Teaching for Numeracy Across the Age Range: An Introduction (SpringerBriefs in Education) 9811637601, 9789811637605

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SPRINGER BRIEFS IN EDUC ATION

Peter Stuart Westwood

Teaching for Numeracy Across the Age Range An Introduction 123

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Peter Stuart Westwood

Teaching for Numeracy Across the Age Range An Introduction

Peter Stuart Westwood Education Researcher Taipa, Macao

ISSN 2211-1921 ISSN 2211-193X (electronic) SpringerBriefs in Education ISBN 978-981-16-3760-5 ISBN 978-981-16-3761-2 (eBook) https://doi.org/10.1007/978-981-16-3761-2 © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

Some years ago, Steen (2007) claimed that ‘Being numerate is one of the few essential skills that students absolutely must master, both for their own good and for the benefit of the nation’s democracy and economic well-being’ (p.16). That statement is as true today as it was in 2007. The digital age and easy access to handheld devices for calculation have not made it any less important for the general population to be competent in understanding and working with numbers. It is widely accepted now that developing the ability of every individual to understand, utilize and create numerical information in different contexts has to be a priority during the school years and beyond. Poor numeracy skills are known to affect a range of everyday competencies such as the ability to understand daily news bulletins, workshop manuals, medical reports, invoices and household expenditure accounts (Thomson et al., 2020a; Tout, 2021). When students leave school with a poor standard of numeracy this can have devastating social and economic consequences, especially for those from disadvantaged backgrounds (Learning Sciences Institute, 2016). Clear evidence from research suggests that there is a large positive effect on students’ achievement in the curriculum and in their quality of life when competence with numbers is well developed (Cason et al., 2019). Today, possession of functional number skills is viewed as absolutely essential for all members of the workforce in every industry (National Numeracy Organization, 2017). Numeracy is regarded as a vital competency in many types of employment, and represents a skill set required not only to gain employment but also to progress further within a given field. Increasingly, digital technology in schools and at home is providing a motivating and engaging medium for working with and mastering essential number skills, from early childhood onwards (Mowafi & Abumuhfouz, 2021; NCTM, 2017a; Serhan & Almeqdadi, 2020). The ability to work confidently, accurately and swiftly with numbers has also been identified as one of the enabling competencies absolutely necessary for success in the STEM subjects—science, technology, engineering and mathematics (Schulz, 2018). In the UK, a policy paper titled Building our industrial strategy has acknowledged the link between numeracy, the STEM subjects and productivity in the workplace (HM Government, 2017). v

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In the current economic, technological and social environment, the need to strengthen numeracy standards in schools and in the general population is more acute than ever before. In the UK, a vision for the immediate future is to produce ‘… a generation of citizens, consumers, students and workers who are as comfortable with numbers as they are with words’ (British Academy, 2015, p.2). Similarly, in Australia, the Education Council (2015) has stressed a need to establish yardsticks indicating the standard of numeracy that all students should attain before leaving school. Although Australia has a national curriculum, the separate states have also produced their own guidelines and policies on ensuring high numeracy standards. A good example is the Department of Education and Training in the state of Victoria, which has produced an online document titled Numeracy for all learners, providing advice on how best to foster students’ conceptual understanding, procedural fluency, reasoning and problem-solving abilities (Victoria State Government, 2020). In the United States, the Common Core State Standards for mathematics express the same intention, with the relevant guidelines stressing that from the start of schooling all students should acquire number concepts and skills through a well-sequenced and well-taught curriculum (CCSSI, 2017a). Chapters in this book explore the nature of numeracy, its component areas of knowledge and skills, and how these can best be taught, applied and maintained in children and adults. The issues covered here range from encouraging basic numeracy development in the preschool years, extending numeracy skills in primary and secondary schools, and adult numeracy. Attention is also given to the learning difficulties that some students experience at any age. The need for higher standards of numeracy is recognized worldwide, and the author of this book has drawn on relevant research and literature from several countries to provide a comprehensive overview. The extensive reference list will help educators wishing to study certain sub-topics in more detail. At the end of each chapter, many links to other sources of online and print information have been provided. Taipa, Macao

Peter Stuart Westwood

Contents

1 Numeracy: Defined, Described and in Context . . . . . . . . . . . . . . . . . . . . Defining Numeracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numeracy in Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numeracy in the Curriculum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Early Years Curriculum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Curriculum in the UK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Curriculum in Australia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Curriculum in the United States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numeracy as an Across-the-Curriculum Capability . . . . . . . . . . . . . . . . . . . Gender Differences in Numeracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Affective Aspects of Numeracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Concerns Over Numeracy Standards in Schools . . . . . . . . . . . . . . . . . . . . . . Systems-Level Response to Poor Numeracy Standards . . . . . . . . . . . . . . . . Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 3 3 4 5 6 6 7 8 9 10 11

2 Early Numeracy Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Number Sense . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Concept Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Theories of Conceptual Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Theories Covering the Development of Procedural Fluency . . . . . . . . . . . . Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13 13 15 16 24 25

3 Numeracy in Preschool and Kindergarten Years . . . . . . . . . . . . . . . . . . Working with Children in Preschool . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Building Firm Foundations in Kindergarten . . . . . . . . . . . . . . . . . . . . . . . . . Counting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numeral Recognition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Using a Number Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . From Counting to Number Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Beginnings of Place Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Data Collection, Representation and Analysis . . . . . . . . . . . . . . . . . . . . . . .

27 27 29 32 33 34 34 34 35

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Contents

Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simple Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

36 36 37

4 Numeracy Development in Primary School . . . . . . . . . . . . . . . . . . . . . . . Transition from Preschool to School . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numeracy in the Primary School Curriculum . . . . . . . . . . . . . . . . . . . . . . . . Addition and Subtraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Place Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bar Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multiplication and Division . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Number Facts: The Importance of Automaticity . . . . . . . . . . . . . . . . . . . . . Other Topics in Primary Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Role of Language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Technology Supporting Numeracy Learning . . . . . . . . . . . . . . . . . . . . . . . . . Teaching Problem-solving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Indigenous Children’s Number Skills . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39 40 41 41 42 43 44 45 47 47 49 49 52 53

5 Secondary School Years . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Secondary School Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Typical Curriculum Content in Secondary Schools . . . . . . . . . . . . . . . . . . . Across the Curriculum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Real-Life Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Role for Technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Students Who Struggle with Secondary Mathematics . . . . . . . . . . . . . . . . . Providing Support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Concerns Over Numeracy Standards of Senior Students . . . . . . . . . . . . . . . The Way Ahead . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55 56 57 58 58 60 60 62 63 63 64

6 Adult Numeracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Concerns Over Adult Numeracy Standards . . . . . . . . . . . . . . . . . . . . . . . . . . Numeracy in the Workplace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Efforts to Improve Standards Adult Numeracy . . . . . . . . . . . . . . . . . . . . . . . Teaching Adults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Way Ahead . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67 67 68 69 70 72 72

7 Approaches to Teaching and Assessment . . . . . . . . . . . . . . . . . . . . . . . . . Effective Teaching Practices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Vital Role of Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Interleaving Knowledge and Skills . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Intervention and Support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Do All Interventions Bring Benefits? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Observing Students at Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75 76 77 78 78 81 82 82

Contents

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Questioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Teacher-Made Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Technology and Formative Assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Diagnostic Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Standards of Teaching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Improving Teachers’ Expertise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . End Note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83 83 84 84 85 86 87 88 88

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

Chapter 1

Numeracy: Defined, Described and in Context

The term numeracy encompasses a set of concepts, understandings and skills that are vitally important for many aspects of daily life––at school, in employment, within the family, and for functioning effectively as a contributor, participant and consumer in the community (NALA, 2016). In addition to the role that numeracy plays in everyday functioning, possession of good numeracy skills has been identified as one of the main factors enabling students to follow a particular career path (Holmes et al., 2018; Nicholes, 2019). For example, Jain and Rogers (2019, p. 23) have written that ‘The development of numeracy skills is a core aspect of university preparation, with many university courses requiring a certain level of mathematical literacy.’

Defining Numeracy The term numeracy appears to have been coined many years ago in the Crowther Report in the UK (Central Advisory Council for Education, 1959). At that time, there was no precise definition of numeracy other than it was the ‘mirror image of literacy’, but it was widely understood to mean proficiency in dealing with numbers and situations requiring measurement, estimation and calculation. Some authorities have described this ability as ‘quantitative reasoning’, and that term is certainly useful because it encapsulates the essence of what functional numeracy really involves (Ramful & Ho, 2015). Later, the Cockcroft Committee in the UK published a report titled Mathematics counts: A report into the teaching of mathematics in schools (1982). Therein it was stated that a ‘numerate’ person possesses two attributes: a confidence in dealing with numbers in order to cope with the practical mathematical demands of everyday life; and a capacity to understand quantitative information such as that provided in tables, graphs, diagrams, charts, medical reports and in the media. A recent definition of numeracy is that it represents one’s ability to ‘use numbers in context to assist decision making’ (Diaz et al., 2020). © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 P. S. Westwood, Teaching for Numeracy Across the Age Range, SpringerBriefs in Education, https://doi.org/10.1007/978-981-16-3761-2_1

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1 Numeracy: Defined, Described and in Context

The original National Numeracy Strategy for schools in the UK (DfEE, 1999), now absorbed into the 2003 Primary National Strategy, recognized that numeracy involves confidence and competence in working with numbers and measures, and requires an understanding of the number system, a repertoire of computation skills, and an ability to apply numbers to solve problems in a variety of contexts. Later, Foster and Beddie (2005) indicated that numeracy includes not only practical arithmetic skills, but also the ability to communicate quantitative information to others in speech and writing, and to apply common sense estimation and approximation when handling numbers. The documents referred to above have all provided very clear and acceptable ways of viewing the characteristics of numeracy. It must be noted, however, that there is still no universally accepted definition––a situation that can lead to confusion (Geiger et al., 2015). For example, this lack of definition has resulted in teachers and the community asking the question, ‘Is numeracy simply the same as being good at math?’ The answer to this question should be ‘no’. The terms ‘numeracy’ and ‘mathematical ability’ are not identical in meaning, even though there has been a tendency to use the terms interchangeably in many articles and books (and even more frequently during school staffroom discussions). Mathematical competence comprises much more than simply acquiring and using number skills, because mathematics is a diverse discipline with many branches both applied and theoretical. Many of the concepts within mathematics go well beyond simply understanding and using numbers. However, proficiency in working with numbers represents a key competency that underpins successful performance within a very broad range of topics in mathematics (Tout, 2021). In this book, the term ‘numeracy’ is used with a deliberately narrow meaning to refer to understanding and applying knowledge and skills involved in measurement, calculation, estimation and quantitative problem-solving. These areas of knowledge and skill are applied not only in mathematics lessons but also across the curriculum and in daily life (Bennison et al., 2020; Forgasz & Hall, 2019). In this respect, the book continues the concept of numeracy that was embodied in the original Cockcroft Report and the National Numeracy Strategy in the UK.

Numeracy in Context Turner (2007, p. 28) suggested that ‘Numeracy has become a personal attribute very much dependent on the context in which the numerate individual is operating [and] numeracy will mean different things to different people according to their interests and lifestyles.’ It is true that numeracy skills are used for different purposes in different settings (Gal et al., 2020), for example, Butcher et al. (2002) referred to ‘numeracy for practical purposes’, ‘numeracy for interpreting society’, ‘numeracy for managing one’s money, budgeting, time and measurement’ and ‘numeracy as an aid to studying other subjects’. The term ‘multiple numeracies’ has therefore emerged in the professional literature (Vacher, 2014).

Numeracy in Context

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There is no doubt that numeracy is a prerequisite for entry into many types of employment, and then for promotion and progression through the ranks (Gravemeijer et al., 2017). While different fields of work may require quite specific types of number knowledge (nursing vs. carpentry vs. engineering vs. accountancy), they all depend on a firm foundation of concepts and skills that need to be established in the primary school years. For this reason, schools are expected to develop young children’s number concepts and skills as early as possible in order to provide a firm foundation for all later learning.

Numeracy in the Curriculum The material below provides a brief introduction and overview of numeracy and mathematics curriculum across early childhood and primary school years. Later chapters expand upon this topic and provide more guidance on teaching.

Early Years Curriculum Beginning in early childhood, math curricula usually focus on three major areas: (i) developing number sense; (ii) establishing counting skills (using the numberword sequence correctly and accurate enumeration with objects); (iii) recognizing number symbols, understanding and using simple addition and subtraction, and written recording of simple operations (3 + 4 = 7). Most of the experiences provided to develop these areas involve the children using real objects and visual representations. Later, children begin to learn the meaning of place value in base-ten and develop basic skills in simple mental and written arithmetic (Aunio & Räsänen, 2016; Cheng et al., 2017). At the same time, these skills and understandings are used in solving age-appropriate problems and when carrying out quantitative investigations. The topics listed above do represent a very reasonable starting point for operationalizing the concept of numeracy, and they highlight areas in which children with learning difficulties may require extra teaching. When children transition to primary school, these skills and concepts will be extended and enriched as new material is taught and as children move from a concrete stage of reasoning to the stage of being able to deal with symbols and abstract ideas. This goal is achieved best when schools use teaching methods, materials and activities that are age-appropriate and that ensure the children are successful in completing the tasks they are required to undertake. It is vital that young learners develop a confident and positive attitude towards working with numbers. It is essential also that the curriculum in the early years has a good sequence and continuity of content (McMahon & Whyte, 2020). In order to become numerate, a young learner needs to develop a conceptual understanding of number and number relationships, and also become fluent in calculating with numbers. Each aspect of numeracy tends to have its own specific vocabulary

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that needs to be mastered (e.g. plus, minus, equals, more than, difference between, multiply, quantity, amount, size, units, tens) (Galligan, 2016). Talking about number relationships and operations must, therefore, be an important accompaniment to all activities, and teachers need to check that every child understands the language that can be used to explain number operations.

Curriculum in the UK In England, teaching numeracy in the primary school years is defined in the National Curriculum as developing students’ conceptual understanding of number and their ability to recall and apply knowledge rapidly and accurately (DfE, 2013a; 2013b). At Key Stage 1 (ages 5 to 7), the principal focus is on developing confidence and mental fluency with whole numbers, counting and place value. It is stated that in the beginning, this should involve learning to work confidently with number words and using the operations of addition and subtraction. Towards the end of the stage, children will be introduced to multiplication and division. These foundation skills should all be taught explicitly by using hands-on practical resources (real objects, blocks, counters) and by providing abundant guided practice in using taught computation processes (DfE, 2013a). With young children, the use of board-games and other informal number activities can also be used as a motivating supplement to structured teaching (Cheung & McBride, 2017; Cohrssen & Niklas, 2019; Scalise et al., 2020). It has also been found that digital devices with apps can provide valuable individualized practice with number facts and processes across the age range from K to 12 (Calder & Campbell, 2015; Haberlah, 2017b; Hilton, 2018). A study involving over 300 children aged 4–5 years in the UK found that interactive apps that combine direct teaching with play activities provide a good system for delivering high-quality early math instruction, and can effectively raise achievement (Outhwaite et al., 2019). The curriculum in England specifies the numeracy skills to be taught for each year level. For example, the guidelines and statutory requirements for Year 1 indicate that students should be able to: • read, write and interpret mathematical statements involving addition (+) subtraction (–); • represent and use addition number facts and related subtraction facts to 20; • add and subtract one-digit and two-digit numbers to 20, including zero; • solve one-step problems that involve addition and subtraction, when necessary using concrete objects and pictorial representations; solve missing number statements such as 9 =  + 7 (DfE, 2013a). The guidelines in England also provide examples of the recommended style and format for written recording of addition, subtraction, multiplication and division. In recent years, in response to a concern in the community that children were relying too much on handheld calculators, greater importance has been placed on developing competence in calculating mentally. It is also clear that students need to practice

Numeracy in the Curriculum

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and apply computation skills until they are confident and fluent. This degree of specificity in the National Curriculum documents is very helpful to schools and is clearly indicative of the importance now placed on developing a firm foundation in numeracy skills in the early school years. For details of mathematics content in the National Curriculum in England, see Online Resources at the end of the chapter.

Curriculum in Australia In Australia, as early as 1998 the Australian Association of Mathematics Teachers published its document titled Policy on numeracy education in schools (AAMT, 1998) making it clear that numeracy comprises a fundamental set of skills that merit particular attention within the curriculum. However, the general teaching approach advocated by AAMT was (and has remained) very much focused on student-centred investigative learning rather than on explicit teacher-led instruction and practice exercises for computation. Number skills were to be acquired almost incidentally as a by-product from exploring real quantitative problems. This approach was (and still is) recommended due to the prevailing influence of the constructivist theory of learning that tends to view direct teaching as less desirable than students using activity and discovery. More recently, in the new Australian Curriculum (ACARA, 2014; 2015), the content that deals with numeracy across the years of schooling is conceived broadly as a continuum, organized under six closely interrelated topics. The topics are: • calculating and estimating with whole numbers, • recognizing and using number patterns and relationships (including algebra, statistics and probability), • using fractions, decimals, percentages, ratios and rates, • spatial reasoning, • interpreting statistical information, • using measurement. Within the guidelines for the Australian Curriculum, these six themes are expanded in finer detail at each year level. For example, in the early school years the basic abilities to be developed are specified as: counting, identifying numbers, understanding the number line, performing simple arithmetic operations, comparing number magnitudes, telling the time, measurement and estimation. For details of mathematics content in the Australian Curriculum see Online Resources at the end of this chapter.

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Curriculum in the United States In the US, the National Council of Teachers of Mathematics (NCTM) had first produced a document titled Curriculum and evaluation standards for school mathematics (NCTM, 1989). That document placed emphasis on using student-centred problem-solving and investigation rather than direct teaching and practice with basic number skills. The document was revised a decade later and published as Principles and standards in school mathematics, in which the explicit teaching of basic numeracy skills was given a somewhat higher profile (NCTM, 2000). Currently, the teaching of numeracy in the US is entirely influenced by the Common Core State Standards (CCSS) for mathematics. The guidelines contain very detailed descriptions of what students are expected to know, from kindergarten to the end of high school. In CCSS the main themes across this age range are identified as: • • • • • • • • • •

counting and cardinality, the number system, number operations in base-10, operations and algebraic thinking, fractions, measurement and data, geometry, ratios and proportional relationships, equations, statistics and probability.

For details of Common Core State Standards, see Online Resources at the end of this chapter.

Numeracy as an Across-the-Curriculum Capability Numeracy skills in school-age students are similar in importance to literacy skills in that they are a means of communication, a source of further learning, and crossdisciplinary in application and both are essential in life beyond the school years. The guidelines in England and in Australia indicate that students should be taught to apply their numeracy skills in all school subjects. It has been stated that ‘… the nature of numeracy suggests that it cannot be developed solely within mathematics lessons’ (Muir, 2016, p. 496). Just as all teachers are said to be teachers of literacy so too all teachers should be teachers of numeracy, taking every opportunity to introduce students to the quantitative and statistical aspects of their subjects and to relate these aspects to the real world (Bennison, 2015; 2016; Forgasz & Hall, 2019; Sweet, 2021). In this context, Perso (2006a) suggests that all subject teachers should ask themselves

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two questions: (i) how can numeracy contribute to learning in this subject? (ii) how can this learning area enhance students’ numeracy? It has been observed that many students do not spontaneously transfer and apply their numeracy skills taught in mathematics lessons to other school subjects, so there is an important role for all teachers in helping facilitate the process of transfer. This is best achieved by making explicit the connections between the content of math lessons and quantitative and statistical information supplied in subjects such as geography, history, health, physical education, civics, science and engineering. It is then necessary to provide opportunities for application and practice in interpreting and working with this type of contextualized information. By integrating mathematical and numerical components into all school subjects, students are helped not only to strengthen and generalize their skills and understandings but also to appreciate the utility of numeracy in the widest sense.

Gender Differences in Numeracy In the past, aptitude for mathematics has often been stereotyped as a capability found mainly in males rather than females. Various theories emerged that seek to explain this possible difference, including genetic and biological causes such as innate differences in brain development that affect spatial and quantitative awareness, or cultural influences that affect how girls are socialized into believing that it is normal for them to do poorly in mathematics because it is a ‘boys’ subject’ (del Rio et al., 2019; OECD, 2016b; Stoet & Geary, 2012). The attitude of parents towards mathematics has certainly been found to influence children’s level of anxiety about engaging with the subject (Szczygiel, 2020a). In recent years however, a debate has flourished over whether boys really are superior to girls in mathematical ability, and the verdict seems to be that differences have been greatly over stated in the past (King, 2019; OECD, 2016b). Lindberg et al. (2010) used a meta-analysis of 242 studies from the period 1990 to 2007. They found that males and females tended to perform similarly in mathematics, and the effect size of gender was negligible. It has also been found that in some countries girls actually score higher than boys. For example, in the US a study using curriculum-based math measures found statistically significant differences in favour of females in Grades 5, 7 and 8 (Yarbrough et al., 2017). The introduction of national and international testing of math and literacy skills has made it possible to explore the issue of gender differences using data from very large samples. For example, Else-Quest, Hyde and Linn (2010) analysed data from the Trends in International Mathematics and Science Study (TIMSS) and the Programme for International Student Assessment (PISA). They discovered that a gender gap favouring boys in mathematics achievement is only evident in some nations. Similarly, a study by Stoet and Geary (2013) explored data produced from 15-year-old students who had taken part in PISA. They also found that boys tended to score higher than girls in mathematics only in some countries, and there was

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considerable variation between nations. This suggests a probable influence of cultural factors and parental expectations, and also the style of teaching and encouragement provided in co-educational classrooms. So, research into gender differences has produced conflicting results, with some studies finding boys outperforming girls, some finding no difference and some even finding girls to be superior. Despite these differences across studies, it is generally reported that boys have a more positive attitude towards math and exhibit stronger self-efficacy than girls when engaging with the subject. Boys also experience less anxiety about mathematics. One study of girls’ math anxiety seems to indicate that it can also generalize to engagement with science because measurement and calculations are often involved. This math anxiety can adversely affect girls’ achievement and interest in the subject (Vakili & Pourrazavy, 2017). Stereotyping girls as lacking in aptitude for number work and mathematics can exert a negative influence on girls’ self-efficacy beliefs, thus discouraging them later from entering a career in areas of mathematics, science or engineering. In most developed countries, senior girls remain much less likely than boys to take higher mathematics courses and science courses and to opt for these as major subjects if they go to university (Finegold, 2016). Buckley (2016) reported that only 31 per cent of eligible females compared to 69 per cent of males enter these courses. To change this situation, it may be necessary to introduce teaching materials and topics that focus deliberately on engaging girls more successfully with the course content, and improving girls’ self-efficacy beliefs in relation to mathematics (Hirshfield & Koretsky, 2018; Hobbs & Earp, 2018). Several countries have taken the initiative to do this, and the next few years will reveal the outcomes. The current view is certainly that we should be aiming to eliminate the stereotypes that have prevailed for too long. There does not appear to be any valid reason why any achievement gap in math should exist between male and female students.

Affective Aspects of Numeracy In recent deliberations on the nature of numeracy, increased importance has been placed on the affective as well as cognitive aspects of ‘being numerate’. The affective aspects include positive attitudes, freedom from anxiety and confidence in one’s own ability that contributes to a person’s willingness to engage with and persevere in quantitative activities. It has been said that numeracy means ‘having the confidence and skill to use numbers and mathematical approaches in all aspects of life’ (National Numeracy Organization, 2020). The words ‘having a positive disposition’ and ‘positive inclination toward mathematics’ are appearing frequently when the characteristics of a numerate person are described. For example, documents relating to the Australian Curriculum state that numeracy encompasses not only knowledge and skills, but also the disposition that students need in order to use mathematics effectively across a wide range of situations (ACARA, 2015). Similarly, the guidelines for numeracy teaching in New Zealand refer to encouraging the formation of

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‘positive attitudes and expectations’ (New Zealand Government, 2016, p. 6). In the US, the Common Core State Standards refer to positive disposition as the ‘habitual inclination to see mathematics as sensible, useful and worthwhile, coupled with a belief in diligence and one’s own efficacy’ (CCSSI, 2017a, p. 1). The teaching of numeracy skills should, from the very beginning, cause young students to develop a positive attitude towards and interest in working with numbers. In the early years of childhood, these affective components of numeracy are either encouraged on entry to school or are snuffed out by lack of success. Effective teaching must, therefore, strengthen children’s self-efficacy beliefs in relation to working with numbers and solving quantitative problems (Mastorodimos & Chatzichristofis, 2019). Despite all the good intentions to develop students’ positive attitude and confidence, there is some doubt that schools are really doing enough in this respect. There is evidence to suggest that the negative attitude towards numeracy and math displayed by some students is shaped by such things as discouraging results in school tests, and teachers’ written remarks in school reports and exercise books. Poor results in tests, plus the accompanying negative comments from teachers can affect some students’ views of their own ability to improve (Parnis & Petocz, 2016). It is clear that the verbal and written feedback teachers provide can enhance or depress a student’s emotional commitment to becoming numerate (Education Endowment Foundation, 2017).

