Teaching Mathematics in Primary Schools: Principles for effective practice

by Robyn Jorgensen

Overview

Since its first publication, this book has established a strong reputation for its valuable insights into the evidence-based approaches to effectively teach primary mathematics and the practical theory behind those teaching strategies. Based on the latest international research, the expert authors focus on learning outcomes and the general principles that underlie educational practices rather than any specific curriculum. Current approaches to mathematics education are explained and critiqued, and insights into why some students have difficulties with mathematics are provided. Teachers are shown how to encourage their students to develop deep learning in mathematics, and to relate mathematics to the rest of the curriculum.

The book first examines the philosophy behind mathematics and its impact on curriculum design, the history of learning outcomes, and theories on how students learn mathematics. The key areas of mathematics teaching are then outlined in detail: number, probability and statistics, measurement, geometry, and algebra. In each area, the emphasis is on problem-solving. Finally, practical classroom issues are discussed such as the trend towards developing students' capacity to think mathematically, broad approaches to teaching mathematics, planning for a whole school approach to mathematics, diversity and access, and assessment, reporting and evaluation. A new focus on numeracy and the integration of digital technologies to enhance mathematics education have been incorporated in this edition.

Featuring practical activities that can be implemented in the classroom, together with diagram, graphs, and other visual aids, this book is an invaluable resource for university lecturers as well as pre-service and in-service teachers.

Product Details

ISBN-13:	9781760529734
Publisher:	Taylor & Francis
Publication date:	10/01/2019
Edition description:	Third Edition
Pages:	432
Product dimensions:	6.88(w) x 9.69(h) x (d)

About the Author

Robyn Jorgensen is an internationally renowned educational researcher. Her work has strongly emphasized how the practices in school mathematics work to support or exclude particular groups of people in society. More recently, she has been working in remote Indigenous communities, rural and farming communities and low SES communities. Shelley Dole is Head of School, School of Education, University of the Sunshine Coast. Over the 25 years she has been in education, she has taught in primary, secondary and tertiary teaching institutions throughout Australia. Kevin Larkin is a senior lecturer in Mathematics Education at Griffith University with over 30 years' teaching experience. Kevin is well established in the field of mathematics education in the use of digital technologies to enhance learning.

Read an Excerpt

CHAPTER 1

THE CHANGING FACE OF SCHOOL MATHEMATICS

There is a need for new approaches to teaching mathematics. Sadly, mathematics is still rated by many students as one of their least favourite subjects at school, partly because mathematics teaching is too often typified by students working individually with their desks in rows facing the board, and the teacher demonstrating procedures from a textbook or an interactive whiteboard (IWB), with students completing worksheet exercises. With the vast amounts of research that have been generated since mathematics education became recognised as a discipline, there is now a strong research base to inform change in teaching school mathematics. Also, wide changes are occurring rapidly in societies, both nationally and internationally. Change impacts students, and there are new theories of how students learn in contemporary times. Research has also contributed to changing perceptions of mathematics as a discipline.

Mathematics education and society

The mathematics curriculum has not been created in a vacuum. Mathematics in schools, and the way it is taught, are the product of broader factors that extend beyond the classroom. Such factors include employers, lobby groups, government policy, parents and professional organisations.

External authorities, such as education departments and other statutory bodies, may develop a curriculum or syllabus that provides guidelines teachers are expected to use to develop their work programs, and to undertake assessment and reporting of students' learning. The development of curriculum documents is influenced by demands from groups including parents, employers and governments. In the Australian context, the Australian federal government developed an agreement among the states and territories for the development and implementation of a national curriculum, which began in mathematics in 2012. Documents relating to mathematics in the Australian national curriculum can be accessed through the Australian Curriculum, Assessment and Reporting Authority (ACARA) website (www.australiancurriculum.edu.au/f-10curriculum/mathematics/). Documents relating to the New Zealand National Curriculum can be found at the New Zealand Curriculum Online website (http://nzcurriculum.tki.org.nz/The-New-Zealand-Curriculum/Mathematics-and-statistics). When all variables are considered, the curriculum guidelines that appear in mathematics classrooms have been created through a highly negotiated (and often hotly contested) process. In addition to curricula, many jurisdictions around the world now utilise some form of high-stakes testing. Examples include, but are not limited to, the Trends in International Mathematics and Science Study (TIMSS), the Programme for International Student Assessment (PISA) and, in Australia, the National Assessment Program — Literacy and Numeracy (NAPLAN). The use of these external, high-stakes assessments affect how teachers teach, and students learn, mathematics.

