Technology-enabled Mathematics Education: Optimising Student Engagement 9780815392996, 9781351189392

Table of contents :
Cover
Half Title
Series Page
Title Page
Copyright Page
Dedication
Contents
List of figures
List of tables
Acknowledgements
1. Why teach mathematics with technology?
2. Teaching mathematics with technology: the current state of play
3. Mathematics teaching, technology,
and student engagement
4. Technology in early years mathematics classrooms
5. Technology in the primary mathematics classroom
6. Technology in the secondary classroom
7. Technology use in mathematics classrooms: what do school leaders, teachers, and students say?
8. Introducing the Technology Integration Pyramid (Mathematics)
Appendix 1: case study 4: lesson documentation
Appendix 2: case study 5: lesson documentation
Appendix 3: case study 6: lesson documentation
Index

Citation preview

Technology-enabled Mathematics Education

Technology-enabled Mathematics Education explores how teachers of mathematics are using digital technologies to enhance student engagement in classrooms, from the early years through to the senior years of school. The research underpinning this book is grounded in real classrooms. The chapters offer ten rich case studies of mathematics teachers who have become exemplary users of technology. Each case study includes the voices of leaders, teachers, and their students, providing insights into their practices, beliefs, and perceptions of mathematics and technology-enabled teaching. These insights inform an exciting new theoretical model, the Technology Integration Pyramid, for guiding teachers and researchers as they endeavour to understand the complexities involved in planning for effective teaching with technology. This book is a unique resource for educational researchers and students study­ ing primary and secondary mathematics teaching, as well as practising mathematics teachers. Catherine Attard is an Associate Professor in primary mathematics education at Western Sydney University. She is a multiple award-winning educator who has transformed teaching and learning in mathematics for pre-service and practising teachers. Kathryn Holmes is a Professor of Education (STEM) at Western Sydney University and began her career as a teacher of secondary mathematics and science. She has a long-standing interest in researching educational technology and its potential for enhancing teaching and learning.

Routledge Research in Teacher Education

The Routledge Research in Teacher Education series presents the latest research on Teacher Education and also provides a forum to discuss the latest practices and challenges in the field.

Values and Professional Knowledge in Teacher Education Nick Mead Professional Development through Mentoring Novice ESL Teachers’ Identity Formation Juliana Othman and Fatiha Senom Research on Becoming an English Teacher Through Lacan’s Looking Glass Tony Brown, Mike Dore and Christopher Hanley Intercultural Competence in the Work of Teachers Confronting Ideologies and Practices Edited by Fred Dervin, Robyn Moloney and Ashley Simpson Teacher Representations in Dramatic Text and Performance Portraying the Teacher on Stage Edited by Melanie Shoffner & Richard St. Peter School-Based Deliberative Partnership as a Platform for Teacher Professionalization and Curriculum Innovation Geraldine Mooney Simmie and Manfred Lang Technology-enabled Mathematics Education Optimising Student Engagement Catherine Attard and Kathryn Holmes For more information about this series, please visit: https://www.routledge. com/Routledge-Research-in-Teacher-Education/book-series/RRTE

Technology-enabled Mathematics Education Optimising Student Engagement

Catherine Attard and Kathryn Holmes

First published 2020 by Routledge 2 Park Square, Milton Park, Abingdon, Oxon OX14 4RN and by Routledge 52 Vanderbilt Avenue, New York, NY 10017 Routledge is an imprint of the Taylor & Francis Group, an informa business © 2020 Catherine Attard and Kathryn Holmes The right of Catherine Attard and Kathryn Holmes to be identified as authors of this work has been asserted by them in accordance with sections 77 and 78 of the Copyright, Designs and Patents Act 1988. All rights reserved. No part of this book may be reprinted or reproduced or utilised in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without permission in writing from the publishers. Trademark notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging-in-Publication Data A catalog record has been requested for this book ISBN: 978-0-8153-9299-6 (hbk) ISBN: 978-1-351-18939-2 (ebk) Typeset in Galliard by Cenveo® Publisher Services

We dedicate this book to all mathematics teachers and their students. The stories within provide an affirmation of the unique contexts within which you work and learn every day.

Contents

List of figures viii ix List of tables Acknowledgementsx 1 Why teach mathematics with technology?

1

2 Teaching mathematics with technology: the current state of play

12

3 Mathematics teaching, technology, and student engagement

29

4 Technology in early years mathematics classrooms

49

5 Technology in the primary mathematics classroom

69

6 Technology in the secondary classroom

93

7 Technology use in mathematics classrooms: what do school leaders, teachers, and students say?

111

8 Introducing the Technology Integration Pyramid (Mathematics)

123

Appendix 1: case study 4: lesson documentation Appendix 2: case study 5: lesson documentation Appendix 3: case study 6: lesson documentation Index

139 140 143 146

Figures

2.1 Relationship between Teacher Positivity Towards Technology Use for Mathematics Teaching and Perceptions of School Support. 2.2 Primary and Secondary Teacher Reported Use of Technology. 6.1 Surface Area and Volume Problem. 8.1 The Technology Integration Pyramid (Mathematics). 8.2 The TIP(M) Base: Influences on Technology Use. 8.3 Circles of Influence and Levels of Control on Technology Integration. 8.4 Mathematics (TIP(M)). 8.5 Pedagogy. 8.6 Tools. 8.7 Engagement.

22 23 104 124 124 129 131 132 134 135

Tables

3.1 The Framework for Engagement with Mathematics (FEM) (Attard, 2014) 7.1 Case Study Details

42 112

Acknowledgements

First, we thank the teachers and students who allowed us into their schools to observe the realities of today’s mathematics classrooms. Without you, this book would not be possible. Your contribution to the broader profession, through your willingness to share your practice, is admirable. Second, we also thank the school leaders who generously gave of their time to give us insight into the complexities of their school contexts and their innovative approaches to leading technology-enabled learning within mathematics classrooms. Third, we thank our colleagues in the mathematics education research community for valuable feedback and inspiration throughout the writing of this book. On a personal note, we thank Mark and Scott for their ongoing support.

1

Why teach mathematics with technology?

When you walk into an average school classroom, you are likely to see students using a range of digital mobile devices such as smartphones, tablets, laptops, or robotics. The students in our classrooms today have never known a life without technology and the Internet, yet technology use in contemporary teaching and learning appears to be inconsistent in quality, quantity, and effectiveness, particularly where mathematics is concerned (Organisation for Economic Cooperation and Development (OECD), 2016). In fact, many teachers in today’s classrooms earned their qualifications at a time when available technologies were significantly different to where they are today and where they will be in the future (Koehler & Mishra, 2009; Orlando & Attard, 2016). Although technology is viewed by some as an educational imperative (Bower, 2017), there still remain many questions about how it should be used and whether its use does, in fact, improve student learning. Many view mathematics as a difficult subject, only accessible to those who are considered “smart” (Boaler, 2009). In addition, many students become progressively disengaged with mathematics as they progress through the school years (Attard, 2014). In part, this disengagement is caused by traditional teaching practices that emphasise memorisation of procedures rather than deep understanding of mathematical concepts (Skemp, 2006). As a result, there is ample evidence (Kennedy, Lyons, & Quinn, 2014; Wang & Degol, 2014) of low participation rates beyond the compulsory years of mathematics leading to potential shortfalls in key skills required in a STEM (Science, Technology, Engineering and Mathematics)-driven workforce. Likewise, mathematics curricula have historically focused on mathematical content rather than mathematical thinking and reasoning. More recent curriculum developments promote a more process-oriented approach to mathematics that includes reasoning, problem solving, and understanding (Australian Curriculum Assessment and Reporting Authority (ACAR A), 2010; National Governors Association Center for Best Practices, Council of Chief State School Officers, 2010). The affordances of current and emerging technologies can provide teachers with the mechanism to broaden their pedagogical repertoires and redefine learning and teaching of mathematics for contemporary students, improving engagement and, ultimately, participation in mathematics.

2  Why teach mathematics with technology? Technology can and should be regarded as a disruptive pedagogy. Hedberg (2011) provides an example of technological disruption caused by the rise of digital photography. The move from chemical processing to instant, high-quality photographs that could be shared immediately and globally was considered a disruptive innovation. Although we expect and hope that the emergence of digital technology would cause significant potential disruption in mathematics education, disruptive innovation does not appear to have kept up with the pace of technology integration into our classrooms. Resistance to innovation, for a range of reasons, is common. Tangney and Bray (2013) suggest that although the affordances of mobile technology align with a social constructivist teaching approach that promotes collaboration, communication, creativity, and problem solving, their use overwhelmingly continues to be restricted to content consumption. Secondary schools, in particular, appear to remain “immune to the transformative potential of mobile learning” (p. 20). Diversity amongst schools, teachers, students, professional development opportunities, resourcing, policy, and funding exists. Tensions relating to the pedagogy of mathematics arise when, for example, standardised testing and school timetabling put pressure on teachers resulting in reduced time for innovation and experimentation with technology. Although research on the use of contemporary digital technologies in education has begun to build momentum, the report Students, Computers and Learning Making the Connection (OECD, 2016), claims that results from the Programme for International Student Assessment (PISA) in 2009 showed that computers were used less frequently during classroom lessons in mathematics than in either language or science classes. Although published in 2015, the significant time lapse between the collection of data in 2009 and its dissemination typifies one of the biggest challenges in educational technology use, that is, the lag between the conduct of research and its dissemination into classrooms. The technological advancements that would have occurred during the six-year gap are significant. Students’ social and home lives would have changed as a result. If research cannot keep up with technological advancements, how can we expect teachers to? How can we equip teachers to deal with emerging technologies as they reach classrooms? The ever-evolving nature of digital technologies coupled with their increasing ubiquity has created a challenge for teachers of mathematics to effectively integrate them into existing practices or use them to create new and innovative practices, and it is a common perception that educators have not yet become good enough at pedagogies that make the most of the available technologies (Attard, 2015; Morsink et al., 2011; OECD, 2016). This sentiment is echoed by Bray and Tangney (2017) who conducted a systematic review of research relating to technology-enhanced mathematics education and found that the usage of technology is mostly confined to an augmentation of existing classroom practices. Furthermore, digital technology challenges teachers of mathematics to conceptualise both the design of new pedagogic approaches and tools, as well as the kinds of knowledge that may be accessed through such tools (Hoyles & Noss, 2009).

Why teach mathematics with technology? 3 If we consider that the OECD report draws on data collected prior to the introduction and popularity of mobile digital devices, it could be argued that the challenges for mathematics teachers continue to become more significant, particularly in light of the increasing popularity of bring your own device (BYOD) programs. Although the structure of BYOD programs varies, many teachers are faced with having to deal with teaching students who are using a range of devices, operating systems, and software applications at any one time. The rate of technological development implies that the technical and pedagogical complexities currently experienced in contemporary mathematics classrooms will continue to increase.

Who are our students? The term “digital natives” was coined by Prensky (2001) to describe those who have been immersed in technology from childhood. Those born prior to personal computers and the Internet are referred to as “digital immigrants”. It would be safe to assume that in most westernised countries we could expect that a large number of students entering formal schooling have had some, if not many interactions with digital technologies. However, as with all other aspects of learning, it would be dangerous to make assumptions about the level of skills and range of experiences each of our students bring to the classroom. Likewise, we cannot assume that those amongst us who are digital immigrants do not have an affinity with technology or that we lack the skills to use them effectively within educational contexts. A common criticism of Prensky’s views is that there is little regard for what is termed the “digital divide” that exists between students’ profiles and preferences (Bower, 2017). We also cannot assume that the “digital natives” in our classrooms have high levels of technological proficiency or engage in critical analysis of information from online sources. This sentiment is echoed by others, who believe that although adoption rates of a limited number of technologies may be high, other technologies have not been used to their full potential (Kennedy, Judd, Churchward, Gray, & Krause, 2008, as cited in Judd, 2018). Let’s consider the educational landscape at the time of Prensky’s work. In 2001, the technology (if any) that appeared in classrooms was limited and immobile (apart from hand-held calculators). Less than ten years later, a new generation of mobile technologies began to appear in classrooms caused by the Internet becoming more accessible than ever. Outside the school, engagement in social media has become the new norm. Access to an unlimited and inexpensive range of software applications (apps) through the use of mobile devices has increased exponentially. The educational landscape has changed dramatically in terms of the range of technological resources that have rapidly become available. Students have also experienced an increase in their access to technology in their personal lives. Similarly, it is arguable that those who were initially termed “digital immigrants” now experience technology as a ubiquitous part of their lives too, and, as Judd (2018) argues, Prensky’s terminology may be of little or no value in contemporary educational contexts.

4  Why teach mathematics with technology? When considering the lives of young people, Gardner and Davis (2013) characterise the current generation as the “app generation”, describing them as being immersed in apps, “seeing their lives as a string of ordered apps, or perhaps, in many cases, a single, extended, cradle-to-grave app” (p. 7). They go on to describe how apps allow us to “take care of ordinary stuff and thereby free us to explore new paths” (p. 9). Gardner and Davis also discuss how apps that allow us to engage in new opportunities can be considered “app-enabling”. Conversely, when the use and reliance on apps restrict and determine procedures, choices, and goals, users can be considered as “app-dependent”. This thinking has implications for the way we utilise technology in mathematics education. Some uses of technology may have limited or no added value to teaching and learning, continuing a dependence on traditional pedagogies. For example, the use of drill and practice apps often simply replicate a textbook or worksheet style of lesson (Attard, 2015). Alternatively, other apps may enable us to reconceptualise the curriculum, our pedagogy, and the way students learn and interact with mathematics. Prior to Gardner and Davis’s research, Hoyles and Noss (2009) had considered the role of digital technologies in mathematics education in a similar manner. They proposed that an important affordance of digital technology is in “outsourcing processing power from being the sole preserve of the human mind, to being capable of being undertaken by a machine” (p. 135). Students often become bogged down in procedures and lose touch with the mathematical problem being tackled, making processing errors and perhaps losing motivation in the task. The “app-enabling” aspect of digital technology can, in this case, have significant potential for the learning of mathematics. However, Hoyles and Noss warn that if the goal is to achieve insight rather than a correct answer, reliance on software for processing may be detrimental, resulting in “app-dependence”. Put simply, students need some understanding of how outcomes are produced and to have some ownership of the actual process.

The digital divide: where does it lie? The digital divide was mentioned in the previous section in regard to the variations in students’ technological proficiency. The concept of a digital divide needs to be acknowledged in the introduction to this book as it is often viewed by teachers as a barrier to the effective use of technology in mathematics classrooms. Dolan proposes a simple definition of the digital divide, “the binary view of the haves and have-nots” (p. 16); however, the divide is not as simplistic as it may seem and extends beyond having or not having a computer or mobile device. There are variations and inequities in access to the Internet, bandwidth capabilities, availability of software, the knowledge and skills of students and teachers in their use of technology, factors such as poverty, teacher professional development and training, and internal and external stakeholder expectations (Dolan, 2016; Orlando & Attard, 2016). Roblyer and Hughes (2019) extend these ideas, referring to technology as a double-edged sword that while potentially providing opportunities to change

Why teach mathematics with technology? 5 education and empower students and teachers, also further divides members of our society based on a number of elements that include socioeconomic status, geographical location, and disability. This belief is echoed by others (Henderson, 2011; Selwyn, Potter, & Cranmer, 2009) who further extended the concept of a digital divide to encompass the disparity between the way young people use digital media outside school and the ways in which digital media is used within the classroom. While writing this book, we specifically selected case studies that drew from a diverse range of schools, teachers, and students; from different socioeconomic areas; geographical locations; and school systems. Our case study students and teachers had access to varying numbers of devices and software applications. This was intentional, as we believe that technology can be used effectively in a range of mathematics classrooms when embedded in high-quality pedagogy and supported from school leadership and the broader school community. We suggest that in the context of mathematics education, there may be an additional divide that is related to students who are engaged and are perceived to be good at mathematics and those who are disengaged and find mathematics challenging. Perhaps the affordances offered by digital technologies in mathematics classrooms can assist in closing this particular divide.

Technology for mathematics education Emerging technologies require mathematical reasoning in order to function in today’s society, yet many adults avoid mathematics, attributing their dislike of the subject to their school experiences (Boaler, 2009; Clarke, 2009). It is believed that an increasingly smaller percentage of students in many countries are pursuing the study of mathematics beyond a lower secondary level, a choice seriously influenced by attitudes towards and performance in mathematics and significantly shaped by mathematics pedagogy. This, coupled with common beliefs that mathematics is a “hard, technical subject where there is an emphasis on learning rules, constant practice and little room for creativity” (McPhan, Moroney, Pegg, Cooksey, & Lynch, 2008, p. 24) and an emphasis on summative assessment has led to serious student disengagement (Bray & Tangney, 2017). We believe that digital technology has the potential to change negative perceptions of mathematics due to the affordances that promote not only the content of mathematics but also the processes of reasoning, communicating, problem solving, fluency, and understanding. The use of digital technologies in mathematics classrooms also offers us an opportunity to rethink how we adapt the teaching and learning tools we currently use and, perhaps, more importantly, how we adapt the mathematics to be learned (Hoyles & Noss, 2009). The use of and access to technology is regarded as a necessity in today’s classrooms, and in some countries, its use is embedded in mandated curricula (e.g., Australian Curriculum Assessment and Reporting Authority (ACAR A), 2010). It is widely agreed that digital technology does have significant potential to disrupt traditional teaching approaches. Although much of the literature on

6  Why teach mathematics with technology? contemporary technology use in education does not relate specifically to mathematics, there are many advantages (and, of course, disadvantages) that can be applied within mathematics classrooms. One of the most significant benefits of technology involves opportunities for teachers to personalise learning and provide differentiation (Hilton, 2018; Robinson & Sebba, 2010). While this can be achieved without technology, teachers can take advantage of the affordances of technology to vary instruction and provide learner-controlled learning paths. Contemporary education apps often provide teachers with frequent formative assessment and progression data aligned to curriculum standards and provide instant feedback to students. In some apps, this then leads to tailored learning pathways that can extend learning or provide intervention (Roblyer & Hughes, 2019). The use of technology has the potential to disrupt the way mathematics education is delivered and assessed, and as a result, improve student engagement with the subject.

Contemporary technologies and their affordances As technology has evolved, devices have become more mobile resulting in a new educational landscape. Although still common in schools today, the traditional computer lab that required timetabled lessons to teach students how to use technology is no longer considered appropriate. Contemporary technologies in mathematics classrooms have progressed from the use of scientific calculators and desktop computers, to interactive white boards and laptops, to an ever-increasing range of mobile devices and software applications that provide both affordances and constraints to influence mathematics teaching and learning (Flegg, 2015). In general, contemporary technologies offer affordances that provide opportunities to create, innovate, redesign learning spaces, and offer deeper learning approaches. Practices that arise from the use of these technologies have often been referred to as mobile learning, or ‘m’learning. The “mobile” element of ‘m’learning refers to more than just the mobility of the device. It also refers to the mobility of people within a physical space, social mobility in how people connect with one another, and learning the occurs both within formal and informal contexts (Sharples, Arnedillo-Sanchez, Milrad, & Vavoula, 2009, as cited in Bower, 2017, p. 263). The increase in popularity of mobile technologies has resulted in many schools investing in tablet devices or requiring students to bring their own tablets or laptops. In some schools, the use of robotics has become a popular way to merge mathematics skills through coding activities. These practices, although not always embedded or integrated into mathematics learning, are considered to be accelerating the rate of technology adoption in education more generally, including the emergent view of coding as a literacy and the rise of integrated STEM learning (Freeman, Adams Becker, Cummins, Davis, & Hall Giesinger, 2017). The relatively low cost coupled with the vast range of affordances offered by mobile technologies has made them appear to be an attractive solution to address

Why teach mathematics with technology? 7 declining student engagement (Attard, 2018). Mobile technologies provide educators with opportunities to interact with students wherever they are located and make it easier to individualise learning, promote collaboration, and provide “just in time” learning (Bower, 2017). This is in contrast to the widely documented traditional approach to teaching that in the past was a typical feature of mathematics classrooms (Boaler, 2009; Even & Tirosh, 2008; Tangney & Bray, 2013) and is significantly challenged by the very nature of mobile technologies. Rather than operating on a “just in case” basis through a teacher-centred didactic, textbook-based approach (Even & Tirosh, 2008; Goos, 2004; Ricks, 2009), there are opportunities to adopt a more student-centred approach that promotes “just in time” learning, thus allowing teachers to explicitly highlight the relevance of mathematics through a significantly broader range of real-life learning contexts and applications. Contemporary digital devices enable learning anywhere and anytime and the ability to capture, annotate, and share multimedia. Coupled with the use of learning management systems, these devices offer entirely new ways for students to learn and engage with mathematics. There are a range of pedagogical opportunities and implications afforded by the use of mobile technologies that extend beyond the nature of the tools and software applications utilised to the learning intentions of the teacher and the ways in which students interact with the technologies and with each other (Calder, Larkin, & Sinclair, 2018). Mobile devices provide tools that are dynamic, graphical, and interactive, providing students with opportunities to “explore mathematical objects from different but interlinked perspectives, where the relationships that are key for mathematical understanding are highlighted, made more tangible and manipulable” (Hoyles & Noss, 2009, p. 132). Students are able to make practical use of technology in mathematics for “genuine and productive purposes, rather than for the application of rote-learned formulae and procedures to contrived scenarios” (Bray & Tangney, 2017, p. 257). Further, mobile devices afford connectivity within and between classrooms and collaboration and promoting collective reflection and manipulation that can be synchronous or asynchronous (Hoyles & Noss, 2009).

The research for this book Through a series of case studies, we present the ways effective teachers use digital technology to enhance the teaching and learning of mathematics, regardless of device, software, or platform. This research explored the effective use of technology in ten mathematics classrooms from pre-school to Year 12 (the final year of schooling in Australia), conducted within eight Australian schools. Each case consisted of a classroom teacher, one member of the school leadership team, and a focus group of five to six students. We examined the practices of teachers of mathematics who were recognised by their peers as effective users of digital technologies, providing examples of how they used technology successfully. We found that there is no one perfect solution. This is a book with a focus on the

8  Why teach mathematics with technology? realities and diversities of classrooms and classroom contexts. We explore mathematics teaching from the early years through to the senior years, in classrooms ranging from government schools in low socioeconomic areas to elite private schools and classrooms where access to technologies ranged from a small number of shared devices to 1:1 programs or BYOD programs. In each of the case studies, data were collected from three participant groups: the case study teacher, a school leader, and a sample of their students. Case study teachers were identified through professional associations, professional teaching networks, and referrals, as teachers who are considered by their peers as effective and innovative users of technology within mathematics classrooms. The ten teachers came from a range of schools in terms of system, geography, and socioeconomic status. Students participating in focus groups were selected by their teachers, from the pool of students who had returned consent forms, as a representative sample of the case study teachers’ students. Where possible, students were chosen to represent a mixture of gender, ability, and attitudes towards mathematics. Data collected from the case study teacher included classroom observations, lesson plans, and interviews. Students participated in a focus group discussion, and a nominated school leader participated in an interview. Using a range of data sources aligns with recommendations by Yin (2008) to provide an in-depth picture of each case with a detailed analysis of each teacher’s classroom practices and their students’ perceptions. We used semi-structured interviews with teachers and leaders as a way of gathering detailed information in each of the cases. This allowed each participant to respond to the same set of prompts and increased the comparability of results while also allowing us to probe deeper into their responses (Cohen, Manion, & Morrison, 2018). The interviews provided an opportunity for us to investigate the teachers’ practices and beliefs and school policies relating to the use of technology. Classroom observations were conducted to give us contextual information about the interactions between teachers, students, devices, and mathematical content (Cohen et al., 2018). The observations were also used to inform our questions for the teacher interviews and student focus groups.

About the book We don’t believe it is reasonable to expect a one-size-fits-all solution to the perplexing problem of effective technology use in mathematics education. After all, today’s technology is likely to be tomorrow’s museum exhibit. Likewise, the diverse contexts in schools, classrooms, amongst students, teachers, cultures, etc., means there can be no perfect solution or recipe for how technology should be used. Deciding what technology is best for which students and how to use it is a continuing challenge. Dolan (2016) posits that “the answer may lie more in understanding the practices inherent in using technology, rather than focusing on specific types of devices or tools” (p. 33). This is the stance that we take. We wrote this book to look beyond the day-to-day

Why teach mathematics with technology? 9 technical issues of technology integration to focus on the core business of improving the teaching and learning of mathematics from the early years to the senior years of schooling, that is, a focus on mathematics and pedagogy and student engagement. Although there are countless publications, websites, frameworks, and resources providing advice on using technologies in educational settings, there is very little guidance specifically designed for teachers of mathematics. The discipline of mathematics education is continually faced with significant challenges arising from student disengagement, declining numbers of students taking up the study of mathematics beyond the compulsory years, and, in western countries, a societal acceptance that it’s acceptable to declare oneself as “not good at mathematics”. The way mathematics is perceived and, indeed, the way it is still taught in many classrooms does not appear to have kept up with the way the world has evolved for our students, where technology pervades all aspects of their lives. To engage today’s learners, we need to make learning relevant by embracing technological tools, taking full advantage of the affordances that they offer for the learning of mathematics. This book is divided into three sections. In Section One, we set the scene for this book by looking at the types of technology that are currently being used in mathematics classrooms and what we know from research about effective ways of teaching mathematics. In Chapter 2, we present the results of a survey of over 400 teachers of mathematics to get a sense of the current technology landscape, which technological tools teachers are using, how they are using them, and the factors that support or impede that use. Then, in Chapter 3, we delve into the research literature related to effective mathematics teaching with technology. We describe and critique existing theoretical frameworks for technology integration and focus on the importance of student engagement, particularly in mathematics. In Section Two, we present ten case studies of teachers ranging from preschool to Year 12. These teachers were identified by their peers as exemplary users of technology for teaching mathematics. In Chapter 4, we introduce Sharon, Ashleigh, and Rebecca, who teach mathematics in early years classrooms using a broad range of technological tools. Then in Chapter 5, we meet Loretta, Bec, and Jessica, who teach in primary school classrooms with varying levels of support provided for technology use. Finally, in Chapter 6, we tell the stories of four secondary schools through an examination of the teaching of John D., Ian, John B., and Joel. In Section Three, we begin with Chapter 7, where we hear the voices of school leaders, case study teachers, and their students, to find out about their experiences of technology use for teaching and learning mathematics. Finally, in Chapter 8, we introduce the Technology Integration Pyramid (Mathematics)—a new, multi-dimensional framework designed not only to illuminate the complexity of technology use when teaching mathematics but also to act as a guide for teachers in making pedagogical decisions that will best support student engagement and learning in mathematics.

10  Why teach mathematics with technology?

References Attard, C. (2014). “I don’t like it, I don’t love it, but I do it and I don’t mind”: Introducing a framework for engagement with mathematics. Curriculum Perspectives, 34, 1–14. Attard, C. (2015). Introducing iPads into primary mathematics classrooms: Teachers’ experiences and pedagogies. In M. Meletiou-Mavrotheris, K. Mavrou, & E. Paparistodemou (Eds.), Integrating touch-enabled and mobile devices into contemporary mathematics education (pp. 193–213). Hershey, PA: IGI Global. doi:10.4018/978-1-4666-8714-1.ch009 Attard, C. (2018). Mobile technologies in the primary mathematics classroom: Engaging or not? In N. Calder, K. Larkin, & N. Sinclair (Eds.), Using mobile technologies in the teaching and learning of mathematics (pp. 51–65). Cham: Springer. doi:10.1007/978-3-319-90179-4_4 Australian Curriculum Assessment and Reporting Authority (ACARA). (2010). The Australian curriculum: Mathematics. https://www.australiancurriculum.edu.au/f-10-curriculum/ mathematics/ Boaler, J. (2009). The elephant in the classroom: Helping children learn and love maths. London, England: Souvenir Press Ltd. Bower, M. (2017). Design of technology-enhanced learning: Integrating research and practice. Retrieved from http://ebookcentral.proquest.com/lib/uwsau/detail.action? docID=4717043 Bray, A., & Tangney, B. (2017). Technology usage in mathematics education research – A systematic review of recent trends. Computers & Education, 114, 255–273. doi:10.1016/j.compedu.2017.07.004 Calder, N., Larkin, K., & Sinclair, N. (2018). Mobile technologies: How might using mobile technologies reshape the learning and teaching of mathematics? In N. Calder, K. Larkin, & N. Sinclair (Eds.), Using mobile technologies in the teaching and learning of mathematics (pp. 1–7). Cham: Springer. doi:10.1007/978-3-319-90179-4_1 Clarke, D. (2009). Mathematics teaching and learning: Where to? Learning Matters, 14, 3–8. Cohen, L., Manion, L., & Morrison, K. (2018). Research methods in education (8th ed.). London, England: Routledge. Dolan, J. E. (2016). Splicing the divide: A review of research on the evolving digital divide among K-12 students. Journal of Research on Technology in Education, 48(1), 16–37. doi:10.1080/15391523.2015.1103147 Even, R., & Tirosh, D. (2008). Teacher knowledge and understanding of students’ mathematical learning and thinking. In L. D. English (Ed.), Handbook of international research in mathematics education (2nd ed., pp. 202–222). New York, NY: Routledge. Flegg, N. (2015). Discussion of innovative ideas to address mathematics anxiety and technology issues for students transitioning to high school. In P. Redmond, J. Lock, & A. D. Danaher (Eds.), Educational innovations and contemporary technologies (pp. 55–68). Hampshire, England: Palgrave Macmillan. Freeman, A., Adams Becker, S., Cummins, M., Davis, A., & Hall Giesinger, C. (2017). NMC/CoSN horizon report: 2017 K-12 edition. Retrieved from https://www.nmc.org/ publication/nmccosn-horizon-report-2017-k-12-edition/ Gardner, H., & Davis, K. (2013). The app generation. New Haven, CT: Yale University Press. Goos, M. (2004). Learning mathematics in a classroom community of inquiry. Journal for Research in Mathematics Education, 35, 258–291. Hedberg, J. G. (2011). Towards a disruptive pedagogy: Changing classroom practice with technologies and digital content. Educational Media International, 48(1), 1–16. doi:10.1080/09523987.2011.549673

Why teach mathematics with technology? 11 Henderson, R. (2011). Classroom pedagogies, digital literacies and the home-school digital divide. International Journal of Pedagogies and Learning, 6(2), 152–161. Hilton, A. (2018). Engaging primary school students in mathematics: Can iPads make a difference? International Journal of Science and Mathematics Education, 16(1), 145– 165. doi:10.1007/s10763-016-9771-5 Hoyles, C., & Noss, R. (2009). The technological mediation of mathematics and its learning. Human Development, 52, 129–147. doi:10.1159/000202730 Judd, T. (2018). The rise and fall (?) of the digital natives. Australasian Journal of Educational Technology, 34(5), 99–119. doi:10.14742/ajet.3821 Kennedy, J., Lyons, T., & Quinn, F. (2014). The continuing decline of science and mathematics enrolments in Australian high schools. Teaching Science, 60(2), 34–46. Koehler, M. J., & Mishra, P. (2009). What is technological pedagogical content knowledge? Contemporary Issues in Technology and Teacher Education, 9, 60–70. McPhan, G., Moroney, W., Pegg, J., Cooksey, R., & Lynch, T. (2008). Maths? Why not? Canberra: Department of Education, Employment and Workplace Relations. Morsink, P., Hagerman, M. S., Heintz, D., Boyer, M., Harris, R., Kereluik, K., & Hartman, D. K. (2011). Professional development to support TPACK technology integration. Journal of Education, 191(2), 3–16. National Governors Association Center for Best Practices, Council of Chief State School Officers. (2010). Common Core State Standards. Retrieved July 12, 2019, from Common Core State Standards Initiative website: http://www.corestandards.org/ Math/ OECD. (2016). Students, computers and learning: Making the connection. France. PISA. Orlando, J., & Attard, C. (2016). Digital natives come of age: The reality of today’s early career teachers using mobile devices to teach mathematics. Mathematics Education Research Journal, 28(1), 107–121. doi:10.1007/s13394-015-0159-6 Prensky, M. (2001). Digital natives, digital immigrants. On the Horizon, 9, 1–6. Ricks, T. E. (2009). Mathematics is motivating. The Mathematics Educator, 19, 2–9. Robinson, C., & Sebba, J. (2010). Personalising learning through the use of technology. Computers & Education, 54(3), 767–775. doi:10.1016/j.compedu.2009.09.021 Roblyer, M. D., & Hughes, J. E. (2019). Integrating educational technology into teaching: Transforming learning across disciplines (8th ed.). New York, NY: Pearson. Selwyn, N., Potter, J., & Cranmer, S. (2009). Primary pupils’ use of information and communication technologies at school and home. British Journal of Educational Technology, 40(5), 919–932. Skemp, R. R. (2006). Relational understanding and instrumental understanding. Mathematics Teaching in the Middle School, 12(2), 88–95. Tangney, B., & Bray, A. (2013). Mobile technology, maths education & 21C learning. Proceedings of the 12th World Conference on Mobile and Contextual Learning (MLearn2013), 2013, 20–27. doi:10.5339/qproc.2013.mlearn.7 Wang, M.-T., & Degol, J. (2014). Staying engaged: Knowledge and research needs in student engagement. Child Development Perspectives, 8(3), 137–143. doi:10.1111/ cdep.12073 Yin, R. K. (2008). Case study research: Design and methods (4th ed.). Thousand Oaks, CA: Sage Publications.

2

Teaching mathematics with technology: the current state of play

Integrating digital technologies effectively into mathematics teaching and learning is a complex task. Although technology and digital trends are now ubiquitous in life and in educational contexts, they can be ineffective, distracting or even dangerous when not integrated into the learning process in ways that are meaningful (Attard, 2015; Freeman, Adams Becker, Cummins, Davis, & Hall Giesinger, 2017). We begin this chapter by exploring what is going on in today’s mathematics classrooms and then consider the future of technology use in mathematics classrooms. We discuss what, how, when and why technology is being used in mathematics classrooms by presenting the results of an international survey of mathematics teachers together with a review of existing literature on technology use in mathematics education. Trends such as the use of bring your own device (BYOD) programs and the evolving set of challenges that this presents to mathematics teachers practically and pedagogically are also explored, as are the barriers to effective technology. Arguably, the teaching of mathematics is different to any other school subject, and as a result, the use of mobile and digital technologies to teach mathematics presents a unique set of challenges for teachers. As mentioned in the previous chapter, teachers of mathematics face specific challenges that include widespread student disengagement. The hierarchical nature of mathematics also leads to challenges in addressing the individual needs of students. Within each mathematics classroom, whether grouped by ability or heterogeneously there are students with a range of abilities and students who have significant gaps in their conceptual knowledge. When gaps do occur in the learning of mathematics, difficulties begin to arise, and misconceptions emerge. Students disengage because they find it difficult to understand more abstract mathematical concepts. Just as you cannot build a house on faulty foundations, you cannot become a confident user of mathematics when foundational concepts are not understood. There is an abundance of research being conducted on the best use of technology in mathematics classrooms to improve understanding and increase engagement. However, Tangney and Bray (2013) argue that when exemplars of good practice are identified in collaboration between research academics and teachers, most new technologies eventually become assimilated into existing practices, stating “many interesting classroom innovations remain at the periphery of

Teaching mathematics with technology 13 practice and do not make their way into the mainstream” (p. 21). The following section provides a brief introduction to some of the ways digital technology is currently reported as being used in mathematics education. We then present findings from a survey we conducted to find out how and why teachers use technology in their mathematics lessons.

Current trends in technology use Gamification Given that almost all young people are actively involved in game playing in either a concrete or digital form, it makes sense to expect that the use of digital games in mathematics education could assist in increasing student engagement with content that may otherwise feel irrelevant to students’ everyday lives. The use of digital games may also assist in bridging the digital divide between how digital technologies are used at home and at school, as described by Selwyn, Potter, and Cranmer (2009). A significant number of consumable apps available for use on digital devices present mathematical content in the form of games. The terms “game-based learning” (GBL) and “gamification” have begun to emerge regularly in academic literature. GBL, defined as the use of video games for educational purposes (Kingsley & Grabner-Hagen, 2015), has been shown in some research to enhance motivation towards learning and academic performance. Interestingly, there are several interpretations of the definition of gamification, which is generally suggested to be the use of game design elements within a non-game context (Brigham, 2015). A teacher might gamify an activity or the teaching of a particular concept by adding achievement badges, rewards, and levels in an attempt to increase student engagement (Goehle & Wagaman, 2016; Kingsley & Grabner-Hagen, 2015). The purpose of gamification within education is the use of game elements such as rewards and game-like activities to promote learning and to engage and motivate students. In the context of this definition, several of the commonly used mathematics digital resources such as Matific, Mathletics, Manga High, Prodigy, and Kahoot can be considered as examples of the gamification of mathematics learning.

Screencasting An important affordance of mobile digital devices is their ability to allow students to record their mathematical work in multimodal forms. Generally referred to in the literature as screencasting, students can capture their work using text, images, and animations that are overlaid with an oral explanation (Calder & Murphy, 2018). Commonly available apps that support screencasting are Explain Everything, Show Me, and Educreations. Research that documents the use of screencasting in mathematics has increasingly highlighted important benefits for student learning. In recent studies, it was found that the use of screencasting

14  Teaching mathematics with technology supports a focus on mathematical thinking and problem solving rather than a focus on speed and accuracy in computation as is typical in many mathematics apps, contributing to students’ understanding of mathematics and allowing them to demonstrate multiple mathematical representations (Calder & Murphy, 2018; Galligan & Hobohm, 2018; Prescott & Maher, 2018). Furthermore, the use of screencasting has been found to promote collaboration and student autonomy, leading to increased engagement with mathematics (Attard, 2018; Ingram, Pratt, & Williamson-Leadley, 2018; Prescott & Maher, 2018).

Flipped classrooms An increasingly popular practice in mathematics classrooms is the use of a flipped classroom approach. This approach has varying definitions that range from simple provision of direct instruction via the use of video lectures through to an approach that is referred to as flipped learning, where learning is individualised according to student needs. At its most basic, the flipped classroom approach is intended to make better use of classroom time. Rather than expose students to new materials within mathematics lessons, students are expected to access pre-prepared materials prior to their lessons (Lo & Hew, 2017). Passive learning occurs outside the classroom, and class time is then used to maximise opportunities for teacher/student interaction, collaboration, provision of remediation, and application of the learning that occurs off site (Bhagat, Chang, & Chang, 2016; Weinhandl, Lavicza, & Süss-Stepancik, 2018). Pre-prepared lesson materials can range from teacher-produced videos and screencasts to the provision of mathematics resources produced with software such as GeoGebra and the use of videos produced by others such as Khan Academy. While typically used in tertiary education, flipped classrooms and flipped learning are fast becoming a popular pedagogical approach in school education and particularly in secondary mathematics classrooms. Flipped classrooms and flipped learning have several benefits that may potentially address some of the major issues in mathematics education. First, emerging research documents an improvement in student engagement due to the “anywhere, anytime” affordance of flipped classrooms. The approach provides students with greater autonomy in their learning and makes mathematics learning accessible for those who may wish to revise challenging or difficult content (Muir & Geiger, 2016). Removing the need for explicit teaching frees up classroom time allowing the teacher to work more effectively to address the learning needs of students. However, as with most approaches, there are potential disadvantages to the flipped approach. An early study into the introduction of iPads in primary mathematics classrooms found that the flipped model did not work well in one Grade 3 classroom (Attard & Curry, 2012). This was due to the difficulty in tailoring the information to the individual needs of students. The success of a flipped approach also relies on the willingness of students to engage with the materials prior to attending their classes as well as their ability to comprehend the information presented. Having unprepared students may result in unproductive

Teaching mathematics with technology 15 classroom time. Another important consideration in a flipped approach is that of equity and access to the appropriate technological tools and the Internet. Students who do not have access may be disadvantaged. The continued development of software has resulted in more opportunities for flipped or blended learning to be implemented. For example, many primary-­ based applications such as Matific and Prodigy allow teachers to differentiate learning by allocating different tasks to different students. Learning management systems such as Google Classroom and applications such as OneNote make it easier for teachers to manage flipped learning and keep track of how and when their students access information. Flipped classrooms and flipped learning approaches provide students with the opportunity to engage with content outside school, providing space and time for them to engage in the processes of mathematics during classroom time with teachers and peers.