Concerns Over Numeracy Standards in Schools In England, Australia and the United States, there has been growing concern over recent years that standards in numeracy are well below those required to function effectively in the twenty-first century. In the US, surveys of adult standards in numeracy have yielded particularly worrying results (Institute of Education Sciences, 2020); and an OECD report in 2016 found that young people in England still lagged behind their counterparts in numeracy in several other countries, particularly in Asia (OECD, 2016a; OECD, 2016c). Similar concerns are evident in Australia, where Baker (2019) citing data from the 2018 Programme for International Student Assessment (PISA), wrote ‘Australian students have recorded their worst results in international tests, failing for the first time to exceed the OECD average in math while also tumbling down global rankings in reading and science.’ The 2018 PISA results indicate that the countries with the highest achievement in mathematics and numeracy continue to be China, Singapore, Macau, Japan, Taiwan and South Korea. These Asian countries outperform Britain, Australia and the US (Factsmaps.com, 2020; OECD, 2016d; Richards, 2020). Data from the 2019 Trends in International Mathematics and Science Study (TIMSS) indicated that Australian students at Year 4 had not improved in mathematics scores since a cohort was tested in 2015, but at Year 8 there had been some modest improvement (Thomson et al., 2020b).

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In Australia, concern has also been raised by results from the regular standardized testing in schools, known as the National Assessment Programme Literacy and Numeracy (NAPLAN). Despite having invested much more money in education to raise achievement levels, standards in mathematics do not seem to be improving (Masters, 2017; Thomson, 2016a; 2016b). A report issued at the end of 2016 stated that achievement level in numeracy had reached a plateau and the conclusion was that at a national level NAPLAN results have indicated no significant improvement in the past few years (ACARA, 2016a; ACARA, 2016b). The lowest achievers at the minimum level of competence comprise an approximately equal number of boys and girls, but a significantly higher proportion of Indigenous students compared to non-indigenous (Australian Government, 2020). Two aspects of the current situation are of particular concern. The first is that in Australia there has been a decrease over time in the number of very high achievers who are talented in mathematics and who go on to study mathematics at tertiary level. The second is that the school system appears to be completely failing students in the lowest achievement bands who quickly give up because they identify themselves as ‘no good at math’. It was also of particular concern in 2018 when it was revealed that many teachers who are teaching math, even at secondary school level, actually have no qualification and training in that subject.

Systems-Level Response to Poor Numeracy Standards England has made a major commitment to strengthening the teaching of numeracy, and has established an independent body referred to as the National Numeracy Organization to help raise levels of numeracy among adults and children, and to promote the importance of everyday mathematics (https://www.nationalnumeracy. org.uk/what-numeracy). In England, there is also the independent body known as Mathematics Education Innovation (MEI), committed to improving mathematics education and curriculum development (https://mei.org.uk/). The National Centre for Excellence in Teaching Mathematics (NCETM), funded by the Department for Education, has established Maths Hubs across the country to help with professional development for teachers (https://www.ncetm.org.uk/ and http://www.mathshubs. org.uk/). The work of Maths Hubs is coordinated by NCETM and takes the form of a national network across England, providing school-led, collaborative professional development for teachers of mathematics. NCETM also trains Teaching for Mastery Specialists deployed across schools. The NCETM programme is based on how basic mathematics is taught in Asia, particularly in Shanghai. A very effective China-England Teacher Exchange programme was implemented to raise teachers’ awareness of Asian teaching strategies (DfE/Sheffield Institute for Education, 2019; Boylan et al., 2016; Leeming, 2018). This ‘mastery approach’ in primary school places emphasis on explicit teaching and a carefully sequenced curriculum, ensuring that students really understand the mathematics they are learning, and maintain high levels of achievement. The approach has been described in the following terms:

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Shanghai practices differ from common English primary mathematics education practices, with an emphasis on whole-class interactive teaching to develop conceptual understanding and procedural fluency, using carefully designed tasks and skillful questioning. To ensure pupils progress together, tasks are designed to allow for extension by deepening understanding of concepts and procedures, and daily intervention is used to support those needing extra tuition (DfE/Sheffield Institute for Education, 2019, p.15).

NCETM also maintains an online website with practical advice, resources and teaching videos, including the Number Blocks PowerPoints that parents can use with their children at home (see: https://www.ncetm.org.uk/classroom-resources/ey-num berblocks-at-home/.) In Australia in recent years, the government has enacted a number of policies and strategies designed to boost numeracy standards, and several early intervention programmes have been also introduced (see later). Each state and territory has also developed its own policies aligned with the Australian Curriculum to ensure that numeracy teaching is given due attention in their own schools. Most recently, the Australian government has invested money to strengthen the capacity of R-12 teachers to teach numeracy more effectively. This initiative is delivered through professional development activities for teachers, and by access to a repository of high-quality teaching and learning resources through an online Mathematics Hub. The resources for numeracy include checklists and assessment tools, as well as teaching resources for students and families (Australian Government, 2019). In the United States, a stress on the importance of ensuring that students acquire proficiency in number skills and mathematics is explicit in the Common Core State Standards (CCSSI, 2017a), and is embodied in the intentions of Every Student Succeeds Act of 2015. This act requires states to provide interventions for the lowest 5% of school performers, for schools with high dropout rates, and for schools with persistent achievement gaps (Dossey et al., 2016). The following chapter explores important issues related to the earliest stages of becoming numerate.

Online Resources Mathematics programmes of study: Key stages 1 and 2 in the National curriculum in England. Department for Education [UK]. https://www.gov.uk/government/uploads/system/uploads/attach ment_data/file/335158/PRIMARY_national_curriculum_-_Mathematics_220714.pdf Australian Curriculum: Mathematics. http://www.australiancurriculum.edu.au/mathematics/curric ulum/f-10?layout=1 Common Core State Standards: Mathematics [US]. http://www.corestandards.org/Math/ Numeracy for all learners. Victoria State Government. (2020). https://www.education.vic.gov.au/ school/teachers/teachingresources/discipline/maths/Pages/numeracy-for-all-learners.aspx National Numeracy Organisation (UK). (2020). Building a numerate nation: Confidence, belief and skills. https://www.nationalnumeracy.org.uk/sites/default/files/building_a_numerate_nation_rep ort.pdf

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Numeracy teaching within domains. Suggestions for how numeracy skills can be taught and applied in subjects across the curriculum. https://www.eduweb.vic.gov.au/edulibrary/public/teachlearn/ student/numeracyteachdomains.pdf

Print Resources DfE/Sheffield Institute of Education. (2019). Longitudinal evaluation of the ‘Mathematics Teacher Exchange: China-England’, Final Report. Sheffield: Sheffield Hallam University. Goos, M., Geiger, V., Dole, S., Forgasz, H., & Bennison, A. (2020). Numeracy across the curriculum: Research-based strategies for enhancing teaching and learning. Routledge. Hughes, N. (2018). Classroom-ready number talks for third, fourth and fifth grade teaches. Ulysses Press. Independently published. (2020). Singapore Math Kindergarten Workbook: Kindergarten and 1st Grade Activity Book Age 5-7 +. Available through Amazon. Sullivan, P. (2020). Leading improvement in mathematics teaching and learning. ACER Press. Tout, D. (2020). Critical connections between numeracy and mathematics. Melbourne: Australian Council for Educational Research.

Chapter 2

Early Numeracy Development

This chapter describes the early foundations upon which later numeracy concepts, skills and strategies are built. Discussion covers some of the theorists and researchers who have contributed to our understanding of numeracy development, and to varying degrees have influenced teaching methods. To begin the discussion attention is given to what is termed number sense––often considered the most fundamental prerequisite for later numeracy development (Anobile et al., 2018).

Number Sense Since the late 1980s, the term ‘number sense’ has gained recognition in cognitive research, and it is now used frequently within curriculum and teaching documents. In early childhood, number sense develops from the intuitive awareness of numerosity that most children seem to possess from birth (Evans & Gold, 2020; Kersey & Cantlon, 2017; Liang et al., 2021). It can be thought of as the innate capacity to perceive differences in quantitative aspects of the child’s physical surroundings. It is reported, for example, that visual attention of infants 4 months old is drawn spontaneously to any marked difference between quantities presented to them (e.g. one cake or three cakes on a plate); and by the time they are 3 years old, most children have developed a definite sense of numerosity, even if they can’t count (Smyth & Ansari, 2020). By age 4, many children can recognize and name groups of up to three or four items without needing to physically count them––an ability referred to as subitizing (Way, 2011). Number sense is the foundation upon which children in the preschool years begin to build the following capabilities: • learning number names; • relating a number name to a given quantity;

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 P. S. Westwood, Teaching for Numeracy Across the Age Range, SpringerBriefs in Education, https://doi.org/10.1007/978-981-16-3761-2_2

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• the ability to count accurately with correct number-to-object correspondence, • numeral recognition. It has been found that young children’s development of number sense can be enhanced through planned activities, for example, playing number games (Praet & Desoete, 2019; Scalise et al., 2020) and using counting cards with dots and numerals (Joswick et al., 2019). This age of technology has also provided teachers and parents with apps and online games designed to develop children’s early number sense (Baccaglini-Frank et al., 2020; Broda et al., 2019). Any online search under ‘number sense activities’ will yield a rich array of materials and games that can be used for this purpose. Children’s early development in this domain is referred to as approximate number sense. It is reported that possession of approximate number sense in preschool children is highly predictive of their later mathematics achievement, even into the secondary school years (Geary & vanMarle, 2016; Wang et al., 2017). As a child gains experience in the physical world and learns to count, approximate number sense evolves into a more accurate understanding that reflects the exact number system (Ivrendi, 2016). Mastering this system enables an individual to cope with increasingly complex number relationships, and to use numbers and number symbols effectively (Lyons et al., 2018). The exact number sense eventually underpins the capability that enables older students to detect when the solution they obtain from a calculation is not plausible and thus needs to be checked (Wong & Odic, 2021). The expansion of approximate number sense into the exact number system follows a fairly predictable developmental path. This occurs most easily in situations, where children are actively involved with materials and where their parents and teachers focus their attention on quantitative aspects through relevant questioning and discussion—a form of ‘guided play’ (Zippert et al., 2020). Several practitioners advocate that all parents and early childhood educators should use simple ‘math talk’ (‘give me three more’; ‘have we got enough’? ‘how many red beads do you have’?) while engaging in appropriate concrete experiences with children (Dulay et al., 2019; Hanner et al., 2019; Nelissen, 2018). Through this informal and natural approach using such things as building blocks, toys, counters and real objects, children are learning the words that will ultimately help them remember, think and reason numerically. In the early primary school years, children continue to reinforce concepts of cardinality (knowing how many elements are in a group or set) and ordinality (knowing the position name for an item within a series … 1st, 3rd, last, etc.). Children also develop an awareness of numerical magnitude (how large a quantity is when represented by a particular numeral). It has been found that these basic understandings of cardinality, ordinality and numerical magnitude are crucial for making progress in all aspects of arithmetic (Orrantia et al., 2019). If children arrive in primary school with a poor sense of numbers, they are extremely likely to have difficulties when formal aspects of computation are introduced. Howell and Kemp (2006) have suggested that assessing young children’s number sense on entry to school may be a valid way of detecting those who are

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likely to have difficulties later. Early detection can lead to appropriate proactive forms of intervention to help these children acquire the understanding and confidence with numbers that they currently lack (Sasanguie et al., 2012). Many cases of mathematical disability are thought to have their roots in poorly developed number sense (Hojnoski et al., 2018; Terry, 2019; Wong et al., 2017a).

Concept Development Children learning arithmetic skills need to understand the composition of numbers and number relationships as well as mastering the steps within algorithms (Prather, 2021). As stated in Chapter 1, acquiring functional numeracy involves both conceptual understanding that underpins number relationships, and procedural fluency as reflected in the easy recall of number facts and the automatic application of skills in calculating (CCSSI, 2017b; Mills, 2019). In the following sections, the focus is on the ways in which number concepts are developed. Concept formation is the means by which we mentally organize our environment into units of information that provide the general understanding of the world, and that can be stored and used flexibly for thinking, reasoning and problem-solving. A concept can be thought of as a ‘mental embodiment’ of all the essential features of an object, situation or idea. For example, the concept we acquire for ‘triangle’ is that it always has three sides, the sides may be of varying lengths, it is a closed figure, its angles always add up to 180 degrees and it remains a triangle no matter its orientation on the page or in space. Some of these features are discovered incidentally but others we learn from instruction. Concept development occurs when new information is taken in by a learner and linked with prior knowledge. New experiences cause concepts to be modified and elaborated over time (Sidney & Alibali, 2015). These processes of taking in new information and adjusting prior knowledge are usually referred to as assimilation and accommodation (see later under the discussion of Piaget’s work). Number concepts are acquired as a result of experiencing and interpreting a variety of quantitative learning situations. In subjects such as mathematics and science, the concepts that are beyond the most basic level require explicit instruction from teachers and exposure to many examples and explanations. For example, it would be very difficult to grasp the concept of improper fractions and how to use them in calculations without first having clear explanations from a more knowledgeable person who can interpret salient features, provide examples and contexts, teach relevant vocabulary such numerator, denominator and common denominator and teach the processes for operating with them. Research has shown conclusively that the most effective initial teaching of basic skills must involve explicit instruction, accompanied by practice with feedback from teachers (Brownell et al., 2021). The Common Core State Standards for mathematics in the US stress the importance of conceptual understanding in acquiring numeracy and mathematical expertise (CCSSI, 2017d). It is also stressed that conceptual understanding needs to accompany

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the learning of all arithmetic processes and problem-solving strategies, so that these procedures are not carried out simply by rote (Willingham, 2010). Lack of conceptual understanding easily occurs when children are just taught rules, procedures and facts without real explanation and discussion. The children are not able to connect (accommodate) what they are being taught with what they already know—they are simply learning new tricks (Gibbs et al., 2018). Problems also arise when teachers attempt to teach concepts that are at a level of complexity and abstraction that is well beyond the children’s current stage of cognitive development. It is pertinent here to note that the teaching of mathematics in most Asian schools (where numeracy achievement levels are usually found to be very high) places heavy emphasis on ensuring that students fully understand how and why a particular algorithm such as long division actually operates. Conceptual understanding is typically achieved through the use of many relevant illustrative examples, accompanied by a discussion that requires students to reflect upon and verbalize their thoughts about the operations they perform (Mills, 2019). The use of language and precise vocabulary by students and teachers is known to facilitate conceptual development in mathematics (Shockey & Pindiprolu, 2015). When students can verbalize their thinking, it is easy for a teacher to detect any who are failing to understand a procedure or concept, and then to re-teach with additional explanations and examples. The Math Hubs project in the UK recognizes the importance of having students verbalize their thinking as an essential part of the learning process that leads to mastery (NCETM, 2016b). The time spent in discussion and clarification of processes during math lessons often takes longer than the traditional methods typically used in Western schools, but it results in a much deeper understanding of subject matter and real mastery of essential skills. Often, Western school mathematics curricula are overloaded with too many different topics to be taught in the time available, leading to very superficial coverage of the content and limited opportunity for deeper discussion (CCSSI, 2017b). Realization of this problem has resulted in a reduction of content in the revised National Curriculum in mathematics in the UK, in the Australian Curriculum and in the Common Core State Standards in the US.

Theories of Conceptual Development It is appropriate here to acknowledge the work of three pioneers whose theories helped illuminate the way in which cognitive development occurs in children. Piaget, Vygotsky and Bruner all made major contributions to our thinking about how numeracy concepts are acquired and how their development can be encouraged in home and school.

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• Jean Piaget We owe much of our understanding of how children appear to develop number concepts to the work of the late Jean Piaget (1942; 1983), a Swiss clinical psychologist who died in 1980. He had been influenced to some extent by the ideas expressed earlier by John Dewey (1902). Dewey was a proponent for education reform based on the principle that students learn best through active and experiential learning, rather than from the passive forms of instruction that were more typical of schools in his time (Tanner, 2016). Piaget’s theory of cognitive development in number was based on close observation of children (mainly his own three children) as they engaged in various tasks involving quantitative and spatial relationships. He was interested to investigate how their thinking and reasoning developed and changed qualitatively over time. His ideas have greatly influenced the currently popular constructivist view of learning––a view that places the learner rather than the teacher at the heart of the learning process. Constructivists believe that true understanding can’t be transmitted to learners directly by a teacher or instructor, instead learners must make meaning (develop concepts and acquire knowledge) for themselves through their own actions, thoughts and interpretations (Shah, 2020). Piaget argued that children continually construct and modify their understanding of phenomena (including number relationships) as a result of their own direct experiences. Cognitive development in childhood through adolescence to adulthood can thus be thought of as the ongoing construction and refinement of meaning based on experience, perception, reflection and memory (Ahmad et al., 2016). In Piaget’s theory, children’s increasing physical and neurological maturation enables them to explore and reflect more deeply upon the quantitative and spatial world, and build and modify their concepts over time (Fowler, 2017). An essential aspect of Piaget’s theory is his notion of schemata (singular: schema). A schema is a connected web of information comprising everything one has learned over time about a particular phenomenon or entity. Learners interpret new information that is taken in and link it to what they already know. As stated above, Piaget used the term assimilation to describe the process of taking in new information and linking it with prior knowledge, and the term accommodation for the process involved in readjusting existing schemata to reflect this new understanding. This process continues throughout life and is the main characteristic of cognitive growth. Piaget considered that children pass through four main stages on their way to mature cognitive functioning. At each stage, they become better able to process information accurately and are less likely to develop misconceptions. The sequence begins at birth and continues into adulthood. An approximate age range for each stage has been suggested by educators, based on Piaget’s writings––but children actually differ significantly in the age at which they pass through each Piagetian stage. These differences are due to factors such as individual rates of maturity, innate mental ability, opportunities to learn through experience, support in the home and exposure to teaching (Case, 1992; Dasen, 1994; Siegal, 1991).

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The four Piagetian stages are briefly described below, with tentative age ranges indicated. • Sensori-motor stage (birth to 18 months). During the early months, a young child takes in a broad range of sensory input, for example, by focusing visual attention, attending to sounds, gaining control over movements, picking up an object and becoming aware of its features, and beginning to develop an awareness of numerosity. When the child is able to crawl and later to walk, he or she is exposed to an ever-increasing range of concrete experiences. At this stage, the child is becoming familiar with his or her immediate environment but is not aware (for example) of object permanence—that physical objects continue to exist even when they are out of sight. • Pre-operational or intuitive stage (from age 18 months to 7+ years). Children at the pre-operational stage tend at first not to be able to manipulate quantitative ideas mentally or reliably deduce links that exist in cause-effect relationships. They tend to focus too much on one feature of a situation and do not consider other aspects that may be important. As a result, they are often misled by what they see and they form wrong conclusions. An important example of this in the numeracy field is the concept of conservation of number. Children at the pre-operation stage do not necessarily understand that the number of items in a group does not change unless you add or remove items. Even when the spatial arrangement of the items in the set may be altered, the cardinal number remains unchanged. It is sometimes argued that until children can understand the conservation of number at around age 6 years, there is little point in attempting any formal teaching of even the simplest level of arithmetic. Piaget believed that preoperational children do not possess stable numerical concepts and, for example, have difficulty imagining an action reversed, for example, if 3 tokens are placed with 2 tokens to make a group of 5, what would be left if 3 tokens were then taken away from 5? This is why hands-on materials such as blocks and counters are used in kindergarten and early primary classrooms to help children perceive and understand grouping, sharing, comparing sets and conservation of number. • Concrete operational stage (7 to 11+ years). During this stage, a child can begin to process more complex quantitative information when it can be experienced and acted upon at first-hand. During this concrete operational stage, a child becomes able to understand symbolic representation (numerals and signs) and how it is used to record operations. For example, being able to grasp that 7 objects can be represented by the numeral 7; and later, that 17 counters can be made from 10 + 7. At this stage, children can reflect upon and carry out age-appropriate operations mentally, but still struggle to grasp purely abstract ideas. The danger at this stage is that they are only taught early arithmetic skills as rote procedures that they must simply memorize. • Formal operational stage (11+ to adulthood). At this stage, a normally developing individual becomes able to operate more easily with abstract ideas and symbols, and to think and reason without the need for objects, pictures or first-hand experience. Adolescents in the formal operational stage become more strategic and

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logical in their problem-solving approach because they can reason things out rather than using a hit-or-miss intuitive approach. Most educators believe that it is important to consider Piaget’s four stages in relation to the types of mathematical experiences normally provided at different ages. For example, it is now accepted that we should not attempt to teach very young children formal written arithmetic algorithms. Abstract reasoning and the use of symbolic representation cannot be forced on children at the sensori-motor and beginning pre-operational stages. According to Piaget (1942), the direct teaching of number knowledge and skills ahead of a child’s cognitive readiness to learn is largely a waste of time and can also have a negative impact on the child’s confidence and future attitude towards working with numbers. Over the years since Piaget’s writings were first translated into English, numerous experimental studies have generally supported his description of the sequence in children’s cognitive development. However, criticisms of his work have emerged. The first is that he underestimated the learning capacity of preschool children, and overstated the role of maturation in controlling readiness. It is now thought that if young children are provided with concrete materials and appropriate guidance (scaffolding) they can learn much more than Piaget predicted (Bukatko & Daehle, 2012; Mandler, 2004; Marchand, 2012). It is now believed that teachers should base their teaching and guidance on a child’s current proven abilities instead of awaiting maturity or so-called ‘readiness’ (Fowler, 2017). More recent work appears to indicate that opportunity, experience and instruction are just as important as maturation in determining what young children can learn. The current view is that with appropriately structured experience and skilled teaching, young children can actually learn more than they would by their own explorations alone. This is particularly true of acquiring number knowledge and skills. Developmental psychologists who have built on Piaget’s earlier work––the NeoPiagetians––assert that the processes needed to learn concepts and to solve problems are teachable, and we do not need to await the biological maturation of the child (Case, 1992; Demetriou et al., 2002). This suggests that instead of opting for what is currently viewed as ‘developmentally appropriate’ activities in early childhood (this usually implies play and incidental discovery) we should be seeking effective teaching methods for accelerating young children’s learning. As a result, it is argued that curriculum typically provided in the early years often underestimates children’s potential ability to learn and is thus insufficiently challenging. Some early childhood educators are now recognizing the importance of supplementing child-initiated learning with high-quality instruction from a teacher (guided and mediated learning) in the early years (National Child Care Information Centre, 2007). In this respect, modern teaching methods in the preschool and early primary years can take much more of value from the theories of Vygotsky than from Piaget (see below). It is also believed that at the other end of the age range Piaget overestimated what the average adolescent can do in terms of abstract reasoning, particularly in mathematics and science. The age of 11+ has often been suggested as the end of the concrete operational stage, but more recent observations suggest that for the

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many students in secondary schools their thinking in subjects such as mathematics may remain at the concrete stage until at least age 15 or 16 years (Cornally, 2013; McLeod, 2010; Oswalt, 2010; Santrock, 2018). Much of the contemporary mathematics teaching in secondary schools, therefore, misses the mark for these students because it is conducted largely through textbooks, whiteboard examples and online exercises. The relative absence of concrete experience and visual representation may well account for many of the learning difficulties evident in some older students and their growing dislike of mathematics. Despite the limitations described above, Piaget’s major contribution to the field of numeracy teaching has been: • to present the view of children as active and constructive learners; • to support the use of concrete and visual materials for hands-on mathematics learning in the primary school (and now into the secondary school); • to remind teachers to consider children’s level of cognitive maturity when introducing particular mathematical topics; • to make teachers more aware that direct teaching and drilling of number skills without a basis of understanding can have a negative impact on a child’s confidence and future attitude toward working with numbers. • Lev Vygotsky Vygotsky was a Soviet psychologist who died in 1934. While agreeing in many respects with Piaget’s views, Vygotsky (1962) places considerably more emphasis than did Piaget on social influences and adult input in shaping children’s cognitive development. He saw adults as ‘more knowledgeable others’ who often provide immediate and proactive guidance that helps advance a child’s awareness and reasoning. His theory led to the notion that each child has a ‘zone of proximal development’ (ZPD) in which learning tasks that are just a little too difficult for him or her to master unaided can be achieved when he or she is provided with guidance from someone. According to Vygotsky, optimum learning occurs when teachers create learning activities that are tailored to be just a shade above a child’s current level of functioning and guidance is supplied. This type of guidance has become known in education as scaffolding and it takes the form of hints, suggestions, thinking aloud, comments, questions, demonstrations, corrective feedback and even direct explanation. In this situation, the use of language at an appropriate level to match a child’s comprehension greatly assists with concept development (Eun, 2019). Research by Cukurova et al. (2018) has indicated that this type of guided learning in subjects such as science and math improves students’ acquisition of knowledge much more effectively than unguided independent work; and proactive guidance reduces significantly the number of misconceptions that students tend to develop. Vygotsky’s form of constructivism is referred to as socio-constructivism and is based on the belief that knowledge is constructed mainly through the language used in interacting with others. His recognition of the importance of students’ discussion with teachers has led now to the principle that ‘talking together’ is at the very heart of students’ knowledge construction (Hanner et al., 2019; Hung, 2015). At the

