Mathematics teachers and digital tools

International audience; This chapter considers mathematics teachers’ appropriation and classroom use of digital tools. The first section considers teachers—who are they, how are they conceived in the literature and what aspects of teachers have been studied? The second section examines twenty-first century research on mathematics teachers using digital tools. This sheds light on the complexity of mathematics teachers’ appropriation and classroom use of digital tools but what we find is that our focus is too narrow and we need to consider digital tools within the range of resources use in planning and realising their lessons, which leads us to the third section, mathematics teachers using re…

Working with Teachers: Context and Culture

This chapter concerns collaborations between teacher educators and teachers in activities involving digital technologies in the teaching and learning of mathematics. In light of the complexity involved in introducing new artefacts into existing cultures of practices, we focus on our attempts to develop ways of working with teachers so that they can become active participants in designing practices and routines appropriate for the particularities of their own classrooms. Three case studies are presented, from three different countries, Norway, Greece and Brazil, each of which describes the participation of teachers in a process of communal design of mathematical tools and activities. Two the…

Tools, Human Development and Mathematics

This chapter raises a number of issues from pre-history and history that one mathematics educator considers ‘worthy of mention’ with regard to tools and mathematics. These issues are: tool use in the development of the human species (phylogenesis); tool use in a mathematical culture, ancient Greek mathematics that goes beyond the obvious tools; an example from ancient Indian mathematics that bears some resemblances to Jon’s experimental mathematics described in Chap. 3; the mutual support of hand, mind and artefact in expert use of an abacus; a consideration of a period (sixteenth-century Europe) where there was a rapid advance in the development of mathematical tools.

Tools and Mathematics in the Real World

This chapter considers, with special regard to tool use, mathematics in out-of-school practices and attempts to replicate these practices in school mathematics. Both foci are important and problematic issues in mathematics education. This chapter has four sections. The two central sections address the two main foci. The opening section sets the scene with an historical account of ways that mathematics has been subdivided with regard to its application(s). The last section considers problem issues.

Discussions of Part I Chapters

This chapter is an opportunity for one of the authors of this book to question the other two authors in the light of issues raised in Chaps. 2– 5. It constitutes both a follow-up to discussions between authors which occurred over the writing process, and emergent issues—new discussions once the book was almost complete. Some fundamental issues are addressed, about the birth of mathematics (and its deep links with the birth of writing), the relationships between mathematics and other sciences, the interactions between conjecture and proof, and the role of visualisation and of gestures. The text is kept short in order to provoke the readers to reflect on these issues rather than for the autho…

Developments Relevant to the Use of Tools in Mathematics

This chapter explores developments in mathematics, computing, mathematics education and scholarship relevant to understanding tools from 1960 to the time of writing. This exploration is biased in accentuating influences relevant to tools and mathematics education. The chapter presents a broad landscape and focuses on selected technological advances, ideas and people that are considered important. The chapter begins with a section charting developments in mathematics, computing and education followed by a section on intellectual trends relevant to understanding tools and tool use. The final section focuses on the development of ideas in mathematics education regarding tools and tool use.

Mathematics and Non-School Gameplay

This chapter investigates the mathematics in the gameplay of three popular games (Angry Birds, Plants vs. Zombies and The Sims) that are unlikely to be played in mathematics lessons. The three games are different but each has been observed to provide opportunity for mathematical activity in gameplay. After describing each game, and the mathematics that can arise in gameplay, the chapter explores two questions: What kind of mathematics is afforded in these games? Can these games be used in/for school mathematics? Issues considered under the first question include: the nature of mathematics and the difficulty of isolating the mathematics in non-school gameplay; players’ strategic actions as m…

Technology in Mathematics Teaching

This chapter introduces the chapter of the book, in situating it in a trajectory of two researchers.

Encouraging students’ problem posing through importing visual images into mathematical software

The Limit Notion at Three Educational Levels in Three Countries

AbstractThis paper documents how the limit concept is treated in high school, at a university and in teacher education in England, France and Sweden. To this end we make use of vignettes, data-grounded accounts of the situation at the three levels in the three countries. These are analysed using the Anthropological Theory of the Didactic (ATD). While university praxeologies are relatively similar across the three countries, greater differences manifest themselves in high school and teacher education. For instance, at the high school level, in France a local praxeology on the limits of sequences is taught, which is not the case in England or Sweden. Results from the analysis of limits are ex…

Doing Mathematics with Tools: One Task, Four Tools

This chapter illustrates a variety of mathematical and educational issues arising from doing a single task with different tools. One task is considered, bisect an angle. The chapter has four sections, each devoted to issues in using one tool to complete this task: a straight edge and compass; a protractor; a dynamic geometry system; and a book.

