Visible Maths

Understanding Numbers Through Multiple Lenses

Mathematics can feel dramatically different depending on how a concept is represented. A fraction might be part of a shaded shape, a point on a number line, or a slice of a bar model. Peter Mattock’s Visible Maths helps teachers understand why those choices matter and how to use them intentionally.

The book begins with a simple idea: the way we think about numbers and operations shapes how easily pupils grasp new ideas. By moving between discrete objects like counters and continuous models like bars or number lines, teachers can expose the underlying mathematical structure that often stays hidden in abstract symbols.

Concrete, Pictorial, Abstract – Brought to Life

Mattock builds on the well-known concrete-pictorial-abstract (CPA) approach, but instead of treating it as a rigid sequence, he shows how different representations reveal different aspects of the same mathematics. The book explores strengths and limitations of tools such as:

• Counters and discrete objects for building number sense
• Bar models and continuous representations for proportional reasoning
• Tallies and vectors for visualising operations
• Virtual manipulatives for whole-class instruction

Each representation is examined both for what it makes easier and what it might obscure, so teachers can choose with confidence rather than relying on habit or glossy resources.

Practical Support for Primary and Secondary Teachers

Whether you teach early number or introduce algebraic thinking, the book offers a framework for planning lessons that build conceptual understanding over time. Mattock draws on classroom experience and research to show how teachers can:

• Introduce new representations separately from new mathematical content
• Help pupils connect different representations of the same concept
• Anticipate common misconceptions linked to particular models
• Move pupils toward abstract proficiency without losing meaning

The aim is not to replace symbols with pictures, but to give pupils a richer mental toolkit they can use when they encounter unfamiliar problems.

Why Representation Matters in Teaching for Mastery

For schools adopting mastery approaches, representation and structure are central ideas. Visible Maths aligns closely with these principles by encouraging teachers to think carefully about the models they use and the connections they want pupils to make. The result is a more coherent, less cluttered approach to mathematics instruction.

A Resource That Respects Teacher Judgment

This is not a scripted scheme or a collection of quick tricks. It is a thoughtful exploration of how mathematical tools work, written for teachers who want to make informed pedagogical decisions. The book encourages reflection, adaptation, and a deeper understanding of why some representations work better than others for particular mathematical ideas.

For anyone responsible for helping pupils see the structure behind the numbers, Visible Maths provides a practical and intellectually satisfying guide. 🧮✨