Running head: Artifacts to Instruments 1 Chapter 23: From Artifacts to Instruments: A Theoretical Framework Behind the Orchestra Metaphor Paul Drijvers Freudenthal Institute, Utrecht University, The Netherlands Luc Trouche LIRDEF, LIRMM & IREM, Université Montpellier II, France 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
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Running head: Artifacts to Instruments 1
Chapter 23: From Artifacts to Instruments:A Theoretical Framework Behind the Orchestra Metaphor
Paul Drijvers
Freudenthal Institute, Utrecht University, The Netherlands
Luc Trouche
LIRDEF, LIRMM & IREM, Université Montpellier II, France
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, & 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)
Running head: Artifacts to Instruments 2
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 pseudo-transparency, 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
(in Guin, Ruthven, & Trouche, 2004)), 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 )sin(10)ln()( xxxf ⋅+= , 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 1); in a paper-and-pencil environment,
only 5% of students answered “no.”
Insert Figure 1 about here.
Running head: Artifacts to Instruments 3
Apparently, the use of cognitive technological tools–in the sense of Lajoie (1993)–for the
learning of mathematics, such as applets, graphing calculators, geometry software, and computer
algebra systems, is not as easy as it might seem. Current research on the integration of
technology in mathematics education, which aims at taking into account the complexity of the
issue, uses a variety of perspectives, such as psychological, didactical and socio-cultural
perspectives. However, the articulation of the different perspectives, and their integration into a
more comprehensive framework is missing. Therefore, we look for a theoretical approach that
allows for:
1. An analysis of the learning process in technological environments of increasing
complexity, which takes into account the non-trivial character of using
technological tools and goes beyond simplistic views on “leaving work to the
tool”;
2. A set of cues for the organization of the teaching in such environments,
concerning both pedagogical resources and classroom settings; and
3. A trajectory for conception, development, and evolution of pedagogical resources
for teachers’ professional development as well as software, and more generally,
computerized learning environments.
Furthermore, such a framework should have predictable as well as explanatory power,
help to organize our thinking, and be applicable to a broad range of phenomena, to cite the
criteria suggested by Arnon and Dubinsky (2001). Our main claim in this chapter is that the
instrumental approach to using technology in mathematics education is a promising “candidate”
for such a comprehensive framework.
Running head: Artifacts to Instruments 4
The aim of this chapter is to present the main ideas of the instrumental approach, to apply
it to the use of technology in mathematics education, to illustrate it by means of some examples,
and to discuss its merits and limitations. Although the examples mainly concern the use of
graphing calculators and computer algebra, we do want to stress the more general character of
the instrumental approach, and that its scope goes beyond these specific kinds of technology. For
example, the instrumental approach has been applied recently to using spreadsheets (Haspekian,
1995; Nemirovsky, 1994; Roth & Tobin, 1997). These approaches stress the dialectic relation
between symbolizing and development of meaning in increasing levels of formalism. This
Running head: Artifacts to Instruments 16
relation may be problematic if the artifact imposes specific symbolizations. For example,
computer algebra environments offer limited possibilities for the student to develop individual
informal symbolizations and related meanings in a process of “symbolic genesis.” Meanwhile,
instrumental genesis includes a signification process of giving meaning to algebraic objects and
procedures. Therefore, we think that an articulation of the instrumental approach with the
perspective of symbolizing, both within and apart from the technological environment, might be
fruitful.
So far, we examined the instrumental genesis of utilization schemes as an individual
process. Different students may develop different schemes for the same type of task, or for using
a similar command in the technological environment. However, instrumental genesis also has a
social dimension. The students develop mental schemes in the context of the classroom
community, in which the guidance of the teacher is one of the factors. The next section,
therefore, addresses the social perspective of the instrumental approach.
The Instrumental Approach: The Conductor’s Teaching Perspective
In contrast to the individual learning perspective of the previous section, this section takes
a more collective, classroom-oriented teaching view. Its main goal is to address criterion 2 in the
introduction, and to illustrate the complexity of the role of the teacher when the use of
technology is an integrated part of his/her teaching.
