John Monaghan

Tools and Mathematics

Preface.- 1.Introduction to the book.- 2.Doing mathematics with tools: one task, four tools.-3.The life of modern Homo Habilis Mathematicus: Experimental computation and visual theorems.- 4.Tools, human development and mathematics .- 5.The development of mathematics practices in the Mesopotamian scribal schools.- 6.Discussions of Part A chapters.- 7. Developments relevant to the use of tools in mathematics.- 8.Constructionism.- 9.Activity theoretic approaches.- 10.Didactics of mathematics: concepts, roots, interactions and dynamics from France.- 11.Discussion of issues in chapters in Part B.- 12.Integrating tools as an ordinary component of the curriculum in mathematics education.- 13.The calculator debate.- 14. Tools and mathematics in the real world.- 15.Mathematics teachers and digital tools.- 16. Tasks and digital tools.- 17.Games: artefacts in gameplay.- 18.Connectivity in mathematics education: Drawing some lessons from the current experiences and questioning the future of the concept.- Epilogue.

Mathematics Teachers and Digital Tools

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 resources. We end with a review of the current state of understanding and an agenda for future research.

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.

Tasks and Digital Tools

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.

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.

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.

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 authors to ‘provide answers’.

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.

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.

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.