Peter Mattock’s* Visible **Maths: Using representations and structure to enhance mathematics teaching in **schools* supports teachers in their use of concrete and pictorial representations to illustrate key mathematical ideas and operations.

Viewing the maths lesson as an opportunity for pupils to develop a deep understanding of mathematical concepts and relationships, rather than simply to follow fixed processes that lead to ‘the answer’, is increasingly recognised as the pinnacle of best practice in maths education.

In this book,* *Peter Mattock builds on this approach and explores – in colourful detail – a variety of visual tools and techniques that can be used in the classroom to deepen pupils’ understanding of mathematical operations. Covering vectors, number lines, algebra tiles, ordered-pair graphs and many other representations, *Visible Maths *equips teachers with the confidence and practical know-how to take their pupils’ learning to the next level.

The book looks at the strengths, and flaws, of each representation so that both primary and secondary school teachers of maths can make informed judgements about which representations will benefit their pupils. The exploration begins at the very basics of number and operation, and extends all the way through to how the representations apply to algebraic expressions and manipulations. As well as sharing his expert knowledge on the subject, Peter draws on relevant research and his own experience of using the representations in order to support teachers in understanding how these representations can be implemented effectively.

*Visible Maths *also includes a glossary covering the key mathematical terms, as well as a chapter dedicated to answering some of the questions that may arise from the reading of the book. Furthermore, the accompanying diagrams and models are displayed in full colour to illustrate the conceptual takeaways and teaching techniques discussed.

Suitable for teachers of maths in primary and secondary school settings.

Click here to view *Visible Maths* featured in *Love Reading 4 Schools* books of the month.

## Reviews

Helen Thompson, Assistant Principal and Head of Maths, Corby Business Academy23rd April 2019This book is a fascinating read for both primary and secondary teachers of mathematics. It explores comprehensively the use of concrete and pictorial approaches such as tallying, counters, the number line, ordered-pair graphs, proportion diagrams, bar models, base ten blocks and vectors in one dimension to represent different types of numbers and how operations using these numbers can be explained.

To begin with, this book looks at the pros and cons of each approach to represent whole numbers, both positive and negative before moving on to fractions and decimals. Addition, subtraction, multiplication and division follows, presented in a colourful, diagrammatic way surrounded with clear, logical written reasoning, allows the reader to make their own informed choice about which representation best suits their students. Once the fundamental concepts are secure, this book moves on to look at more complex ideas such as powers and roots, irrational numbers, laws of arithmetic and order of operations before moving into the abstract world of algebra, yet still applying the same concrete and pictorial approaches as before.

Primary teachers are able to appreciate the mathematics that they will teach from a variety of angles. Secondary teachers are given a valuable insight into approaches taught in feeder primaries. Both sectors can consider how the concepts are extended and how these concrete and pictorial representations can be used to demonstrate these concepts at secondary school.

Furthermore, throughout the book and in the final ‘frequently asked and anticipated questions’ chapter, ideas and appropriate questions to consider are given to support teachers in developing their own understanding of each approach and provide the structure needed to build confidence as to how they can implement these representations in their own classrooms.

Visible Mathshas given me food for thought and opened my eyes to new pictorial representations that I hadn’t considered as well as making me evaluate some of the teaching approaches I currently use. I thoroughly recommend this book, especially for those adopting a teaching for mastery approach. It is also a ‘must read’ for all KS2/KS3 mathematics teachers enabling a smooth transition in the teaching of these key concepts between primary and secondary school to be achieved for the benefit of every student.Click here to read the review on

LoveReading 4 Kidswebsite.Lisa Coe, Readings and Musings blog19th March 2019”

My reading viewpointAs someone who works for a mathematics organisation which puts conceptual understanding at the heart of all that we do, reading about representations in mathematics was a natural step for me. Having been educated in the ‘it just does, learn it’ field of mathematics, I am always fascinated by representations which help me, and pupils, see the why of the mathematics and recognise the underlying structure.

I should also mention that I was given a review copy of this book by the lovely folks at Crown House Publishing. I am incredibly grateful to them for this (and they have a great selection of edu-books!).

SummaryThe title of this book utterly sums up its purpose – to support teachers in making mathematics visible to learners. Visible Maths takes a series of representations – from counters and Cuisenaire to vectors on number lines – and, beginning with representation of number, leads the reader through the ways in which these representations can support the understanding of a huge number of concepts of mathematics, making connections between these concepts. It supports the idea of developing these representations from the beginning of primary to enable them to be built upon in secondary, helping pupils recognise the underlying mathematical structures. Visible Maths covers a huge number of aspects from positive and negative numbers, to irrational numbers and surds and is a great way to develop understanding of these as well as considering how representations can help support pupils’ understanding.

First off, I have to say the whole book is beautiful, visually. Colour illustrations are used throughout to explicitly show examples of these representations being used. These representations thread throughout the book, allowing the reader to flip back and forth and really see the connections and links between them. The illustrations really make the book, for me.