From a Set to an Orchestra of Instruments
We agree with the point of view of Hollebrands, Laborde and Straeser (this volume,
2005): “We do not take the learner as an isolated individual facing the word, but take the learner
Running head: Artifacts to Instruments 17
as deeply embedded in his/her environment which is highly structured and defines the ways the
individual is learning” (page number will be needed in the final copy). The way the environment,
within the classroom, is structured hardly depends on the teacher. As stated by Zbiek and
Hollebrands (2005), “teachers play a central role in the technology-based mathematics learning
experiences of children of all ages” (page number will be needed in the final copy). The role of a
teacher, in such an environment, is rather complex, because s/he always has to manage a set of
instruments, from a two points of view:
Each student builds a set of instruments for her/himself (for example, in a CAS
environment, an instrument for solving equations, an instrument for studying
function behavior, etc.); and
In a classroom seen as a community of practice (Wenger, 1998), mobilized
instruments are built by each student for each task; these instruments are not
necessarily the same: we have shown (Trouche, in Guin et al., 2004) that the more
complex the environment, the greater the diversity of the instruments.
Therefore the question is: How can the teacher help each student as well as the class as a
whole to articulate or fine-tune these sets of instruments, namely, to build coherent systems of
instruments, working as an orchestra, with each student building his or her own orchestra?
Asking this question means conceiving of instrumental geneses as individual as well as social
processes. As a result, utilization schemes also acquire the character of social schemes: “schemes
are elaborated and shared in communities of practice and may give rise to an appropriation by
subjects, or even result from explicit training processes” (Rabardel & Samurçay, 2001, p. 20).
We do insist here on one aspect of the answer to the question above, often a blind spot in
research studies: the constitution of systems of instruments strongly depends on the organization
Running head: Artifacts to Instruments 18
of the artifactual environment the teacher establishes. In order to describe this organization, we
have introduced the notion of instrumental orchestration1 (Trouche, 2004). An instrumental
orchestration is the intentional and systematic organization of the various artifacts available in a
computerized learning environment by the teacher for a given mathematical situation, in order to
guide students’ instrumental genesis. An instrumental orchestration is defined by
didactic configurations (i.e., arrangements of the artifactual environment, according to
various stages of the mathematical situation); and
exploitation modes of these configurations.
Example: Configuration and Exploitation Modes in a Calculator Environment
Designing a configuration first depends on the given technological environment. For
example, in a calculator environment, the small screen of this kind of artifact particularly raises
the issue of the socialization of students’ actions and productions. There is a particular artifact–a
viewscreen or a data projector–which allows one to project the calculator’s small screen onto a
big screen, which the entire class can see. This device is probably designed to project the
teacher’s calculator screen (the cable linking this artifact to a calculator is rather short and the
connection requires a special plug on the calculator). Although in some instances students’
calculators have this special plug, the plug often is reserved for the teacher’s calculator.
In the “Artifact and Instrument” section we explained that a subject, while using an
artifact, always transforms it through a process of instrumentalization. This transformation can
sometimes be in directions that are unplanned by the designer. This is all the more true in a
teaching environment: teachers have to organize the use of artifacts according to their
1 The word “orchestration” is quite natural when speaking of a set of instruments, in the sense of “the art to put in action various sonorities of the collective instrument which one names orchestra by means of infinitely varying combinations” (Lavignac, French musicographer, 1900).
Running head: Artifacts to Instruments 19
pedagogical goals. We have thus presented (Trouche, 2004) a configuration integrating the
viewscreen and students’ calculators (instead of teacher’s calculator) with the main objective of
socializing–to a certain extent–students’ instrumental genesis.
This configuration (Figure 6) rests on the devolution of a particular role to one student:
this student, called the sherpa-student2, handles the overhead-projected calculator. This
configuration has several advantages:
It favors the collective management of a part of the instrumentation and
instrumentalization processes: what a student does with her/his calculator–traces
of her/his activity–is seen by all and can be the subject of classroom discussions.
The teacher can guide, through the student’s calculator, the calculators of the whole
class (the teacher does not perform the instrumented gesture but checks how it is
performed by the sherpa-student). The teacher thus fulfils the functions of an
orchestra conductor rather than a one-person band.3
For his/her teaching, the teacher can combine paper-and-pencil results obtained on the
blackboard, and results obtained by the sherpa-student’s calculator on the class
screen. For the student, this facilitates the combination of paper-and-pencil work
and calculator work at his or her own desk, as well as the articulation of the
different sets of instruments.
It favors debates within the class and the elucidation of procedures: the existence of
another point of reference distinct from the teacher’s allows new relationships to
2 On the one hand, the word sherpa refers to the person who guides and who carries the load during expeditions in the Himalaya, and on the other hand, to diplomats who prepare international conferences.
3 This advantage is not a minor one. Teachers, in complex technological environments, are strongly prone to perform alone all mathematical and technical tasks linked to the problem solving in the class.
