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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)
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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 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 1); in a paper-and-pencil environment, only 5% of
students answered “no.”
Insert Figure 1 about here.
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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.
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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,
2003) and dynamical geometry software (this volume, Hollebrands, Laborde, & Straesser, 2005).
Furthermore, we explain how the approach encompasses both an individual and a collective
perspective.
We consider the metaphor of the students in a classroom as instrumentalists, and the
teacher as the conductor of this classroom orchestra, and we first address the learning process of
the soloist musician. The main ideas of the instrumental approach concerning the individual
learner using tools and developing instruments are explained in the next section, thus addressing
criterion 1 in the preceding list. In the third section, which corresponds to criterion 2, we take the
“orchestra perspective” and consider the notion of instrumental orchestration. The fourth section
addresses the question of resources for professional development mentioned in item 3.
The Instrumental Approach: The Soloist Learning Perspective
As stated previously, we believe that the theoretical framework of the instrumental
approach allows for an analysis of the learning process in technological environments of
increasing complexity, and takes into account the non-trivial character of using computerized
environments. In this section, we review the principles of the instrumental approach to the
individual learner who uses technological tools for mathematical tasks.
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Artifact and Instrument
The principles of the instrumental approach can be characterized by the following
keywords: artifact, instrument, instrumental genesis, instrumentation, instrumentalization, and
scheme. These keywords are addressed in the general description that follows.
The basis of the instrumental approach is formed by the ideas of Vygotsky (1930/1985).
He noted that an instrument constitutes “a new intermediary element situated between the object
and the psychic operation directed at it” (p. 42) and thus mediates the activity. A tool, which has
been developed in a specific cultural and historical context, can be a material artifact, such as a
violin, a calculator or a computer, but also a non-material cognitive tool such as language or an
algebraic symbol. An “instrumental act” for Vygotsky includes a problem that needs to be solved,
the mental processes involved in solving, and the (psychological) tools that are used to carry out
and to coordinate these processes. Interesting here is the active role attributed to the tools, which
influence the mental processes. The notion that tools are not just passively “waiting to be used” is
an important one when considering tool use and learning.
To illustrate with an example of the role of the tool in the instrumental act, we can think
of a hammer. A hammer can be considered as an extension of the body, which allows us to hit a
nail much more efficiently and with less effort and pain than would be the case if we did so by
hand. However, the availability of a hammer may lead us to try to solve all kinds of problems–
including problems for which a screwdriver would be more appropriate–by using the hammer. In
some cases, this may be adequate, but in others it may just be a matter of using the tools at hand.
In that sense, the availability of the tool guides the choice of the problem-solving strategy. An
effective use of a hammer requires skills, experience, and insight into the way it can contribute to
solving the problem or carrying out the task at hand.
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Rabardel and his colleagues elaborate on Vygotsky by distinguishing an artifact from an
instrument (Rabardel, 2002; Verillon & Rabardel, 1995). The artifact is the “bare tool,” the
material or abstract object, which is available to the user to sustain a certain kind of activity, but
which may be a meaningless object to the user as long he does not know what kind of tasks the
“thing” can support in which ways. Only after the user has become aware of how the artifact can
extend his capacities for a given kind of relevant task, and after he has developed means of using
the artifact for this specific purpose, does the artifact becomes part of a valuable and useful
instrument that mediates the activity.
If the artifact is the tool that is used, such as a hammer, a violin, a calculator or a
computer, it is not automatically a mediating instrument. What is the instrument? Following
Rabardel, we speak of an instrument when there exists a meaningful relationship between the
artifact and the user for dealing with a certain type of task–in our case a mathematical task–which
the user has the intention to solve. As the interaction between user and artifact requires mental
processes, we see that the main “players” here, the mental processes of the user, the artifact, and
the task, are the same as was the case for Vygotsky’s previously described instrumental act.
Particularly for mathematical tools, which can be considered to be “extensions of the mind”
rather than extensions of the body, these mental processes are essential. Therefore, the instrument
consists of both the artifact and the accompanying mental schemes that the user develops to use it
for performing specific kinds of tasks. This notion of instrument is illustrated in Figure 2 and can
be summarized as:
Instrument = Artifact + Scheme for a class of tasks.
Insert Figure 2 about here.
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Reflecting on Figure 2, one can wonder whether the type of task should be included in the
oval that represents the instrument. Furthermore, we point out that the artifact is not always the
material object as a whole. For example, the graphing module of a symbolic calculator can be
seen as an artifact on its own, or even the sub-module for setting the dimensions of the viewing
window of the graph can be seen as an artifact. Finally, it is worthwhile to stress that the meaning
of the word “instrument” here is more subtle than it is in daily life: The artifact develops into an
instrument only in combination with the development of mental schemes.
