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 21^st 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. 

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!”

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.

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.

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.

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.

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. 

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.

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.