Similarly, Mattock’s writing style is great. Straightforward and clear, with some delightful maths tangents. Visible Maths is designed for both primary and secondary educators, meaning that some of the mathematics and terms may be unfamiliar to some teachers, but a clear glossary and concise explanations support understanding without it feeling patronising. It also assumes little in the use of representations – clearly explaining, for example, how negative numbers may be represented.

As you will see from my key takeaways (below) I found this book enthralling. As a primary educator, it helped me to fully appreciate the importance of deep conceptual understanding and using representations effectively to help pupils ‘see the maths’. As a blueprint for providing some consistency between primary and secondary, Visible Maths has serious power. It really helps with the ‘joined up thinking’ concept too – that, as a primary educator, I need to see where pupils are going in order to fully appreciate the importance of building those foundations.

My key takeaways1.

Pupils need different representations, and not all representations work for all maths.I am guilty of having my favourite representations (Cuisenaire, anyone?) and have been known to try to shoehorn the maths to fit the representation. What I love about Mattock’s writing is that he is open and honest about the strengths as well as the drawbacks and limitations of different representations. While he maintains the thread of repeating representations throughout, he acknowledges the limitations of the representations but crucially (see next point) makes connections between representations to help the transition.2.

Making explicit connections between representations is something teachers should know how to do and have secure knowledge about.As I have said, Mattock uses a range of representations – counters, bars, Cuisenaire, vectors on a number line – to represent different mathematical concepts. In doing so, and in the way in which the book is laid out, it forces you to think about the relationship between the representations and the way in which these can be introduced to make connections. As he says himself, there is a huge amount of writing on different representations, but this is the first time I have come across so many in one place.3.

Rounding ‘rules’ can be represented using counters.This blew my mind, if I am honest! While I have often used number lines for rounding, to deepen conceptual understanding, I am the first to admit I used ‘because you do’ when explaining why 5 (or 5 units e.g. 0.5, 5 tens) rounds ‘up’ to the next multiple of 10/100 etc. I have always been uncomfortable with depictions of hills or bus stops or any other explanation. On p.184, Mattock provides a diagram of rounding which made it so clear I goggled at its simplicity. Essentially, when considering rounding of 2.5, if we show all the possible tenths between 2 and 3 using place value counters, we can see we have ten possible options. Dividing these into two equal groups (the halfway point) actually places 2.5 in the group with 2.6 onwards, and therefore it should be treated the same. Take a look – he explains it much better than me!4.

Algebra genuinely is every teacher’s responsibility.Through linking representations, Visible Maths shows the connections between algebraic thinking and other mathematical concepts often deemed ‘simpler’. Mattock says, “If we see algebra as a generalisation of the relationships we observe in number, then this representation [bars] exemplifies the emergence of the generalisation – the idea that, regardless of value, the relationship between a single square and the orange bar remains the same.” This, for me, was a really powerful moment. In KS1 we should be promoting this idea, just hinting at it, so that when pupils encounter algebra it is not a ‘new’ concept.5.

Don’t wait until you need the representation to use it.One of the representations Mattock uses that was new to me was that of vectors (essentially arrows, but a bit more nuanced than that!) both with and without a number line. He suggests these are particularly useful for algebra however if a pupil has never seen this before they are going to find it challenging to apply to the concept (p.202). This is really important for any representation and teachers should again engage in ‘joined up thinking’ to ensure exposure to representations throughout school.I think you should read this book if…- You teach maths. Seriously. If you are a teacher who, at any point, to pupils of any age, teaches them mathematics, then read this book. You may choose to skip over the more complex concepts if you don’t teach them, but I think as an explanation of different representations, why they should be used and - when, you can’t go wrong with this book.

- You are a Maths lead/HOD looking to get some consistency in representations.”

Click here to read the review on Lisa’s blog.

UKEdChat22nd February 2019Getting mathematical concepts to stick with students is one of the more challenging aspects of teaching the subjects, especially to pupils whose own belief is that they’re not good at maths, and give up early on in their mathematical adventures. Being able to make mathematical concepts relevant to students as well as making them relatable to real-world problems is essential in ensuring that skills learned can be applied once faced with the world of work.

Usually, being able to visualise mathematical concepts to students is an important step in helping them understand techniques to illustrate connections with previous learning, helping them master maths notions to progress their skills. The importance of visualising concepts is clearly integral for Peter Mattock who has collected together a valued resource of mathematical activities that can be used with manipulative across the age and ability range.

