Mastery Mathematics for Primary Teachers

Robert Newell

• Publisher: SAGE Publications Ltd
• Year: 2019
• ISBN:
  9781526482471
• Edition: 1
• Topic: Education
• Subtopic: Elementary Education

This book explores how mathematical mastery, influenced by East Asian teaching approaches, can be developed in a UK context to enhance teaching and to deepen children?s mathematical knowledge. It gives guidance on using physical resources to demonstrate key concepts, extended examples on how to teach different curriculum topics and how to plan for small-step progression. Key coverage includes: - Key terminology in mastery-style teaching
- The challenges in implementing a mastery approach
- The use of manipulative resources for deeper understanding
- An analysis of mastery and related schemes of work currently available
- Assessing mastery
- How to apply mastery concepts in the early years

Chapter 1: The context for mathematics teaching

What you will learn from this chapter:

• The historical issues surrounding why teaching procedures have tended to be at the expense of teaching for deeper understanding
• The strengths and weaknesses of primary maths teaching in the UK
• What a mastery approach might bring to our teaching
• The challenges and the possibilities that pursuing mastery teaching will face
• Clarity about how teachers need to understand their mathematical knowledge to teach effectively and how we can best support them

Background

As a lecturer in primary mathematics for the last ten years, I have seen a number of changes involving the National Curriculum and the extent to which teachers are told how and what to teach. The interactive whiteboard and the visualiser have changed the nature of class-led discussion significantly. The process of discussing maths has become more visual. At times this has been accompanied by an increase in the use of concrete resources. There have also been real attempts in some quarters to relate taught maths ideas to the real world.

Still there remain a number of truisms related to teaching and learning maths in the UK. It is often taught by very capable teachers, attempting complex pedagogical approaches with considerable success. Many, however, when it comes to teaching maths, are anxious and lack confidence. They seek to teach almost from a script, as it seems safer than attempting lessons that may invite contributions from children that they may feel ill-equipped to deal with. Only in a few cases is such a limiting approach the result of disinterest on the teacher’s part.

Setting and testing

A lot of primary and secondary schools have employed a setting system for teaching maths, grouping by some form of measured attainment. It is not clear how much impact this has on different attainment groups. Indeed, in many cases, it may relate to the quality or range of confidence among the teaching staff. Other schools favour a mixed attainment approach, either random groupings or specific arrangements of children whose attainment is at different stages. The broad findings from research indicate that only the highest grouping is likely to benefit from a setting approach and that this isn’t always guaranteed. Social constructivist learning theorists such as Vygotsky (1978), and more recent writers regarding effective development in attitudes towards learning maths such as Jo Boaler (2009), would be unlikely to be surprised by such outcomes.

Primary teacher confidence in maths

The search for an effective learning pedagogy takes place against the backdrop of quite specific testing at ages seven and 11 where an interpretation of the curriculum content is evaluated. The mixed experience that many school children have means that in the later stages of Key Stage 2 there are often sustained, maybe belated, attempts to secure necessary strategies for solving problems that have not been developed successfully at earlier stages. Perhaps as many as half the teachers teaching maths in primary school would not list maths as a strength in their teaching. There is no shame here. It is, though, worthy of real attention as we seek a sound way forward; we need to increase confidence and achievement levels in maths among our school children.

My belief, similar to other writers on the pedagogy of primary maths teaching, is that it is a subtle skill that all teachers can excel in and yet a number have not yet learnt to. The fact that a sizeable minority of those who feared they would never teach maths effectively now do so, leads me to believe that this situation can and must change.

Pedagogy of effective primary maths teaching

Alongside this, we have the conundrum about the performance of teachers who are also skilled in general primary teaching pedagogy and who have confidence and belief in their maths knowledge and ability to structure maths learning. What are their achievements and what is the impact of them? Well, many of the Shanghai teachers who have visited England to model their style of maths teaching have been in awe of our maths teachers who teach to add value to current levels of attainment across a wide range of achievement in our standard primary classes.

Our most confident primary maths teachers have, for many years, excelled at a number of things:

• They have modelled clearly in a way that has made lessons accessible to all children in classes with different learning needs and levels of attainment.
• They have allowed lessons to be accessed at different levels.
• They have managed independent learning whilst providing focused teaching to increase progress amongst an identified group of learners.

