Luc Trouche

The Complex Process of Converting Tools into Mathematical Instruments: The Case of Calculators

Transforming any tool into a mathematical instrument for students involves a complex ‘instrumentation’ process and does not necessarily lead to better mathematical understanding. Analysis of the constraints and potential of the artefact are necessary in order to point out the mathematical knowledge involved in using a calculator. Results of this analysis have an influence on the design of problem situations. Observations of students using graphic and symbolic calculators were analysed and categorised into profiles, illustrating that transforming the calculator into an efficient mathematical instrument varies from student to student, a factor which has to be included in the teaching process.

Managing the Complexity of Human/Machine Interactions in Computerized Learning Environments: Guiding Students' Command Process through Instrumental Orchestrations.

After an introduction which addresses some basic questions, this article is organized around three points: (1) The theoretical framework of the so-called “instrumental approach” which has been a theme in the last two CAME symposia; (2) A consideration of two processes (instrumentalization and instrumentation) which interact in the instrumental genesis; and (3) The introduction of the idea of instrumental orchestration as a way of allowing the teacher to assist the student’s instrumental genesis.

Technology and Mathematics Education: A Multidimensional Study of the Evolution of Research and Innovation

This chapter will highlight the interest and necessity of considering a plurality of perspectives (or dimensions) when addressing the issue of the integration of information and communication technologies (JCT) into the teaching and learning of mathematics. It will also show how this multidimensional perspective can be efficient for an analysis of the existing literature.

Instrumental Genesis, Individual and Social Aspects

In Chapter 6, we analyzed didactic phenomena occurring during experiments in integrating symbolic calculators. We then showed how adopting an instrumental approach to analyzing these phenomena helped in understanding the influence of such tools upon mathematical activity and upon knowledge building. It is during the process of instrumental genesis that a calculator becomes a mathematical instrument.

From artifact to instrument: mathematics teaching mediated by symbolic calculators

The evolution of calculation tools available for the learning of mathematics has been quick and profound. After the first illusions on a naturally positive integration of these tools, new theoretical approaches have emerged. They take into account individual and social processes of the mathematical instrument construction from a given artifact. In this article we show how analyzing constraints of the tool allows the understanding of its influence on the knowledge construction. We propose the concept of instrumental orchestration to design different devices which may be built in class and thus strengthen the socialized part of the instrumental genesis: instrumental orchestration is defined by objectives, configuration and exploitation modes. It acts at the same time on the artifact, on the subject, on the relationship the subject has with the artifact and on the way the subject considers this relation.

An Instrumental Approach to Mathematics Learning in Symbolic Calculator Environments

A rapid technological evolution (Chapter 1), linked to profound changes within the professional field of mathematics (Chapter 3), brings into question the place of techniques in mathematics teaching (Chapter 5). These changes have created serious difficulties for teachers; obliged to question their professional practices, they make different choices regarding integration of new technologies and techniques (Chapter 4), choices that are linked to their mathematical conceptions and to their teaching styles.

Calculators in Mathematics Education: A Rapid Evolution of Tools, with Differential Effects

The appearance of more and more complex tools in mathematics classes is not a response to an institutional need of school. It is, rather, the expression within this institution of a huge social phenomenon (the increase in the number of screens and machines) arising from the utilization of computerized tools by certain branches of mathematics and science.

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.

From Text to 'Lived' Resources

Nowadays, mathematics teachers are confronted with a myriad of resources that are available through technological means and through the internet in particular. However, teachers may perceive difficulties in orchestrating education which makes use of technological tools and resources. With instrumental orchestration as a theoretical lens, the main question addressed in this chapter is into which types of orchestrations teachers transform the technological resources. In the setting of a pilot teaching sequence in grade 12 on using applets for practicing algebraic skills, this question is investigated, through a case study of one participating mathematics teacher, through questionnaires among the 69 participating teachers, and through interviews with six of them. The results show that teachers privilege orchestrations in which students work individually or in pairs, at the cost of whole-class orchestration types. Within the performance of these student-centered orchestrations, we recognize elements already known from previously identified orchestrations. Compared to their regular teaching practices and their expectations before the pilot, the involvement in the pilot causes teachers to adapt their orchestrations during the pilot.