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Children's Mathematics

Children's Mathematics
Making Marks, Making Meaning

Second Edition
  • Elizabeth Carruthers - Early Years Researcher, Head of Children's Centre and formerly NNS Consultant
  • Maulfry Worthington - Doctoral Research: Free University, Amsterdam and former NNS Consultant

August 2006 | 280 pages | SAGE Publications Ltd

'In Case Study 5 (a grassroots 'Children's Mathematics Network group') the initiative supported the participants in their professional change by giving them a space for the detailed and joint consideration of children's mathematical thinking. Another significant feature of this initiative is its focus on careful consideration and analysis of children's mathematics, and the ways in which professionals can support and encourage the children's mathematical thinking and reasoning… The standard of the mathematical understanding, thinking and reasoning that the displays revealed was far higher than the specified curriculum objectives for children of this age…' - Researching Effective CPD in Mathematics Education (RECME) project: (NCETM, 2009)

'The review also plays great score by play-based learning of a mathematical nature, and makes specific recommendations regarding early mark-making as a precursor to abstract mathematical symbolism'. Section 115 features children's mathematical graphics and emphasises: 'The role of mark-making in children's cognitive development is set out in the taxonomy (Carruthers and Worthington, 2006)'. The report recommends that 'local authorities, leaders, managers and head teachers should provide a culture with a significant focus on mathematical mark-making' and 'a learning environment that encourages children to choose to use their own mathematical graphics to support their mathematical thinking and processes' - The Williams Maths Review: (DCSF, 2008)

`At the very heart of the success of the book is the authors' ability to see mathematics through young children's eyes by listening to and reflecting on the constant efforts made by children to make sense of their world. This is a liberating book which proposes that the teaching of mathematics could and should be a highly creative and enjoyable proceess' - Branwen Llewelyn Jones, Early Years Consultant at PACE Ltd / TACTYC

'Ground breaking... To single out any one chapter would be unfair because there is something thought-provoking and inspirational throughout. If you want to expand your understanding upwards and outwards then get a copy soon' - Times Educational Supplement

'I first read Children's Mathematics, Making Marks, Making Meaning a couple of years ago and it had an immediate impact on my own thinking and teaching, and the work I do with trainee teachers. I'm sure you will find it compelling reading too. I think it has the potential to change, in a fundamental way, how we think about early mathematical development' - Lynne McClure, Editor, Math Co-ordiator's File, Mathematics Association

'In their exceptionally readable and informative book, Children's Mathematics, Making Marks, Making Meaning Carruthers and Worthington (2006) draw attention to one of the main goals of early years teaching, that is, to help children make links between the mathematics they have already encountered (and continue to engage with) at home and the more abstract mathematics of the school. These authors suggest that by encouraging children to represent mathematical ideas in their own ways and, crucially, by talking to the pupils about the marks they have made, we are given a "window" onto their thinking that may otherwise be inaccessible' - Liz Pumphrey, NRICH

This book draws on the authors' many years of teaching children aged three to eight years and also on their extensive research with children in the home, nursery and school. The authors explain the development and range of young children's mathematical marks and visual representations, showing how children make mental connections between their own early marks and subsequent abstract mathematical symbolism, and go on to develop their own written methods.

Combining theory and practice, this acclaimed book demonstrates how children's own mathematical graphics are highly creative and show deep levels of thinking. The authors show how this is the key to success in school mathematics and to higher levels of achievement.

The authors are winners of TACTYC's (2003) Jenefer Joseph Award for the Creative Arts (3 - 8) - awarded for their innovative work with children on mathematical graphics.

Who Takes Notice of Children's Own 'Written' Mathematics?
Children's Mathematical Graphics

International Findings

Studies That Relate To Mathematical Literacy

Enquiring into Children's Mathematics

Making Marks, Making Meaning
Children Making Meaning with Marks

Different Literacies: Mathematical Literacy

Children Represent Their Mathematical Actions and Understanding On Paper

Learning Theories

Reading and Using Mathematical Graphics

Sociocultural Perspectives Contexts in Early Years Settings

Teachers' Beliefs

Creativity in Mathematics


Mathematical Schemas
What Is a Schema?

