Learning and Teaching Mathematics 08
Publication Year: 2014
DOI: http://dx.doi.org/10.4135/9781526401601
Subject: Elementary Mathematics, Early Childhood Mathematics
 Chapters
 Front Matter
 Back Matter
 Subject Index

Part 1: Issues in Mathematical Learning and Teaching
 Chapter 1: How Children Learn Mathematics and the Implications for Teaching
 Chapter 2: Children Talking about Mathematics
 Chapter 3: Play and Mathematics
 Chapter 4: Children Representing their Mathematics
 Chapter 5: Mathematical Learning Outside the Classroom
 Chapter 6: Mathematics and Display
 Chapter 7: Assessing Children's Mathematics
 Chapter 8: Working with Parents
Part 2: Learning and Teaching Mathematics

Copyright
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Editorial and selection matter © Helen Taylor & Andrew Harris 2014
Chapter 1 and Chapter 5 © Helen Taylor 2014
Chapter 2 © Jill Matthews 2014
Chapter 3 © Clare Wiseman and Karen Vincent 2014
Chapter 4 © Clare Wiseman 2014
Chapter 6 © Bridie Price 2014
Chapter 7 © Helen Taylor and Karen Vincent 2014
Chapter 8 © Helen Taylor and Jill Matthews 2014
Chapter 9 © Andrew Harris 2014
Chapter 10 © Andrew Harris 2014
Chapter 11 © Jon Wild 2014
Chapter 12 and Chapter 15 © Gina Donaldson 2014
Chapter 13 © Louise O'Sullivan 2014
Chapter 14 © Paula Stone 2014
First edition published 2014
Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act, 1988, this publication may be reproduced, stored or transmitted in any form, or by any means, only with the prior permission in writing of the publishers, or in the case of reprographic reproduction, in accordance with the terms of licences issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers.
Library of Congress Control Number: 2013935225
British Library Cataloguing in Publication data
A catalogue record for this book is available from the British Library
ISBN 9781446253311
ISBN 9781446253328 (pbk)
Commissioning editor: Jude Bowen
Associate editor: Miriam Davey
Production editor: Nicola Marshall
Copyeditor: Rosemary Campbell
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Indexer: Martin Hargreaves
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List of Figures
 1.1Oliver's drawings of shapes of different sizes 7
 1.2Bead string, Cuisenaire rods and interlocking cubes 8
 1.3Hundred square 9
 1.4Themes and mathematical activities 13
 1.5Problemsolving opportunities 16
 3.1Mathematical learning in different play activities 35
 3.2Amy plays with the bricks 36
 3.3Amy's solution 39
 3.4Play activities for older children and mathematical links 41
 4.1Bobby's tractor 50
 4.2Bobby's dominoes 51
 4.3Isobel's calculations 52
 4.4Jose's numbers 53
 4.5Sam's graph 55
 4.6Role of adults in supporting mathematical markmaking 56
 4.7Variations of the numeral for ‘four’ 58
 5.1‘Can I squeeze through?’ 63
 5.2‘Who is heavier?’ 64
 5.3Developing mathematical learning using resources for outside areas 66
 5.4Developing mathematical learning within the school grounds 67
 5.5Measuring trees 69
 5.6‘The big stick is heavy’ 70
 5.7‘Three heavy and five light things’ 71
 6.1Factor bugs – ‘Larger numbers always have more factors: true or false?’ 79
 6.2Different ways to make 10 79
 6.3‘Maths is all around us! What maths can you see?’ 80
 6.4Information display 82
 6.5Children's writing to show how they used the Empty Number Line 83
 6.6Properties of twodimensional shapes 84
 6.7‘We have been exploring all the ways that Numicon can support our learning and had some fun too!’ 86
 6.8Interactive spring display 87
 6.9Town planning 89
 7.1An example of a completed narrative observation form 97
 7.2Example of a sticky note record 99
 7.3Exemplar recording sheet for group activity 103
 8.1One, two, three, four, five 108
 8.2Examples of slips sent home to parents at the end of a week 114
 9.1Onetoone correspondence: matching one flag to each sandcastle 123
 9.2Modelling the cardinality of ‘three’ 129
 9.3Examples of models for ordinal number 130
 9.4Number line 130
