Mastery and Depth in Primary Mathematics

Enriching Children's Mathematical Thinking

Fay Lewis, Amanda Wilkinson, Marcus Witt

• Publisher: Routledge
• Year: 2022
• Print ISBN:
  9780367407452
• eBook ISBN:
  9781000517934
• Topic: Education
• Subtopic: Education General

The UK National Curriculum is clear about the importance of reasoning and problem-solving in mathematics. Mastery and Depth in Primary Mathematics aims to support trainee and established teachers to embed mathematical thinking into their lessons. The authors focus on practical and actionable ways that primary teachers can develop their children's mathematical thinking, reasoning and problem-solving: ideas which are at the heart of the UK National Curriculum.

Covering a range of areas in mathematical thinking such as reasoning, problem-solving and pattern-spotting, as well as systematic and investigative thinking, each chapter provides clear examples of how teachers can make small, manageable 'rich tweaks' to their existing lessons to increase the opportunities for children to develop their mathematical thinking. Teachers will be able to dip into the book and find inspiration and ideas that they can use immediately and, importantly, develop a set of principles and skills which will enable them to take any mathematical activity and tweak it to develop their pupils' thinking skills.

This practical guide will be invaluable to all trainee teachers and early-career teachers that wish to enhance their primary mathematics teaching.

1

The national curriculum, ‘mastery’, depth and making sense of the problem at the heart of primary mathematics teaching

DOI: 10.4324/9780367808860-2

Figure 1.1

The problem

The picture in Figure 1.1 is a summary of the first four pages of the Primary National Curriculum for Mathematics. The size of each word reflects the number of times that word is mentioned. We include this here, at the start of the book, because we think that it illustrates one of the fundamental difficulties that teachers, like you, have with using the curriculum document.

The National Curriculum for primary mathematics (DfE, 2013) has, at its heart, a problem for pre-service and Early Career mathematics teachers. The introductory pages state clearly the vision and aspiration for a deep and comprehensive mathematics education for children up to the age of 11. Such an education should include opportunities to increase fluency, to develop reasoning in mathematics and to engage in problem-solving. The curriculum document reminds teachers that children need to engage with ‘rich and sophisticated problems’ and to make connections between mathematics and other areas of the curriculum. However, and here’s the problem, the subsequent forty-three pages of the document provide little, or no guidance to teachers about how to live up to these ideals. There is a lot of guidance about the content of the curriculum, but very little about how new teachers can incorporate the need to develop fluency, reasoning and problem-solving skills into their teaching so that the children are both covering the required content and developing those skills. The Advisory Committee on Mathematics Education, in a publication in 2014, acknowledged the challenges faced by teachers implementing the (then) ‘new’ National Curriculum and saw ‘a need to develop a broad understanding of core curricular aims amongst teachers’ (p. 2),

If you are training to become a primary school teacher or are newly or recently qualified or are a more experienced teacher who wishes to develop your maths teaching further, we hope that this book will be of interest to you. The aim of this book is to support you to be able to achieve the aims of the National Curriculum document by suggesting ways in which mathematical skills (such as reasoning and problem-solving) might be embedded into your lessons. This is no easy task but one that is essential if children in primary classrooms are to have an experience of mathematics that is as rich as the one suggested in the first four pages of the curriculum document. We begin with a consideration of the context in which these wider mathematical skills are being taught.

Mastery

The Programme of International Student Assessment (PISA) assesses 15-year-old children from a variety of countries (or other jurisdictions) around the world with a view to providing an international comparison of mathematics achievement. This is done every three years, with the results usually published the year after the data is collected. The results of the 2012 data (published in 2013) showed the city of Shanghai clearly outperforming all other countries or jurisdictions. In 2014, the UK government launched a number of ‘Maths Hubs’ around the country, with the intention to disseminate best practices in mathematics teaching. As part of this, an exchange programme with Shanghai was established in which UK teachers went to Shanghai to observe teaching and Chinese teachers came to the UK to share ideas and teach demonstration lessons.

From this exchange programme, and from looking at curriculum documents from other high-performing countries, the notion of maths ‘mastery’ arose. This has now become part of the ‘conversation’ about good practice in primary mathematics and is relevant to the problem identified earlier.

