Conceptual Maths

Teaching 'about' (rather than just 'how to do') mathematics in schools

By: Peter Mattock


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Size297 x 210mm
PublishedMay 2023

Written by Peter Mattock, Conceptual Maths: Teaching ‘about’ (rather than just ‘how to do’) mathematics in schools aims to empower teachers to support students on a comprehensive and coherent journey through school mathematics. Showcasing the best models, metaphors and representations, it provides excellent examples, explanations and exercises that can be used across the curriculum.

Concepts are at the heart of the study of mathematics. They are the ideas that remain constant whenever they are encountered, but which combine and build upon each other to create the mathematical universe. It is the structure of each concept that gives rise to the procedures that are used in calculation and problem-solving – and, by learning about these structures, a learner can make sense of how different processes work and use them flexibly as need demands.

In his first book, Visible Maths, Peter Mattock focused on the use of representations and manipulatives as images and tools and how this can provide a window into some of these mathematical structures. His aim in Conceptual Maths is to go deeper, beyond the procedures, and to shed greater light on the structures of the subject’s different concepts. The book explores how a variety of visual tools and techniques can be used in the classroom to deepen pupils’ understanding of mathematical structures, concepts and operations, including: number; addition and subtraction; multiplication and multiples; division and factors; proportionality; functionality; measures; accuracy; probability; shape and transformation; and vectors, among many others. In so doing, Peter equips teachers with the confidence and practical know-how to help learners assimilate knowledge of mathematical concepts into their schema and take their learning to the next level.

Containing numerous full-colour diagrams and models to illustrate the conceptual takeaways and teaching techniques discussed, Conceptual Maths also includes a glossary covering the key mathematical terms.

Suitable for teachers of maths in primary, secondary and post-16 settings.

Picture for author Peter Mattock

Peter Mattock

Peter Mattock has been teaching and leading maths for over 15 years. He is a specialist leader of education (SLE) and an accredited secondary maths professional development lead who regularly presents at conferences across the country. Peter also develops teaching for mastery in the secondary school classroom, having been part of the first cohort of specialists trained in mastery approaches by the National Centre for Excellence in the Teaching of Mathematics (NCETM).

Read Peter's features in TES:

Why our SLT need to be as accountable to us as we are to them'

The government and Ofsted have cost more people their careers than I can count'

Four ways to make sure maths teachers stay happy (and stay put)'

Take it from the task-master: easy ways to get pupils thinking'

Ten steps to surviving as a new head of department'

Five ways maths teachers can persuade secondary colleagues to embed numeracy'

How to grade questions in the new GCSE system'

Click here to read Peter Mattock’s blog.


  1. Very early in my career, I was asked what my philosophy was in the teaching of mathematics. My reply was to create enthusiasm for and excitement in the subject for students of all ages and abilities. The other question – what is maths? – has always been a more challenging one. Mattock asks this question and answers it by looking at the subject as ‘definitely … not ... a collection of procedures’ but being deeper than just procedures, it is about the structure of its concepts.

    Conceptual Maths is a tour de force. It is written to provide ideas of methodologies for teachers in primary, secondary and post-16 settings. This substantial book with over fourteen chapters covers number, addition and subtraction, multiplication and multiples, division and fractions, equality, equivalence and congruence, proportionality, functionality, measures, accuracy, transformation and vectors, chance, charting and graphing and data handling.

    Mattock is an excellent communicator. He has clearly given a lot of thought and analysis in developing his methodologies. He writes in a fluent, conversational style which is easy to read and holds the reader’s attention. Most teachers become used to introducing ideas and concepts using a method with which they are most familiar or with which they are most comfortable. Mattock provides a variety of ways of thinking about and exploring each concept which will make every teacher think about their own methods and explanations and then adapt them to embrace those described. His writing is littered with clear illustrations and diagrams which make good use of colour to reinforce the ideas and methodologies. Mattock progresses through mathematics starting with the basics of numerical calculations and on to GCSE level concepts. There is a wealth of ideas for teachers to explore individually or to use as a very valuable material for training and development.