Running head: Artifacts to Instruments 20
develop between the students in the class and the teacher as well as between this
sherpa-student and the teacher–concerning a mathematical result, a conjecture, a
gesture or a technique–and it can give the teacher means through which to
reintegrate lower-achieving students into the class. The sherpa-student function
actually gives lower-achieving students a different status and forces the teacher to
tune his/her teaching procedures with the work of the student who is supposed to
follow her/his guidelines. Follow-up of the work by this student shown on the big
work-screen allows very fast feedback from both teacher and class.
Insert Figure 6 about here.
Several exploitation modes of this structure may be considered. The teacher may organize
work phases of different kinds:
Sometimes calculators are shut off (and so is the overhead projector); it is then a
matter of work in a paper-and-pencil environment.
Sometimes calculators as well as the overhead projector are on and work is strictly
guided by the sherpa-student under the guidance of the teacher (students are
supposed to have exactly the same thing on their calculator screens as is on the big
screen in front of the class). Instrumentation and instrumentalization processes are
then strongly constrained.
Sometimes calculators are on as well as the overhead projector and work is free over a
given time. Instrumentation and instrumentalization processes are then relatively
constrained (by the type of activities and by referring to the sherpa-student’s
calculator which remains visible on the big screen) and sometimes calculators are
Running head: Artifacts to Instruments 21
on and the projector is off. Instrumentation and instrumentalization processes are
then only weakly constrained.
These various modes seem to illustrate what Healy (2002) named filling out and filling
in,4 during classroom social interaction: when the sherpa-student’s initiative is free, it is possible
for mathematically significant issues to arise out of the student’s own constructive efforts (this is
a filling out approach); when the teacher guides the sherpa-student, it is possible for
mathematically significant issues to be appropriated during students’own constructive efforts
(filling in approach).
Other questions must also be answered: will the same student play the role of the sherpa-
student during the whole lesson or, depending on the results, should another student’s calculator
be connected to the projector? Do all students have to play this role in turn or must only some of
them be privileged?
In the frame of this configuration, teachers and students play new roles. The sherpa-
student can be considered, for both class and teacher, as a reference, a guide, an auxiliary or a
mediator; the function of orchestra conductor, for the teacher, combines the various roles
pinpointed by Zbiek and Hollebrands (2005): technical assistant, resource, catalyst and
facilitator, explainer, task setter, counselor, collaborator, evaluator, planner and conductor,
allocator of time, and manager. The predominant role, for the sherpa-student as well for the
teacher and for the other students, strongly depends on the exploitation modes chosen for this
4 Healy (2002) identified a major difference between instructional theories drawing from constructivist perspectives and those guided by sociocultural ideologies, which related to the primacy assigned to the individual or the cultural in the learning process. Constructivist approaches emphasize a filling-outwards (FO) flow in which personal understandings are moved gradually towards institutionalized knowledge. A reverse filling-inwards (FI) flow of instruction described in sociocultural accounts stresses moving from institutionalized knowledge to connect with learners’ understandings. Teaching interventions in Healy’s study were therefore designed to allow investigation of these two different instructional approaches: the FO approach aimed to encourage the development of general mathematical models from learners’ activities; and the FI approach intended to support learners in appropriating general mathematical models previously introduced.
Running head: Artifacts to Instruments 22
configuration. Following the orchestra metaphor, the relationships between musicians and
between conductor and musicians are not the same in a jazz band and in a symphonic orchestra.
In the same environment, different configurations can be conceived, based on different
relationships between students, teacher, and artifacts. More generally, we (Trouche, in Guin et
al., 2004) gave examples of configurations at several levels: the level of the artifacts itself
(concerning internal arrangement of software itself), the level of the instruments (as the sherpa-
student configuration), and the meta-level of the relationship a subject maintains with an
instrument (aiming to develop self analysis of subjects’ activity).
Orchestration and Mathematical Situations
Following the metaphor once more, we can say that designing an orchestration obviously
requires a musical frame. Actually, an instrumental orchestration is to be designed related to both
a particular environment and a mathematical situation (Brousseau, 1997). The choice of the
situations is crucial: as stated by Rabardel (2001), “activity mediated by instruments is always
situated and situations have a determining influence on activity” (p. 18). For the case of a CAS
environment, Artigue (in Guin et al., 2004) gives several examples of mathematical situations
aiming “to manage jointly and coherently the development of both mathematical and
instrumental knowledge” (p. 233).