In some cases, curriculum shaping is a reciprocal process, where the benefits of change are two-way. Consider the impacts of technology within society, and of numeracy expectations. In the current context, employers are demanding that students exit schools with high levels of numeracy (and literacy). As a result, there is a much heavier emphasis on numeracy in education. Similarly, schools recognise the value of technology as a learning tool that enables students to exit schools with a strong appreciation of how technology can be used to enhance mathematical work and thinking.

Teaching mathematics in modern society

We use the term 'modern society' throughout this book to refer to the contemporary context of education. 'Modern' is more appropriate than terms such as 'Western' because it does not support a notion that Western views and approaches are more valid than those of Eastern or indigenous cultures. Instead, it suggests that the curriculum reflects a contemporary view of education that embraces new approaches to teaching, such as the use of digital resources (the internet, computers, tablet devices, and so on).

In the times in which our students live, technology, globalisation, the information age and very different patterns of family, leisure and work have brought changes to society, work, schools and life. We use the term 'modern society' to provoke thinking about the age in which we live and the quite different lives of contemporary young people, and to consider how these have changed since 'the old days'. Educational researchers have underscored that modern society is different. Many cultures — such as those that embrace Eastern philosophies, or indigenous cultures seeking to gain access to contemporary ways of thinking and learning — also live in the modern world. Curricula in schools must reflect the changes occurring in the wider society to ensure that schools adequately prepare students for the world beyond compulsory schooling.

Mathematics classrooms

Most young people are now growing up in technology-rich environments. They do not remember a time when you had to physically get up to change the television station (remote controls do that) — indeed, many are now watching television using streaming technologies. Cooking in pre-programmed microwaves happens at the touch of a button. Today's young people are generally technologically savvy.

One of the biggest growth areas in employment is self-employment, which means that many young people will be creating jobs for themselves in positions that will not even exist when they exit school. Our students are growing up immersed in an information-rich society — they no longer have to search through the relatively few books in a school or a local library, but can instead undertake searches on computers or handheld devices that may yield thousands or even millions of hits. The skills they require in order to be able to search for and identify key information are very different from those they needed when only 'page-based' texts were available.

Students growing up in this technology-rich world have become used to multiple sources of information input — they are constantly bombarded with short bursts of infotainment, as well as brief snippets of information from television and other media. They are able to fragment and reconstruct images (such as maps) in ways unimaginable in former times (Lowrie 2003), as well as practise algebraic thinking through the study of curves made possible by dynamic modelling software (Padula 2014). Commonly used terms to describe contemporary students, such as 'cyberkids' or 'digital natives', although contested, recognise that their dispositions towards learning have been formed by the wider social conditions in which they have grown up. Traditional models of teaching and learning need to reflect, and sometimes to challenge, these changed circumstances.

The mathematics education of the students of modern society must be considered in light of this. Students need to develop mathematical ways of seeing and interpreting the world; they need to develop strong problem-solving skills; they need to be numerate; and, most importantly, they must have a disposition towards using mathematics to solve the problems they confront. School mathematics needs to adopt pedagogies that will cater for diversity within a classroom. The old models of seated individual work — found in what might be termed 'traditional mathematics teaching'— are possibly contributing to the problems that emerge as students progress through school. For considerable numbers of students in the years of upper primary and lower secondary school, the teaching that they encounter can lead to many negative feelings and misleading learnings about mathematics (Larkin & Jorgensen 2016).

Eastern philosophies that focus more on the wellbeing of learners are impacting the pedagogies used in Western classrooms. Notions of happiness and mindfulness, as per Buddhist philosophy, are gaining traction in teaching approaches. Yoga is finding a place in the classroom, too. These approaches are assisting learners to focus and be 'in the moment', rather than be distracted by the numerous stimuli that abound in the contemporary world.