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same time, listening to students talking together and asking each other questions can reveal to a teacher any misunderstandings that may have occurred (Urquhart, 2009; Wagganer, 2015). The concept of ZPD is not limited to learning of number skills but applies in all areas of the curriculum (Fithriani, 2019). The major messages for numeracy teaching that stem from Vygotsky’s work can be summarized as: • concept development can be enhanced and refined through discussion; • a teacher has a responsibility to guide children towards better understanding by mediating between their actions, observations and thinking; • a teacher must identify a child’s present level of understanding in order to provide tasks that will help him or her progress further within their own zone of proximal development. • Jerome Bruner Jerome Bruner (1960; 1966) was an American cognitive psychologist who died in 2016. He was instrumental in raising educators’ awareness of the important active role that learners themselves must play in constructing knowledge. He is therefore regarded, along with Piaget, as having a constructivist orientation. A major theme in his work is that learning is an active process in which learners use current experiences to construct and refine concepts, always building upon their prior knowledge. In the domain of mathematics teaching, Bruner stressed the need for creating problems that cause students to think mathematically, rather than simply performing calculation routines that they have been taught without real understanding. However, like Vygotsky, he saw the role of the teacher to be more than simply a facilitator in an activity-based approach. Children need to interact positively with more knowledgeable adults and peers who can support their efforts, challenge them, provide information and assist them in interpreting new discoveries and acquiring new strategies for calculating (Hopkins & Bayliss, 2017). In this context, it was Bruner who first coined the term ‘scaffolding’. According to Bruner, concept development progresses from a hands-on ‘enactive’ stage, through an ‘iconic’ stage (where pictorial and other graphic representations can be used to move beyond the purely concrete) to a final ‘symbolic’ stage where abstract symbols and notation alone can convey meaning to the learner. Applying Bruner’s three stages to early numeracy development, the first step that most children take is interacting with the real world and using real objects (concrete materials). Their move to the iconic stage may first involve using exact pictorial representations instead of real objects (for example, a drawing of three goldfish in a tank, or six balloons in a picture). Children at his stage are able to interpret these pictorial representations of objects and can count them. This is why the early math books and computer materials designed for beginners in preschool contain mainly pictures of familiar objects for counting and comparing. When situations are presented to children in pictorial form, or are drawn by them as pictures, they can easily interpret what they are seeing,

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even though they are not the real objects. At the next stage, three wooden blocks or counters can stand for three cars moving along an imaginary road. The blocks or counters don’t look like cars, but the notion that one thing can be represented in a different way is established. This stage might be called the ‘semi-concrete’ stage. Later, three tally-marks /// (looking even less like the real object) can be used at the ‘semi-abstract’ stage, with an understanding of their one-to-one correspondence with the original objects (●●● + ●● = ●●●●●; /// + // = /////). It is not until a child has had these intermediate experiences of translating reality into different forms of semi-concrete and semi-abstract representation that they really understand signs and symbols at the abstract stage (3 + 2 = 5). It is believed that some children begin to experience difficulty in learning number relationships and mastering algorithms because they have been taken too quickly from the concrete stage to the abstract symbolic level of recording. A gap is created in children’s schemata related to numbers if they are forced to operate too soon with symbols and mathematical notation. The Singapore Ministry of Education advocates the use of the heuristic known as the Concrete-Pictorial-Abstract (CPA) sequence as part of the mastery approach to teaching and learning mathematics. In the USA, this approach is often referred to as ‘Concrete-Representational-Abstract (CRA), and CRA as a teaching principle has been found effective for increasing students’ conceptual knowledge and fluency in computation (Flores et al., 2019). The application of CRA is highly recommended for establishing a firm foundation for all learners, and has proved of particular value when teaching students with learning difficulties (Yakubova et al., 2020). This CPA/CRA sequence is based on Bruner’s conception of the enactive, iconic and symbolic stages (see https://www.ncetm.org. uk/resources/42194).video). The use of concrete materials such as Dienes’ Multibase Arithmetic Blocks (MAB), Cuisenaire Rods, Montessori Number Blocks or Unifix (often called manipulatives) can help children bridge semi-concrete to the abstract level by providing a visual link between objects and the symbols that can eventually represent them (Morin & Samelson, 2015; Jones & Tiller, 2017). Ross et al. (2020) have found that being able to interact with manipulatives can result in improved mental arithmetic performance in children aged 7 to 9 years. Today, online learning media and digital apps can make use of virtual manipulatives that children can move around on interactive touch screens or iPad to experiment with base-ten patterns, number lines, fraction bars, geo-boards and solid figures (Litser et al., 2019; Moyer-Packenham, 2016; Shin et al., 2017; Tucker et al., 2017). Again, this approach has proved very useful when working with students with learning difficulties and disabilities (Bouck et al., 2020). Bruner’s views are clearly compatible with those of Piaget, but like Vygotsky, he is much less concerned with issues such as readiness and maturation. Instead, Bruner supports the view that young children can be taught many things if the quality of instruction is good and the teaching follows the sequence of concrete, through semiconcrete, to the abstract levels. His claim is that any subject can be taught effectively in some intellectually honest form to a normal child at any stage of development––if the method is right.

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In summary, Bruner’s influences on numeracy teaching include: • the need to create learning environments that provide materials and situations to investigate; • the need for learner’s to be actively involved in exploring quantitative situations and real problems; • the need for children to work through concrete and semi-concrete experiences before they are ready for abstraction; • the advantages of planning the most effective sequence in which to present curriculum material (concrete to pictorial to abstract). • Variation theory Variation theory attempts to explain how students may experience the same learning event differently because each has focused attention on a different aspect. For example, when learning about right-angles one student has perceived the vertical orientation of the L diagram on the whiteboard as the key feature, whereas another student has attended to the relevant concept of a 90° angle. The latter student is immediately able to recognize a right-angle on a worksheet when the examples are presented in any orientation. The former is confused if the example is not vertical L(e.g. or ├). Teachers need to ensure that each individual recognizes relevant and irrelevant features of what is being presented as an example. In terms of numeracy, students need to grasp the commutativity law (or principle) by recognizing that the procedure for recording simple addition can be written as 3 + 5 = 8 or 5 + 3 = 8, and for multiplication 7 × 5 can be written as 5 × 7. This law does not apply to division or subtraction, because 8 ÷ 3 can’t be written as 3 ÷ 8, and 9 – 6 is not the same as 6 – 9. [Some useful examples of variation theory in the domain of number concepts can be found at https: //thirdspacelearning.com/blog/variation-theory/#11number-number-and-place-value]. In other words, variation theory explains why all learners need to experience and reflect upon variations in any phenomenon by exposure to examples and nonexamples. They need to appreciate both what something is and what it is not. This requires that teachers design and sequencing mathematical tasks using multiple representations and drawing attention to what is relevant and what is irrelevant in a particular concept. This notion of multiple representations of a concept is entirely compatible with principles underpinning universal design for learning (UDL). UDL frameworks recommend embodying a concept in many different forms to provide students with different pathways to reach the same level of understanding (Hall et al., 2012). The creation of variation theory is usually credited to Marton (Ling & Marton, 2011) and the theory has become important in influencing how numeracy and mathematical concepts are taught. For example, Trundley and Williams (2020) found variation theory can contribute usefully to the better application of manipulatives in 

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the teaching of ‘counting all’ and ‘counting on’ to children aged 5–6 years. Similarly, Björklund et al. (2021) have embodied principles of variation theory in an intervention to help preschoolers learn basic number concepts and skills. More can be discovered about variation theory by following the link: https://var iationtheory.com/category/number/

Theories Covering the Development of Procedural Fluency Constructivist theories reviewed above help to explain concept development––but they have contributed very little to our understanding of how procedural fluency in computation and in mental arithmetic develops. From the constructivist and socioconstructivist perspective, automated procedural skills are simply acquired naturally, through ongoing successful interactions with numbers in the everyday surroundings, and by engaging in solving problems––referred to as acquiring ‘numeracy by participation’ (Atweh et al., 2014). The notion of spending specific time and effort on what used to be called ‘speed and accuracy exercises’ in arithmetic is abhorrent to most constructivists––yet, fluency in carrying out any learned skill can only come from vast amounts of deliberate sustained practice (DeKeyser, 2007; Farkota, 2017; Stocker & Kubina, 2017). Once attained, this fluency enables an individual to work quickly and efficiently, and reduces the cognitive load when attempting to perform calculations required within a math problem. Constructivist views thus tend to overlook the role of deliberate practice in acquiring number skills to the required level of mastery and automaticity. There are several theories and beliefs that relate to how learners become fluent in the recall of number facts and in the smooth and efficient use of calculation skills (Atweh et al., 2014). All these theories contribute something to our understanding of the learning processes involved. First, from a behavioural perspective the theory of operant conditioning and reinforcement (Skinner, 1969) clearly applies, because each time a student performs a calculation correctly (mentally, written recording or by using a calculator) the correct result intrinsically rewards and reinforces the performance, and the likelihood of performing the same process correctly next time is strengthened (Law of Effect: Thorndike, 1911). In the early stages, this reward may be in the form of a tick in the exercise book, a word of praise from the teacher, or a visible reward on the computer screen. Later, the reinforcement is more likely to come from an intrinsically satisfying feeling of competence and self-efficacy. In other words, procedural fluency relies on confidence that accrues from each success building on all earlier successes. The role of frequent and successful practice is clearly important here. As Farkota (2017, p. 15) correctly observed, ‘… genuine mastery of both basic skills and problem solving can only come about with constant practice.’ Similarly, Morkunas (2020) has presented a convincing case in support of daily practice and review of knowledge and skills in all areas of the curriculum, and particularly mathematics.

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Closely related to theories that support the value of frequent practice for skill acquisition is practice engagement theory (Reder et al., 2020). In simple terms, this theory posits that the more that a student engages in meaningful work with numbers, the stronger is that individual’s motivation to achieve in numeracy. There is also a three-stage theory of skill learning that suggests any skill, such as calculating with numbers, develops through distinct stages (Anderson, 1982; Taie, 2014). The three stages are: • presentation––a skill is demonstrated and taught explicitly; • imitation and practice––the learner then rehearses the process he or she has been taught, and receives corrective feedback; • automaticity––frequent practice results in effortless and highly skilled behaviour. The notion of automaticity achieved through practice is absolutely fundamental to appreciating what procedural fluency in number skills really involves (Datchuk & Hier, 2019). It is to be greatly regretted that modern reforms in mathematics education quite clearly ignore the essential role that practice plays in mastering computation skills. When practice is devalued and replaced entirely by problem-based inquiry methods, many students will fail to develop fluency in the most basic arithmetic skills (Westwood, 2003; 2011). The following chapter raises some of the issues involved in encouraging numeracy in the preschool and kindergarten years. During this formative period, concept development and the foundations for later procedural fluency need to be firmly established.

Online Resources Wiggins, G. (2014). Conceptual understanding in mathematics. https://grantwiggins.wordpress. com/2014/04/23/conceptual-understanding-in-mathematics/. Williams, H. (2020). Mathematics in the early years: What matters? https://impact.chartered.col lege/article/mathematics-in-early-years/. Piaget’s theory of cognitive development. http://www.explorepsychology.com/piagets-theory-cog nitive-development/. National Council of Teachers of Mathematics. Position paper on procedural fluency in mathematics. http://www.nctm.org/Standards-and-Positions/Position-Statements/Procedural-Fluencyin-Mathematics/. Briars, D. J. (2015). Tasks and strategies to develop procedural fluency from conceptual understanding. Presentation made at the NCTM Atlantic City Regional Conference, October 22, 2015. [email protected]. McClure, L. (2014). Developing number fluency: What, why and how. http://nrich.maths.org/10624.

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Print Resources Björklund, C., van den Heuvel-Panhuizen, M., & Kullberg, A. (2020). Research on early childhood mathematics teaching and learning. ZDM - The International Journal on Mathematics Education, 52(4), 607–619. Carpenter, T. P., Franke, M. L., Johnson, N. C., & Turrou, A. C. (2016). Young children’s mathematics: Cognitively guided instruction in early childhood education. Heinemann. Datchuk, S. M., & Hier, B. O. (2019). Fluency practice: Techniques for building automaticity in foundational knowledge and skills. Teaching Exceptional Children, 51(6), 424–435. Hattie, J. et al. (2016). Visible learning in mathematics K-12. Thousand Oaks, CA: Corwin. Kanter, P., & Leinwand, S. (2018). Developing numerical fluency: Making numbers, facts, and computation meaningful. Heinemann. Moyer-Packenham, P. S. (Ed.) (2016). Teaching and learning mathematics with virtual manipulatives. New York: Springer International.

Chapter 3

Numeracy in Preschool and Kindergarten Years

In the United States, the Principles and standards for school mathematics included children in preschool for the first time––a clear indication that acquisition of numeracy is now acknowledged to begin long before a child enters school (NCTM, 2000). The early years of childhood are extremely important because they represent a critical period for establishing the foundations for later development (Clerkin & Gilligan, 2018; Jordan et al., 2015). Bernadette Donald (2016, p. 18) has stated that ‘Research shows that focusing on mathematics particularly in the early years has major benefits throughout a child’s life, helping them not only in their later education but also all the way through life.’ Children’s mathematical knowledge in the early years has proved to be predictive of their math grades much later in high school and even college (Kiss et al., 2019). Research has also indicated that it is vitally important that young children do not develop anxiety in connection with using numbers, because this can seriously interfere with their later learning (Tomasetto et al., 2021). As stated in Chap. 2, even before formal schooling begins children have acquired basic awareness of quantitative relationships, and many have developed their own simple number strategies such as counting on fingers (Björklund et al., 2019; Kullberg et al., 2020) for situations they encounter in daily life. Some children appear to possess a particularly strong innate number sense that enables them to grasp numerical concepts from a young age and then build upon them over time (Emerson & Cantlon, 2015). This chapter considers some of the issues involved in providing young children with the optimum experiences to enable this development to occur.

Working with Children in Preschool Clements (2001) has suggested that preschools should capitalize fully on young children’s natural motivation to learn through interacting with their environment. The foundations for mathematical ability can be set down by creating enjoyable

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 P. S. Westwood, Teaching for Numeracy Across the Age Range, SpringerBriefs in Education, https://doi.org/10.1007/978-981-16-3761-2_3

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and exploratory situations that involve numbers. Preschool settings need to establish an abundance of materials that motivate children and invite them to construct and explore. In particular, every early childhood setting should have a ready supply of building blocks, boxes, counters, tiles, shapes, pattern boards, measuring tapes, calculators, squared paper, jars of beads, egg cartons and so on. In recent years, digital technology in the form of interactive math apps delivered using iPads in a play-based learning environment has proved to be engaging and effective in developing young children’s early number skills (Miller, 2018). Adequate time needs to be made available for children to work with these apps while also receiving input and suggestions from an adult (Baccaglini-Frank et al., 2020). Parents, caregivers and preschool teachers have an important mediating (interpretive) role to play when encouraging children’s exploration of quantities (Dulay et al., 2019; Zippert & Ramani, 2017). Encouraging young children to work physically, orally and mentally with small numbers is much more important at this stage than introducing written recording of number operations too soon. Almost all children in preschool will be at a pre-operational or early concrete operational stage as described by Piaget (see Chap. 2). The teacher’s role in children’s activities is, as Vygotsky (1962) suggested, to support their number skill development by helping them advance to the next stage. This is best achieved by talking with them, asking focusing questions and challenging their thinking at an age-appropriate level. Much of this important early learning can also be accomplished by using games, rather than formal exercises (Cohrssen & Niklas, 2019; Evans & Gold, 2020; Hendrix et al., 2020). Before entering kindergarten at age 4, most children already possess a basic awareness of numbers that enables them to tackle simple everyday quantitative tasks such as sharing 4 biscuits equally with a friend or counting to page 5 in a story book (Tobia et al., 2016). Studies by Barth et al. (2006) and Gilmore et al. (2007) revealed that young children can carry out simple quantitative comparisons (‘the smallest cake’; ‘the biggest ice-cream’; ‘the middle-size bear’). There is also evidence to suggest that the ability to discriminate between quantities and to count accurately are predictive of later ability to cope well with mathematics in primary school (Magargee & Beauford, 2016; Tobia et al., 2016). Developmentally appropriate mathematics experiences in the preschool should, therefore, include activities that help develop these basic skills. Some children in the preschool years can represent numbers in their early drawings (e.g. ‘My Mum and her 3 friends’; ‘Our 2 hens and 3 chicks’). A few may also perform very simple mental addition and subtraction with numbers below 10, long before these skills are taught in school. They may have acquired these abilities from brothers or sisters, or a parent may have deliberately taught the child. As indicated in Chap. 2, the language used by adults is very important in young children’s development of number knowledge. For example, adults automatically expose children to language that accompanies everyday quantitative experiences, such as serving food at the table––‘Give me two potatoes today please, because they are rather small’; ‘Just take half a cup of juice’. Or at the supermarket––‘Can you get three tins of baked beans please, not the big tins, the smaller ones’; ‘Apples are

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three for $5.00 today’. Young children also hear and use number names as they join in with rhymes and songs: ‘One potato, two potatoes, three potatoes, four ….’. They hear sister counting the steps as she walks upstairs or uses a skipping rope, and they hear big brother say, ‘Mum, I have four pages of homework to do tonight, but I have already finished half.’ These incidental encounters with the language of everyday numbers add to the existing preschool number sense. Through these informal quantitative experiences, most children in early childhood begin to develop confidence with numbers. Studies across cultures have found that the quality of the home learning environment is strongly associated with numeracy development in preschool, and that this advantage is maintained at later ages (Anders et al., 2012; Visser et al., 2019). Many Chinese parents, for example, tend to take a very active role in teaching their children about numbers at an early age (Liu et al., 2019). Ng (2014) suggests that this strong parental support at home, plus more frequent exposure to basic mathematics in preschools, may explain why Chinese school children tend to perform so well in mathematics when compared to children from many other cultures. Parents can do much more to encourage children’s curiosity about numbers by simply drawing attention in an interesting way to relevant quantitative situations by asking questions, making comments or interpretations and making more explicit their own daily use of numbers (Zippert et al., 2019; Zippert & Ramani, 2017). As indicated in Chap. 2, Vygotsky saw such verbal interactions as providing a major contribution to children’s cognitive development. Research indicates that through specific training, parents can be helped to increase their verbal interactions with children in situations involving numbers (Dulay et al., 2019; Hendrix et al., 2019; Niklas et al., 2016). Advice for parents on supporting their children’s numeracy growth is available online (see: http://www.teachingyour child.org.uk/maths.htm).

Building Firm Foundations in Kindergarten It can be seen in the following sections that there is reasonably close agreement across the UK, Australia and the United States in what should constitute numeracy learning objectives for the kindergarten years. The generally agreed core knowledge and skills to be taught can be summarized as: • • • • • • •

accurate counting to 10 and then to 20, connecting number names to numerals, recognizing or counting accurately ‘how many’ in a small group, ordinal numbers (1st, 2nd, 3rd), recognizing number patterns and sequences, naming basic shapes, sorting and classifying objects by size, number, shape.

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In the UK, the Early Years Foundation Stage (EYFS) now exists as a framework for numeracy teaching (DfE, 2017). The ‘early years’ are officially deemed to be from birth to age 5 but are interpreted for teaching purposes as ages 3–5. Within the framework, it is suggested that early number concepts and skills should be developed by providing structured opportunities for children to use counters and other materials for counting, comparing groups, simple addition and subtraction, and working with shapes and measurement. The approach recommended is to use ‘planned, purposeful play and through a mix of adult-led and child-initiated activity’ (DfE, 2017, p. 9). The use of an ‘adult-led’ approach represents a major shift from the earlier ideology that supported only free-play methods and informal learning, with no adult intervention or explicit teaching. According to the Early Years Foundation Stage in the UK, children who complete kindergarten education at age 4 + to 5 years should be able to: • • • • • • • • • • •

count reliably with numbers from 1 to 20, place numbers in order, say which number is one more or one less than a given number, use objects (e.g. counters, blocks) to add and subtract single-digit numbers, count on or back to find an answer, double, halve and share with numbers to 20, use everyday language to talk about numbers and relative size, compare quantities, solve simple age-appropriate number problems, recognize and create number patterns, explore characteristics of common shapes and use correct language to describe them.

In Australia, the Early Years Learning Framework serves much the same purpose as EYFS in the UK, providing guiding principles for educating children in the period from birth to five years and through transition to school (DET, 2009). Within this framework, numeracy is regarded as an important capability essential for successful learning across the curriculum. Numeracy in that document is defined as the ‘capacity, confidence and disposition to use mathematics in daily life’, and includes understandings about numbers, counting, patterns, measurement, spatial awareness and quantitative reasoning. Most states and territories produced their own guidelines based on the framework. For example, the Department for Education and Child Development in South Australia provided ‘indicators’ for numeracy to assist teachers in assessing children’s current competencies (DECD, 2015). The Australian Association of Mathematics Teachers joined forces with Early Childhood Australia to issue a joint position statement on Early childhood mathematics (AAMT/ECA, 2006). This document still tended to favour a child-centred approach and makes no mention of direct teaching or the need for abundant practice to strengthen basic skills. Since the publication of the original Early childhood mathematics, there has been a slight increase in the amount of attention given now to teacher-led learning in the early years. This increase is in keeping with the recognition that early numeracy learning can be greatly enhanced if adults help young

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children interpret their learning experiences (Education Review Office New Zealand, 2016; Hardy & Hemmeter, 2019). Even the National Council of Teachers of Mathematics in the US has moved away from its earlier calls for entirely child-centred informal methods in preschools and schools, and now advocates the use of ‘effective, research-based curricula and teaching practices’. More specifically NCTM (2017b) states: Early childhood educators should actively introduce mathematical concepts, methods, and language through a variety of appropriate experiences and research-based teaching strategies. Teachers should guide children in seeing connections of ideas within mathematics as well as with other subjects, developing their mathematical knowledge throughout the day and across the curriculum. They must encourage children to communicate, explaining their thinking as they interact with important mathematics in deep and sustained ways (p. 1)

The terms ‘mediated learning’, ‘guided participation’ and ‘teacher-led enquiry’ have become popular and are appropriate when describing effective learning interactions between adults and children. This implies more input by the teacher and less unguided or unstructured play. In mediated learning, the role of the early childhood educator (or parent) is to supply information and ask questions at the ‘teachable moment’, and to build bridges between the real world and abstract ideas. Mediated learning is really the practical application of Vygotsky’s and Bruner’s concept of scaffolded teaching within a child’s zone of proximal development. The National Council of Teachers of Mathematics in the US has now endeavoured to forge closer links to the Common Core State Standards for mathematics (Koestler et al., 2013). The State Standards for this age group cover the teaching and learning of counting, cardinality, simple number operations in base 10, age-appropriate data collection and depiction and simple geometry. As an example of the counting standard, it is suggested that by the time children leave kindergarten they should be able to count accurately the number of objects in a group (up to 20), make quantitative comparisons between sets of objects and know the sequence of numbers up to 100. The Technical Education Research Center (TERC, 2012) in the US has presented a very clear framework for the kindergarten mathematics curriculum (see Online Resources). This website has useful visual representations and recordings that can be used to teach the recommended content at concrete and semi-abstract levels. Criteria for assessing children’s knowledge and skills in basic numeracy are also included. The ideas easily complement the curriculum content set within the national curricula of the UK and Australia, as well as in the Common Core State Standards in the US. The kindergarten content suggested by TERC (2012) includes the following key numeracy competencies: • counting, • whole number operations: simple addition and subtraction with small numbers using blocks and counters, • using data: collecting, sorting, classifying and depicting simple data, • linear measurement, • patterns and shapes: constructing patterns and shapes; recognizing patterns, • identifying, composing, comparing and sorting 2-D and 3-D shapes.