 9.5Typical counting errors 133
 9.6Progression in counting contexts 135
 9.7Numicon number track showing the numbers 10–20 136
 9.8Counting stick 136
 9.9Representing 43 (using Multilink, Numicon, Dienes' Base 10 materials and a bead string) 138
 9.10Using arrow cards and Gattegno chart to model partitioning and combining 140
 9.11Number cards on a washing line 142
 10.1Forms of addition and subtraction (for 3 + 2 = 5 and 5 − 2 = 3) 146
 10.2Models for 7 + 3 = 10 and 10 − 3 = 7 149
 10.3Principal mental strategies for twodigit addition and subtraction 151
 10.4Progression in ‘adding on with bridging’ strategy 153
 10.5Expanded and standard written methods for addition and subtraction 154
 10.6Forms of multiplication and division (for 2 × 3 = 6 and 8 ÷ 2 = 4) 155
 10.7Two different representations of 3 × 2 = 6 155
 10.8Modelling the commutative law for multiplication 160
 10.9Modelling the distributive law using chocolate bars and Cuisenaire rods 160
 10.10Mental strategies for multiplication and division 161
 10.11Expanded and standard written methods for multiplication and division by a singledigit number 162
 11.1‘The biggest half’ 165
 11.2‘Part–whole’ misconceptions 170
 11.3Composition of a packet of sweets 172
 11.4Incorrectly marking ½ on a number line 173
 11.5Fraction walls 175
 11.6Modelling $\raisebox{1ex}{$4$}\!\left/ \!\raisebox{1ex}{$8$}\right.=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{1ex}{$2$}\right.$ 176
 11.7A bar of chocolate showing $\raisebox{1ex}{$4$}\!\left/ \!\raisebox{1ex}{$10$}\right.$ or 0.4 178
 11.8Square shaded to represent 0.23 178
 11.9Representations of 0.4 and 0.15 179
 12.1The first star 182
 12.2The second star 182
 12.3Recording cubes used for each star 183
 12.4Predicting cubes needed for larger stars 184
 12.5Visual pattern of number bonds to 5 187
 12.6Step pattern 190
 13.1Incorrectly comparing the length of two items 199
 13.2Rulers and tape measures 203
 13.3Incorrect and correct positioning of hands on clock faces 208
 14.1Bethan playing with wooden blocks 216
 14.2Developing children's understanding of two and threedimensional shapes 218
 14.3What is the same and what is different? 220
 14.4Single criterion Venn diagram 220
 14.5Mathematical visualisation activity 220
 14.6PeekaBoo 221
 14.7Freddie playing with the train set 222
 14.8Developing children's understanding of position and direction 223
 14.9Shelving unit 224
 14.10Translating patterns 225
 14.11Different transformations of a triangle 226
 15.1Venn diagram for a one criterion sort 233
 15.2Venn diagram for a nonintersecting two criteria sort 233
 15.3Venn diagram for an intersecting two criteria sort 234
 15.4Venn diagram for a two criteria sort involving a subset 234
 15.5Carroll diagram for a one criterion sort 234
 15.6Carroll diagram for a two criteria sort 235
 15.7Pictogram showing children's choices of toppings for pancakes 237
 15.8Block graph showing children's choices of toppings for pancakes 237
 15.9Tally chart showing children's choices of toppings for pancakes 238
 15.10Bar chart showing children's choice of toppings for pancakes 239
Acknowledgements
We would like to thank family members, friends and colleagues of the authors, including Maria Elsam. Special thanks go to the children, staff and parents of the following Kent primary schools: Godinton Primary (especially Rachel Taylor), Ightham Primary, Wingham Primary, Lunsford Primary, Capel Primary, Slade Primary and Hadlow Primary. All have kindly given their permission to include photographs, our observations and their work.
About the Editors and Contributors
The EditorsHelen Taylor is the Primary Lead Tutor for Teach First at Canterbury Christ Church University. Her work also involves teaching primary mathematics to student teachers and mentor training for experienced teachers supporting students during their professional placements. She has taught across the age range in primary schools in Kent and was a deputy head teacher.
