Solving Rate of Change Tasks with a Graphing Calculator: a Case Study on Instrumental Genesis (original) (raw)

Graphing calculator supported instrumentation schemes for the concept of derivative: a case study

2014

This paper reports on the role of the graphing calculator (GC) in the learning of derivatives and instantaneous rate of change. In a longitudinal study, we administered task based interviews before and after the introduction of calculus. We analyzed students’ use of the GC in these interviews. This paper reports on the case of one student, Andy, who is a resilient user of the GC while he develops into a flexible solver of problems on instantaneous rate of change. His case demonstrates that, although the GC is meant to promote the integration of symbolical, graphical and numerical techniques, it can facilitate a learning process in which symbolical techniques develop separately from other techniques.

Learning about Functions with a Geometrical and Symbolic Software Environment: a Study of Students' Instrumental Genesis along Two Years

atcm.mathandtech.org

Software learning environments, especially those offering extended multi representational capabilities, are more and more complex. That is why researchers are now sensitive to the process of instrumental genesis that transforms this kind of artefact into an instrument for students' mathematical work. The study reported here deals with Casyopée, a geometrical and symbolic learning environment dedicated to functions at upper secondary level. The same students have been observed along their 11th and 12th grade. Learning situations have been designed with the teacher, an experienced user of Casyopée. Consistent with the curriculum, these situations aimed at approaching functions by modelling geometrical dependencies, a task for which Casyopée offers special capabilities. The observation tried to capture how students developed together their use of the software and their mathematical knowledge. Although the situations had been prepared with students' instrumental genesis in mind, the observation in the first year was relatively deceiving: students had little initiative and did not identify clearly key functionalities of the software in relationship with aspects of functions. In the second year, there was a clear improvement: while students used Casyopée more freely, they developed a flexible understanding of functions, associating the objects in the software to mathematical representations. The study suggests that such an instrumental genesis can be a real attainment, but needs to be achieved as a long term process.

Construction of mathematical knowledge using graphic calculators (CAS) in the mathematics classroom

International Journal of Mathematical Education in Science and Technology, 2011

Our research project aimed at understanding the complexity of the construction of knowledge in a CAS environment. Basing our work on the French instrumental approach, in particular the Task-Technique-Theory (T-T-T) theoretical frame as adapted from Chevallard's Anthropological Theory of Didactics, we were mindful that a careful task design process was needed in order to promote in students rich and meaningful learning. In this paper, we explore further conjecture that the learning of techniques can foster conceptual understanding by investigating at close range the taskbased activity of a pair of 10th grade students-activity that illustrates the ways in which the use of symbolic calculators along with appropriate tasks can stimulate the emergence of epistemic actions within technique-oriented algebraic activity.

Teaching derivative with graphic calculators: The role of a representational perspective

2002

There is now a considerable literature on the value of an approach to learning mathematics which incorporates multiple representations of concepts. A widespread application of this approach will require the full support and appreciation of teachers, along with suitable professional development. However, it is not entirely clear yet how assisting classroom teachers to build the theoretical ideas into successful teaching approaches relates to their content knowledge and pedagogical content knowledge. In our current research project we are working with a group of teachers using graphic and CAS calculators in the classroom teaching of processes and concepts associated with derivative. This paper describes the way in which interviews and classroom observation confirm how the representation-based content knowledge and pedagogical content knowledge of derivative of one teacher influenced her use of the calculator and her teaching of derivative.

Teaching and learning elementary calculus concepts with a graphics calculator

The availability of ICT offers opportunities to reconsider teaching and learning in the calculus curriculum. In this paper, some potential contributions of one form of ICT, the hand-held graphics calculator, are described and evaluated. Although algebraic calculators, graphics calculators with symbolic manipulation capabilities, have been available for some years now, attention in the paper is restricted to calculators without algebraic capabilities. These are more likely to be available on a wide scale in many East Asian countries than are algebraic calculators, and significant experience with them in schools has now accumulated. The paper will consider a number of key calculus concepts, such as the derivative of a function at a point, the derivative function, continuity, asymptotic behaviour, convergence, limits, integration and differential equations, to develop the argument that ICT offers an opportunity to help students understand the concepts behind the calculus, upon which st...

Graphing Calculators and Students' Conceptions of the Derivative

Perceptual and Motor Skills, 2002

Sut?zmary.-This study compared the effect of using graphing calculators on college students' concept~~al understanding of the derivative with the traditional teaching approach in Calculus I course. Students (49 men and 52 women) in four classes at a large public university parcicipated. O n a posrtest, women who were taught calculus using the graphing calculator had signiFicantly higher scores than those taught by the traditional method.

From Artifact to Instrument: Mathematics Teaching Mediated by Symbolic Calculators

2003

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

The Introduction of Calculus in 12th Grade: The Role of Artefacts

International Group for the Psychology of Mathematics Education, 2004

The paper concerns the analysis of the role of artefacts and instruments in approaching calculus by graphic-symbolic calculator at high school level. We focus on an element of the introduction of calculus: the global/local game. We discus the hypothesis that the zoom-controls of calculator support the production of gestures and metaphors that foster the shift from a global to a local point of view. The analysis of protocols confirms that the exploration of several functions through the zooming process was supported by gestures and language. They appeared during the zooming process and in pupils' answers to the tests when the calculator was no more available and when the task concerned mathematical objects.