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Some of these areas of competence are discussed in more detail below.

Counting Perhaps the most vital competency for young children to acquire is the ability to count (Gibbs et al., 2018; Turrou et al., 2017). This fundamental skill includes rote counting (reciting the number names in correct order), counting on fingers, accurate counting of small groups using one-to-one correspondence and more advanced counting strategies such as ‘counting on’ from a given number, counting back and counting by 2 s, 5 s and 10 s. Nguyen et al. (2016) have found that counting strategies in preschoolers are highly predictive of their later mathematics achievement, so counting must be recognized as fundamental for further numeracy development. Activities in the early years that involve counting can be very enjoyable and motivating for children, providing the main pathway to understanding cardinality (Jacobi-Vessels et al., 2016; Throndsen et al., 2017). For example, mobile apps have been found useful for developing the counting and quantifying skills of children aged between 3 and 5 years (Mowafi & Abumuhfouz, 2021). Physically counting items in a set also helps the youngest children reinforce the concept of number conservation— the understanding that no matter how items in a group are arranged, the number of items does not change unless you put more in or take some out. Conservation of number was a focus of much attention in Piaget’s studies of children’s concept development (see Chap. 2). Most parents realize it is well worth cultivating their child’s ability to count even before he or she enters school. In their daily lives, children are surrounded by situations where they hear others count and that require them to count and compare quantities. These activities are often a feature in children’s preschool television programmes, where counting may be paired with pictures and numeral recognition. Numbers, counting and quantity words also occur quite often in rhymes that are read to young children in the home and in preschool—‘Ten in the bed and the little one said, roll over. They all rolled over and one fell out. There were nine in the bed …’ (Hassinger-Das et al., 2015). When children enter kindergarten, many can already rote-count to 10 or even 20, but they may not always be accurate when counting a group of objects. They can recite number-names in the correct sequence, but they don’t necessarily match each word to a separate item in a group. To count items in a group accurately and meaningfully children must understand at least the following principles: • • • •

number names must be said in their correct order, each item in the set must be counted once and only once, each number word must be matched to a separate item, the final number reached identifies the total number of items in the group.

Counting

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The ease with which most children learn to count and how they refine this skill over time has been the focus of much research on emergent numeracy (e.g. HannulaSormunen et al., 2015; Posid & Cordes, 2015). It appears that there are several different counting strategies that children employ once they have moved beyond the level of rote counting a group of objects by touching each one. For example, when given two sets of counters on the desk and told ‘This group has 4 counters; if we add 3 more, how many altogether?’ Some young children will count all the objects in both groups (the ‘counting all’ strategy) while others will count on from the number already known ‘4 … 5, 6, 7’ (the ‘counting on’ strategy). It is considered that ‘counting on’ is a more insightful and effective strategy than ‘counting all’ in most situations. After gaining experience with counting all and counting on, most learners begin to store commonly occurring number facts in memory (e.g. 3 + 3 = 6) and can then retrieve them easily without counting. However, Hopkins and Bayliss (2017) have reported that around 50 per cent of students in Year 7 don’t always use retrieval of number facts from memory but instead still solve even simple single-digit addition problems by counting (often on fingers). When students cling to these slow counting procedures, they need to engage in practice activities and speeded mental tests that will help them commit number facts to memory. The Prekinders website contains very useful activities for teaching and practising counting skills. The site is well illustrated and provides information of practical value (see Online Resources).

Numeral Recognition Alongside counting, the ability to recognize instantly the numerals from 1 to 10 (and later from 11 to 20) is an important step in increasing a preschool child’s exact number sense. It is helpful that numerals are very frequently on display in preschool children’s educational environment, in television programmes and in picture books (Perger & Major, 2018). The correct writing of numerals should also be taught at the same time, as part of normal handwriting instruction. Often the youngest children tend at first to reverse numerals like 3, 5 and 7. Later, some children reverse place values (e.g. 61 instead of 16) which causes difficulties in calculation. A tendency to reverse a figure or place value when writing needs to be remedied before it becomes stored in muscle memory as an incorrect motor habit. Testing young children’s ability to recognize numerals to 10 or 20 is part of any preschool assessment procedure, and a note can be made of any numerals that are not known instantly by sight or that cannot be matched to a correct group of objects.

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Using a Number Line A number line or number track is an exact visual representation of numbers spaced out in the correct linear sequence. It is a very useful aid for supporting students’ counting skills and for making clear processes of simple addition (counting on from a given number) and subtraction (counting back) (Woods et al., 2018). The number line can also be used later to teach ‘interval counting’ in twos, fives and tens (Cramer et al., 2019). Clarke et al. (2020) suggest that age-appropriate number line tasks could usefully be included in any assessment process for children who may be candidates for early intervention. Many students discover the value of a number line for themselves without formal instruction; and even older students will often be seen using calibrations on a ruler to count forward or back as an aid to mental calculation (Drake, 2014). In primary school, teachers can use a number line to demonstrate basic operations such as multiplication and division. The teaching material known as Unifix has a 1-to-100 number track divided into 10-unit sections. For beginners, one would use only the section for 1 to 10 or 1 to 20. In primary school, the sections from 20 to 100 can be added. to enable children to place blocks in the track to model operations such as 22 + 13, 45 + 17, 21 × 3, 26 ÷ 4 (showing the remainder), etc. Basically, a number line helps children build mental representations of number relationships and operations (Yuan et al., 2020).

From Counting to Number Operations ‘Operations’ are the manipulations that can be carried out with numbers mentally, in writing or with a calculator. The common operations are, of course, adding, subtracting, multiplying and dividing. Much of this work will be done later in primary school, but even in the preschool years young children need to understand addition as ‘putting together’ and subtraction as ‘taking from’. They need to encounter these operations many times in concrete ways, using everyday objects and also blocks and counters. It is at this stage that progress is made from concrete to the semi-concrete stage of development (see Chap. 2). Language to accompany and guide these operations also needs to be taught: ‘4, take away 3, leaves 1’; or ‘4 add 3 add 1 makes 8.’ Simple addition and subtraction of numbers below 10 are the most common number facts that must be remembered.

The Beginnings of Place Value By the end of kindergarten, most children will begin moving on to numbers from 11–20. This stage signals beginning awareness of place value. Place value refers to

The Beginnings of Place Value

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the understanding that ‘1’ in a written number does not always mean a single unit—in a number like 16, the 1 represents 10. Establishing this concept for children can be achieved by using apparatus such as Dienes’ Base-ten Arithmetic Blocks (MAB), Unifix Cubes or Cuisenaire Rods. But everyday materials such as bundles of drinking straws or ice-block sticks can also be used to help children store visual images of 14 being ‘a bundle of 10 with 4 more’. In computer-based programmes, it is also possible now to use touchscreen devices to manipulate visual groups of 10 s and units to experiment with place value (Kondys, 2017). Most of the work on place value will be covered later in the primary school years, where children will learn that the numeral ‘1’ may also represent 100 or 1000, depending on its position within a whole number. However, the foundation for this work should be laid in kindergarten by involving children in constructing and deconstructing groups of more than 10 objects, and from these groups making separate piles or bundles of 10. Some children transitioning out of kindergarten will not have reached this stage, so it still needs to be revisited in the first year of primary school.

Data Collection, Representation and Analysis Working with data that young children can collect for themselves facilitates the application of six key skills that are components of numeracy—observation, categorization, comparison, classification, order arrangement and measurement (Chumark & Puncreobutr, 2016). Collecting data is also an interesting and motivating activity for students of this age—it represents working with numbers in a meaningful context and for a real purpose. In the kindergarten, children can collect data at first-hand, such as the number and types of pets owned by children in the class, or the number of cars, trucks, buses and bicycles passing on the road outside the school in a given period. At first, these quantities are represented on paper or whiteboard as simple drawings of the objects arranged vertically or horizontally (8 cars, 9 trucks, 4 bicycles). Later, when the children are making good progress and can fully understand the process of recording, the teacher can introduce tally marks as substitutes for pictures: cars //////// trucks ///////// motorcycles // bicycles //// buses // cars 8. trucks 9.

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motorcycles 2. bicycles 4. buses 2.

Measurement Measurement can be defined as the assignment of a numerical value to some attribute (e.g. length, height, weight, speed, time, temperature). Carrying out simple measurements helps children develop skills that will be of benefit in several subject areas in school and in aspects of adult life (Claessens & Engel, 2013). Research in Australia, conducted by MacDonald (2010) suggests that many children enter kindergarten with a basic awareness of what simple measurement involves. The role of kindergarten teachers is to provide age-appropriate experiences that involve simple measurements and recording these as data (Kotsopoulos et al., 2017). Kindergarten children can be introduced to measurement first by using such natural units as hand spans, steps or paces. Later they will begin to use rulers and tape-measures for linear measurement and will engage in simple weighing tasks. In primary school, students will build significantly on these early experiences by being taught how to measure length, weight, time and area more accurately.

Simple Geometry Shapes can be regarded as a component of numeracy because they have measurable dimensions, areas and angles. An introduction to shapes is included in all typical preschool and kindergarten math curricula in England, Australia and the US. In England, the Early Years Foundation Stage requires children below age 5 to have experiences with 2-D and 3-D shapes, and with measuring and constructing lines and angles (Standards and Testing Agency, 2014). Similarly, the Common Core State Standards in the US recommend that young children should be taught to identify and construct common shapes (CCSSI, 2017c). In Australia, the Early Years Framework refers to children naming and describing familiar 2-D and 3-D shapes and recognizing these shapes within objects in the environment. They also need to learn the correct vocabulary to describe relative positions in space (above, behind, inside, next to, etc.) (Perry et al., 2012). The next chapter explores the further development of numeracy once children enter primary school. It is at this stage that mastery of basic calculating procedures should be achieved alongside problem-solving. These skills need to be firmly established with adequate automaticity before entry to secondary school.

Online Resources

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Online Resources Data analysis. Erikson early math collaborative. http://earlymath.erikson.edu/foundational-con cepts/data-analysis/. Education Review Office New Zealand. Early mathematics: A guide to improving teaching and learning. https://www.ero.govt.nz/assets/Uploads/ERO-Early-mathematics-March-2016.pdf. Prekinders website. Ways to teach counting. http://www.prekinders.com/teach-counting/. Simple games for number recognition. Reading Confetti website. http://www.readingconfetti.com/ 2013/02/5-simple-games-for-teaching-number.html. Technical Education Research Center (TERC). (2012). Kindergarten math content Cambridge, MA: TERC. https://investigations.terc.edu/library/curric-gl/math_content_gk_2ed.pdf. Teaching preschool math. https://www.preschool-plan-it.com/teaching-preschool-math.html. Teaching measurement in the preschool. https://www.teachpreschool.org/2013/02/07/exploringmeasurement-in-preschool/.

Print Resources Aunio, P. (2019). Early numeracy skills learning and learning difficulties: Evidence-based assessment and intervention. In D. C. Geary, D. B. Berch & K. Mann Koepke (Eds.), Cognitive foundations for improving mathematical learning (vol. 5, pp.195–214). Amsterdam: Elsevier-Academic Press. Blevins-Knabe, B., & Austin, A. M. (2016). Early childhood mathematics skill development in the home environment. Cham, Switzerland: Springer International. Harris, B., & Petersen, D. (2019). Developing math skills in early childhood. Mathematica Policy Research. Pluckrose, H. (2018). Numbers (Math Counts). Children’s Press. Snow, K. (2016). Preschool math at home: Simple activities to build the best possible foundation for your child. The Well-Trained Mind Press.

Chapter 4

Numeracy Development in Primary School

By their fifth year of life, children seem to have developed two cognitive systems, one for making comparisons of quantities (bigger than…, less than…, more…, not the same) and another for counting. By age 6 years, these two systems combine to provide a more powerful grasp of the exact number system upon which computation skills can begin to develop. The general expectation is that most students will develop firm and functional arithmetic skills by the end of their primary school years. These skills include the ability to calculate using addition, subtraction, multiplication and division, measure length, weight and time and use appropriate words when discussing or explaining these processes and number relationships. Together, number and measurement skills are considered functional when they can be used fluently and confidently to tackle age-appropriate everyday tasks and routine and non-routine problems (Estapa et al., 2018). It is unnecessary here to provide details of all the topics covered within primary school mathematics curricula. This information can be accessed easily on the websites that provide guidelines for the National Curriculum in the UK, the Australian Curriculum in Mathematics and the Common Core State Standards in the United States. Links to these important websites are listed at the end of this chapter. It is easy to appreciate that proficiency with numbers and measurement is a fundamental capability in everyday life and in most forms of work. For this reason, developing students’ basic numeracy alongside reading and writing has always been a priority in the primary school curriculum. The great importance of numeracy is fully supported by research, as well as by common sense (Gravemeijer et al., 2017; Haberlah, 2017a; Westwood, 2019). In this chapter, we will explore some of the selected curriculum content that can help reinforce and expand the development of numeracy in school Years 1–7. The discussion will focus in particular on the effective teaching of computation skills and of strategies for problem-solving.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 P. S. Westwood, Teaching for Numeracy Across the Age Range, SpringerBriefs in Education, https://doi.org/10.1007/978-981-16-3761-2_4

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Transition from Preschool to School From the beginning of the primary school years, a teacher’s responsibility is to help all children understand key math concepts, calculate accurately with numbers, reason logically, solve age-appropriate problems and engage confidently with basic math. The primary school years are vital for achieving this goal. Children entering the first year of school will possess widely different levels of conceptual development and very different degrees of proficiency in working with numbers. This reflects the combined effect of their innate ability, their diverse range of preschool experiences, and the amount of guidance they have received in home and preschool. What happens to them over the next few years will either strengthen their skills and confidence in working with numbers or will cause some to develop a life-long distaste for mathematics. A priority in primary schools will always be to ensure that students develop confidence in working with numbers and a positive attitude towards mathematics. Their success from Year 1 onwards will ensure that they do not begin to feel that they have no aptitude for the subject. Any students who have not achieved mastery of basic computation processes by the end of primary school and who still lack easy recall of number facts will encounter many difficulties and frustrations when they meet more complex mathematics in secondary school (Nicholas and Fletcher, 2017; Pearn, 2014). For some of these students, early difficulties cause the onset of ‘math anxiety’ and an accompanying loss of confidence and motivation (Gunderson et al., 2018; Mutlu, 2019). Difficulties with math in school is a major obstacle later to a student opting for this important subject in tertiary studies and eventually pursuing a career in fields such as science and engineering. Transition from kindergarten into the reception or ‘prep class’ in primary school needs to be smooth. Observers would not usually see a dramatic change in teaching approach for the first 6 months. Teachers in most Prep and Year 1 classes still tend to use a predominantly child-centred activity approach, rather than formal teacherdirected instruction. When children enter the second half of Year 1 and then transition to Year 2, the methods they encounter will become a little more structured and more teacher-directed to ensure that basic number skills are firmly established alongside learning with understanding. That is certainly the case within the ‘mastery approach’ described later. When moving beyond kindergarten, Aunio and Räsänen (2016) have proposed that children in the age range 5–8 years should continue to strengthen the following aspects of numeracy: • symbolic and non-symbolic number sense, • basic skills in arithmetic, • understanding of place value in base-ten.

Numeracy in the Primary School Curriculum

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Numeracy in the Primary School Curriculum Much of the mathematics teaching covered in the first year of primary school simply extend skills and understandings that were established less formally in the kindergarten. In the next few years, students need to become fluent in using the four basic processes in arithmetic and applying them to problem-solving. Often these arithmetic skills and measurement abilities can and should be used within other areas of the primary school curriculum, thus assuring reinforcement, generalization and transfer of learning. Inspection of most primary mathematics courses—for example, those in the National Curriculum in England and in the Australian Curriculum—suggests that the following strands are typical content for teaching students aged 6–10 years. • Recognizing, reading, writing and using number symbols to 100 and then to 1000. • Becoming proficient at calculating, beginning with simple addition and subtraction and later multiplication and division. • Interpreting place value to 1000 in base-10. • Using computation skills to solve age-appropriate routine and non-routine simple problems. • Carrying out activities involving measurement. • Collecting, recording and working with data. • Familiarity with the properties and construction of shapes.

Addition and Subtraction If students have attended kindergarten, many entering Year 1 can perform simple addition and subtraction using fingers, blocks, pictures, tally marks—and in a few cases even by mental calculation alone—but may not be fully familiar with recording these operations in writing (3 + 3 = 6). As indicated in Chap. 2, it is now recognized that the transition from concrete operations to the use of symbolic representation with numerals has intermediate stages of semi-concrete and semi-abstract. At first, written recording is taught alongside the use of countable materials such as counters, blocks or tally marks. Later these props are faded out and simple arithmetic is practised using only the written form on paper or on the computer screen. Every opportunity must be taken during lessons (not just mathematics lessons) to make use of real-life situations that involve counting, measuring, comparing, sharing and manipulating materials and numbers. Numeracy skills should never be taught in isolation, but always connected with realistic contexts in age-appropriate ways. Learning and applying number skills in real contexts is particularly important for students with learning difficulties (Schreiber-Barsch et al., 2020). More is said later about real-life mathematics in the following chapter on the secondary school curriculum.

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While still operating with small numbers, the youngest children in primary school are explicitly taught the conventional use of elementary mathematical notation, signs and symbols. At this stage, children need to become fully aware of the ways in which number relationships can be written. Activities still need to involve concrete materials at times, for example, to show how a group of 8 can be deconstructed and recorded in many different ways (e.g. 8 = 4 + 4; 8 = 5 + 3; 8 = 6 + 2; 8 = 7 + 1; etc.). Children also need to learn the ‘complement principle’ that applies in addition and subtraction (if a – b = c, then c + b = a) (Torbeyns et al., 2016). To ensure that students understand and interpret recordings, McNeil et al. (2015) suggest that practice activities in Year 1 should also vary the position of the equal sign (4 + 1 = 5; 7 = 5 + 2) and include several ‘empty box’ items (e.g. 3 +  = 8;  + 5 = 8; 10 –  = 3). Systematic exposure to both horizontal and vertical formats for addition and subtraction is extremely important. 12 + 5

12 + 5 =

The horizontal left-to-right form (6 + 4 = 10) has usually evolved naturally from recording of concrete operations using counters (oooooo + oooo = 10) or tallies (////// + //// = 10) and finally using only numerals. For a few children, the transition from horizontal to vertical format can create temporary difficulties. Some seem to abandon the sound thinking they used when processing numbers horizontally as soon as they are confronted with a vertical algorithm. This is particularly the case when place value is involved and they encounter examples with columns headed Tens and Units. The two different systems for setting out addition and subtraction need to be taught together for a while, and adequate teaching time must be spent to establish the equivalence of the two recordings.

Place Value In the UK, the Mathematics Programmes of Study: Key stages 1 and 2 (DfE, 2013a, p. 5) states that ‘By the end of Year 2, pupils should be precise in using and understanding place value.’ In any classroom, the students will be at different stages in acquisition of this concept. Explicit teaching with many examples and with discussion is necessary to help them all make progress (Howe, 2019; Young-Loveridge & Bicknell, 2016). Without place-value concept, it is impossible for students entering Year 3 to comprehend algorithms that rely on place-value manipulations (e.g. subtraction with borrowing; long multiplication; moving a decimal point) (Bicknell et al., 2017; Fraivillig, 2018). Understanding place value is often very weak in students with learning difficulties, and this can be one of the reasons why they fail to acquire fluency in calculating with larger numbers. Finding concrete and visual methods for illustrating and teaching place value is, therefore, a high priority when working with these students (Lu et al., 2021). Base-10 materials in Dienes Multi-base Arithmetic Blocks (MAB)

Place Value

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are excellent for this purpose. It is also useful to have students gain experience in partitioning numbers and materials in different ways (for example, 24 = 10 + 10 + 4). Research with Year 4 students has found that using concrete manipulatives together with technology-assisted virtual manipulatives increases the students’ place-value understanding and arithmetic performance (Sari & Aydogdu, 2020). As they progress to the next level of understanding by working with hundreds, tens and units, students need to learn more complicated step-by-step calculation procedures. All too often, this is the stage at which rote teaching and rote learning replace learning with understanding. Rather than teaching a set of ‘tricks’ to obtain the correct answer (e.g. ‘just cross that number out and carry a 10’), teachers need instead to demonstrate visually (with materials and the whiteboard) procedures involving ‘borrowing 10’ and regrouping. This is best achieved best by demonstrating the processes with materials such as ‘bundles of 10’ from Unifix or MAB, while ‘thinking aloud’ and making the appropriate recordings on the whiteboard. A teacher ‘thinking aloud’ during the demonstration of a process provides children with verbal cues to regulate their own thinking and action at each step when working independently (e.g. ‘I must borrow 10 and write it here with the 4 units to make 14. Now I can take 6 from 14’). The children then engage with the same materials to practice the steps involved in, for example, subtracting 17 from 31. Confidence and fluency in executing these basic computation procedures will only come from abundant successful practice.

Bar Modelling One approach that has proved very useful for representing number relationships visually is the Bar Model. This approach helps to span the Concrete-Pictorial-Abstract (CPA) stages, and at the same time teaches students a strategy they can use independently. Bar Modelling can also be applied to addition, subtraction, multiplication and division problems that are much more complex than the example given below. Students are taught to represent quantities stated in word problems as ‘bars’ to help them visualize what they need to do. Examples of this approach (including a video clip) can be found at: http://www.mathsnoproblem.com/en/the-maths/teaching-met hods/bar-modelling/ A simple example of bar modelling: A small greengrocery shop sold 15 cabbages on Sunday. On Monday, fewer shoppers wanted cabbages, so only 10 were purchased. Tuesday saw an even greater fall in sales, with only 3 cabbages sold. How many cabbages were sold altogether over that 3-day period?

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15

10

3

15 + 10 + 3 = 28 cabbages How many more cabbages were sold on Sunday than on Tuesday? 15 3

The Bar Model strategy is an important component within the Singapore Math Method® now adopted for use in the UK. Bar modelling requires students to draw and visualize number relationships and mathematical concepts as an aid to solving problems. A study with primary school children using the Bar Model found it to be an effective method to increase students’ confidence and accuracy when solving word problems (Kaur, 2019; Morin et al., 2017). The approach has also been found effective for students with learning difficulties who benefit most when a math problem is made visible (Preston, 2016; Sharp & Shih-Dennis, 2017). A full explanation of the Bar Model method can be found online at/http://www.hmhco.com/~/media/sites/home/education/global/pdf/whitepapers/mathematics/elementary/math-in-focus/mif_model_drawing_lr.pdf?la=en. An excellent well-illustrated item on the Singapore Math Approach and how it relates to Bar Modelling can be found at: http://www.bbc.co.uk/skillswise/0/249 25787.

Multiplication and Division In the National Curriculum in England, multiplication and division processes are introduced at a simple level in Year 1 and then extended in the following years. The ultimate aim is to foster multiplicative thinking, which can be interpreted as the capacity to recognize and work flexibly with representations of multiplication and division in a wide range of situations. It is stated in the guidelines that in the early years, students should use arrays of counters and pictorial representations to solve one-step multiplication or division problems. An array as used in math teaching means a visual depiction of relevant number relationships. For example: 3 × 3, 9 ÷ 3: ••• ••• ••• The array method is well illustrated in a paper by Salmah et al. (2015) and on the website at: https://nrich.maths.org/8773.

Multiplication and Division

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Day and Hurrell (2015) suggest that students should experience multiplication in several different ways, particularly through creating repeat patterns and arrays of blocks as above. This use of arrays is also strongly advocated by NCETM as part of the mastery approach to mathematics to help students build up a mental image of what, for example, 7 × 4 or 8 × 3 ‘looks like’. Making division ‘visual’ is also important, because students need to see what happens when larger groups are subdivided, and what it really means to have a ‘remainder’ (Pratt et al., 2015). Later these visual arrays lead naturally to teaching the concept of fractions. Good examples of arrays can be found on the NCETM website at: https://www.ncetm.org.uk/public/files/177 15283/Appendix+2+-++Calculation+Policy+Multiplication+and+Division.pdf. Mills (2019) has pointed out that when teachers use manipulatives, relate word problems to children’s real lives and promote discussion in groups, children develop a much deeper understanding of multiplication. However, in the typical rush to get children in upper primary school to the abstract reasoning and symbolic stage, instruction in multiplication and division often tends to concentrate only on procedural fluency. A much more efficient approach recognizes that procedural fluency and conceptual understanding must be developed in tandem because each supports development of the other (NCETM, 2016a). Efficient teaching is achieved when teachers spend time illustrating and discussing what happens in reality when combining groups of equal size, exchanging and regrouping. However, many teachers (particularly those who are not math specialists) find it simpler to teach the ‘tricks’ in multiplication and division algorithms that will yield a correct answer. It is common to find that students in upper primary and secondary schools have the greatest difficulty in mastering long multiplication and long division (Pearn, 2014), probably because these processes are taught only as computation tricks to be mastered by rote learning. This failure to use concrete or visual materials may be partly because teachers are not aware of their importance and value, or that they feel the students are too mature to need them as an aid to learning and may reject their use in lessons. It is certainly possible to perform algorithms correctly as a result of rote-learning tricks, but this does not provide a secure foundation on which to develop the flexible mathematical thinking required in solving problems. When concrete materials and visual aids have been phased out too soon, many students lack a real understanding of the relationship between multiplication and division. Applying what we know of the cognitive development of typical adolescents, it is evident that visual representation is still needed by students when learning these concepts and processes (Education Endowment Foundation, 2017; Flores et al., 2014; Zhang et al., 2014).