The National Centre for Excellence in the Teaching of Mathematics (NCETM), an organisation, which has enthusiastically supported the introduction of ‘mastery’ principles into UK classrooms has published what it considers to be the five underpinning ideas in ‘mastery’ (NCETM, 2015). The five principles are as follows:

• Coherence (breaking new learning up into a series of small but logically connected smaller steps)
• Representation and Structure (the use in lessons of different representations to illustrate and expose underlying mathematical structures and concepts)
• Mathematical Thinking (the importance of children gaining more than an ability to carry out processes, or procedures, but that they understand why they are doing so. This has echoes Skemp’s (1976) ideas about instrumental and relational understanding.
• Fluency (children should have quick and accurate recall of number facts but should also be able to move fluently and flexibly between different representations of the maths and mathematical contexts)
• Variation (attention is given to the sequencing of the different lessons and the way that examples and questions are organised so that children’s attention is focused on what changes [varies] and what stays the same. Through this attention, the underlying mathematical structure is made clearer)

The NCETM has also published a number of guides to ‘mastery’ (Askew et al., 2015) which explore what ‘mastery’ might look like in different year groups. These publications also provide some guidance as we try to embed their ideas into our lessons. The guides suggest that a child has mastered a mathematical concept if they are able to explain it in their own words, give examples and counter-examples to illustrate it, represent it in a variety of ways, apply it to different contexts and problems and make connections between it and other facts or ideas. We refer to these as ‘markers of mastery’. Therefore, as we explore ways to support you in embedding ideas about mastery into your lessons, we consider how you might create opportunities for children to exhibit (and therefore to develop) some of the ‘markers of mastery’ outlined earlier.

Although ‘mastery’ is not mentioned explicitly in the National Curriculum, many of the principles outlined earlier are present in the picture of mathematics painted in its first few pages. Therefore, as we come to think about ways to embed some of the National Curriculum ideals into mathematics teaching, we draw on ideas about mastery as presented by the NCETM.

Depth – enrichment rather than acceleration

Drury (2018) suggests that one of the key ideas underpinning mastery is the belief that all children are able to understand and do mathematics, given sufficient time. She states (p. 1) ‘to teach for mastery is to teach with the highest expectation for every learner, so that their understanding is deepened’. She further breaks down the idea of ‘depth’ into three ‘dimensions of depth’. These are the deepening of conceptual understanding by seeing connections between concepts and between different representations of concepts, encouraging children to think like mathematicians by seeing patterns and rules, posing and answering mathematical questions and developing mathematical communication skills. This is again echoed in some of the ideas in the first part of the National Curriculum, which states:

Pupils who grasp concepts rapidly should be challenged through rich and sophisticated problems before any acceleration through new content. Those pupils who are not sufficiently fluent with earlier material should consolidate their understanding, including through additional practice, before moving on.

(p. 3)

There is agreement that children should be challenged to develop a deep understanding of the mathematical concepts they are learning before they are moved to new ideas. While this aim is very welcome (as is the shift from an approach that advocated ‘acceleration’ through the curriculum), it presents pre-service or Early Career teachers with a problem. Where are these ‘rich and sophisticated’ problems? How can they be incorporated into lessons so that the children described previously are challenged in mathematics lessons and are developing the depth of understanding that Drury and the National Curriculum seem to demand?

Ideas to help solve ‘the problem’

The good news is that you are not alone in trying to find ways to enable your children to have the kind of mathematical experiences imagined by the National Curriculum and by those advocating a ‘mastery’ approach to mathematical learning. In this next section, we consider some of the places (other than the rest of this book) where you could look for inspiration and some of the key principles which inform the suggestions that we make in the subsequent chapters of the book.

Nrich (see https://nrich.maths.org/) is an organisation, which seeks explicitly to enrich and deepen children’s mathematical experiences. It contains a huge number of rich tasks, designed with problem-solving and deepening experience in mind. However, your job is to do a little more than hunt around for other people’s lesson ideas. While we thoroughly endorse everything that Nrich does and stands for, we want you to be able to make ‘rich tweaks’ to existing tasks or activities so as to deepen your pupils’ mathematical experience. We would also like you to be able to design your own rich tasks so that you are able to embed them successfully into your lessons. We will therefore look at some of the key principles behind the Nrich tasks.