Aimed at teachers of maths in primary and secondary school settings, the book is split into 16 chapters exploring familiar mathematical strands such as the basic functions, but then expands onto powers and routes, fractions, irrational numbers, and concluding chapters all looking at the friendly algebraic problems. Exploring the ‘Laws of arithmetic’ is also explored, as many of the rules explained need understanding by pupils – so Peter showcases strategies that can be used with students to help embed their understanding. The use of Cuisenaire rods (see https://amzn.to/2Xg5CZR) is advocated throughout, and manipulating these resources is a great way to support learning about complex mathematical concepts. Furthermore, using Algebra Tiles (such as https://amzn.to/2Xebe6K) also helps to explain many difficult concepts that students struggle with when this topic is introduced, and maths counters (such as https://amzn.to/2XftfBS) also support many topic areas explained in the book.

Each chapter is introduced with a clear explanation to the topic being covered, but quickly gets into how to use manipulatives and representations to help secure knowledge with the reader, and then (more importantly) the students. Peter keeps his language throughout accessible, and will appeal to experienced and newer teachers in offer different ways of teaching (often) complex mathematical notions. Some pupils seem to easily pick up maths by the ‘talk and chalk’ method of teaching, but the creative use of manipulatives and representations in this book can help more students become more confident with their mathematical learning, building connections with their previous learning to the concept being taught.

This is certainly a book that should be regularly referred to by teachers and would be ideal within the staffroom CPD library, to help teachers become more confident in their own teaching of some complex mathematical concepts, as well as inspiring a shift to more practical, relevant and linkable mathematical learning for all students.

PROS:-Great ideas to use manipulatives and visual representations to help understand complex concepts.- Relevant for anyone who teaches maths in primary or secondary schools.

- Filled with easy-to-understand images to help explain concepts.

- Covering vectors, number lines, algebra, graphs and so much more.

- Helps teachers and students understand the processes and links to previous mathematical learning

Click here to read the review on

UKEdChat‘s website.Steve Lomax, mathematics adviser and national teaching for mastery lead24th January 2019What a great read this is!

Visible Mathswill quickly become essential reading for teachers of maths interested in teaching for secure and deep understanding rather than chasing quick wins or thinking they have to follow the latest educational fads.An ideal guide for those educators who want to help their pupils ‘see’ and ‘feel’ the mathematics.

Lucy Rycroft-Smith, Research and Framework Design, Cambridge Mathematics24th January 2019Visible Mathsis a thoughtful, careful and thorough exploration of some of the most useful visual models we can use to teach mathematics. Considering and critiquing the representations in this book will no doubt help teachers of maths at any stage of their career and will encourage them to engage with not only some of the deepest ideas, but also the deepest connections that make our maths classrooms so powerful. We ‘see’ mathematics in many different ways at different times in our teaching of the subject, and Peter has captured much of that interesting and shifting variety here.Jonathan Hall, Lead Maths Practitioner, Leeds City Academy and creator of mathsbot.com24th January 2019Visible Mathsis an essential read for any teacher of maths looking to improve their use of manipulatives and visual representations in order to help develop their pupils’ conceptual understanding. Throughout the book Peter demonstrates his superb pedagogical and subject knowledge to show how, through the use of representations, typically ‘difficult’ topics can be made more accessible for all pupils.An invaluable resource to refer to again and again, this book deserves a place on the shelf in every school’s maths department.

Geoff Wake, Professor of Mathematics Education and Convenor of the Centre for Research in Mathematics Education, University of Nottingham24th January 2019In

VisibleMaths, Peter Mattock provides a very thorough overview of how a variety of different representations can be developed to explore working with number and algebra in primary and secondary school mathematics education. I am sure that the book will become a must-have resource for all teachers of maths, and equally should be central to the discussions of any maths department as they plan their approach to the use of representations and models.Peter presents a carefully argued approach to the use of the different representations, clearly setting out how and when different approaches might be most beneficial and when they are less helpful. The copious supply of examples provided illustrate, in colour, the different approaches – offering clear insights into mathematical structure and how this might best be revealed in the classroom. Although some readers may have met some of the approaches before, I feel sure that all will find things here that will challenge their thinking – for example, the suggested use of vectors at much earlier stages in the development of working with number.

I recommend

VisibleMathsto all those who constantly consider different ways of supporting their teaching and their pupils’ learning.Emma McCrea, teacher trainer and author of Making Every Maths Lesson Count24th January 2019Covering various areas in the subject of mathematics, from counting to simultaneous equations,

VisibleMathsprovides a practical guide to using representations and manipulatives in the classroom. Counters, bars (including Cuisenaire rods and algebra tiles), vectors and number lines feature throughout, demonstrating how we can offer pupils coherence in the representations we choose to use, irrespective of the complexity of the topic we are studying.Overall, the book is a great resource for professional development and is a must-read for all teachers of maths.

Dr Colin Foster, Reader in Mathematics Education, Mathematics Education Centre, Loughborough University24th January 2019Visible Mathsis a beautifully presented, comprehensive manual containing detailed guidance on how a wide variety of visual representations can be creatively used in the classroom to support pupils’ deep understanding of mathematics. There is no other book like this, as it contains so much wisdom and practical advice on how to get the very best out of visual representations.