New models of teaching mathematics

The mathematics curriculum encountered by students prior to the 1960s focused on arithmetic and operations. Most of the mathematics education developed after that time in Western countries emerged post-Sputnik, when the race to the moon had become a race for intellectual superiority, with mathematics seen as the linchpin of success. The 'New Mathematics' contributed to a lock-step approach to teaching mathematics, with hierarchies in orders and sequences of teaching (Brown et al. 1998). The 1970s witnessed a boom in the ways in which mathematics curricula were organised; most were not research-based, but rather were influenced by arguments of logic and reason. A hierarchical approach to mathematics teaching ('skill, drill and kill', before application and problem-solving) was implemented in most Western classrooms, and for many teachers and systems, such an approach has become a way of life. Brown and colleagues (1998) argue that much of what was written in terms of mathematics curriculum reform had very little research base, thus raising questions about the validity of the curriculum itself.

In more recent times, there has been a growing awareness that such approaches are not resulting in positive learning outcomes. Indeed, as Clements (1989) argues, all that students learn from ten years of compulsory schooling (which, in most countries, may be extended to twelve to thirteen years) is that they cannot do mathematics! However, not all countries have bought into this approach to teaching. Research, while somewhat dated, emanating from these countries, particularly the Netherlands (see Anghileri 2001, 2006; Beishuizen 1999; Buys 2001; Treffers & Beishuizen 1999; van den Heuvel-Panhuizen 2001), has focused on developing new methods and approaches to teaching mathematics. This research is now being adopted in many countries, as it has been shown to enhance the understandings of numbers, particularly seeing numbers, in a very flexible way (Anghileri 2006; Revina & Leung 2018). The program, referred to as 'Realistic Mathematics' (van den Heuvel-Panhuizen & Drijvers 2014), has sought to develop deep number sense from real-world examples.

The Netherlands did not embrace the New Mathematics movement, instead focusing its efforts on how students think mathematically (Treffers 1991). There is now a substantive body of knowledge drawing on students' thinking that has not been constrained by New Mathematics. From this, curricula have been developed that draw on students' understandings, build on them, and move progressively towards abstract and formal mathematical processes. Dutch mathematics reformers call this 'progressive mathematisation'. Occurring in parallel with this work has been the development of constructivist theory and a general awareness that students actively construct meaning from their experiences. This latter work has had a powerful influence on mathematics education, with it increasingly being recognised that students' individual understandings are based on their lived experiences.

These twin movements have emerged at a time when it was being recognised that many of the old, 'traditional' methods of teaching mathematics were failing too many students. This made the moment ripe for identifying more valid methods of teaching mathematics, and many countries, states and provinces adopted new methods of mathematics teaching and learning. More recently, methods such as direct or explicit instruction have seen a resurgence in traditional behaviourist approaches to teaching, and bring their own set of challenges for the authentic learning of mathematics. The use of approaches such as direct instruction, which is founded on a strong behaviourist approach to teaching, is guided by the strong influence of conservative politics advocating a return to 'the basics', as well as the ongoing questioning of teacher quality. Teacher standards and curriculum/assessment requirements are now placing unprecedented demands on teachers. This makes the work of the mathematics teacher very complex and bound by many compliancy regulations.

Content and pedagogy

Contemporary approaches to teaching mathematics need to encourage two aspects: content and pedagogy.

Content is the intellectual integrity of the subject. It is where students learn, apply and appreciate mathematics, and where deep learning and deep knowledge are paramount to learning experiences. Importantly, students are able to make connections between the mathematics they learn and other curriculum areas, as well as with the world beyond school. It is important for them to develop an appreciation of how mathematics is an informing discipline that has importance and relevance to many spheres of life.

Pedagogy relates to developing supportive environments where student diversity is recognised, and practices are developed that value and build upon the different backgrounds and knowledges that students bring to the mathematics classroom. Good pedagogy is about teachers developing inclusive practices to build and extend their students' knowledge and confidence in using and applying mathematics. Classrooms are places where students understand the expectations teachers have of them, and the work they are to undertake. In developing inclusive practices, intellectual integrity should also be preserved. Teachers need to value students and to believe that all students can learn mathematics.

