Teaching Mathematics in Primary Schools: Principles for effective practice
432
Teaching Mathematics in Primary Schools: Principles for effective practice
432eBook
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Overview
'This is an outstanding book: it should be high on the list of any primary school teacher's set of references and a required text for pre-service teachers.'
Australian Primary Mathematics Classroom
In our technology-rich world, numeracy is just as important as the smartphone in your pocket. Students need to develop mathematical ways of seeing the world and strong problem-solving skills, and those foundations are taught in the primary school classroom.
Teaching Mathematics in Primary Schools covers the mathematical content taught in primary and middle years, always emphasising how students can connect what they learn in mathematics with other curriculum areas and with the world beyond the classroom. The authors draw on the latest international research to show how teachers can develop a rich repertoire of classroom teaching techniques, and effective planning, assessment and reporting methods. They outline approaches to creating supportive learning environments for all students, and to building their knowledge and confidence in using mathematics.
This third edition has been updated throughout and includes a new chapter on numeracy. Evidence-based uses of digital technologies to support learning and teaching are included in every chapter. With practical strategies that can be implemented in the classroom, this book is an invaluable resource for pre-service and early career primary and middle years mathematics teachers.
Product Details
| ISBN-13: | 9781000247701 |
|---|---|
| Publisher: | Taylor & Francis |
| Publication date: | 08/12/2020 |
| Sold by: | Barnes & Noble |
| Format: | eBook |
| Pages: | 432 |
| File size: | 7 MB |
About the Author
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…)
Excerpted from "Teaching Mathematics in Primary Schools"
by .
Copyright © 2020 Robyn Jorgensen, Shelley Dole and Kevin Larkin.
Excerpted by permission of Allen & Unwin.
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Table of Contents
1. The changing face of school mathematics
2. Theories of learning mathematics
3. Language and mathematics
4. Diversity and equity
5. Planning for teaching
6: Assessment
7. Working mathematically
8. Numeracy
9. Early number
10. Basic facts, mental computation and algorithms
11. Rational number
12. Patterns and algebra
13. Measurement
14. Statistics and probability
15. Geometry
Index