  2. Conceptual Maths by Peter Mattock is one of the most powerful texts on the teaching of mathematics that I have ever seen. As a young trainee teacher, I regarded my ‘maths bible’ as being ‘Primary Mathematics Today’ by Shuard & Williams, but if I was going to recommend a tome on which to base my learning in the 21st century and all that the mathematics curriculum demands, Mattock’s opus magnum would undeniably be the one to choose.

    In an age when there are many books available on ‘how to teach maths’, or ‘mathematical knowledge for teachers’, Peter Mattock uses his own extensive knowledge and experience to bring everything together in a book which should be a ‘given’ on the reading lists for pre-service teachers and a must for every primary staffroom and secondary mathematics department. 

  3. Overall, I look forward to sharing the book with colleagues who work in teacher professional development.

    The most joyful and thought-provoking aspects of the book are illustrations of visual representations suggested to help assimilate new conceptual understanding.

    The book made me reflect on my own practice and students who are likely to use prepotent strategies when faced with a calculation following a procedure without visualising the underlying mathematical structure they have created. The book invites us to reframe ‘solving questions’ as ‘manipulating mathematical structure’ and share that with our pupils – “Look how you changed the structure of that mathematical object!”

  4. I liked the idea behind this book as soon as it landed on my doorstep.

    As a maths teacher, I am always looking for new ways to help my students learn, and regularly go back to the basic concepts to build up an understanding of the processes required.

    The idea of teaching about the background concepts of maths, rather than just ‘doing’ maths, is the fundamental goal of this book.

    It contains lots of explanations supported by established and sound models, and demonstrated with excellent examples.

    This book is big but well structured and organised into chapters based on the main areas of study.

    It is clearly aimed at educators rather than students – but there is no reason why parents or older students could not benefit from it themselves.

    It is aimed at all stages of education and would be a useful resource for any further education (FE) maths teacher – especially where working with learners who have struggled with maths GCSE or functional maths.

    The author is an experienced teacher and presenter who is clearly passionate about education and the mastery of his subject. While this book has clear links to the author’s previous publication, it is a very useful resource.

  5. Conceptual Maths, which is suitable for both primary and secondary mathematics teachers, offers practical guidance for developing students' understanding of key mathematical concepts. The author demonstrates how teachers can use manipulatives, visual representations, and relevant tasks to support students' conceptual understanding of core ideas in primary and secondary mathematics curricula. This book will serve as a most helpful companion for teachers seeking to teach mathematics for understanding.

  6. Peter Mattock focuses on the development of robust conceptual knowledge in school mathematics – where do core mathematical ideas come from and where might they lead? This knowledge does not come about by chance through repetitive practice but through tasks and discussion with knowledgeable teachers whose own conceptual knowledge is coherent and connected. Peter provides the background to this aspect of professional wisdom using ideas from literature and extensive experience of teaching mathematics – his own and that of others. Primary teachers will see how the conceptual foundations for further learning can be laid using learners’ natural responses to ideas, and secondary teachers will see how mathematical foundations have developed that can be used to teach more complex and abstract mathematics. New and experienced teachers will find plenty to support their thinking and maybe some moments of insight.

  7. All of compulsory school mathematics tied up elegantly in a detailed and comprehensive narrative. Concepts are explained with great clarity of language, beautifully clear representations and plenty of examples. New life is breathed into ideas – like surds, for example – with a fresh and rarely seen viewpoint.

    This book will become a central reference for anyone teaching mathematics – not just for teaching but for the pure joy in understanding the structures of mathematics from various new viewpoints. I know I will refer to it often in years to come.

  8. Conceptual Maths is a superbly detailed exploration of the structure of key maths concepts which will be useful for anyone who is interested in the teaching and learning of school-level mathematics. Whether used to inform curriculum-level decisions or day-to-day lessons, this book will be invaluable for both.