Schemas and Mathematics

Schemas and Mark-Making

Observing Schemas in a School Setting

Mapping Patterns of Schema Exploration

Early Writing, Early Mathematics
The Significance of Emergent Writing

Young Children Explore Symbols

Early Writing and Early Mathematical Marks

Early (Emergent) Literacy Is Often Misunderstood


Bridging the Gap between Home and School Mathematics

Understanding Symbols

Mathematics as a Foreign Language

Becoming Binumerate

Teachers' Difficulties


Making Sense of Children's Mathematical Graphics
The Evolution of Children's Early Marks

Categories of Children's Mathematical Graphics

Common Forms of Graphical Marks

Early Development of Mathematical Meaning

Early Explorations with Marks

'The Beginning Is Everything'

Early Written Numerals

Numerals as Labels

Representations of Quantities and Counting

The Development of Early Written Number, Quantities and Counting

Understanding Children's Developing Calculations
Practical Mathematics

The Fifth Dimension: Written Calculations

Representations of Early Operations

Counting Continuously

Narrative Actions

Supporting Children's Own Mathematical Marks

Separating Sets

Exploring Symbols

Standard Symbolic Calculations with Small Numbers

Calculations with Larger Numbers Supported By Jottings

The Development of Children's Mathematical Graphics: Becoming Binumerate


Environments That Support Children's Mathematical Graphics
Rich Mathematical Environments for Learning

The Balance between Adult-Led and Child-Initiated Learning

Role-Play and Mark-Making

The Physical Environment

Practical Steps

Graphics Areas


Case Studies from Early Childhood Settings
The Birthday Cards

A Number Line

'No Entry'

Carl's Garage

Children's Centres: The Cambridge Learning Network Project

Spontaneous Dice Game

Young Children Think Division

A Zoo Visit

Mathematics and Literacy in Role-Play: The Library Van

Aaron and the Train

Multiplying Larger Numbers

Nectarines for a Picnic


Developing Children's Written Methods
The Assessment of Children's Mathematical Representations On Paper

The Problem with Worksheets

Assessing Samples of Children's Own Mathematics

Examples of Assessment of Children's Mathematics

The Pedagogy of Children's Mathematical Graphics

Modelling Mathematics

Involving Parents and Families
Children's First and Continuing Educators

The Home As a Rich Learning Environment

What Mathematics Do Young Children Do At Home?

What Mathematics Do Parents Notice At Home?

Parents Observe a Wealth of Mathematics

Helping Parents Recognise Children's Mathematical Marks

Parents' Questions about Children's Mathematical Graphics


Children, Teachers and Possibilities

Children's Questions

Teachers' Questions

It's All Very Well - But What About Test Scores?



' I purchased a copy of your book over the holidays-it is my bebtime reading at the moment but it is far too stimulating'- Philippa Cook, Nursery Teacher, Clifton High School, Bristol

'Children's Mathematics: Making Marks, Making Meaning is essential reading for students of Education and Early Childhood Studies, as well as for those practitioners working in the Foundation and Primary stages of education' - Anne Cooper, Mathematics Association

'Thought-provoking and inspirational throughout. If you want to expand your understanding upwards and outwards then get a copy soon' - John Dabell, Education Consultant, Former Numeracy Consultant & Ofsted Inspector

Praise for the First Edition:

'Children's Mathematics: Making Marks, Making Meaning is essential reading for students of Education and Early Childhood Studies, as well as for those practitioners working in the Foundation and Primary stages of education' - Anne Cooper, Mathematics Association

'The writers make a very convincing case for the usefulness of exploring children's marks in order to understand their mathematical cognition. The examples of children's written representations provide fascinating insights into how different children think about mathematics' - Katherine Canobi, University of Melbourne

This was already an adopted book, in its previous edition. The new edition build upon this, and appears to be the only reliable text on the ground once occupied by Hughes' excellent "Children and Number".

Mr Ralph Manning
Sch of Education & Lifelong Learning, University of East Anglia
October 12, 2010

Sample Materials & Chapters

Foreword and preface (143kb)

For instructors

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ISBN: 9781412922838
ISBN: 9781412922821

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