How are mathematics teachers using technology? The mathematics being accessed through digital technologies is often dependent on the beliefs about mathematics and mathematics pedagogy held by teachers. Olive and Makar (2010) conducted an extensive literature review on the influence of technology on contexts for learning mathematics. In their discussion they use Piaget’s theory of assimilation and accommodation to describe how technology has influenced the teaching of mathematics. When teachers assimilate technology, they essentially fit the technology and its affordances into their existing practices. For example, they may replace paper-based resources such as textbooks or workshops with online materials that essentially have the same functions. When they accommodate technology, they adapt and evolve their practices to form new ways of teaching and learning mathematics. When investigating how teachers are using technology, it is also important to consider curriculum. The way teachers perceive mandated curricula influences how they use technology and how they either assimilate or accommodate them into existing practices. An example of this can be explored using the Australian Curriculum (Australian Curriculum Assessment and Reporting Authority (ACAR A), 2010), where the mathematics to be taught is articulated as a set of processes named as proficiencies (fluency, understanding, reasoning, and problem solving) through which the content strands are accessed (number and algebra, measurement and geometry, statistics and probability). Likewise, there are teachers who elect to teach mathematics using a topic-by-topic approach, compared to a more integrated approach that draws on the relationships between and amongst the strands of mathematics. It is likely that there are teachers whose beliefs about mathematics influence them to prioritise content over processes, and teachers who teach content through a process-based approach. Teachers’ combined beliefs about technology and mathematics also influence how students are required to interact with technology. If we consider a flipped learning approach, students may be passive recipients of information. In other words, they are consumers. The thousands of drill and practice apps available to

16  Teaching mathematics with technology students also require students to be consumers. On the other hand, the use of screen-casting or productivity apps such as Explain Everything rely on the student being an active creator or producer. In this way tasks move from closed to more open-ended; teacher-centred to student-centred. Teachers’ beliefs regarding assessment in mathematics also influence if and how they use technology. Traditional mathematics assessments focus more on mathematical content rather than mathematical processes, as discussed above. Bower (2017) believes using mobile technologies for assessment can be problematic due to a misalignment between traditional summative assessments that are a typical feature of mathematics education, and more contemporary approaches to assessment that feature tasks that are more open-ended. However, some mobile technologies may not have the appropriate functionality to allow students to complete intended assessment tasks (Hwang et al 2016 as cited in Bower, 2017), posing another challenge for teachers to address. Perhaps one of the biggest influences on how teachers use technology in the mathematics classroom is accessibility to devices. Despite the increasing expectations for technology to be embedded in teaching and learning, disparities still exist due to funding constraints from one classroom to another. One way that schools are dealing with this is to introduce programs that require students to bring their own devices.

BYOD programs Before we look to the future we must first acknowledge and explore the current directions being taken by schools and teachers in regard to the ways they are currently gaining access to digital devices. As would be expected, each school context is unique, and this results in a broad range of accessibility to and use of digital devices that is dependent on a number of variables including the school system, policy, funding, location, technology-related infrastructure, and socio-economic status. While many schools still function with dedicated computer labs or sets of mobile devices to be shared across class groups, BYOD programs and strategies have evolved over the past two decades as an attractive way for schools to address the constraints of educational budgets while providing access to digital technology for students. The emergence of BYOD programs is also a drive towards making digital tools available as an integral part of education rather than a series of unconnected interactions in computer labs. In addition to financial motivations, there are other reasons BYOD programs have become increasingly popular. Maher and Twining (2017) cite the 2011 Ofsted report that claims the introduction of BYOD programs not only lowers school costs but also assist in strengthening school–home connections. Others claim that BYOD programs have been adopted due to a recognition that education needs to adapt to technological changes in wider society (Parsons & Adhikari, 2016). Contemporary students have never known a world without digital devices, and in order to remain relevant, schools need to integrate digital technologies in a manner that aligns with students’ personal use. A review

Teaching mathematics with technology 17 of literature conducted in 2013 also claimed that BYOD programs emerged from increased pressure from teachers and students to use their own personal devices, making the introduction of BYOD inevitable (Technology for Learning Program - Information Technology Directorate, 2013).

What do BYOD programs look like? There are varied definitions of what constitutes a BYOD program. Maher and Twining (2017) believe BYOD is not a 1:1 strategy where schools expect students to purchase and use a specific device as a requirement of attendance. However, this is contradictory to the way many schools regard BYOD programs. Strictly speaking, a true BYOD program allows students to bring any device to school whether it be a laptop, a tablet, or a mobile phone, using any operating platform (e.g., iOS or Windows). There are many variations of this model where schools do specify a particular device for parents to purchase or lease and have requirements in regard to particular software or operating system specifications (Janssen & Phillipson, 2015). Others have distinguished between BYOD and Bring Your Own Technology (BYOT) (Lee, 2012, as cited in Maher & Twining, 2017). However, both versions do require students to bring in their personal devices to school. Lee distinguishes BYOT as allowing the learner to have more control over the device, as opposed to BYOD programs that prescribe the device/platform required. For the purpose of this chapter, BYOD is defined as any program that requires students to provide their own device. There are documented benefits and challenges relating to BYOD programs that have emerged from research. At a basic level, students are generally more familiar with their personal devices and this removes the need for technical familiarisation on the part of the student. Conversely, teachers may not be fluent with the range of devices, their platforms, and their affordances. This scenario is described by Rae, Dabner, & Mackey (2017), who conducted a case study of three New Zealand primary teachers introducing BYOD for the first time: “this variety required teachers to have a working knowledge of a range of different platforms, programs and applications given technical help was only available on a fortnightly basis”. The literature review conducted by the Technology for Learning Program - Information Technology Directorate (2013) warns that when faced with a range of devices within a true BYOD program, teachers may plan for the least powerful device because students may not have access to the same range of software. This issue has been partially overcome in recent years with the introduction of web-based applications such as Google Apps and Microsoft Online products that offer accessible productivity software. In a case study of two Australian primary schools, Maher and Twining (2017) discuss the notable increase in student engagement as a result of the implementation of a BYOD (iPad) program. Although there were challenges relating to students consistently bringing a device to school, these were overcome through the implementation of small group work where devices were shared and with

18  Teaching mathematics with technology activities that required groups to swap tablets with other groups for peer review. Observations indicated increased levels of collaboration amongst the students. Maher and Twining also note that the use of BYOD in these classrooms supported a constructivist approach to learning. However, this research is limited to one case study and there is no evidence that these observations were related to mathematics teaching and learning. One of the more commonly cited benefits of BYOD programs is that they promote a shift from teacher-centred practice to student-centred practice, bridging formal and informal learning and allowing learners to construct their own learning environments (Maher & Twining, 2017; Rae et al., 2017). BYOD programs increase student agency and have the potential to support student learning needs and encourage deeper learning. They are also claimed to provide opportunities for teachers to develop new approaches. However, these particular benefits may be a perceived as a disadvantage within some mathematics classrooms where a traditional approach to teaching may be the favoured practice. This aligns with findings from a study conducted in a New Zealand secondary school by Parsons and Adhikari (2016). They noted that teachers of different subjects utilised mobile devices in very subject specific ways. They also noted that the devices were not ideal for every classroom situation. Much has been written about a digital divide that encompasses the complex disparities relating to access and use of technology in schools and at home (Dolan, 2016). BYOD programs have the potential to both narrow and widen that divide. Mandating the purchase of devices in areas of economic disadvantage has the benefit of putting a device in every child’s hands yet puts pressure on family budgets (Maher & Twining, 2017; Rae et al., 2017). BYOD programs that allow students to bring any device to school can produce a lack of inclusivity and a have/have not culture that detracts from teaching and learning. In an audit of BYOD programs in government schools from the state of Victoria, Australia, it was found that BYOD (any device) models are more prevalent in socio-educational advantaged government schools than in socio-educational disadvantaged government schools. From the 115 schools surveyed, 53 had compulsory BYOD programs, and of these schools 69.8% did not have provision for those students who did not have devices. Another element of BYOD programs and technology integration more generally that could potentially widen the digital divide are the digital capabilities of teachers and their students. While many teachers appear to want to give students more agency in the classroom, it appears that there are several barriers. One is that teachers have found that many of their students are not “digital natives” and cannot naturally work effectively with technology without considerable guidance. (Parsons & Adhikari, 2016, p. 72) Similarly, we cannot assume teachers, regardless of experience or age, have the desired capabilities to teach well with technology. In fact, it is suggested that the

Teaching mathematics with technology 19 continual developments in technology may result in teachers being “perpetual novices” in the process of technology integration. This sentiment is reflected in a discussion by Orlando and Attard (2015) about young teachers and the assumptions made about their affinity with and ability to effectively teach with technology simply because they are regarded as “digital natives”.

Challenges and barriers in technology integration in mathematics In many countries, it is an expectation that technology is used in the teaching of mathematics, yet the challenges experienced by many teachers are often the result of a range of commonly experienced barriers. These barriers to technology integration are often discussed in literature in an attempt to understand the different ways technology is integrated, or not, among teachers who have relevant knowledge (Kim, Kim, Lee, Spector, & De Meester, 2013). As early as 1999, Ertmer (as cited in Ertmer, Ottenbreit-Leftwich, Sadik, Sendurur, & Sendurur, 2012) distinguished between two types of barriers to technology use. Ertmer’s barriers continue to feature in contemporary literature. Firstorder barriers are described as those that are external to the teachers, including resources, professional development, and support. Ertmer et al. (2012) describe second-order barriers as those that are internal, including the teachers’ confidence, beliefs relating to learning, and beliefs relating to the perceived value of technology to the process of teaching and learning. Although Ertmer et al., believe that the increase in access to technology in recent decades has contributed to fewer, and in some cases no first-order barriers, others disagree. It is claimed that a range of external or environmental barriers continue to change and evolve as a result of rapid advances of technology (Mueller, Wood, Willoughby, Ross, & Specht, 2008). There is agreement within the literature that second-order barriers appear to be much more difficult to shift and are a greater challenge to effective technology integration, particularly in the area of mathematics education (Inan & Lowther, 2009; Kim et al., 2013; Mueller et al., 2008). Teacher beliefs play a critical role in a teacher’s decision to use technology for instructional purposes, potentially delaying or inhibiting technology integration. Kim et al. (2013) discuss different ways that teacher beliefs are regarded by researchers. There are beliefs about the value of technology to enhance learning, beliefs regarding self-efficacy in using technology, and a combination of beliefs about the value of technology, self-efficacy, and teaching and learning with technology. Arguably, there may also be specific beliefs about the place of technology within mathematics pedagogy as opposed to other subject areas, given the traditional approach to teaching mathematics is still common in many of today’s classrooms (Bray & Tangney, 2017). A traditional view of mathematics that is content-driven rather than process-focused (problem solving, reasoning, and communication) is also a significant barrier to technology use.

20  Teaching mathematics with technology It is believed by some that effective integration requires teachers to envision new ways of seeing and doing things (Ertmer, 2005, as cited in Kim et al., 2013). Tsai and Chai (2012) suggest this may be considered a third-order barrier for technology-integrated instruction that extends beyond issues relating to understanding how and when to use technology. They posit that such knowledge “lies in the dynamic creation of knowledge and practice by teachers when they are confronted with the advancement of ICT and its associated pedagogical affordances” (p. 1058), believing the capacity for “design thinking” is the new barrier to technology use in education. This third-order barrier was evidenced in a study that evaluated an online digital resource and ways in which it was used by a group of 16 teachers (Attard, 2016). Although each of the teachers had been provided with the same orientation to the specific software, resulting practices appeared to be influenced by the teachers’ abilities to embed the software within their existing practises in new and innovative ways. First- and second-order barriers may be decreasing as technology becomes more ubiquitous in our classrooms; however, the advancement of technology does appear to be contributing to further new barriers that contribute to a range of dilemmas for teachers. While the challenge for teachers of mathematics to redesign teaching and learning in ways that take advantage of the range of affordances offered by the devices and their associated applications has been identified, there are others. The sheer volume of available apps has created barriers related to teachers choosing the appropriate software. Some studies have identified the challenge of time restrictions (Attard, 2015), while others have identified issues in relation to the expertise required for teachers to be able to accurately evaluate mathematics apps and their use in classrooms (Larkin, 2015). In a paper exploring the use of technology by early career teachers (first five years of teaching), Orlando and Attard (2015) adapt Windschitl’s (2002) framework of dilemmas by applying it to the use of technology in mathematics classrooms. These dilemmas arise as a result of the barriers we have described. The framework illustrates four categories of dilemmas: 1 Pedagogical dilemmas that arise from the need to develop deeper subject knowledge; challenges in the management of integrating technology into the classroom environment; and the orchestration of meaningful use of technology that supports student learning. 2 Cultural dilemmas that arise when teachers become conscious of using technology for different purpose, within different contexts and with varying bodies of technological and content knowledge; and questions relating to assumptions and discourses about the types of activities that should be valued. 3 Political dilemmas arising from issues of accountability with stakeholders (leadership, community, curriculum authorities) regarding the value of technology use. 4 Conceptual dilemmas that arise from teachers’ attempts to understand how technology can support their understanding of the process of learning and the role that technology can play in that process.

Teaching mathematics with technology 21 It is clear that there is a strong awareness amongst educational researchers of the barriers to technology integration and the resulting dilemmas that teachers face, but what do teachers think about this? In the next section, we explore data gathered directly from teachers about how they use technology in their mathematics classrooms.

Teachers’ views on technology use in mathematics classrooms In 2018, we invited teachers to tell us about their attitudes towards and practices with technological tools when teaching mathematics. We contacted the teachers via Twitter and Facebook, two social media platforms popular with educators (Holmes, Preston, Shaw, & Buchanan, 2013) and so it is probable that the teachers who responded to our questions were technologically savvy to some degree and likely to have positive dispositions towards the use of technology. We heard from 406 teachers from a range of locations, with 73% of the sample from Australia and 27% from a wide range of other countries around the world. Just over 40% of the sample were primary school teachers, with the remainder teaching in secondary schools. A total of 53% of the respondents taught in schools in a metropolitan area, 33% in regional areas, 12% rural, and 1% in remote schools. In terms of school sector, 67% of survey respondents taught in government schools, 20% in independent schools, and 10% in Catholic schools. In line with the typical teaching workforce 72% of the teachers were female. In general, these teachers held positive attitudes towards technology use with 87% responding that they felt comfortable using technology and 84% agreeing that technology can help students to understand mathematics concepts more effectively than traditional instruction. Despite these positive dispositions towards the use of technology in the mathematics classroom, first-order barriers to technology integration still appeared to dominate teachers’ use (Ertmer et al., 2012). Only half of the teachers felt that they had sufficient technical support to assist with technology integration, and even fewer (37%) agreed that there was sufficient instructional support for technology use in their schools, corresponding with reports from other research and our stance that there is a significant lag between research and its dissemination into classrooms. We found a statistically significant positive correlation between teachers’ general positivity towards technology use for mathematics learning and their perceptions of the level of support provided by their schools (Figure 2.1). This relationship reinforces the importance of school context and school leadership commitment as key variables influencing teacher attitudes towards technology use for teaching and learning (Holmes, Clement, & Albright, 2013). One teacher explained the benefits of using technology as a means of allowing students to work at their own pace, while emphasising the critical importance of effective technology infrastructure by stating that their best lessons occurred “when all students have a charged laptop, when the wifi works, and when students are then able to work at their own pace on the next concept that they need” (secondary teacher, government school).

Teacher positivity towards technology use

22  Teaching mathematics with technology 5

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So, what types of technology do these mathematics teachers use and what do they do with technology in the mathematics classroom? Almost 90% of the teachers used a computer or laptop, 62% made use of scientific calculators, 53% had used a tablet, and a smaller proportion of teachers employed cameras (37%), graphing calculators (27%), and data loggers (10%). Websites were popular tools for 70% of these mathematics teachers primarily for accessing online tutorials and exercises for students to practice their proficiency with mathematics problems. Desmos and GeoGebra were popular tools, primarily with secondary teachers, for producing mathematics visualisations and for exploring mathematics in a dynamic way. Spreadsheets were also very popular tools used by almost 60% of mathematics teachers. The proportion of primary and secondary teachers using various technological approaches is displayed in Figure 2.2. Almost half of the teachers also made use of technology in a flipped classroom model, allowing students to access recorded explanations of mathematics concepts or procedures outside of the classroom setting so that time in the classroom was spent actively engaging with the teacher, their peers, and the mathematics content rather than passively listening to the teacher. Such an approach also facilitated a more student-centred, individually paced form of instruction aligning with reports from research. One teacher explained as follows: My course materials are all posted so that students can access materials and work somewhat at an individual pace, as long as they come together for the test date. I include videos for review or instruction for students who wish to work ahead. I include on-line games for review/reinforcement. (primary teacher, independent school)

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Technological approaches Primary

Secondary

Figure 2.2  Primary and Secondary Teacher Reported Use of Technology.

Several teachers mentioned the pedagogical value inherent with students making their own video and audio recordings, or screencasts, to demonstrate their understanding of mathematics concepts. As one teacher explained, “I use technology as a tool in most maths lessons. It is more engaging and easier to see where kids are having difficulties as they verbalise their learning through video and audio recording” (primary teacher, government school). In this case, technology is used as a tool to engage students, particularly those who may have difficulty with writing, and also to assist the teacher to assess and understand student thinking and to diagnose student difficulties and misconceptions. This teacher appears to promote a process-focused approach that promotes reasoning while providing students with the opportunity to create content rather than simply consume it. Other teachers saw the potential for technology to enable student input into how they learned mathematics, “I think any lesson that gives students choice in relation to mathematics and technology is a good one. Allowing students freedom to see how technology can help them in maths and allowing them to discover that themselves is valuable” (primary teacher, government). Although the teachers in our sample were generally confident with using technology for teaching mathematics, they also recognised that these tools could sit comfortably alongside more traditional mathematics manipulatives or with pen

24  Teaching mathematics with technology and paper techniques. The need to include a variety of techniques was emphasised by one teacher as follows: Technology is a great tool but all methods are useful. One method I don’t feel is more superior than another. You can have excellent engaging lessons with technology and you can have excellent fantastic lessons without technology. If students use one strategy everyday every lesson then it no longer is engaging and becomes boring - so mixing it up is the key I feel. (Secondary teacher, Catholic school) Another teacher reinforced this sentiment in relation to the use of a variety of tools, explaining that she uses a blended approach when using technology. In this case, the teacher uses a blend of digital tools: My most effective mathematics lesson using technology has been where the technology is part of a blended learning approach where students have explored mathematical concepts and integrated a variety of learning resources to discover concepts for themselves. Eg. Used dice simulations, recorded results on a class google sheet and then graphs the results in GeoGebra. The benefit of the technology has been to speed up the dice throwing so more time can be dedicated to students predicting and discussing their work. (secondary teacher, government school) This teacher provides an illustration of Gardner and Davis’ (2013) conception of being “app enabled”, where the use of a range of apps has allowed the students to focus on deeper learning. The teacher has recognised the affordances and limitations of the tools she has access to and has designed a range of tasks that are designed to maximise student learning with minimum distraction. Many teachers we surveyed were inherently aware of the need to integrate technology in the classroom alongside other pedagogical approaches. Their responses indicated that they sought to use technology judicially and purposefully, to optimise student learning, and to promote the development of conceptual understanding in mathematics. The survey also revealed an underlying concern about the level of school support for technology use, both in terms of school leadership and infrastructure. Lack of support at the school level clearly has the potential to derail teachers’ willingness to experiment, design, and embed technology as a pedagogical tool in mathematics, thus resulting in third-order barrier of design thinking as suggested by Tsai and Chai (2012). The survey responses from teachers support the view that technology use in schools should not be left to individual teacher discretion or to those with the motivation to investigate novel pedagogical approaches. The broader school community, in terms of teacher colleagues, school leaders, and parents, has a vital role to play in ensuring that conditions are optimal to support technology use in the classroom (Holmes, Bourke, Preston, Shaw, & Smith, 2013). It is vital that first-order structural barriers to technology use (e.g., poor Internet access,

Teaching mathematics with technology 25 inadequate devices) are minimised so that teachers can focus on the pedagogical affordances of technologies and how they can be leveraged to enhance student learning in mathematics.

The future of technology in mathematics classrooms Research has clearly indicated that there are many challenges and barriers to effective technology integration in mathematics classrooms, and this was confirmed by the participants of our survey. The ways in which technology is integrated into mathematics lessons is extremely diverse, as a result, it can be argued that we can expect increasing challenges and inconsistencies as new technologies continue to emerge within school systems that are clearly diverse in terms of funding, access, and support. If teachers feel that they are not supported technologically and pedagogically with integrating digital technologies into their practices, what will occur in the future as new technologies emerge? What are the technologies of the future? The international movement to improve participation in science, technology, engineering, and mathematics (STEM) based career paths has resulted in an increase in the technological devices being used in classrooms for STEM-based activities, many of which are underpinned by mathematical concepts. These devices already include a range of robotics, augmented realty, and virtual reality, but may eventually include the use of artificial intelligence and technologies that have not yet evolved. In their report on existing and emerging digital technologies, Southgate, Smith, and Cheers (2016) present a roadmap to highlight contemporary and future digital technologies in education because of their potential to create “deeper and authentic learning opportunities in schools” (p. 9). If we are to take advantage of such opportunities, questions must be asked regarding how, what, when, and where mathematics will be taught in the future. With this in mind, we return to the intention of this book to provide some guidance to assist teachers of mathematics, regardless of what type and how many devices they have, and the types of applications they have access to. To do this, in the following chapter, we consider what effective mathematics teaching looks like, what student engagement looks like, and what we need to know in order to teach mathematics effectively with digital technologies.

References Attard, C. (2015). Introducing iPads into primary mathematics classrooms: Teachers’ experiences and pedagogies. In M. Meletiou-Mavrotheris, K. Mavrou, & E. Paparistodemou (Eds.), Integrating touch-enabled and mobile devices into contemporary mathematics education (pp. 193–213). doi:10.4018/978-1-4666-8714-1.ch009 Attard, C. (2016). Research evaluation of matific mathematics learning resources: Project report. doi:10.4225/35/57f2f391015a4 Attard, C. (2018). Mobile technologies in the primary mathematics classroom: Engaging or not? In N. Calder, K. Larkin, & N. Sinclair (Eds.), Using mobile technologies in the teaching and learning of mathematics (pp. 51–65). doi:10.1007/978-3-319-90179-4_4

26  Teaching mathematics with technology Attard, C., & Curry, C. (2012). Exploring the use of iPads to engage young students with mathematics. In J. Dindyal, L. P. Cheng, & S. F. Ng (Eds.), Mathematics education: Expanding horizons (Proceedings of the 35th annual conference of the Mathematics Education Research Group of Australasia, pp. 75–82). Singapore: MERGA. Australian Curriculum Assessment and Reporting Authority (ACARA). (2010). The Australian curriculum: Mathematics. Bhagat, K. K., Chang, C.-N., & Chang, C.-Y. (2016). The impact of the flipped classroom on mathematics concept learning in high school. Journal of Educational Technology & Society, 19(3), 134–142. Bower, M. (2017). Design of technology-enhanced learning: Integrating research and practice. Retrieved from http://ebookcentral.proquest.com/lib/uwsau/detail.action? docID=4717043 Bray, A., & Tangney, B. (2017). Technology usage in mathematics education research – A systematic review of recent trends. Computers & Education, 114, 255–273. doi:10.1016/j.compedu.2017.07.004 Brigham, T. J. (2015). An introduction to gamification: Adding game elements for engagement. Medical References Services Quarterly, 34(4), 471–480. Calder, N., & Murphy, C. (2018). Using apps for teaching and learning mathematics: A sociotechnological assemblage. In J. Hunter, P. Perger, & L. Darragh (Eds.), Making waves, opening spaces (Proceedings of the 41st annual conference of the Mathematics Education Research Group of Australasia, pp. 194–201). Auckland, New Zealand: MERGA. Dolan, J. E. (2016). Splicing the divide: A review of research on the evolving digital divide among K-12 students. Journal of Research on Technology in Education, 48(1), 16–37. doi:10.1080/15391523.2015.1103–147 Ertmer, P., Ottenbreit-Leftwich, A., Sadik, O., Sendurur, E., & Sendurur, P. (2012). Teacher beliefs and technology integration practices: A critical relationship. Computers & Education, 59(2), 423–435. Freeman, A., Adams Becker, S., Cummins, M., Davis, A., & Hall Giesinger, C. (2017). NMC/CoSN horizon report: 2017 K-12 eEdition. Retrieved from https://www.nmc. org/publication/nmccosn-horizon-report-2017-k-12-edition/ Galligan, L., & Hobohm, C. (2018). Mathematics screencasts for teaching and learning. In N. Calder, K. Larkin, & N. Sinclair (Eds.), Using mobile technologies in the teaching and learning of mathematics (pp. 265–282). Cham, Switzerland: Springer. Gardner, H., & Davis, K. (2013). The app generation. New Haven, CT: Yale University Press. Goehle, G., & Wagaman, J. (2016). The impact of gamification in web based homework. Problems, Resources, and Issues in Mathematics Undergraduate Studies, 26(6), 557–569. Holmes, K., Bourke, S., Preston, G., Shaw, K., & Smith, M. (2013). Supporting innovation in teaching: What are the key contextual factors? International Journal of Quantitative Research in Education, 1(1), 85. doi:10.1504/IJQRE.2013.055644 Holmes, K., Clement, J., & Albright, J. (2013). The complex task of leading educational change in schools. School Leadership & Management, 33(3), 1–14. Holmes, K., Preston, G., Shaw, K., & Buchanan, R. (2013). “Follow” me: Networked professional learning for teachers. Australian Journal of Teacher Education, 38(12). Retrieved from https://eric.ed.gov/?id=EJ1016021 Inan, F., & Lowther, D. (2009). Factors affecting technology integration in K-12 classrooms: A path model. Educational Technology Research and Development, 58(2), 137–154.

Teaching mathematics with technology 27 Ingram, N., Pratt, K., & Williamson-Leadley, S. (2018). Using show and tell apps to engage students in problem-solving in the mathematics classroom. In N. Calder, K. Larkin, & N. Sinclair (Eds.), Using mobile technologies in the teaching and learning of mathematics (pp. 301–315). Cham, Switzerland: Springer. Janssen, K. C., & Phillipson, S. (2015). Are we ready for BYOD? A systematic review of the implementation and communication of BYOD programs in Australian schools. Australian Educational Computing, 30(2). Retrieved from http://journal.acce.edu. au/index.php/AEC/article/view/54 Kim, C., Kim, M., Lee, C., Spector, J., & De Meester, K. (2013). Teacher beliefs and technology integration. Teaching and Teacher Education, 29, 76–85. Kingsley, T. L., & Grabner-Hagen, M. M. (2015). Gamification: Questing to integrate content knowledge, literacy, and 21st-century learning. Journal of Adolescent and Adult Literacy, 59(1), 51–61. Larkin, K. (2015). “An App! An App! My kingdom for an app”: An 18-month quest to determine whether apps support mathematical knowledge building. In T. Lowrie & R. Jorgensen (Eds.), Digital games and mathematics learning: Potential, promises and pitfalls (pp. 251–276). doi:10.1007/978-94-017-9517-3_13 Lo, C. K., & Hew, K. F. (2017). A critical review of flipped classroom challenges in K-12 education: Possible solutions and recommendations for future research. Research and Practice in Technology Enhanced Learning, 12(1), 4. doi:10.1186/s41039-016-0044-2 Maher, D., & Twining, P. (2017). Bring your own device – A snapshot of two Australian primary schools. Educational Research, 59(1), 73–88. doi:10.1080/00131881.2016. 1239509 Mueller, J., Wood, E., Willoughby, T., Ross, C., & Specht, J. (2008). Identifying discriminating variables between teachers who fully integrate computers and teachers with limited integration. Computers & Education, 51(4), 1523–1537. Muir, T., & Geiger, V. (2016). The affordances of using a flipped classroom approach in the teaching of mathematics: A case study of a grade 10 mathematics class. Mathematics Education Research Journal, 28(1), 149–171. doi:10.1007/s13394-015-0165-8 Olive, J., & Makar, K. (2010). Mathematical knowledge and practices resulting from access to digital technologies. In C. Hoyles & J. B. Lagrange (Eds.), Mathematics education and technology - Rethinking the terrain (pp. 133–177). London, England: Springer. Orlando, J., & Attard, C. (2015). Digital natives come of age: The reality of today’s early career teachers using mobile devices to teach mathematics. Mathematics Education Research Journal, 1–15. doi:10.1007/s13394-015-0159-6 Parsons, D., & Adhikari, J. (2016). Bring your own device to secondary school : The perceptions of teachers, students and parents. Retrieved from http://unitec.researchbank. ac.nz/handle/10652/3682 Prescott, A., & Maher, D. (2018). The use of mobile technologies in the primary school mathematics classroom - Developing “CreateAlouds.” In N. Calder, K. Larkin, & N. Sinclair (Eds.), Using mobile technologies in the teaching and learning of mathematics (pp. 283–300). Cham, Switzerland: Springer. Rae, G., Dabner, N., & Mackey, J. (2017). Bring your own device (BYOD) and teacher pedagogy in a New Zealand primary school. doi:10.15663/Tandc.V17i2.160. Retrieved from https://ir.canterbury.ac.nz/handle/10092/15384 Selwyn, N., Potter, J., & Cranmer, S. (2009). Primary pupils’ use of information and communication technologies at school and home. British Journal of Educational Technology, 40(5), 919–932.

28  Teaching mathematics with technology Southgate, E., Smith, S. P., & Cheers, H. (2016). Immersed in the future: A roadmap of existing and emerging technologies for career exploration (No. 3). Retrieved from DICE Research website: http://dice.newcastle.edu.au/DRS_3_2016.pdf Tangney, B., & Bray, A. (2013). Mobile technology, maths education & 21C learning. Proceedings of the 12th World Conference on Mobile and Contextual Learning (MLearn2013), 2013, 20–27. doi:10.5339/qproc.2013.mlearn.7 Technology for Learning Program - Information Technology Directorate. (2013). Bring your own device (BYOD) in schools: 2013 Literature review. Sydney, Australia: NSW Government Education and Communities. Tsai, C., & Chai, C. S. (2012). The “third”-order barrier for technology-integration instruction: Implications for teacher education. Australasian Journal of Educational Technology, 28(6), 1057–1060. Weinhandl, R., Lavicza, Z., & Süss-Stepancik, E. (2018). Technology-enhanced flipped mathematics education in secondary schools: A synopsis of theory and practice. K-12 STEM Education, 4(3), 377–389. doi:10.14456/k12stemed.2018.9 Windschitl, M. (2002). Framing constructivism in practice as the negotiation of dilemmas: An analysis of the conceptual, pedagogical, cultural, and political challenges facing teachers. Review of Educational Research, 72(2), 131–175. doi:10.3102/00346543072002131

3

Mathematics teaching, technology, and student engagement

As technology continues to develop and evolve, the role and expectations of the teacher also continue to change (Armstrong, 2014). The New Media Consortium/ Consortium for School Networking (NMC/CoSN) Horizon Report (Freeman, Adams Becker, Cummins, Davis, & Hall Giesinger, 2017) expresses concern over the challenges that have resulted from technological developments: Teachers now address social and emotional factors affecting student learning, mentor students, model responsible global citizenship, and motivate students to adopt lifelong learning habits. These evolving expectations are changing the ways teacher engage in their own continuing professional development, much of which involves collaboration with other educators and the use of new digital tools and resources. (p. 30) The report stresses that appropriate teacher training, ongoing professional development, research about student learning, and teacher collaboration are critical for the improvement of teacher practice in this digital age (Freeman et al., 2017). Livingstone (2012) also suggests that perhaps digital learning tools require us to rethink “the relations between pedagogy and society, teacher and pupil, knowledge and participation” (p. 20). These concerns are reflected in the calls of mathematics education researchers who have found that teachers particularly struggle when using technology in the mathematics classroom when compared to other subject areas (Attard, 2015; Attard & Curry, 2012; OECD, 2016a; Orlando & Attard, 2015). In this chapter, we interrogate the literature to explore what effective teaching in mathematics classrooms looks like. Are we using technology to improve student engagement or outcomes, or both? The emergence of new technologies offers opportunities to disrupt the traditional delivery of mathematics education and can influence the practices of teachers in both positive and negative ways. For example, the abundance of tablet applications that promote drill and practice conflicts with what we understand to be effective practices that promote deep understanding of mathematical concepts. Conversely, the use of productivity apps that require students to problem solve and explain their thinking can promote creativity, critical thinking, and reasoning, thus enhancing teaching practice.

30  Mathematics teaching and student engagement As mentioned in the introductory chapter, the rapid pace of technology development has resulted in a lag between the conduct and dissemination of research into the effective use of technology in classrooms. This often leads to teachers having to utilise new technologies through a trial and error approach. In this chapter, we analyse existing generic frameworks for technology integration such as SAMR (substitution, augmentation, modification, and redefinition) (Puentedura, 2006) and the Technology Information Matrix (Florida Center for Instructional Technology, 2018) along with the technological pedagogical content knowledge (TPACK) model (Koehler & Mishra, 2009) to explore if and how they can inform or support technology-integrated mathematics teaching and learning. We argue that there is an urgent need to “future proof” mathematics educators and their students against the ongoing emergence of new technologies that we have not yet imagined. Teachers need to be prepared, willing, and able to confidently use future technologies to effectively teach mathematics and improve student engagement and learning outcomes.

What is effective teaching in mathematics? Teachers are a powerful influence on students’ engagement with mathematics (Anthony & Walshaw, 2009; Askew, 2012; Attard, 2013; Hattie, 2003), and they are cited in research as a major supporting or limiting factor influencing how students use technology (Schnellert & Keengwe, 2012; Thomas, 2007). Classroom pedagogy is one aspect of a broader perspective of the knowledge a teacher requires in order to be effective. The knowledge of what to teach, how to teach it, and how students learn is often referred to as pedagogical content knowledge (PCK), a construct originally introduced by Shulman (1986). Substantial research since Shulman’s work has seen a strong focus on PCK in relation to mathematics teaching and learning (Delaney, Ball, Hill, Schilling, & Zopf, 2008; Hill, Ball, & Schilling, 2008; Hurrell, 2013; Neubrand, Seago, Agudelo-Valderrama, DeBlois, & Leikin, 2009). PCK in mathematics consists of three domains: mathematics content knowledge (understanding the mathematics being taught); knowledge of how students learn mathematics (common misconceptions and knowledge of the continuum of progression); and knowledge of approaches to mathematics teaching (understanding appropriate mathematics-specific pedagogies that lead to student learning) (Askew, Brown, Rhodes, Johnson, & Wiliam, 1997; Neubrand et al., 2009). Let’s first consider mathematics teaching and learning separately from the use of digital technologies. One of the most obvious factors that often set mathematics education apart from other school subjects is the teaching approach that students often experience. Teaching practices in mathematics classrooms cover a broad spectrum from the traditional, text book-based lesson to the contemporary or reform approaches of problem solving and investigation-based lessons, or a combination of both. When asked to recall a typical mathematics lesson, many describe a traditional, teacher-centred approach in which a routine of teacher demonstration, student practice using multiple examples from a text book and

Mathematics teaching and student engagement 31 then further multiple, text book generated questions are provided for homework (Even & Tirosh, 2008; OECD, 2016b; Ricks, 2009; Stigler & Hiebert, 1999). The traditional approach to teaching mathematics reflects a behaviourist perspective in which facts are transmitted and regurgitated, there is an emphasis on procedural knowledge, and the focus is on individual students. The classroom is viewed as a collective of individuals, discourse is limited, and power and control are held by the teacher. An alternate approach to teaching mathematics reflects a social constructivist perspective where students are provided with opportunities to construct their own knowledge with a focus on relational understanding rather than instrumental understanding (Skemp, 2006). Such an approach promotes problem solving and reasoning and is consistent with frameworks for quality teaching (Australian Association of Mathematics Teachers, 2006; Newmann, Marks, & Gamoran, 1996) and recent curriculum requirements in many countries. Within a reform classroom, the traditional roles and responsibilities of the teacher and students are altered and lessons are viewed as “student-centred”, with power and control shared amongst students and teacher. The traditional teacher-centred approach to teaching mathematics often results in feelings of anxiety towards mathematics leading to lower levels of engagement (Boaler, 1998, 2003, 2009). Teachers sometimes contribute to mathematics anxiety by placing too much emphasis on memorising formulae, learning mathematics through drill and practice, applying rote-memorised rules, and setting out work in a set, traditional way (Geist, 2010). In addition to these concerns, the pressure of timed tests also contributes to high anxiety for many students (Boaler, 2015). The traditional approach of presenting learners with an algorithmic task before expecting them to construct their own understandings mystifies the mathematics behind the algorithm and obscures the power of reasoning that lies within mathematics, perpetuating an image of mathematics as an elitist, remote, and inaccessible subject (Nardi & Steward, 2003). There are obvious benefits of incorporating a contemporary, alternate pedagogy within the mathematics classroom; yet, the practicalities often involve substantial challenge for students, teachers, and schools, particularly when digital technologies are considered. Although mathematics reform movements during the 1980s were based on the idea that open-ended problems would provide productive learning experiences and there has since been a plethora of material based on this, there have been objections from mathematicians and others who “gained extensive understandings through more traditional routes” (Boaler, 2002, p. 239). Some objectors (often the parents of students in our classrooms) prefer to maintain the traditions of the past and view changes to school presentations of mathematics as a challenge to the social order. There are, however, many researchers who support an investigative, contemporary approach to teaching and learning mathematics (Anthony & Walshaw, 2009; Boaler, 2015; Clarke, 2003; Lovitt, 2000). Open-ended, rich tasks can transform students’ beliefs about problem solving and alter the culture of mathematical engagement.

32  Mathematics teaching and student engagement A seminal study on effective teachers of numeracy in the United Kingdom was conducted by Askew et al. (1997). In this study, effectiveness was measured by measuring academic improvement. Askew et al., were able to identify “key factors that enable teachers to put effective teaching of numeracy into practice in the primary phase” and “strategies which would enable those factors to be more widely applied” (p. 4). Two aspects of effective teaching that informed the work of Askew et al. (1997) were the teacher’s set of beliefs about teaching mathematics and the collection of knowledge and understanding held by teachers about numeracy and its teaching (PCK). Askew et al., described three beliefs that have significant influence over the teaching of numeracy: beliefs about what it is to be numerate, about how students learn to become numerate; and about how best to teach students to become numerate. Although these beliefs are pivotal in informing and influencing lessons, the different aspects of teachers’ knowledge are also a significant contributor to student learning (Askew, 2012). Interestingly, findings from the Askew et al., study (1997) highlighted that no single style of classroom organisation for mathematics teaching and no specific level of teacher qualification predicted the effectiveness of teaching. Rather, the amount of continuing professional development undertaken by individual teachers appeared to be a better predictor of effective teaching. A distinguishing feature of effective teachers was the beliefs held by teachers about “how best to teach mathematics that took into account children’s learning” (Askew, 2012, p. 35). The specific set of beliefs is referred to by Askew et al., as a connectionist orientation to teaching, encompassing the following elements: • •

making connections within and between mathematics content, concepts and representations making connections with the range of children’s methods and having flexibility in valuing and using various methods.

Obviously, the work of Askew et al. (1997) was conducted prior to the digital revolution and did not consider digital technologies. Classrooms and resources are now vastly different, and teachers with a connectionist orientation have more opportunity than ever before to promote powerful connections within and across mathematical concepts, content, and representations through the use of contemporary digital technologies. In summary, there is a range of approaches used to teach mathematics in schools. Some teachers may carry a strong orientation towards one or the other, and some use a combination of the two approaches. The traditional, behaviourist approach incorporates a teacher-centred approach with the incorporation of rote learning, drill, and repetition. This approach is said to suit learners who have certain expectations of mathematics lessons and sometimes these students who come from a diversity of backgrounds and experiences or have specific learning needs. At the other end of the spectrum, the alternate approach is social constructivist in nature, utilises open-ended type tasks, is student-centred, and

Mathematics teaching and student engagement 33 allows students to access mathematics in diverse ways, allowing them to see the relevance of the mathematics. Regardless of the teaching approaches that are undertaken, a discussion of mathematics pedagogy at this point in time cannot ignore the issue of digital technology and so we now turn to some of the more commonly used frameworks that attempt to bring together the elements of effective teaching as we have described above, taking into account the affordances and constraints of contemporary classroom digital technologies. The teaching of mathematics is different to any other school subject, and as a result, the use of mobile and digital technologies to teach mathematics presents a unique set of challenges for teachers. As mentioned in the introductory chapter, teachers of mathematics face specific challenges that include significant levels of student disengagement. Within each mathematics classroom, whether grouped by ability or heterogeneously, there are students with a range of abilities and students who have gaps in their conceptual knowledge. The hierarchical nature of mathematics compounds the challenge to address the individual needs of students. Engagement with mathematics becomes an additional challenge because inattention to elements of a lesson within a hierarchical subject may hamper students’ accumulation of knowledge, creating gaps in learning (Yair, 2000). When gaps do occur in the learning of mathematics, difficulties begin to arise. Students disengage and find it difficult to understand more abstract concepts. As mentioned in the previous chapter, you cannot build a house on faulty foundations. Similarly, you cannot become a confident user of mathematics when foundational concepts have not been understood.