Research on productive pedagogies (Education Queensland 2001; Mills & McGregor 2016) states that good pedagogy is about high intellectual engagement and helping students to see and make connections; it is learner-centred, with each individual's knowledge and culture valued, and students feeling supported in their learning. Many teachers bemoan issues of behaviour management in mathematics classes, but in a large longitudinal study on teaching (Education Queensland 2001), it was found that many of the elements of productive pedagogies were absent from the 2000-plus classrooms observed. Students were not engaged in deep learning about and through mathematics. Often, it was found that students undertook busywork during mathematics lessons (e.g. sticking butterflies on paper, rather than engaging in discussion about area) but did not engage in much, if any, deep mathematical learning. It must be asked whether students in mathematics classes engage in behaviour that is subsequently construed as 'misbehaviour' because they are bored or because the pedagogy is unsound.

Teachers can make a difference

Teachers and teaching can make a significant difference to students' learning outcomes in school mathematics. In a large study of effective teachers, Hattie (2014) reported that it was not the school that made the difference to students, but individual teachers. Teachers have a powerful influence over what and how students learn. Through the provision of an appropriate learning environment in which content and pedagogy match the backgrounds, needs and interests of individual students, all students can learn mathematics.

The power of teachers' beliefs

In a study exploring the characteristics of effective teachers of numeracy, Askew and colleagues (1997) concluded that one of the most important influences on learning was the teacher's belief that all students could learn mathematics. Often, values and stereotypes influence how behaviours and actions are interpreted and implemented. Teachers who believe that some students, due to their backgrounds or behaviours, are unable to learn mathematics will ultimately create learning environments that construct the expected outcomes. This has been shown to be the case in many studies. For example, in a seminal study (Rosenthal & Jacobsen 1969), it was shown that when students were assigned scores randomly as they commenced study in a new class at the beginning of term, the teacher, believing that these were the students' academic scores, interacted with different students in particular ways. By the end of the teaching term, the students' results for that class correlated closely with the scores they had been assigned randomly at the term's beginning.

This study (and many subsequent ones) highlights how powerfully teachers' views of their students influence the ways in which they organise learning experiences. In an influential study undertaken in New Zealand, Bishop and Berryman (2006) investigated how to improve the educational achievement of Maori students, finding that if a teacher believes that all students can learn mathematics, then learning environments are likely to reinforce this belief. The important work undertaken by Sarra (2012) and others (e.g. Buxton 2017) in relation to Aboriginal and Torres Strait Islander learners has focused on teachers having high expectations of learners. Similarly, if teachers believe that the best way to learn mathematics is by making authentic connections with real-life examples, or by incorporating digital technologies to support students' mathematical understanding, then their teaching and learning environment will reflect these beliefs. The classroom is likely to be peppered with equipment and displays demonstrating the links between mathematics and the world beyond schools. In good teaching, teachers must be mindful of the influence of their beliefs about teaching and learning mathematics, and of the influence these have on their students' learning outcomes.

(Continues…)

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Excerpted from "Teaching Mathematics in Primary Schools"
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Copyright © 2020 Robyn Jorgensen, Shelley Dole and Kevin Larkin.
Excerpted by permission of Allen & Unwin.
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Table of Contents