Chevallard (1992) distinguishes, within a computerized learning environment, three kinds
of elements, whose interaction is essential to successfully integrate artifacts in the teaching
process:
environment components: various artifacts (calculators, overhead projectors, teaching
software…), but also instructions for use, technical sheets,and so on;
mathematical situations; and
Running head: Artifacts to Instruments 23
didactic exploitation system: an essential element concerned with making relevant use
of the potential resources of a given environment and with achieving both the
coordination and integration of the environment components and the mathematical
situations.
Instrumental orchestrations can be positioned in this schema: a didactic exploitation
system can be described as a set of didactical exploitation scenarios (one for a given
environment and for each mathematical situation). A didactical exploitation scenario (Figure 7)
contains both the mathematical management of different stages of the situation and an
instrumental orchestration (with successive configurations and their exploitation modes,
according to the mathematical treatment and to the teacher’s pedagogical goals). From this
perspective, teachers have to build scenarios that are fitting for their personal teaching
environment and for the mathematical situations they want to introduce.
Insert Figure 7 about here.
Obviously, building such new pedagogical resources requires time and experience. We
agree with the conclusion of Zbiek and Hollebrands (2005): “If we give teachers mathematical
technology as nets, but provide no personal learning experiences and no support, we should not
be surprised when they prefer to catch their mathematical and pedagogical fish by hand” (the
page number for this quote will need to be provided by the editors in the final copy). The
question of how to generate personal experiences and how to support teachers will be addressed
in the next section.
Running head: Artifacts to Instruments 24
Instrumental Approach for Professional Development
In this section, we address the question of professional development-criterion 3 in the
introduction-, and particularly the evolution of professional practices in computerized learning
environments, from a single person’s band practice into an orchestra conductor practice. In a
sense, we will take a meta-perspective here by using the instrumental approach as a framework
for the learning of the teacher.
New Pedagogical Resources, New Teachers’ Communities of Practice
In a previous section, we stressed the crucial point of pedagogical resources helping the
teacher to organize the technological environment. For this purpose, the idea of conceiving usage
scenarios (Vivet, 1991) has proved particularly relevant: These scenarios consist of the
presentation of a unit with its objectives, student materials, and supporting notes for teachers to
help put the unit into practice. This idea acknowledges the necessity of taking into account the
available artifacts, the pedagogical organization of a class, and the role of the teacher. Such usage
scenarios may be considered as a first approach of didactical exploitation scenarios, as previously
described. Usage scenarios have also been developed for teachers wanting to produce teaching
units integrating dynamical geometry software (Laborde, 1999).
Using and, moreover, conceiving such scenarios requires teachers’ communities of
practice to exist or to be built. The idea of building an evolving network of teachers to develop
usage scenarios for geometry software was introduced in the United States (Allen, Wallace,
Cederberg, & Pearson, 1996). Similar training mechanisms have been developed around units
integrating a lesson presentation, a usage scenario, and reports of experimentations with these
units by a group of teachers in training, aiming at assisting management of the unit by the teacher
Running head: Artifacts to Instruments 25
and at promoting collaborative work both in the class around a scientific debate and within the
group of teachers (Guin, Delgoulet, & Salles, 2000).
This approach to organization has been extended through the use of a distance platform to
conduct both collaborative workshops aimed at providing pedagogical resources and continuous
long-term support for integration by teachers (Guin, Joab, & Trouche, 2003). The structure of
these pedagogical resources was devised with the aim of facilitating both their implementation in
classrooms and their evolution in response to teachers’ ideas, experiments and experiences. Thus
pedagogical resources evolve through usages in the classrooms and discussions in the community
of teachers (Figure 8). Such an approach aims at creating learning and training conditions for
teachers in which technological environments can provide spaces for discovery with flexible
tutorial assistance. It appears that this mechanism may help teachers to make the transition to
pedagogical action.
Insert Figure 8 about here.
An Instrumental Approach to Creating Pedagogical Resources
In fact, the process shown in Figure 8 is more complex than it might seem. It is not a
linear process, and what makes it interesting is precisely that not only the resources but also
teachers’ practices evolve through the process. The instrumental approach, as presented earlier,
affords a better description of such a process. In order to use this approach, we have to consider
not only one teacher and one resource, because the interaction in that case is quite limited: One
can not say that one resource deeply modifies one teacher’s practice. Moreover, there is not,
between a given teacher and a given resource, a cycle of interactions.