Frameworks to support the effective integration of digital technologies Current literature relating to the use of digital technologies in educational contexts offers a range of frameworks that attempt to describe how technology can be integrated into teaching and learning. They also attempt to detail the knowledge required by teachers to use technology effectively. Given that the digital tools currently available in contemporary classrooms are often not designed for educational purposes, it is critical that any technology integration framework provides educators with guidance for using digital tools to advance student learning and understanding. Harris, Mishra, and Koehler (2009) cite a range of studies that indicate instructional applications of educational technologies are “limited in breadth, variety, and depth and are not well integrated into curriculum-based teaching and learning” (p. 393). This claim aligns with findings from studies conducted in Australian mathematics classrooms during the early years of tablet integration (Attard, 2013a, 2013b; Attard & Curry, 2012). Although it is reasonable to expect that teachers may now be more comfortable with using digital technologies due to their increasing presence in classrooms, there are always going to be challenges caused by the constant development of new technologies. The frameworks described in this section are attempts to

34  Mathematics teaching and student engagement redirect the default “technocentric” approach (Papert, 1987, as cited in Harris et al., 2009) that focuses on the technology being used rather than curriculum and students’ learning needs. Typically, these frameworks attempt to detail the knowledge teachers require to teach with technology or the various levels of technology integration in classrooms from both teaching and learning perspectives. This section will explore some of the frameworks that have attracted popular attention in order to investigate whether they are useful in assisting teachers in day to day decision-making regarding when, where and how to use digital devices in the teaching of mathematics.

The TPACK framework Koehler and Mishra (2009) acknowledge that teaching, in general, is an “ill-structured” (p. 61) discipline that requires the application of complex knowledge structures and flexible knowledge. They also argue that digital technology (as opposed to analog) has made the integration of technologies into teaching more complex and less straightforward than traditional technologies due to their inherent properties and affordances. Building on Shulman’s construct of PCK (Shulman, 1986), Koehler and Mishra developed the TPACK model as a professional knowledge construct and posit that there are three components that lie at the heart of effective teaching with technology: content, pedagogy, and technology (Koehler & Mishra, 2009). CK relates to the subject matter that is to be taught or learned; pedagogical knowledge (PK) is the deep understanding about the practices, processes, or methods related to the content; and technological knowledge (TK), which is difficult to define as it is in a constant state of flux, relates to the mastery of and productive application of technology. TPACK emerges from the interactions amongst the three components of the model. These interactions are identified as PCK as described in Shulman’s work (1986); technological content knowledge (TCK), an understanding of the ways a technology’s affordances can be used to teach specific content; and technological pedagogical knowledge (TPK), an understanding of how teaching and learning varies when a variety of technologies are used. These interactions are considered to be equally important to the TPACK model. It is the interactions amongst the three components of TPACK that result in wide variations in the extent and quality of technology integration. According to Krauskopf, Foulger, and Williams (2018), the application of the TPACK framework to technology infused teaching scenarios potentially provides a richness to teaching conversations, providing a theoretical vocabulary to help understand the required pedagogical considerations of technology integration (Koh, 2018). It also provides opportunities for educators to move beyond treating technology as an “add-on”, promoting a greater focus on the connections among content, pedagogy, and technology within classroom contexts. However, there are limitations of the TPACK framework. Although it is helpful in identifying specific knowledge domains for consideration when integrating technology, TPACK is regarded as a pedagogically neutral model by

Mathematics teaching and student engagement 35 Harris, Grandgenett, and Hofer (2010, as cited in Bower, 2017) and makes no suggestions about specific technologies and pedagogies that would be appropriate for a particular content area such as mathematics. In addition, TPACK does not consider the contextual elements that may influence task design, teacher practice, and student learning, nor does it consider the variety of barriers and dilemmas that are typical to technology integration such as a lack of technical support or issues of access.

TPACK and mathematics It has been agreed that a deficiency of the TPACK framework is its lack of attention to specific disciplines and the belief that each discipline requires a different level or type of TPACK (Bower, 2017; Guerrero, 2010; Koh, 2018; McGrath, Karabas, & Willis, 2011). In attempting to address this deficit, Guerrero (2010) extends the TPACK model to ways it can be used to improve the teaching and learning of mathematics. Guerrero proposes four components of mathematical TPACK: 1 Conception and use of technology: teacher’s beliefs about mathematics and how it can be best addressed through the use of technology and what aspects of mathematics students should learn about through the use of technology. 2 Technology-based instruction: the teacher’s ability to adapt pedagogy and recognise the need for flexibility in instruction that results from the use of technology. 3 Management: the range of issues relating to the implementation of technology including maintenance of student engagement and dealing with behaviour management. Management also deals with the technical aspects of using technology including dealing with the physical environment, hardware and software issues. 4 Depth and breadth of mathematics content: the increased responsibilities associated with having a deep understanding of mathematics content and a willingness to allow students to explore mathematical content that may arise as a result of students’ use of technology. Although Guerrero’s extension of TPACK provides more detail regarding the use of technology in the mathematics classroom, it does not acknowledge the guidance teachers may require when faced with incorporating new technologies into their existing PCK. Bower (2017) suggests TPACK is valuable to teachers as a framework to identify elements of practice that require consideration but does not support them in how to use technology effectively within specific discipline areas or contexts. Although the TPACK framework provides some explanation of the types of knowledge teachers require to integrate technology, it fails to explain why teachers who have the appropriate knowledge utilise technology in different ways (Kim, Kim, Lee, Spector, & De Meester, 2013), or the gap between the

36  Mathematics teaching and student engagement amount of technology available and teachers’ use of that technology within their instructional practices (Kopcha, 2012). This was evidenced in a study that explored how a group of 16 teachers used the same digital resource after having received the same professional learning and introduction to the resource (Attard, 2016). The variation in the ways in which the resource was integrated into their mathematics pedagogy resulted in differing levels of success in relation to student engagement and achievement. The teachers’ beliefs about the value of digital technologies and the specific resource itself may have influenced they ways they embedded the resource into their teaching. As discussed in Chapter 2, teachers’ beliefs are considered to be a second-order barrier to successful technology integration and more difficult to resolve than first-order barriers, which are external to the teacher (Ertmer, Ottenbreit-Leftwich, Sadik, Sendurur, & Sendurur, 2012). Teacher beliefs, regardless of their ability to use technology, are a major influence on their instructional and technology integration practices. Although it would be typical for teachers who hold student-centred beliefs to integrate technology in student-centred ways, it is suggested that this is not always the case (Ottenbreit-Leftwich, Glazewski, Newby, & Ertmer, 2010). In a study conducted by Judson (2006, as cited by Ottenbreit-Leftwich et al., 2010), teachers who claimed to have student-centred beliefs were not observed to be implementing student-centred practice. However, the affordances of mobile technologies have advanced significantly since Judson’s study and an exploration of changes in the correlation between beliefs and actual practices in today’s classrooms would be beneficial. The challenge with mobile technologies is that there are many applications that do not promote student-centred practice. For example, apps that simply require the users to consume, such as question/answer-based apps, rather than apps that require students to be producers, do not support a student-centred approach. Similarly, the use of an interactive whiteboard can be seen as reinforcing a teacher-centred approach due to its positioning in the classroom, limiting the capacity for shared control of the device and of learning (Attard & Orlando, 2014). If a teacher does not see the potential of a device to transform teaching practices from teacher- to student-centred, then he or she will not spend time, energy, or resources learning about the tool or incorporating it into new or current practices (Ottenbreit-Leftwich et al., 2010). Another significant influence on teachers’ beliefs regarding technology integration in the mathematics classroom relates to the quality and quantity of professional development they experience. For example, Kopcha (2012) discusses the effectiveness of situated professional development such as mentoring. Research has shown that mentored teachers report more positive attitudes towards technology use and tend to integrate technology into their teaching practices more than non-mentored teachers (Kopcha, 2012). As discussed in Chapter 1, there is often a lag between research on best practice with emerging technologies and the design and delivery of professional development to assist teachers in their implementation.

Mathematics teaching and student engagement 37 A further influence on teachers’ beliefs relates to the time taken to learn, plan, teach, and manage a classroom when a new technology is introduced (Kopcha, 2012; Prieto-Rodriguez, 2016). This was a significant issue when iPads were introduced and fast became a common feature in many classrooms. The introduction of the tablets and the vast range of ready-made mathematics-based apps caused teachers to spend more time than usual on planning instructional activities, with little support from professional development or colleagues due to the newness of the devices and their affordances (Attard, 2013a, 2013b). In an effort to incorporate teachers’ beliefs, Niess et al. (2009) developed a framework that identifies five steps that mathematics teachers progress through when introducing a new technology into existing PCK, resulting in the development of TPACK: 1 Recognising: teachers have the ability to use the digital technology and recognise its alignment with mathematics content but do not integrate it into mathematics teaching and learning. 2 Accepting: teachers form either a positive or negative attitude toward teaching mathematics with appropriate digital technologies. 3 Adapting: teachers engage in activities that lead to a decision to either adopt or reject teaching with appropriate digital technologies. 4 Exploring: teachers actively integrate their teaching practices and learning activities with technology. 5 Advancing: teachers evaluate the results of their integration of digital technologies on teaching and learning. The progression that Niess et al. (2009) describe is not necessarily linear. Each time a teacher considers the incorporation of a new digital technology, he or she is required to rethink content and pedagogy and ways the technology can enhance student learning. It could also be argued that teachers who present mathematics in a traditional, topic by topic manner may experience the progression when moving from one mathematical topic to another. Although a more contemporary, inquiry-based approach to the teaching of mathematics is increasing in popularity, studies have shown teachers’ lesson plans show their use of technology predominantly supports recall and application as opposed to analysis, comparison, or evaluation of mathematical concepts (Attard, 2015; Dawson, Ritzhaupt, Liu, Rodriguez, & Frey, 2013). The TPACK framework helps us to understand the types of knowledge that teachers require to effectively integrate technologies, and the Niess et al., progression describe how teachers develop their TPACK. The turn-around technology integration pedagogy and planning (TTIPP) model (Roblyer & Hughes, 2019) presents three phases or prompts that add a more useful, reflective dimension of TPACK for teachers. The intention of TTIPP is to provide a general approach to identifying and addressing when planning for technology use. TTIPP is different to the Niess et al., model in that teachers self-assess their ability to carry out the chosen technology integration, questioning each element of the TPACK

38  Mathematics teaching and student engagement framework, and incorporating task design, resources, objectives, assessments, and evaluation. As with TPACK, TTIPP is agnostic to the subject matter and does not prompt reflection and planning that is discipline specific. The following section explores a range of frameworks that attempt to describe the levels and ways in which teachers incorporate digital technologies into their practice.

SAMR: a lens for technology use The range of frameworks that describe how technologies can or should be used in classrooms is diverse; yet, there are many similarities amongst them. Kimmons and Hall (2018) claim the fact that so many frameworks or theoretical models exist suggests that no single model is “universally valuable, understandable, or useful to all shareholders” (p. 31). In their survey of K-12 teachers and teacher candidates, results indicated that although the use of a model such as TPACK may be widespread, this does not necessarily reflect usefulness. A widespread model that describes the levels of technology integration that may be more useful for teachers is SAMR (Puentedura, 2006). This model represents four levels of incremental technology integration within educational contexts: 1 Substitution: technology is used as a direct substitute for traditional resources with no functional change. 2 Augmentation: technology is used as a direct substitute with some functional improvement. 3 Modification: technology allows for significant task redesign. 4 Redefinition: technology allows for the creation of new tasks that were previously inconceivable. The SAMR model can be a useful lens for teachers to reflect on how they are using technology. However, it should be noted that SAMR is task-focused rather than having a more holistic focus on broader teacher practices. We say this because it is important to recognise the strengths and limitations of any framework designed for technology-infused education. Mathematics teaching and learning is far more complex than task design. For example, the use of productivity technologies, such as learning management systems, can provide affordances that enhance learning and teaching, independent of the task. A similar construct that identifies the ways in which technology is used within teacher practice is presented by Goos, Galbraith, Renshaw, and Geiger (2000) and when aligned with the SAMR model may be of further assistance to teachers. The Goos et al., model conceptualises four roles of technology: technology as master, occurring when the technology is imposed on the teacher or its use is limited due to the teacher’s beliefs about teaching; technology as servant, where the teacher is knowledgeable with regard to using the technology but limits its use to the teacher’s preferred methods; technology as partner, describing the creative integration of technology that results in improved quality of student

Mathematics teaching and student engagement 39 learning; and technology as extension of self, which occurs when technology forms a natural part of the teacher’s practice and is used in highly creative ways. When viewed together, the frameworks imply that the level of technology integration influences the quality of the task and the quality of student learning. But how does this translate to the teaching and learning of mathematics? A framework that extends SAMR by identifying a range of learning environments and presenting exemplars of technology use according to discipline is the technology integration matrix (TIM) (Florida Center for Instructional Technology, 2018). Originally published in 2006, the matrix was designed to assist in evaluating technology integration within instructional settings. The TIM provides five characteristics of learning environments: active, constructive, goal directed, authentic, and collaborative and associates each of the characteristics with five levels of technology integration: entry, adoption, adaptation, infusion, and transformation, resulting in 25 different cells. Each cell of the matrix provides examples of practice from four different disciplines, including mathematics. To date, this may be the most helpful and practical framework for teachers of mathematics, providing more detailed information that could be applied to a range of contexts. Studies that have compared technology use in mathematics using SAMR as a lens have found that teachers using mobile devices are most likely to use them as enhancement tools that Augment their existing practices. “Substitution of tablets for books may well reduce the weight of school bags but is unlikely to result in a move into the Transformation space” (Tangney & Bray, 2013). Similarly, previous studies by Attard (2015) have found that teachers are more likely to design tasks that enhance existing practice and this may be directly related to the nature of the affordances and apps they utilise, and the ways in which the students are required to interact with the technology. Consumable apps tend to replace existing practices while apps that require students to become authors or producers tend to result in more transformational practices. While understanding how technology can influence instructional practices is important, an element that appears to be missing from each of the frameworks and models discussed above is that of student engagement. We now explore the construct of engagement, and we investigate whether and how the incorporation of technologies does, in fact, improve engagement with mathematics.

Student engagement and mathematics A significant deficit of current frameworks relating to technology use is a lack of regard for student engagement. Student engagement with mathematics plays a significant role in influencing student achievement and the decision to continue the study of mathematics beyond the compulsory years and beyond into tertiary education (Attard, 2013). One of the major reasons many schools invest in technologies is to boost student engagement based on the assumption that when students are deeply engaged, they are more likely to be more motivated and develop positive attitudes towards mathematics thus promoting learning

40  Mathematics teaching and student engagement (Attard, 2018). Expectations of improved engagement through the integration of digital technologies is reflected widely in literature (Beavis, Muspratt, & Thompson, 2015; Bray & Tangney, 2015; Pierce & Ball, 2009) so it is evident that we need to seriously consider if and how digital technology use contributes to student engagement.

Defining engagement In order to consider the influence of digital technologies on student engagement, it is important to provide a definition of engagement that will be useful in helping us review the case studies presented in this book. The constructs of engagement and motivation are often used interchangeably and although they are strongly related, they are quite separate. This is highlighted in the Motivation and Engagement Framework (Munns & Martin, 2005), which was developed to highlight the interrelated components of student motivation and engagement. Knowledge of student motivation can impact on teachers and their expectations of students, the way they structure their classes, and how they introduce resources including digital technologies. When teachers are able to adapt their practices in order to improve or maintain motivation, this effects student engagement and achievement levels (Martin, 2005). The difference between motivation and engagement is made clear by Ryan (2000, p. 102): “the distinction between motivation and engagement is between student cognition underlying involvement in schoolwork (i.e., beliefs) and actual involvement in schoolwork (i.e., behaviour).” In summary, a student’s motivation can influence his or her engagement. In analysing the case studies for inclusion in this book, we focus on engagement (observable behaviours) rather than motivation. The Motivation and Engagement Framework (Munns & Martin, 2005) provides two clear levels of engagement. The first, small “e” engagement (‘e’ngagement) is characterised by students being “in-task” as opposed to “on-task”. A joint effect of the individual and relational levels of the Framework is termed big “E” engagement (‘E’ngagement). The combination of all three facets (motivation, ‘e’ngagement, and ‘E’ngagement) leads to students feeling good, thinking hard, and actively participating in school, leading to an overall enduring engagement. Engagement that occurs in the context of a subject area or within an individual classroom is related to small “e” engagement, and it is this engagement that we are concerned with in relation to mathematics education and the use of technology. The construct of engagement can be characterised as meaningful participation in an educational context where knowledge and learning are valued and used. In terms of engagement with mathematics within the context of this book, ‘e’ngagement can be seen when students are procedurally engaged within the classroom, actively participating in tasks, and “doing” the mathematics with the view that learning mathematics is worthwhile, valuable and useful within and beyond the classroom. We draw on the seminal review conducted by Fredricks, Blumenfeld, and Paris (2004) to further define engagement as a multidimensional

Mathematics teaching and student engagement 41 construct operating at behavioural, cognitive, and affective levels, resulting in a deeper student relationship with mathematics. Viewing engagement as the combination of behaviour, emotion, and cognition provides a characterisation of children that is potentially more valuable that researching the individual components, as in reality these factors are dynamically interrelated (Fredricks et al., 2004). This perspective informs the definition of engagement applied in this book: the coming together of affective, cognitive, and operative facets, leading to students valuing and enjoying school mathematics and seeing connections between schools mathematics and their own lives (Attard, 2013; Fair Go Team NSW Department of Education and Training, 2006; Munns & Martin, 2005). This definition includes the individual’s thoughts that are projected outwards in relation to a person’s investment and effort towards learning and relational behaviours as they are observed in the mathematics classroom (Attard, 2014). The Framework for Engagement with Mathematics (FEM) (Table 3.1) (Attard, 2014) draws on research and literature pertaining to student engagement and mathematics pedagogy to provide a tool that was originally intended to assist in planning for engaging mathematics experiences. The framework emerged from a longitudinal study of the influences on student engagement during the middle years of schooling (Grades 5 to 8). More recently, it has been used as an framework to assist in analysing qualitative data based on classroom observations, interviews, and focus group discussions in mathematics classrooms to determine how the use of technologies assist in increasing (or decreasing) student engagement (Attard, 2018; Hilton, 2018). Importantly, the FEM takes student voice seriously in its consideration of influences on engagement with mathematics; hence, student voice is a feature of this book (Chapter 7). The FEM describes the influences on student engagement as consisting of two separate yet inter-related elements: pedagogical relationships and pedagogical repertoires. Pedagogical relationships refer to the interpersonal teaching and learning relationships between teachers and students that optimise the learning of and engagement with mathematics. Pedagogical repertoires refer to the teaching practices that are employed by the teacher in day-to-day teaching. The development of positive pedagogical relationships is considered to set the foundations for substantive student engagement. It is challenging to engage students without the establishment of such relationships, regardless of the quality of pedagogical repertoires. For example, if a teacher is unaware of individual student needs (TA) that result from continuous interactions (CI), or the teacher does not consider students’ pre-existing knowledge (PK), it would be difficult to plan lessons that provide an appropriate challenge (CT) or are relevant to the lives of his or her students (RT). When we add technology to the FEM as an additional instructional consideration, teachers also need to consider students’ prior experiences with technology use to ensure they are able to design effective technology enhanced learning experiences in mathematics. Once positive pedagogical relationships are established, the implementation of engaging pedagogical repertoires can be considered. The FEM details six different elements that assume aspects of more traditionally recognised frameworks

42  Mathematics teaching and student engagement and constructs such as Shulmans PCK (1986) and TPACK (Koehler & Mishra, 2009). The elements also address non-content specific practices that have been found to directly influence student engagement with mathematics such as the provision of choice (PC) and a variety of tasks (VT). Although the FEM contains a technology-specific element within its list of engaging pedagogical repertoires, in this book the entire framework will act as one lens through which we explore and analyse technology-enhanced mathematics lessons and teacher practices. Table 3.1  The Framework for Engagement with Mathematics (FEM) (Attard, 2014) Aspect

Code

Element

Pedagogical Relationships

In an engaging mathematics classroom, positive pedagogical relationships exist where these elements occur: PK

Pedagogical Repertoires

Pre-existing Knowledge: students’ backgrounds and preexisting knowledge are acknowledged and contribute to the learning of others CI Continuous Interaction: interaction amongst students and between teacher and students is continuous PCK Pedagogical Content Knowledge: the teacher models enthusiasm and an enjoyment of mathematics and has a strong Pedagogical Content Knowledge TA Teacher Awareness: the teacher is aware of each student’s mathematical abilities and learning needs CF Constructive Feedback: feedback to students is constructive, purposeful and timely Pedagogical repertoires include the following aspects: SC CT PC ST RT

VT Students are engaged

Substantive Conversation: there is substantive conversation about mathematical concepts and their applications to life Challenging Tasks: tasks are positive, provide opportunity for all students to achieve a level of success and are challenging for all Provision of Choice: students are provided an element of choice Student-centred Technology: Technology is embedded and used to enhance mathematical understanding through a student-centred approach to learning Relevant Tasks: the relevance of the mathematics curriculum is explicitly linked to students’ lives outside the classroom and empowers students with the capacity to transform and reform their lives Variety of Tasks: mathematics lessons regularly include a variety of tasks that cater to the diverse needs of learners with mathematics when:

• Mathematics is a subject they enjoy learning (affective) • They value mathematics learning and see its relevance in their current and future lives, and • They see connections between the mathematics learned at school and the mathematics used beyond the classroom

Mathematics teaching and student engagement 43

Technology and engagement with mathematics “All too often mathematics is a black box that is kept closed, either as there is no reason to try to open it, or it is deemed as too complicated to even try” (Hoyles, 2016, p. 227). Hoyles claims that a major challenge for increasing engagement with mathematics is to address its current invisibility, suggesting that one way to do this is to harness digital technology. While student engagement continues to be a dominant reason for using technology in today’s classrooms, research relating specifically to the influence of technology on student engagement in mathematics is still limited. Some examples of research specifically investigating engagement as a result of technology use are included here to provide some insight into the success of digital technologies in improving engagement with mathematics. Several studies on the use of iPads in primary mathematics classrooms have reported some levels of improved engagement (Attard, 2015; Attard & Curry, 2012). In a synthesis of these studies, Attard used the FEM to analyse the findings of three separate studies. The first was an exploratory case study conducted in the early days of iPad integration. The second was a multiple case study that investigated the pedagogies of four classroom teachers from one school in their first six months of iPad integration, and the third was a multiple case study involving 16 teachers and their students from eight Australian schools. All three studies employed qualitative methods. When comparing the data from the three studies, Attard found that there were great variances in the levels of improvement in student engagement, and this was due to the use of a range of applications (with a variety of affordances), variations in teacher confidence and experience, and a great diversity in the ways in which the devices were used. Rather than the device or software making a difference to student engagement, it is suggested that it is how they are used, the purpose for their use and the “pedagogical practices that embed their use that determine how engaging they are” (2018, p. 63). A longitudinal study conducted in Australia by Hilton (2018) also used the FEM alongside a quantitative survey to explore increases in engagement during the first two years of a three-year phased program of iPad integration in a primary school. The FEM was specifically used to code data collected from the teachers and students. Hilton’s findings indicated that there were high levels of student engagement resulting from iPad use, and in particular, increased engagement in students who may not otherwise have been engaged. The ability for students to learn through the use of multimodalities “and to use multiple modes when creating their own products is a strongly engaging factor” (p. 156). Hilton also found that students were highly engaged through the use of drill and practice apps. In addition, the use of iPads improved the engagement of a diverse range of students including those with special needs. Students’ attitudes to technology use for learning mathematics have also been studied in secondary classrooms. The Mathematics and Technology Attitudes Scale (MTAS) was developed in 2007 and measures students’ mathematics confidence, confidence with technology, attitudes towards use of technology for learning mathematics, affective engagement, and behavioural engagement

44  Mathematics teaching and student engagement (Pierce, Stacey, & Barkatsas, 2007). When trialing the scale with 350 students in grades 8–10 in six secondary schools, the authors found that in every school, most students agreed rather than disagreed that it was better to learn mathematics with technology. In a follow-up study using MTAS with 1089 Year 9 and 10 students, the researchers found that positive attitudes towards learning mathematics with computers, which was linked to mathematics confidence and affective engagement (Barkatsas, Kasimatis, & Gialamas, 2009). As discussed in Chapter 2, the flipped classroom is an approach, facilitated by technology, where passive forms of instruction that generally take place in the classroom are “flipped” to occur at home. Typically, students will watch a video of their teacher explaining a new concept before they go to class, so that time spent in class can be used for discussion, clarification, and for actively engaging in mathematics. There is evidence that the use of the flipped classroom with secondary students can improve their levels of engagement with mathematics (Clark, 2015; Muir & Geiger, 2016). Similarly, in a research synthesis of 60 papers, positive effects on engagement were noted with the use of mobile technologies for learning mathematics (Fabian, Topping, & Barron, 2016). Clearly technology has the potential to have a positive impact on student disengagement in mathematics, which is particularly acute in secondary school classrooms. This chapter has provided an examination of effective teaching in mathematics and popular frameworks for integrating technology in classroom settings. We have also outlined the importance of considering student engagement and the role that technology can play, given that student disengagement with mathematics continues to be a significant issue for teachers to address. However, questions still remain about what is happening in the reality of today’s classrooms that vary considerably in terms of access, equity, and practices with technology. Can teachers use technology effectively with minimal access to devices? What does the effective use of technology look like in an early years’ classroom when compared to a senior mathematics classroom? What approaches, at the school level, are most effective in supporting teachers of mathematics to use technology to optimise student engagement and learning? What do teachers, leaders, and students actually think about technology use in mathematics classrooms? To answer these questions, in Section two, we present ten case studies of teachers who have been identified by their peers as effective users of technology. The case studies range from early childhood through to senior secondary classrooms, allowing us to investigate the common elements of effective teaching practices regardless of school system, context, and grade level. We also explored the differences in practices that arose due to variations in resourcing, approaches to professional development, and leadership perspectives.

References Anthony, G., & Walshaw, M. (2009). Effective pedagogy in mathematics. Belley, France: International Academy of Education. Armstrong, A. (2014). Technology in the classroom: It’s not a matter of “If,” but “When” and “How.” The Education Digest; Ann Arbor, 79(5), 39–46.

104  Technology in the secondary classroom

Figure 6.1  Surface Area and Volume Problem.

The students (Figure 6.1) were then asked to consider the following problem displayed on the IWB. The students were encouraged to use the cubes to examine the problem. John then drew different cross-sectional representations on the whiteboards and demonstrated alternative methods of approaching the problem to assist the students in their visual reasoning. After this lesson starter, John asked the students: “What is a prism?” and proceeded to illustrate student suggestions on the board, leading them to consider the general rule for finding the volume of a prism (the area of the cross section multiplied by the length). John then used this prompt to talk to students about how all area formulae for two-dimensional shapes are related, showing a video linking the area of a circle to the area of a rectangle. John again paused the video at crucial points to allow the students to conduct their own calculations before proceeding to the solution revealed in the video. A student then asked about the origins of the formula for the circumference of a circle, indicating a high level of cognitive engagement. John praised the student for asking the question and demonstrated the relationship between the diameter of a circle, pi, and the circumference using one student’s water bottle. This exchange provided evidence of several elements of the FEM, including continuous interaction and explicit linking between abstract mathematical content and real-life examples (Attard, 2014). Continuing to emphasise the relationships between various 3D shapes, John then asked the students how cones and pyramids relate to prisms in terms of their volume. Students estimated that the volume of a cone or pyramid with the same base area would be one-half of the related prism. John then played a video which demonstrated that the volume of a cone is one-third of the volume of the related prism. This judicious use of video provided students with engaging ways of accessing mathematics concepts that are difficult to visualise and require developed spatial skills. John’s use of multiple representations to reinforce the connections between 2D shapes and 3D solids, moving seamlessly between drawings on whiteboards, textbook exercises, and video clips provided multiple points of access for students to engage with the content from numerous

46  Mathematics teaching and student engagement Clark, K. R. (2015). The effects of the flipped model of instruction on student engagement and performance in the secondary mathematics classroom. Journal of Educators Online, 12(1), 91–115. Clarke, D. (2003, December 4). Challenging and engaging students in worthwhile mathematics in the middle years. In B. Clarke, A. Bishop, R. Cameron, H. Forgasz, & W. T. Seah (Eds.), Making mathematicians (pp. 98–109). Melbourne, Australia: The Mathematics Association of Victoria. Dawson, K., Ritzhaupt, A., Liu, F., Rodriguez, P., & Frey, C. (2013). Using TPCK as a lens to study the practices of math and science teachers involved in a year-long technology integration initiative. Journal of Computers in Mathematics and Science Teaching, 32(4), 395–422. Delaney, S., Ball, D. L., Hill, H. C., Schilling, S. G., & Zopf, D. (2008). “Mathematical knowledge for teaching”: Adapting U.S. measures for use in Ireland. Journal for Mathematics Teacher Education, 11, 171–197. Ertmer, P., Ottenbreit-Leftwich, A., Sadik, O., Sendurur, E., & Sendurur, P. (2012). Teacher beliefs and technology integration practices: A critical relationship. Computers & Education, 59(2), 423–435. Even, R., & Tirosh, D. (2008). Teacher knowledge and understanding of students’ mathematical learning and thinking. In L. D. English (Ed.), Handbook of international research in mathematics education (2nd ed., pp. 202–222). New York, NY: Routledge. Fabian, K., Topping, K. J., & Barron, I. G. (2016). Mobile technology and mathematics: Effects on students’ attitudes, engagement, and achievement. Journal of Computers in Education, 3(1), 77–104. doi:10.1007/s40692-015-0048-8 Fair Go Team NSW Department of Education and Training. (2006). School is for me: Pathways to student engagement. Sydney, Australia: Author. Florida Center for Instructional Technology. (2018). Retrieved July 27, 2018, from The Technology Integration Matrix website: https://fcit.usf.edu/matrix/matrix/ Fredricks, J. A., Blumenfeld, P. C., & Paris, A. H. (2004). School engagement: Potential of the concept, state of the evidence. Review of Educational Research, 74, 59–110. Freeman, A., Adams Becker, S., Cummins, M., Davis, A., & Hall Giesinger, C. (2017). NMC/CoSN horizon report: 2017 K-12 edition. Retrieved from https://www.nmc.org/ publication/nmccosn-horizon-report-2017-k-12-edition/ Geist, E. (2010). The anti-anxiety curriculum: Combating math anxiety in the classroom. Journal of Instructional Psychology, 37, 24–31. Goos, M., Galbraith, P., Renshaw, P., & Geiger, V. (2000). Reshaping teacher and student roles in technology-enriched classrooms. Mathematics Education Research Journal, 12(3), 303–320. doi:10.1007/BF03217091 Guerrero, S. (2010). Technological pedagogical content knowledge in the mathematics classroom. Journal of Digital Learning in Teacher Education, 26(4), 132–139. Harris, J., Mishra, P., & Koehler, M. (2009). Teachers’ technological pedagogical content knowledge and learning activity types. Journal of Research on Technology in Education, 41(4), 393–416. doi:10.1080/15391523.2009.10782536 Hattie, J. (2003). Teachers make a difference: What is the research evidence? Presented at the building teacher quality: The ACER annual conference, Melbourne, Australia. Hill, H. C., Ball, D. L., & Schilling, S. G. (2008). Unpacking pedagogical content knowledge: Conceptualising and measuring teachers’ topic-specific knowledge of students. Journal for Research in Mathematics Education, 39, 372–400. Hilton, A. (2018). Engaging primary school students in mathematics: Can iPads make a difference? International Journal of Science and Mathematics Education, 16(1), 145– 165. doi:10.1007/s10763-016-9771-5

Mathematics teaching and student engagement 47 Hoyles, C. (2016). Engaging with mathematics in the digital age. Cuadernos de Investigación y Formación En Educación Matemática, 225–236. Retrieved from http://www.centroedumatematica.com/Cuadernos/CuadernosCompletos/ Cuaderno15.pdf Hurrell, D. (2013). What teachers need to know to teach mathematics: An argument for a reconceptualised model. Australian Journal of Teacher Education, 38(11). doi:10.14221/ajte.2013v38n11.3 Kim, C., Kim, M., Lee, C., Spector, J., & De Meester, K. (2013). Teacher beliefs and technology integration. Teaching and Teacher Education, 29, 76–85. Kimmons, R., & Hall, C. (2018). How useful are our models? Pre-service and practicing teacher evaluations of technology integration models. TechTrends, 62(1), 29–36. doi:10.1007/s11528-017-0227-8 Koehler, M. J., & Mishra, P. (2009). What is technological pedagogical content knowledge? Contemporary Issues in Technology and Teacher Education, 9, 60–70. Koh, J. H. L. (2018). Articulating teachers’ creation of technological pedagogical mathematical knowledge (TPMK) for supporting mathematical inquiry with authentic problems. International Journal of Science and Mathematics Education, 1–18. doi:10.1007/ s10763-018-9914-y Kopcha, T. J. (2012). Teachers’ perceptions of the barriers to technology integration and practices with technology under situated professional development. Computers & Education, 59(4), 1109–1121. Krauskopf, K., Foulger, T. S., & Williams, M. K. (2018). Prompting teachers’ reflection of their professional knowledge. A proof-of-concept study of the graphic assessment of TPACK instrument. Teacher Development, 22(2), 153–174. doi:10.1080/13664530. 2017.1367717 Livingstone, S. (2012). Critical reflections on the benefits of ICT in education. Oxford Review of Education, 38(1), 9–24. doi:10.1080/03054985.2011.577938 Lovitt, C. (2000). Investigations: A central focus for mathematics. Australian Primary Mathematics Classroom, 5, 8–11. Martin, A. J. (2005). Can students’ motivation and engagement change? Findings from two intervention studies. Presented at the association for research in education focus conference, Cairns, Queensland, Australia. McGrath, J., Karabas, G., & Willis, J. (2011). From TPACK concept to TPACK practice: An analysis of the suitability and usefulness of the concept as a guide in the real world of teacher development. International Journal of Technology in Teaching and Learning, 7(1), 1–23. Muir, T., & Geiger, V. (2016). The affordances of using a flipped classroom approach in the teaching of mathematics: A case study of a grade 10 mathematics class. Mathematics Education Research Journal, 28(1), 149–171. doi:10.1007/s13394-015-0165-8 Munns, G., & Martin, A. J. (2005). It’s all about MeE: A motivation and engagement framework. Presented at the Australian association for academic research focus conference, Cairns, Queensland, Australia. Nardi, E., & Steward, S. (2003). Is mathematics T.I.R.E.D? A profile of quiet disaffection in the secondary mathematics classroom. British Educational Research Journal, 29(3), 345–367. Neubrand, M., Seago, N., Agudelo-Valderrama, C., DeBlois, L., & Leikin, R. (2009). The balance of teacher knowledge: Mathematics and pedagogy. In T. Wood (Ed.), The professional education and development of teachers of mathematics: The 15th ICMI study (pp. 211–225). New York, NY: Springer.

48  Mathematics teaching and student engagement Newmann, F. M., Marks, H. M., & Gamoran, A. (1996). Authentic pedagogy and student performance. American Journal of Education, 104(4), 28–312. Niess, M. L., Ronau, R. N., Shafer, K. G., Driskell, S. O., Harper, S. R., Johnston, C., & Kersaint, G. (2009). Mathematics teacher TPACK standards and development model. Contemporary Issues in Technology and Teacher Education, 9(1), 4–24. OECD. (2016a). Students, computers and learning: Making the connection. France: PISA. OECD. (2016b). Ten questions for mathematics teachers…and how PISA can help answer them. Retrieved from http://www.oecd.org/publications/ten-questions-for-mathematics-teachers-and-how-pisa-can-help-answer-them-9789264265387-en.htm Orlando, J., & Attard, C. (2015). Digital natives come of age: The reality of today’s early career teachers using mobile devices to teach mathematics. Mathematics Education Research Journal, 1–15. doi:10.1007/s13394-015-0159-6 Ottenbreit-Leftwich, A. T., Glazewski, K. D., Newby, T. J., & Ertmer, P. A. (2010). Teacher value beliefs associated with using technology: Addressing professional and student needs. Computers & Education, 55(3), 1321–1335. doi:10.1016/j. compedu.2010.06.002 Pierce, R., & Ball, L. (2009). Perceptions that may affect teachers’ intention to use technology in secondary mathematics classes. Educational Studies in Mathematics, 71, 299–317. Pierce, R., Stacey, K., & Barkatsas, A. (2007). A scale for monitoring students’ attitudes to learning mathematics with technology. Computers & Education, 48(2), 285–300. doi:10.1016/j.compedu.2005.01.006 Prieto-Rodriguez, E. (2016). “It just takes so much time!” A study of teachers’ use of ICT to convey relevance of mathematical content. International Journal for Technology in Mathematics Education, 23(1), 16. Puentedura, R. (2006). SAMR: A brief introduction. Retrieved from http://www. hippasus.com/rrpweblog/archives/2013/10/02/SAMR_ABriefIntroduction.pdf Ricks, T. E. (2009). Mathematics is motivating. The Mathematics Educator, 19, 2–9. Roblyer, M. D., & Hughes, J. E. (2019). Integrating educational technology into teaching: Transforming learning across disciplines (8th ed.). New York, NY: Pearson. Ryan, A. M. (2000). Peer groups as a context for the socialisation of adolescents’ motivation, engagement and achievement in school. Educational Psychologist, 25, 101–111. Schnellert, G., & Keengwe, J. (2012). Digital technology integration in American public schools. International Journal of Information and Communication Technology Education (IJICTE), 8(3), 36–44. doi:10.4018/jicte.2012070105 Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. American Educational Research Journal, 15, 4–14. Skemp, R. R. (2006). Relational understanding and instrumental understanding. Mathematics Teaching in the Middle School, 12(2), 88–95. Stigler, J. W. & Hiebert, J. (1999) The teaching gap: Best ideas from the world’s teachers for improving education in the classroom. New York, NY: The Free Press. Tangney, B., & Bray, A. (2013). Mobile technology, maths education & 21C learning. Proceedings of the 12th World Conference on Mobile and Contextual Learning (MLearn2013), 2013, 20–27. doi:10.5339/qproc.2013.mlearn.7 Thomas, D. (2007). Teaching technology in low socioeconomic areas. The Technology Teacher; Reston, 67(3), 4–8. Yair, G. (2000). Educational battlefields in America: The tug-of-war over students’ engagement with instruction. Sociology of Education, 73, 247–269.

4

Technology in early years mathematics classrooms

The increasing popularity and ubiquitous nature of mobile devices has resulted in very young children becoming adept at their use prior to entering formal education. The engaging and game-based nature of many software applications lend themselves well to the existing pedagogies within early mathematics classrooms, given that almost all young children are actively involved in game playing in either a digital or concrete form. It is also during the early years of formal schooling that critical conceptual foundations are formed in the mathematics classroom. These foundations play a significant role in determining future mathematical success for students. In this chapter, we present three case studies situated in early years mathematics classrooms where technology use significantly enhances teaching and learning. The first case study takes us to a pre-school setting. The second is located in a Year 1 classroom, and the third is a classroom at the same school, situated within a special support unit. This classroom caters to students who are unable to participate in mainstream education due to a range of conditions including autism spectrum disorder (ASD) and specific learning disabilities (SLD). The chapter will give a big picture view of the structures of the two schools in relation to how they support technology integration in early years classrooms, the teachers’ practices that incorporate the technologies, and the ways the students are accessing and learning from the technologies. Observations of technology enhanced lessons will be provided alongside accompanying detailed lesson plans. Data derived from interviews with school leaders and the classroom teachers is included to provide a holistic picture of successful technology enhanced mathematics classrooms.

Case study 1: pre-school In Australia, many children begin their education in pre-school settings. The pre-school featured in Case Study 1 is part of a public primary school. Students generally attend the preschool two to three days per week, and in this classroom, all students are approximately four years old and are expecting to begin formal schooling the following year.

50  Early years mathematics classrooms

School context Sharon’s school is located in a small, mid-socio-economic suburban area of Canberra, Australia’s capital, and was established in the late 1980s. The school is a public primary school that has a pre-school attached and a combined population of 322 students (2018 data). Children attending the pre-school generally go on to enrol in the primary school. There is a close relationship between the pre-school and primary classrooms at this school, and the teachers work collaboratively to ensure consistency in the technology-related practices across the school. Sharon, our case study teacher, is a technology coach, working across the pre-school and lower primary classrooms.

