**The book**

**Mattock, P..** *Visible Maths: Using representations and structure to enhance mathematics teaching in schools. ***2019. Crown House Publishing.**

**My reading viewpoint**

As 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!).

**Summary**

The 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 takeaways**

1. **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.

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