Andrew Harris is a Senior Lecturer at Canterbury Christ Church University, teaching primary mathematics on undergraduate and postgraduate programmes. Previously, he was the Joint Programme Leader for the PGCE Parttime/Modular programme at University College of St Martin and taught in primary schools in Derbyshire and Gloucestershire. His research focuses on mathematical knowledge for teaching, the selection, sequencing and use of examples in teaching mathematics and progression in calculation strategies.
The ContributorsGina Donaldson was a primary classroom teacher for 11 years, teaching across the primary phase. She is now a Senior Lecturer at Canterbury Christ Church University, teaching primary mathematics to student teachers. She leads a Mathematics programme at Masters level, working with practising teachers across Kent, Medway and Essex seeking the status of Primary Mathematics Specialist Teacher.
Jill Matthews has recently retired from her Senior Lecturer post at Canterbury Christ Church University, where she coordinated the Year 3 BA (Hons) in Primary Education. She taught primary mathematics to undergraduate student teachers. She held the role of BA Year 3 Partnership Placement Tutor for many years, supporting students on their professional placements. Her research interests relate to language development and ‘talk for learning’, particularly within the context of mathematics, and she is currently completing her PhD.
Louise O'Sullivan is Head of School Partnership at Canterbury Christ Church University. She is particularly involved in supporting student teachers on their professional placements. Louise is interested in supporting teachers, both qualified and unqualified, in improving their mathematics teaching. She also teaches mathematics on PGCE and undergraduate programmes and is particularly interested in children's approaches to number, calculation and measures.
Bridie Price is a Senior Lecturer at Canterbury Christ Church University teaching both primary mathematics and primary art. She also has a special interest in the use of display in primary schools. Bridie has taught in primary schools in London and Kent and has been a deputy head teacher and acting head teacher.
Paula Stone is a Senior Lecturer in Primary Education. She teaches professional studies and primary mathematics on undergraduate and postgraduate programmes, and employmentbased routes into teaching. Paula specialises in teaching enhanced primary mathematics courses and encourages her students to contribute to the mathematics community through publication in subject association magazines. She has made regular contributions to the NCETM Primary Magazine.
Karen Vincent worked as a teacher in early years, primary and secondary education for 17 years before taking up a post as a Senior Lecturer in the Department of Primary Education at Canterbury Christ Church University in 2010. She teaches across a range of primary teacher training programmes specialising in Early Years education. Her research interests include young children's perceptions of learning and the transition between Year R and Year 1. She is Programme Director for the Primary Education Progression Route.
Jon Wild has worked in the primary education sector for over 25 years, coordinating mathematics, science and ICT, before moving successfully into leadership and management as a deputy head teacher and head teacher. His interest in initial teacher education and continuing professional development led him to Canterbury Christ Church University where he teaches in the primary mathematics team. Jon has a special interest in mathematics and ICT, particularly in using ICT to enhance effective teaching and management within schools.
Clare Wiseman is a Senior Lecturer in the Department of Childhood Studies at Canterbury Christ Church University. She previously lectured on both the undergraduate and postgraduate initial teacher education programmes. Prior to joining Canterbury Christ Church University, Clare was a Primary Mathematics Consultant for Kent Local Authority. Previously, she had taught across Foundation Stage, Key Stage 1 and Key Stage 2 in Kent primary schools for 12 years.
Introduction
Children encounter mathematical ideas in everyday life from birth onwards. Babies quickly learn to distinguish differences in numerical quantities and begin to explore the shapes of the objects and spaces around them. As children develop, mathematics increasingly provides skills, models and ways of thinking which can be used to interpret and describe the contexts and objects they experience and to solve problems. Developing a good understanding of early mathematical ideas provides a foundation for success in everyday tasks and in increased employment and education prospects when reaching adulthood.
To fully understand the mathematical teaching and learning for the 0–8 age group, we need both a secure personal knowledge of the mathematics involved and an understanding of how children learn mathematics and of the range of teaching approaches (pedagogy) which will best support their learning. This book is written to help you to develop your understanding of teaching and learning of mathematics for the 0–8 age group. As you read the book, you will also encounter a range of effective, interesting and engaging approaches to promote mathematical learning for young children. Fostering an enthusiasm for mathematics and mathematical confidence in young children is a vital part of supporting their mathematical development.