Whole-school commitment to technology use To gain insight into the school context, we interviewed Sandra, the executive leader of the pre-school, kindergarten, and two ASD units at the school. Sandra explained that the whole school pedagogical lens is based on an inquiry approach, which she believes lends itself to the use of technology. She explained the school’s philosophy in regard to the use of technology: Technology is a tool that is used to support the teaching and learning for students to be able to demonstrate their understanding, to consolidate any understanding, and also as an avenue to be able to communicate their learning of different concepts. Technology is embedded across all learning areas, so it’s used in every aspect of teaching and learning where it is authentic and most useful to support students’ learning. The school has a commitment to use technology in mathematics teaching and learning in a way that is consistent across all grades. Sandra explained that when it comes to numeracy, the teachers at this school seek to understand students’ prior knowledge. This aligns with the Framework for Engagement with Mathematics (FEM) (Attard, 2014), which recognises this practice as an element important for establishing student engagement. Sandra explained that the teachers use a range of thinking routines, mathematical problems, or what she termed as inquiry provocations to determine students’ prior knowledge. This is then followed by what she terms “a finding out phase”, where teachers seek different ways for students to use technology. She explains this strategy: We always find out first what the students bring to the table, things that they already know about a particular concept, and then we look for different ways for students to - in the finding out phase - to use technology as a way to find out information as researchers so they can go on and research about particular mathematical concepts, as well as using different apps to be able to consolidate and practice understanding of concepts, and then even as an approach to demonstrating that understanding of how they produce something through technology to show that.

Early years mathematics classrooms 51 Following the process described by Sandra, teachers then plan technologyenriched teaching and learning activities. These technology-related practices have been incorporated into the culture of the school, including the students, and the broader community of parents and caregivers, and the ways they engage in these practices.

Building community Schools that operate with an emphasis on technology-rich learning environments sometimes need to convince the broader community that their practices are going to add value to their children’s education. A common concern amongst parents and caregivers of young children is the amount of screen time that children spend on digital devices, and whether schools are providing an appropriate balance between physical activity, interactions with others, and digital screen time. At Sharon’s school, communication with the parent community is prioritised and takes several forms. In order to gain the continued support of parents, Sharon and the other teachers at her school communicate teaching and learning strategies through traditional methods as well as through digital platforms such as Google Communities and Facebook. Google Communities is a tool that allows people with the same interests to communicate with one another. The school uses Google Communities to share school news and students’ work in real-time. Sandra spoke about the use of Google Communities: I mean our parents are very much tech-savvy and that as well. I mean we use it as a way of communicating learning to families through the Google communities, so families have a lot of access to seeing learning in real time. They can see what their kids are doing in their - while they’re at work in their busy lives. There’s always that balance between do kids spend too much time on technology or should they be spending more time writing, but it’s us communicating to the families. Sharon also spoke about the ways she uses Google Classroom with her class and how it has promoted interactions between her and her students’ parents, who often comment on what their children are doing in the classroom. She also talks about the importance of having community support in relation to the use of technology in the pre-school classroom, saying that it is the school’s responsibility to inform parents about how technology is being used in teaching and learning.

Developing culture through professional development The leadership team at Sharon’s school has a strong commitment to building teacher capacity in regard to technology-related practice. The school has a team of “digital champions” who lead their colleagues in building technical and

52  Early years mathematics classrooms pedagogical expertise in alignment with the school and system plans. The digital champions run a coaching program that is accessed by teaching and administration staff as well as teaching support staff. This is how Sandra described the coaching program: Teachers can enlist for the coaching program and then we look at the best match and when a program - that is - Initially their digital champion will meet with the teacher that they’re going to be coaching with, talk about what they want to get out of it, their goal, if it’s for themselves, and then for the classroom. Then the teacher - the coach will come in and demonstrate, work with the teacher and the students, and that can go for as long as they feel that is needed. The level of individualised support for teachers is an important influence on how technology-related practice plays out in the classroom. This strategy, as opposed to a one-size-fits-all approach to professional development, recognises the diversity of teacher expertise and doesn’t make assumptions relating to teachers’ age, experience or ability (Garet, Porter, Desimone, Birman, & Yoon, 2001; Higgins & Parsons, 2011; Orlando & Attard, 2016). In addition to providing internal support, the teachers at Sharon’s school also attend external professional development. This is usually in the form of an annual digital technology conference where the teachers are encouraged to network with educators from across the country.

About Sharon Sharon is an award-winning early childhood teacher. She was originally trained as a primary teacher and has been teaching for over 30 years. She began her career in the primary setting but after having children of her own, returned to university and completed her qualifications in Early Childhood. When we met Sharon, she was in the midst of participating in a significant research study, the Early Learning STEM Australia (ELSA) project (Lowrie & Larkin, 2019). ELSA is an initiative of the Australian government aimed at promoting curiosity and engagement in STEM, through the use of a set of four apps designed for pre-school children, their teachers, and their parents. Sharon spoke to us about how her involvement in the ELSA project had heavily influenced her teaching, saying: “I look at everything very differently. I do things with far more intent than I’ve ever done before. I think that’s exciting”. Sharon perceives herself as a lifelong learner, and this is evidenced by her learning in relation to technology use. Prior to her school winning a digital literacy grant, Sharon did not actively use technology and did not feel confident as a user of technology. However, since that earlier experience and her participation in the ELSA initiative, she has become a technology leader within her school and uses technology daily in her own teaching. Sharon has taken her own professional learning experiences and

Early years mathematics classrooms 53 transitioned from being a novice to leader using a combination of digital technology and an inquiry-based approach to learning. The inquiry approach adopted by Sharon and her school has aligned well with her work within the ELSA initiative, which, according to Lowrie and Larkin (2019), adopts a heuristic that involves an experience, represent, apply (ER A) framework. In this framework, students’ prior experiences (E) are the basis for concept development and involve play-based (off-app) experiences. Children then engage with the app to represent the relevant concepts (R). Within the R phase, children are able to create their own representations by “creating images, interpreting pictures, visualising and using symbols” (p. 6). These affordances embedded within the ELSA apps align with the redefinition phase of the SAMR (substitution, augmentation, modification, and redefinition) framework (Puentedura, 2006), while simultaneously communicating engaging messages of knowledge, place, and control (Munns, 2007) to the children. Following the representation phase, children then engage in further off-app activities through play-based experiences. Sharon spoke about her experiences using the ELSA apps, which have become a regular feature in her students’ numeracy experiences: So, they get to play the app. Then they get to go and apply that learning and you see that. The reason we’ve gone back to patterns was because - I thought we were doing quite well. But there’s still a few - a bunch that really are struggling to get there. So, I thought - and it’s great for everybody to keep going back and having that extra practice with all sorts of patterns and that sort of thing. So, we’ve come - coming back around and doing that. Sharon uses the apps to develop conceptual understanding, collect formative assessment data, and provide additional experiences for the children to build fluency. She is passionate about teaching mathematics with technology, and her use of technology is carefully planned to ensure a balance of indoor and outdoor play, physical activity and interactions, and screen time.

Sharon’s approach to technology use In Australia, pre-school settings are required to align children’s experiences with the Early Years Learning Framework (EYLF) (Department of Education, Employment and Workplace Relations, 2009). Rather than being discipline specific, the EYLF identifies the following broad goals to be addressed: Belong, Being, and Becoming. Sharon plans her teaching with these goals in mind and believes that the use of technology assists in addressing them. While the ELSA apps are a feature of many of the children’s numeracy experiences, Sharon also uses a range of other digital technologies to make learning engaging, and some of these were seen during her lesson observations that are described next. As with several of our other case studies, Sharon uses videos sourced from YouTube on a regular basis in her classroom. Sharon commented how the children

54  Early years mathematics classrooms are familiar with such platforms (particularly recognition of their logos) and how this familiarity has led her to believe that technology should be a regular feature in pre-schools and school classrooms, saying: “We have a responsibility to make this part of every day. It’s every day for them at home, it needs to be”. Along with YouTube clips, Sharon also regularly uses PowerPoint presentations and a range of free online games. For example, she spoke about two games; Starfall and Top Marks: “They were just free - they’re great apps. Matching the numbers or there’s something about patterns or something about opposites. The children just have to match”. Sharon also makes use of the Internet and Google searches within her inquirybased learning approach. She has found that the natural curiosity of young children can be promoted in this way, implementing a “just in time” rather than “just in case” approach to learning. Children in Sharon’s classroom are encouraged to wonder, and when they do have a “wondering”, it is common for the children to request that Sharon “google” the information: We’re always looking for answers. It’s nice that the kids can see that we don’t have it all - I don’t have it all worked out, myself. But - so, we can find out together. I often write their wonderings on the board and I might put a bubble. Like, get Mason to stand out there with a bubble and we - how are we going to work this out? Sharon also uses a range of devices in her classroom. She has access to a small set of iPads that the children share during activities that are conducted either individually or collaboratively. IPad use is generally supervised by an adult. The importance of supervision was evident to Sharon when one of her students experienced an issue with an app that did not recognise a pattern she generated as it was a more complex pattern than what would typically be expected. In this case, Sharon had to reassure the student that her answer was correct, even though the app did not recognise it as such. Supervision of iPad use also enables teachers to step in when they observe children making repeated errors or when the technology fails to work. During whole-class activities and group activities, Sharon uses a SmartScreen, which is mounted at a height that allows the children to interact with it during their activity time. Sharon is a teacher who has embraced technology use. She recognises its ubiquity in the children’s lives now, the role it will play in their future lives, and the skills that that the affordances of contemporary technologies promote: So, this learning that - this idea that we don’t - really, we don’t know what the future will hold. We’re not sure what our - the kid’s jobs will have in the future. But we do know that if they’re going to be successful in for whatever they go into, then they need to be a team player. They need to collaborate. That’s going to be a life skill. They need to be a thinker. They need to be creative. They need to problem-solve. They need to be able to communicate. They need to be able to explain that pattern back so that language - and they

Early years mathematics classrooms 55 need to self-manage themselves. Some people find it a bit tricky. But we use that language and I think that’s great. It goes - it starts here but it goes right through the whole school.

Lesson 1: making patterns Technology: Touch Screen Monitor, iPads, ELSA pattern app Mathematical Content: Recognising, continuing, and creating patterns Lesson Duration: 45 minutes During our two observations in Sharon’s classroom (conducted on consecutive days), she was supported by a teacher’s aide and a pre-service teacher who was in her final week of practicum. Nineteen students were in attendance for this lesson (three were absent). The mathematics experiences began as soon as the children arrived at school. This lesson appeared to be loosely structured around the ER A framework (Lowrie & Larkin, 2019) as described earlier. Sharon and her teacher’s aide had set up materials and activities. As the children arrived, they were guided to the activities where they found baskets of Unifix cubes and were instructed to make a pattern with the cubes. There was lots of dialogue about patterns between the children and adults, with the adults making comments like “You’ve made an AAB pattern”, and “Can you tell me about your pattern?” aligning with the FEM in relation to continuous interaction (CI). The intentional teaching part of the lesson began after 15 minutes of guided informal concrete experiences. The children sat in front of the touch screen monitor that was displaying a spreadsheet of the class roll. Sharon moved to the screen and asked the children to choose from cars, balloons, or butterflies. Balloons were selected, and the children’s names were then represented in balloons on the monitor. The students were guided, one at a time, to find their name in a balloon and touch it on the screen, recording their attendance. Although word recognition is not a mathematical skill, Sharon incorporated mathematics in a conversation with students about the number of students present, and the number of children away. This led to some brief counting-on practice. The children were then asked to stand and form a horse-shoe arrangement, facing the monitor. Sharon used the monitor to mirror her iPad screen. The students were told they would be going to use one of the ELSA songs to show how patterns can be made with bodies. The children were told they would be copying the dance movements in the animation. The students and adults all danced and sang along with the character in the app and all appeared to be highly engaged. The kinesthetic element of this activity reinforced a range of repeating patterns and aligned well with the Unifix activity and discussions that had occurred at the start of the day. Following the dance, the children were instructed to sit down in a circle on the perimeter of the mat. Sharon led the singing of the song that incorporated patterns in words and actions. At this point, there was no further use of technology. Sharon continued with some more short, focused activities using hands-on materials, pattern cards, and a picture book provide multiple representations to convey the concept of pattern.

56  Early years mathematics classrooms After 30 minutes, the children were then provided with a range of tasks to choose from. One of the options was the use of a pattern app on iPads, which were located in a dedicated iPad area of the classroom. Five students took an iPad each and immediately began to use the apps. The app they were using required them to make a pattern using sounds. Although they were working individually, the children were sharing what they were doing and enjoyed a giggle when playing their patterns. Some of the children switched to different apps, selecting from the range available on the iPads. During this time, Sharon walked around the class, checking the iPad group every few minutes to provide support if required. The students worked in this way for approximately ten minutes. The remainder of mathematics time was conducted outside the classroom. The students were given a paper bag and instructed to go outside into the playground to collect ten things. Sharon suggested they could collect sticks, rocks, or leaves. The students were then required to make a pattern out of their objects. Once outside, the teachers used chalk to draw a circle on the concrete path for each student to place their pattern. This part of the lesson aligned with the representation (R) phase of the ER A heuristic described earlier (Lowrie & Larkin, 2019).

Lesson 2: ordering length and sequencing Technology: Touch Screen Monitor, iPads, YouTube (Mat Man), Letter School app Mathematical Content: Counting, number recognition Lesson Duration: 45 minutes As students entered the classroom they moved directly to the floor and got started on the pre-prepared activity. The activity required the students to cut a strip of paper that had lines drawn on it, resulting in strips of paper of different lengths. Students were then required to glue them onto another piece of paper in order from the smallest to the largest. The children then decorated their bits of paper and were encouraged to make patterns, reinforcing the learning from the previous day’s focus. After approximately 15 minutes, the children were asked to sit in a circle on the floor for show and tell (work samples). This was followed by the same roll call routine that occurred the previous day. During the intentional teaching section of the lesson, Sharon announced that the children would be talking about order. She asked the children about things they do when they get to preschool (e.g., fruit break, lunch). To focus the students, Sharon showed them the necklace she was wearing, which was a small Babushka (nesting doll) and earrings. She then showed them an actual Babushka doll, getting the dolls out and putting them in order to demonstrate the concept of order. The children were then led in the two order-related songs; Days of the Week and Seven Steps. Sharon then played a video, Mat Man (https://www.youtube.com/ watch?v=ec7J57YewxQ). Mat Man is a video clip, accompanied by a song that incorporates counting and order. Mat Man is an outline of a man, with body

Early years mathematics classrooms 57 parts compiled step by step. The children sang along with the clip and this was followed by Sharon introducing a replica of all the body parts from the Mat Man song. Each child was provided with one part and the children then replicated the song, building their own Mat Man. This part of the lesson appeared to be a seamless integration of technology. These students were clearly used to this integration and appeared to almost expect its use. Following Mat Man, Sharon mirrored one of the ELSA apps that the children had engaged with on a previous occasion. The app required the children to take a series of photographs to build a picture story which they would then put into the correct order. Sharon showed a story completed earlier in the year by one of the students and the children appeared to be highly engaged when watching their own stories back and hearing their own voices. This strategy conveyed strong messages of engagement with regard to place and knowledge. Using resources constructed by the children for future learning experiences gave the students a voice in their classroom and was an acknowledgement of their role as a member of the class group. The strategy also enhanced the relevance of the lesson and personalised the learning for the children, which are important elements in building positive pedagogical relationships (FEM). In the final part of the lesson, the children were split into three groups, each with an adult for supervision. Each adult was equipped with an iPad. The groups were to create new photo stories within the particular ELSA app. They would then order their photographs. After approximately 15 minutes, the children were called back together to share each group’s work on the large monitor.

Case studies 2 and 3: year 1/2, support class School context In this section, we visit the classrooms of Ashleigh (Year 1/2) and Rebecca (Support Class). The case studies were conducted in the same school, a public primary school situated in a young, low-socio-economic suburb of what had previously been semi-rural land in the outer Sydney suburbs. In 2018, the school had 356 students. When we first visited the school, it was in its fourth year of operation. The school was established in 2016 and was still experiencing significant growth in population and teaching staff each year. This meant that many school policies, including those regarding technology-related practice, were still in their development phase. At the time of our visit, the teachers were trialling a whole-school approach to their mathematics scope and sequence. This approach required the teachers to teach a different strand of the mathematics curriculum each day. This strategy influenced what happened in mathematics lessons, including their structure and the types and duration of activities that could reasonably fit within individual lessons. For example, it would be challenging for teachers to introduce rich mathematical tasks that allow students to delve deeply into the mathematics, as tasks could not be carried over from one lesson to the next. Limitations such as this type of scope and sequence

58  Early years mathematics classrooms highlight how significant the influence of school policy is on day-to-day classroom teaching. It can also be considered as a political dilemma, requiring the teachers to address issues of accountability demanded by various stakeholders, which then creates a pedagogical dilemma for teachers in terms of how they make mathematics meaningful within the boundaries set by those stakeholders (Windschitl, 2002).

Commitment to technology-enriched teaching To gain insight into the school’s developing philosophy regarding the place of technology in teaching and learning we interviewed Fran, the Assistant Principal responsible for the school’s support unit. Rebecca, the second case study teacher from this school, also held an informal leadership role in the implementation of technology across the school and was able to speak to us about the school’s philosophies. Fran spoke about how the school was forward-thinking in its approach to technology use. Each of the classrooms is equipped with a SMART board (interactive whiteboard, IWB). The school had also begun investing in iPads, and at the time of our visit, had a few sets that were shared amongst various class groups. Due to financial constraints and the newness of the school, it was taking time to build what Rebecca referred to as the “technology profile” of the school. Rebecca explained how the school managed with limited funds and the donations of old computers from a neighbouring school: …we do have one class set of stand-alone computers and we’ve got that in the library. We were able to do five iPads per class with the budget that we had. We have now expanded that. We were able to purchase 50 more for across the school. We’ve made sure that those iPads are going to be one-toone in the [support] unit. The school had formed a technology committee to assist in guiding the purchase of devices and software and the technology-related practices within the school. Fran made this comment about the general attitudes towards technology use: “they are quite open. It’s very young staff and forward thinking and definitely a twenty-first century learning approach”. The overall approach to technology at the school was organic, and at the time, involved some trial and error. This might be considered by some as an inefficient way to develop shared practices across the school. However, the challenges of developing a brand-new school in a brandnew community cannot be underestimated. Allowing the teachers freedom to undertake a trial and error approach to technology use can be viewed from a positive perspective. Teachers at this school were trusted for their professional judgement, and successes and failures were shared with colleagues. According to Fran, “someone will sign up for that and everyone will get access to it and then that also helps guide whether that’s something that people would like to do”. An example of this is highlighted in this quote from Rebecca: “I really like Prodigy.

Early years mathematics classrooms 59 These guys don’t use it, but I’ve used it with the MC (multi-categorical support) class last year and championed it across the school”. Although the school’s approach to technology-related teaching and learning was in an early stage of development, it did have a strategic approach to developing a sense of community through the use of digital technology. The school has a Facebook page and this is used to post photographs and videos of students engaged in learning. According to Fran, Rebecca, and Ashleigh, the reactions of parents to seeing their children in action at school have been positive. Arguably, the positive reactions would also then act as a further incentive for teachers to use devices to capture student activities to share with the broader community.

About Ashleigh When we met Ashleigh, she had just begun her fourth year of teaching. Most of her prior teaching experience had been in early years classrooms and she had been teaching at this school for three years. In contrast to the majority of school contexts in this book, Ashleigh was not working in a technology-rich environment. Her classroom was equipped with one IWB and at the time of our observation, she had no access to any other mobile devices. The small group of iPads that were normally available were inaccessible at the time of our visits. The only other digital device available to Ashleigh was her personal mobile phone.

Technology-enriched lessons with limited technology When planning her lessons, Ashleigh said she always includes elements of digital technology embedded within students’ learning experiences. Her reasons for doing so are varied. She believes the use of digital technology aligns with her students’ interests in the technology itself. This recognition of students’ lives and experiences outside the classroom is an important element when it comes to engagement and bridging the digital divide as described by Selwyn, Potter, and Cranmer (2009). Ashleigh also uses the IWB because it promotes movement, interactivity, and manipulation of mathematical representations through the use of apps designed for mathematics. The limitation of having one IWB does not phase Ashleigh. When she plans how students will access the technology, she structures her lessons around wholeclass and group activities. Her intention with group activities that involve the IWB is to develop leadership skills: “I try and share that leadership role around”. When groups are using the IWB, one student is given the role of leader. In the lessons we observed, the leader of the group ensured all of the other group members took turns to respond to questions posed with the app that was being used. If we consider this simple strategy through the lens of engagement, allowing students to take leadership roles within their groups sends powerful messages of place, knowledge, and control about the classroom space. The technology-related limitations that Ashleigh experienced at the time of this case study resulted in the development of a range of strategies that although

60  Early years mathematics classrooms based on technology use were not reliant on having lots of devices in the classroom. One strategy that was observed was the use of the Plickers app. Plickers replicates the use of a classroom response system that uses individual student codes printed on card rather than personal devices or clicker devices. Plickers allowed Ashleigh to gather formative data using multiple choice questions, an IWB and her personal mobile phone. Ashleigh, like all the other teachers featured in this book, has an obvious affinity with technology and it is embedded in all aspects of her teaching, from lesson planning to lesson delivery and assessment of student work. She uses technology as an organisational tool through productivity apps such as Google Drive. Although on the surface this may not be considered to influence the student experience with mathematics, we need to consider how the ease and access of resources and assessment data improve the flow and planning of the lessons due to the evidence able to be captured and reflected upon seamlessly throughout.

Assessment through digital technology One of Ashleigh’s strategies for collecting assessment evidence is through the use of photographs and videos. During lessons she uses her personal mobile phone to capture evidence of student learning. Ashleigh talked about how this saves time and also allows her to remember the fine detail of what students are saying and doing during lessons. This evidence is also used to demonstrate student achievement to parents during interviews, providing real confirmation of alignment with Ashleigh’s professional judgement about student achievement. Ashleigh’s use of technology as an assessment tool allows her to focus on teaching and interacting with the students rather than writing notes and is a move away from traditional assessment in mathematics that has always been reliant on pen and paper assessment. Such traditional assessments often lead to mathematics anxiety (Henschel & Roick, 2017). An additional benefit of digital assessments in a class where students are still learning to write is that it allows students and teacher to retain their focus on the mathematics on the lesson. Digital assessment also promotes an emphasis on the processes of mathematical reasoning and communication alongside mathematical content, giving a deeper insight into student abilities. When asked about her philosophy regarding teaching mathematics with the use of digital technologies, Ashleigh talked about how she predominantly uses it because it engages her students. However, she does admit that not all students are engaged all of the time, and this is something she has to “let go”. Ashleigh believes the hands-on and interactive activities afforded by the use of her IWB contribute to student engagement and enriches mathematics learning. When asked about her thoughts about which topics are better suited to the use of technology, Ashleigh talked about measurement and geometry. Many of the required skills for students in Grades 1 and 2, such as comparing, sorting, and organising, are enhanced through mathematics-based apps. Ashleigh also discussed how she would teach most topics with technology but is wary of using

Early years mathematics classrooms 61 apps that are just “busy work” such as drill-based games. Her preference is to use apps that require students to apply their knowledge and make mathematical connections. The following two lessons were observed on consecutive days within the first few weeks of school, when students were still settling in to their new classroom.

Lesson 1: representing numbers Technology: IWB, Mobile Phone, Plickers app, Rekenrek app Mathematical Content: addition and subtraction Lesson Duration: 75 minutes There were 21 students in Ashleigh’s class when we observed her lessons. These students were a combination of Year 1 and Year 2. The first lesson observed began with the children forming three groups. A range of rotation-based activities was prepared by Ashleigh. One group began working with Ashleigh using hundreds charts and counters. The second group was working with laminated A4 sheets. Each sheet had one number on it and the students used dominoes to represent that number. The third group of eight students was working on the IWB, using the Nelson Maths Hundreds Chart activity. The app represented numbers up to 100 in words. The students were required to find the numbers in figures on the chart. One student was appointed as the “leader” and would choose another to find the number. As each child took a turn, he or she would use the IWB pen to colour it on the hundreds chart. The program provided instant feedback, indicating whether the response was correct or incorrect. During this task, most of the students appeared to be engaged. While working with the group using physical hundreds charts and counters, Ashleigh promoted high levels of discussion, encouraging students to share their thinking. During this time, we observed Ashleigh video recording her students’ responses. As she did so, the students did not appear distracted and seemed confident when discussing their thinking. When considering this practice through the lens of the SAMR model (Puentedura, 2006), it would be difficult to place this use of technology. The SAMR model is limited to student activities rather than teacher practices, which we consider to be a limitation. However, this seamless use of technology could be regarded as an extension of self, forming a natural part of Ashleigh’s day-to-day practice (Goos, Galbraith, Renshaw, & Geiger, 2000). Approximately one hour following the start of the lesson, after each group had attempted each task, the group work was packed away and Ashleigh asked the students to collect their Plicker codes and sit on the floor facing the IWB. Each of the students had a laminated card with a personalised code printed on it. The Plickers app was on the IWB, and Ashleigh had written a range of questions based on subtraction, which had been the main focus of this lesson. As each multiple-choice question was posed, Ashleigh instructed the students to hold up their card to show their answer (each side of the personalised code was marked with either an A, B, C, or D).

62  Early years mathematics classrooms To gather the students’ answers, Ashleigh scanned the codes with her mobile phone, which instantly provided her with data relating to the students’ understandings. To gain further insight, she asked the students about the computation strategies they used to arrive at their answers. This assessment practice is important in the context of multiple-choice questions, where students could simply guess an answer and be correct, without any understanding of the mathematics. Ashleigh made comments like “this is going to get me ready for our next lesson”, indicating the value of gathering formative assessment data and using it to direct practice that caters to students’ needs. During the Plickers assessment, the students appeared to be very excited, demonstrating affective engagement. The additional questions posed by Ashleigh ensured that cognitive engagement was also a feature of this part of the lesson. An important benefit of the Plickers app was that although Ashleigh could gather a record of her students’ knowledge, the screen view only showed the number of students who selected each response, highlighting the correct one. While subtle, this affordance avoided embarrassment for those students whose answers were incorrect. Such public embarrassment has the potential to lead to mathematics anxiety, restricting students’ ability to engage with mathematics (Boaler, 2015). Following the Plickers task, Ashleigh then used a digital representation of a Rekenrek, an arithmetic bead rack designed to develop number sense. She arranged the beads and conducted a number talk about the number of beads displayed, and the strategies the students used to come to their conclusions. While this activity could have been conducted with the use of a physical Rekenrek, this would have made it difficult for all students to see the rack, and the cost of purchasing the physical materials may have been prohibitive in a school that had minimal funding. In this case, the technology enhanced the task by providing an enlarged image of the bead rack at no cost, allowing all students a direct view. The lesson ended after this activity, and the students went out to recess.

Lesson 2: number (fractions) Lesson Focus: Number Technology: IWB, Rekenrek, Nelson Maths Mathematical Content: Fractions Lesson Duration: 1 hour This lesson began with the students sitting in a circle on the floor, in front of the IWB. The lesson began with Ashleigh organising the students in groups of two and three to play a game of double snap. During this time, Ashleigh had the Class Dojo app displayed on the IWB. She used this to reward students for good behaviour. The students spent five minutes playing snap and were directed to pack up and get ready for maths groups. The group activities were explained and demonstrated to the whole group prior to students separating into groups. Part of this explanation involved modelling the IWB activity to ensure the students understood how to use it. This also included lots of dialogue between Ashleigh and the students about the mathematical concepts within the app.

Early years mathematics classrooms 63 The three groups were organised as they had been in the previous lesson, with one teacher-led group working on a paper-folding fraction activity, an independent group playing a spinner game with counters and laminated gameboards, and a third group working on a Nelson Maths activity sorting 3D objects on the IWB. As in the previous lesson, Ashleigh works with one group at time. During this time, she conducted small group reflections and recorded these with her iPhone. Forty-five minutes after the lesson had begun and all the students had worked on the three different activities, Ashleigh directed them to pack up and come to the floor at the front of the class. The lesson concluded in a similar manner as the previous day. This time, the Rekenrek number talk occurred first, followed by a formative assessment using the Plickers app. This time the focus of the revision was based on the fractions concept covered in the teacher-led group task.

Technology use in a support class Rebecca’s school hosts a support unit comprised of three classes. Each class has a small number of students who are unable to join mainstream classrooms due to significant intellectual disabilities. The students within each class are grouped according to needs rather than age. For example, Fran teaches students from Kindergarten to Year 6. Our case study teacher, Rebecca, came to the school to assist in establishing the support unit and teaches a small class of six students who range in age from six- to eight year old.

About Rebecca Rebecca originally trained as a Kindergarten to Year 12 music teacher and has over 12 years of teaching experience. She explained that she “fell into special education” when she started teaching across all subject areas. Rebecca has a special interest in educational technology, working as an ICT coordinator at her previous school: I had some understanding and I took over the role just to help initially. But realised that I had some skill and I had a lot of passion for using technology and using it well. Integrating it and making sure that it wasn’t just being used for the sake of being used because that is detrimental. Rebecca’s past experiences with technology included the incorporation of coding lessons through apps such as Minecraft, in order to engage students and improve their attendance rates: “the kids were disengaged, and they didn’t want to do bookwork. I mean they didn’t want to write; they didn’t want to be at school, they didn’t see the purpose of school”. Although the context at her current school and classroom is significantly different to her previous contexts, Rebecca still regards technology as critical in engaging those hard-to-engage students. This is particularly relevant in her current classroom, where her students’ needs are vastly different, due to their disabilities, than the needs of students in regular classrooms.

64  Early years mathematics classrooms

Technology and engagement in a high-needs classroom Due to the small number of students in the support unit with high learning needs, access to technology is significantly better than in the mainstream classes at this school. Each support class has an IWB and access to one iPad per student, and technology use forms an integral part of everything that happens in her classroom: Because we don’t just use technology for doing part of your numeracy rotation, doing part of your literacy rotation. We have things on there that include timetabling, choice, timers, all stuff that supports the autism aspect of our students. The use of technology in Rebecca’s mathematics lesson appears to play a critical role in focusing, engaging, and communicating with her students. While some of her strategies, such as the use of game-based mathematics apps, reflect what you would observe in any mainstream classroom, others assist in addressing the challenge of simply communicating with her students and opening up opportunities for learning to occur. For example, the use of YouTube videos of songs based on counting is used as a hook for students: “We have always used the smartboard and songs on the smartboard as a focuser. To the extent that now that they’re able to engage with a routine”. Rebecca also relies on using mathematics-based apps to engage her students. However, due to their learning needs, she has to be careful in selecting apps that allow enough time for her students to respond. She describes the challenge in finding such apps: If I have a complaint, it’s that there aren’t enough apps and games out there that slow the process down for these guys. Like that addition practice game on Studyladder, it’s just meant to be for fun, the kids just do that after they do it. But it’s perfect for these guys because the game waits for them to process the problem, figure out what they need to be doing and then enter it. Whereas a lot of games are time-based. Rebecca spends a significant amount of time trawling through the App Store for suitable apps. She also explained how important it is for her to trial the apps she intends on using with her students. However, her search is limited to free apps due to a lack of school funding. Reflecting the school’s current trial and error approach, Rebecca’s app choices do not always work for all of her students, and due to the nature of her students, this often depends on the current disposition of the students. She explains this challenge in the following quote: I will trial them and then you put them in front of the kids. Some of them don’t work. Like (student) - too it depends on the day. (Student) yesterday didn’t want anything to do with the number formation app, he just wanted

Early years mathematics classrooms 65 to do the letter formation. But today you saw I gave him the letters as the reward, and he went back to numbers. So it depends on the kids as well. One of the most significant benefits to the use of technology in Rebecca’s classroom appears to be the ways in which it promotes communication. In a room where some students’ disabilities severely restrict their capacity to communicate at a most basic level, technology use is critical in relation to laying the foundations for learning. Although they may have difficulty communicating in a traditional manner, this does not mean that these students don’t understand some mathematical concepts. The use of technology has provided Rebecca with opportunities to gain better insight into her students’ mathematical capabilities. Due to the nature of Rebecca’s students and our desire to avoid disrupting their learning, we only observed one of Rebecca’s lessons, which we describe below.

Lesson: number recognition Technology: IWB, iPads, YouTube, Addition Battle (Mathematics), Study Ladder, Hundred Rap Mathematical Content: Number recognition, counting, tracing numbers Lesson Duration: 50 minutes Rebecca’s class has an enrolment of six students. However, on the day of our observation, three were absent due to illness. Rebecca was assisted by a teacher’s aide. Rebecca generally timetables mathematics lessons to occur first thing in the morning following the school assembly. The mathematics lesson began with the three students sitting in chairs facing the IWB. Rebecca asked one of the students to select a YouTube clip from their playlist. The student selected the clip and they sat and counted along. Although two of the children were rocking in their chairs and appeared quite restless, it was also clear that they were engaging with the content of the clip as they counted along. The clip involved a great deal of repetition. At the conclusion of the first video clip, Rebecca played the Hundred Rap clip, which included an animated rap. Rebecca and the teacher’s aide sang along to the clip to encourage the students to count. At this point, the three students appeared to be very focused on the clip as the song counted from one to hundred. The three students no longer rocked in their chairs and appeared to be tapping to the beat of the song. Following the Hundred Rap, Rebecca directed two of the children, one at a time, to take their chairs back to their desks. She provided each student with individual instructions as they each completed different tasks. One student remained sitting in front of the IWB. Rebecca worked with this student to practice the skill of counting on, using the Study Ladder app. Rebecca divided her time between two of the students providing assistance, while the teacher’s aide assisted the third student. After approximately ten minutes, the children swapped places so that a second child was invited to take a turn using the IWB. This student requested to

66  Early years mathematics classrooms use an iPad rather than the IWB, so Rebecca sat with him at her desk. This student has a severe intellectual disability, and Rebecca had to provide a high level of support to enable him to engage with the app. After a short period, the student appeared to become upset and walked away, and Rebecca was unable to get him to return to her desk. She eventually got the student to sit at his desk where they worked together on the iPad. During the remainder of the lesson, the students appeared to move between their iPads and the IWB, engaging in a range of apps. Although this was a small class of only three students, the nature of their disabilities made it a challenging class. Throughout the lesson, Rebecca was constantly on her feet, actively supporting the students. Although on the surface, the lesson appeared disjointed, Rebecca worked hard to ensure each student’s needs was catered to. Technology was used throughout the lesson, and apart from the introductory session where the students watched video clips, each child accessed different apps that were selected by Rebecca to specifically meet their individual needs. Throughout the lesson, Rebecca offered the children choice of device and app, an important element in terms of student engagement. The lesson was a strong reflection of the philosophies articulated by Rebecca during her interview. The technology clearly focused the children early on in the lesson; however, due to the challenges of having disabilities, two of the children’s interactions with technology were observed to require high levels of assistance from either Rebecca or the teacher’s aide.

Summary In this chapter, we have detailed how two schools use technology for teaching and learning in mathematics in the early years of schooling. The two schools vary considerably, with one well-established in terms of technology-related policies and practices compared to the other, which is a new school still developing their approach to technology across the school. In Sharon’s school, there was a shared philosophical commitment to inquiry-based learning, which included embedded technology across all subject areas. To support teachers in delivering this vision, the school took a whole-school approach to professional development, by appointing technology leaders (digital champions) to work in a personalised way with individual teachers as required. Both schools considered it important to build their school communities through structured communication with parents, with Sharon’s school seeing it as their responsibility to educate parents about technology use in the school. In both schools, the teachers took care to deliver lessons where traditional activities, often involving student movement and interaction, took place alongside screen-time activities, which focussed on carefully chosen apps. Where young children are concerned, there is substantial community concern about the overuse of screens, and therefore it is important for schools and parents to understand how digital devices are being used and both the benefits and limitations

Early years mathematics classrooms 67 of this use. Despite this qualification, the teachers at both schools were strongly committed to technology use in the classroom, as they saw it as an integral part of the world we live in and of their students’ futures. At Rebecca and Ashleigh’s school, where a whole-school approach was not yet in place, there was still an ethos of encouragement of technology use, albeit through a “trial and error” method, rather than through an agreed systematic approach. Both teachers were very inventive with how they used technology, possibly out of necessity, as there were few resources reliably available for every lesson aside from an IWB. Despite these resource limitations, the purposes for technology use in each school were similar. In both schools, we saw the teachers use technology to build conceptual understanding of mathematical concepts, fluency in mathematical processes, and to collect rich formative assessment data to inform their teaching. As the lessons observed in this chapter took place in early years classrooms, the activities observed were relatively short in timespan to cater for the attention span of young children, requiring the teachers to manage multiple transitions within each lesson. The teachers made purposeful decisions to vary their pedagogical practices between digital and non-digital activities, to maintain and enhance student engagement. In this way, the technological tools at their disposal were used deliberately alongside complementary “analog” approaches so as to maximise student engagement and learning.

References Attard, C. (2014). “I don’t like it, I don’t love it, but I do it and I don’t mind”: Introducing a framework for engagement with mathematics. Curriculum Perspectives, 34, 1–14. Boaler, J. (2015). Mathematical mindsets: Unleashing students’ potential through creative math, inspiring messages and innovative teaching. Hoboken, NJ: John Wiley & Sons. Department of Education, Employment and Workplace Relations. (2009). Belonging, being and becoming: The early years learning framework for Australia. Retrieved from https://docs.education.gov.au/system/files/doc/other/belonging_being_and_ becoming_the_early_years_learning_framework_for_australia.pdf Garet, M., Porter, A., Desimone, L., Birman, B., & Yoon, K. (2001). What makes professional development effective? Results from a national sample of teachers. American Educational Research Journal, 38(4), 915–945. Goos, M., Galbraith, P., Renshaw, P., & Geiger, V. (2000). Reshaping teacher and student roles in technology-enriched classrooms. Mathematics Education Research Journal, 12(3), 303–320. doi:10.1007/BF03217091 Henschel, S., & Roick, T. (2017). Relationships of mathematics performance, control and value beliefs with cognitive and affective math anxiety. Learning and Individual Differences, 55, 97–107. doi:10.1016/j.lindif.2017.03.009 Higgins, J., & Parsons, R. (2011). Professional learning opportunities in the classroom: Implications for scaling up system-level professional development in mathematics. Mathematics Teacher Education and Development, 13(1), 54–76. Lowrie, T., & Larkin, K. (2019). Experience, represent, apply (ERA): A heuristic for digital engagement in the early years. British Journal of Educational Technology, 1–17. doi:10.1111/bjet.12789

68  Early years mathematics classrooms Munns, G. (2007). A sense of wonder: Pedagogies to engage students who live in poverty. International Journal of Inclusive Education, 11, 301–315. Orlando, J., & Attard, C. (2016). Digital natives come of age: The reality of today’s early career teachers using mobile devices to teach mathematics. Mathematics Education Research Journal, 28, 107–121. Puentedura, R. (2006) SAMR: A brief introduction. Retrieved from http://www. hippasus.com/rrpweblog/archives/2013/10/02/SAMR_ABriefIntroduction.pdf Selwyn, N., Potter, J., & Cranmer, S. (2009). Primary pupils’ use of information and communication technologies at school and home. British Journal of Educational Technology, 40(5), 919–932. Windschitl, M. (2002). Framing constructivism in practice as the negotiation of dilemmas: An analysis of the conceptual, pedagogical, cultural, and political challenges facing teachers. Review of Educational Research, 72(2), 131–175. doi:10.3102/00346543072002131

5

Technology in the primary mathematics classroom

The upper primary/elementary years are a critical time in relation to maintaining student engagement with mathematics. When students disengage, they are likely to make the decision that they are simply “not good at maths” (Attard, 2014) and often discontinue its study beyond the compulsory years. The effective use of technology could potentially improve student engagement, assist in developing deep conceptual understanding, and bridge the digital divide (van Deursen & van Dijk, 2014) between how technology is used at home and at school. In this chapter, we provide three case studies of exemplary practice in the integration of technology within primary/elementary classrooms. We now present three teachers, Loretta, Bec, and Jessica. Loretta and Bec teach at the same school. Loretta teaches Year 3, and Bec teaches a composite Year 3 and 4 class. Jessica teaches Year 6 class. Details of the specific school policies and philosophies that support technology integration at each of the schools are provided before we delve into each individual teachers’ beliefs and practices relating to technology in the mathematics classroom.