Number Facts: The Importance of Automaticity Number facts are all the simple relationships that exist among small numbers— examples are 7 + 3 = 10; 10 – 3 = 7; or 2 × 7 = 14; 14 ÷ 2 = 7. Number facts are so fundamental that children should eventually have them in memory without having to work them out each time they need to apply them. Knowing number facts ‘by heart’

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is partly a matter of learning them through practice, and partly a matter of grasping a principle (e.g. if 7 + 3 = 10 then, 7 + 4 must be ‘one more than ten’, etc.). By the end of primary school, children should recall all basic number facts with a high degree of automaticity, because this allows them to deal swiftly and effectively with each step within a calculation (Morano et al., 2020; NCETM, 2016a; Riccomini et al., 2017). The students are able to focus attention entirely on the higher-order processes involved in working through a problem. The website for the National Center for Excellence in Teaching of Mathematics has a useful table indicating the facts that should be taught in Year 1 and Year 2 (see Online Resources); and a publication from the Association for Supervision and Curriculum Development, titled Math fact fluency: 60 + games and assessment tools to support learning and retention (ASCD, 2019) is also of value. Fluent computation skills rely on swift and accurate recall of basic number facts to 20 and multiplication facts to × 12 (Garza-Kling, 2011; Gross et al., 2016). Memorizing multiplication tables is still vitally important, because easy retrieval of these facts is essential for success in most secondary school mathematics topics (Poast et al., 2021). The mastery mathematics curriculum used in Shanghai, where students’ achievement levels are very high, suggests that tables should be taught in this order: × 10 × 5 × 2 × 4 × 8 × 3 × 6 × 9 × 7 (NCETM, 2015). By the middle of Year 2 students are expected to recall and use multiplication and division facts for the ×2, ×5 and ×10 multiplication tables. They can solve problems involving these processes using recall of table facts. At this stage, they learn that, as with addition, multiplication of two numbers (e.g. 5 × 3) can be performed as 3 × 5 (commutative principle). No doubt, a few students acquire mastery of number facts and computation skills simply through engaging in a discovery learning approach, but it is not at all certain that they achieve the same degree of automaticity in recall as those who devote specific time to deliberate practice (Dennis et al., 2016; Morano et al., 2020). Students with learning difficulties often struggle to memorize number facts, and they require an intervention that includes abundant successful practice (Farkota, 2017; Leach, 2016). In days gone by, children usually achieved this practice through using books in series like Beacon Arithmetic (Ginn, 1950) and ‘speed and accuracy worksheets’, but today it is easy to find online programmes and apps designed for precisely this ‘drill and practice’ purpose. An interesting simple approach to improving fluency with multiplication facts was used by Sides and Cuevas (2020) with students in 3rd and 4th grade. They required the children to set themselves achievement goals each week (the number of facts to be completed accurately within five minutes) and to plot their results on a graph. They found that students involved in setting goals showed an increase in their scores for multiplication facts. There is evidence that fluency in recall can be improved significantly when students undertake additional practice through online number games and exercises (Cozad & Riccomini, 2016; Hawkins et al., 2017). It also appears that self-initiated practice—that is practice that students themselves decide to do to improve their own skills—is more effective than rote practice imposed by the teacher (Lehtinen et al., 2017).

Other Topics in Primary Mathematics

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Other Topics in Primary Mathematics During Key Stage 1 in England, students are also introduced to common fractions (½, ¼, ¾) and simple decimal fractions. They learn how to calculate with money (+, –, x, ÷), how to work with time and how to interpret and construct simple data tables and charts. Measurement skills are further refined, and students use measurement data to calculate and solve problems. Students at Key Stage 2 (age range 7–11 years) are taught to use their number skills to solve an increasingly broad range of word problems, including some that involve simple fractions and decimals. When students reach the end of Year 6 (age 11+), a considerable amount of learning will have taken place and they can now attempt problems that require several steps to obtain a solution. Students will be proficient in multiplying a 4-digit number by a 2-digit whole number (long multiplication), dividing numbers up to 4 digits by a 2-digit whole number (long division) and interpreting any remainders as fractions. By the end of primary school students should also be able to use common factors to simplify fractions, use common multiples to express fractions in the same denomination, compare and order fractions and mixed numbers, add and subtract fractions with different denominators, and identify the place value of each digit in numbers with up to three decimal places. They should also understand how decimal equivalents can be substituted for a common fraction (e.g. 0.75 for ¾) or a percentage (75%), use written calculation methods for division where the answer has decimal places; and can solve problems that require answers to be rounded to a specified degree of accuracy. Research over many years has discovered that working with fractions is one area of numeracy that many students find very difficult (Braithwaite et al., 2019); but they can be helped significantly when the teaching deliberately involves engaging with concrete and visual representations and is accompanied by discussion (Roesslein & Codding, 2019). For a more detailed summary of expected outcomes from primary mathematics, see Mathematics programmes of study:Key stages 1 and 2 (DfE, 2013a). The Australian Curriculum has very similar content and expectations for children up to age 12. Readers are referred to the Australian Curriculum: Mathematics for details (ACARA, 2015).

The Role of Language It has been stressed already that language input from adults and peers is essential for helping students interpret learning experiences. It is stated in the preamble to the Programme of study for mathematics in England that the vocabulary children hear and speak at home and in school is a key factor in developing their mathematical thinking (DfE, 2013a). For example, language input from knowledgeable others introduces new math-related vocabulary in a meaningful context. When that new vocabulary is

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understood, internalized and used in communication by a student it becomes the raw material for thinking and reasoning (Donlan et al., 2007; PDST, 2017). Concepts and procedures that are stored as linguistic representations in memory, as well as memories of concrete experiences and visual representations, provide the foundation for mathematical reasoning (Negen & Sarnecka, 2012). The National Centre for Excellence in Teaching of Mathematics (NCETM, 2016b) indicates that children should not only hear mathematical vocabulary used in context by their teacher but should also be encouraged to use that language themselves when explaining their thinking and when answering questions. NCETM even advocates requiring children at first to answer oral questions in full statements, rather than with one word or a number. For example, the teacher asks ‘How many 10s make 60?’ The child should answer ‘6 tens make 60’. ‘How many degrees are there in a right angle?’ The child replies ‘There are 90 degrees in a right angle’. This helps children store that information most effectively in memory and can then recall it easily when solving problems. Requiring students to use mathematical language in this way is an important component in the Mastery Approach. A video from NCETM illustrating the approach is listed under Online Resources. The method used for teaching mathematics in Asian countries places great importance on oral questioning, discussion and precise use of vocabulary by students and their teachers (Shockey & Pindiprolu, 2015). Asian teachers require their students to verbalize their thinking when attempting to solve a problem as part of the ongoing learning process. This appears to be a very effective way of developing their mathematical thinking (Powell et al., 2019). Given that clear communication is a key factor in building students’ understanding, it is important for teachers to evaluate the clarity and accessibility of their own use of language when explaining, questioning, answering and talking with students. Boulet (2007) has observed: … it is not sufficient to tell teachers to be more sensitive to the language used in mathematical conversations; addressing specific instances of language commonly used in the mathematics classroom helps teachers identify more clearly what sort of language can be a source of difficulties and helps them understand why such language must be adapted in order to make it more meaningful (p.11).

In Japan, it is common practice for teachers to collaborate in a professional development activity known as ‘lesson study’ (Elliott, 2019; Murata & Takahashi, 2002). Clarity of explanations by the teacher is a common focus in such lesson study, alongside suitability of material and learning activities. The teachers observe each other teaching a particular math lesson and then provide detailed feedback on how the lesson could be improved next time it is taught. Lesson Study has also been introduced as a professional development activity in several other countries, and for other subject areas (Lee, 2019; Leeming, 2018; Yalcin, 2019). An approach with some similar characteristics in the US is the Better math teaching network with a focus on topics such as algebra for students in grades 9, 10 and 11 (Bayerl, 2020). The aim is to help practitioners develop and test instructional routines together to improve the effectiveness of their approach (see Chap. 7).

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Technology Supporting Numeracy Learning Many teachers are allowing students to use handheld devices and apps as a supplement to regular exercises in math to increase practice in calculating and problemsolving. Using technology along with well-designed worksheets and textbook exercises constitutes an effective way of reinforcing computation processes (Dostal & Robinson, 2018; Hawkins et al., 2017; Kaur et al., 2017). It has been confirmed that technology used in this way can be very successful in holding learners’ engagement and motivation, and can reduce the achievement gap that exists between students who struggle with basic math and their higher-achieving peers (Cullen et al., 2020; Lysenko et al., 2016; Ng, 2016). Findings from existing studies on mathematics teaching with technology have suggested that it yields good overall achievement gains and is most effective when combined with teacher-led instruction (Outhwaite et al., 2019; Slavin et al., 2009; Zaranis, 2016). The most common form of technology in the mathematics classroom is, of course, the handheld calculator or calculator contained within other devices. In recent years, more math textbooks are incorporating calculator use in the tasks and problems they set for students (Sherman et al., 2020). The calculator has proved to be a boon for many students, allowing them to complete more work and spend more time rather than less time on problem-solving (Yakubova & Bouck, 2014; Yang & Lin, 2015). Calculators have been of particular value for students of high ability, enabling them to tackle complex and challenging problems and explore mathematical ideas more deeply (Kissane, 2017). Calculators with voice-input and audio-output have proved invaluable for students with sensory impairment (Bouck et al., 2011). All these benefits need to be weighed against fear that if calculators become too commonplace in primary classrooms, students may feel that there is no need to understand the complex operations because they can so easily perform them by pressing buttons. For this reason, the Department for Education in England has stated: Calculators should not be used as a substitute for good written and mental arithmetic. They should therefore only be introduced near the end of Key Stage 2 to support pupils’ conceptual understanding and exploration of more complex number problems, if written and mental arithmetic are secure. In both primary and secondary schools, teachers should use their judgement about when ICT tools should be used (DfE, 2013a).

NCETM has produced a useful online set of notes on appropriate use of calculators: www.ncetm.org.uk/public/files/5050113/pri_frmwrk_ma_gd_calculator.pdf.

Teaching Problem-solving Numeracy requires that an individual develop fluency in carrying out calculations— but it goes without saying that proficiency in calculating is only valuable if it is used for authentic purposes in problem-solving and investigation (Education Endowment Foundation, 2017; Wong & Ho, 2017b). Studies have shown that individuals with

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higher numeracy competencies are more likely to have a higher level of problemsolving skills (Xiao et al., 2019). It has also been found that when students lack adequate proficiency in number skills, they are much more likely to develop stress and anxiety when faced with problem-solving in mathematics (Khoshaim, 2020). In the early part of this chapter, coverage of computation and number facts may have created the impression that children need to acquire a full range of arithmetic skills before they can even begin to engage in any ‘real’ problem-solving. This is certainly not the case, and arithmetic skill development needs to go hand in hand from the start with the immediate application of those skills for solving age-appropriate problems (Diamond, 2018). The topic of problem-solving and how to teach it is vast, so it is beyond the scope of this book to go into problem-solving strategies in great detail. It can also be argued that higher-order problem solving often involves concepts and strategies that go beyond the generally accepted boundaries of what constitutes everyday numeracy. The sections below provide a brief summary of the general issues involved in helping children approach math problems strategically and with confidence. First, it is necessary to differentiate between ‘routine’ and ‘non-routine’ problems. Routine problems are usually not complex and simply require the application of a pre-taught algorithm to obtain a solution. An example would be to calculate change from a $20 note after buying three small items at the supermarket. Non-routine problems are ‘messy’ in the sense that the steps required to reach a solution are not immediately obvious, one is not sure at first which bits of available information are important and which are not. Solving a non-routine problem is rarely as easy as simply applying a pre-taught algorithm; often it will require multistep calculations to obtain a final result. An example of a non-routine problem requiring several steps and calculations for the solution would be: Jane and Frank have just moved into an old flat. They need to replace the wall-to-wall carpet which is torn and stained, but they think it may be cheaper to use floor tiles instead. They measure the size of the room and find that it is 5m x 6.5m. Carpet costs $66 per sq.m. Floor tiles each measure 25cm x 25cm and cost $6.00 each. Will it be cheaper to buy new carpet or use floor tiles instead? Compare the prices.

There are logical steps that can be taught to students so that they approach a problem like this with a self-regulated strategy in mind (Johnson et al., 2021; Losinski et al., 2021; Popham et al., 2020). First, the problem needs to be read carefully to identify what is being asked and where to begin a solution. Reading comprehension is, therefore, involved in interpreting math problems presented in books, so any intervention for problem-solving needs to include comprehension strategies such as looking for keywords, finding connections, ignoring redundant information, concentrating on the main idea and re-reading several times (Fuchs et al., 2018). For many students, drawing a rough sketch of the room and the tiles described in the problem above may be of assistance. Students then need to select and perform the appropriate algorithms. These steps involve mainly cognitive processes, but students also need to use metacognitive skills such as self-monitoring and self-correction. Even though mathematical problem-solving today often involves the use of a calculator, students must still be able to judge the feasibility of any result they obtain.

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The list below identifies some of the self-directing questions that an individual could ask when approaching a problem. • • • • • • •

Can I picture this problem in my mind? Can I draw it? (visualization) What needs to be worked out? How will I try to do this? (identify the operations and steps required) Is this working out OK? (self-monitoring) How will I check if my solution is correct? (reflection and evaluation) Is my answer reasonable? (judgment and common sense) Perhaps I need to correct this error and then try again (self-correction).

Students who find this process difficult need help in sifting the relevant information from the irrelevant, identifying exactly what the problem requires and deciding the best way of obtaining and checking the result. To achieve this outcome, direct teaching using routine problems in the early stages is a necessary first step toward later independence (Morgan et al., 2015). Much of what is known about effective evidence-based teaching has direct implications for teaching problem-solving strategies; in other words, a teacher or tutor needs to provide students with the following forms of direct guidance: • modelling and demonstrating effective steps in solving a particular routine or non-routine problem; • ‘thinking aloud’ while identifying and analysing key aspects of the problem; • ‘thinking aloud’ while selecting and applying appropriate procedures to obtain a solution; • reflecting upon the effectiveness of the procedure and the plausibility of the solution obtained. Once students have been shown an effective strategy for a particular type of problem, they need many opportunities to apply it themselves under teacher guidance with feedback. Finally, they are able to use the strategy independently and to generalize its use to other problems. If all students are to develop effective problem-solving skills, time must be made available for discussing their ideas, comparing suggestions and reflecting on the most feasible methods of solution. Group work in problem-solving can be an effective approach to use quite frequently, because students can share ideas and learn from one another. Students should be encouraged to reflect upon and justify verbally the procedures they use, not merely carry out the steps by rote. The use of video tapes of realistic situations that involve measurement, spatial reasoning and calculation can be of value when teaching high school students (Rasiman et al., 2020). It is said that students who learn to ‘think mathematically’ and monitor their own problemsolving strategies show the most improvement over time. This approach is typical of the teaching that is highly successful in Asian classrooms.

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Indigenous Children’s Number Skills In Australia, it is well established that a gap exists between the basic number skills of Indigenous and non-Indigenous children (Leder & Forgasz, 2014; Warren & Miller, 2010). This gap tends to widen as the children grow older and have to cope with the content of conventional school mathematics curricula. The same appears to be true of indigenous children in most other cultures, with a similar situation noted for example in the United States, Papua New Guinea, Canada and Chile. In many countries, steps have been taken by researchers and teachers to develop what are termed culture-based materials and strategies for mathematics teaching (Aikenhead, 2017; Yao, 2016). This involves creating real-life number situations with which the children are already familiar, and then building concepts and skills carefully and sequentially from what they already know. Some of the difficulties Indigenous children experience in the early years can be due to differences in the way that processes for counting, measuring, sorting and comparing are traditionally carried out in that particular culture (Edmonds-Wathen et al., 2019). There is then a significant mismatch between what preschools and primary schools are trying to teach and the out-of-school knowledge of the children (Pinxten, 2016). Other problems can be related to teaching that uses mathematical language not easily understood within the children’s own vernacular vocabulary (Edmonds-Wathen et al., 2019; Watts et al., 2019). When teaching Indigenous children, it is clearly essential that teachers and classroom assistants use ‘local’ terms and examples that children can relate to easily, and at the same time carefully teach the terms that will be encountered in the curriculum. Using familiar materials, context and language is particularly important also when conducting an assessment of the children’s learning, if accurate evaluation is to occur (Siemon et al., 2004). An intervention programme in Australia (QuickSmart Numeracy) has proved to be effective in helping Indigenous students improve their number skills and confidence (Graham & Pegg, 2010). The QuickSmart program (see Chap. 7) uses an explicit teaching approach, with many practice opportunities for students to increase fluency (automaticity) in applying number skills. The designers’ guiding principle is that building fluency and confidence enables students to devote more cognitive effort to the higher-order processes involved in solving mathematical problems. In all forms of intervention, there is a priority need to help Indigenous learners feel more successful, because increased confidence fosters increased motivation and self-efficacy. The next chapter considers those aspects of secondary school mathematics that continue to build a student’s numeracy. Attention then turns to issues associated with adult numeracy.

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Online Resources Mathematics programmes of study: Key stages 1 and 2 in the National curriculum in England. Department for Education [UK] (2013). https://www.gov.uk/government/uploads/system/upl oads/attachment_data/file/335158/PRIMARY_national_curriculum_-_Mathematics_220714. pdf. Australian Curriculum: Mathematics. http://www.australiancurriculum.edu.au/mathematics/curric ulum/f-10?layout=1. Common Core State Standards: Mathematics [US]. http://www.corestandards.org/Math/. Details of the Maths Teaching for Mastery approach promoted by the Centre for Excellence in Teaching of Mathematics in England can be located at: https://www.ncetm.org.uk/files/37086535/ The+Essence+of+Maths+Teaching+for+Mastery+june+2016.pdf. NCETM website also has many invaluable teaching resources at: https://www.ncetm.org.uk/ and at: https://www.ncetm.org.uk/resources/50639?utm_source=NCETM%20Newsletters&utm_ campaign=221deba1e4National%20Newsletter%20February%202018&utm_medium=email& utm_term=0_13f8d631f4-221deba1e4-221395505. Ideas for learning multiplication tables: http://www.australiancurriculumlessons.com.au/2017/ 01/20/learning-multiplication-table-lesson-mastering-multiplication-facts/. http://www.australia ncurriculumlessons.com.au/2014/08/23/multiplication-division-teaching-ideas-activities/. A video that illustrates children answering oral questions with complete sentences can be found on the NCETM website at: https://www.ncetm.org.uk/resources/49824.

Print Resources Barrett, J., Cullen, C., Behnke, D., & Klanderman, D. (2017). A pleasure to measure! Tasks for teaching measurement in the elementary grades. Reston, VA: National Council of Teachers of Mathematics. Cotton, T. (2021). Understanding and teaching primary mathematics (4th ed.). Routledge. NCETM (National Centre for Excellence in Teaching Mathematics). . (2015). Calculation guidance for primary schools. NCETM. NCETM (National Centre for Excellence in Teaching Mathematics). . (2017). Five big ideas to develop mastery. NCETM. Neagoy, M. (2017). Unpacking fractions: Classroom-tested strategies to build students’ mathematical understanding. Reston, VA: National Council of Teachers of Mathematics. Newton, N., Record, A. E., & Mello, A. J. (2019). Fluency doesn’t just happen with addition and subtraction: Strategies and models for teaching the basic facts. Routledge. Small, M. (2019). Math that matters: Targeted assessment and feedback for Grades 3–8. Teachers College Press.

Chapter 5

Secondary School Years

It was indicated in the previous chapter that primary school is where mathematics lessons should have ensured that students develop confidence and fluency in basic numeracy skills and competence in age-appropriate problem-solving. These are the capabilities that will serve them well in their secondary school years and beyond. The majority of students transitioning from primary to secondary school will have mastered the essentials of arithmetic and measurement, and can solve routine and non-routine problems. They can also interpret data—for example, graphs, bar charts and tables—and can work with weights, measures and shapes. These areas of proficiency are necessary prerequisites for successful entry into secondary school mathematics. Secondary school students are of course now introduced to more challenging content that helps advance their conceptual understanding and their application of procedural skills for negotiating more complex mathematical problems. It is now widely agreed that, while effective direct teaching is still essential, involving secondary school students in math-focused activities, interactive games, discussions and reflection within math lessons needs to become a major component in classrooms (Winters, 2020). The role of teachers in secondary schools is to continue to encourage mathematical thinking, rather than teaching tricks that will simply help students obtain correct answers in examinations. The National Numeracy Organization website includes the important statement: ‘Being numerate is as much about thinking and reasoning logically as about doing sums’ (National Numeracy Organization, 2020). It appears that a strong predictor of success in secondary school mathematics is whether or not a student has good ‘relational reasoning’ and can ‘think mathematically’ (Blums et al., 2017; Dumas et al., 2013). Relational reasoning can be defined as the ability to discern meaningful patterns and connections within a set of quantitative information. There will, of course, be some students who transition from primary school with poorly developed numeracy skills and who will experience difficulties when they encounter new concepts and processes in secondary school. These students will require extra support. This difficulty often relates particularly to learning algebra and © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 P. S. Westwood, Teaching for Numeracy Across the Age Range, SpringerBriefs in Education, https://doi.org/10.1007/978-981-16-3761-2_5

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solving algebraic equations (Ferretti, 2020). A later section of this chapter addresses issues associated with learning difficulties and how best to provide support.

Secondary School Mathematics Mathematics is defined as the discipline that deals with the logic of quantity, arrangement and shape. As explained in Chap. 1, mathematics is a much broader field of study than basic numeracy and arithmetic, but these number skills are obviously essential for engaging effectively with all aspects of secondary school mathematics in both applied and theoretical forms (Xin, 2019). In England, the Department for Education suggests that: Mathematics is a creative and highly interconnected discipline that has been developed over centuries, providing the solution to some of history’s most intriguing problems. It is essential to everyday life, critical to science, technology and engineering, and necessary for financial literacy and most forms of employment (DfE, 2014a, p. 1).

The main purpose of teaching mathematics in secondary schools is to build firmly on the concepts and skills established in the primary school and to equip learners with useful new strategies for thinking mathematically and applying mathematics across other school subjects and in real-life contexts (Forgasz & Hall, 2019). Many of the mathematics concepts taught in the secondary school curriculum have direct application for working in areas such as commerce, science, engineering and technology. It is, therefore, important to enhance every student’s understanding of the relevance of mathematics to their future life (Gijsbers et al., 2020). In the UK, the Mathematics Programmes of Study (DfE, 2014a) set out in detail the content that is to be covered in Key Stages 1 to 4 (ages 5–16 years). Similarly, Australian schools are now guided by the Australian Curriculum: Mathematics. In the United States, the Common Core Mathematics Standards provide an outline of what every student should know and be able to do at the end of each school year. Links to these documents can be found under Online Resources at the end of the previous chapter. In the UK, the guidelines for secondary school mathematics Key Stage 3 (ages 12–14) and Key Stage 4 (ages 14–16) indicate that students should be able to apply arithmetic skills fluently to solve problems, understand and use measures, make estimates and check the results of their calculations (DfE, 2014b). In particular, they need to become increasingly skilled in seeing connections within a complex problem, and breaking the problem down into manageable steps. The National Centre for Excellence in Teaching Mathematics stresses the need to foster relational reasoning in ways that are compatible with the students’ cognitive ability level (NCETM, 2017). In this respect, as indicated previously, many adolescents still function at a concrete operational level when faced with non-routine problems in mathematics, and teaching methods need to take this into account. Even at this age level, it is important to make use of hands-on materials, visual representations and whiteboard

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diagrams when introducing new concepts and making connections with prior knowledge. This ensures that even these older students reach genuine understanding by following the necessary learning progression from concrete-to-pictorial-to-abstract. Concrete materials such as models or structural apparatus, and pictorial representations, diagrams, fraction charts, number lines and sketches provide an opportunity for learners to store key information in visual memory and to recall it at appropriate times (Nikula et al., 2020). These visual images, alongside math vocabulary stored in memory, assist students’ recall, thinking and reasoning when solving a variety of problems (Morin et al., 2017; Ozdemir, 2017).