Now that we distinguished artifacts and instruments, the question is how the availability
of an artifact can lead to the development of an instrument. To do so, the user has to develop
mental schemes, which involve skills to use the artifact in a proficient manner and knowledge
about the circumstances in which the artifact is useful. The “birth” of an instrument requires a
process of appropriation, which allows the artifact to mediate the activity. This complex process
is called the instrumental genesis. The example in the first section on the interpretation of a graph
(Figure 1) indicates that the instrumental genesis is a far from trivial process, which requires time
and effort of both students and teachers. As the double arrows in Figure 2 indicate, instrumental
genesis is a bi-directional process. On the one hand, the possibilities and constraints of the artifact
shape the techniques and the conceptual understanding of the user. Some approaches are quite
natural in a specific environment, while others are discouraged because of the peculiarities of the
artifact. This is called the instrumentation process: the artifact shapes the thinking of the user. On
the other hand, the conceptions and preferences of the user change the ways in which he or she
uses the artifact, and may even lead to changing or customizing it. For example, one may update
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the version of the word-processing software on a computer, and in this way change its
functionality. A student who enters into the graphing calculator a program that calculates the
zeros of a quadratic function, also extends the artifact. If a game is downloaded, the primary
functionality of the artifact–at least as the teacher perceives it–is changed. The artifact is shaped
by the user, and this is called instrumentalization. The difference between instrumentation and
instrumentalization, therefore, consists of the bi-directional interaction, in which the student’s
thinking is shaped by the artifact, but also shapes the artifact (Hoyles & Noss, 2003).
The instrumental genesis thus involves the development of mental schemes, which
organize the problem-solving strategy, and induce the concepts that form the basis of the strategy.
Meanwhile, techniques co-evolve, consisting of means for using the artifact in an efficient way to
complete the intended types of tasks. The questions now are: What is such a scheme? and How
can we identify it and observe its development? As a first approach to the notion of scheme, we
follow Vergnaud, who elaborated on Piaget. Vergnaud defines a scheme as “une organisation
invariante de la conduite pour une classe donnée de situations,” (Vergnaud 1996, p.177). In the
context of tool use, we speak of a utilization scheme, which we consider as a more or less stable
mental organization, including both technical skills and supporting concepts for a way of using
the artifact for a given class of tasks. In the case of a mathematical problem, a mental scheme
involves the global solution strategy, the technical means that the artifact offers, and the
mathematical concepts that underpin the strategy. It contains operational invariants that consist
of–explicit or implicit–knowledge in the form of concepts-in-action or theorems-in-action
(Trouche, in Guin et al., 2004). The relationship of the co-evolving technical and conceptual
elements in a mental scheme is characteristic for instrumental genesis. As a consequence, the
technical work with the artifact–in our case technological tools for mathematics–is connected to
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conceptual insights. The “art” for the teacher is to exploit this relationship for the sake of
learning.
Let us consider an example outside of mathematics. Moving a text block while writing a
chapter like this in a word-processing environment can be done with a technique that we call
“cut-and-paste.” An experienced user applies this cut-and-paste scheme quickly, accurately and
without thinking by means of a sequence of keystrokes and/or mouse clicks. But do you
remember the first time you did this? You had to find your way through some menus, and also
accept the somewhat frightening fact that the text block that needed to be moved elsewhere,
seemed to have disappeared after it had been cut. Some insight into the difference between what
is on the screen and what is in the memory of the computer is a conceptual aspect of the
accompanying scheme. Without that notion, the instrumental genesis is not completed and
applying the technique will remain difficult; meanwhile, the technique may evoke this insight, as
the text block re-appears after the command “Paste.”
Two types of utilization schemes are distinguished, usage schemes and instrumented
action schemes. The cut-and-paste scheme described above is a basic, elementary scheme,
directly related to the artifact. Such schemes are called usage schemes. The usage schemes can
serve as building blocks for schemes of a higher order, the instrumented action schemes.
Instrumented action schemes focus on carrying out specific kinds of transformations on the
objects of activity, which in our case are mathematical objects such as formulas, graphs, and so
on. Instrumented action schemes are coherent and meaningful mental schemes, and are built up
from elementary usage schemes by means of instrumental genesis. This articulation of usage
schemes may involve new technical and conceptual aspects, which are integrated in the scheme.
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A well-known example of an instrumented action scheme concerns scaling the viewing
window of a graphing calculator (Goldenberg, 1988). The technical skills that the instrumental
genesis of this instrumented action scheme requires are not very hard: it is a matter of finding the
window-setting menu and knowing the meaning of the different fields that need to be filled in.
The use of negative numbers and the corresponding difference between the minus sign for
negative numbers and the minus sign for subtractions can be considered as one of the component
usage schemes. However, more is needed. The student should be able to perceive the calculator
screen as a relatively small viewing window, through which we look at part of an infinite plane,
theoretically speaking. The position and the dimensions of our window determine whether the
window includes some part of the graph. In fact, in most cases no window can show the entire
graph. The student needs skills to determine appropriate window settings. Based on our
classroom observations, we claim that it is the incompleteness of these conceptual aspects of the
scheme rather than the technical aspects that cause the difficulties that many students experience
with graphing on a graphing calculator.