Case studies 4 and 5: year 3 School context Loretta and Bec’s public primary school is situated in a mid-socio-economic, suburban area of Sydney. There were 478 students enrolled in the school during 2018. The leadership team in the school actively promotes the use of technology in teaching and learning and for external communication with the broader school community via social media tools (Twitter and Facebook). The school has been running a bring your own device (BYOD) program from Years 3 to 6 for approximately three years where each student is required to purchase an iPad for use at school and at home. Strictly speaking, true BYOD programs do not prescribe the device or platform that students bring to school (Maher & Twining, 2017); however, many schools opt for a consistent approach for a range of reasons, including equity and ease of implementation and planning. To assist in making the purchases, the school organised a discounted price and payment program through Apple. The school assists

70  Primary mathematics classrooms families who are unable to afford to purchase an iPad by lending one to those students (there were three students in this position at the time the case study was conducted). Students in the lower grades have access to sets of iPads that are shared across the classes. The school also owns sets of laptops that can be shared amongst the classes. There is no whole-school approach to how technology is used at Bec and Loretta’s school although the Assistant Principal, Amy, indicated that all teachers were asked to open a SeeSaw account but were not mandated to use it. SeeSaw is a digital portfolio that stores work samples in a range of forms including video, photograph, PDF, and drawings. Work captured in Seesaw can be shared with peers, teachers, and families and can be carried over from year to year, providing a long-term record of student growth and achievement. Each of the teachers also use Google Classroom, a tool for managing coursework. When interviewed, the School Principal, Rachel, who had only been at the school since the beginning of the year, talked about the program saying: The different teachers across different stages decide how it will be used. They’ll negotiate and talk with parents about which apps and things they should [use]. They [the teachers] have a list of apps that they would like the parents to load up onto the device. That will look different in different years. The leadership team at the school encourages teachers to focus on pedagogy rather than technology, and this is one of the reasons they allow teachers flexibility in how they integrate technology into their mathematics teaching. However, it appears that inconsistencies in the way technology is being used across the school may be caused by a variety of reasons, as we now explore.

Teachers as digital natives or digital immigrants The introduction of a BYOD program brought about significant changes in practice for the teachers and students at this school, particularly in Years 3 to 6. Prior to the program, practice in the mathematics classrooms was focused on the use of textbooks: “…we were a textbook school. I got here and I was told we teach this page on this day…” (Amy). Amy talked about the impact the technology has had, saying: “…we have had a shift in understanding how technology is used to support teaching and learning as opposed to just something they pick up and do”. Although the school leadership team encourages the creative use of technology, Amy discussed the differences in technology use between the early years classrooms (no BYOD) and the Years 3 to 6 classrooms (BYOD), expressing a diversity of attitudes and views with regard to teaching with technology: “I think 3 – 6 definitely see technology and are very open to it and can use it quite well. Where K-2, not so much. I think they are more sceptical as to the ability it has to support learning”. It is interesting that there is diversity within the culture of this pro-technology school. This may be explained by the way the leadership has allocated teachers to various grades.

Primary mathematics classrooms 71 Amy described the K-2 teachers as being of a different generation, alluding to Prensky’s (2001) digital divide: K-2 has some very experienced teachers who haven’t been exposed to how technology can be used, whereas 3–6, while we are experienced, we are of the younger generation which means we are probably more adaptable to using it. But that doesn’t mean that quality things aren’t happening in K-2, they are just probably much more planned. (Amy) Another reason for the differences in attitudes could be the barriers that are caused by having to share only eight iPads amongst up to 150 students in their K-2 classrooms. The challenges for teachers in planning for the use of the devices are much greater than those experienced by the Years 3–6 teachers whose students have uninterrupted access to devices. It appears that at this school, there are pedagogical, cultural, and political dilemmas (Orlando & Attard, 2016) experienced by the K-2 teachers due to a range of reasons that are both within and beyond the control of individual teachers. Although in this chapter we will be focusing on the use of technology within Bec and Loretta’s Year 3 classrooms, it is important to acknowledge that contexts can vary within an individual school, and this is influenced by individual beliefs, group beliefs, and variances in the amount of access to technology. Amy also makes an assumption regarding the age of the teachers and their status as either digital natives or digital immigrants (Prensky’s, 2001). This is an assumption that may not be accurate due to the developing pedagogical and mathematical content knowledge of early career teachers (Orlando & Attard, 2016). An earlier study (Attard, 2015) also found that older teachers who were not confident with the use of technology were still able to design effective learning experiences due to their depth of mathematics pedagogical and content knowledge.

Challenges and barriers in a BYOD school Apart from an obvious divide caused by having only half the school implementing a BYOD program, other challenges and barriers to technology integration were discussed by Amy. She talked about how some of the teachers don’t own their own personal iPads, even though they are expected to use them in their day to day teaching. Rather than owning their own devices, these teachers use devices that are owned by the school. Amy feels this influences the teachers’ beliefs about the potential of the devices due to the limitations placed on the school-owned devices. For example, teachers are unable to download apps or change any of the settings on the iPads due to Department of Education restrictions. Another barrier that has impact on classroom practice relates to the licencing of apps. Amy explained the issue: There is some kind of licence thing to do with the amount of times it has been loaded on and off, and with schools…it makes it really hard because you want these kids to access the same learning and produce the same learning artefacts. There is no space for them to do that, because their parents can’t afford to buy them.

72  Primary mathematics classrooms Amy has addressed this issue to some degree by purchasing her own iPad and allowing her students to use this when required. Although this is not ideal, it is evidence of a strong commitment to the use of technology at the school. A final challenge to the use of technology at this school is the variation in TPACK (technological pedagogical content knowledge) (Koehler & Mishra, 2009) amongst the teachers. This poses potential problems within a school where the use of digital technology is strongly encouraged. For example, some teachers require support with basic technical issues, while others were able to focus more directly on pedagogy as they were familiar with the technical tools at hand. Despite running technology sessions for her colleagues, Amy found that some were still concerned they and their students did not know enough about how to use the various apps. Some of the teachers felt they needed to be experts at using the technologies. Amy’s position on this is that students and teachers learn together, which is arguably a sensible position to take give the fast pace of technological development: …it was getting over the fact that they didn’t need to know everything. Which is not a teaching thought that is probably that they’re comfortable with, like, I am okay not being an expert in technology. Someone, a kid will fix it before I will and I am okay with that. As long as I know why I am using it, what I want out of it, and the benefit of it I am okay. But they didn’t see that. It was more about well why won’t it click in the box? It was more that sort of barrier to them. Despite providing opportunities to share teaching ideas with technologies, Amy has found that there are still teachers at the school who are resistant, saying “they’re still sitting over here not comfortable with it but not willing to get uncomfortable”. These teachers do not appear to move beyond Niess et al.’s accepting phase, where either a positive or negative attitude towards the teaching of mathematics with digital technologies is formed (2009).

Technology transforming mathematical practices When discussing the school’s philosophy regarding the teaching of mathematics with technology, both Rachel and Amy discussed how mathematics and numeracy professional learning were a recent focus at the school. Amy talked about how the use of technology in mathematics classrooms opens up new opportunities, however, she cautioned: …if you don’t use it the right way it is a hindrance at that point. I honestly believe that you need to plan for how you want them [the students] to use the technology and really critically think about whether it is going to support the teaching and learning of maths because ultimately the end goal is what are they going to learn and how are they going to learn it and how is this piece of technology going to support that?

Primary mathematics classrooms 73 Amy talked about how the use of generic apps such as Google Maps, the use of digital cameras and audio capture can be used to support mathematical learning. This, along with the affordance of being able to share digital work with peers and between students and the teacher, enhances mathematics learning. However, there are challenges associated with such a heavy emphasis on technology in primary mathematics. Amy talked about how the perceptions of parents have evolved with regard to school mathematics. When the BYOD program began, there was initial concern about the type of mathematics now being learned. Having only experienced textbook mathematics and student work that was recorded in hard copy notebooks, and mathematics that was content based and focused on algorithms rather than mathematical reasoning, parents had to be educated on the approach now being taken by the school. Regular parent and community workshops are run at the school and this has resulted in a better understanding of contemporary practice as well as a reduction in the use of external mathematics tutors, “before they were hiring tutors because they didn’t think what we were doing was successful” (Amy). The use of technology in mathematics at this school has resulted in a stronger focus on mathematical processes rather than the traditional focus on learning content. The affordances of screen capturing alongside a heavier use of problem solving and investigation have enhanced the students’ abilities to articulate their thinking and use non-traditional methods to solve problems. A surprising challenge that has emerged as a result of this is a mismatch between the practice in primary classrooms and the practice in some secondary mathematics classrooms. Students who have graduated from this school since the introduction of the BYOD program have had some difficulties adjusting to a more traditional approach at the local secondary school, where teachers place an emphasis on the correct answers rather than methods, reasoning and correct answers. The school is currently working with the secondary teachers on a transition program in an attempt to alleviate transitions such as this. Collaboration between primary and secondary teachers in mathematics has been shown to decrease the divide in practices across this transition point (Albright, Clement, & Holmes, 2012).

About Loretta When we met Loretta, she was in her eighth year of teaching. She had been working at this particular school for seven of those years. Loretta was trained as an Early Childhood teacher and has worked as a Kindergarten teacher for four years prior to teaching Year 3 for three years. Loretta also has a Master of Education (Leadership) degree and is the Computer Coordinator at the school. This role requires her to support her colleagues with integrating technology into their teaching. Loretta is passionate about the use of technology in teaching and learning mathematics. When asked how she plans for technology use Loretta talked about how she actually doesn’t plan much at all. Rather, she uses digital technology in

74  Primary mathematics classrooms an organic way, treating it as one tool amongst a repertoire of tools, retaining her focus on the teaching, and learning of the mathematics within the task, “I don’t really plan to use technology a lot, quite often It just happens”. Loretta also described how her students have an affinity with technology, often reaching for their iPads to assist them in a non-technology-based task. For example, if they are curious about how to spell a word, they access a dictionary app, “…if they need their iPad they can go and get it and it is not like we have technology times”. Loretta believes that her students’ iPads “should be like having a pencil…it should be something that they just can go and use if they need it”. It appears that Loretta and her students use technology as an extension of self, as described by Goos, Galbraith, Renshaw, and Geiger (2000). The use of technology in this classroom is seamless rather than viewed as an “event”. Loretta believes technology plays an important role in engaging students with mathematics learning. She has noticed her students are much more engaged in what she calls “simple” mathematics concepts when using digital technologies compared to the traditional pen and paper activities. She also thinks it is important that students are able to use technologies fluently, saying: “the way society is changing they need to be exposed to it in ways beyond just knowing how to type a Word document or how to use an Internet browser”. She also confirmed her stance on how technology needs to be an integrated element of learning rather than something that is just added on to existing practices. When discussing the suitability of using technology for the range of mathematics topics and concepts, Loretta claimed that there were no topics she wouldn’t teach without technology. However, the way the technology is used may be an influence on this belief. For example, Loretta uses generic apps to capture student reflection, or QR Codes in her task design. These tools could be used with any mathematics content or any other subject area. When it comes to using technology to teach and learn mathematical content, Loretta still values the use of concrete manipulatives, believing the opportunity to feel, touch, and manipulate objects when learning about topics such as position or 3D objects is important in developing conceptual understanding. She does use tools such as Google Sheets to teach data, and Kahoot, an app that allows the teacher to design a multiple choice, timed quiz, to assess or revise work. However, there is less of an emphasis on using technology to provide representations of mathematical concepts. This is interesting given the affordances of digital devices and the vast range of applications available. The following two lessons were observed in Loretta’s classroom. In these lessons, Loretta attempted to highlight her use of a range of technologies.

Lesson 1: attributes of polygons Technology: IWB, iPads, BeeBots, BlueBot app, SeeSaw app, QR Code reader Mathematical Content: Properties of 2-Dimensional shapes, using and following positional and directional language Lesson Duration: 75 minutes

Primary mathematics classrooms 75 The lessons we observed in Loretta’s class were conducted on consecutive days. There were 26 students in attendance for the first observation. Loretta began the lesson by mirroring her iPad onto the IWB (interactive whiteboard). All students were seated on the floor with their iPads and were instructed to download the BlueBot app. Loretta announced that the students would be going to use the BeeBots to find polygons. Because the class only had access to six BeeBots, the iPads would be used in conjunction with the BlueBot app by those students who did not have access to the actual BeeBot. To ensure equity, Loretta had uploaded a file that replicated the physical shape mats that would be used with the physical BeeBots. At the start of the Lesson, Loretta shared this file with the students to be uploaded into their BlueBot apps. Although this procedure sounds complex, these students followed the instructions and there were no apparent issues, indicating a high level of technology experience in this classroom. This was made easier through Loretta’s clear, step-by-step instructions. Loretta then explained the task (the full task is documented in Appendix 1). She had designed a set of clue cards that linked to the “road map” (BeeBot mat) she had designed. The children had to find polygons that fit the criteria in the clues. Loretta placed the large road maps on the floor and the children appeared very excited by the prospect of using the physical BeeBots. After explaining the task, Loretta conducted a quick revision of the understandings that the students needed in order to complete the task. Rather than giving the information, Loretta allowed the students to contribute their understandings, and students made comments such as: “It’s a closed shape”, “Has more than 3 sides”, “There’s no curved sides”, and “Polygons with four sides and 10 sides”. Loretta asked lots of open-ended questions that promoted continuous interaction, an element of the Framework for Engagement with Mathematics (FEM) (Attard, 2014), a contributor to student engagement through the development of positive pedagogical relationships. Another element of Loretta’s practice that promoted engagement was the provision of choice. The students were permitted to choose the partners they wanted to work with. Students who did not have their iPads were given priority to work with the physical BeeBots. The students did not appear to mind whether they worked with physical or digital BeeBots. This may be due to the nature of the task and its game-like element of having to respond to clues. As Loretta finished explaining the requirements of the tasks she directed students to work collaboratively on their iPads, with three students working in each group. Recording sheets were handed out and students were set to work. During the task the students appeared to be engaged. They did not appear to be distracted by the physical BeeBots, and conversations were focused on the mathematics of the task, demonstrating cognitive and operative engagement. The flexibility demonstrated by Loretta in adapting the task to suit the number of devices/robotics available appeared to be an advantage for these students, allowing all students to engage with the same task at the same time. However, those students who worked on iPads did have some difficulties caused by the size

76  Primary mathematics classrooms of the screen. One of the shapes was a nonagon (11-sided), and it was difficult for the students to accurately count the number of sides. Following the students’ return to class after a lunch break; they conducted a reflection on the Puzzling Polygons activity, using the SeeSaw app to record their thinking. To access the reflection, Loretta provided students with a QR Code. Students were required to scan the code, then follow instructions that led them to an online journal. This was demonstrated on the IWB by mirroring one of the student’s iPads. During reflection time, the students spread out across the classroom, with some sitting on the floor, and some sitting at tables, taking advantage of the mobility of their devices. Although the students could have easily conducted their reflections verbally or in their work books, the use of the app allowed Loretta to access the student responses anytime and anywhere, without the need to carry a pile of workbooks. This also meant that work samples could be shared with parents.

Lesson 2: grouping numbers Technology: IWB, iPads, QR Code Reader Mathematical Content: Multiplication, Division Lesson Duration: 40 minutes This lesson began with the students sitting on the floor in a circle. Loretta placed 20 plastic cups in the circle on the floor. This seemed to build excitement amongst the students, promoting affective engagement. Loretta then conducted a digital technology-free number activity by asking the students to try and work out the total number of cups, and the different ways they could be grouped to facilitate efficient counting. During this time there was lots of discussion and mathematical reasoning, and all students played a role. While Loretta could have designed this task using digital technology, the use of physical materials and the incorporation of movement provided a balance and was evidence that Loretta had reflected on the most effective way to demonstrate the mathematical concepts for her students. Following the task, Loretta made a seamless transition to the body of the lesson, saying this to the students: “There’s a reason that I got you to start off with this activity. Because today, you’re going on a treasure hunt”. She then went on to explain that the students would be working in groups and would require an iPad, a whiteboard, a pencil, and a container of counters. Each group was allocated a different letter of the alphabet. Students would be required to scan a QR Code, which would lead them to a problem to solve collaboratively. This strategy allowed Loretta to provide tasks that were differentiated to the needs of each group without the groups being aware that the tasks were offered at different levels. Loretta then made an explicit link to the introductory activity, to ask students to think about the kind of activity they would be doing (in terms of mathematics). This alerted the students to the focus of the lesson, which was multiplication and division. Students were instructed to have their worked checked by a teacher before they could progress to the next QR Code. The students appeared excited to

Primary mathematics classrooms 77 begin the task, and during the course of the lesson, they were observed to work collaboratively, using the counters provided to assist in solving the problems. The lesson concluded when the students were directed to sit on the floor in front of the IWB. The students were instructed to have their iPads ready to play Kahoot. Loretta displayed Kahoot on the IWB and provided a range of multiple-choice questions intended to build fluency in multiplication. Students were asked to respond individually if they had their own iPads. If not, they were allowed to work as a team and share iPads. Although it took Loretta a few minutes to set up the app, the students did not seem to become restless. This may have been because they had experienced the app on prior occasions and were highly engaged due to the competitive nature of the activity. As soon as the task began, the students became very quiet as they responded to the questions. However, the task was not as seamless as it could have been due to connectivity issues for some of the students. This was a slight disruption to the lesson and some students began chatting. Once the app was rebooted, the students completed the questions. While the students were highly engaged, this app differed from the Plickers app used in Ashleigh’s classroom (Chapter 4). In this case, students were identifiable by their peers, and Loretta used this to ask specific students to explain their thinking when responding to specific questions. The students were also able to see who was in the lead, and this competitive element seemed to engage this particular group of students. The lesson concluded when all of the Kahoot questions had been responded to and the lunch bell had rung.

About Bec Bec was in her second year of teaching and had a combined Year 3 and 4 classes. During her first year, Bec had worked as a casual (relief) teacher prior to being employed in a temporary full-time basis at her current school. Although very new to teaching, Bec embraces technology as a regular element in her teaching across all subject areas. One of her major reasons for using technology in mathematics lessons is to provide differentiation for her students: “I try and differentiate as much as possible and I find that I use technology a lot to do that”. One of the ways Bec does this is by designing inquiry-based tasks that are open-ended. For example, she spoke about a recent mathematics task that required her students to plan a trip around Australia. This required them to access maps via Google Maps, information regarding major landmarks via the Internet, and apply measurement skills and a range of number-related concepts included statistics. Bec said the task allowed students to “get a real sense of themselves in the world” and provided opportunities for students to apply their mathematical knowledge: “they had to figure out how long it would take to get them from here to there”. Bec believes open-ended tasks such as this, along with the use of digital technology, provides opportunities for students to learn concepts independently and build on previously learned concepts. Bec’s classroom is equipped with an IWB. Each child in the class has a personal iPad as part of the school’s BYOD program. Bec is in an unusual position

78  Primary mathematics classrooms as she has never taught outside a BYOD program, so has always worked with the expectation that technology is embedded throughout the curriculum, unlike many of her colleagues. The inclusion of digital technologies is an integral part of Bec’s planning process, and this has resulted in her using a broad range of affordances to achieve her goal of differentiating tasks and ensuring high engagement of her students. An affordance that Bec takes advantage of is the flexibility devices can offer in relation to capturing student work samples and formative assessment data: …there’s different ways of presenting ideas and information. I like to get them to take videos or take voice recordings because sometimes it is not always about writing it down, and more about finding out what they know. Because with some of my kids it takes them forever to write things down in a book. It is just a different way to cater for all the needs in the classroom and being able to get them to show me their knowledge… The focus on capturing mathematical thinking rather than writing skills during mathematics lessons provides opportunities for all students to demonstrate their mathematical reasoning skills and their understanding of mathematical concepts. There are multiple benefits to this approach. First, students are able to maintain their focus on the mathematics of the task. Students who may struggle with converting their thinking into written text that is comprehensible to others are able to demonstrate their understanding easily and are more likely to form positive attitudes towards mathematics. Finally, primary students are notoriously slow when asked to record their thoughts in writing. The affordances of still and video cameras and the mobility of devices such as iPads save precious classroom time. This is not to say that the development of writing skills isn’t important. The majority of formal external examinations are still conducted in pen and paper format, and students still need to be able to communicate their thinking in written form, so we recommend a balance of approaches that require the teacher to understand the purpose and intention of each task and the desired assessment goal of each lesson. Another benefit of using technology for assessment in Bec’s classroom is that it allows her to respond to students’ needs in a timely manner, an important element of student engagement as described in the FEM: “…I can go and look at those results and see okay, well this person didn’t get any questions right and they need help”. Bec also believes the use of technology to capture assessment evidence is a way to address the test anxiety that is often experienced in mathematics classrooms, resulting in negative attitudes and disengagement (Henschel & Roick, 2017). One of the apps that Bec uses regularly to capture her assessment data is Seesaw. Bec uses this tool as a way of managing student work and a way to communicate with parents, building stronger home–school connections, and providing parents with the opportunity to assist their children by increasing their awareness of strategies that are being taught in the classroom. Bec also finds Seesaw provides an advantage when she is writing school reports as she is

Primary mathematics classrooms 79 able to access the students’ work anywhere and anytime. Other managementrelated tools that Bec uses include Google Drive and Google Team Drive. These are often used to curate tasks that are accessible by the students within and beyond their mathematics lessons. One of Bec’s strategies when it comes to technology in the mathematics classroom is to use a range of resources to provide variety, in alignment with the FEM, for her students and to ensure she is tailoring the pedagogy to the content of the lesson and the context of the task. An app Bec uses regularly is Prodigy, a digital mathematics resource that embeds mathematics questions within the context of a game. Prodigy addresses content across the mathematics curriculum and collects diagnostic and formative assessment data that allow the teacher to tailor the level of difficulty to each individual student, providing the differentiation that Bec strives for in her lessons and providing opportunities for students to apply mathematics skills and build fluency. Another app that Bec uses is Explain Everything. An example of how she has used this app is when she required her students to create a mind-map of everything they knew about a specific number. This included the students adding pictures, recording audio, and other media to demonstrate their understanding. Bec spoke about how she often uses a trial and error strategy to experiment with various uses of digital technology in her classroom. This indicates that she is willing to take small risks in her practice, aligning with the adapting phase of technology integration described by Niess et al. (2009). This step occurs when teachers engage in activities that lead to a decision to either adopt or reject teaching with the specific digital technology application. When discussing her status as an early career teacher who would be considered to be a “digital native”, she recommends that teachers who are unsure of how to use technology try incorporating it in small steps, such as using apps like Prodigy. Although Bec uses technology in every lesson, she also uses a range of non-digital resources and provides important opportunities for students to work in a more traditional pen and paper format as well as the use of concrete manipulatives. She also stressed that she plans for the use of technology by considering the capabilities of her students, a critical element in the development of positive pedagogical relationships that is articulated in the FEM. Above all, she believes that the use of digital technologies keeps her students engaged and enjoying mathematics, making them more confident and developing positive attitudes towards the subject.

Lesson 1: triangle inequalities Technology: iPads, IWB, Kahoot, Explain Everything, Google Drive, Google Classroom Mathematical Content: Properties of Triangles Lesson Duration: 60 minutes This lesson began with a quick warm-up activity on Kahoot. Students responded to the questions using a device and their answers were collated in real time,

80  Primary mathematics classrooms allowing the teacher to assess the level of knowledge immediately. In this lesson, Kahoot was used to revise the properties of 2D shapes in preparation for the main task of the lesson. When students answered questions correctly in Kahoot, they earned points and, in this classroom, the competitive nature of the game appeared to promote students’ affective engagement and tune them in to the lesson content. Once the quiz was completed, Bec introduced the main task of the lesson that was projected onto the IWB via PowerPoint slides. The lesson was focused on an investigation using 20 matchsticks to create a range of triangles. Bec’s purpose in setting this task was to encourage her students to explore how the sum of the lengths of two sides of a triangle must be greater than the length of the third side (Appendix 2). Students were set to work in groups of two or three, using their iPads, matchsticks and small whiteboards. As detailed in the lesson plan, the students were to record their findings using photographs and audio recording on the Explain Everything app. Students worked at desks and on the floor, taking advantage of the mobility of their devices. They appeared to be engaged while they were working on the task and this was evidenced through their conversations, their body language, and the sustained period of time spent on the task (approximately 40 minutes). Bec called the students to the floor ten minutes prior to the end of the lesson. Bec had drawn a table on the whiteboard to allow the class to collate their data and discuss any observations or hypotheses that emerged. During this time, the students were given the opportunity to talk to their group members to identify and describe any “rules” they could see within the table. This continued with a whole-class discussion about the patterns observed. Although this part of the lesson did not involve the use of any technology, the opportunities for peer to peer and student/teacher discussions appeared to maintain the students’ engagement with the mathematics of the task. Opportunities for continuous interaction such as this are a critical element in establishing positive pedagogical relationships, as featured in the FEM. The lesson was concluded when students were directed to conduct an individual reflection by responding to the reflection prompt “what have you learned today?” that Bec had posted on Google Classroom. Using a digital platform for reflection rather than pen and paper allowed Bec to see her students’ responses immediately.

Lesson 2: computation Technology: iPads, Google Sheets, Google Drive, Kahoot, QR code reader, Woolworths app, Seesaw Mathematical Content: Addition, subtraction, multiplication, division Lesson Duration: 50 minutes This lesson began with a number talk focusing on multiplication mental computation strategies using a traditional whiteboard, with students sitting on the floor. Students shared their strategies as Bec recorded them. Bec explained to the students that the warm-up was to get students’ brains ready for multiplication,

Primary mathematics classrooms 81 which was a feature of this lesson along with addition, subtraction, and division. She then moved to the IWB to explain the main lesson activity, where students would be required to apply their knowledge of the four operations to the real-life scenario of converting and pricing recipes for blueberry muffins for the school canteen (Appendix 2). Bec designed three versions of the task to allow for three different ability levels. She projected the task details via PowerPoint and explained the steps the students were required to take to complete the task. The students were then directed to select a partner to work with from their mathematics group. Each set of instructions had been uploaded to the Google Team Drive to allow students access to the task. Bec introduced the task by asking students if they have ever had muffins from the school canteen. This strategy immediately provided a relevant context for students that increased the potential for high levels of student engagement with the task and resulted in the students reacting positively, with one student asking: “is this real?” Evidence of immediate engagement with the task was when they begin to ask questions relating to the mathematics, such as, “how much is in a cup?” and “how do we know how much it’s going to cost?” The students were directed to use a spreadsheet to record the prices of their ingredients and to create a Google Slideshow that incorporated screenshots of the tables created using their spreadsheets. An additional use of technology within this lesson was the use of a QR Code to link students directly to the Woolworths (grocery store) online site. This strategy allowed students to retain their focus on the mathematics of the task rather than wasting time finding the site or being distracted by having to navigate a search engine. Once the task explanation was complete, the students began to work in pairs, using their iPads. Rather than sitting at desks, many of the students sat on the floor. During this time, Bec moved around the room, providing support where required. The students continued to work, appearing engaged in the task. We observed evidence of operative engagement due to the high levels of interaction. Cognitive engagement was evident by the questions being asked, and affective engagement was apparent in the body language (students leaning in towards each other and their devices) and the sustained time on task. As the task was complex, the students did not finish it within one lesson and Bec continued the following day.

Case study 6: year 6 School context The school where Jessica was located was an independent, K-12 independent school in an urban, high socio-economic area. The school had 1,658 students enrolled in 2018. It is well resourced with devices, infrastructure, and support from the early years through to senior years. Students from Kindergarten to Grade 2 have access to banks of iPads. In Grades 3 and 4, the school provides an iPad for each child to use during school hours, and in Grades 4 and 6, the

82  Primary mathematics classrooms students are required to purchase a personal iPad that they are able to take to and from school. Students in the senior school participate in a BYOD program and are able to use any PC or Mac laptop that meets the minimum specifications required by the school including the ability access the Office 365 software suite. A full-time technology coach is employed to ensure that the digital technology curriculum is embedded in key learning areas as opposed to being viewed in isolation, as well as a dedicated Head Teacher of ICT and Innovation, who also looks after the logistical and technological aspects related to digital technology use in the school. The technology coach’s task is to build capacity amongst all of the teachers rather than perform as a specialist teacher. The coach works with the primary teachers as they are planning and often when they are teaching. We interviewed the coach (David), who spoke about the way he works with the teachers: …when we are doing curriculum planning I am there…we are suggesting what ways that they could incorporate both the digital technologies curriculum into their curriculum that they are designing and also the General Capability (the ICT capability). Part of our job is to have our heads around all of that and be able to see where that can slot into the curriculum…. (David, technology coach) The school encourages the use of ICT and the development of ICT skills throughout the school years in order to promote consistency. In the primary classrooms, the coach works with the teachers to integrate and embed digital technologies into all curriculum areas. In the early secondary years (Years 7 and 8), students receive 40 hours of lessons specifically devoted to digital technology education to address the requirements of the digital technology curriculum. The technology coach assists the secondary teachers when required. As it is the school’s policy to provide a consistent approach to the use of digital technologies, teachers are encouraged to focus on students’ skill development rather than the use of specific apps. If teachers request a particular app to be downloaded on his or her classroom devices, they are encouraged to reflect on the purpose, potential benefits, and potential distractions of introducing new software. David gave the following reason for minimising the number of apps used in the primary classrooms: …to lessen their (the students’) cognitive load it is important that they get some sort of consistent approach and that we are always teaching skills rather than software…throwing a different app at them every six months isn’t going to do that because that is not helping them understand the concepts behind the products… An additional reason for encouraging a limited number of software applications was the need to minimise the number of times students need to share personal information through the creation of accounts and a desire to ensure student activity remains within the safety of the school domain: “We don’t want every

Primary mathematics classrooms 83 kid getting accounts created all over the place and basically selling their information to the highest bidder” (David, technology coach).

Technology and mathematics at Jessica’s school When asked about the school philosophy regarding the use of digital technology in mathematics classrooms, David expressed that, in general, the school promotes the use of technology to build conceptual understanding rather than simply providing opportunities for drill and practice. The teachers use technology to enhance existing practices as well as enhance student engagement with mathematics, aligning with the lower two “enhancement” levels of the SAMR (substitution, augmentation, modification, and redefinition) framework (Puentedura, 2006) and the use of digital technology as “partner”, as described by Goos et al. (2000).

Professional learning and technology When discussing the professional learning opportunities provided to teachers at Jessica’s school, David expressed the belief that the provision of time to explore, discuss, demonstrate, and share ideas relating to technology use is critical. Rather than seeking external professional learning, the teachers at this school find it more effective to support each other through short internal sessions and the coaching that David is able to provide. David stressed that it is important for any professional learning relating to technology is contextualised due to the broad variations on infrastructure, number of devices, school policies, and student and teacher capabilities. These sentiments are reflected in literature on effective professional learning. The provision of in situ learning offers a contextually responsive approach, allowing the facilitator to contextualise to the teacher’s site of practice (Garet, Porter, Desimone, Birman, & Yoon, 2001). Extending this idea, Higgins and Parsons (2011) argue for “situated professional learning opportunities in the teachers’ classroom” enabling “facilitators to engage teachers in the PD core ideas and enact these in practice” (p. 55). David expressed a desire to see a bigger emphasis on coding and programming within the primary mathematics lessons at the school. It is his belief that the use of robotics has the potential to improve students’ understanding of algebraic, geometric, and measurement concepts through exploration and problem solving. Arguably, the depth and breadth of technology integration does depend on each individual teacher’s confidence, capability, and willingness to innovate. For some, this also involves a level of risk taking through adapting or renewing existing practices, which is exactly what was observed in Jessica’s classroom.

About Jessica At the time of data collection, Jessica had been teaching for five years in upper primary at the same school. During this time, she taught within the International Baccalaureate Primary Years Program (IB PYP) framework, which encourages

84  Primary mathematics classrooms lifelong learning, taking action, and problem solving through inquiry. Jessica’s passions lie in using real and meaningful open-ended tasks to challenge students thinking and help them make connections. Her enjoyment for teaching and learning mathematics came from a colleague who encouraged her to complete a Graduate Certificate of Primary Mathematics, which provided her with expertise in the leadership of mathematics teaching. During this experience, Jessica claims she learnt about the power of rich mathematical tasks to ignite curiosity, encourage problem finding and problem solving, and shape critical thinkers within a mathematics context. Since participating as a case study for this book, Jessica is now working as a Science, Technology, Engineering, Art, and Mathematics (STEAM) teacher at a different independent school. Jessica is an enthusiastic user of technology when it comes to the teaching of mathematics. In fact, digital technology is deeply embedded in almost every lesson in a variety of ways. Her philosophy regarding the use of technology in mathematics lessons focuses on making mathematics meaningful for her students: “I think that when the maths is meaningful and relevant that’s when connections are made and that sometimes it is not necessarily linear”. When Jessica plans her mathematics lessons, she has a long-term purpose in mind, and she is aware that often it is difficult to predict where her tasks may lead her students mathematically, “I can’t quite predict where my lessons are going. I think I am just content with…knowing the long-term purpose”. Jessica’s willingness to allow her students to explore mathematical content that arises because of the use of technology reflects one of the four components of mathematical TPACK, depth and breadth of mathematical content, as articulated by Guerrero (2010). Jessica’s Year 6 classroom is equipped with five desktop computers, and each of her students owns a personal iPad mini. At the front of her room is a digital LCD touchscreen, alongside two traditional whiteboards. The touchscreen is positioned to the right of the whiteboards, rather than in the centre. All of Jessica’s teaching and learning experiences are managed through the Microsoft OneNote application, which acts as a digital notebook for individuals or multi-user collaborations, accessible on any device and using a number of different operating systems. The OneNote app allows users to collect drawings, texts, videos, audio files, and photographs. Jessica takes advantage of the many affordances of OneNote. First, the mathematics tasks she creates are uploaded for students to access anywhere, anytime. Jessica’s students also use OneNote to curate their learning in a variety of formats from text, to photographs, to videos. As her students have access to devices at home and at school, Jessica also uses OneNote to share the students’ work with their parents, promoting strong home–school connections, stating “it just means that learning is accessible at home, parents can see what we are doing and stuff isn’t lost. OneNote is a resource that they can take with them and continue to grow with that.” The benefits of home–school connections are also appreciated by Jessica’s students, who talked about how they use OneNote in

Primary mathematics classrooms 85 conjunction with another app, Padlet, during our focus group discussion. One student made this comment: I think technology is a very good way to learn maths, because if we do it on a Padlet we can show our parents all the different ideas people in the class have and then that gives them ideas too. Another use of OneNote is to provide opportunities for Jessica’s students to engage in a flipped learning approach. Rather than using classroom time to view instructional videos, Jessica embeds videos into her OneNote files with the expectation that her students view the videos prior to attending class. Jessica explained why and how she uses a flipped learning approach: The flipped learning happens for home learning. I don’t really like sitting in class watching another teaching explain…that is not very exciting but at the same time for some kids that can be a helpful confidence boost before they are coming into a lesson. So, for example, something that I looked at was – I found a video for four different ways to multiply for home learning, put those on the OneNote, [students] reviewed them, [and were asked to] come to school the next day ready to share which one they made a connection with, why? Why not? And we go from there. The Padlet app mentioned above is another of Jessica’s favourite digital tools. Although not specifically designed for mathematics, Padlet is a productivity tool that Jessica uses extensively to collate student work, assess, evaluate, and provide feedback. One of Jessica’s more common uses of Padlet and OneNote is to capture assessment data. One of the most significant benefits of digital technology is that it allows real-time assessment that allows the teacher to react instantly to the needs of students, as was evidenced in Jessica’s observed lessons. Another benefit, according to Jessica, is that digital tools can be used alongside concrete manipulatives to document evidence of learning that was previously difficult to capture. The use of still photographs, videos, and screencasting has provided opportunities for teachers to move away from the traditional pen and paper assessments. Jessica also regularly uses a range of other technological devices including robotics. Her school had recently purchased Spheros, small spherical robots that can be programmed via Bluetooth. Jessica likes to use programming within her mathematics lessons and this was demonstrated within the observed lessons below. She articulated her reasons for using them in mathematics: I think that developing spatial reasoning is something particularly these students and the students that are coming through need to develop through technology because that is going to be the future jobs that they are going to be coding and being on computers.

86  Primary mathematics classrooms When talking about how she decides whether and what technology to use in mathematics, Jessica selects tools that allow her to gauge the understanding of her students. I think that we teach in a really fast paced way and that often getting through each individual book can be beneficial but time consuming so often just having all the thinking from that session there, I can comment and give feedback on that Padlet or I can just get a sense of what groups understood. Students don’t feel like they are being assessed; students feel like they are just sharing their learning. Jessica’s affinity with technology has meant that she is willing to spend time experimenting with the various devices and apps that were available to her. She also admits that other teachers at the school do not use the technologies in the same ways and there are varying capabilities amongst the teachers. However, the school does provide support and the teachers appear to be willing to share ideas and learn from each other.

Lesson 1: number patterns Technology: iPads, Padlet, OneNote, Wheeldecide.com, Solstice, Edmodo Mathematical Content: Prime numbers Lesson Duration: 70 minutes The Number Patterns Snap lesson began with the students entering the classroom for their first lesson of the day, iPads in hand, and instructed to go to Edmodo (a content sharing and collaboration platform for educators). Jessica had prepared a resource based on using web-based spinners (wheeldecide.com) that resulted in each student spinning a “condition” based on the results of their spin. To demonstrate the warm-up task, Jessica used Solstice, an application that allows screen sharing. This meant that students were able to see the task on the large LCD screen as well as on their individual iPad screens. Students were instructed to find other students who had something in common between the “conditions” of their spins. For example, if one student’s condition was “multiple of 7”, and another’s was “odd number”, they would record information regarding the commonalities. The students used small individual whiteboards to record their responses to the task. During this part of the lesson, the students were actively discussing the task and moving around the room to find others who had something in common. Although at this point of the lesson, technology was not required, the use of the digital spinner was an effective way of beginning the lesson in an engaging manner and from a logistical perspective, allowing the teacher to create a tailored resource with minimum effort. The alternative way to run this lesson without the use of technology would have required Jessica to physically create some way of allowing her students to randomly receive individual tasks. The use of technology saved time and created an engaging task. After allowing the students approximately ten minutes to explore commonalities, Jessica called them back to the front of the room for a quick reflection on the warm-up task. Levels of cognitive, affective, and operative engagement were

Primary mathematics classrooms 87 evident in the student responses to the reflection prompt “why would I get you to do that [task]?” Some of the responses were as follows: “To get us thinking, using our times tables, and multiplication and all the things we know”; “So that it will be easier for us to find patterns”; and “I get really excited when I find this multiple is a multiple of…”. These quotes indicate high levels of engagement with the mathematics of the task rather than the use of digital spinners, indicating that the use of technology in this classroom was a regular occurrence, was seamless, and was not a distraction for students. Jessica then introduced the main task of the lesson, saying “we’re going to be curious and we’re going to seek to find connections between numbers”. She explained that she had uploaded the Number Pattern Snap task (Appendix 3) on the class OneNote file. Although this task was similar to the introductory task, students were now required to come up with a conjecture based on their findings. Students were instructed to record their work on their small whiteboards, photograph their conjectures, and upload them to the Padlet link that had been set up for that specific lesson. The majority of students sat in pairs on the floor, spread across the room. Again, there were high levels of student to student and student to teacher discussions that were all focused on the task, indicating substantive engagement. During this time, student work samples began to appear in the Padlet on the large LCD screen. Jessica noticed immediately that some students had difficulty expressing their mathematical patterns and was able to provide immediate assistance and intervention, explaining: “What I was able to tell from what they were writing on their whiteboards was that they couldn’t actually write conjectures, they were just actually writing connections which is okay…so really [this] has captured what the next step is”. Throughout the lesson, students were deeply engaged in the task, with lots of focussed discussion. Jessica brought the students back together at the front of the room approximately seven minutes prior to the end of the lesson. During the final minutes, she led a discussion based on the things she had noticed through her discussions and through the work samples that were uploaded on the Padlet. She used a specific example on the Padlet and asked the authors of that sample to explain their working to the class. The use of the Padlet enhanced the discussion by allowing the students to reflect on each other’s work. It also allowed Jessica to evaluate the task design real time and allowed her to tailor the reflection to the students’ immediate needs. Further evidence of the high levels of engagement was observed when some students continued to work on their conjectures after the lesson was over.

