Mathematics Education in the Digital Era Alison Clark-Wilson Ornella Robutti Nathalie Sinclair   Editors The Mathematics Teacher in the Digital Era International Research on Professional Learning and Practice Second Edition Mathematics Education in the Digital Era Volume 16 Series Editors Dragana Martinovic, University of Windsor, Windsor, ON, Canada Viktor Freiman, Faculté des sciences de l’éducation, Université de Moncton,  Moncton, NB, Canada Editorial Board Members Marcelo Borba, State University of São Paulo, São Paulo, Brazil Rosa Maria Bottino, CNR – Istituto Tecnologie Didattiche, Genova, Italy Paul Drijvers, Utrecht University, Utrecht, The Netherlands Celia Hoyles, University of London, London, UK Zekeriya Karadag, Giresun Üniversitesi, Giresun, Turkey Stephen Lerman, London South Bank University, London, UK Richard Lesh, Indiana University, Bloomington, USA Allen Leung, Hong Kong Baptist University, Kowloon Tong, Hong Kong Tom Lowrie, University of Canberra, Bruce, Australia John Mason, The Open University, Buckinghamshire, UK Sergey Pozdnyakov, Saint Petersburg Electrotechnical University,  Saint Petersburg, Russia Ornella Robutti, Dipartimento di Matematica, Università di Torino, Torino, Italy Anna Sfard, University of Haifa, Haifa, Israel Bharath Sriraman, University of Montana, Missoula, USA Eleonora Faggiano, University of Bari Aldo Moro, Bari, Italy The Mathematics Education in the Digital Era (MEDE) series explores ways in which digital technologies support mathematics teaching and the learning of Net Gen’ers, paying attention also to educational debates. Each volume will address one specific issue in mathematics education (e.g., visual mathematics and cyber- learning; inclusive and community based e-learning; teaching in the digital era), in an attempt to explore fundamental assumptions about teaching and learning mathematics in the presence of digital technologies. This series aims to attract diverse readers including researchers in mathematics education, mathematicians, cognitive scientists and computer scientists, graduate students in education, policy- makers, educational software developers, administrators and teacher-practitioners. Among other things, the high-quality scientific work published in this series will address questions related to the suitability of pedagogies and digital technologies for new generations of mathematics students. The series will also provide readers with deeper insight into how innovative teaching and assessment practices emerge, make their way into the classroom, and shape the learning of young students who have grown up with technology. The series will also look at how to bridge theory and practice to enhance the different learning styles of today’s students and turn their motivation and natural interest in technology into an additional support for meaningful mathematics learning. The series provides the opportunity for the dissemination of findings that address the effects of digital technologies on learning outcomes and their integration into effective teaching practices; the potential of mathematics educational software for the transformation of instruction and curricula; and the power of the e-learning of mathematics, as inclusive and community-based, yet personalized and hands-on. Submit your proposal: Please contact the Series Editors, Dragana Martinovic ([email protected]) and Viktor Freiman ([email protected]) as well as the Publishing Editor, Marianna Georgouli ([email protected]). Forthcoming volume: • The Evolution of Research on Teaching Mathematics: A. Manizade, N. Buchholtz, K. Beswick (Eds.) Alison Clark-Wilson • Ornella Robutti Nathalie Sinclair Editors The Mathematics Teacher in the Digital Era International Research on Professional Learning and Practice Second Edition Editors Alison Clark-Wilson Ornella Robutti UCL Institute of Education Dipartimento di Matematica University College London Università di Torino London, UK Torino, Italy Nathalie Sinclair Faculty of Education Simon Fraser University Burnaby, BC, Canada This work contains media enhancements, which are displayed with a “play” icon. Material in the print book can be viewed on a mobile device by downloading the Springer Nature “More Media” app available in the major app stores. The media enhancements in the online version of the work can be accessed directly by authorized users. ISSN 2211-8136 ISSN 2211-8144 (electronic) Mathematics Education in the Digital Era ISBN 978-3-031-05253-8 ISBN 978-3-031-05254-5 (eBook) https://doi.org/10.1007/978-3-031-05254-5 1st edition: © Springer Science+Business Media Dordrecht 2014 2nd edition: © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. 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This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Introduction The eight intervening years between this second edition of The Mathematics Teacher in the Digital Era and the first edition have seen increased attention on the role of the teacher within technology-enhanced educational contexts, leading to a more developed understanding of the components of related teacher education pro- grammes and initiatives for both pre- and in-service teachers. The shock to the edu- cation system caused by the global coronavirus pandemic simultaneously highlighted the key role that teachers and lecturers play in the nurturing of generations of learn- ers, alongside increased global attention to the role that (educational) technology plays as a mediator of teaching and learning. Studies that have taken place during the pandemic have provided insights into how teachers’ practices have had to evolve, whilst also highlighting theoretical and methodological gaps in our under- standing of the relatively new phenomena of “hybrid”, “at distance” or “remote” teaching in school and university settings (Bretscher et al., 2021; Clark-Wilson et al., 2021; Crisan et al., 2021; Drijvers et al., 2021; Maciejewski, 2021). As we reflect on the academic impacts of the first edition of the book, the chap- ters within have offered theoretical constructs and methodological approaches, which have provided other researchers in the field with research tools that are con- tinuing to advance our collective understandings of the field. In this second edition, we invited all of the authors who had contributed to the first edition to submit new research that evidenced advances in their experiences, knowledge and practices. We also invited new authors, whose research had emerged in the intervening years, to offer new critical perspectives that broaden the international commentary, with con- tributions from Argentina, Australia, Canada, France, Germany, Hong Kong, Iceland, Italy, Mexico, Turkey and the United Kingdom. vii viii Introduction A Journey Through the Text The evolution of the research on technology in mathematics education has enabled a more nuanced understanding of the teacher’s perspective to take account of their trajectories of development from pre-service contexts through to in-service prac- tices over time. Hence, we have chosen to loosely organise the text body in accor- dance with teachers’ trajectories of experience with technology use. These experiences concern those within: university undergraduate courses as learners of mathematics; university-based pre-service teacher education courses; university- based teacher education courses and research projects with in-service teachers as participants. We begin with chapters by Thurm, Ebers and Barzel, and Bozkurt and Koyunkaya that address more practical considerations regarding the provision of support and training for both in-service and pre-service teachers of mathematics. The growth of large-scale, online professional development initiatives aimed at teachers has resulted in new research that seeks to develop theoretical understand- ing of the design and impact of such initiatives alongside the development of appro- priate methodologies to inform both aspects. The chapter by Thurm, Ebers and Barzel addresses aspects of the design of professional development for mathematics teachers in Germany with a particular focus on the role of the professional develop- ment facilitators within a regional professional development programme for 30 par- ticipants who are all such facilitators. The programme was conducted online (due to the Covid-19 pandemic) and Thurm and colleagues’ findings focus on the impact of a module of the programme that supported participants’ understanding (and use) of video-based case studies of mathematics teaching that embed multi-representational technology. They use Prediger, Roesken-Winter and Leuders’ Three-Tetrahedron Model as a framework to highlight the complexities of PD design that has a class- room level, teacher PD level and facilitators’ PD level (Prediger et al., 2019). Their findings, which highlight aspects of facilitators’ noticing, emphasise the need for carefully structured prompts to support the analysis of video-based activities that serve the dual needs of the facilitators and the teachers with whom they are working. A pre-service teacher education context in Turkey is the subject of the qualitative action research reported by Bozkurt and Koyunkaya in which they study the impacts of a redesigned practicum course informed by the Instrumental Orchestration model (Drijvers et al., 2010; Trouche, 2004). The course design emphasises the pre-service mathematics teachers’ (PSTs, n=4) developing use of a dynamic mathematics soft- ware (GeoGebra) from the university setting (through micro teaching to their peers) as their practices move to school classrooms. Their study adopts a cyclical research method that draws on data from the PSTs’ lesson plans, supported by analyses of their teaching and associated interviews. The research findings offer insights into how the PSTs initially overlooked the exploitation modes for the technology in their planning but became more systematic in their approach through both the processes of micro teaching and during the practicum itself. Given that many pre-service pro- grammes stop short of requiring PSTs to apply their learning about mathematical Introduction ix technologies within authentic teaching situations, this chapter provides valuable insights on the design decisions taken by the teacher educators to develop such an approach. The majority of the remaining chapters in the book report studies that involve in-service teachers as participants within a range of research settings, each with a different focus. We order these chapters according to teachers’ trajectories of devel- opment with novel to them technologies. We adopt this phrase from Ng and Leung (Chap. 10) as it better reflects our experience and expectation that it is not possible for all teachers to be cognisant of all available (and educationally relevant) tech- nologies at any point in time, irrespective of how mature the wider community considers these technologies to be. The study by Bakos explores how a novel multi-touch tablet technology, TouchTimes, is used by two primary teachers in British Columbia, Canada, through a lens that considers the teacher, the tool and the mathematical concept as an ensem- ble. Rooted in the instrumental approach, and in particular Haspekian’s elaboration of double instrumental genesis (2011, 2014), Bakos uses her case studies to reveal three new orchestration types alongside sharing insights on how the agency exerted by the tool extends our existing understandings of the nature