Typical Curriculum Content in Secondary Schools A detailed summary of the content for students in the age range 12–14 years can be found in theMathematics Programmes of Study: Key Stage 3 of the National Curriculum in England (DfE, 2013f). Similar information can be found on the websites for the Australian Curriculum and for the Common Core State Standards (see Online Resources for Chap. 4). In many topics, students will move well beyond what is commonly regarded as basic numeracy—but numeracy still underpins areas of knowledge involved in equations and graphs, calculating with fractions, linear and simple quadratic functions, ratio and proportion and probability and statistics. In the domain of statistics, students in secondary school must become proficient in constructing and interpreting line or bar graphs, frequency tables, diagrams, pie charts and pictograms. At Key Stage 4 (approximate age range 14–16 years), students will continue to build on mathematical knowledge from Stage 3. The detailed curriculum content for this senior secondary age group can be found in Mathematics Programmes of study: Key Stage 4 (DfE, 2014c). In particular, students are expected to extend their understanding of the number system to include powers, roots and fractional indices; and they will frequently use multistep calculations to solve increasingly complex problems. They should extend their knowledge of ratio and proportion (including trigonometric ratios) and begin to use algebra to support and construct arguments and proofs. In all these areas they should be able to demonstrate their ability to reason mathematically. In the US, the Common Core State Standards (CCSS) provide detailed goals for high school mathematics, covering number, algebra, geometry, statistics, probability, linear, quadratic and exponential models, trigonometry and modelling (CCSSI, 2017d). The guidelines for CCSS indicate that successful transition from high school to post-secondary education and later to a career path, particularly in science and engineering, depends heavily on mathematical knowledge and skills acquired in both the elementary school and high school classes. It is stated: … some of the highest priority content for college and career readiness comes from Grades 6-8. This body of material includes powerfully useful proficiencies such as applying ratio reasoning in real-world and mathematical problems, computing fluently with positive

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The Australian Curriculum covers much the same overall content in the secondary school years, but it also has a component known as ‘specialist mathematics’. This segment enables teachers to demonstrate application of mathematical ideas, strategies and models in many other subject areas.

Across the Curriculum It has already been argued that numeracy is an essential capability needed in all subjects across the curriculum. Specialist teachers in areas such as geography, history, economics, civics, the sciences, sport and art must seize every opportunity to present examples of how math skills are often applied in these domains (ACARA, 2016c; DfE, 2013f). Problems and situations involving data, symbols and calculations will be encountered most particularly in the sciences and engineering. Quantitative data in these subjects are often contained within workshop manuals that contain specifications, calculations, scale diagrams, data tables and formulae. Students need to gain experience in interpreting such information.

Real-Life Applications Effective teachers of mathematics at all age levels take every opportunity to use many events occurring regularly in school and the community to provide authentic opportunities to apply numeracy skills. For example, engaging students in budgeting for school trips, ticket sales for concerts, summer camp expenses, profits from bringand-buy sales, fundraising events, sports days, inter-school matches and many more. Teaching for numeracy in motivating ways will make full use of all such naturally occurring events. Secondary school mathematics introduces many new concepts that are not easy to absorb at first; and this situation is not helped by the fact that from the first year of high school math is often taught with too much dependence on textbooks and rote memorization. As Zawojewski & McCarthy (2007) have pointed out that there is often a very serious mismatch between the type of mathematics content taught in many secondary schools and the abilities that are really needed beyond school. Rather than teaching ‘pure math’ at secondary school level (particularly to students of below-average ability) much of the effort should be devoted to teaching the application of numeracy capabilities for real-life purposes (Benson-O’Connor et al., 2019; FitzSimons & Björklund-Boistrup, 2017). There is a place for using project-based learning as part of the secondary mathematics curriculum, to provide

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authentic opportunities for students to seek out, interpret and use math concepts and numbers skills (Viro & Joutsenlahti, 2020). A key focus of projects such as the International Mathematical Modelling Challenge (IM2 C) is making mathematics more relevant and challenging for students. This event involves teams of secondary students from a number of countries attempting to find solutions to real-world mathematical scenarios by using freely available material from the web and other sources (see Online Resources). Over a period of 5 days, teams tackle the given problem, develop relevant hypotheses, test a working solution and finally submit a report. In 2019, the challenge was for teams to use mathematical modelling to determine the Earth’s carrying capacity for human life under today’s conditions and technology. The task was attempted by 57 teams from 33 countries or regions—with teams from Hong Kong, Australia, New Zealand, Poland and Netherlands being judged most successful. An approach known as Enhanced Anchored Instruction (EAI) has gained attention in recent years. The approach presents students with a real-life quantitative problem via a video clip or similar input, followed by a group discussion that leads to suggestions for possible ways of obtaining a solution. During attempted solutions, additional interpretation is provided by the teacher when necessary, leading ultimately to students calculating and checking. The approach has proved to be of value for students across a wide ability range (Bottge et al., 2018; Saunders et al., 2018). Another relevant approach is Realistic Mathematics Education (RME). This approach began in The Netherlands in the 1970s and was taken up later in many countries (Hidayat & Iksan, 2015; Stemn, 2017; Venkat & Matthews, 2019). The approach uses everyday problems to develop students’ mathematical skills. In RME, little or no emphasis is placed on practising calculating skills out of context (it is assumed these will develop through solving the problems), and much more emphasis is placed on ‘thinking mathematically’. It is suggested that RME can be further enhanced by integrating the use of technology to aid collaborative investigation (Bray & Tangney, 2016; Cerratto Pargman et al., 2018). The approach is used mainly in secondary and tertiary education settings, but has also found a place in some primary schools. While there is some evidence that RME in the hands of a skilled teacher can have benefits for many students (Dickinson & Hough, 2012; Hirza et al., 2014), there are also some criticisms that can be made of this form of open-ended problem-solving. The first difficulty is the almost total reliance on non-routine problems as the focus of all learning activities. Students who have not yet acquired basic number skills to an adequate level of automaticity (and this can include some students in secondary schools) are often totally confused when faced with open-ended and challenging quantitative problems (Murphy, 2016). They tend to leave the ‘hard work’ to other students in the group, and learn nothing from the activity. In an effort to facilitate more effective use of RME, the textbook series Making Sense of Maths (Hough et al., 2012) includes suggestions for providing more structure to lessons. A second difficulty with RME arises from the fact that many teachers do not find open-ended approaches easy to use. This is particularly the case with teachers who are having to teach math but who are actually trained for other areas of the

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curriculum. Some teachers feel that their own depth of knowledge in mathematics and problem-solving strategies is inadequate to tackle this work confidently. It is disquieting to learn from statistics released by the Department for Education in the UK that many mathematics classes in secondary school are taught by teachers with no qualification in the subject (Cassidy, 2015). This situation is evident also in Australia, and is likely to be similar in other countries. These teachers are simply not equipped to teach mathematics effectively using an open-ended problem-based approach.

A Role for Technology It has been argued that one improvement that can be made in the teaching of secondary school mathematics is better use of technology, not only by the teacher in the delivery of the curriculum but also by the students within the learning process (Clark-Wilson et al., 2020; Murphy, 2016). The use of technology as a supplement in high-quality instruction has been shown to have a positive effect on students’ engagement in the subject and in improving their understanding (Fokides, 2018; Zaranis & Synodi, 2017). Technology can also be a means of linking school mathematics with the real world through its ability to model problems, provide steps to solve and give immediate corrective feedback (Budinski & Milinkovic, 2017; Xiao et al., 2019). As mentioned previously, the calculator is the most widely used form of technology, but apps available on computer and handheld devices are also readily available for use across the age range (Mowafi & Abumuhfouz, 2021), as are video clips that depict real-life problems needing solution. Teachers’ willingness to use technology appears to depend upon their own technology self-efficacy and the style of teaching they normally employ (Li et al., 2019). If teachers are to be truly effective and up-to-date in using technology in their math lessons, they usually require regular in-service courses and practical advice from advisory teachers. It has been found that a useful strategy for staff development is to show teachers video clips of good practice using technology in the math classroom, rather than merely describing it in lectures and discussing it in workshop sessions (Radloff & Guzey, 2017). Teacher education programs also need to include a component on the use of technology when training teachers of mathematics.

Students Who Struggle with Secondary Mathematics There are many students who find mathematics difficult to learn, but not as a result of any specific disability or learning disorder. They have simply fallen behind for a variety of reasons, such as irregular attendance at school, too many changes of school or poor-quality teaching that results in lack of interest and motivation. Most of these students have to rely entirely on rote learning in an attempt to cope with the secondary school mathematics course and examinations. In large classes, their

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difficulties may remain undetected for many months or even years. This is particularly the case where students have become skilled at hiding their difficulties by copying work from others, failing to hand in homework or not attending school on days when tests are given. These behaviours may conceal a high level of anxiety in relation to learning mathematics (Szczygiel, 2020b). In addition to these students with general learning difficulties, a few other students do present with problems that are due to a specific disability affecting their learning of mathematics. This disability has been termed dyscalculia and is regarded as a disorder similar to dyslexia in the literacy domain. The estimated prevalence rate for dyscalculia is no more than 3 per cent of the school population, but for these students the negative effects of a mathematics disability can be very significant, affecting their self-esteem and limiting their choice of career path. Dyscalculia in younger children causes them to be much slower to develop number sense; and in primary school, it results in difficulties acquiring computation skills. As they get older, they have a problem remembering algorithms, switching from one process to another and thinking mathematically. Their bookwork is usually slow, inaccurate, often untidy and incomplete. In secondary school, they struggle most in classrooms where mathematics teaching lacks clear guidance and explicit instruction and proceeds at a fast pace. Students with a learning disability in mathematics have been shown to experience math anxiety at rates nearly double those of their typically-developing peers (Johnson et al., 2021). In some high schools, these students with a specific disability now find themselves in mixed-ability inclusive classes, where it is difficult for their teachers to provide the additional individualized support they require. The students are constantly frustrated and humiliated by their learning difficulty, and it is not surprising that many give up hope of improving. However, despite the severity of their learning problem these students can improve very significantly if provided with intensive systematic instruction (Witzel & Mize, 2018). Although students with a specific disability in mathematics have just been distinguished from those who are simply low achievers in the subject, it is important to indicate now that there is a significant overlap between the profile of difficulties exhibited by both groups. There is, therefore, no ‘special’ approach to teaching number skills to students with a disability that is markedly different from that needed by others who are low achievers—both groups need explicit instruction, explanations and demonstrations that lead to understanding, and the opportunity to practice basic skills to the point of mastery. In all cases of difficulty in learning, the planning of effective intervention should first involve the assessment of each student’s existing number knowledge and functional skills. This is necessary in order to detect any gaps and misconceptions in prior learning, and as an aid to planning individualized teaching so that it can begin at an appropriate level (Karagiannakis et al., 2016). It is always important to identify not only the student’s current math abilities but also affective and environmental factors that may be influencing his or her motivation (Petersen & Hyde, 2017). These factors include: students’ intrinsic interest in mathematics, their self-efficacy beliefs for learning math successfully, anxiety, their understanding of the potential value of

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math in everyday life and in employment and the amount of support available from home. This individual diagnostic assessment approach does not come easily to some mathematics teachers in secondary schools who are often under much pressure to take the whole class of students through the curriculum at a brisk pace, with examinations in mind as an endpoint. They do not find time for individualized assessment or support, and tend to simply accept that some learners will ‘fall by the wayside’.

Providing Support Most students who struggle with mathematics will not improve unless additional tutoring is given. This type of one-on-one or small-group instruction may be provided by a teacher, a paraprofessional or a volunteer, and has been described as ‘one of the most versatile and potentially transformative educational tools in use today’ (Nickow et al., 2020, p.1). Peer tutoring in mathematics (students teaching students) has also produced academic benefits for school students and particularly for those in higher education (Alegre et al., 2020). Tutoring may be organized within school hours or after school, and is usually regarded as Tier 2 (small group) or Tier 3 (individual intensive) intervention. In some countries, parents who can afford the expense often employ a private tutor. Tutoring programs of all types have been found to yield a very positive impact, with strong improvements for both reading and math in early primary school, and mathematics in secondary school years. Experience over many years has indicated that positive effects of tutoring and intervention are most evident when parents are also willing and able to support students’ homework efforts and encourage extra study at home (Kalena, 2018; Silinskas & Kikas, 2019). According to Ngo & Kosiewicz (2017) the variables that determine students’ rate of improvement in tutorial groups (and in inclusive classrooms) appear to be: • quality of instruction, as reflected in the clarity, sequence and structuring of lessons, and to the constructive feedback and support provided to students; • ability of the teacher to assess the needs of individual students and then tailor instruction to meet these needs; • amount of time available each week for guided and independent practice. Teachers need to be able to look at each new math topic they are teaching from the perspective of a learner meeting it for the first time, and then breaking lessons into manageable steps to help students learn the content most easily and avoid misconceptions. Teachers’ ability to organize curriculum content into an effective sequence for learning in this way has long been referred to as ‘pedagogical content knowledge’ (Shulman, 1987). More recently in the mathematics domain, it is termed ‘mathematical knowledge for teaching’ (MKT) (Edelman, 2017). Unfortunately, MKT is frequently lacking in any teacher who is obliged to teach mathematics but is not trained in that specific subject, and where math is not even a personal interest.

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When tutoring is not available, students who struggle with math in secondary school still tend to find themselves in a low-ability group, and they follow a curriculum often described as ‘developmental’, ‘remedial’ or ‘catch up’. In these groups, instruction tends to rely on simply re-teaching and rote memorization of arithmetic procedures, rather than seeking to improve students’ deeper mathematical understanding and interest (Bickerstaff & Edgecombe, 2019). Many secondary schools still use ability grouping of this type for mathematics, even when all other subjects in the curriculum are now taught to mixed-ability inclusive classes.

Concerns Over Numeracy Standards of Senior Students Numeracy standards were discussed briefly in Chap. 1, but here it is important to address the growing concern regarding numeracy of senior students and schoolleavers. This concern was spurred by disquieting data from surveys by the Organization for Economic Co-operation and Development (OECD, 2001, 2016c; Thomson et al., 2017) and a report called A fresh start: Improving literacy and numeracy (Moser, 1999). Other data came from the 2003 Skills for Life Survey (Grinyer, 2006), the National Assessment of Educational Progress data in the United States (Institute of Education Sciences, 2007), in the UK the OfSTED (2011) report, Tackling the challenge of low numeracy skills in young people and adults. An Australian study conducted in 2017 found that too many adolescents have a poor understanding of such things as interest rates and how long it would take to pay off a credit card debt. It is also relevant to note that in Australia the number of students who now take intermediate and advanced mathematics courses in senior school is in serious decline, yet these subjects provide the foundation for studying science and engineering at university and then opting for a career in these fields (Miller et al., 2018; Proctor, 2015). Clearly, the teaching of mathematics in secondary schools still needs to improve. It is now recognized that assessment methods used in secondary schools need to be strengthened, so that each learner’s progress is monitored effectively and individuals who struggle with mathematics are identified early and provided with well-targeted instruction and support (Education Endowment Foundation, 2017).

The Way Ahead The teaching of numeracy to learners of secondary school age requires a careful balance between ensuring that computation skills are mastered and that individuals also become competent in applying these skills to problem-solving and to real demands of everyday life. Some of the reforms suggested for mathematics teaching in recent years have tended to devalue computation and instead advocate for a more open-ended, investigative and problem-based approach. The ongoing poor standards

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of many school-leavers would surely indicate that a suitable balance in school mathematics teaching is yet to be achieved. The ‘math wars’ (Schoenfeld, 2004) still continue between those who firmly believe that computation skills need to be taught explicitly and practiced to mastery level (sometimes referred to as the ‘mechanistic approach’) and those who believe that such skills will be acquired simply as a result of engaging in interesting problem solving and investigation (the ‘immersion’ or ‘whole math approach’). As Schoenfeld (2004) indicated, using a teaching approach that does not combine both mastery of computation skills and everyday application of those skills, makes no sense at all. The number of students still leaving school with poor basic numeracy has implications for providing help to adults. The following chapter addresses this issue and considers how best to teach these older learners.

Online Resources A paper on the National Center for Excellence in the Teaching of Mathematics website explains the importance of retaining use of concrete and pictorial materials in secondary school mathematics lessons to help bridge the transition to abstract reasoning: https://www.ncetm.org.uk/resources/ 50199. For five ‘big ideas’ that should guide the components included in interventions see Essentials of numeracy for all at: https://www.nationalnumeracy.org.uk/sites/default/files/essentials_of_nume racy_download_v2.pdf. International Mathematical Modelling Challenge (IM2 C). For details see: https://www.immchalle nge.org.au/about-the-immc. For useful practical advice on helping secondary school students and adults master the basics of number see: https://www.nala.ie/tutors/top-tips/learning-materials/numeracy Mathematics Interventions: What strategies work for struggling learners or students with learning disabilities? https://educationnorthwest.org/resources/mathematics-interventionswhat-strategies-work-struggling-learners-or-students-learning. Standards Unit (UK): Improving Learning in Mathematics. Online at: https://www.stem.org.uk/eli brary/collection/2933.

Print Resources Bickerstaff, S., & Edgecombe, N. (2019). Teaching matters and so does curriculum: How CUNY Start reshaped instruction for students referred to developmental mathematics. CCRC Working Paper No. 110. New York, NY: Community College Research Center, Columbia University. Chinn, S. (2021). The trouble with maths (4th ed.). Routledge. Lee, C., & Ward-Penny, R. (Eds.). (2019). A practical guide to teaching mathematics in the secondary school (2nd ed.). Routledge. Reys, R., et al. (2017). Helping children learn mathematics (2nd ed.). Wiley. Sherman, H. J., Richardson, L., & Yard, G. J. (2019). Teaching learners who struggle with mathematics (4th ed.). Waveland Press.

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Swan, M. (2006). Collaborative learning in mathematics: A challenge to our beliefs and practices. National Institute for Advanced and Continuing Education. White, J. (2018). Mental math: How to develop a mind for numbers, rapid calculation and creative math tricks. Amazon Digital Services LLC.

Chapter 6

Adult Numeracy

Concerns Over Adult Numeracy Standards Surveys in several countries have revealed that a surprisingly large number of adults lack adequate numeracy skills (OECD, 2013a; Mamedova & Pawlowski, 2020; Steinke, 2015; Yu et al., 2019). This appears to be particularly true of younger adults, who are often not as competent in working with numbers as those who are significantly older (OECD, 2016e). Data from an OECD survey suggests that England is the only country in the developed world where the older generation approaching retirement is more numerate than younger adults. This must surely raise concerns about the effectiveness of numeracy teaching in schools today, and the need to strengthen the professional skills of teachers at pre-service and in-service levels. In the UK, alarm was triggered by evidence indicating that some 7 million individuals aged 16 to 65 exhibit very poor numeracy skills, and that this had major impact on their employability (Moser, 1999; Parsons & Bynner, 2005). In 2014, it was confirmed that an alarming number of adults in the UK still have low numeracy skills, despite efforts to provide them with opportunities to ‘catch up’ through additional teaching (House of Commons Business, Innovations and Skills Committee, 2014). In 2020, it was reported that still only one in four Britons of working age are ‘functionally numerate’ (Barrett, 2020). In the US, it has been found that 35 per cent of senior students leaving school were scoring ‘below basic level’ in the math component of the National Assessment of Educational Progress test (NAEP) (American Institutes for Research, 2006). The situation does not appear to be improving much, because in 2017 it was reported that two out of every six adults in the US cannot handle everyday number situations effectively. These everyday demands include such basic skills as understanding bus timetables, interpreting credit card conditions, working a cash register and giving change (COABE, 2017). Similarly, in 2020, the National Center for Education Statistics reported that one in three US adults has difficulty completing such tasks as calculating with whole numbers and percentages, estimating numbers or quantity and interpreting simple statistics (Mamedova & Pawlowski, 2020). Also in the US, © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 P. S. Westwood, Teaching for Numeracy Across the Age Range, SpringerBriefs in Education, https://doi.org/10.1007/978-981-16-3761-2_6

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Bickerstaff and Edgecombe (2019) have remarked that even in the higher education sector, universities are struggling to cope with the sizeable proportion of students who are deemed ‘academically underprepared in mathematics’. The international survey known as Adult Literacy and Life Skills (ALLS) introduced in 2003 was first applied in the United States and several other countries. The survey conducted in 2006 found that 49.8 per cent of individuals aged 15 to 64 in Australia (approximately 6.8 million persons) had not attained skill Level 3 in numeracy. Level 3 is considered to be the minimum level needed by individuals to meet the demands of work and life in modern economies. Tasks at this level require the adult to demonstrate an understanding of mathematical information represented in a range of different forms such as numbers and symbols, maps, graphs, texts and drawings. They must also demonstrate an ability to interpret data and statistics as embedded in the everyday text (e.g. newspaper reports). At Level 3, these tasks commonly involve no more than basic number processing to solve very routine problems—something that is supposed to be achieved by the end of primary school. These national and international surveys reveal less than satisfactory numeracy standards in the general adult population, with some groups such as migrants with a history of interrupted schooling or who lack English language being particularly vulnerable (Gal et al., 2020; Jurdak, 2020). This is despite the fact that most developed countries are now devoting more resources to improving literacy and numeracy standards. The Programme for the International Assessment of Adult Competencies (PIAAC) was introduced in various countries in 2011 to assess the literacy and numeracy skills of working-age adults. The survey collects information on how often adults undertake a range of specified numeracy-related activities in their work setting and in everyday life (Coben et al., 2016; OECD, 2013b; Yu et al., 2019). PIAAC is conducted at 10-year intervals, the first survey having been undertaken in 2011. Some countries involved in the survey also included within the PIAAC survey a specific component on ‘problem solving in technology-rich environments’. Data from the survey was analysed by Coben et al. (2016) and provides interesting insights into which particular math concepts and skills are used most in the working world. This information can be useful to tutors who are providing adults with additional career-related math teaching or are embedding such instruction within vocational courses (Braaten, 2017; Coben & Alkema, 2017; Ginsburg, 2017).

Numeracy in the Workplace It is important to highlight the role that numeracy plays in many types of employment (Braaten, 2017; Duchhardt et al., 2017; Russell, 2018). In relation to this issue Foster and Beddie (2005, p.4) remarked: The workplace affords opportunities for workers to develop and use numeracy skills. Ensuring that people have the appropriate level of numeracy is particularly important in jobs that involve a risk to public safety and the environment.

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A study by Lane and Conlon (2016) found that when workers’ numeracy skills are improved there is a narrowing of the gap in opportunities for promotion that typically exists between individuals with different levels of formally education. There are also positive signs that numeracy courses for adults can yield good results, not only in terms of skill acquisition but also in social and personal gains (Dymock, 2007; Wright, 2016).

Efforts to Improve Standards Adult Numeracy The problems associated with poor numeracy in adults were discussed briefly in the first chapter. The discussion here will focus on strategies for teaching and supporting adult learners. It has been stated by the House of Commons Business, Innovations and Skills Committee in England (2014) that: The ability to gain literacy and numeracy skills should be considered a fundamental right of all adults. Improved skill levels contribute to the social and economic wellbeing of individuals and the country as a whole (p. 4).

During the first decade after the year 2000, many new strategies were introduced in UK, Australia and the US in an endeavour to raise the standard of numeracy in the adult workforce. In the UK, the Qualifications and Curriculum Authority (QCA, 2000) introduced the National Standards for Adult Literacy and Numeracy; and in 2001 the DfEE prepared a policy Skills for Life: The national strategy for improving adult literacy and numeracy skills. The Basic Skills Agency developed a guide titled Adult Numeracy: Core Curriculum (2001). The goal of this curriculum was to ensure that adults could interpret quantitative data and use basic mathematical information, calculate and manipulate numbers accurately, and communicate mathematical information to others. In addition to basic arithmetic skills, attention was to be given in the curriculum to working with fractions, decimals, percentages, measures, shapes and space. These areas of numeracy are also the focus in online materials available from the BBC in England (see Online Resources). In Australia, the urgent need to address adult standards in numeracy has been the focus of the National foundation skills strategy for adults (COAG/SCOTESE, 2012). The strategy set a target that by 2022 at least two-thirds of working-age Australians will have the literacy and numeracy skills needed to operate effectively in the workforce. It is recognized that competence in numeracy is increasingly important in this technological world and is an important influential factor for individuals’ employability and life chances (Duchhardt et al., 2017; Sulak et al., 2020). The authorities in New Zealand introduced the Workplace Literacy and Numeracy Fund which enables around 7000 employees a year to complete a 25- to 80-h numeracy programme, usually in their own workplace and during working hours. The focus is on the use of mathematics in everyday life and includes digital skills to engage workers with Information and Communication Technologies (Alkema, 2020).