The book is divided into two main parts. Part 1 addresses specific issues associated with the learning and teaching of mathematics from birth to age 8. In Part 2 we explore the progression in learning about different areas of mathematics encountered by young children and discuss possible approaches to teaching and suggested activities to support learning. Throughout Parts 1 and 2 of the book we have discussed the role of problem solving as a central aspect of mathematical learning.
Each chapter begins with an overview of the chapter material. Case studies provide illustrations of particular aspects of learning or teaching which are then analysed in order to offer insights about key principles for effective practice. Significant research is highlighted in each chapter to help you apply the findings to your own professional context and practice. In each chapter, prompts are provided to encourage you to reflect about the practice you have experienced and to consider ways in which practice may be enhanced. Each chapter ends with a summary and suggestions for further reading which you can explore to extend your understanding. You can find a glossary of mathematical terms used in each chapter at the end of the book.

Glossary
Accountability
Being answerable to others and taking responsibility for decisions and actions for children's learning.
Agency
Having a sense of control over actions and choices.
Algorithm
A procedure for performing a calculation.
Analogue clock
A clock with a dial where the position of the hands indicates the time.
Associative Law
The associative law holds for addition and for multiplication only. It states that the outcome of an addition or multiplication is unchanged when operations are performed in a different order, for example, (3 + 4) + 5 = 3 + (4 + 5) and (2 × 5) × 4 = 2 × (5 × 4).
Axes
Vertical and horizontal lines used to frame graphs and charts.
Bar chart
A graphical data representation in which the length of bars is used to indicate the frequency of discrete data categories.
Base
The number of different digits used for representing numbers within a counting system.
Block graph
A data representation showing the frequency of discrete data categories by using blocks arranged in rows or columns. One block is shown for each piece of data.
Capacity
A measure of the space within a container or of the liquid or material poured into it.
Cardinal number
Using a number as a descriptor of the number of items in a set.
Cardinality
Knowing the last number in a count indicates the cardinal value of the set.
Carroll diagram
A sorting diagram with rectangular areas labelled with relevant criteria and their corresponding negations (for example, ‘red’ and ‘not red’).
Commutative law
The commutative law holds for addition and for multiplication only. It states that the answer to an addition or multiplication is unchanged when the numbers on either side of the operation sign are interchanged, for example 4 + 2 = 2 + 4 and 5 × 7 = 7 × 5.
Common denominators
Fractions with the same denominator, for example 4/5, 7/5, 1/5.
Concept
An abstract, generalised idea.
Conjecture
A proposed general statement about a pattern which has yet to be verified.
Consecutive numbers
Whole numbers which follow one another in the counting sequence.
Conservation (of number or measures)
Knowing that a quantity is still the same (without recounting or remeasuring) regardless of its arrangement.
Continuous data
Data which can take any value on a continuum, usually collected by measuring.
Criterion
An attribute used to decide whether an item is included in, or excluded from, a set when sorting.
Data (plural), datum (singular)
Pieces of information, often displayed in graphs, lists or tables.
Decimal
A fractional number expressed using the base 10 place value system extended to include tenths, hundredths, thousandths, and so on.
Denominator
The bottom number in a written fraction, representing the number of parts into which a whole has been divided.
Diagnostic assessment
Assessment designed to diagnose the cause of a difficulty experienced by a child.
Digits
The set of ten symbols (09) used to write whole numbers (for example, in the number 25 the digits are 2 and 5).
Digital clock
A clock which uses digits to display the number of hours and minutes in either 12hour or 24hour format.
Discrete data
Data which belong to one of a number of distinct categories and are normally collected by counting.
Distributive law
The distribution of one operation over another. The distributive law for multiplication over addition or subtraction states that partitioning a number and multiplying its component parts separately has the same outcome as multiplying the number, for example we could calculate 3 × 25 using 3 × (20 + 5) = (3 × 20) + (3 × 5) and 4 × 19 using 4 × (20 − 1) = (4 × 20)  (4 × 1).
The distributive law also holds for division over addition and subtraction (provided the division is to the right of the addition or subtraction), for example for 36 ÷ 2 we can calculate (30 + 6) ÷ 2 = (30 ÷ 2) + (6 ÷ 2) and for 56 ÷ 2 we can calculate (60 − 4) ÷ 2 = (60 ÷ 2)  (4 ÷ 2).
Equivalent
Equal in value but may be represented differently, for example 1/2 = 2/4 = 0.5.