of multiplication, and the role of haptic devices within young children’s development. Ng, Liang and Leung’s study also focuses on a more novel technology, 3D pens, which enable 3-dimensional models to be drawn as physical objects. The 3D pen warms and extrudes a plastic filament to produce a model that then hardens as it cools. Ng, Liang and Leung’s method adopts the use of video-aided reflection with a group of four in-service secondary school teachers in Hong Kong to support their realisations of the affordances of such technologies as a potential teaching tool. In their findings, Ng, Liang and Leung provide evidence for how the videos operate as a boundary object between the teachers and researchers in the study (Robutti et al., 2019). Although the concept of silent animated films to show mathematical concepts dates back to the early twentieth century and was further developed in the 1950s by Nicolet, the design-based research developed by Kristinsdóttir examines aspects of their design and use in her case study in an upper secondary mathematics classroom in Iceland. Kristinsdóttir describes silent videos as short (< 2 min) videos that do not pose a mathematical problem to be solved but rather invite the viewer to wonder, to experience dynamically changing mathematical objects such that they might dis- cover something new or consolidate previous thoughts about the mathematics shown in the video. Each associated silent video task invites students to work in pairs to prepare and record a voice-over for the video clip, which is then shared with the class during a whole-class discussion that is led by the teacher. Framed by a lens that focuses on the formative assessment dimension of such discussions, Kristinsdóttir adapted Schoenfeld’s Teaching for Robust Understanding framework (2018) to identify opportunities and challenges associated with such discussions. McAlindon, Ball and Chang’s study also explores an innovative technology- enhanced pedagogic approach, the flipped classroom, through a case study involving an experienced teacher in an Australian secondary school. Defining the flipped x Introduction classroom as one in which the activities that would normally be conducted in the classroom are flipped with those that would normally be conducted as homework, they explore their case study teacher’s experiences and perceptions of a first imple- mentation for the teaching of linear equations. This exploratory study, which involves the teacher making qualitative comparisons with a parallel class that she taught using her traditional approach, concludes positive outcomes such as improved student engagement and improved formative assessment practices. Although the design pro- cess for the teacher requires new technology skills and is time consuming, the authors offer some guidelines to inform professional development initiatives that have the goal to support mathematics teachers’ flipped classroom pedagogies. Gueudet, Besnier, Bueno-Ravel and Poisard extend earlier research that featured in the first edition of the book, which shone a theoretical lens on teachers’ classroom practices at the kindergarten level from a Documentational Approach to Didactics perspective (Gueudet et al., 2014). In the intervening years, evolutions of this theory and its associated research methods have enabled the authors to consider a kinder- garten teacher’s development as evidenced by both one of her documents (a micro view) and the encompassing resource system (a macro view). The authors conclude that both the micro and macro views are necessary to fully appreciate a teacher’s design capacity within the context of long-term professional development concern- ing digital technologies for education. Staying in France, Abboud-Blanchard and Vanderbrouck report findings from a study in France that explores the implementation of tablet computers in the French primary school setting. Although tablets are no longer widely considered a new technology, the authors’ contribution extends ideas reported in the first edition of the book, which concludes three axes (cognitive, pragmatic and temporal) through which to consider teachers’ adoption of new technologies within their mathematics classrooms (Abboud-Blanchard, 2014). Abboud-Blanchard and Vanderbrouck introduce the additional constructs of tensions and proximities, which they argue align more specifically to classroom uses of tablet computers. In their chapter, the authors articulate how these two new constructs evolve from Activity Theory, and elaborations of Vygotsky’s and Valsiner’s respective Zone Theories. Sandoval and Trigueros’ chapter is also situated in a primary school setting, this time in Mexico. They offer new perspectives on the teaching of mathematics in primary schools, with an emphasis on how two teachers integrate digital technolo- gies to particularly meet the needs of learners from challenging socio-economic contexts. In common with their contribution to the first edition of the book (Trigueros et al., 2014), they adopt an enactivist approach to characterise teachers’ actions and the resulting student activities that reveal high levels of participation in immersive environments for learners who are commonly disenfranchised by education systems. We move from primary school contexts to the secondary phase in the next two chapters, which both follow teachers over a period of time with the aim to identify aspects of their evolving practices. The first, by Simsek, Bretscher, Clark-Wilson and Hoyles, is situated in England and focuses on three in-service teachers’ evolv- ing use of a dynamic mathematical technology (Cornerstone Maths) for the teach- ing of geometric similarity to 11–14 year olds over a period of months. The chapter