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The focus of all these initiatives over the years was to provide opportunities beyond school for individuals to acquire knowledge and skills that they failed to gain while at school. The effects of this additional teaching and tutoring have varied, ranging from very good to poor (Billington et al., 2017). Often the expertise of the instructors employed to deliver adult classes has not been adequate, and it is now recognized that professional training is essential for instructors and tutors in this field (OfSTED, 2011). It is necessary for them to be highly familiar with the required curriculum content, and also deeply aware of how adults learn best. One aspect that they often need to address is the potential emotional barrier to learning created by the adult’s previous failures, frustrations and anxiety. Many adults have experienced very negative encounters with math over many years, and as the National Adult Literacy Agency states: ‘Tutors need to recognize […] the emotional baggage that some of their learners might bring to their learning’ (NALA, 2017). In the US, the need for better numeracy standards was addressed by the Department of Education when the Adult Numeracy Initiative (ANI) was established. The aim was to improve research and practice for enhancing adult numeracy (American Institutes on Research, 2006). In the US, a major purpose of ANI was to identify instructional practices that appear to be most effective in raising numeracy skills in older persons. Prior to that time, most of the research into how individuals become numerate was conducted with children rather than adults, so it was not clear whether the teaching methods used in school (pedagogy) were likely to be appropriate for instructing adults (andragogy).

Teaching Adults A debate has long existed over whether adults need to be taught by methods that are markedly different from those used for teaching school students, or whether the methods can be the same but the material presented and the emotional support provided in tutoring needs to be different. It is worth considering this issue in more detail. Pappas (2015) has suggested that there are five main ways in which adults differ from younger learners: • adults are potentially in a very receptive state for instruction because they are motivated and know what knowledge they must acquire and why they need it; • adults are more self-directed and can have control over their learning; • adults need to be involved in setting their own achievement goals and monitoring their own progress;

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• adult learners seek information that is immediately useful in their work environment and personal lives; • adults have much more background experience to draw upon and can more easily relate what they are learning to real life. The House of Commons Business, Innovations and Skills Committee in England (2014) concluded that there is no right or wrong way of teaching numeracy skills to adults. All learning programmes need to be flexible enough to address the characteristics of different learners, such as their motivation level, study goals, employment situation and prior skillset. Studies conducted to date indicate that adult learners are most likely to remain engaged and to achieve success when their learning programme presents content that links directly with their personal goals for improvement, for example, math related to their future or current employment and promotion prospects (NCLNA, 2012; OfSTED, 2011). In terms of service delivery, some adults are happy to participate in classes along with others who acknowledge their need to increase their math skills, while a few prefer one-to-one individual tutoring. In terms of methodology, it has been found that in addition to face-to-face instruction the use of e-learning in the form of computerassisted instruction and online mobile learning offers many opportunities for adults to practice and apply their numeracy skills (Bauer, 2018; Kaczorowski et al., 2019; Seage & Türegün, 2020). E-learning has the great advantage of being a mature and socially acceptable medium for adults, and it can be engaged in at home. It is generally agreed that embedding reality-based numeracy learning within vocational courses and relating it directly to the workplace or a particular career will increase adults’ motivation (e.g. numeracy for nurses; math for the electrical trades). As stated above, there have been efforts made to determine numeracy needs associated with particular areas of employment to facilitate linking of numeracy to real life (Coben et al., 2016; Hagston & Marr, 2007; OECD, 2013b). The term ‘functional mathematics’ is used in the UK to identify courses designed to help learners apply mathematical skills in the workplace (Brooks, 2014). A report by Gal et al. (2020) suggests that adult competencies must include digital numeracy, financial numeracy, health numeracy, civic numeracy and workplace numeracy. The National Adult Literacy Agency (NALA) has a useful website that presents hints for working on numeracy with adults. In the US, the Adult Numeracy Network (ANN, 2015) has also produced a list of guiding principles for teaching basic math skills to adults (cited here in Online Resources). Brief video clips on teaching can be found with an online search under ‘teaching math to adults.’ In the UK, a charity was established in 2012 under the banner National Numeracy, with the stated aim of improving everyday mathematical skills in people of all ages.

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The Way Ahead It is evident that the teaching of numeracy to learners of any age must achieve a careful balance between ensuring that computation skills are mastered and that individuals become competent in applying these skills to problem-solving and the real demands of the workplace. As stated already, some of the reforms suggested for mathematics teaching in recent years have tended to devalue computation, and instead advocate for a more open-ended, investigative and problem-based approach. The ongoing poor standards of many school-leavers, and the continuing need for adult basic numeracy classes would surely indicate that a suitable balance in school mathematics teaching is yet to be achieved (VanDerHeyden & Codding, 2020). The ‘math wars’ (Schoenfeld, 2004) still continue between those who firmly believe that computation skills need to be taught explicitly and practiced to mastery (sometimes referred to as the ‘mechanistic approach’) and those who believe that such skills will be acquired simply as a result of engaging in problem-solving and investigation (the ‘immersion’ or ‘whole math approach’). As Schoenfeld (2004) indicated, using a teaching approach that does not combine skills teaching with the application of those skills makes no sense at all. It is also recognized that assessment methods used in secondary schools and with adults needed to be strengthened so that each learner’s progress is monitored effectively, and individuals who struggle with mathematics are identified early and provided with well-targeted instruction and support (Education Endowment Foundation, 2017). The following chapter provides more detail concerning effective teaching and the assessment of learning.

Online Resources For information on workplace numeracy visit the National Numeracy website. www.nationalnume racy.org.uk. (See particularly the 2017 paper titled ‘Industrial strategy consultation: National Numeracy calls for different approach to improving workplace numeracy’). For information on workplace numeracy visit the National Numeracy website. www.nationalnume racy.org.uk. (See particularly the 2017 paper titled Industrial strategy consultation: National Numeracy calls for different approach to improving workplace numeracy). In the US, the Adult Numeracy Network (ANN) has an informative website. http://adultnumerac ynetwork.org/. (See particularly: ‘Teaching and learning principles’). In New Zealand, the National Centre of Literacy and Numeracy for Adults provides information in a section titled ‘Learning progressions for adult numeracy’. http://www.literacyandnumeracyfo radults.com/resources/354913. In New Zealand, the National Centre of Literacy and Numeracy for Adults provides information in a section titled ‘Learning progressions for adult numeracy’. http://www.literacyandnumeracyfo radults.com/resources/354913. Mathematics Education Innovation [England] (MEI). Core maths qualifications for post-16 students. https://mei.org.uk/core-maths. The BBC has useful online resources under the title Skillswise: Maths for adults. https://www.bbc. co.uk/teach/skillswise/maths/zfdymfr.

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Print Resources Fisher, R. W. (2018). Math refresher for adults. Los Gatos, CA: Math Essentials. Griffiths, G., Ashton, J., & Creese, B., (2015). Training to teach adults mathematics. Leicester: National Institute of Adult Continuing Education. Griffiths, G., & Stone, R. (2013). Teaching adult numeracy: Principles and Practice. Open University Press. Haighton, J., Holder, D., & Thomas, V. (2020). Maths the basics: Functional skills (3rd ed.). Oxford University Press. Kuczera, M., Field, S., & Windisch, H. C. (2016). Building skills for all: Policy insights from the survey of adult skills. OECD.

Chapter 7

Approaches to Teaching and Assessment

Teaching principles and methods have been mentioned already in most chapters of this book. In particular, it has been indicated throughout that direct teaching followed by opportunities for practice with feedback are necessary throughout the school years and beyond (Brownell et al., 2021). Such an approach in primary schools provides the right environment for developing proficiency in basic arithmetic skills, alongside gaining experience in age-appropriate problem solving (DfCSF, 2007; Farkota, 2017). A similar approach is also needed in secondary schools for students who are still weak in skills needed for calculation, and for all students when new and complex mathematical processes are taught. Unfortunately, over the past few decades, teacher educators in the mathematics discipline have argued strongly for an approach that plays down the importance of practising arithmetic procedures and skills to develop fluency, and favours instead an immersion approach to problemsolving at all ages. The argument that children will become numerate simply by immersion in problems and creating their own mental strategies is attractive, but not entirely convincing. It should be noted too that this ‘immersion approach’ is not necessarily in keeping with the views held by most professional mathematicians, who recognize the vital importance of all young students mastering basic number skills in the early stages. Perso (2006b) has compared the debate over the relative importance of developing automaticity in basic number skills to the previous longstanding debate over the teaching of phonics in the literacy domain. Just as phonic skills have proved to be essential in the early stages of reading and spelling, so too it should be recognized that basic skills in arithmetic are essential for making good progress in other areas of mathematics. Meta-analyses of studies that relate proficiency in number skills to achievement in mathematics across the age range have found an overall effect size of 0.88—a very large positive effect (Cason et al., 2019).

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 P. S. Westwood, Teaching for Numeracy Across the Age Range, SpringerBriefs in Education, https://doi.org/10.1007/978-981-16-3761-2_7

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Effective Teaching Practices Research over several decades has yielded hard evidence to support a view that highquality learning in mathematics, just as in all other curriculum subjects, requires enthusiastic and skilled teachers with deep subject knowledge—in this case, mathematical knowledge for teaching (MKT) (Edelman, 2017; Livy et al., 2019). As indicated in Chapter 5, effective instruction also needs teachers with the ability to anticipate in advance any difficulties that students may have when meeting a new math concept or process for the first time. They then require the expertise to plan lesson content and materials to minimize these potential difficulties (Chapman, 2015; Leong et al., 2019). Increasingly, effective teaching also requires that teachers know how and when to integrate technology into their math lessons to enhance students’ motivation, participation and understanding (Koh, 2019). In the UK, several requirements set out in the Teachers’ Standards document of 2011 have direct implications for numeracy teaching. For example, teachers are expected to demonstrate: • • • •

a secure and deep knowledge of the relevant subject matter; an understanding of how students learn and how this should influence teaching; a clear understanding of appropriate teaching methods and strategies; use of relevant assessment data to monitor students’ progress, to set targets and to plan or adapt subsequent lessons; • the importance of providing students with regular feedback, both orally and through marking of work, and encouraging students to respond to that feedback. A study in the United States involving students in the K to 12 age range set out to discover what makes a difference in the quality of mathematics instruction (Weiss & Pasley, 2004). Classroom observations were conducted during 364 mathematics lessons, and the findings revealed that the following variables were significant in ensuring that students make good progress: • • • •

using interesting subject matter; maintaining a high level of student engagement; using effective questioning to encourage children to think mathematically; assisting students to make complete sense of subject matter—with the teacher as mediator and guide.

Another much earlier study of mathematics teaching in the United States (Good & Grouws, 1977) supported what they termed an ‘active teaching’ model. This research was one of the first examples of identifying ‘evidence-based practice’ in mathematics teaching. Good and Grouws found that effective learning in mathematics could be best achieved by teacher-led instruction within a structured and sequential curriculum. Lessons were found to be maximally effective if the teacher introduced each new topic by linking it clearly with previous work and providing process explanations and demonstrations. Effective teachers also used many illustrative examples, engaged students in guided and independent practice and checked very frequently that

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all students could understand the material (Good et al., 1983). These findings were powerful—but they were at odds with the ‘official’ view at the time favouring a more open-ended, constructivist, and student-centred approach. More recently, Masters (2014) and ACARA (2020) have suggested that teaching approaches producing the best results embody many of the same features identified many years ago by Good and Grouws. These features include: • • • •

clearly stated learning objectives within a well-sequenced programme; new material deliberately connected with prior learning; explicit teaching, clear modelling and practice of new skills; multiple embodiments of new concepts through using a variety of different examples; • formative assessment used regularly for checking on students’ understanding; • immediate corrective (but encouraging) feedback given to learners to guide their progress. Many of the effective tactics described above are also evident in what was discovered about highly effective teachers of mathematics in Japan and other parts of Asia (Benjamin, 1997). In the TIMSS research, it was noted that these effective teachers appear to provide systematic instruction in a way that students not only master arithmetic skills and problem-solving strategies but also develop a genuine understanding of the subject matter. These teachers operate lessons in an interactive way and are seen to encourage their students’ mathematical thinking at all points during the lesson. This achieves both conceptual understanding as well as procedural fluency in relevant skills (Shimizu, 1995; Stigler & Hiebert, 1997). The teaching sequence respects and applies the principle of concrete-pictorial-abstract already described in an earlier chapter, and the active involvement and brisk pace of a lesson help motivate students and keeps them on task and productive.

The Vital Role of Practice The importance of providing students with abundant opportunities for practice in calculating and problem-solving on their way to numeracy was explained in Chapter 2; and the role of practice is stressed by the National Center on Intensive Intervention (2016, p.14): Students need effective strategies and ample practice to increase their fluency in basic mathematics skills such as operational facts. The only way to truly increase fluency is to combine timed activities with additional practice opportunities.

These timed activities (often called ‘speed and accuracy games’) can motivate students to increase their computation fluency; and in the primary school years they are a vital part of a comprehensive teaching approach. They can also have an important place in remedial sessions with older students.

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While practice in basic number skills is vitally important for all children, it must be recognized that as individuals they differ in terms of the amount of practice required to achieve mastery. This is one area of teaching where a differentiated approach is required (Deringöl & Davasligil, 2020; VanTassel-Baska et al., 2020). Gifted children who are talented in mathematics usually require rather less time spent in routine practice—and may easily become bored if they are already competent in the particular process or strategy. Instead, they benefit more from opportunities to select and apply a given procedure to solve a wider variety of problems. However, it is equally important to acknowledge that a few academically gifted students actually have a specific weakness in mathematics (dyscalculia: see Chapter 5) and therefore do require additional practice to overcome their difficulties.

Interleaving Knowledge and Skills One particular teaching strategy for mathematics that has received much support has been termed ‘interleaving’ (Agarwal & Agostinelli, 2020; Emeny, 2015). Interleaving in this context means that exercises and activities are deliberately planned to mix new content together with the application of older knowledge, for example, using an already-familiar long division process as one step towards solving a new type of problem. This results in a more cumulative and consolidated progression in the growth of math knowledge, rather than the piecemeal accumulation of concepts and skills that can occur with a solely problem-solving method. As Carey (2014) suggests, ‘…you’re essentially surrounding the new material or new skill set with older stuff you already know but haven’t revisited in a while’ (p. 170). The same principle can be applied to homework assignments where problems requiring a solution deliberately draw on prior knowledge or skill to obtain a solution—using what you know already to solve a problem that is new. In many ways, interleaving has much in common with Bruner’s notion of a spiral curriculum, where the overall direction of learning is forward and upward but previously taught material is deliberately reviewed and applied in a new context. The National Center on Intensive Intervention (2016) states that teachers should provide many opportunities for students to work through multistep problems in order to develop the higher-level thinking skills and confidence needed in tackling complicated math problems at the secondary school level and beyond. Clear modelling by the teacher of each step in completing a complex problem is an essential approach.

Intervention and Support It is widely acknowledged that some students find even the most basic numeracy skills difficult to acquire, and it is recognized that these students do need extra support, tutoring and re-teaching. Students who are struggling should be identified as soon as

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possible, even as early as the kindergarten years (Jordan et al., 2015) and certainly during the primary school years (Powell & Fuchs, 2012). Research has indicated that students who received early intervention do benefit, but may also need longerterm monitoring and support (Bailey et al., 2020). Students with learning difficulties usually need intensive help with mastering computation processes, retaining number facts in memory and developing problem-solving strategies. Without such intervention, the students’ difficulties will go with them into secondary school and present a major barrier to coping with secondary school mathematics. Most of the required re-teaching and support for these students must come in via individual or small group tutoring. A few of these students may also require some degree of personal counselling and cognitive therapy if they have developed anxiety and learned helplessness concerning their engagement in mathematics (LaGue et al., 2019). Until recently, intensive tutorial assistance in mathematics was far less common at any level of schooling than providing assistance for literacy difficulties. As indicated already, the weakest students in math were often placed in a low-ability group and received a watered-down curriculum rather than individualized intervention to improve their skills. Under the now popular Response to Intervention Model, it is becoming a little more common to find remedial help being provided in math at Tier 3 (individual intensive tutoring) (Riccomini & Witzel, 2010). The tutors providing this support need to know: • how to give positive corrective feedback and encourage a student to keep him or her motivated and engaged; • how to be supportive rather than critical or too ‘didactic’; • how to demonstrate number processes clearly and how to use ‘thinking aloud’ as an accompaniment to the teaching of problem-solving; • how to break a math problem down into easy steps; • how to provide multiple examples of any new process or concept; • how to monitor the progress their student is making; • how to adapt the math programme and reteach when necessary. There are several individual or small-group tutoring interventions designed to help struggling learners. The best-known examples are summarized below. In all cases, the efficacy of an intervention depends on how well the specified teaching and assessment processes are conducted (Bouck & Cosby, 2019; Nelson et al., 2019). QuickSmart Numeracy: This is a very well-researched approach involving up to 30 weeks of intensive tutoring of individuals or very small groups. Tutoring is conducted as three 30-min sessions per week (Bellert, 2009; Pegg & Graham, 2013). The target groups are students who are falling behind in primary school math or with serious learning difficulties in secondary school. The main aim is to use explicit instruction and abundant successful practice to increase fluency and automaticity in basic number skills. It has been of particular benefit for low-achieving students from socially disadvantaged or culturally different backgrounds. Number Worlds: This is an intervention associated with Griffin (2004) and has been extensively evaluated with children from low-income populations. It has proved

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to be effective in enhancing students’ number sense, computational fluency, mathematical reasoning and performance on standardized tests. Number Worlds has now been mapped on to the Common Core State Standards in the US. The programme is reported to bring significant gains in children’s number knowledge; and the activities seem to be particularly beneficial for children who have not had positive experiences with a number before entry to school. Mathematics Recovery ® : This is another well-respected numeracy intervention that uses an individualized assessment-based approach for children aged 6 to 7 years. It is implemented daily for 30 min and has an emphasis on formative assessment to guide instruction (Wright, 2003). The programme focuses on developing facility with number words, recognizing and writing numerals, strategies for adding and subtracting, and place value. A fourth book in the series provides suitable interventions for students age 7–11 years (Wright et al., 2011; Wright & Ellemor-Collins, 2018). Mathematics Recovery ® has proved to be very useful in working with students who have learning difficulties or disabilities (Grindle et al., 2020; Tabor et al., 2020). Numeracy Recovery: This scheme involves working with children who have been identified as having problems with arithmetic. The content covers: counting; use of written symbols; place value; solution of word problems; translation between concrete, verbal and numerical formats; use of derived fact strategies for calculation; estimation; memory for number facts. The children receive weekly individual intervention in the particular components with which they have difficulty (Dowker, 2001). Odyssey® Math: This is a web-based programme in the US, developed by Compass Learning® for supporting mathematics instruction in grades K–8. It can be used via desktop PC, laptop or mobile device. It comprises a mathematics curriculum and formative assessments designed to support individualized data-driven instruction. An evaluation by What Works Clearing House (WWC, 2017) reached the conclusion that Odyssey® Math has potentially positive effects on mathematics achievement for primary school students. In 2016, features of Odyssey® Math were incorporated into Compass Learning Pathblazer ® Math and Hybridge® Math. GRIN (Getting Ready in Numeracy): This small group or individual tutoring program is designed to prepare primary and secondary school students who are weak at math and lacking confidence for upcoming mathematics lessons back in their mainstream classroom (Kalogeropoulos et al., 2020). The program is delivered by trained GRIN tutors who conduct the 15-to 25-min sessions at least three times per week for 6 months. The classroom math teacher plans sequences of future lessons and communicates in advance the details of each lesson to the tutor. The tutor then implements intervention sessions out of the classroom that prepare the targeted students for what is to come. Relevant concepts and skills are taught at an appropriate level and pace. The sessions include three elements: developing familiarity with pre-requisite concepts, targeted fluency practice and language development. During classroom mathematics lessons, the math teacher deliberately draws upon the work of the tutor by calling upon the students to answer questions and provide explanations or suggestions. The aim is to recognize their prior learning and to build

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their confidence. Evaluation studies indicate that students participating in GRIN engage more willingly and successfully in math lessons and exhibit a more positive attitude.

Do All Interventions Bring Benefits? A review of 19 studies of early intervention for math with at-risk children (Mononen et al., 2014) reached a conclusion that all programmes show a positive effect in improving numeracy skills through games playing, explicit instruction, computerassisted instruction and activities that deliberately set out to bridge understanding from concrete materials to abstract representation. The interventions also led to improvements in attitude and motivation. An earlier review by Xin and Jitendra (1999) had looked not only at the immediate effect of an intervention but also any evidence that progress was maintained over time and generalized to other contexts. Their review identified the following strategies as effective in improving students’ numeracy skills and their ability to solve problems: • • • • • • •

using concrete materials to model number relationships, providing guided practice, estimating a possible answer in advance (e.g. ‘It should be less than 15’), in word problems, looking for keywords that may suggest a process to use, visualizing, and drawing a diagram or table, planning the necessary steps for calculating a solution, utilizing computer-aided instruction (CAI) and appropriate use of a calculator.

While the overall evaluation of numeracy interventions has been positive (e.g. Fuchs et al., 2019; Moser-Opitz et al., 2017; Roesslein & Codding, 2019), it is important to note that some doubts have been raised that they are cost-effective when they involve daily individual or small group tutoring (Ministerial Advisory Group on Literacy and Numeracy, 2013). Of course, the same criticism is frequently levelled in relation to individual and small-group literacy intervention in the early years; but the conclusion is that long-term benefits do accrue for at-risk young children who participate in them, and therefore the cost should not be seen as a prohibitive factor (Strand-Cary et al., 2017; Toll & Van Luit, 2014). It was suggested by Xin and Jitendra (1999) that students vary in their response to an intervention, due mainly to differences in their motivation, engagement, prior knowledge and existing skill level on entry, and to the fidelity with which particular teaching strategies or programme features are implemented. More recently, Nelson et al. (2020) have also cited implementation fidelity as a factor that must be taken into account when evaluating and intervention method. Student response is also influenced by the quality of teaching, and the ability of the teacher to adapt instruction to meet their particular needs (Karagiannakis et al., 2016). For interventions to be truly effective, they need to be based on a careful assessment of students’ current knowledge, skills, confidence and motivation. Accurate evaluation of students’ existing

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competencies must, therefore, be the starting point for any intervention (Education Endowment Foundation, 2017; Karagiannakis et al., 2016). Appropriate approaches to assessment are described below, and additional resources are listed at the end of the chapter.

Assessment Assessment in the domain of numeracy serves the same basic functions as assessment in all other areas of the curriculum. These functions are: • determining a student’s current stage of development, • evaluating affective factors such as the student’s attitude, confidence, self-efficacy beliefs, anxiety and resilience, • gaining information about the student’s specific strengths, weaknesses and instructional needs, • identifying concepts or procedures which may need to be re-taught, reviewed or practised again, • checking regularly on the efficacy of the teaching programme and making adjustments when necessary. Effective teaching of numeracy thus requires that the programme include regular monitoring of students’ progress through daily observation, questioning and teachermade and curriculum-based tests. Information gained from this ongoing assessment ensures greater precision in programme planning and delivery (ACARA, 2020; Education Endowment Foundation, 2017; Ngware et al., 2019). In keeping with the principle that numeracy is important across the curriculum, the Queensland Curriculum and Assessment Authority (QCAA, 2013; 2017) produced useful guidelines for assessing numeracy in the context of several subject areas (e.g. The Arts, Science, Health and Physical Education). The material is referenced against the Australian Curriculum and presents examples of appropriate test items. The notion of embedding numeracy tests using real-life contexts is valuable and should be taken into account when teachers devise their own tests and assessment tasks.

