the fundamentals of mathematics’, to ‘reason mathematically’ and to ‘solve problems by applying their mathematics’ (DfE, 2014, p4). If learners can successfully achieve these aims, then they would be expected to have mastered the subject to some extent. The importance of developing fluency ‘through a deep understanding of mathematical ideas and processes’ when attempting to master the subject is noted by Haylock and Cockburn (2017), who also emphasise the need to understand ‘mathematical structures’ and ‘make connections’ (p10).

The National Centre for Excellence in the Teaching of Mathematics (NCETM) is a government-funded organisation working to encourage teachers to adopt the ideas of teaching for mastery. It defines mastery in mathematics as a ‘deep, long-term, secure and adaptable understanding of the subject’, enabling ‘pupils to move on to more advanced material’ (NCETM, 2016a). In their Five Big Ideas in Teaching for Mastery (NCETM, 2019), shown in Figure 1.1, they promote an approach to teaching mathematics based on teaching practices in Shanghai.

Coherence

Coherence is set out as a main component of the teaching for mastery process, enabling learners to realise the potential of connections between mathematical topics previously learnt and working to build on this to develop knowledge. To support this, lessons are planned to contain small episodes of learning, which then connect, leading to generalisations that can be used to support problem-solving in a variety of contexts. This is not a new idea. In fact, it has been fixed at the heart of the National Curriculum since its first version in 1988, where ‘Attainment Target 1: Using and Applying Mathematics’ contained a variety of ideas within the statements of attainment that described this process. These aimed to lead to learners being able to demonstrate the ability to ‘use number, algebra and measures in practical tasks in real-life problems, and to investigate within mathematics itself’ (DES and Welsh Office, 1989, p3).

Figure 1.1 Five Big Ideas in Teaching for Mastery (NCETM, 2019)

Representation and structure

This puts great emphasis on how learners should be presented with a new concept or topic and the importance of considering different ways in which this can be achieved. Planning needs to recognise both the academic potential of the learners as well as their age, always aiming to extend learners while ensuring there is depth of understanding underpinning the ongoing development of knowledge. Internalising the learning is essential if learners are to be able to demonstrate their knowledge without requiring specific examples of concrete representation.

Mathematical thinking

Teaching for mastery includes the intention that learners will be encouraged to develop appropriate mathematical thinking skills leading to deep and sustainable learning. Learners’ active involvement is essential, and teaching should include careful questioning alongside opportunities for them to discuss and explore ideas, opening up opportunities for them to reason and construct learning with guidance from the teacher...