Lesson 2: problem solving Mathematical Content: Location, Number Patterns Lesson Duration: 45 minutes Resources: Metre square hundred charts In this lesson, Jessica’s purpose was to combine the teaching of mathematics with digital technology skills. This lesson built on the previous lesson, where

88  Primary mathematics classrooms students investigated number patterns and conjectures. Prior to the lesson, the students had built metre square hundreds charts using paper, which were spread out on the floor of the classroom. Jessica began the lesson by introducing the task (Appendix 3). As detailed in the task instructions, students were required to form groups of four. Each group would then split into two opposing teams. In order to complete the task, each pair of students had to write three clues to lead to a number on the chart. They were also given the option of using the spinner from Lesson 1 to assist them. Jessica had set up a OneNote collaboration space for groups to record their work, which allowed her to monitor their progress as well as the timing of the lesson. You will notice as you read through this description that the lesson did not follow the task exactly as set out in the description (Appendix 3). The ability to be flexible and respond to student needs in the moment is an important element of effective teaching (Askew, 2012) and reflects the level of Jessica’s TPACK (Niess et al., 2009). Once they had two different conditions, the opposing students were to identify a number on the hundreds chart, place a counter on that number and program their Sphero to move to the number. The team would then score the number according to the scoring scale as detailed in the task description. While Jessica was explaining the task, she made a point of asking the students why they are doing the task, and what mathematics was involved. This strategy links to an important aspect of the FEM in terms of highlighting the relevance of the mathematics and providing transparency about the purpose of the task. When this is missing, students could arguably mistake the task for a game, without acknowledgement of the mathematics and the mathematical thinking involved. The complexity of the task soon became obvious to Jessica as the students began to work in their groups. Many of the students took much longer to write their clues as they were focussed on being competitive and wanted to design complicated clues to trick their opponents. Several times during the lesson, Jessica stopped the students to provide clarification and to give feedback in relation to the types of clues that were being written. Having the work samples appearing in real time on the OneNote collaboration space that was projected on the large LCD screen enhanced Jessica’s ability to understand what the students were up to and how she could assist them effectively. An additional challenge for the students was the programming of the Spheros as they had to measure the distance and directions required to move the Sphero to the correct space on the hundreds chart. Although challenging in a number of ways, the students appeared to be engaged throughout this task and this was evident through observation of their body language, with students leaning in towards each other, interacting with their iPads and engaging in animated discussions that were focused on the mathematics of the task. After 40 minutes, Jessica called the students together for a reflection, asking students to think about what was tricky or interesting about the mathematics they were doing. Students spoke about how they were still trying to make their clues too complicated, and how the Spheros were challenging to program in order for them to land on the correct space on the hundreds chart. The lesson ended prior to the students completing a written reflection due to

Primary mathematics classrooms 89 the need for them to attend chapel. When discussing this lesson, Jessica reflected that the overall complexity of the task may not have resulted in the highest quality of mathematics, saying: “I think there was a lot of maths going on in their brains which meant that the amount of coding that got done wasn’t a lot and the amount of maths that got done wasn’t necessarily the best quality.”

Lesson 3: rich task Technology: iPads, Spheros, Padlet, OneNote, Google Earth, Google Maps Mathematical Content: ratio, location, length and number Lesson Duration: 1 hour, 45 minutes Task documentation provided to the students on OneNote and via the large LCD screen in the classroom. This lesson could be described as the beginning of an extended mathematical inquiry. The inquiry was embedded in a context that Jessica designed with the interests of her students in mind. It also fit with the students’ current unit of inquiry, which was focused on causation. The overarching question driving this task was “What is the cause and effect of designing and driving driverless cars?” As described in the task description (Appendix 3), the students were given a hypothetical task to program a driverless car (Sphero) that could travel to local destinations using a map. The task required detailed discussion to ensure the students understood the range of mathematical concepts they would need. For example, the students needed to use their knowledge of scale, location, position, length, and number. They also needed to use computational thinking when coding to ensure the Spheros were able to navigate the chosen routes. To introduce the task, Jessica announced that the students would be applying for a job at Tesla to become software engineers, creating a prototype. Immediately, the students appeared excited, and it was obvious the task context was well matched to the students’ interests, an important element in developing positive pedagogical relationships as described in the FEM. This led to a discussion of the definition of prototype and a Google search of the definition. This seamless use of technology provided “just in time” access to the knowledge required for students to complete the task. Of course, this action could have been executed without technology, but would have required someone to physically find a dictionary, look up the word, and read it out loud to the class. In this scenario, Jessica had her laptop and literally took seconds to project the definition onto the screen, resulting in the students maintaining their focus and an uninterrupted flow to the lesson. This was a clear example of the role of technology as extension of self (Goos et al., 2000). Although this element of technology use is not in itself highly creative, it does demonstrate Jessica using technology as a natural part of her practice. The remainder of the lesson introduction included a detailed discussion of the mathematics that would need to be applied during the course of the task. There were many questions regarding scale and the need to consider the size of the Sphero in relation to the size of the map to be drawn. When asked if they had used Google Maps when travelling with their parents, all of the students indicated they had, with

90  Primary mathematics classrooms one girl stating that they could use a GPS. These students had a strong affinity with technology both within school and at home, possibly due to the high socio-economic status of their families. This same task in a different school of different socio-economic status may have resulted in very different discussions and questions and possibly would have been more challenging because of the technology involved. This is another indication of positive pedagogical relationships where Jessica understood the backgrounds and abilities of her students and set the challenges within the task accordingly. The students appeared to have high cognitive engagement at this stage, with many questions relating directly to the mathematics of the task. The body language of the students, who were sitting on the floor in a circle and leaning inwards, was another indication of high student engagement. Students were instructed to draw their routes to somewhere within the local vicinity, ensuring the task had relevance to students’ lives within and outside the school. Following a detailed discussion on scale, conversion of lengths from kilometres to centimetres and speed (relating to the local speed limits), the students began to work using their iPads to access Google Maps to assist in working out scale and distance. The students worked enthusiastically in their groups. Some were working on iPads, while others were recording their thinking on small, individual whiteboards. After a period of approximately ten minutes, Jessica called the students to the floor to regroup. Through her observations, Jessica noticed that the students had become distracted when selecting the destinations of their driverless cars. Each of the groups was given an opportunity to report on their progress and on their chosen destinations and the distance that needed to be travelled. Students went back to their groups and continued working, with lots of task-focussed discussion happening. One hour after the lesson had begun, most students appeared to be working well and on task; however, a small number appeared to have become restless. Jessica then called the students back to the floor to regroup again, saying: “I’ve called you in because I think we’re getting a bit overwhelmed and I’m going to try and bring it back a bit. We’ve figured out our maps, what was challenging about this?” This question was followed with a brief discussion about some of the challenges the students were experiencing. One student commented that a tricky part of creating a map was ensuring that the measurements were consistent so that when the Sphero was coded, it would be able to travel the planned route. In other words, the students would need to program according to the relationship between time and distance. Jessica commented to the class that this student had made a connection to the “big purpose” of the task, and it was evident that the student was also making connections within and amongst mathematical concepts. Students were then sent back to work in their groups and continue working on the task for a further 15 minutes. They are then invited to post an individual reflection to the task Padlet (embedded within Edmodo). The reflection requirement was for students to list: “three mathematics connections, two questions you have for your group, and one next step”. The lesson ended with a short discussion of “what did you notice?” This discussion was focused on the importance of accuracy. This task was continued in further lessons that were not observed.

Primary mathematics classrooms 91

Summary The two school settings examined in this chapter vary in context with one being a government school and the other a very well-resourced independent school. The additional resources allowed the independent school to employ a technology coach and a head teacher dedicated to technology and innovation. The extra support staff were invaluable in assisting teachers to address any technical concerns and in supporting staff by linking innovative practices with the curriculum. In the government school, teachers were expected to be far more independent when developing and implementing technology-based lessons and units of work. The independent school took a whole-school approach to technology integration, ensuring a one-to-one iPad ratio and providing professional development for teachers, largely in situ, allowing for a highly contextualised approach. The teachers were expected to consistently reflect on the proposed purpose when thinking about using a new technological tool or app. Teachers in this school were actively encouraged to limit the number of apps used during teaching, only adding new ones when there was a clear pedagogical purpose for doing so. In contrast, the government school had a different approach for the early years (K-2) and the primary years (3–6). All students in Years 3 to 6 were required to have their own iPad, which the school facilitated through an Apple purchase plan. Students in the lower years had a small number of iPads to share in the classroom, but the teachers of these years were perceived as being more sceptical about the value of technology for learning. Rather than taking a whole-school approach, the technology divide in this school between older and younger students was quite embedded and unlikely to change with current teaching staff. When the iPad plan was introduced for Years 3 to 6, the school faced considerable backlash from parents, concerned about how the technology might change the teaching and learning practices at the school. The school then increased communication with parents to ensure that support for the technology was present at home as well as at school. Interestingly, such concerns were not raised at the independent school where even very young learners were expected to have their own devices. The teachers at both schools generally saw the benefit of using technology in the mathematics classroom as a way to shift the focus from learning content to developing conceptual understanding and mathematical reasoning. They saw increased opportunities for students to explore mathematics content and to communicate their mathematical understanding in a variety of modes using digital cameras, audio and video recording, and screen capture. In both schools, Google Sheets was used for learning about data, and basic programming using Beebots and/or Spheros was also used to enhance spatial reasoning. Both schools used a learning management system (One Note, SeeSaw) as a means of tracking student progress and to share student work with parents. Kahoot was also employed in both schools to check on student progress both from the teachers’ perspectives and as a means for students to gain immediate feedback on their understanding. In all observed lessons, we saw high level of student engagement. The technology was being used seamlessly with few technical difficulties. This level of familiarity

92  Primary mathematics classrooms with the technology allowed students to focus on the mathematical task under consideration rather than being distracted by the tools involved. It was clear that the students had moved well beyond experiencing the novelty value that technology might have when first introduced and were simply using it as a learning and communication tool, much like they would use a pen and paper in the classroom. The technology sat comfortably alongside more traditional hands-on manipulatives in all classrooms, adding further affordances to enhance mathematics understanding.

References Albright, J., Clement, J., & Holmes, J. (2012). School change and the challenge of presentism. Leading & Managing, 18(1), 78–90. Askew, M. (2012). Transforming primary mathematics. London, England: Routledge. Attard, C. (2014). “I don’t like it, I don’t love it, but I do it and I don’t mind”: Introducing a Framework for Engagement with Mathematics. Curriculum Perspectives, 34, 1–14. Attard, C. (2015). Introducing iPads into primary mathematics classrooms: Teachers’ experiences and pedagogies. In C. Attard (Ed.), Integrating touch-enabled and mobile devices into contemporary mathematics education (pp. 193–213). doi:10.4018/978-1-46668714-1.ch009 van Deursen, A. J. A. M., & van Dijk, J. A. G. M. (2014). The digital divide shifts to differences in usage. New Media & Society, 16(3), 507–526. doi:10.1177/1461444813487959 Garet, M., Porter, A., Desimone, L., Birman, B., & Yoon, K. (2001). What makes professional development effective? Results from a national sample of teachers. American Educational Research Journal, 38(4), 915–945. Goos, M., Galbraith, P., Renshaw, P., & Geiger, V. (2000). Reshaping teacher and student roles in technology-enriched classrooms. Mathematics Education Research Journal, 12(3), 303–320. doi:10.1007/BF03217091 Guerrero, S. (2010). Technological pedagogical content knowledge in the mathematics classroom. Journal of Digital Learning in Teacher Education, 26, 132–139. Henschel, S., & Roick, T. (2017). Relationships of mathematics performance, control and value beliefs with cognitive and affective math anxiety. Learning and Individual Differences, 55, 97–107. doi:10.1016/j.lindif.2017.03.009 Higgins, J., & Parsons, R. (2011). Professional learning opportunities in the classroom: Implications for scaling up system-level professional development in mathematics. Mathematics Teacher Education and Development, 13(1), 54–76. Koehler, M. J., & Mishra, P. (2009). What is technological pedagogical content knowledge? Contemporary Issues in Technology and Teacher Education, 9, 60–70. Maher, D., & Twining, P. (2017). Bring your own device – A snapshot of two Australian primary schools. Educational Research, 59(1), 73–88. doi:10.1080/00131881.2016. 1239509 Niess, M. L., Ronau, R. N., Shafer, K. G., Driskell, S. O., Harper, S. R., Johnston, C., & Kersaint, G. (2009). Mathematics teacher TPACK standards and development model. Contemporary Issues in Technology and Teacher Education, 9, 4–24. Orlando, J., & Attard, C. (2016). Digital natives come of age: The reality of today’s early career teachers using mobile devices to teach mathematics. Mathematics Education Research Journal, 28, 107–121. Prensky, M. (2001). Digital natives, digital immigrants. On the Horizon, 9, 1–6. Puentedura, R. (2006) SAMR: A brief introduction. Retrieved from http://www. hippasus.com/rrpweblog/archives/2013/10/02/SAMR_ABriefIntroduction.pdf

6

Technology in the secondary classroom

Many students struggle with the transition to secondary school as it can be a time when they experience a loss of confidence, motivation, and engagement (Attard, 2011). In mathematics, the impact can be felt in the first year of secondary school (Martin, Way, Bobis, & Anderson, 2015) when students experience less personal support for their learning and the subject matter increases in difficulty and is more formally structured (Tytler, Williams, Hobbs, & Anderson, 2019). Given the importance of maintaining student interest in mathematics, as it is the foundation for many STEM disciplines, technology may provide a means of personalising and differentiating instruction to mediate the negative impacts of the primary-secondary transition. In this chapter, we provide four case studies featuring a range of secondary mathematics teachers who have been recognised by their colleagues as effective users of technology. The teachers, John D, Ian, John B, and Joel, come from a range of school contexts, using technology for a variety of purposes and in diverse ways. Given the nature of secondary schooling, we observed each teacher conduct lessons with a range of different grade groups.

Case study 7: years 9–12 School context John’s school is a metropolitan secondary school for boys in the Catholic system with an enrolment of 1,021 students in 2018. The school prides itself on outstanding academic outcomes and as a leading school for the integration of technology and innovative technological approaches. The school employs project-based learning from Years 7 to 10 and problem-based learning in Years 11 and 12, drawing on a flipped approach to classroom instruction. Taking a forward-focussed approach, the school endeavours to prepare its students for an uncertain future, with transferable skills that will serve them well in a broad range of circumstances. The Head Teacher of Mathematics, Trent, explains that the school has taken a progressive attitude to technology for a long time, embracing mobile devices, phones, and iPads, rather than seeing them as a distraction from learning. Now, every student has an iPad from Years 7 to 12, but they are currently moving

94  Technology in the secondary classroom towards laptops for greater functionality. Trent sees value in technology as a means to provoke mathematical discussions between students and teachers, stating: “we want them to discuss it and be able to explain it, and, as Albert Einstein says, if you cannot teach it you don’t truly understand it. I guess that’s a mantra that we use throughout our mathematics department”. He attributes the school’s success to innovative approaches to pedagogy and technology, explaining that: …we have had exceptional results in mathematics and I believe it is through the engaging use of IT and also problem based/project based learning and flipped learning, which has led our kids to thoroughly enjoying mathematics and actually wanting to learn more. In terms of apps for either laptops or iPads, Trent explained that the school tries to choose products that are free, to reduce the cost to families. In particular, they frequently use Youtube, Desmos, and Geogebra for this purpose. Given their focus on flipped learning, the school is also considering constructing a video room, to enhance the look and functionality of the videos that the mathematics staff produce. Trent sees the flipped approach to learning as a means to accelerate student learning and believes this has contributed to superior results in comparison to similar schools, aligning with the findings of Clark (2015). He also heralds the high levels of technology use with improving student engagement in mathematics. He states that: …kids are engaged once you put the TV on, kids are engaged by iPads, kids are engaged by their laptop, their video games, phones. This IT will engage kids in mathematics, in something that can be quite dry and boring if it’s taught in the traditional sense.

About John D John has been teaching for 12 years and is recognised as a leader in the use of technology for teaching mathematics. When planning a lesson involving technology, John believes that “technology shouldn’t be the focus of the lesson. The focus should always be the content, that is, the mathematics. The technology should always be there to support it.” John has completed university studies in technology and has taught software as well as mathematics. He admits that he always thinks about how to use technology when he has disengaged students, aligning with student-centred technology element within the Framework for Engagement with Mathematics (FEM) (Attard, 2014). He uses technology for demonstrating mathematics concepts and finds it to be a quicker option than drawing on a whiteboard at the front of the classroom. Beyond using the technology for demonstration purposes, he sees it as a powerful tool for students to explore mathematics. He believes that “when students can actually interact with mathematics, where they can do something, and something comes back to them, then there’s a lot more meaning”.

Technology in the secondary classroom 95 In terms of mathematics content, he sees that technology is particularly beneficial for “anything involving graphs” and “probability and statistics”. John is able to recognise both affordances and the constraints of specific technologies, recognising that some areas of mathematics are not necessarily enhanced by technology use, for example, mental arithmetic and algebraic manipulation. Student engagement is the key reason for his technology use as he contends that “the modern student tends to engage with technology” and that if they are engaged then they are “more open to learning things”.

Lesson 1: financial mathematics Year level: Year 9 Advanced Technology: iPads, laptops, Desmos, Youtube, scientific calculators, Wootube, Khan Academy Mathematical Content: Commission Lesson Duration: 90 minutes John’s classroom is spacious and visible to adjacent classrooms via glass walls. The 29 students (boys) sat on mobile chairs with individual desks attached to their chairs. The lesson started when John directed the students to go to the Echo learning management system (LMS) to view mathematics content from Desmos and YouTube focused on the topic of commission in financial mathematics. He directed the boys to interact with the graphs through their touchscreens enabling them to examine the coordinates behind the graphs more closely. The students worked their way through a structured series of tasks and short mathematics exercises via the LMS. Although the students were encouraged to work in groups to discuss the mathematics content, many students initially appeared to be focused on getting the right answers and were checking with their peers to see if that was the case. While this appeared to replicate students working from a traditional textbook and checking answers from the back of the book, the affordances of the technology did allow John to use the Desmos platform to monitor students’ progress. He was then able to intervene with particular students, or with the whole class, when he noticed misconceptions as they arose. This is a variation on what we observed in Case Study 6 (Chapter 5), where Jessica used Padlet to provide “just in time” intervention. Interestingly, the boys were using their handheld calculators alongside their laptops, perhaps in preparation for their traditional pen and paper test at the end of the topic. This highlights one of the tensions that continues to exist between traditional pen and paper assessment practices that are often external and beyond the influence of teachers, and progressive teaching and assessment approaches made possible by the affordances of contemporary technologies. As the lesson progressed and the students worked through the content, John continually used the LMS to monitor student progress and at times called on specific students to re-examine their work. He then called for the whole class to work on the same problem, demonstrating its solution on the whiteboard. The lesson had now been proceeding for approximately one hour, and a small

96  Technology in the secondary classroom number of students appeared to be off task. This indicated that the structure of this lesson required students to be self-regulating, and the lesson duration may have presented a challenge for some students; however, John was able to monitor the situation in real time via the LMS.

Lesson 2: complex numbers Year level: Year 12 Advanced Technology: Canvas, Complexly, Projector, and Screen Mathematical Content: Complex numbers Lesson Duration: 90 minutes This lesson was attended by six students undertaking the highest level of school mathematics in Year 12. The lesson started with John asking the students if they had watched the flipped learning videos as preparation for the lesson. In this case, the videos were hosted on Canvas, a different LMS, which allowed John to track the number of video views and allowed students to annotate the videos with questions. This type of technology-infused practice aligns with the redefinition level of the SAMR (substitution, augmentation, modification and redefinition) framework (Puentedura, 2006). That is, this affordance would have been inconceivable without the use of technology. Most students replied that they hadn’t watched the videos, highlighting a potential issue for flipped learning in general. Student disengagement with the out-of-class component is a recognised issue within a flipped learning approach (Lo & Hew, 2017). John then proceeded to demonstrate a teacher-made tool Complexly (complexly.mrdrake.com) to demonstrate different representations of complex numbers. The lesson was focused on revision for an upcoming topic test, and after watching the demonstration, students worked individually on textbook exercises to consolidate their knowledge. At this level of mathematics, many different topics come together, requiring students to make links between them when they may not be easily apparent. John referred to Pythagoras’ theorem to link to the modular form of complex numbers and then to the unit circle, bringing together mathematical strands which may have seemed disconnected until this point. The Complexly tool enabled quick transitions between the various representations of complex numbers that would not be possible without technology. Again, this is an example of the redefinition level of SAMR (Puentedura, 2006). The tool can also be controlled via the Scratch programming platform allowing for easy customisation for particular mathematical problems.

Case study 8: years 7–10 School context Ian’s school is a K-12 public school in a regional city that delivers a comprehensive curriculum with a performing arts specialty. The school catered for an enrolment of 1,155 students in 2018. Students are selected for the school via

Technology in the secondary classroom 97 auditions, and as a result, the student body is drawn from a broad geographical area. The school encourages and nurtures students’ creativity and accommodates their interests and capacities in traditionally non-academic pursuits. In terms of technology, the school expects students to work on their own devices. We interviewed the head teacher of mathematics (Carly) who explained that: …our school is a BYOD school with a very close to one-to-one ratio…our students have been bringing devices for the last three years but it has gradually built up to that one-to-one ratio, it wasn’t one-to-one at the drop of a hat. The school takes a broad view of the types of devices allowed with mobile phones permitted in class for accessing or photographing learning materials. This view is at odds with recent moves to ban or minimise the use of phones in this school jurisdiction. The mathematics department standardises their approach to technology through the consistent use of OneNote and Canvas as the main delivery vehicles for disseminating content and lessons. Digital textbooks are used via this platform in conjunction with teacher-produced presentations, notes, and exercises. The teachers collaborate closely and share the work of producing the digital resources for each year level. Carly described the process for each year level as follows: …we have a course coordinator who writes a base OneNote which is the nuts and bolts of what any teacher is going to deliver, put into some sort of logical order or sequence…it must start with the dot points from the syllabus that are being addressed in each lesson….from that base OneNote all the teachers that are teaching the course can simply pick up the lesson material and drop it into their class OneNote [with] whatever else they would like to incorporate into their classroom. This process ensures some degree of consistency in delivery across all classes. This consistency is also supported through all teachers using a Surface Pro device or another device with the Windows 10 platform. Also, given that the school is a performing arts school, many of the students regularly miss classes due to rehearsals, auditions, and performances. The OneNote base files accommodate their absences by providing a consistent, comprehensive bank of resources for students to access when they have missed a class. In this way, the school is using a technology to facilitate the broader mission and ethos of the school without compromising academic expectations. Beyond the mathematics department there is a school technology committee with representation from each faculty. All major decisions about technology infrastructure and support are made via this committee, and expertise within each faculty is shared across the school through connections facilitated by the committee. Full staff development days are organised around particular skills

98  Technology in the secondary classroom when the need emerges, generally drawing on in-house expertise rather than external consultants. The benefits of an in situ approach to professional development are well documented (Garet, Porter, Desimone, Birman, & Yoon, 2001; Higgins & Parsons, 2011). As Carly clarifies, if the suitable expertise is not found within the school, they tend to look to other local schools for additional support: We do most of our stuff in-house…. I can’t think of the last time we sent someone off to something…our network here is very strong. If they don’t know how to do it there is another school around here that is doing it. This underlying collaborative approach is quite deliberate, and Carly sees staff collaboration as vital to success: “In the same way that I want children to collaborate I want my staff collaborating”. Although the technology use in the school is ubiquitous and operates in a seemingly efficient manner, Carly recognises the limitations it can impose for the teaching and learning of mathematics. Ideally, she would like all students to have a touchscreen device with a digital pen so that mathematics can be handwritten into devices. Unlike other subjects, the symbolic and diagrammatic nature of mathematics notation is problematic when students are entering work into digital formats. As a result, the school still requires all students to have exercise books and to handwrite mathematics solutions. In addition, most classrooms have whiteboards on all walls, and students are encouraged to work on the whiteboards during class, either individually or in groups, using photographs to keep a copy of their work. In these ways, the school is acknowledging the constraints inherent with most students’ devices by providing alternative non-technological means for them to communicate mathematically.

About Ian When we visited Ian, he was in his third year of teaching secondary mathematics. Prior to training as a teacher, Ian ran his own business for 20 years focussing on IT and web design, and he holds a PhD in physics. Alongside these pursuits, Ian had always coached junior sporting teams and enjoyed this engagement with young people. Looking for a change in focus, he decided to retrain as a teacher completing a Master of Teaching at his local regional university. Ian teaches all levels of mathematics from Years 7 to 12 and aims to help all students engage with the subject, particularly during the formative lower secondary years when many students decide that mathematics is not for them (Attard, 2011). Ian is an avid technology user in the classroom, mainly as a means to differentiate his teaching so that students at diverse levels are accommodated. He noticed that many students were disengaged because they were finding the work to be either too challenging or not challenging at all. The work he was expecting students to complete was really only addressing the learning needs of a small number of students. Ian decided to use the technological tools at his disposal to

Technology in the secondary classroom 99 tailor his students’ experience so that they were better matched to the diversity of learning needs. This approach, which empowers students to work at an appropriate level of difficulty, led to marked changes in student attitudes to mathematics as Ian explains: …they come out of class with a sense of achievement and accomplishment because they have actually achieved something as opposed to getting halfway through class, getting one or two [problems] right and then being thrown into the harder problems, getting confused and ultimately frustrated. Ian has noticed a significant increase in student engagement levels and willingness to work consistently in mathematics. Ian arranges his OneNote file in three levels: Foundation, Mastery, and Challenge. Students are able to choose the level that they attempt in the first instance, moving between levels in accordance with their success, or otherwise, on the set work. In his interview, Ian reinforced the importance of student agency in determining the level of work that they were attempting: So, the technology allows students to seamlessly move between [levels] and also, surreptitiously move between them. So, they are not announcing to the class they are on one level or another, they can just work on what they are comfortable with. Ian sees this aspect of the approach as being particularly beneficial to the females in his class, as they generally like to complete their work in a less conspicuous way compared to the males. Ian sees great value in using technology for visual demonstrations in mathematics, particularly through the use of vibrant digital representations of mathematical phenomena. Easy access to dynamic visualisations can provide insights into mathematics, which may otherwise appear dry and incomprehensible. Ian decides on the educational value of each tool or application on an individual basis trying to avoid the use of technology for technology’s sake: “So, I am always on the lookout for new technology and I will have a look at it and say …. is there just a whiz bang factor to this or can I see an educational benefit?”

Lesson: fractions Year level: Year 7 Technology: iPads, OneNote, Canvas, Geogebra, IWB, Excel Mathematical Content: Simplification of Fractions Lesson Duration: 50 minutes The lesson was conducted in a regular classroom with students seated at desks placed in rows. A small number of students stood at standing desks arranged towards the back of the classroom. A whiteboard at the front of the room acted as a screen for a data projector, linked to the teacher’s iPad, while whiteboards

100  Technology in the secondary classroom on two of the other walls were available to students to write on throughout the lesson. Twenty-three students entered the classroom in an orderly fashion, quickly taking their seats. The lesson started with the projection of an image created in Geogebra (a freely available dynamic geometry application) representing two partially shaded circles. The students were asked “Which is bigger? 3/8 or 2/5?” They offered speculative responses before the teacher changed the number of divisions in each circle using a slider created in Geogebra. By doing so, he was able to quickly and dynamically change the denominator of each fraction to 40, allowing for a direct comparison of the numerators, in order to determine the relative size of each fraction. This visual demonstration reinforced the ways equivalent fractions can be visualised with varying numerators and denominators without changing their size. The teacher then moved to the standard fraction notation for 12/18, asking students how the fraction might be reduced to its simplest form. Students were asked to determine all factors of the numerator and denominator and to look for the highest common factor, which was subsequently used to reduce the fraction to its simplest form. After a brief whole-class discussion, the students were asked to work, either in groups or independently, on similar questions of varying levels of difficulty. The students were able to choose from three sets of questions: foundation, mastery, or challenge. However, they were only permitted to move on to the challenge questions after consultation and approval from the teacher. One of the challenge questions required students to use Excel. The students chose where to conduct their work with roughly half of the class choosing to work on the static whiteboards around the room. Some students read the questions from the data projector display while others accessed the questions via OneNote on their own devices. Most students were engaged for the entire lesson and seemed intent on working diligently through the question sets while the teacher circulated, interacting with students where required, demonstrating high levels of cognitive engagement. In an interview following the lesson, Ian explained that the lesson had gone to plan. He was pleased with the students’ level of engagement and had noted that four groups of students were working on the Excel challenge questions by the end of the lesson. He found that most of the students were working at the mastery level, while only two students remained working on the foundation questions. Ian was confident that most students would now be ready to progress to the next topic: I was quite happy with how that went and I think that will set up well then for addition and subtraction [of fractions] because they are getting quite fluent in getting a common denominator, so I feel that moving to addition and subtraction now will be fairly straightforward. The lesson was a success in achieving the desired mathematical objectives. The technology use, while effective in achieving the desired levels of differentiation,

Technology in the secondary classroom 101 was also discreet and non-distracting from the mathematics under consideration. The technology allowed students to access mathematical work at the desired level of challenge and facilitated the use of a dynamic visual representation of fractions. This interactive visual provided a tangible link between students’ understanding of fractions, numerators, and denominators and the abstract manipulation required to conduct mathematical operations on fractions such as addition and subtraction. Fractions concepts are inherently challenging to learn (Gabriel et al., 2013), and Ian’s use of technology supported students in building their understanding through dynamic representations and differentiated tasks.

Case Study 9: 9–12 School context John’s school is a Catholic secondary school in an outer-regional setting. The school had 561 students enrolled in 2018. The school is committed to pursuing excellence in student achievement alongside fostering a sense of community and understanding of social responsibility. The school is currently transitioning to a BYOD (bring your own device) school, but as the head teacher of mathematics (Rowena) notes, there are significant barriers that they are working through. Inadequate and unreliable Internet access is an ongoing issue, which is often addressed with trusted, older students by allowing them to use their mobile phones in class. The other significant issue is that not all students have their own devices, and although the school has laptop trolleys and computer labs that can be booked by teachers, there are not enough to ensure easy access at all times. The pressure on family budgets resulting from mandated BYOD programs is a common challenge for schools to address (Maher & Twining, 2017). In her interview, Rowena reported that all teachers use data projectors in class, some in conjunction with interactive whiteboards (IWBs). She sees particular benefit in using visuals in mathematics, taking advantage of colour as a means of highlighting various mathematical features. Major decisions about technology infrastructure are made by the school’s executive committee and professional development is often carried out in-house by a small group of technology leaders who meet every three weeks. Up until the time of our visit to the school the technology leaders had primarily focussed on issues relating to computer hardware and technical specifications. They were yet to put a whole-school focus on pedagogical changes that might accompany the move to BYOD.

About John B John’s interest in becoming a teacher was seeded as a school student in the United Kingdom. He often helped his peers with mathematics in the senior years of schooling, and when one of them commented that he would make a good teacher, he began to take the notion seriously. Following a gap year after

102  Technology in the secondary classroom finishing school, John completed a four-year teaching degree, with mathematics as a major and physical education (PE) as a second teaching area. Interestingly, he claims that his experience as a PE teacher taught him a lot about physical presence when teaching and how to relate to students. He also credits his own struggle with mathematics as being crucial to informing how he relates to students: “So, I think that my empathy with students has always been much stronger because I didn’t find maths naturally easy, I had to work at it”. Since starting his teaching career, John has held various positions in the United Kingdom, Asia, and in various locations in Australia. His planning approach when using technology in the classroom starts with a consideration of the mathematics concepts that are central to the lesson. The key question that he asks is: “How can [technology] be used to strengthen a student’s understanding of something?” He emphasises that it is important for teachers to have time to experience and play with the technology so that they are aware of all of the appropriate affordances at their disposal. He sees two main advantages of technology used in mathematics: first, the tedious calculations can be avoided to some extent, so that more time can be spent on the rich mathematical ideas, and second, the opportunities for visual representations of mathematics that can make the subject come alive for students. John’s use of technology can be regarded as “app-enabled” as he uses technology to replace mundane tasks and replaces them with tasks that require higher-order thinking (Gardner & Davis, 2013). John sees a myriad of opportunities for mathematics to be more dynamic through the use of technology: “I think you are not getting the most out of your subject if you are not using technology. I think your subject is becoming very dry, bland and boring”. He believes that teacher reluctance is possibly the biggest barrier with technology use in schools with many teachers reluctant to use a tool if they are not totally confident with it: I think more traditional teachers struggle if they don’t [appear to] know everything and therefore I think that is the biggest thing with technology – they are not willing to go, you know, that it is OK if I don’t know everything. In the recent years, John has started to introduce programming to his students. He notes differences in students’ willingness to persist with difficult problems when programming in comparison to their readiness to give up more easily when attempting a textbook mathematics problem. He finds that his students are less likely to ask for his help when confronted with a difficult programming challenge, so he considers that it could be advantageous to teach some mathematics concepts though a programming lens, to capitalise on students’ interest, resilience, and engagement. There is a significant history linking computer programming and mathematics, beginning with Papert (1980), and there is evidence that multiple mathematics topics can be taught through programming (Hickmott, Prieto-Rodriguez, & Holmes, 2018).

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Lesson 1: De Moivre’s theorem Year level: Year 12 Technology: Graphics calculators, IWB, YouTube Mathematical Content: Complex numbers, De Moivre’s theorem Lesson Duration: 50 minutes This lesson was conducted in a standard classroom with rows of tables and chairs facing the front of the room where there were two traditional whiteboards on either side of an IWB. All students possessed graphics calculators, which they were encouraged to use regularly. Only six students were present on the day of the observation, and the lesson began with John making connections to link previous learning with this lesson and checking student responses to assigned questions. John encouraged discussion in the classroom, and students were confident to explain their thinking, even when they produced the incorrect response. Over the previous weekend, John had sent a screencast video to the students’ mobile phones, and he began the lesson proper by playing the video and pausing at opportune times to check student understanding. Following the video, students were directed by John to turn to “page 10 squared” of their textbook to the section titled “De Moivre’s theorem”. John then used the traditional whiteboard to lead the students from the specific to the general case represented by De Moivre’s theorem. The students were then directed to work on textbook exercises using this theorem with a reminder to access YouTube for support if needed. John then worked through the solutions to the exercises on the traditional whiteboard encouraging students to be able to justify their approaches to each problem and to use technology wherever possible to check the answers. The IWB was then used to display the screen of John’s graphic calculator, to model a successful approach, and also to check the answers. John then used YouTube to access a video on the “nth root method”, however, rather than play the video in full, John regularly stopped the video to ask students questions, thereby checking on their understanding. While it should be noted that the six students in this class were advanced Year 12 students, the lesson itself was relatively low-tech and traditional in many respects; however, the students were all very engaged and motivated to truly understand the content, rather than just being able to get the right answers.

Lesson 2: measurement (volume) Year level: Year 10 Technology: Graphics calculators, IWB, YouTube Mathematical Content: Volume of 3D solids Lesson Duration: 50 minutes The lesson began with students being provided with a handful of coloured cubes, with the instruction that they could use them for assistance if needed.

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Figure 6.1  Surface Area and Volume Problem.

The students (Figure 6.1) were then asked to consider the following problem displayed on the IWB. The students were encouraged to use the cubes to examine the problem. John then drew different cross-sectional representations on the whiteboards and demonstrated alternative methods of approaching the problem to assist the students in their visual reasoning. After this lesson starter, John asked the students: “What is a prism?” and proceeded to illustrate student suggestions on the board, leading them to consider the general rule for finding the volume of a prism (the area of the cross section multiplied by the length). John then used this prompt to talk to students about how all area formulae for two-dimensional shapes are related, showing a video linking the area of a circle to the area of a rectangle. John again paused the video at crucial points to allow the students to conduct their own calculations before proceeding to the solution revealed in the video. A student then asked about the origins of the formula for the circumference of a circle, indicating a high level of cognitive engagement. John praised the student for asking the question and demonstrated the relationship between the diameter of a circle, pi, and the circumference using one student’s water bottle. This exchange provided evidence of several elements of the FEM, including continuous interaction and explicit linking between abstract mathematical content and real-life examples (Attard, 2014). Continuing to emphasise the relationships between various 3D shapes, John then asked the students how cones and pyramids relate to prisms in terms of their volume. Students estimated that the volume of a cone or pyramid with the same base area would be one-half of the related prism. John then played a video which demonstrated that the volume of a cone is one-third of the volume of the related prism. This judicious use of video provided students with engaging ways of accessing mathematics concepts that are difficult to visualise and require developed spatial skills. John’s use of multiple representations to reinforce the connections between 2D shapes and 3D solids, moving seamlessly between drawings on whiteboards, textbook exercises, and video clips provided multiple points of access for students to engage with the content from numerous

Technology in the secondary classroom 105 perspectives. In this circumstance, John used technology as an extension of self (Goos, Galbraith, Renshaw, & Geiger, 2000). John finished the lesson by challenging students to work out the formula for the volume of a six-sided pyramid, drawing on the generalisations about volume and area that had arisen during the lesson. A student volunteered to come to the whiteboard to explain his reasoning with an explanation of how the area of the base of the pyramid could be calculated. John encouraged students to come up with their own methods rather than suggesting a “best” method.

Case 10: Years 10–12 School context Joel’s school was an independent K-12 school in a high socioeconomic urban area. The school had an enrolment of 1,327 students in 2018. At the broader school level, Tina, Head of Academics and Innovation, explained that the school had been focusing on the meaningful use of ICT for a number of years. The school was well resourced in terms of digital devices and emerging technologies. In addition, it ran a BYOD program, “we lease the laptops to the students and we just roll them over every three years. So, we have - do have a common device…”. Tina also spoke about the school policy regarding the use of students’ personal devices such as mobile phones: Mobile phones - so, no, the students don’t use - unless they have - the teacher has actually given permission. Given permission there to be actually able to use their devices. But we try and provide them with all the technology that they require in the classroom without having to go back to a personal device. We have looked at that and sometimes when they go out on excursions, they will be able to use their device because we need certain apps. But here at the college, we actually have enough of the technology to be able to support that in class. The school uses internal “teach meets” as a delivery vehicle for professional learning where teachers informally showcase new technologies and share their expertise. In addition, the school has a leadership team to support departments and teachers with implementing technology in the classroom. A current focus for staff development, at the time of our visit, was on mixed reality, looking at the use of the Microsoft Hololens with an application called Dig, which is similar to Minecraft. Tina stressed the importance of building a school “culture of innovation”. Technology is at the forefront of all school activities and is built into each Head of Department’s annual operational plan and into each teacher’s professional development plan. The staff also develop action research questions which they put into action and evaluate as a means of facilitating constant reflection and improvement. The school leaders recognise that all of their teachers are on a continuum and need different levels of support for integrating technology.

106  Technology in the secondary classroom For some, group learning and workshops may work but others may need intensive one-on-one support.

About Joel Joel has been teaching mathematics for over 10 years and has also taught secondary science, beginning his career at a government school catering for academically able students. He then moved to a Catholic boys’ school where he took on the role of technology coach for other teachers and also started to produce teaching videos that were noticed by other teachers in the school. He was attracted to move to his current school because of their culture of innovation and progressive approaches to technology integration. All students at this school have laptops (owned by the school and leased by the students), and when Joel arrived, most teachers were familiar and proficient with using OneNote as an LMS. Joel’s lessons are not the typical format followed by most teachers. Rather than planning technology integration around individual lessons, Joel states that “I’m much more about planning a whole sequence [of lessons] and making it… as individualised as I can”. He begins each school term with an idea of the content, processes, and skills that students need to learn by the end of that term. He then uses technology to provide resources so that there are multiple pathways for students to progress towards the planned outcomes. This approach allows some students to progress through the content very quickly, but in order to ensure that they are learning the content beyond a superficial level, extension activities are built into his planning. Joel uses technology in ways that allow him access to his students’ progress, resulting in “just in time” assistance and intervention during class time. He also uses a flipped learning approach to provide resources, assistance, and personalised feedback to students. This strategy promotes positive pedagogical relationships, the foundation for student engagement with mathematics. Joel recognises that explicit instruction is needed at times, but he finds that constructive discovery is a more engaging way to structure lessons. Like I just think it really, like I’ve tried to, I’ve tried to make explicit instruction based around function transformations, but I find it’s the most boring thing to tell someone about but it’s actually quite interesting to play with. He sees technology as a means to differentiate instruction for students, but he always makes decisions based on what is most important for learning mathematics, and he emphasises that teachers have to know the constraints of the tools that they are using. Also, he recognises that some mathematics topics lend themselves more readily to technology as a means of developing understanding.

Technology in the secondary classroom 107 …probability, if you’re going to teach probability then then it probably makes a lot of sense to be able to use something like Excel, … to be able to show okay, this is what happens if I flip a coin 10 times, 100 times, 1000 times, 10,000 times. If you’re going to do something like function transformations and you’re not opening up something like GeoGebra or Desmos or something like that then I’m not sure what you’re doing.