Commitment to inclusive teaching

In addition to this, they have enthused, guided and managed ‘low threshold, high ceiling’ tasks which allow a range of children to work at different speeds within common problems, tasks and investigations. Within this structure such teachers have created breakthrough moments that allow children to connect with more complex ideas and thinking within the class. Following on from the above point, they have and do show a belief in the potential of all children and allow many opportunities that might move learning forward.

These teaching skills and beliefs have been shown in sometimes challenging environments where some children are in fact resistant to learning.

Cultural attitude in the UK towards maths and confidence levels

Such teaching skills also take place against the backdrop of home environments where maths is not a central part of life. Nor is confidence often there among parents who themselves may have suffered from a cycle of underachievement over several generations. In short, not enough engagement of all children has taken place to allow deeper, more-confident mathematical understanding to emerge. Not enough teachers have been able to plan and deliver lessons that allow most, if not all, children, to face and conquer questions and nuances around the maths being explored.

For example, younger children sometimes struggle to become secure around counting beyond ten, they confuse 30 and 13, likewise 14 and 40, even 12 and 20. Meeting this issue head on can assist development. Fourteen sounds as though the four should be written first but we write the ten first.

Multiplication is also a hard concept for young children to understand:

• 3 + 2 means we get three and two and put them together.
• 3 × 2 actually means we repeat two three times as in 2 + 2 + 2 or that we are finding three lots of two, easier for a young child to understand through known contexts such as shoes, socks or partners for lining up. It can also mean 3 (two times) as in 3 + 3.

Handling mixed attainment groups so that real value added is achieved

Lessons with such emphasis are in fact quite common in our classrooms; identifying key points for learning and driving them home through reference to common experiences and real life, concrete situations. Yet, the class gets spread out so quickly in terms of achievement that understanding and learning maths throughout the school becomes an experience where confidence among children (and adults) is the domain of the minority.

Content-led curriculum or deeper mathematical discussion?

In short, the field becomes too spread out too early among our children to allow a sound understanding to be the basis of all lessons being taught in Years 1 to 6 of primary school. Coupled with this, not all primary teachers who teach a range of subjects (as many as eight or more in a single week) have become confident enough to manage learning experiences in maths that prevent the gap between children in the class becoming too wide. The result of this is that the whole class delivery in mixed attainment lessons is not ensuring something equating to parity in value added. The stronger are surviving, the less strong are condemned to a slower rate of progress. The accompanying confidence levels among children and staff exacerbate this. Teachers who themselves learnt, as Richard Skemp (1978) articulated, ‘rules without reasons’ or ‘instrumentally’ are often ill equipped to delve underneath the surface to tackle, confront and lead further discussion that would make learning in maths more robust and ready for more rigorous challenges. Adult and child alike are inadvertently colluding to avoid deeper discussions that would be beneficial.

It is through the set of circumstances captured here that we have arrived at the following point. Some of our most promising young mathematicians really thrive. They achieve very highly. They do not all come from private sector schooling. Some achieve their full potential because enough state sector schooling has allowed them to thrive. They then continue the journey into higher education and beyond. The rest of the story is a variation on this, with students dropping out along the way and many passing GCSE but without a secure, deeper understanding and not seeking to expose themselves to the rigours of maths A level. Teaching to the tests at Key Stages 1–4 can sometimes limit the depth of understanding that takes place.

Thus the spread of achievement in maths across the nation needs close scrutiny. A basic qualification is often achieved but not deeper understanding. This process starts early and has to change. In our primary schools, the teachers whose knowledge is not yet secure enough to teach effectively for deeper understanding need support. Their pedagogical skills in general are often very sound.

Sound pedagogy needs to become sound maths pedagogy

It seems that the government response has been to look at the countries who are achieving more rigorous understanding across a wider range of children. Amongst other countries, the spotlight has fallen on China and Singapore. Serious consideration has been given to the style of teaching and learning that forms the basis of maths education in primary and secondary schools. Money has been invested in allowing groups of teachers and organisations from the UK to become familiar with the methods and approaches being used there. These are examined closely in subsequent chapters.

However, before analysing the style of teaching that has come to be known as maths mastery, I would like to consider a range of initiatives undertaken in England that are linked to the circumstances outlined above. They are part of the journey in primary maths teaching in England and they have had real impact.