Observing Students at Work The most natural method of assessing students’ learning is direct observation of what they actually do when engaged in numeracy tasks. This type of informal assessment should be part of any teacher’s approach from kindergarten and throughout the primary and secondary school years (Komara & Herron, 2012; Purpura & Lonigan, 2015). Through careful observation and questioning, a teacher obtains a fairly clear picture of which students are progressing well and those who require additional

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follow-up. Often teachers will notice that certain students have not understood the lesson content or are practising an incorrect procedure without realizing their error. Observing students as they work often reveals the ‘teachable moment’ to step in and reteach, explain again and revise key points from a lesson. Often minor problems can be dealt with at the time, rather than later when workbooks are passed in for marking. During lessons, supportive comments and prompts can be used to keep students on-task and to help them focus on key aspects of the work. Direct observation also provides a teacher with information related to the affective aspects of students’ engagement in the lesson. For example, it will be evident when students are interested in and motivated by the tasks, and if they are confident or hesitant. It will also be evident whether students are monitoring their own performance and self-correcting where necessary. Teachers and tutors working with at-risk students should keep reasonably detailed records of what they have observed during lessons, particularly where the evidence suggests that something may need to be taught again, or that additional practice with different examples may be required.

Questioning The National Center on Intensive Intervention (2016) suggests that teachers need to ask many well-focused questions during any math class or tutorial session to increase students’ engagement and help stimulate higher levels of thinking and understanding. Active and frequent questioning of this type is a key feature of teaching approaches that lead to higher levels of student achievement in math. Questioning of an openended nature is also useful because it can stimulate discussion and sharing of ideas during problem-solving activities (Smith & Sherin, 2019). Asking students to explain and justify their solutions and calculations is an important way in which teachers probe for depth of understanding and mathematical thinking (Barton, 2018). Questioning is the focus of a publication titled Teaching for mastery: Questioning, tasks and activities to support assessment (Askew et al., 2015). This material can be found online at www.ncetm.org.uk/public/files/23305581/Mastery_Assessment_ Y3_Low_Res.pdf.

Teacher-Made Tests The informal tests that teachers regularly construct serve one of two main purposes: (i) to determine how much students have understood and mastered in a particular unit of work (curriculum-based assessment), or (ii) to find the skill level already achieved by students, and to detect any persistent errors or misconceptions (diagnostic testing).

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Teacher-made tests of the first type should be directly linked to the learning objectives set for each unit of study and should embody the following features: • in most situations, test items should not be confined to routine computation but should also assess students’ ability to apply these skills to new problems and topics; • a test should begin with a few easy items to allow even the weakest students to experience some success; • a variety of question formats can be used to make a test more interesting (e.g. some multiple-choice, some ‘missing numbers’, some items involving drawing or measuring, plus some computation questions); • where computation skills are being assessed, at least three test items at the same level of difficulty should be provided to enable careless errors to be differentiated from those that are persistent. This principle certainly applies to the design of diagnostic tests.

Technology and Formative Assessment It is not surprising to find that technology has contributed additional ways of assessing and monitoring students’ math skills. For example, Tamargo and Johnston (2019) describe how they use Math Learning Center apps with Grade 2 students. These apps include a range of manipulatives and models and enable a teacher to closely observe and evaluate students’ mathematical thinking (see: https://www.mathlearningcenter. org/resources/apps). Similarly, Confrey, Toutkoushian and Shah (2019) report on the use of Math-Mapper 6–8 with middle-school students to monitor their learning trajectory as a part of formative assessment (see https://ced.ncsu.edu/news/2017/01/ 10/mapping-out-middle-school-math/. Several online programmes contain within them formative assessments that provide student and teacher with immediate information on how well they are performing, and also indicate areas needing more attention (e.g. Odyssey® Math referred to earlier in this chapter). An online search using ‘online math programmes with built-in assessments test’ will yield many other digital resources that may be used across the age range.

Error Analysis Error analysis is not a process that a teacher would carry out with all students in the class, because it is time consuming and not always rewarding. Analysis of errors can be used on the bookwork of a student who is regularly performing poorly. It involves close inspection of a student’s calculations to identify any tendency to repeatedly make specific types of errors (e.g. failure to exchange tens correctly in subtraction; inaccurate recall of multiplication facts). Appropriate re-teaching can then have a

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clear focus in order to rectify the error (Ashlock, 2009; Kingsdorf & Krawec, 2014). Although error analysis can be valuable in identifying specific aspects of computation that require attention, reteaching should not simply replace one rote-learned trick with another; the aim should be to help the student really understand the process more clearly. In order to conduct a valid and reliable error analysis, it is necessary to collect a fairly large sample of the student’s bookwork. A pattern sometimes emerges indicating a point of confusion that always leads to an incorrect response when executing one step in an algorithm. However, sometimes errors appear to be quite random and may simply reflect carelessness or a tendency to be distracted easily while working. When error-analysis is well conducted on an adequate sample of work it may reveal for example: • • • • • • •

errors due to carelessness (e.g. misalignment of figures on the page), poor attention to process signs + − × ÷ , inaccurate recall of multiplication or division facts, place-value difficulties, selection of an incorrect operation, regrouping errors within a calculation, difficulty manipulating common fractions and decimals.

Murchan and Oldham (2017) have indicated how error analysis can be included in the data generated from the computer-based assessment with its automated scoring of responses. They recommend this as a helpful procedure for use with middle grades in primary schools.

Diagnostic Testing In diagnostic assessment, a teacher is endeavouring to find answers to the following questions: • What can the student do already without assistance? This reflects the student’s existing knowledge, skills and strategies. • What can the student do if given some degree of support or guidance? This is a clear indication of which concepts and skills can soon be developed within that student’s zone of proximal development. • What gaps exist in the student’s previous learning? Often specific gaps can be detected, due perhaps to absences from school, changing schools or having a less than effective teacher during one year. • What does the student need to be taught next as top priority in the programme? Identify the starting point for any intervention that is planned. Diagnostic assessment in numeracy must always take account of the student’s current developmental level. For this reason, with young learners or older students with learning difficulties, it may need to involve the use of concrete materials, pictorial

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representations and oral answering of questions, rather than paper and pencil testing. During the diagnostic session, the teacher will move up or down the sequence from concrete to abstract in an attempt to determine the level at which the student is functioning in different areas of numeracy. A thorough diagnosis of a student’s learning difficulties may need also to consider the cognitive processes that are involved in numeracy operations (Harvey & Miller, 2017; Raghubar et al., 2010). For example, working memory—the ability to retain information in the mind long enough to operate on it (e.g. remembering the result at each step in a complex calculation). Working memory differs from one individual to another, but it is greatly influenced by the amount of focused attention that a student is giving to a particular problem or task, and also by anxiety. For this reason, teaching methods and materials in math should always be as interesting and relevant as possible. Attention span is also highly relevant because solving problems and completing calculations requires a student to stay focused and shift attention easily from one process to another when necessary. Difficulty in maintaining attention to task and in switching from one process to another is often typical of students who struggle with mathematics (Cragg & Gilmore, 2014; Raghubar et al., 2009). It is also often relevant to asses a student’s knowledge of math vocabulary, because limited vocabulary often causes students difficulty in understanding teachers’ explanations and the questions they ask. It is useful for a teacher to compile an age-appropriate list of math-related words, to be used as an assessment tool with any individual who is struggling with the math course. Words that are not known can then be taught and practiced. It should always be remembered that all forms of assessment should lead to action. For example, diagnostic assessment of an individual should guide a teacher or tutor in how best to tailor a teaching programme for that student. Formative and ongoing assessment should tell a teacher whether his or her teaching approach is being effective or whether something needs to change in the programme.

Standards of Teaching It is notable that in Asian countries where students perform extremely well in mathematics their teachers are usually subject specialists, even in primary school. They have deep subject knowledge and enthusiasm and have become highly skilled in teaching their subject (Stigler & Hiebert, 2004). They are also highly proficient in regular formative assessments of students’ learning. In contrast, countries such as Britain, United States, Canada, Australia and New Zealand, all tend to have many teachers who are teaching math as part of their total workload but have no specialist qualification in the subject. Concern has been expressed in recent years regarding the lack of deep subject knowledge in these nonspecialist teachers, and their limited range of skills for ensuring that students become truly numerate. One aspect of teaching in which they are particularly weak is in using data from formative testing to guide them in planning help for individual students

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(Lamberg et al., 2020). There is growing awareness that too many general-subject teachers in primary schools are ill-prepared to teach number work and mathematics effectively (Burghes, 2012; New Zealand Government, 2016; Yang et al., 2015). For example, a headline in an Ontario newspaper read ‘Elementary teachers’ weak math skills spark mandatory crash courses’ (Brown, 2016, p. 1). In Australia, the year 2017 saw the introduction of compulsory testing of numeracy standards in graduating student teachers, in order to ensure that those with poor math skills are identified and required to up-skill. A major problem is that many teacher-education courses do not devote sufficient time to increasing trainee teachers’ subject knowledge in maths, instead being more concerned with education policies, current constructivist theories, and using groupwork and activity methods. Many trainee teachers graduate from their university courses with not much more math content knowledge than they possessed when they left their secondary schools—and many confess that math had been their weakest subject.

Improving Teachers’ Expertise There have been several initiatives taken to improve teachers’ knowledge and skills in teaching for numeracy. In the UK, the government has endeavoured to improve the quality of teacher training with its 2015 policy on teaching and school leadership (UK Government, 2015). In that country, there was also a unique attempt to increase teachers’ math teaching expertise by creating the National Collaborative Project, bringing exemplary teachers of mathematics from Asia to Britain to provide classroom input and teacher training. In exchange, teachers from the UK have been sent to Asia to observe mathematics lessons. It is reported that this project has produced definite benefits (Stripp, 2016). As mentioned in Chapter 1, the Department for Education in England has also supported the creation of the National Centre for Excellence in Teaching Mathematics (NCETM) (https://www.ncetm.org.uk/) and the establishment of Maths Hubs across the country. The Hubs, together with an online website, provide practical advice, resources and teaching videos to help with professional development (http://www.mathshubs.org.uk/). On the other side of the world, New Zealand ran a series of professional development activities for teachers within the Numeracy Development Project (NDP). It was reported that over the period of operation almost every teacher of Years 1–6, and the majority of teachers of Years 7 and 8, had the opportunity to participate (New Zealand Ministry of Education, 2008; Young-Loveridge, 2009). Evaluation of the project indicated that many school students made substantial gains as a result of their teachers’ participation in NDP (effect size around 0.40). Somewhat earlier, Australia had introduced a professional development package in New South Wales titled Count Me in Too (1997–2002); and in Victoria, a scheme was introduced to employ ‘coaches’ who are experts in mathematics to work with staff (DEECD, 2010). The conclusion was that coaching is a highly effective in situ

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method for the professional development of math teachers and is probably much more effective than off-campus short courses and lectures. More recently, the Teach for Australia programme was introduced in an attempt to tackle teachers’ lack of subject knowledge by recruiting university graduates with backgrounds in mathematics (but not in teaching) to become mentors in disadvantaged schools (Weldon et al., 2013). This initiative is part of the Teach for All Network that now extends across 26 countries—other significant programmes being Teach First (UK) and Teach for America (US). Evaluation of the Teach for America programme has found that teachers’ expertise in teaching math has increased as a result of participation (Chiang et al., 2017). In the US, in addition to ongoing efforts to improve teacher education courses for teachers of mathematics, a Better Math Teaching Network has been established as a practical model for teacher learning (Bayerl, 2020). Members of the network are high school math teachers. They are working with each other, and with support from a team of researchers, to improve the effectiveness of their own teaching strategies. The network provides frequent in-person and virtual meetings to enable members to share ideas as they test and evaluate new instructional routines in their classrooms. Most recently, additional training of teachers in early childhood math has been attempted successfully through online programs (Sheridan & Wen, 2021). The topics included were number sense, patterns, geometry, measurement, data collection, math processes and math literacy. Evaluation of the effectiveness used data from 2332 participants indicated that engagement in the program was effective in positively impacting their attitudes, confidence, beliefs and knowledge in teaching early math.

End Note I hope that this introduction to teaching for numeracy across the age range has provided readers with a firm foundation for pursuing the subject in greater depth. The reference list and the resources listed at the end of each chapter should assist in this search for information.

Online Resources Bayerl, K. (2020). Better Math Teaching Network: Deepening practice in community. Years 1 and 2 Summary. Quincy, MA: Nellie Mae Education Foundation. https://files.eric.ed.gov/fulltext/ED6 08794.pdf. Department for Education and Skills (UK). (2016). Numeracy diagnostic assessment materials: Entry 1. Designed for assessing adults with poor numeracy skills; but this material also provides teachers at any grade level with excellent examples of test material that uses pictorial representation. http://www.excellencegateway.org.uk/content/etf1262. Gittens, C. A. (2015). Assessing numeracy in the upper elementary and middle school years. Numeracy, 8(1), Article 3. http://scholarcommons.usf.edu/numeracy/vol8/iss1/art3.

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Information on the numeracy test for graduating teachers in Australia can be found at: https:// teacheredtest.acer.edu.au/files/Literacy-and-Numeracy-Test-for-Initial-Teacher-Education-Stu dents-Assessment-Framework.pdf. Ministry of Education New Zealand. Individual Knowledge Assessment of Number (IKAN). http://assessment.tki.org.nz/Assessment-tools-resources/Commonly-used-assessments/ Numeracy-assessment. Sample test material for Key Stage 1 Mathematics (arithmetic) can be seen online at: www.gov.uk/ government/uploads/system/uploads/attachment_data/file/616508/STA177727e_2017_KS1_ mathematics_paper_1_arithmetic.pdf. State Government of Victoria (Australia). Learning and Assessment Framework (LAF). http://www. education.vic.gov.au/school/teachers/teachingresources/discipline/maths/assessment/Pages/lea rnassess.aspx.

Print Resources Askew, M., Bishop, S., Christie, C., Eaton, S., Griffin, P., & Morgan, D. (2015). Teaching for mastery: Questions, tasks and activities to support assessment. Oxford University Press. Chinn, S. (2020). More trouble with maths (3rd ed.). Routledge. Hansen, A., et al. (2020). Children’s errors in mathematics. Sage. Hattie, J., et al. (2016). Visible learning in mathematics K-12. Corwin Press. Small, M. (2019). Math that matters: Targeted assessment and feedback for Grades 3–8. Teachers College Press. Smith, A. (2017). Report of Professor Sir Adrian Smith’s review of post-16 mathematics. Information Policy Team, The National Archives, Kew, London, TW9 4DU. Tabor, P. D., Dibley, D., Hackenberg, A. J., & Norton, A. (2020). Numeracy for all learners: Teaching mathematics to students with special needs. Math Recovery series. SAGE. Wright, R.J., & Ellemor-Collins, D. (2018). The learning framework in number: Pedagogical tools for assessment and instruction. [Math Recovery]. Thousand Oaks, CA: Sage.

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© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 P. S. Westwood, Teaching for Numeracy Across the Age Range, SpringerBriefs in Education, https://doi.org/10.1007/978-981-16-3761-2

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Index

A Accommodation, 15, 17 Active teaching, 76 Addition teaching of, 4 Adult literacy and life skills, 68 Adult numeracy standards, 67, 69 teaching of, 67, 69 Adult Numeracy Initiative (ANI), 70 Affective aspects of numeracy anxiety, 8 attitude, 9 disposition, 8 inclination, 8 motivation, 9 self-efficacy beliefs, 9 Algorithms, 15, 16, 19, 22, 42, 45, 50, 61, 85 Anxiety, 7, 8, 27, 40, 50, 61, 70, 79, 82, 86 Approximate number system, 14 Aptitude, 7, 8, 40 Arithmetic, 2, 3, 5, 14–16, 18, 19, 24, 25, 39–41, 43, 46, 50, 55, 56, 63, 69, 75, 77, 80 Asian classrooms, 51 Assessment diagnostic, 62, 85, 86 formative, 77, 80, 84, 86 purpose of, 83 Assimilation, 15, 17 Australian curriculum, 5, 8, 10, 11, 16, 39, 41, 47, 56–58, 77, 82 Automaticity in calculation processes, 46 recall of number facts, 45 tables, 46

B Bar drawing model, 22. See also Singapore Math Behavioral perspective, 24 Better math teaching network, 48, 88 Bruner, 16, 21–23, 31, 78 Building our industrial strategy [UK], v

C Calculators, 4, 24, 28, 34, 49, 50, 60, 81 Cardinality, 6, 14, 31, 32 Child-centered approach, 30 Cockcroft report, 2 Cognitive and perceptual foundations, 17 Cognitive constructivist view, 17, 21, 24 Cognitive load, 24 Cognitive therapy, 79 Common Core State Standards (US), 6, 9, 11, 15, 16, 31, 36, 39, 57, 80 Commutativity law, 23 Computation skills automaticity in, 36 learning and teaching of, 39 Computer-assisted learning (CAL), 3, 5 Concepts development of, 1, 3 stages in development, 19, 51 Conceptual understanding, 3, 4, 11, 15, 16, 45, 49, 55, 77 Concrete experience, 14, 18, 20, 23, 48 Concrete operational stage, 18, 19, 28 Concrete-to-Pictorial-to-Abstract (CPA) sequence, 22, 43, 57, 77 Conservation of number, 18, 32 Constructivist theories, 5, 24, 87 Corrective feedback, 20, 25, 60, 79

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 P. S. Westwood, Teaching for Numeracy Across the Age Range, SpringerBriefs in Education, https://doi.org/10.1007/978-981-16-3761-2

115

116 Counting difficulties with, 40 as a foundation skill, 4 teaching of, 24 Count Me in Too, 87 Critical thinking, 56 Crowther report, 1 Cuisenaire rods, 22, 35 Curriculum-based assessment, 83

D Data collection and analysis, 35 Design and technology, 11 Diagnostic assessment, 62, 85, 86 Dienes MAB, 22, 35, 42 Digital technology, 28 Division, 4, 16, 23, 34, 39, 41, 43–47, 78, 85 Dyscalculia, 61, 78

E Early intervention effectiveness of, 88 programs for, 88 Early Years Foundation Stage (EYFS), 30, 36 Early Years Learning Framework (ELYF), 30 E-learning, 71 Enhanced anchored instruction (EAI). See (enhanced anchored instruction) Error analysis, 84, 85 Exact number system, 14, 39 Executive functions, 43 Explicit teaching, 6, 10, 30, 42, 52, 77

F Feedback from teacher, 15 Formal operational stage, 18 Formative assessment, 77, 80, 84, 86 Fractions, 5, 6, 15, 22, 45, 47, 57, 58, 69, 85 Functional mathematics, 71

G Genders differences in mathematics, 7 stereotyping, 8 Geometry, 6, 31, 36, 57, 88 Getting Ready for Numeracy (GRIN), 80, 81 Gifted (high ability) students, 49, 78 Guided participation, 31

Index H Hand-held digital devices and apps, 4, 49, 60 High achievers, 10

I Indigenous children, 52 Interleaving as a curriculum feature, 78 International Mathematical Modelling 2 Challenge (IM 2 C), 59 Investigative problem solving, 63

K Kindergarten, 6, 18, 25, 28–32, 34–36, 40, 41, 79, 82

L Language its role in learning, 30 its role in mathematical thinking, 47 Learning difficulties, 3, 20, 22, 41, 42, 44, 46, 56, 61, 79, 80, 85, 86 Learning disability, 61. See also dyscalculia

M Mastery approach, 10, 22, 40, 45, 48 Math anxiety, 8, 40, 61 Mathematical ability, 2, 7, 27 Mathematics big ideas, 31 defined, 56 in preschool schools, 27, 29 in other subjects, 49 in primary schools, 87 in secondary schools, 20 numeracy and, 2 Mathematics counts, 1 Mathematics difficulties and disabilities, 60, 61 Mathematics programmes of study [UK], 42, 47, 56, 57 Mathematics recovery, 80 Math Hubs project [UK], 16 Math wars, 64, 72 Measurement learning to measure, 36 STEM, v Mechanistic approach, 64, 72 Mediated learning, 19, 31 Mediated play, 28

Index Mental arithmetic, 22, 24, 49 Modeling as a teaching tactic, 77 Motivation, 25, 27, 40, 49, 52, 60, 61, 71, 76, 81 Multiple numeracies, 2 Multiplication, 4, 23, 34, 39, 41–47, 84, 85 Multiplicative thinking, 44 Multi-step problems, 78 N National Assessment Programme Literacy and Numeracy (NAPLAN) [Australia], 10 National Collaborative Project [UK], 87 National Curriculum [England], 4, 5, 16, 39, 41, 44, 57 National Foundation Skills Strategy for Adults [Australia], 69 National Numeracy Strategy [UK], 2 New Zealand, 8, 9, 31, 59, 69, 86, 87 Next Generation Science Standards [US], 16 Number facts, 4, 15, 24, 33, 34, 40, 45, 46, 50, 79, 80 Number line, 5, 22, 34, 57 Number sense approximate, 14 assessment of, 14 defined, 13 development of, 14 Number worlds, 79, 80 Numeracy across the curriculum, 30, 82 adult numeracy, 52, 67, 69, 70 and employability, 67 assessment of, 72 defined, 1, 30 described, 1 standards, 9–11, 63, 68–70, 87 standards, in the workforce, 69 Numeracy recovery, 80 Numeracy standards, 9–11, 63, 67, 68, 70, 87 Numeral recognition, 14, 32, 33 Numerosity, 13, 18 O Object permanence, 18 Observation as an assessment strategy, 82 Operant conditioning, 24 Ordinality, 14

117 P Parents involvement, 77 part-to-whole relationships, 61 role in teaching, 29 Pedagogical content knowledge, 62 Pedagogy, 70. See also teaching Piaget, 15–22, 28, 32 Place-value, 3, 4, 33–35, 40–43, 47, 80, 85 Practice importance of, 75 Pre-operational or intuitive stage, 18, 19, 28 Preschool numeracy, 25, 29 Primary national strategy, 2 Primary school, 3, 4, 10, 14, 20, 28, 34–36, 39–42, 44–47, 52, 55, 56, 59, 61, 62, 68, 75, 77, 79, 80, 85–87 Principles and standards in school mathematics [US], 6 Prior knowledge its importance, 15, 17, 21, 57, 78, 81 Problem solving teaching how, 49, 50 types of, 78 Procedural fluency, 11, 15, 24, 25, 45, 77 Programme for International Student Assessment (PISA), 7, 9 Programme for the International Assessment of Adult Competencies (PIAAC), 68 Proportionality, 6, 10, 57, 68

Q QuickSmart mathematics, 52

R Realistic Mathematics Education (RME), 59 Relational reasoning, 55, 56 Research-based practice, 76 Response toiIntervention model, 79 Reversal of numerals, 33 Rote learning, 43, 45, 60

S Scaffolding, 19–21 Schema (plural schemata), 17, 22 Secondary school, 10, 14, 20, 36, 40, 41, 45, 46, 49, 52, 55–63, 72, 75, 78–80, 82, 87 Self-correction, 50, 51

118 Sensori-motor stage, 18 Singapore Math, 44 Socio-economic disadvantage, 79 Spiral curriculum, 78 Standards adult, 9, 69 of numeracy, 69 of teaching, 86 Strategies for solving problems, 39 teaching of, 39 Structural apparatus, 57 Subitizing, 13 Subtraction, 3, 4, 23, 28, 30, 31, 34, 39, 41–43, 84

T Teachers expertise, 87, 88 subject knowledge, 76, 87, 88 record keeping, 30 training of, 88 Teachers’ Standards, 76 Teach for America, 88 Teach for Australia, 88 Teaching direct, 4–6, 19, 20, 30, 51, 55, 75, 76 evidence-based, 51, 76 explicit, 6, 10, 30, 42, 52, 77 interactive, 11 methods, 3, 13, 19, 56, 70, 86 pre-school, 19, 28 primary, 41 secondary, 10, 20, 55, 60, 62 of adults, 71 Technical Education Research Center [US], 31 Technology

Index its role in numeracy, 49 its role in teaching, 60 Thinking mathematically, 56, 59, 61 Thorndike’s Law of Effect and Law of Exercise, 24 Three-stage theory of skill learning, 25 Trends in International Mathematics and Science Study (TIMSS), 7, 9, 77 Tutoring, 62, 63, 70, 71, 78–81

U Unifix, 22, 34, 35, 43 Universal Design for Learning (UDL), 23

V Variation theory, 23, 24 Virtual manipulatives, 22, 43 Vision-impaired students, 49 Visual perception, 17 Visual representation, 3, 20, 31, 34, 45, 47, 48, 56 Vocabulary acquiring, 15 importance of, 16 in mathematics, 86 Vocational courses, 68, 71 Vygotsky, 16, 19–22, 28, 29, 31

W Workplace numeracy, 71 Writing, 2, 17, 19, 33, 34, 39, 41, 80

Z Zone of proximal development, 20, 21, 31, 85