Monaghan J., Trouche L., Borwein J.M. Tools and Mathematics: Instruments for learning

Springer International Publishing, Switzerland, 2016. — 497 p. — (Mathematics Education Library 110). — ISBN: 9783319023953, 978-3-319-02396-0This book is an exploration of tools and mathematics and issues in mathematics education related to tool use. The book has five parts. The first part reflects on doing a mathematical task with different tools, followed by a mathematician's account of tool use in his work. The second considers prehistory and history: tools in the development from ape to human; tools and mathematics in the ancient world; tools for calculating; and tools in mathematics instruction. The third part opens with a broad review of technology and intellectual trends, circa 1970, and continues with three case studies of approaches in mathematics education and the place of tools in these approaches. The fourth part considers issues related to mathematics instructions: curriculum, assessment and policy; the calculator debate; mathematics in the real world; and teachers' use of technology. The final part looks to the future: task and tool design and new forms of activity via connectivity and computer games.
This book originates in discussions at conferences between John and Luc. We (meaning John and Luc at the moment) had similar backgrounds: a number of years in high school mathematics teaching and then university work in mathematics education. In both sites we’d done a lot of experimenting (both practical and research) using digital technology. We shared prejudices for the types of tools/systems we liked to use with our students: graph plotters, Logo, computer algebra systems and similar tools/systems which our students could use to express mathematical relationships rather than tools/systems that might go under the name of ‘computer-aided instruction’. When we first met, in the later 1990s, we’d read papers of the other where we’d being trying to ‘get a theoretical handle’ on what went on in high school mathematics classrooms when teachers and students used these tools/systems. Our theoretical frameworks were not identical (John was influenced by activity theory and Luc by instrumentation theory), but there was plenty of room for productive dialogue; we were both also influenced by reading papers of and talking to Miche`le Artigue and Jean-baptiste Lagrange who were, in turn, influenced by Yves Chevallard’s anthropological theory of the didactic and Brousseau’s theory of didactical situations.Setting the Scene
Introduction to the Book
Doing Mathematics with Tools: One Task, Four Tools
The Life of Modern Homo Habilis Mathematicus: Experimental Computation and Visual Theorems
Tools, Human Development and Mathematics
The Development of Mathematics Practices in the Mesopotamian Scribal Schools.
Discussions of Part I Chapters
The Development and Interpretation of Tools
Developments Relevant to the Use of Tools in Mathematics
Constructionism
Activity Theoretic Approaches
Didactics of Mathematics: Concepts, Roots, Interactions and Dynamics from France
Discussion of Issues in Chapters in Part II
Selected Issues with Regard to Tool Use in Mathematics
Integrating Tools as an Ordinary Component of the Curriculum in Mathematics Education
The Calculator Debate
Tools and Mathematics in the Real World
Mathematics Teachers and Digital Tools
The Future?
Tasks and Digital Tools
Games: Artefacts in Gameplay
Connectivity in Mathematics Education: Drawing Some Lessons from the Current Experiences and Questioning the Future of the Concept