The national interest in maths mastery is a worthy one. However, even if we implement it full on, with no adaptation, that system of teaching would still have to contend with the variables that have brought about the circumstances that exist in our schools and society regarding maths. We have already initiated some very interesting ideas and strategies in the last 40 years that have contributed heavily to some of the achievements that have been made in developing students’ understanding in maths. Primary teachers in general have contributed to this, at least in part.

Extended INSET, BEAM and NRICH

The notion of developing teacher confidence and belief in the idea of teaching for understanding is not new. The debate has raged for 40 years or more as to how to make teacher maths knowledge in the primary age range more robust.

Extended INSET (in-service training) development in maths was a feature of most borough and educational authority provision. Through this approach, cautious teachers with traditional and formal knowledge became enthused with the ideas that lay behind mathematical concepts and how children could be supported in their learning in ways that were useful and interesting as well as relevant. The staggered nature of the course allowed for ‘in school’ development that was then built on when the course resumed.

The ‘Be a Mathematician’ (BEAM) publications bridged a gap in the 1980s between a formal, procedural style of teaching and a more open-ended pursuit of problem solving skills. It then supported the National Curriculum which included Attainment Target 1 (AT1) whereby children needed to apply knowledge as well as learn skills and memorise information. This helped to maintain an awareness of the importance of understanding maths itself rather than merely the procedures involved. BEAM publications were produced on a range of curricular-specific themes such as geometry, space, measure and number as well as child- or topic-centred themes such as ‘wheels’ or ‘shops’. The emphasis was always on problem solving and investigation. The resources proved invaluable as a means of keeping ‘mathematical thinking’ and ‘understanding’ at the heart of the experiences that children needed to receive. This was at a time when content was sometimes dominating through a transmission of knowledge (Askew et al., 1997).

The NRICH organisation and website (https://nrich.maths.org) seems to me to be an extension of some of the work that BEAM championed for many years. Accessible swathes of mathematical ideas including many opportunities to problem solve, investigate and apply knowledge, support teachers very well. Accompanying solutions, comments and deviations provide food for thought as ways to continue and adjust ideas and problems. The discerning teacher may adjust tasks in the light of experience and knowledge of children’s learning needs. The new iteration of an idea or investigation is submitted, thus widening and deepening the pedagogical and mathematical possibilities further. Finally, it is important to add that the interactive opportunities extend learning possibilities further in a way that is very different to earlier periods. The emphasis on ‘low threshold, high ceiling’ activities and investigations means that children can feed off ideas that others have, as the teacher scaffolds the learning experience and helps to empower children by assisting their ability to articulate their reasoning, a key requirement of anyone seeking to become a robust mathematician. More immediately relevant is that it is, commendably, a National Curriculum requirement.

However this approach does not achieve the goal that it is hoped that a mastery approach will achieve.

Official maths teaching requirements in primary schools

The first iteration of the National Curriculum (in 1989) identified a body of mathematical knowledge to be taught and learnt for the primary age phase with national testing linked to this content. The attainment targets within the curriculum were mainly knowledge based, although Attainment Target 1 (AT1) was very much related to a child’s ability to apply knowledge. Initially there was pleasing scope to evaluate understanding of their problem solving and investigative qualities. Time constraints and cost of evaluation meant that this was quickly streamlined to some application of knowledge through more formal standardised written tests. The sheer bulk of an entire document folder for each curriculum area was quite overwhelming. Thankfully, this was reduced in size quite quickly.

The introduction of the National Numeracy Strategy (DfEE, 1999) at the turn of the millennium was pleasin...

Table of contents

• Cover
• Half Title
• Publisher Note
• Title Page
• Copyright Page
• Contents
• About the Author
• Acknowledgements
• Chapter 1: The context for mathematics teaching
• Chapter 2: What is mastery teaching at primary level?
• Chapter 3: Key terms and features of mastery teaching
• Chapter 4: Manipulatives: Their purpose and use in a mastery approach to teaching
• Chapter 5: Planning for small-step progression
• Chapter 6: Mastery in the UK
• Chapter 7: Teaching for mastery
• Chapter 8: Planning and schemes of work
• Chapter 9: School models for teaching mastery
• Chapter 10: Assessment and mastery
• Chapter 11: Early years mastery
• References
• Index