Lessons: multiple topics Year 12 Trigonometry Year 12 Dynamics (vectors) Year 10 3D Design

Lesson: structure Regardless of the year level or the mathematical content under consideration, Joel’s lessons began in the same way with students demonstrating high levels of self-regulation by sitting down and beginning work without any instruction from the teacher. All students had identical laptops, and for each of his classes, Joel prepared a highly structured sequence of learning materials that the students accessed online and at their own pace. He used OneNote as a delivery vehicle for the materials, with students able to access and re-access the resources as needed. The materials housed on OneNote consisted of carefully sequenced mathematics questions for students to complete, interspersed with short videos where Joel explained mathematical concepts and completed worked examples with commentary. Each video consisted of an insert of Joel’s face using screencast technology or working with pen and paper to explain a mathematics example. In addition, Joel would annotate learning tasks and provide hand-written worked examples within OneNote. This provided his students with permanent access to his explanations that they could return to if required. In the lessons observed, the use of technology was seamless, providing not only a focal point for students to access and work with the mathematics content but also a platform for conversations with the teacher and fellow students. During each lesson, we observed the teacher connecting with almost all students in a one-on-one manner and at times bringing small groups of students together at a classroom whiteboard to explain particular aspects of the topic under consideration. Throughout the lessons, student engagement was high at all times, with no need for the teacher to comment on student behaviour at any time. If compared to a traditional mathematics lesson where students might work through textbook content, the technology in Joel’s lessons allowed for students to interact with more dynamic content through teacher-made videos, thereby increasing student engagement and reinforcing the pedagogical relationships.

108  Technology in the secondary classroom

Summary Each of the secondary schools examined in this chapter varies in the degree to which students and teachers had access to technological tools and the ways in which the use of these tools were supported within the school communities. Despite these differences, there were common threads that emerged in relation to how teachers utilised the affordances of technology to promote student learning and engagement with mathematics. All four secondary schools had either achieved or were moving towards a one-to-one device ratio for all students. Joel’s school provided all students with a leased laptop ensuring that all students were using the same device, while John D’s school ensured that all students had purchased an iPad and were soon to move to laptops. Ian’s school, with relatively fewer resources had achieved a one-to-one ratio over a period of time, although there was no standardisation in terms of the devices that students used and John B’s school was not yet in a position where all students had devices and regularly experienced the extra challenge of dealing with unreliable Internet access. All schools had elevated the responsibility for technology use to a school level committee ensuring that the responsibility for leading technology integration did not lie with the subset of tech-enthusiast teachers. In all schools there was a culture of innovation in teaching with technology which was generally supported and developed through in situ professional development. Teacher collaboration and sharing was encouraged and where possible, staff were supported according to their individual needs. Teachers shared the time-consuming responsibility for creating digital resources and used LMSs to share and develop the resources further. In all schools, there was a heavy reliance on LMSs as a key point of connection between teachers, students, and mathematics content. Having teacher-­ made and/or teacher-curated resources readily available to students allowed the teachers to make pedagogical choices, which enabled effective differentiation of content and supported student agency when engaging with that content. In most cases, the technology supported personalised learning for each student that gave the students choice about how quickly they wanted to progress through the content, where and when they accessed it, and the level of the content that was appropriate for them. During class time, this approach minimised the amount of time that each teacher was lecturing the class in traditional style and maximised the time spent engaging with individual students or small groups of students. In the lessons, we observed we did not see an effective use of group work using technology, which may indicate a downside to one-to-one device ratios if the devices are always in use. In Ian’s classroom, the use of vertical whiteboards around the classroom did encourage some students to work with their peers, indicating that space for “old” technology can be accommodated in a technology-­rich classroom. There was general agreement expressed by the teachers about the affordances of technology best suited for mathematics teaching. All agreed that

Technology in the secondary classroom 109 the ease of presenting and manipulating visual representations was of great benefit for student learning, particularly any topics requiring graphing. In this realm we saw numerous applications of Geogebra, Desmos, and Excel. Also, replacing tedious calculations was seen to be of value, as was the ease of collecting data for probability experiments or when examining statistical concepts. Videos were also used liberally to illustrate mathematical concepts across all topics. The ways that we saw technology used in these secondary classrooms certainly contributed to student engagement with mathematics. Using LMSs to facilitate student choice of difficulty level and pace of learning ensured that all students were able to engage with mathematics at the appropriate level. These systems also supported the teachers to track student progress, to intervene, and to provide feedback when required. Coupled with the ability for technology to present mathematics more dynamically and visually than static textbooks, we saw evidence of the capacity of technology to engage students in multi-faceted ways.

References Attard, C. (2011). “My favourite subject is maths. For some reason no-one really agrees with me”: Student perspectives of mathematics teaching and learning in the upper primary classroom. Mathematics Education Research Journal, 23, 363–377. Attard, C. (2014). “I don’t like it, I don’t love it, but I do it and I don’t mind”: Introducing a framework for engagement with mathematics. Curriculum Perspectives, 34, 1–14. Clark, K. R. (2015). The effects of the flipped model of instruction on student engagement and performance in the secondary mathematics classroom. Journal of Educators Online, 12(1), 91–115. Gabriel, F. C., Coché, F., Szucs, D., Carette, V., Rey, B., & Content, A. (2013). A componential view of children’s difficulties in learning fractions. Frontiers in Psychology, 4. doi:10.3389/fpsyg.2013.00715 Gardner, H., & Davis, K. (2013). The app generation. New Haven, CT: Yale University Press. Garet, M., Porter, A., Desimone, L., Birman, B., & Yoon, K. (2001). What makes professional development effective? Results from a national sample of teachers. American Educational Research Journal, 38(4), 915–945. Goos, M., Galbraith, P., Renshaw, P., & Geiger, V. (2000). Reshaping teacher and student roles in technology-enriched classrooms. Mathematics Education Research Journal, 12(3), 303–320. doi:10.1007/BF03217091 Hickmott, D., Prieto-Rodriguez, E., & Holmes, K. (2018). A scoping review of studies on computational thinking in K-12 mathematics classrooms. Digital Experiences in Mathematics Education, 4(1), 48–69. doi:10.1007/s40751-017-0038-8 Higgins, J., & Parsons, R. (2011). Professional learning opportunities in the classroom: Implications for scaling up system-level professional development in mathematics. Mathematics Teacher Education and Development, 13(1), 54–76. Lo, C. K., & Hew, K. F. (2017). A critical review of flipped classroom challenges in K-12 education: Possible solutions and recommendations for future research. Research and Practice in Technology Enhanced Learning, 12(1), 4. doi:10.1186/s41039-016-0044-2

110  Technology in the secondary classroom Maher, D., & Twining, P. (2017). Bring your own device – A snapshot of two Australian primary schools. Educational Research, 59(1), 73–88. doi:10.1080/00131881.2016. 1239509 Martin, A., Way, J., Bobis, J., & Anderson, J. (2015). Exploring the ups and downs of mathematics engagement in the middle years of school. The Journal of Early Adolescence, 35(2), 199–244. Papert, S. (1980). Mindstorms: Children, computers, and powerful ideas. New York, NY: Basic Books. Puentedura, R. (2006) SAMR: A brief introduction. Retrieved from http://www. hippasus.com/rrpweblog/archives/2013/10/02/SAMR_ABriefIntroduction.pdf Tytler, R., Williams, G., Hobbs, L., & Anderson, J. (2019). Challenges and opportunities for a STEM interdisciplinary agenda. In B. Doig, J. Williams, D. Swanson, R. Borromeo Ferri, & P. Drake (Eds.), Interdisciplinary mathematics education (pp. 51–81). ICME13 Monographs. Cham, Switzerland: Springer.

7

Technology use in mathematics classrooms: what do school leaders, teachers, and students say?

School leaders, teachers, and their students all contribute to the unique and complex context of each mathematics classroom and more broadly, the school site. These contexts have a strong influence on, how, or if technology is used in mathematics teaching and learning. While there are aspects of context that cannot be influenced by the practices of those within the school site (geography and socio-economic status), there are other inter-related and often inter-dependent aspects of school such as classroom culture, commitment to the use of technology, and community support of technology use, that contribute to and shape each context in relation to how students experience technology use in mathematics classrooms. In this chapter, we include the perspectives of the three critical groups that influence, and are influenced by, what happens in mathematics classrooms: school leaders, teachers, and students. We have organised our discussions regarding each group of participants according to themes that emerged from our cross-case analysis. These findings provide additional insight into how the views and beliefs held by the case study teachers and their leaders have influenced the technology-related practices at their school and how these practices influenced their students’ experiences of mathematics. An understanding of the commonalities and differences that emerged across each of the three groups may be of assistance to other teachers and school leaders wishing to implement whole-school or individual classroom approaches and may provide pathways for future research in technology-related mathematics practice. To provide an overview of the varying contexts and the case study voices within those contexts, we list the details of each of the schools, the teachers, and their leaders in Table 7.1.

School leaders For the purpose of this study, we identified school leaders as those who had a formal leadership role within each of the case study schools. In some cases, the leaders were School Principals, in other cases, they were Deputy Principals, designated Technology Leaders, or Heads of Mathematics.

112  What do school leaders, teachers, and students say? Table 7.1  Case Study Details Case Leader*

Teacher

1 2

Sharon Rebecca

3 4 5 6 7 8 9 10

Sandra Fran

Grade

Pre Special unit Fran Ashleigh 1-2 Rachel Loretta 3 Amy Bec 3 David Jessica 6 Trent John D 9-12 Carly Ian 7-10 Rowena John B 9-12 Tina Joel 10-12

Students (2018)

Experience**

School Type

Location

Late-career Mid-career

Public Public

Metropolitan Metropolitan

322 356

Public

Metropolitan

478

Independent Catholic Public Catholic Independent

Metropolitan Metropolitan Regional Remote Metropolitan

1,658 1,021 1,155 561 1,327

Early-career Mid-career Early-career Early-career Mid Early-career Mid-career Mid-career

* Pseudonyms. ** Early-career = 0–5 years; Mid-career = 5–10 years; Late-career = 10+ years.

School commitment to technology use: Leading through policy It is clear from these case studies that leadership in its many and varied forms plays a critical role in supporting the effective use of technology in mathematics classrooms. The common thread across conversations held with leaders in this study is a commitment to providing continued professional learning, to developing a community of practice and working collaboratively with the broader school community, and trust in the professional judgement of individual teachers. Within each of the schools, teachers were given freedom to use technologies in ways that they were comfortable with. This enabled our case study teachers to innovate, and others to use technology in ways that were aligned with their individual capabilities. As would be expected, each of the eight schools featured in the ten case studies was unique, representing a broad range of geographical and socio-economic contexts and school systems. These contextual differences alone influenced the teachers’ and students’ access to technology, the support they were able to provide their teachers, and ultimately, the level to which digital technology was embedded in classroom practice. Despite this, each of the teachers in the study was able to take advantage of the affordances offered by the available technologies to enhance the teaching and learning of mathematics. In some cases, the support from the school leadership was central to the success that occurred in the classroom. In other cases, the teachers’ success was independent of school policy and support. In several cases, the teachers themselves also held either formal or informal leadership roles in terms of providing professional learning and for colleagues. An example of a case study where leadership played a significant role in influencing the technology-related practices was Case 10. At the time of data collection, the school had recently won a national award that recognised the school’s innovative practices in the integration of technology. This school has clearly defined technology-related policies, strong support from the leadership team that includes time for teacher professional learning, and a technology-infused

What do school leaders, teachers, and students say? 113 teaching and learning framework. It is also an expectation that all teachers showcase their technology-related practices to peers at regular intervals through the school year. It should also be noted that being a private school, there are more funds available for professional learning and investment in technology than other less affluent public schools. Tina, the school’s Head of Academics and Innovation spoke about the schoolwide commitment to innovative technology use: So, as an example, the staff - to learn, we have a big showcase of new and emerging technologies. We get our colleagues to present that, so we do it like a teach-meet session. They’ll go around and they’ll brainstorm about what it is that they might be able to use in their classroom. Then we have another leadership team there that then go in and support the departments and the teachers then implementing that technology in the classroom. So, we always provide the support there. I guess, we run a - very much a differentiated professional development model, so that our staff feel supported in what they’re doing. While the school in Case 10 has formal policies and high expectations of all teachers regarding their technology use, the ways in which technology is used in Case 10 are left up to individual teachers. Allowing teachers to explore new technologies at their own pace with the support of additional professional learning aligns with frameworks such as that articulated by Niess et al. (2009) and acknowledges the steps involved in assimilating (or not) a new technology into one’s practices. The range of technology use amongst the mathematics faculty provided evidence that each teacher could adapt their technology use according to their personal abilities, beliefs, and student needs. Joel, our case study teacher, admitted that he avoided using technology just for the sake of it. Rather, he used a limited range of software in ways that resulted in enhanced learning experiences for his students, resulting in learning experiences that aligned with the “redefinition” level of the SAMR (substitution, augmentation, modification and redefinition) framework (Puentedura, 2006). The leadership in Case 7, similarly to the school in Case 10, promoted a whole-school approach to the use of technology. In this case, the technology related policies were closely related to the whole-school pedagogical approach of using problem-based learning across all disciplines. Trent, the Head Teacher: Mathematics, spoke about the commitment and culture of the school in relation to technology use: We’re already a member of the New Tech Network here, which very much embraces innovative mathematics teaching. We’re also a member of the Singapore Republic Polytechnic and we are still looking for what the next one is. But we have had exceptional results in mathematics, and I believe it is through the engaging use of IT and also problem based, project-based learning and the flipped learning, which has led to our kids thoroughly enjoying mathematics and actually wanting to learn more.

114  What do school leaders, teachers, and students say? The levels of formal school policies and technology-related school strategies amongst the schools varied significantly and appeared to be heavily influenced by the beliefs of either the school principals or the members of the school executive teams. While supportive of the use of technology, the two schools in Cases 2/3 and 4/5 did not have formal policies in place but were in the process of developing them. Despite this, the teachers were supported and encouraged in their technology use. In Cases 4 and 5, the Principal, Rachel, spoke about technology use at the school: They have BYOD policy from Years 3 to 6. The different teachers across the different stages they decide how it will be used. They’ll negotiate and talk with parents about which apps and things they should - we use iPads. They use iPads as part of the Bring Your Own Device. They have a list of apps that they would like the parents to load up onto the device. That will look different in different years. Then, also we have the use of laptops.

Community approaches to technology use A strong commonality across all of the case study schools was a community approach, heavily supported by school leadership, to the use of technology both within the school (teachers and students) and beyond (parent community). Communities of practice (Wenger, 2000) appeared to have developed within the schools where teachers shared their ideas either formally through coaching or organised professional learning opportunities or informally through day-today interactions and sharing of ideas. Regardless of policy, all the school leaders did appear to prioritise professional learning, which took different forms in each context, although was generally in situ and was often influenced by the amount of funding available. Collaborative decision making with regard to how technology is used, the devices and software to be purchased, and expectations of teachers were a common theme that emerged in varying degrees across the schools. This ranged from having teams of designated technology leaders to whole-school approaches. In some of the cases, consultation with the parent/caregiver community occurred, where in others, it didn’t. Some parents have said to me that they weren’t that happy about it and they weren’t - they feel they weren’t given a lot of choice but then I hear that there was a bit of consultation, so you’re always going to hear that from parents

Challenges faced by leadership While all the participants in this study strongly supported the use of technology in mathematics classrooms, its use was not without challenges. One common challenge for the leaders in this study was having to deal with staff attrition

What do school leaders, teachers, and students say? 115 (teachers and leaners). Two of the leaders interviewed were very new to their positions at the time of data collection, and although they were highly supportive and prioritised the use of technology across their schools, they had not had time to make any significant changes or formalise technology-related policies. This is a challenge experienced by many schools. For example, when a new school principal begins at a school, priorities tend to change according to the beliefs held by that individual. This could result in inconsistencies arising from differing priorities. In the case of the school featured in Case Studies 2 and 3, the new principal, Rachel, had begun her term in a school where technology use was prioritised but not formalised through school policy. A lack of policy combined with differing personal beliefs about technology and learning, resulted in an ad hoc approach by teachers, and inconsistencies in technology use across the school. The Assistant Principal, Amy, spoke about the issue: In essence, I think there is a divide. I think 3-6 definitely see technology and are very open to it and can use it quite well. Where K-2 not so much, I think they are more sceptical as to the ability it has to support learning. While the inconsistencies could be viewed as a deficiency, the school leadership supported teachers as they used technology at a level each individual teacher was comfortable with. Similarly, the turnover in teaching staff or the increase in staff due to growing schools presented challenges relating to the amount and consistency of technology-related practice in mathematics. In these cases, the support of leadership in relation to the provision of professional development opportunities both in situ and externally was critical. A third challenge for the leaders and teachers related to the students and their devices. Six out of the 10 case studies were running BYOD (bring your own device) programs in various forms. This resulted in the students being responsible for bringing their devices to school each day. Equity issues also arose here, presenting a first-order barrier that impacted the number of resources available for the teachers to use (Ertmer, Ottenbreit-Leftwich, Sadik, Sendurur, & Sendurur, 2012). In each of the case studies, the leadership of the school ensured that there were spare devices, where possible, available for students who did not have devices at school. The success of this strategy was dependent on the number of students who did not have a device at any given time. In some of the schools, another strategy to address this issue was the provision of desktop computers, which again posed a different challenge for teachers and students and influenced task design.

Case study teachers Several themes emerged as a result of the cross-case analysis of teacher interviews and observations. These themes were teacher beliefs, teachers’ pedagogical decisions, and teacher engagement.

116  What do school leaders, teachers, and students say?

Teacher beliefs There were several clear commonalities in the beliefs of the case study teachers. First, each showed a strong commitment to the use of technology in mathematics teaching, and all of them were open to learning about new technologies, experimenting, and innovating with them. They also all showed a willingness to share their expertise with their colleagues, with some of the teachers taking on technology-related leadership roles within their schools. This positive attitude towards learning to use new technologies was evident in the teachers from Sharon (Case Study 1, Chapter 4) in her pre-school setting, through to Joel (Case Study 10, Chapter 6) in his senior secondary teaching: “I’ve had to bring myself up to speed. So, the year before last, I was asked to be part of a digital technology team and I’m like, okay” (Sharon). Sharon has been teaching for over 30 years and was still open to new teaching innovations. Joel, who had been teaching for ten years, expressed his beliefs regarding technology, teaching, and innovation: I think quite often when a big, big, big shift like that happens perhaps we underestimate the difference that it causes in a classroom. We just kind of go well everyone’s got a computer, so everyone knows how to use it and we’ll just move forward with that. But it really, really should change the way that you teach. When discussing their beliefs about the place of technology in mathematics teaching and learning, all the teachers felt that almost all mathematics topics could be taught using a variety of devices. For example, Loretta (Case 4, Chapter 5) said “I haven’t come across a maths topic that I haven’t been able to incorporate technology into yet”. However, she also commented that: “…there are some activities that sometimes you just need the concrete materials and they need to be able to feel it and touch it and manipulate it, but I suppose that is on a task-by-task basis”. Loretta’s sentiments with regard to the use of hands-on, concrete materials were echoed by all of the early years and primary case study teachers. Typically, secondary teachers rely less on hands-on materials yet these secondary teachers made similar comments to those of the primary teachers: “….you can use a lot of static props and hand things out and sometimes that is great to get hands-on experience and I still like to use dice and things like that in the classroom” (Ian, Case 8, Chapter 6). One difference between the primary and secondary teachers’ beliefs with regard to technology and mathematics was that the primary teachers’ use of technology was focused more on promoting mathematical reasoning and communication, rather than using technology to deepen conceptual understanding of mathematics content. For example, when teaching fraction concepts, the technology was used to elicit student reasoning rather than to demonstrate or introduce fraction representations, which was the case in Ian’s lesson (Case 8, Chapter 6).

What do school leaders, teachers, and students say? 117

Teachers’ pedagogical decisions Tsai and Chai (2012) believe the capacity for design thinking in relation to finding new ways of practice when confronted with new technologies is a third-order barrier to effective technology use. However, the teachers in this study did not appear to experience such a barrier. While some of their practices were not necessarily considered to be at the level of redefinition (Puentedura, 2006), others were, and all of their uses of technology-enhanced learning and teaching in mathematics in one way or another. The pedagogical decision-making process is a complex one, particularly when faced with the plethora of affordances offered by contemporary technologies within the unique context of each school site. During our interviews with the 10 case study teachers, they spoke about the thinking behind their decision-making and planning, revealing important insights. The following comments illustrate some of their thinking behind their choices to use (or not use) technology: We’re looking for the educational - I guess, how we can use the app to really better what we do in the classroom. So, they can represent that understanding. (Sharon, Case Study 1, Chapter 4) I suppose it is finding the app or the website or the resources or whatever it is that kind of fits the need and I suppose I may teach something one way with my Year 3 class and then next year I wouldn’t teach the same lesson in the same way because it is different with kids. So, I suppose the way I use technology is very adaptive, depending on the cohort of the kids as well. (Loretta, Case Study 4, Chapter 5) I just use technology to support us as we would in any subject we have in the school. Our learning management system, randomized questioning, flipped videos. That goes throughout all the subjects we - that’s like the underlying technology. It’s not an additional one. (John D, Case Study 7, Chapter 6) I am always on the lookout for new technology and I will have a look at it and say sort of what we were saying earlier, is there just a gee whiz factor to this or can I see an educational benefit? And sometimes, you know, certainly Friday afternoon sometimes the gee whiz factors benefit and you go well there is not a lot mathematical learning to be had from this but the kids will enjoy it and it will reinforce a couple of concepts we have been working on, so for a Friday afternoon, guess what, it is probably worthwhile. Whereas I wouldn’t use that earlier in the week period 2 for example, to um demonstrate a concept and get them familiar with it. (Ian, Case Study 8, Chapter 6) I don’t plan to use technology. I plan on how I am going to get a concept across. (John B, Case Study 9, Chapter 6) I guess you’ve got to know what the limitations of your tools are and what your tools can do. Then you’ve got to say well, once you’re that comfortable with them you could say oh, okay, usually I would just draw it on

118  What do school leaders, teachers, and students say? the board but is there a better way to represent those things? So there is but that said, I don’t think it’s particularly exciting, particularly whiz-bang. But I do think that it sort of lowers the cognitive load of a student if you don’t say “now, picture this”. (Joel, Case Study 10, Chapter 6) An important commonality amongst each of these teachers is the critical stance they take when making decisions regarding whether to use technology and which affordances to use. Each of these teachers prioritises the mathematics and student learning over the novelty of using technology for the sake of it.

Teacher engagement In education, we are often concerned about lowered levels of student engagement with mathematics (Attard, 2013). However, we rarely consider how student engagement might be heavily influenced by the engagement of their teachers in the process of teaching mathematics. If student engagement is conceptualised as a multi-dimensional construct consisting of cognitive, operative, and affective engagement (Attard, 2014; Fredricks, Blumenfeld, & Paris, 2004), then we would argue that teacher engagement can be conceptualised the same way: teachers thinking hard, working hard, and feeling good about teaching mathematics. In this study, it was evident that the case study teachers were all highly engaged with teaching mathematics, and hence they were deemed by their peers to be effective in their use of technology as a result of their engagement with teaching and their commitment to their students. We found evidence of this through our observations of their teaching and the comments they made during interviews. Some of these comments are presented here: I think quite often when a big, big, big shift like that happens perhaps we underestimate the difference that it causes in a classroom. We just kind of go well everyone’s got a computer, so everyone knows how to use it and we’ll just move forward with that. But it really, really should change the way that you teach. (Joel, Case 10, Chapter 6) I suppose I am always trying to improve myself and that naturally I can see how I can use the technology more. (John B, Case 9, Chapter 6) I think it is kind of our responsibility as teachers to expose them to so much technology that it isn’t a new thing to them, it is just commonplace that if they need something they know that there are so many different things they can use. (Loretta, Case 4, Chapter 5) But I realised that I had some skill and I had a lot of passion for using technology and using it well. Integrating it and making sure that it wasn’t just being used for the sake of being used because that is detrimental. (Rebecca, Case 3, Chapter 4) Each of the teachers featured in this book demonstrated cognitive engagement in terms of the thinking behind their technology-related pedagogical

What do school leaders, teachers, and students say? 119 decisions. They were operatively engaged from the planning process through to the teaching of mathematics. In addition, the use of technology, where flipped learning approaches were featured, promoted teacher operative engagement by freeing them to interact more with their students face-to-face and through digital formats. Finally, it was clear that each of these teachers was affectively engaged with their teaching. They valued their contribution and the contribution of technology in terms of its enhancement of the mathematics education of their students.

The influence of technology on students It is important to include student voice in educational research to fully appreciate the effects of technology-related pedagogy on today’s students (Attard, 2013; Hayes, Mills, Christie, & Lingard, 2006) and inform and improve teaching practice (van den Heuvel-Panhuizen, 2008). As explained in Chapter 1, we conducted focus groups of students from each of the case study teachers’ classes in Cases 4 to 10 (student from Year 3 and higher). We did this to help us understand their perceptions of technology use in mathematics learning and whether their experiences aligned with their teachers’ intentions. Each focus group consisted of students who were selected by the case study teachers as a representative sample of their class population/s. In the secondary case studies, students were selected from the combined class groups that were observed by us. Where possible, each focus group was made up of mixed gender, ability, and attitudes towards mathematics. Themes that emerged from the analysis of the focus group discussions were focused on the students’ perceptions of the benefits of technology use, the amount of use, and, to a lesser degree, the challenges related to their use of technology in mathematics classrooms. The most dominant theme highlighted the perceived benefits and was a clear indication that students were more engaged with mathematics as a result of the use of technology. The following series of quotes provide further detail, highlighting the practical benefits: Instead of bringing a physical maths textbook to look at questions and answer in our books we can just bring an iPad and download it on the iBooks. It’s a lot easier and more convenient because we don’t have to - it’s all in your iPad and you don’t have to bring a massive copy of the book in your bag. It’s going to put more strain on you. (Year 9 student, Case 7) Technology really helps us because we use a platform called Echo and teachers can post videos on there, put all our homework onto there. Without those platforms it would be hard for students to actually go home and do what they did in class. If they don’t understand something they can go home and redo it. Without Echo and the technology that won’t be able to happen. (Year 9 student, Case 7) Basically everything we need for learning in school is there. (Year 9 student, Case 7)

120  What do school leaders, teachers, and students say? Well I … technology in that way ‘cause like it helps you like – it tells you the questions like and you have to answer it and then like – I like how you can tape it, like record yourself like saying stuff and stuff like that. (Year 3 student, Case 5) The affordance offered by learning management systems (LMS) in the secondary case studies had a significant influence on the students in each school, allowing them to access resources in a variety of multimedia, within and outside the classroom. Although some could argue that textbooks perform a similar role, the use of an LMS provides substantially more affordances, linking students to a broader range of resources, including their teacher. Another dominating feature of the students’ conversations focused on the use of videos in their secondary mathematics classrooms: I like YouTube videos. YouTube videos, they step-by-step explain the task. Sometimes if you don’t understand it from a teacher’s point of view, the equation or question, someone on YouTube might do the same question and they’ll have a different angle to it. That’s really good. (Year 9 student, Case 7) …it’s more stimulating than just a black and white text book. You kind of have to try and - you have to work really hard to stay focused when it’s just a text book and your book or whatever. But it’s a lot different. You’ve got the video in front of you. You just flick the switch and focus I find. (Year 10 student, Case 10) The primary students focused on the fun aspects of technology use, indicating high levels of affective engagement resulting from the use of robotics and apps based on gamification (Kingsley & Grabner-Hagen, 2015) of learning, which featured more heavily in the primary and early years case studies: Well on the Sphero I enjoy using it because usually with maths you have to work really, really hard and as much as you are working you have fun with it. And it is a good way to like calculating the speed and everything. (Year 6 student, Case 6) I like to play games, like maths games and an example is Maths Jump, like it helps me think of equations like off the top of my head and it gets me thinking on my feet. And I also like how with Dr D we get to write on the whiteboards while we the page on the screen. (Year 7 student, Case 8) Well we were playing a game called Prodigy and it helps a lot in maths ‘cause it is like a wizard game where to attack you need to work out a maths problem and then – but if you get it wrong you only have – you can only try twice and then it will show you the answer then you have got to remember it again the next time when it asks that question. (Year 3 student, Case 5) However, while the students perceived the gamification of learning to be a positive aspect of technology use in their mathematics learning, it appeared that

What do school leaders, teachers, and students say? 121 some students did not equate fun or technology use with serious learning. When asked to comment about the amount of technology use in mathematics classes, the following comments typified the sentiments of the students: I think we need to get some new learning games that will teach us. (Year 3 student, Case 4) I think we use enough because sometimes you can use technology for other things, not just for math and sometimes when you use too much technology you might just not get on with learning. (Year 3 student, Case 4) Well sometimes I think we might use a bit too much. Because we are always on iPad and I think it is also good to just use your brain and not use it. (Year 6 student, Case 6) Interestingly, none of the students suggested they would like more technology use in their mathematics lessons, claiming the technology they were currently using was the right amount. While they were satisfied with the amount of technology, the students also felt that there were some barriers and distractions related to their use of technology: Sometimes I find with the YouTube videos it’s easier to get off task. That’s my only issue with it and I find some days if I’m not in the right head space for math then it’s easier to get off task because you’re on your internet or you’re on YouTube. (Year 11 student, Case 10) The school wi-fi is – sometimes for some people it is a problem where they will type in their user name and their password and they will ask them 15 more times. (Year 8 student, Case 8) I think it would be easier with um, is with technologies, like it is a lot more expensive. It can become a problem if there is a system where you actually have to have a device and some can’t afford it, then it becomes a major problem. (Year 7 student, Case 8)

Influences on and of technology use Overall, the voices that have informed this chapter, combined with our observations of their practice as detailed in Chapters 4, 5, and 6, have provided us with a confirmation that effective teachers of technology-infused mathematics instruction provide enhanced learning experiences for their students, increase their access to mathematics resources, and promote student engagement with mathematics, regardless of the number of devices and types of software available to them. This occurs as the result of supportive leadership, professional learning that is tailored to teachers’ needs and contexts, and engaged, passionate teachers who recognise the value of using technology in mathematics education. These teachers make the most of the technologies they have at hand to design learning experiences that are differentiated, promote mathematical communication, present dynamic representations of mathematical content, and build connections in

122  What do school leaders, teachers, and students say? mathematics and a strong sense of connection between students, teachers, and parents and carers. We conclude this chapter with quotes from two students: Without technology I would feel less motivated to do maths because instead of - we have step-by-step guides that we use and interactive activities on the technology. If I needed help with maths then I’d rely on technology the most but if I didn’t have the technology I would struggle a lot and I’d lose motivation to continue doing my homework. (Year 9 student, Case 7) It gives you that sense of hope, which is good. (Year 11 student, Case 10)

References Attard, C. (2013). “If I had to pick any subject, it wouldn’t be maths”: Foundations for engagement with mathematics during the middle years. Mathematics Education Research Journal, 25(4), 569–587. doi:10.1007/s13394-013-0081-8 Attard, C. (2014). “I don’t like it, I don’t love it, but I do it and I don’t mind”: Introducing a framework for engagement with mathematics. Curriculum Perspectives, 34, 1–14. Ertmer, P., Ottenbreit-Leftwich, A., Sadik, O., Sendurur, E., & Sendurur, P. (2012). Teacher beliefs and technology integration practices: A critical relationship. Computers & Education, 59(2), 423–435. Fredricks, J. A., Blumenfeld, P. C., & Paris, A. H. (2004). School engagement: Potential of the concept, state of the evidence. Review of Educational Research, 74, 59–110. Hayes, D., Mills, M., Christie, P., & Lingard, B. (2006). Teachers and schooling making a difference. Sydney, Australia: Allan & Unwin. van den Heuvel-Panhuizen, M. (2008). Learning from “didactikids”: An impetus for revisiting the empty number line. Mathematics Education Research Journal, 20, 6–31. Kingsley, T. L., & Grabner-Hagen, M. M. (2015). Gamification: Questing to integrate content knowledge, literacy, and 21st-century learning. Journal of Adolescent and Adult Literacy, 59(1), 51–61. Niess, M. L., Ronau, R. N., Shafer, K. G., Driskell, S. O., Harper, S. R., Johnston, C., & Kersaint, G. (2009). Mathematics teacher TPACK standards and development model. Contemporary Issues in Technology and Teacher Education, 9(1), 4–24. Puentedura, R. (2006). SAMR: A brief introduction. Retrieved from http://www. hippasus.com/rrpweblog/archives/2013/10/02/SAMR_ABriefIntroduction.pdf Tsai, C., & Chai, C. S. (2012). The “third”-order barrier for technology-integration instruction: Implications for teacher education. Australasian Journal of Educational Technology, 28(6), 1057–1060. Wenger, E. (2000). Communities of practice and social learning systems. Organization, 7(2), 225–246. doi:10.1177/135050840072002

8

Introducing the Technology Integration Pyramid (Mathematics)

In this final chapter, we draw on literature and existing frameworks, coupled with findings from ten case studies presented in Chapters 4, 5, and 6, to present a model for effective integration of digital technologies in mathematics classrooms. While existing frameworks each go some way in helping us understand the types of knowledge, tasks, practices, and tools teachers require to teach effectively with technology, there is no one framework that brings all of the elements together. Further, we do not believe that existing frameworks account for the realities of schools and classrooms and the diversity of contexts, beliefs, abilities, and access to technology. It is these realities that we now attempt to capture in our model, the Technology Integration Pyramid (Mathematics) [TIP(M)] (Figure 8.1). The TIP(M), has been intentionally conceptualised as a three-dimensional model to illustrate the connections and inter-related elements within it that teachers need to consider when planning for the use of any technology, regardless of device, software, access, and school context. In other words, the TIP(M) model will assist in future-proofing technology-infused teaching and learning as new technologies continue to emerge. This model does not present a step-by-step guide. Rather, it presents holistic means of understanding the parameters within which teachers operate and a recognition that student engagement with mathematics is a critical element for learning to occur in contemporary classrooms. We begin the chapter by acknowledging the complex influences (context, culture, commitment, and community) that must be understood before planning for teaching with technology. These influences form the base of the TIP(M) (Figure 8.2). We then follow with the individual elements that form the sides of the model, that is, mathematics, pedagogy, tools, and student engagement. We believe that these individual elements, when considered together, combine to result in effective mathematics teaching with technology.

The pyramid base: influences on technology use A range of factors from within and beyond the classroom have considerable influence on how technology use is actualised for teaching and learning mathematics. Some of these factors can be controlled by individual teachers, but others are beyond their control. Likewise, school leaders may have the power to impact

124  Technology Integration Pyramid

Figure 8.1  The Technology Integration Pyramid (Mathematics).

Figure 8.2  The TIP(M) Base: Influences on Technology Use.

Technology Integration Pyramid 125 or change these influences, but they too operate within a larger school system and within the background of broader political and societal pressures. In this section, we examine the influences in the educational landscape and the ways in which they can impact technology-infused practices in mathematics classrooms. While we present the four discrete influences of context, community, culture, and commitment, there are complex interactions amongst and within them. The categories serve as a guide for teachers to understand the opportunities and challenges within their school setting and to work effectively and realistically within those boundaries.

Context We know that schools operate in unique contexts; varying in terms of location, socio-economic status, school system, and technology-related resourcing. These factors can be beyond the control of individual teachers and school leaders. However, it is important for educators to understand the influence of their contexts on teaching and learning so that the maximum benefit can be gained from the available affordances of the technologies at hand. Findings from the ten case studies highlighted how varying school contexts determined the amount of access to technology, yet also revealed that learning and teaching can still be enhanced with limited resources. For example, in Chapter 4, Ashleigh (Case Study 2, Chapter 4) worked in a low-socioeconomic school that had minimum access to devices. She was able to leverage the affordances of a single mobile phone and an interactive whiteboard (IWB) to effectively conduct technology-enriched lessons leading to high levels of student engagement. In contrast, Jessica (Case Study 6, Chapter 5) operated within a high-socioeconomic and highly resourced teaching environment that enabled her students to directly interact with devices throughout all lessons. This technology-­r ich environment supported a greater range of pedagogical choices for Jessica, yet each teacher was successful within their individual contexts. We also recognise that the students within each of these contexts came to school with very different levels of experience and exposure to technology in their homes. For example, the students in Ashleigh’s class may not have had reliable access, if any, to Internet or devices outside the school environment. In contrast, the students in Jessica’s class each owned their own iPad and are likely to have had access to other devices, within their own homes. This was evidenced when one of Jessica’s students brought in his own Sphero robot to school. A similar issue to socioeconomic status is the school location. In Case Study 9 (Chapter 6), the school was located in an outer-regional district where the population experienced difficulty in accessing reliable Internet. This restricted student access to flipped learning resources both at school and at home, and the teacher, John B, used a variety of resources including traditional textbooks and graphics calculators to ensure continuity of learning for students despite the challenges. Another strategy employed by John B was to distribute learning resources via the mobile phone network, which was more reliable than Internet.

126  Technology Integration Pyramid The variety of schools within the case studies also highlighted differences in the amount of funding available for technology use. For example, the independent schools typically employed technology support staff to provide technical and instructional support, while the public schools relied more heavily on the expertise of individual teachers. These varying levels of support inevitably impact on classroom practices with technology; however, as the case studies revealed, teachers in schools, without heavily funded support, were still able to develop innovative practices within the constraints of their contexts. In summary, technology-infused mathematics teaching is influenced by school contexts that vary by: • • • •

Socio-economic status Location (regional/rural/remote/metro/per-urban) Funding System (policy, type, restrictions)

Community Schools operate within and are influenced by their local communities. These communities comprise of parents and carers, local businesses, media, government, and other stakeholders. School leaders, teachers, and their students also form a community within each school site. The external community can play a significant role in how schools operate and where technology is concerned, attitudes can influence school policy and practices. For example, community angst in Australia in relation to the use of mobile phones at school has led to some schools and systems banning their use in the classroom. This has also been fuelled by media attention. Such bans have the potential to limit teachers, such as John B and Ashleigh, who use mobile technologies as a resource to overcome connectivity and funding constraints within their contexts. Negative community attitudes towards technology use, particularly for younger students, can also impact technology-related school initiatives such as BYOD (bring your own device) programs due to fears related to excess screen time. In mathematics in particular, where traditional practices are often seen as preferable within the community, technology-based teaching approaches can be viewed with scepticism. As we saw at Loretta and Bec’s school (Case Studies 4 and 5, Chapter 5), considerable effort was taken to educate parents who were initially resistant to the introduction of a BYOD program. At this school and others, the use of learning management systems as communication tools between parents and teachers assisted in changing attitudes to technology use. In addition, the technology provided parents with insights into contemporary teaching practices and expanded their views of what constitutes students’ mathematical understanding through sharing of student work through multimedia. These examples highlight the need to include the broader community with technology-based decisions within schools. BYOD programs, in particular, require the support of parents in terms of finance and management of the

Technology Integration Pyramid 127 devices. The introduction of mobile devices and resulting changes in teaching and learning often challenge parents’ perceptions and expectations of how mathematics classrooms should operate. School leaders, teachers, and students form the immediate school community, and these relationships influence and are influenced by technology use in mathematics classrooms as highlighted in Chapter 7. Our case studies revealed school communities, where a consistent approach to technology was demonstrated through formal school policy and a shared understanding of its purpose and value in mathematics education. Given the considerable investment required for technology programs in schools and the potential impact on student outcomes, the perceptions and support of all community stakeholders are vital. In summary, consultation regarding technology use for teaching mathematics should include the views of: • • • •

Parents Other local stakeholders (business, media, government) Colleagues Students

Culture The community within a school, its collective practices, beliefs, and customs, all contribute to the school culture, and can determine how technology-infused practices develop and evolve over time. In several of the case studies, the teachers had clearly developed a community of practice (Wenger, 2000) in relation to technology-use in mathematics teaching. For example, in Ian’s school (Case Study 8, Chapter 6), the mathematics faculty worked collaboratively to produce digital resources for students that were shared amongst the mathematics teachers and resulted in a consistent approach. This community of practice was facilitated through the faculty leadership taking an active role and providing the opportunity for collaborative meeting times and professional development. In the majority of the case study schools, teacher professional development was conducted in situ, thereby promoting an innovative culture within the teachers’ immediate teaching contexts. This approach ensured that the professional development was tailored to the local environment, immediately applicable, and appropriate to the specific classroom environments within each context. Some schools did supplement this internal professional development with external conference attendance, allowing the space for the introduction of new innovative practices. In summary, school culture assists in supporting technology use in mathematics through: • • • •

School leadership Professional development Collaboration (teachers working in teams/individual) Innovation

128  Technology Integration Pyramid

Commitment In order for any technological innovations to take hold and be successful within schools, teachers and their school leaders need to demonstrate ongoing commitment. For school leaders, that is commitment to supporting teacher efforts to integrate and innovate with technology, both in terms of technical and instructional support. In some of the case studies, the school leadership illustrated commitment by providing mechanisms to encourage a culture of innovative practices. This was evidenced at John D’s school (Case Study 7, Chapter 6), where the leadership was planning the construction of a video recording studio to enhance the quality of teacher-made videos and to encourage all the teachers to participate in this innovation. At other schools, the leadership provided space and time for teachers to share and showcase practice through formal staff meetings and more informal sharing sessions. Individual teachers also need to possess a willingness to continually learn about new technologies with a focus on the ways in which they can increase student engagement and enhance learning in mathematics. As discussed in Chapter 7, all of our case study teachers spoke about their belief that technology has the potential to improve student engagement with mathematics. In particular, the primary teachers believed that technology provided new and innovative ways to assess student work and provide feedback. In the secondary case studies, the teachers used technology to provide alternate representations of mathematical concepts and improved students’ access to learning resources through the use of learning management systems. Regardless of the strength and support provided by the school leadership in terms of technology use, individual teachers and their beliefs regarding mathematics, technology, teaching, and learning are important in determining the ongoing success of technological initiatives. For example, teachers who genuinely believe in the power of technology to transform representations of mathematics are more likely to investigate the affordances of new technologies as they become available. Such teachers are happy to take pedagogical risks that often represent the stages of technology integration described by Niess et al. (2009) in terms of allowing time to explore and experiment with new technologies. Others who may have low self-efficacy regarding technology may be less likely to persist and commit to ongoing professional development and the use of technology within their mathematics classrooms. The provision of tailored, in situ professional development along with a school culture that promotes teacher collaboration can assist all teachers to develop the confidence to experiment and explore the use of technology in mathematics teaching. In summary, effective technology use for the teaching of mathematics requires commitment in the form of: • • • •

Support (technical and instructional) Individual beliefs (mathematics, technology, teaching, and learning) Teacher self-efficacy with technology Willingness to innovate

Technology Integration Pyramid 129

What does this mean for school leaders and teachers? The base of the TIP(M) (Figure 8.2) outlines four categories of influences that can affect the degree to which technology integration can occur within mathematics classrooms. Some of these influences can be controlled from within the school by school leaders and teachers, but others are either controlled externally or cannot be controlled (location and socio-economic status). The diagram in Figure 8.3 illustrates four levels of stakeholders who, in different ways, can change the influences on technology use in mathematics classrooms. For example, external stakeholders could be government bodies and curriculum authorities, media, or technology developers. Figure 8.3 illustrates the amount of control that each level of stakeholder has in relation to the influences. For example, teachers do have significant control over school culture and commitment to technology use the ways they collaborate with colleagues and through their beliefs and practices. Teachers can also work with school leaders to influence the school community to promote innovative practices. It is important for educators to understand these influences and their level of control over them to assist in understanding the possibilities for technology integration within individual, unique contexts. When teachers gain clarity regarding their contextual affordances and constraints, they are then well-placed to begin planning for technology-infused mathematics instruction. We now move to the four major components to be considered when for technology use in mathematics classrooms.