Running head: Artifacts to Instruments 26
So, consider (Figure 9) a database of resources (seen as a “collective artifact”) and a
community of teachers. This community will use this database in order to perform a particular
type of task (for example, teaching algebra at a given school level in a given technological
environment). When integrating this artifact, teachers develop individual and social schemes.
Through a process of instrumental geneses instruments develop. The interaction between teachers
and resources can be analyzed as the two components of the instrumental genesis:
Teachers, when experimenting with resources in their classes, modify these resources,
incorporating in them their own experiences (Guin & Trouche, 2005). This is the
instrumentalization process.
Resources, when implemented by teachers in their classes, contribute to modify their
practices. This is the instrumentation process.
Insert Figure 9 about here.
There are obviously some conditions for such a process to succeed:
The resources have to be rather flexible, putting in evidence possible didactical
choices for the teacher.
The resources have to include a scenario in use, in order to assist teachers when
putting this situation in place in their classes.
The resources have to allow experimentation reports to be fulfilled, in order to
transmit and socialize each teacher’s experience.
Last but not least, there is a need for an instrumental orchestration (as introduced in a
previous section), organizing the relationships within the community and the
interaction between teachers and resources.
Running head: Artifacts to Instruments 27
In the context of a distance training organization, Guin and Trouche (2005) thus stress the
importance of making rights and duties explicit for all actors (trainers and trainees) involved in
this organization through the use of charts. These charts are reference texts explaining in detail
tasks and working modes in the community, both for trainers and trainees, interacting modes
with others and modes of using resources. Charts illustrate that distance working modes require
agreement to a strict schedule and the unavoidable act of writing down (and consequently,
making explicit) didactical choices which usually remain tacit for teachers. Under these
conditions, the database of resources may give birth to instruments integrated into each teacher’s
practice. This is an endless process: usages as well as technologies evolve, and no database of
resources is completely and definitively closed. New resources (through the Internet, for
example) always can be added and can enter into this process of instrumental genesis.
An Instrumental Approach to Software Design
The process we described in the previous section for conception, development and
evolution of pedagogical resources may be applied to the design of software and computerized
learning environments. As stated by Sarama & Clements (this volume, 2005): “[…] curriculum
and software are not only based on research a priori. Research also must be conducted
throughout the development process” (the page number for this quote will need to be provided by
the editors in the final copy). Linking research and software development is not only a matter of
a technical, sequential procedure, in which computer scientists, mathematicians, and researchers
in mathematics education first conceive a product, and teachers use it later. Rather, from an
instrumental point of view, this requires multidisciplinary teams, in which researchers and
teachers collaborate in an iterative and bi-directional way. Such an approach is needed in order to
conceptualize and develop software that takes into account users' experiments and experiences.
Running head: Artifacts to Instruments 28
Conclusion
In the introduction to this chapter, three essential issues were raised: the need for the
analysis of the learning process in technological environments, the question of how the teacher
can organize the teaching in such an environment, and the need to describe the development of
resources for professional development of new teaching practices. As a conclusion, we would
like to stress three essential points.
First, the transformation of an artifact–particularly if it is a complex one–into an
instrument for mathematics requires time. The instrumental genesis, the construction of an
instrument, depends on several factors, such as the affordances and constraints of the artifact, the
type of tasks, the learning ecology as a whole, and the inventiveness of the student. A general
feature of the instrumental genesis, however, is that it is a time-consuming and laborious process.
Second, the notion of system of instruments is essential, on the one hand to describe the
“ensemble” of artifacts that a student has to integrate during the learning process, and on the
other hand as the set of instruments that are developed within a class of students, which the
teacher has to manage in her/his teaching.
Third, the metaphor of orchestration is appropriate to describe the organization of the
available artifacts by the teacher. The importance of the metaphor is that it expresses the idea of
articulation or fine-tuning of a system of instruments, including the guidance by the conductor as
well as the improvisations by the soloist players and adaptability for different styles of music.
In this chapter we first applied the instrumental approach to technological artifacts, but
also to pedagogical resources and even to the development of software, indicating that this
development can only take place effectively if it is related to its use in the classroom. Finally,
one could apply this perspective to the instrumental approach itself: if we apply the instrumental
Running head: Artifacts to Instruments 29
approach to different situations, and confront it with other theoretical frameworks, it will further
develop within the community of researchers. In fact, this chapter illustrates this process. The
diversity of the situations in which the instrumental approach can be used illustrates its potential.
Running head: Artifacts to Instruments 30
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Paper presented at the Fourth Conference of the European Society for Research in
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Paper presented at the Third Conference of the European society for Research in Mathematics