# Learning and Teaching Mathematics 0-8

Books

### Edited by: Helen Taylor & Andrew Harris

• Chapters
• Front Matter
• Back Matter
• Subject Index

## 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 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 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.

## 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 re-measuring) 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 (0-9) 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 12-hour or 24-hour 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.

Free-flow 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 ‘top-heavy’ 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 over-generalisations.

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 three-dimensional 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.

One-to-one 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 two-dimensional 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.

Non-routine tasks that promote mathematical thinking and talk.

Role-play

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 (non-zero) digits.

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