Learning and Teaching Mathematics 0-8
Publication Year: 2014
‘What a super book! It is absolutely packed with practical ideas and activities to help you love maths, and love teaching and/or learning it. It certainly helps to develop an enthusiasm for a subject most adults tend to say “I'm no good at…”’
- 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
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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 978-1-4462-5332-8 (pbk)
Commissioning editor: Jude Bowen
Associate editor: Miriam Davey
Production editor: Nicola Marshall
Copyeditor: Rosemary Campbell
Proofreader: Fabienne Pedroletti Gray
Indexer: Martin Hargreaves
Marketing manager: Catherine Slinn
Cover design: Wendy Scott
Typeset by: C&M Digitals (P) Ltd, Chennai, India
Printed and bound in Great Britain by Ashford Colour Press Ltd
List of Figures[Page vii]
- 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
- 1.5Problem-solving 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 mark-making 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
- [Page viii]6.5Children's writing to show how they used the Empty Number Line 83
- 6.6Properties of two-dimensional 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
- 9.1One-to-one 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 two-digit 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 single-digit number 162
- 11.1‘The biggest half’ 165
- 11.2‘Part–whole’ misconceptions 170
- [Page ix]11.3Composition of a packet of sweets 172
- 11.4Incorrectly marking ½ on a number line 173
- 11.5Fraction walls 175
- 11.6Modelling 176
- 11.7A bar of chocolate showing 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 three-dimensional shapes 218
- 14.3What is the same and what is different? 220
- 14.4Single criterion Venn diagram 220
- 14.5Mathematical visualisation activity 220
- 14.6Peek-a-Boo 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 non-intersecting 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
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[Page xi]The Editors
Helen 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 Part-time/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 Contributors
Gina 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 [Page xii]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 employment-based 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, co-ordinating 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.
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.[Page xiv]
Being answerable to others and taking responsibility for decisions and actions for children's learning.
Having a sense of control over actions and choices.
A procedure for performing a calculation.
A clock with a dial where the position of the hands indicates the time.
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).
Vertical and horizontal lines used to frame graphs and charts.
A graphical data representation in which the length of bars is used to indicate the frequency of discrete data categories.
The number of different digits used for representing numbers within a counting system.
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.
A measure of the space within a container or of the liquid or material poured into it.
Using a number as a descriptor of the number of items in a set.
Knowing the last number in a count indicates the cardinal value of the set.
A sorting diagram with rectangular areas labelled with relevant criteria and their corresponding negations (for example, ‘red’ and ‘not red’).
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.
Fractions with the same denominator, for example 4/5, 7/5, 1/5.
An abstract, generalised idea.
A proposed general statement about a pattern which has yet to be verified.
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 re-measuring) regardless of its arrangement.
Data which can take any value on a continuum, usually collected by measuring.
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.
A fractional number expressed using the base 10 place value system extended to include tenths, hundredths, thousandths, and so on.
The bottom number in a written fraction, representing the number of parts into which a whole has been divided.
Assessment designed to diagnose the cause of a difficulty experienced by a child.
The set of ten symbols (0-9) used to write whole numbers (for example, in the number 25 the digits are 2 and 5).
A clock which uses digits to display the number of hours and minutes in either 12-hour or 24-hour format.
Data which belong to one of a number of distinct categories and are normally collected by counting.
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).
Equal in value but may be represented differently, for example 1/2 = 2/4 = 0.5.
Assessment to judge the quality of teaching and learning.
A number that divides exactly into another number.
Writing a number as a multiplication of its factors.
[Page 244]Formative assessment
Assessment at the time of teaching which enables an adult to identify the next steps in learning.
A way of representing part of a whole, or part of a set.
Play in which children select activities and resources.
A table showing how many times each data category occurs.
Forming a general idea, conclusion or concept by recognising common properties of instances.
Peculiar, or particular, to the individual.
Fractions where the numerator is greater than the denominator (sometimes called ‘top-heavy’ fractions), such as 3/2.
A mathematical operation which reverses the effect of another operation.
A representation of continuous data shown by a continuous line.
The measure of the quantity of matter in an object.
The average of a set of data found by adding all the data values and dividing by the number of data values.
The middle data value in an ordered set of data. If there are two middle values, the median is the mean of the two.
Partial, faulty or alternative conceptions based on incomplete or immature reasoning or under- or over-generalisations.
A fraction between two whole numbers which is expressed as a whole number and a fractional part, for example 4 1/2.
The most frequent data value.
The repeating unit within a pattern.
A number that can be made by multiplying a given number by another, for example 10, 15 and 20 are multiples of 5.
The positive whole numbers 1, 2, 3 …
A flat shape which can be folded up into a three-dimensional solid.
The use of a number as a label.
The symbols (1, 2, 3 …) used to represent numbers.
The top part of a written fraction, representing the number of equal parts taken from a whole.
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.
A mathematical function which produces an output value from one or more input values (for example addition, subtraction, multiplication and division).
Using numbers to indicate position or order.
Knowing the order of the numbers.
Splitting a number into parts, for example hundreds, tens and ones.
A data representation using pictures or icons displayed in vertical or horizontal lines.
A data representation showing data categories proportionally as sectors of a circle.
A two-dimensional closed shape with only straight sides.
A polygon in which all the sides are equal in length and all the angles are the same size.
Non-routine tasks that promote mathematical thinking and talk.
Playful imitation of the roles of others (for example teacher, dog, policeman).
A data representation for two variables showing a set of data plotted as points to explore possible relationships.
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.
Instant recognition of small quantities without explicitly counting.
Assessment designed to make a judgement about what children know, understand or are able to do at a certain point in time.
A counting record in which vertical marks are made for each item, with every fifth mark made diagonally.
A classification into categories, based on similarities.
Covering a flat surface using repeated shapes without overlaps or gaps between them.
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.
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
[Page 246]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.
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.
The amount of space occupied by an object.
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 (non-zero) digits.
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