About this book xi

About the authors xiii

1 The changing face of school mathematics 1

Mathematics education and society 1

Teaching mathematics in modern society 3

Mathematics classrooms 3

New models of teaching mathematics 5

Content and pedagogy 6

What is mathematics? 9

Where does mathematics come from? 12

The mathematics curriculum 12

Technology for thinking mathematically 17

Integrating curriculum 20

Integrating mathematics with other curriculum areas 20

Review questions 22

Further reading 23

References 23

2 Theories of learning mathematics 26

Why study theories of learning mathematics? 26

Behaviourist theories 28

Cognitive theories 29

Sociocultural theories: the influence of Vygotsky 33

Self-concept theories 35

Modern society 38

Theory into practice 40

Review questions 40

Further reading 41

References 42

3 Language and mathematics 44

The language of mathematics 44

Word problems 50

Language error analysis 54

Mathematical literacies 55

Making language explicit 56

Review questions 57

Further reading 58

References 58

4 Diversity and equity 61

Equity and equality 62

Cognitive models of difference 63

Social models of difference 64

Teachers' beliefs 65

Mathematical mindsets 67

Home-school differences 68

Knowledge and worldviews 72

Implications for school mathematics 75

Review questions 77

Further reading 77

References 78

5 Planning for teaching 80

Planning 81

Why plan? 82

Planning: what do I want to teach? 83

Teaching: how will I teach? 87

Evaluating teaching: how can I determine success? 91

Planning the learning environment 92

Planning for substantive learning 97

Review questions 97

Further reading 98

References 98

Assessment 100

Why assess? 100

Models of assessment 103

What to assess 106

Planning assessment 108

Assessment tools 110

Student writing 120

Tests 121

Rubrics 121

Opening up assessment 122

Communicating results 124

Data-driven pedagogy 126

Review questions 127

Further reading 127

References 128

7 Working mathematically 130

Working and thinking mathematically 130

Problem-solving 132

Problem-posing 139

Making connections 140

Thinking mathematically 141

Communicating mathematically 142

Review questions 145

Further reading 145

References 145

8 Numeracy 147

The numerate person 150

Defining numeracy 152

Numeracy across the curriculum 154

Numeracy is everyone's business 156

Building numeracy knowledge in the prior-to-school years 157

Building teachers' knowledge of numeracy 157

Review questions 159

Further reading 160

References 160

9 Early number 162

Number sense 162

Pre-number 164

Subitising: group recognition 166

Counting 167

Number benchmarks 173

Types of numbers 175

Numeration and place value 178

Addition and subtraction 187

Multiplication and division 188

Review questions 191

Further reading 191

References 192

10 Basic facts, mental computation and algorithms 193

Working flexibly with number 194

Basic facts as the foundation for computation 196

Basic facts of addition 197

Basic facts of subtraction 207

Addition and subtraction: mental computation 207

Basic facts of multiplication 208

Basic facts of division 216

Mental computation and estimation 217

Promoting mental computation in the middle years 220

Written computation and standard algorithms 222

The value of concrete materials 223

Principles for using concrete materials 223

The addition algorithm 224

The subtraction algorithm 227

The multiplication algorithm 231

The division algorithm 238

Teaching written algorithms in the middle years 242

Summary 242

Review questions 243

Further reading 244

References 244

11 Rational number 246

Common and decimal fractions 246

Whole number and rational number connections 247

Fractions 248

Decimals 259

Ratio and proportion 264

Promoting language and mental images of ratio and proportion 270

Rate 276

Per cent 277

Teaching rational number in the middle years 284

Summary 285

Review questions 285

Further reading 286

References 286

12 Patterns and algebra 288

Patterning 289

Number, arithmetic and algebra 297

The importance of algebra in the primary school 301

Teaching algebra in the middle years 302

Review questions 303

Further reading 303

References 303

13 Measurement 305

What is measurement? 305

Teaching measurement 311

Length 319

Area 321

Volume and capacity 324

Mass 326

Time 327

Temperature 331

Money 331

Making connections between sub-strands 333

Teaching measurement in the middle years 336

Review questions 337

Further reading 337

References 337

14 Statistics and probability 340

What are probability and statistics? 340

Teaching probability (chance) 343

Data collection, representation and interpretation 351

Measures of central tendency 362

Statistical literacy: interpretation of data representations 365

Teaching notes for graphing 366

Teaching probability and statistics in the middle years 369

Review questions 370

Further reading 371

References 371

15 Geometry 373

Topics in the geometry strand 375

Visualisation 375

The van Hiele levels of geometric thinking 377

Teaching notes 378

Spatial concepts: early activities 379

Shape and structure 380

Transformation and symmetry 387

Location and arrangement 396

Teaching geometry in the middle years 401

Review questions 403

Further reading 403

References 404

Index 407