Chapter 14: From Artifacts to Instruments: A Theoretical Framework Behind the Orchestra Metaphor

The large-scale distribution of PCs and handheld devices made software for use in mathematics education available to both students and teachers. Currently, programming languages, graphing software, spreadsheets, geometry software, computer algebra systems, and other kinds of new tools for the learning of mathematics are widely disseminated. Originally, optimism dominated the debate: Technology would free the student from calculation and procedural drudgery, and would enable mathematics education to focus on more relevant issues such as realistic applications, modeling, conceptual understanding, and higher-order skills. An—often implicit—underlying idea was that technical skills and conceptual understanding could be separated in the learning.

At present, the optimism has taken on additional nuances. The research survey of Lagrange, Artigue, Laborde, and Trouche (2003) indicates that difficulties arising while using technology for learning mathematics have gained considerable attention. These difficulties on the one hand recognize the complexity of teaching and learning in general, but on the other hand reveal the subtlety of using tools for educational purposes. For example, Drijvers (2002) addresses obstacles that students encountered while working in a computer algebra environment. Balacheff (1994) sees computational transposition as part of the complexity of using computerized environments. He describes computational transposition as the “work on knowledge which offers a symbolic representation and the implementation of this representation on a computer-based device” (p. 16). Artigue (1997) brings to light two phenomena linked to this process, the phenomenon of pseudotransparency, linked to the gap between what a student writes on the keyboard and what appears on the screen, and the phenomenon of double reference. The latter refers to the double interpretation that students and teachers may have of tasks. Whereas teachers want the task to address the mathematical concepts involved, students may perceive the task as one of finding the typical way in which the computerized learning environment deals with these concepts and represents them. Techniques that are used within the computer algebra environment differ from the traditional paper-and-pencil techniques (Lagrange, 2005), a phenomenon that again may lead to conceptual difficulties. As an example of the non-trivial character of the use of technological tools for mathematics, we refer to an example presented by Guin and Trouche (1999). Students were asked to answer the question: Does the function f, defined by f(x) = ln(x) + 10 • sin(x), have an infinite limit as x tends to +∞? The answers depended strongly on the working environment (even though elementary theorems make it possible to answer “yes” to this question). In a non-CAS graphing calculator environment, 25% of students answered “no,” appealing to the oscillation of the observed graphic representation (Figure 14.1); in a paper-and-pencil environment, only 5% of students answered “no.”