Kindergarten teachers’ orchestration of mathematical activities afforded by technology: agency and mediation

This paper focuses on kindergarten teachers’ interactions with young children during mathematical learning activities involving the use of digital tools. We aim to characterise the teachers’ roles and actions in these activities and extend considerations of teachers’ orchestrations current in the research literature with regard to agency and mediation. Our analysis of teacher-children-digital tool interaction reveals that the kindergarten teachers took three roles in their work with young children, which we call Assistant, Mediator and Teacher roles. These roles were used interchangeably and purposefully by the kindergarten teachers. With regard to agency and mediation, we argue that agency…

Themes within lecturers' views on the teaching of linear algebra

Author's accepted manuscript (postprint). This is an Accepted Manuscript of an article published by Taylor & Francis in International Journal of Mathematical Education in Science and Technology on 25/09/2019, available online: http://www.tandfonline.com/10.1080/0020739X.2019.1668976. Available from 26/09/2020. This paper reports on themes that arose in an investigation of university lecturers’ views on the teaching of linear algebra. This focus on themes was the initial part of a study concentrating on four areas: What is important to teach in a first course in linear algebra? Are there teaching methods which are particularly suited for such a course? Are there tools that should/should not …

Discussion of Issues in Chapters in Part II

This chapter is a second opportunity for one of the authors of this book to question the other authors (and a guest, Richard Noss) about matters raised in Chaps. 7– 10. It is designed as a series of questions from John, and Jon, Luc and Richard were free to answer (or not) as they deemed appropriate. The chapter ends with a short review.

Introduction to the Book

This chapter sets the scene for the book. The three sections, respectively: state the purpose and scope of the book; present two attempts at answering the question ‘what is a tool?’; and outline the structure of the book.

Perspectives and reflections on teaching linear algebra

Abstract This paper presents ‘expert opinions’ on what should be taught in a first-year linear algebra course at university; the aim is to gain a generic picture and general guiding principles for such a course. Drawing on a Delphi method, 14 university professors—called ‘experts’ in this study—addressed the following questions: What should be on a first-year linear algebra undergraduate course for engineering and/or mathematics students? How could such courses be taught? What tools (if any) are essential to these two groups of students? The results of the investigation, these experts’ opinions, mainly concern what should be in a linear algebra course (e.g. problem-solving and applications)…

The Calculator Debate

The ‘calculator debate’ refers to an (often heated) exchange of ideas, on the ‘proper’ use of specific forms of hand-held technology in mathematics instruction and the assessment of learning, that has been ongoing for decades. This chapter considers issues in this debate in three sections. The first section positions ‘the calculator’ within ‘portable hand-held computational technology’ and reviews calculator use, the research literature on the use of the calculator and the ‘calculator debate’ itself. The second section considers the calculator with regard to Wertsch’s (Mind as action, Oxford, England, 1998) ten properties of mediated action. The last section speculates on a possible future …

Activity Theoretic Approaches

Activity theory is an approach to the study of human practices—including mathematical and educational practices—in which mediation, including mediation by artefacts/tools, is a central construct. The chapter is of four sections. The first section provides an overview of AT. Section 9.2 traces early influences of AT in mathematics education research. Section 9.3 considers foci of a set of mathematics education papers recent at the time of writing. Section 9.4 explores emphases and tensions in papers considered in Sects. 9.2 and 9.3.

Games: Artefacts in Gameplay

This chapter reviews the past and looks to the future of the potential for games and gameplay to provide opportunity for engaging in mathematical activity. This review a glimpse into a possible future is conducted with a specific focus on the role of artefacts in gameplay. The chapter is in four sections. The first section considers the range of games; the second section considers artefacts in games and gameplay; the third section addresses games in mathematics education; and the final section looks to future development.

Tool use in mathematics: A framework

International audience; In the course of research into the interpretation of tools in the didactics of mathematics I found both voids and conflicts. This paper presents the results of my research and a resultant statement on tool use in mathematics education. The statement incorporates constructs from several theoretical frameworks and I consider the consistency of my statement on tool use with regard to activity theory.

Tasks and Digital Tools

International audience; This chapter considers scholastic tasks with digital tools. The first two sections consider tasks in ‘ordinary’ classrooms (tasks for learning) and issues relating to tasks using mathematical software. The first section presents examples of tasks with digital tools to highlight potential problems and opportunities for learning. The second section considers issues arising from the literature on tasks design with and without digital tools. The final section looks at task-tool issues in larger-than-the-individual classroom research and in assessment; it also comments of avenues for further development