One of the key principles, which informs many of the Nrich tasks, is the idea of ‘low threshold, high ceiling’ (LTHC). The Nrich team (2013) sum up LTHC as meaning that ‘everyone can get started, everyone can get stuck.’ We understand this to mean that the task has an easy entry point so that all learners, almost irrespective of their current mathematical understanding, can make a start on the task. Lee and Johnson-Wilder (2018) suggest that developing resilience when doing mathematics is an important part of learning. We feel that learners are more likely to persist with a task if they have already experienced some success with it. Indeed, several writers (Barton, 2018) suggest that the relationship between mathematical success and motivation is bidirectional and that success may be a stronger predictor of motivation than motivation is of success. Designing tasks that have an easy entry point (low threshold) will increase the likelihood that your learners will persist. The notion that ‘everyone can get stuck’ is not intended to be off-putting. The Nrich Team (2013) note that getting stuck (and then getting unstuck) develops resilience and that allowing children to experience this as part of their mathematical education normalises it.

An analogy that many teachers find useful here is James Nottingham’s (2017) ‘Learning Pit’. This encourages children to ‘get in the pit’ and then get out again, thus building resilience in their mathematical approach. Children can often find encouragement when they are told it’s okay, in fact it is desirable, to get things wrong and not be sure what to do next. Put more simply, this is the idea of being stuck on a problem. The Learning Pit actually encourages children to recognise when they are in this stuck state and, to some extent, celebrate it. When children recognise that they are stuck in the pit, they can then consider what tools they have to get out of the pit. They can also see that their current situation is an important part of learning. They then might ask themselves the question, ‘How did I get out of this last time? Or ‘Where can I find resources to help?’ Nottingham (2017) argues that all of this can not only develop the children resilience but also lead to a deeper more permanent level of learning.

For many children, the idea of mathematical learning is that the teacher smooths the path for them to the point that they never experience any difficulties or ‘bumps in the road’. A feature of these LTHC tasks is that there is no fixed end-point but that different children can take them in different directions and may reach different stopping points, or points at which they get stuck. As we consider ways to embed opportunities to embed mathematical thinking skills (see p. 1) into your maths lessons, we will return to ideas about LTHC.

The Nrich Team (2013) note that some tasks may appear to be mathematically ‘low threshold’ but have other barriers, possibly psychological ones, to getting started. If a child has the required amount of mathematical understanding to begin a task but feels overwhelmed by the context or the structure of the task, they may feel unable to begin. They note that ‘some problems may require a very basic understanding of the mathematical content to solve, and yet still be extremely challenging’. In designing tasks that are genuinely LTHC, we need to consider more than the mathematical content of the task. In considering what LTHC tasks might ‘look like’ as you seek to create an LTHC classroom, the Nrich team emphasise the need for questions that draw learners’ attention to what is the same and what is different and to encourage learners to represent their mathematical thinking in different ways. The teacher might then ask specific children, who have chosen to represent their thinking differently from each other, to show and explain their thinking to the rest of the children. Encouraging the asking of questions as a result of these explanations might further encourage learners to explore avenues of interest for themselves.

‘Big ideas’ in primary mathematics teaching

Barclay and Barnes (2013) identified a number of ‘big ideas’ in primary mathematics teaching. Their aim in identifying them was to enable teachers and children to ‘develop connection and coherence in a curriculum which comprises micro-level objectives’ (p. 19). This seems very much in line with the aims of this book, namely to enable you to embed opportunities for children to engage in ‘bigger’ mathematical thinking into your lessons....

Table of contents

• Cover
• Half Title Page
• Title Page
• Copyright Page
• Contents
• Introduction
• 1 The national curriculum, ‘mastery’, depth and making sense of the problem at the heart of primary mathematics teaching
• 2 Reasoning with calculations
• 3 Reasoning in geometry and statistics
• 4 Problem-solving
• 5 Patterns and variation
• 6 Mathematical investigations, systematic thinking and finding all the possibilities
• 7 Planning for mathematical thinking
• Conclusion
• Index