  9. Conceptual Maths offers that lightbulb moment for any teacher wanting to make sense of the key mathematical concepts they teach. The book takes the reader through time-saving walkthroughs, supporting them to understand links between concepts and subconcepts, strengthening their mathematical understanding and signposting potential misconceptions. Each chapter gives a blueprint for thinking that builds the pedagogical habits required to support excellent conceptual teaching and understanding. The book earns its must-read position on the bookshelf with the choice and range of concepts and subconcepts considered. Clearly written by a teacher with huge experience in the classroom, Conceptual Mathsprovides easy-to-follow advice on how concepts link together and how they can be used to inform mathematical processes. With its immersive writing style and timely, well-chosen visual representations, this book can be enjoyed cover to cover and as a troubleshooting toolkit to dip in and out of. A superb book for any maths teacher who is wanting to improve their teaching practice and deepen their own mathematical understanding, Conceptual Maths is the new go-to book on the bookshelf for taking maths lessons to the next level.

  10. The approach that Conceptual Maths takes encourages teachers to develop their own conceptual understanding in order to promote teaching for understanding in the classroom. Congruence of shapes is considered alongside algebraic equations and equivalent fractions, for instance, highlighting the concept of the equivalence relation and showing that three ‘topics’ that are taught separately are actually closely linked. This focus on concepts over processes makes Conceptual Maths a valuable read for anyone teaching mathematics.

  11. Mattock’s Conceptual Maths provides a real focus on the understanding of concepts and the need for teaching for understanding. Key to this are the connections made between different concepts, subconcepts and procedures. Representations repeat to support connecting ideas and these are repeated across numerous concepts. For example, repeated representations show clear connections between multiplication of integers and multiplication of surds and how these ideas link to those of factors, multiples and equality. As a primary practitioner, I do not understand surds. However, I can clearly see how the ideas connect and how what I teach at a primary level supports and is developed by maths teaching at secondary level.

    I also particularly like Mattock’s focus beyond number and calculation. While Conceptual Maths naturally begins with these building blocks, chapters on measures, shape and data allow explicit opportunities for teachers both new and experienced to think more deeply about how to teach these areas for understanding – something that is certainly less of a focus in primary mathematics literature. The discussion of measuring turn is particularly fascinating and has supported me in thinking more deeply about how we introduce degrees and protractors in Key Stage 2. Clear connections are made between proportion in an early chapter and conversion of units of measure in a later chapter, meaning the reader can pinpoint and explore the linked concepts with ease. The same representations are used to allow connections to be made and these have made me think even more deeply about how we teach units of measure in primary mathematics.

    The layout of Conceptual Maths allows for clear and easy reading: concepts are linked to subconcepts, which are in turn connected to procedures rooted in the concepts. Each concept starts with prerequisites and link concepts, providing a clear path and supporting teachers in seeing relationships. Within this, Mattock puts emphasis on alternative methods and strategies meaning that more novice teachers of maths can gain a depth of understanding. For example, column addition is brilliantly explained through representations, but Mattock also emphasises the importance of other methods and strategies, depending on the values.

    In a time when we want teachers to be thinking deeply about concepts and promoting depth over breadth when it comes to challenging our quicker-grasping learners, this book provides the stepping-stones for this. As I have mentioned, more novice teachers can quickly develop an appreciation for the connectedness of mathematics and gain a depth of understanding to support teaching. More expert teachers have opportunities to think more deeply about a concept through a raft of excellent, carefully chosen example tasks peppered throughout Conceptual Maths.

    Overall, I think that any teacher of mathematics at any level of experience will benefit from Conceptual Maths. Trainee teachers would understand the complex connections between mathematical ideas, and the links between those taught at primary and secondary. More expert teachers can gain further insight into how one concept connects to another and how deep conceptual understanding is key if we want learners to apply the principles of, say, integer addition to working with fractions or surds. The depth activities woven throughout provide excellent thinking opportunities for new and experienced teachers alike, whatever phase is taught. I cannot wait to share this book with colleagues from all areas of mathematics teaching.

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