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Children's Mathematics
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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
Visit the author's own website here! Children's Mathematics Network

'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

 
Summary

 
 
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

 
Conclusion

 
 
Bridging the Gap between Home and School Mathematics
Disconnections

 
Understanding Symbols

 
Mathematics as a Foreign Language

 
Becoming Binumerate

 
Teachers' Difficulties

 
Conclusion

 
 
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

 
Conclusion

 
 
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

 
Conclusion

 
 
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

 
Conclusion

 
 
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

 
Conclusion

 
 
Children, Teachers and Possibilities
Inclusion

 
Children's Questions

 
Teachers' Questions

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

 
Reflections

 

' 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)


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ISBN: 9781412922838
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