This 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 Maths has 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.
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