Evaluative assessment
Assessment to judge the quality of teaching and learning.
Factor
A number that divides exactly into another number.
Factorise
Writing a number as a multiplication of its factors.
Formative assessment
Assessment at the time of teaching which enables an adult to identify the next steps in learning.
Fractions
A way of representing part of a whole, or part of a set.
Freeflow play
Play in which children select activities and resources.
Frequency table
A table showing how many times each data category occurs.
Generalising
Forming a general idea, conclusion or concept by recognising common properties of instances.
Idiosyncratic
Peculiar, or particular, to the individual.
Improper fractions
Fractions where the numerator is greater than the denominator (sometimes called ‘topheavy’ fractions), such as 3/2.
Inverse operation
A mathematical operation which reverses the effect of another operation.
Line graph
A representation of continuous data shown by a continuous line.
Mass
The measure of the quantity of matter in an object.
Mean
The average of a set of data found by adding all the data values and dividing by the number of data values.
Median
The middle data value in an ordered set of data. If there are two middle values, the median is the mean of the two.
Misconception
Partial, faulty or alternative conceptions based on incomplete or immature reasoning or under or overgeneralisations.
Mixed numbers
A fraction between two whole numbers which is expressed as a whole number and a fractional part, for example 4 1/2.
Mode
The most frequent data value.
Motif
The repeating unit within a pattern.
Multiple
A number that can be made by multiplying a given number by another, for example 10, 15 and 20 are multiples of 5.
Natural numbers
The positive whole numbers 1, 2, 3 …
Net
A flat shape which can be folded up into a threedimensional solid.
Nominal number
The use of a number as a label.
Numerals
The symbols (1, 2, 3 …) used to represent numbers.
Numerator
The top part of a written fraction, representing the number of equal parts taken from a whole.
Onetoone correspondence
Matching or pairing the contents of two sets (a set may be a set of items, symbols, or counting numbers) such that every element of one set is paired with a corresponding element of the second set and no elements of either set are left unpaired.
Operation
A mathematical function which produces an output value from one or more input values (for example addition, subtraction, multiplication and division).
Ordinal number
Using numbers to indicate position or order.
Ordinality
Knowing the order of the numbers.
Partitioning
Splitting a number into parts, for example hundreds, tens and ones.
Pictogram
A data representation using pictures or icons displayed in vertical or horizontal lines.
Pie chart
A data representation showing data categories proportionally as sectors of a circle.
Polygon
A twodimensional closed shape with only straight sides.
Regular polygon
A polygon in which all the sides are equal in length and all the angles are the same size.
Rich task
Nonroutine tasks that promote mathematical thinking and talk.
Roleplay
Playful imitation of the roles of others (for example teacher, dog, policeman).
Scattergram
A data representation for two variables showing a set of data plotted as points to explore possible relationships.
Schemas
Repeated behaviours that characterise children's exploration of particular ideas and concepts (for example enclosing, rotating or transforming).
Small world play
Using toys such as trains, farm animals or people to symbolise characters or objects.
Subitising
Instant recognition of small quantities without explicitly counting.
Summative assessment
Assessment designed to make a judgement about what children know, understand or are able to do at a certain point in time.
Tally
A counting record in which vertical marks are made for each item, with every fifth mark made diagonally.
Taxonomy
A classification into categories, based on similarities.
Tessellation
Covering a flat surface using repeated shapes without overlaps or gaps between them.
Topology
The study of properties that are preserved when objects are deformed, twisted, and stretched. For example, in topology a square and a circle are said to be equivalent because one can be stretched or squeezed to look like the other.
Transformation
Changing a shape by applying the same process to each point in the shape, for example, through:
Translation: sliding a shape in a straight line from one position to another, without turning
Rotation: turning a shape through an angle about a point (known as the centre of rotation)
Reflection: reflecting a shape in a mirror line
Enlargement: enlarging a shape by a scale factor.
Venn diagram
A sorting diagram, with the whole or universal set enclosed by a rectangle and, within this, subsets, usually shown as circles, each labelled by the relevant criterion.
Volume
The amount of space occupied by an object.
Weight
The force exerted on an object by gravity.
Zero as a placeholder
The use of zero digits to mark empty places within numbers and so ‘hold’ the place of the other (nonzero) digits.
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