Figure 8.3  Circles of Influence and Levels of Control on Technology Integration.

130  Technology Integration Pyramid

The pyramid sides: four considerations for effective technology use When planning for effective mathematics teaching with technology, there are four fundamental considerations that are necessarily inter-related. As recognised in the TPACK (technological pedagogical content knowledge) framework (Koehler & Mishra, 2009), teachers’ knowledge of the combination and interactions between mathematics, pedagogy, and technology is vital for teachers’ decision making. However, we argue that this knowledge must be able to be actioned with the explicit goal of improving student engagement. Likewise, we draw on the SAMR (substitution, augmentation, modification and redefinition) model as it provides a means of understanding the hierarchy of task design that teachers may use as their proficiency with technology develops. We draw on these existing frameworks, literature pertaining to effective mathematics teaching, and the findings from our ten case studies to construct the four sides of the TIP(M). In this section, we will examine these considerations: mathematics, pedagogy, tools, and engagement, in turn, outlining the key principles which can guide teachers’ decision making.

Mathematics In many of the case studies, the teachers expressed the view that the mathematics itself was the prime consideration when planning for technology use. In other words, they started by identifying the mathematics to be taught and then moved to considerations relating to pedagogy and technology, ensuring that the latter considerations supported the mathematics. All of the teachers agreed that technology could be used to support more than computation. In fact, they felt it could be used to teach almost any area of mathematics. However, we did notice some variations across the levels of schooling in relation to how technology was used to enhance the learning of mathematics. For example, in the early years classrooms, we noticed a heavy use of mathematicsbased apps, which provided visual representations of early mathematics concepts, enhancing the students’ conceptual understanding. In the primary years, we noticed less of an emphasis on mathematical representations via technology. Instead, the emphasis was on productivity-based apps that facilitated mathematical communication and reasoning and alternate assessments that allowed teachers better access to data on student learning. In the secondary classrooms, we noticed a combination of both approaches, with a heavier emphasis on the use of mathematical videos and dynamic technological tools that supported mathematical investigation. We argue that there is a need to strengthen the focus on using the affordances of technology to strengthen all aspects of mathematics, including the provision of alternate representations alongside the processes of problem-solving, reasoning, and communication. Regardless of the chosen technologies, identifying the mathematical purpose of any lesson is a key principle when teachers are planning instruction. Selecting

Technology Integration Pyramid 131

Figure 8.4  Mathematics (TIP(M)).

pedagogical approaches and technological tools that are focused on the identified mathematical purpose, along with clearly identified learning objectives, is critical (Figure 8.4). The affordances and constraints of each element of technology to be used should be considered in relation to the mathematical purpose of teaching individual task or lesson. For example, John D (Case Study 7, Chapter 6) developed a purpose-built tool to convey a graphical representation of complex numbers. Using this tool, he was able to cognitively engage students with abstract mathematics in a visual manner that supported their mathematical reasoning. This use of technology aligns with the SAMR level of redefinition, providing a task that would have been inconceivable without the use of technology (Puentedura, 2006). Conversely, all of the case study primary teachers demonstrated a judicious use of technology depending on the mathematical content to be taught. They often supplemented or replaced technology use with hands-on, concrete materials when the technology might constrain rather than support learning. In all cases, the teachers’ decisions were also made with student engagement in mind. That is, substantive engagement with the mathematics being taught rather than simply interacting with the technological tools. In summary, being clear about the mathematical purpose of any lesson is the essential starting point in planning for mathematics teaching and learning, and the following aspects of mathematics should guide lesson and task design: • • • •

Content: topics in isolation vs. connected Process: problem solving vs. fluency Representations: dynamic, making connections between Computation and/or higher-order thinking

132  Technology Integration Pyramid

Figure 8.5  Pedagogy.

Pedagogy Pedagogy is complex and is underpinned by the teachers’ philosophy and beliefs about mathematics education. Arguably, the inclusion of technology adds a further layer of complexity to teachers’ pedagogical decision making, depending on one’s philosophy, and where it sits on the spectrum between a social-constructivist, student-centred approach, and a behaviourist, teacher-­ centred approach (Figure 8.5). For example, if a teacher believes that mathematics is a series of computational processes for students to replicate and practice, then they are likely to plan technology-infused lessons using drill and practice apps or other tools that replicate traditional mathematics lessons. In cases such as this, technology renders the users as “app-dependent” (Gardner & Davis, 2013), and students are positioned in the role of mathematical consumers. On the other hand, when teachers have a connectionist approach (Askew, Brown, Rhodes, Wiliam, & Johnson, 1997), viewing mathematics as dynamic interconnected discipline, they are likely to plan lessons through a range of different technological tools that provide opportunities for inquiry, problem-solving and reasoning. This “app-enabled” stance positions the students as producers rather than consumers. Many of the case study teachers took advantages of the technological affordances of learning management systems as a pedagogical tool. We observed teachers using the LMS (learning management system) as a tool to differentiate learning for students of different abilities, as a communication tool for both students and parents, as a vehicle to support a flipped learning approach, and as a portal for learning resources. While not specifically designed for mathematics, these productivity apps did have a profound effect on how students were able to engage with and access mathematics. For example, in Ian’s classroom (Case Study 8, Chapter 6), student engagement was supported through

Technology Integration Pyramid 133 the differentiated learning materials available to students via the LMS. In Joel’s classroom (Case Study 10, Chapter 6), the students were able to receive feedback via recorded videos outside the timetabled mathematics lessons, allowing them to continue their learning anywhere, anytime. When considering technology use, teachers should reflect on how the affordances of the available devices can enhance teaching as separate to learning. For example, the use of specific apps that allow teachers to collect assessment data, can change how teachers operate, and how they respond to students’ needs. Collecting, assessing, and providing feedback to students’ work have traditionally been labour intensive, but technology can reduce administrative tasks to allow teachers more time to focus on modifying their teaching to better meet students’ needs. Pedagogical decisions can be impacted by the number of devices available within an individual school context (Guerrero, 2010). This also influences student grouping and task design. For example, in a BYOD classroom where all students have devices, a teacher has to decide whether to plan individual tasks or collaborative tasks. If the collaboration is important to the identified learning outcomes, then individual devices may impede the quality of the collaboration. On the other hand, if the technology is being used for assessment purposes, then it may be vital for students to have their own devices. Teachers who do not operate within a 1:1 program also need to consider the management of fewer devices. For example, teachers should consider whether the students should share one device amongst several, or whether the tasks should be rotated to allow small groups to have better access to devices. Group size is also a critical consideration. For example, when there are more than three students sharing a device, it may be challenging for all students to engage with the mathematics. In summary, pedagogical decisions regarding mathematics teaching with technology should encompass considerations of: • • • • • • • • •

Social constructivist vs. behaviourist Student-centred vs. teacher-centred Tech for teaching and/or learning Student as consumer/producer Differentiation Assessment Grouping Number of devices (shared or 1-1) Flipped

Tools It is reasonable to assume that the number and types of devices and software applications available in classrooms will continue to increase. This abundance of technology will require teachers to critically evaluate their options in order

134  Technology Integration Pyramid

Figure 8.6  Tools.

to ensure that decisions made result in improved learning and engagement with mathematics (Figure 8.6). While teachers often do not have choice in the number and type of devices in their classrooms, they are able to make informed pedagogical decisions based on the affordances and constraints of those devices. However, in some circumstances where BYOD programs allow students to bring in any device, using any operating platform, teachers’ decisions become more complex and require more technological knowledge. In schools without dedicated technical support, these decisions can be overwhelming for teachers. In order to cope with this situation, teachers need to maintain a clear focus on the affordances that they are seeking from the technology, regardless of the type of device. For example, in Ian’s classroom (Case 8, Chapter 6) where students did work on different devices, the key consideration was their access to learning materials within the LMS and this was enabled across both laptops, mobile devices, and via the classroom data projector. The dynamic software used by Ian was demonstrated via the data project and not reliant on individual student devices. While devices in a classroom may be outside the teacher’s control, they do generally have some control over the software or apps that are used or made available to students. Arguably, the software choices and how they are embedded within pedagogical practices are a significant factor in the effective use of technology. For example, a simple drill and practice app on its own can merely replicate the functions of a paper-based textbook. However, when that app is embedded in other pedagogical practices such as student reflection and collaborative participation, the value of the app is increased. An ordinary app can be made extraordinary through good pedagogy. In contrast, a high-quality app with multiple affordances can be used in an ineffective way when the affordances are not adequately recognised and pedagogically supported. For example, there are game-based apps that provide opportunities for teachers to differentiate

Technology Integration Pyramid 135 for differing student needs by allowing them to set the topic and level for each student, capturing student progress. When teachers use these apps in a non-differentiated manner, they will not necessarily enhance learning for any students. An extraordinary app can be rendered ordinary through poor pedagogy. Rebecca (Case Study 3, Chapter 4) demonstrated knowledge of the importance of software choice. Rebecca’s students each had unique learning difficulties, and as a result, Rebecca spent considerable time selecting apps that had the appropriate affordances to address these learning needs. In summary, when making decisions about technological tools for mathematics instruction, teachers should consider the following technical aspects: • • •

Devices (BYOD, number, type, affordances, constraints) Software (type, affordances) Administration (connectivity, updates, downloads)

Engagement Given that one of the major reasons we use technology in mathematics classrooms is to improve student engagement, it is important for teachers to make this goal explicit in their decision making and planning (Figure 8.7). Each dimension of engagement should be addressed; that is, cognitive, operative, and affective (students thinking and working hard and feeling good about learning mathematics). Tasks need to be challenging, interactive, and valued by students. The framework for engagement with mathematics (FEM) (Attard, 2014) details two separate yet inter-related elements that are fundamental for supporting student engagement: pedagogical relationships and pedagogical repertoires. In each of the case studies, the teachers’ technology-related practices resulted in improved pedagogical relationships. That is, they used the technology in ways

Figure 8.7  Engagement.

136  Technology Integration Pyramid that allowed them to recognise and address the differences between and abilities of their students. For example, in Sharon’s classroom (Case Study 1, Chapter 4), she used social media and other apps in two-way communication with parents to enhance her knowledge of her students. This knowledge allowed her to better meet individual learning needs and therefore promote student engagement. Similarly, across all primary and secondary case studies, positive pedagogical relationships were supported by strategies that included flipped learning approaches. These approaches enhanced student engagement by extending the teacher–student relationships through the creation of teacher-made videos, allowing communications beyond the confines of timetabled lessons, and providing personalised learning pathways. The many affordances accessed by the case study teachers resulted in new and innovative pedagogical repertoires. These repertoires, as detailed in the FEM (see Chapter 3), include the opportunities for substantive conversations, the provision of challenging tasks (see Jessica, Case 6, Chapter 5), the provision of student choice (see Ian, Case 8 and Joel, Case 10, Chapter 6), and relevant tasks that make explicit links to students’ lives (see Bec, Case 5, Chapter 5). In all cases, the technological affordances extended the teachers’ pedagogical repertoires through the redefinition of mathematical tasks and learning spaces, narrowing the home–school divide. In summary, student engagement should be a prominent concern in teachers’ decision making for technology use in mathematics classrooms. Engagement is more than simple on-task behavior and encompasses the following aspects: • • • • •

Operative engagement Affective engagement Cognitive engagement Develops positive pedagogical relationships Expands pedagogical repertoires

What does TIP(M) mean for mathematics educators? The TIP(M) provides a holistic, inter-connected model of the influences on technology use in schools and the related considerations that are required for effective mathematics teaching with technology. The base of the pyramid provides the starting point for understanding the technology landscape in any school. The four influences of context, community, culture, and commitment provide a lens through which to develop a nuanced understanding of the potential spaces for innovation within any school setting. Then, the four sides of the pyramid provide the focus for teacher decision making, ensuring that student engagement is paramount throughout this process. The congruency of the pyramid’s sides represents the equal importance of the four considerations (mathematics, pedagogy, tools, and engagement). The TIP(M) model as a whole can be considered as a metaphor for effective mathematics teaching with technology. The TIP(M) model acknowledges the

Technology Integration Pyramid 137 realities and diversities between classrooms, schools, and communities. In doing so, it enables school leaders and teachers to plan realistically so that any technological innovations will build on existing school practices. All parts of the pyramid are essential for providing teachers with a multi-faceted understanding of technology use in contemporary schools. Such understanding should equip teachers to teach mathematics effectively with any new technologies as they emerge, but with a deep understanding of the reality of each unique classroom and school context.

What does TIP(M) mean for research in mathematics education? We contend that the TIP(M) model incorporates and extends existing literature and frameworks on technology use for effective teaching in mathematics classrooms. While these frameworks have provided researchers with an understanding of the elements related to technology use for teaching, we argue that the TIP(M) grounds the research in the realities of contemporary classrooms, regardless of the setting. In doing so, it provides researchers with a systematic means of examining school settings and the influences that impact on them, so that successes or failures in technology use can be better understood and analysed. As Dylan Wiliam, the UK education researcher, often states in relation to education initiatives, “everything works somewhere, and nothing works everywhere”. In other words, school context matters. The TIP(M) model provides a means to address this concern through the influences detailed on its base which recognise the uniqueness of each school setting. The teaching considerations that form the sides of the pyramid are inextricably linked to the unique school setting addressed through the base of the pyramid and can only be truly understood from this perspective. The TIP(M) was developed through our analysis of ten unique case studies. It should be acknowledged that these case studies were all conducted in Australia, albeit in a diverse range of school settings. We encourage other researchers to use the model to build a greater bank of evidence in relation to technology-integrated practices, but evidence that acknowledges the complex realities of school-settings across the world.

References Askew, M., Brown, M., Rhodes, V., Wiliam, D., & Johnson, D. (1997, September). Effective teachers of numeracy in primary schools: Teachers’ beliefs, practices and pupils’ learning. Presented at the British Educational Research Association annual conference, University of York, York, England. Attard, C. (2014). “I don’t like it, I don’t love it, but I do it and I don’t mind”: Introducing a framework for engagement with mathematics. Curriculum Perspectives, 34, 1–14. Gardner, H., & Davis, K. (2013). The app generation. New Haven, CT: Yale University Press.

138  Technology Integration Pyramid Guerrero, S. (2010). Technological pedagogical content knowledge in the mathematics classroom. Journal of Digital Learning in Teacher Education, 26, 132–139. Koehler, M. J., & Mishra, P. (2009). What is technological pedagogical content knowledge? Contemporary Issues in Technology and Teacher Education, 9, 60–70. Niess, M. L., Ronau, R. N., Shafer, K. G., Driskell, S. O., Harper, S. R., Johnston, C., & Kersaint, G. (2009). Mathematics teacher TPACK standards and development model. Contemporary Issues in Technology and Teacher Education, 9, 4–24. Puentedura, R. (2006). SAMR: A brief introduction. Retrieved from http://www. hippasus.com/rrpweblog/archives/2013/10/02/SAMR_ABriefIntroduction.pdf Wenger, E. (2000). Communities of practice and social learning systems. Organization, 7(2), 225–246. doi:10.1177/135050840072002

Appendix 1 Case study 4: lesson documentation

Puzzling polygons Outcome MA2-15MG → manipulates, identifies, and sketches two-dimensional shapes, including special quadrilaterals, and describes their features - identify and describe two-dimensional shapes as either “regular” or “irregular” - identify regular shapes in a group that includes irregular shapes - explain the difference between regular and irregular two-dimensional shapes - name a shape, given a written or verbal description of its features Tuning in (10 minutes) Quick revision of what we know about polygons and their properties. Use Airdrop to give students the polygons map to upload into the Bluebot app. Getting involved: (20 minutes) Students will use the Beebots (or Bluebot software) to identify different polygons on the map. They draw a card from the pile and have to navigate their way to the correct polygon. They must record their answers on the answer sheet. Assessment/Reflecting on Learning: (15 minutes) Students use the Seesaw Activity to reflect on the lesson. A → Using the image provided, trace around the irregular polygons and put a cross on the regular polygons. B → Add a voice recording answering the following questions. 1 What are polygons? 2 How can we identify regular polygons? 3 How can we identify irregular polygons? 4 What is one thing you enjoyed about the lesson? Lesson evaluation Drawing the cards and recording on the sheet was too much. Next time I would just provide students with the recording sheet and have them locate and identify the polygons from there. Blue Bot software worked very well, some students found this easier than the physical Beebot as it was more precise and not influenced by external factors (students leaning on the map, crashing with other Beebots, etc.).

Appendix 2 Case study 5: lesson documentation

Lesson 1: triangles Goal/learning intention and success criteria Lesson overview Students use 20 matchsticks to create different triangles. Using 20 sticks, it is only possible to make eight different triangles. Students will learn that in any triangle, the sum of the lengths of any two sides must be greater than the length of the third side. Students will need to take photos of their triangles and prove that their edges are straight (using a ruler or edge of a book). Students will also need to use the voice over feature on Explain Everything to describe their processes and reasoning. Extension activity - Where do we see the triangle inequality theorem in action? Learning Intention - To find out: what makes a triangle a triangle? Success Criteria - Develop rules to describe what makes a successful triangle - Use Explain Everything effectively - Take pictures to show your working - Describing your reasons for your answers

Curriculum connections/maths KLAs • uses appropriate terminology to describe and, symbols to represent, mathematical ideas • selects and uses appropriate mental or written strategies, or technology, to solve problems • checks the accuracy of a statement and explains the reasoning used • manipulates, identifies, and sketches two-dimensional shapes, including special quadrilaterals, and describes their features.

Case study 5: lesson documentation 141 Content - Construct regular and irregular two-dimensional shapes from a variety of materials, e.g., cardboard, straws, and pattern blocks - Determine that a triangle cannot be constructed from three straws if the sum of the lengths of the two shorter straws is less than the length of the longest straw (Reasoning)

Problem/question 1 I made a triangle using exactly 20 sticks. The sticks were all joined together so there were no gaps. None of the sticks were broken. What might my triangle look like? 2 How many different triangles can you make with 20 sticks? How will you know if you have found all of them? 3 Can you explain why it is possible to make some triangles and why it is impossible to make other triangles?

How technology will be used 1 Kahoot as introductory game/warm up (2D Shapes) https://play.kahoot. it/#/k/41637bea-afef-410e-9dc3-05d7cb3ae5a6 2 Camera feature of iPad to document triangles made 3 App: Explain everything - to create a video that includes images and voice recording of reasoning 4 Google Drive: Students have a copy of the questions on our team drive so they can refer to them at their own pace 5 Google Classroom: Reflection question will be posted and students are to answer about one thing they have learnt

Teacher questions - Can you see the commutative property in action? - What is the longest side that can be made with 20 toothpicks? Why? - What is the shortest side? Why?

Assessment opportunities - Communication skills: Can students communicate their thinking? - R easoning: Can students explain the reasons behind their solutions and outcomes? - K nowledge of 2D Shapes and their properties (from the Kahoot and Explain Everything activity)

Lesson 2: Blueberry muffins Goal/learning intention and success criteria Lesson overview - Students will be applying skills of addition, subtraction, multiplication and division to adapt a recipe for blueberry muffins to sell at our school canteen. - They will need to increase quantities of ingredients, research the cost of ingredients, and decide on a price to sell the muffins at.

142  Case study 5: lesson documentation - Students will need to identify how they will make a profit by increasing their selling price to cover the cost of ingredients. - Students will be in three different groups and given differentiated recipes and instructions to cater to the needs of the class.

Learning intention - To apply knowledge of operations to a real-life problem—creating a new food item for the canteen to sell, with the intention of making money for the school Success Criteria - Accurately changing quantities of ingredients to make more muffins - Accurately adding up cost of ingredients and profit - Successful completion of the Google Sheets table - Posting table and images of any working out on Seesaw

Curriculum connections - uses appropriate terminology to describe, and symbols to represent, mathematical ideas - selects and uses appropriate mental or written strategies, or technology, to solve problems - checks the accuracy of a statement and explains the reasoning used - uses mental and written strategies for addition and subtraction involving two-, three-, four-, and five-digit numbers - uses mental and informal written strategies for multiplication and division

Problem/question After the success of the homemade cookies, the canteen has asked us to plan some new additions to their menu. We will be testing three different blueberry muffin recipes, comparing how much it costs to bake each, and seeing how much profit can be made.

How technology will be used - Google Sheets: Creating a table, students will make a copy of the template I have provided and fill it in - Google Drive: Differentiated instructions to cater to the needs in the classroom - Google Sheets: Pairs are able to both collaboratively work on their slideshow - Kahoot: Warm up - QR Codes: Easy access to Woolworths website - Seesaw: To share collaborative results and working out

Assessment opportunities - Observation of problem solving-skills - Finding out students’ knowledge of the four operations and whether they can apply them to real-life scenarios - Communication skills: Can students communicate their thinking?

Appendix 3 Case study 6: lesson documentation

Number patterns snap Learning intention: To find patterns and connections between numbers Words you will need to be familiar with: even number, odd number, less than, more than, multiple, factor square number, prime number, triangle number, square number Spinner Link: https://goo.gl/kTNmRR Steps to success 1 The teacher will spin the Spinner twice 2 Your group will need to find as many numbers as you can that match both number patterns (you may use your 100s chart to help you) 3 Upload your answer to the Padlet (link provided) a Score 1 point if you find a number that satisfies both spinner choices b Score 2 points if you think of the highest number that satisfies both choices c Score 3 points if you get a number that another group doesn’t get Impossible pairs! Sometimes there might not be any numbers that satisfy both statements! If this happens, you can replace one of the cards with a new one. Reflection questions • How many impossible pairs can you find? • Can you find a number that satisfies 3 cards? Or 4 cards? Or…? • What new number patterns would you add to the spinner?

Sphero races to 100! Learning intention: You are going to challenge yourself, your team, and Sphero in the ultimate race to 100, where you are identifying different number patterns and connections between numbers. Steps to success 1 Decide your groups 2 Group 1 spins the spinner twice and identifies a number that matches both conditions (if there is no match, you miss a turn!).

144  Case study 6: lesson documentation 3 This group moves Sphero from 0 to this number and earns points (see scoring table below). 4 Place a counter on this number as it is no longer available, Sphero must go around it. 5 Group 2 spins the spinner twice and identifies a number that matches both conditions. 6 This group moves Sphero from its current positions to the appropriate spot avoiding any counters in the way. 7 The game repeats until a group reaches exactly 100 points! Triangle numbers – 6 points Prime numbers – 5 points Square numbers – 6 points Composite numbers – 1 point Fibonacci numbers – 4 points Divisible by 5 – 2 points Divisible by 3 – 2 points Divisible by 12 – 4 points Numbers 1 and 100 – 0 points Reflection questions 1 2 3 4

What key knowledge did you need to know prior to completing this challenge? What knowledge did you learn throughout the challenge? What strategy did your group use? How did you know you were correct? Did Sphero help or hinder your strategy?

Home Learning Challenge: Design a challenge similar to this that your buddies could complete.

“Driverless Car prototype” Congratulations! You are applying for a job to be a software engineer at Tesla. Your role will be coding the routes that the Tesla cars have to drive. To apply for this job, you need to provide a prototype to prove your coding and mathematical ability! To prepare for this job interview your group will need to • Choose a real-life experience driving experience, for example, driving from school to (insert name of local café) • Design and draw a map to scale of your area that includes the roads • Design the code that Sphero will need to go • Take a time lapse of Sphero completing the challenge • Make any adjustments to your code • Design another route using the same map

Case study 6: lesson documentation 145 Things to think about and discuss in your group (record your answers in OneNote) 1 2 3 4

What would be the best “scale” to use? What is the total area that your map represents? Why is it important for the map to be accurate? What is the difference in speed that Sphero is going compared to an electric car? 5 Why is prototyping important? 6 How would this code for Sphero be converted into real time/real km? Sharing with the interview panel • Upload a screenshot of your code to the Padlet link • Turn your time lapse into a QR code and upload to the Padlet link Reflection on task This contained a hyperlink to a Google form that required students to respond to reflection prompts.

Index

Note: Page numbers in bold and italics refer to tables and figures, respectively. academic improvement measurement 32 adapting phase of technology integration 79 affective engagement 136 annual digital technology conference 52 apps: app-dependent aspect 4; app-­enabling aspect 4; based on gamification 120; contemporary education apps 6; enabled concept 24; generation 4; Google Apps 17; licensing 71–72 artificial intelligence 25 attitudes 70–71 augmentation 38 Australian Curriculum Assessment and Reporting Authority (ACAR A) 1, 5–6, 15 autism spectrum disorder (ASD) 49, 50 bandwidth 4 Beebot (app) 74, 75, 91, 139 blended learning 15, 24 BlueBot 74, 75, 139 Bring Your Own Device (BYOD) programs 3, 8, 16–19, 69, 73, 82, 101, 114, 115, 126; school, challenges and barriers 71–72 Bring Your Own Technology (BYOT) 17 Canvas (learning management system) 96, 97, 99 challenging tasks 42 choices, provision of 42 Class Dojo (app) 62

classroom: contemporary 33; culture 111; pedagogy 30 coaching program 52 cognitive engagement 136 collaborative decision making 114 commitment 12, 128; to innovative technology 113 community 126–127; approaches to technology use 114; attitudes 126; support of technology 111 Complexly tool 96 complex numbers 96 computation 80–81 conceptual dilemmas 20 connectionist orientation 32 constructive feedback 42 consultation, technology use 127 consumable apps 39 contemporary classrooms 33 contemporary digital technologies 2, 6–7, 32 contemporary education apps 6 content knowledge 71 context 125–126 continuous interactions (CI) 41, 42, 55 cultural dilemmas 20 culture 127 curriculum-based teaching and learning 33 default “technocentric” approach 34 De Moivre’s theorem. 103 design thinking 20 Desmos 22, 94, 95, 109 digital cameras 2, 73 digital capabilities 18 digital champions 51–52 digital divide 3, 4–5, 18, 71

Index 147 digital immigrants 3; teachers as 70–71 digital natives 3, 18; teachers as 70–71 digital revolution 32 digital technologies 30, 72, 112; app-­ enabling 4; assessment 60–63; challenges 2; contemporary 6–7; curriculum 82; education 82; integration frameworks 33–39 dilemmas 20–21 disability 5 disruptive pedagogy 2 distribute learning resources 125 diversity and culture 70–71 division 76–77 Early Learning STEM Australia (ELSA) project 52 Early Years Learning Framework (EYLF) 53 Echo learning management system 95, 119 Edmodo (collaboration platform) 86, 90 educational landscape 3 educators 125 Educreations 13 ELSA pattern app 55–56 embed technology 24 engagement 33, 135–136 Excel 109; see also PowerPoint experience, represent, apply (ER A) framework 53 Explain Everything (app) 13, 16, 80 extension of self 74 external community 126 EYLF See Early Years Learning Framework (EYLF) Facebook 51 FEM see Framework for Engagement with Mathematics (FEM) financial mathematics 95–96 flipped learning approaches 119; classrooms 14–15, 44; resources 125 flipped videos 117 formal school policies 114 fractions 62–63 Framework for Engagement with Mathematics (FEM) 41, 42, 50, 75, 79, 94, 104 “game-based learning” (GBL) 13 gamification 13; of learning 120–121

Gardner, H. 4, 24 GeoGebra 14, 22, 94, 100, 109 geographical location 5 Google 54 Google Apps 17 Google Classroom 15, 51, 70, 80 Google Communities 51 Google Drive 60, 79 Google Maps 73, 77, 90 Google Sheets 74, 91 Google Slideshow 81 Google Team Drive 79, 81 Graduate Certificate of Primary Mathematics 83 group activities 54 grouping numbers 76–77 high-needs classrooms 64–65 home–school connections 78, 84, 136 iBooks 119 immediate school community 127 inconsistencies 115 individual student needs 41 innovative mathematics teaching 113 interactive whiteboards (IWB) 58, 59, 77, 101, 103, 125; see also specific IWBs International Baccalaureate Primary Years Program (IB PYP) framework 83 investigation-based lessons 30–31 iPad mini 84 iPads 55–57, 74, 81, 114, 119; Department of Education restrictions 71; one-to-one iPad ratio 91; in primary mathematics classrooms 14, 43; supervision 54; for use at school and at home 69–70 IWB see interactive whiteboards (IWB) Kahoot (app) 13, 74, 77, 79–80 K-2 classrooms and teachers 71 Khan Academy 14 knowledge and participation 29 K-12 schools 81–82 leadership 113, 115 learning: and academic performance 13; blended 15, 24; flipped 14–15, 44, 119, 125; game-based 13; passive 14; screencasting 13–14; variety of 42 learning management systems (LMS) 15, 38, 95, 117, 120, 126, 128

148  Index learning spaces, redesign 6 lesson documentation: number patterns 143–145; polygons 139; triangles 140–142 Letter School app 56–57 lifelong learning habits 29 literature as screencasting 13 LMS see learning management systems (LMS) Manga High 13 mathematical content 55–56, 56–57 mathematical reasoning 5; and communication 116 mathematical thinking 13; and reasoning 1 Mathematics and Technology Attitudes Scale (MTAS) 43–44 mathematics digital resources 13 mathematics pedagogy 15, 30, 33, 71 mathematics teaching and student engagement: definition 40–42; effective teaching in mathematics 30–33; integration of digital technologies 33–39; technology and engagement with mathematics 43–44 mathematics with technology: contemporary technologies and their affordances 6–7; digital divide 4–5; mathematics education, technology for 5–6; research 7–8; students 3–4 Mathletics 13 Maths Jump 120 Matific (app) 13, 15 mentor students 29 Microsoft Hololens 105 Microsoft Online 17 Minecraft 63, 105 mobile learning (‘m’learning) 6 mobile technology 2 model responsible global citizenship 29 modification 38 Motivation and Engagement Framework 40 multi-categorical support (MC) 59 multiplication 76–77 Nelson Maths Hundreds Charts (app) 61, 62, 63 New Media Consortium/Consortium for School Networking (NMC/CoSN) 29

New Tech Network 113 Niess, M. L. 37, 72, 79, 113, 128 Number Patterns Snap lesson 86–87 numbers: complex 96; financial mathematics 95–96; fractions 62–63; grouping 76–77; patterns 86–87 Office 365 82 OneNote 15, 84, 85, 87, 91, 97, 99, 106, 107 one-to-one device ratio 108 on-line games 22 open-ended type tasks 32 operative engagement 136 Organisation for Economic Cooperation and Development (OECD) 1 organised professional learning opportunities 114 Padlet (app) 85, 87, 90, 95 parent community, communication 51 passive learning 14 patterns: making 55–56; numbers 86–87 pedagogical content knowledge (PCK) 30, 42 pedagogical decision-making process 117 pedagogical dilemmas 20 pedagogical knowledge (PK) 34 pedagogical relationships 41–42, 42, 57, 136 pedagogical repertoires 41–42, 42, 136 pedagogical risks 128 pedagogical tool in mathematics 24 pedagogy of mathematics 2, 132–133 physical education (PE) 102 Plicker (app) 60, 61, 62, 77 political dilemmas 20 polygons, attributes of 74–76 position 74 PowerPoint 54, 80, 81; see also Excel pre-existing knowledge (PK) 41–42 pre-prepared lesson materials 14 pre-school: building community 51; classroom 51; culture through professional development 51–52; school context 50; whole-school commitment to technology use 50–51 primary and secondary teacher 23 primary-based applications 15

Index 149 primary mathematics classrooms, technology and 69–70; BYOD school, challenges and barriers 71–72; digital immigrants 70–71; digital natives 70–71; mathematical practices 72–73; school context 69–70; years 3, 69–81; years 6, 81–90 primary teachers’ beliefs 116 problem solving 14, 30–31, 87–89 process-oriented approach 1, 15, 23 Prodigy (app) 13, 15, 79, 120 productivity apps 16 professional development model 113 professional learning 112, 121; and technology 83 proficiencies 15 Programme for International Student Assessment (PISA) 2 project-based learning 93 pro-technology schools 70 QR Codes 74, 76, 81 randomized questioning 117 redefinition 38 Rekenrek number talk 63 relevant tasks 42 research about student learning 29 resistance to innovation 2 rewards 13 risk task 89–90 robotics 120 SAMR (substitution, augmentation, modification, and redefinition) framework 83, 96 scepticism 126 school culture 127; and technology use 113 school–home connections 16 school leaders 112, 123–124; challenges faced by leadership 114–115; community approaches 114; influences of technology use and 129; school commitment to technology use 112–114; and teachers 129 school leadership and infrastructure 24 school wi-fi 121 science, technology, engineering, and mathematics (STEM) 1, 25 science, technology, engineering, art, and mathematics (STEAM) 84 Scratch (programming platform) 96

screencasting 13–14, 16 secondary mathematics classrooms 73, 120 secondary teachers’ beliefs 116 SeeSaw (digital portfolio) 70, 76, 91 self-efficacy 19, 128 ShowMe 13 SLD see specific learning disabilities (SLD) SmartScreen 54 social constructivist teaching approach 2, 31, 32 socioeconomic status 5 Solstice (app) 86 specific learning disabilities (SLD) 49 Sphero 90 staff attrition 114–115 stakeholder expectations 4 Starfall (game) 54 student-centred approach 7, 36 student engagement and mathematics: defining 40–42; expectations 40; framework for 42; student achievement 39; technology and 43–44 students: digital technologies and 3–4; disengagement 5, 69; influence of technology on 119–121; learning 29; numeracy experiences 53; thinking 23 Students, Computers and Learning Making the Connection (OECD report) 2 Studyladder (game) 64–65 substantive conversation 42 substantive student engagement 41 substitution 38 substitution, augmentation, modification, and redefinition (SAMR) 30, 38–39, 61, 113 support class 57–59, 63–66 supportive leadership 121 Surface Pro device 97 tasks: challenging 42; relevant 42; rich 89–90; variety of 42 TCK see technological content knowledge (TCK) teachers: awareness 42; beliefs 19, 36, 37, 116; collaboration 29; as digital immigrants 70–71; as digital natives 70–71; engagement 118–119; formal or informal leadership roles 112; formal policies and expectations 113;

150  Index influences of technology use and 129; pedagogical decisions 117–118; positivity towards technology 22; professional learning 112–113; and pupil 29; training 29; use of technology 15–16; views, in mathematics classrooms 21–25; views on technology use 21–25 teacher-centred approach 30–31, 36 teacher-centred didactic 7 teaching: connectionist orientation to 32; and learning mathematics 15; and learning with technology 19; of numeracy 32 teaching mathematics with technology: BYOD programs 16–19; challenges and barriers in 19–21; flipped classrooms 14–15; future 25; gamification 13; screencasting 13–14; teachers’ views, in mathematics classrooms 21–25 teaching staff, turnover in 115 technological content knowledge (TCK) 34 technological knowledge (TK) 34 technological pedagogical content knowledge (TPACK) model 30, 34–35, 42, 72, 88; components 35; and mathematics 35–38, 84 technological pedagogical knowledge (TPK) 34 technologies 50; for assessment 78; disruptive pedagogy 2; as double-edged sword 4–5; as extension of self 39, 89; as master 38–39; for mathematics education 5–6, 73–74, 83; as partner 38–39; roles 38–39; as servant 38–39; student-centered 42; and their affordances 6–7 technology-enhanced mathematics education 2, 58–59, 117, 126 technology-enriched lessons 125; with limited technology 59–60 technology in early years mathematics classrooms: case studies 2 and 3: year 1/2, support class 57–66; case study 1: pre-school 49–57; in support class 63 Technology Information Matrix 30 technology-infused mathematics 121; teaching 126 technology-infused teaching and learning framework 113

technology in secondary classroom: complex numbers 96; 3D design 107; De Moivre’s theorem 103; dynamics (vectors) 107; financial mathematics 95–96; fractions 99–101; measurement (volume) 103–105; school context 93–94, 96–98; structure 107; trigonometry 107; years 7–10, 96–101; years 9–12, 93–96, 101–105; years 10–12 105–107 technology-integrated instruction 20 technology-integrated mathematics teaching 30 technology integration 19 technology integration matrix (TIM) 39 Technology Integration Pyramid (Mathematics) [TIP(M)] 124; base 124; considerations for effective technology use 130–136 (engagement 135–136; mathematics 130–131; pedagogy 132–133; tools 133–135); future-proofing technology-infused teaching and learning 123; influences on technology use 123–129 (commitment 128; community 126–127; context 125–126; culture 127; school leaders and teachers 129, 129); for mathematics educators 136–137; for research in mathematics education 137; student engagement with mathematics 123 technology profile of school 58 technology-related pedagogical decisions 118–119 technology-related practices 50, 51, 52, 58, 114, 126 technology-rich environment 125 technology transforming mathematical practices 72–73 technology use in mathematics classrooms: influence on students 119–121; influences on and of technology use 121–122; school leaders (challenges faced by leadership 114–115; community approaches 114; school commitment to technology use 112–114); teacher beliefs 116; teacher engagement 118–119; teachers’ pedagogical decisions 117–118

Index 151 textbook-based approach 7 3D objects 74, 103–104 TIM see technology integration matrix (TIM) topic-by-topic approach 15 Top Marks (game) 54 touch screen monitor 55–57 TPACK model see technological pedagogical content knowledge (TPACK) model TPK see technological pedagogical knowledge (TPK) traditional mathematics assessments 16 traditional pedagogies 4 traditional teacher-centred approach 31 traditional teaching practices 1 triangle inequalities 79–80

turn-around technology integration pedagogy and planning (TTIPP) model 37–38 unproductive classroom time 14–15 video lectures 14 volume 103–105 Wheeldecide.com 86 whole-class activities 54 whole-school pedagogical approach 113 Windows 10, 97 Wootube 95 YouTube 53–54, 56–57, 64, 94, 95, 103, 120