The difference between elementary usage schemes and higher-order instrumented action
schemes is not always obvious. Sometimes, it is merely a matter of the level of the user and the
level of observation: what at first may seem an instrumented action scheme for a particular user,
may later act as a building block in the genesis of a higher-order scheme.
The examples of the cut-and-paste scheme and the viewing window scheme illustrate that
a utilization scheme involves an interplay between acting and thinking, and that it integrates
machine techniques and mental concepts. In the case of mathematical information technology
tools, the conceptual part of utilization schemes, therefore, includes both mathematical objects
and insight into the “mathematics of the machine.” As a consequence, seemingly technical
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obstacles that students experience while using a computerized environment for mathematics often
turn out to have an important conceptual background.
A difficulty, here, is that we cannot observe mental schemes directly. Our observations are
limited to techniques that students carry out with the artifact, and to the way they report on this in
a written or oral form. From these data we try to reconstruct the schemes, but it is important to
keep in mind that they are no more than our reconstructions.
We already argued that the construction of schemes, the instrumental genesis, is not
straightforward in practice. Students may construct schemes that are not appropriate, not efficient,
or that are based on inadequate conceptions. The elaborated example in the next section
illustrates the difficulty of the instrumental genesis of a scheme for solving equations.
Example: Solving Equations With a Computer Algebra Device
As an example, described in more detail in Drijvers (2003), we now consider a scheme for
solving equations in a computer algebra environment. At a first glance, this seems to be a trivial
task: one just types in “Solve” and everything works. However, further examination shows that
things are more complex.
The following assignment is presented to a tenth-grade, high achieving student called
Maria (Figure 3).
Insert Figure 3 about here.
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Let us consider Task ii. Maria uses a TI-89 symbolic calculator, a handheld device, which
offers facilities for graphing as well as for symbolic manipulation. She first enters
Solve(x2+b*x+1, x), which results in an error message from the TI-89, because the first argument
of the Solve command should be an equation instead of an expression. She comments:
M:
“It hates me, that calculator.”
Then she considers substituting a value for x with the substitution bar: Solve(x2+b*x+1, x |
. However, she clears this and enters Solve(x2+b*x+1, b). Apparently, she is not sure about the
unknown with respect to which the equation should be solved, and about the role of the different
literal symbols in the problem situation.
Next she enters Solve(x2+b*x+1 = y, b), which results in the solution b  
x2  y 1
. She
x
seems to think that this is not a valid result, and wishes to substitute the value 5 for b. Besides the
substitution operator, |, she also uses the Solve command: Solve(b = -(x^2-y+1)/x | b=5). Once
more, this leads to an error message.
The observer passes by and Maria complains:
M:
“So you cannot solve that, you don’t have x and you don’t have y. Only b.”
In fact, this is not true, we do know that y = 0.
M:
“Maybe x is zero, it should pass by the zeros, do you have to fill in zero for x?”
The observer indicates that y = 0. Maria enters Solve(b=-(x^2-0+1)/x, x), which expresses
x in b instead of the other way around.
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M:
13
“But then you have to fill in a value for b?”
She seems to think that an equation should have a numerical solution, and that an
expression cannot be considered as a solution. Furthermore, she confuses “express in b” with
“solving with respect to b”’ The observer suggests solving x2+b*x+1=0 with respect to x, but
Maria comments:
M:
“But it is of no use if it lies between two zeros if you cannot calculate it?”
Once more, solutions to her should be numerical. However, after some more discussion
with the participating observer, she solves the equation with respect to x and seems to understand
why. However, while copying the result into her notebook, she does not realize the scope of the
square root symbol, which results in an error (Figure 4).
Insert Figure 4 about here.
How do we interpret these observations in light of the instrumental approach? The tasks
here involve solving parametric quadratic equations. The artifact is the algebraic application
within the symbolic calculator. The instrumented action consists of solving the parametric
equation with the artifact. The mental scheme that Maria is developing can be seen as an
instrumented action scheme, as it involves the transformation of mathematical formulas. In this
case, however, the scheme is not built up from other elementary usage schemes. As such, it can
be seen as an elementary usage scheme itself, but because of the conceptual components in the
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instrumental genesis that obviously cause many difficulties, we prefer to see it as an instrumented
action scheme here.
Our claim now is that the instrumental genesis of the scheme for the application of the
Solve technique in the handheld computer algebra device interacts with several conceptual
aspects and therefore illustrates how technical and conceptual aspects interfere with each other
and co-develop during the instrumental genesis. In fact, the problem situation and the artifact use
force Maria to sharpen and to extend her conception of solving equations in several ways, which
are part of the instrumental genesis of an appropriate instrumented action scheme for solving
parameterized equations. We distinguish the following aspects, which are depicted in Figure 5:
1. Knowing that the Solve command can be used to express one of the variables in a
parameterized equation in other variables. This is an extension of the notion of solving,
and requires expressions to be considered as solutions.
2. Knowing where to find the Solve command on the TI-89, remembering its syntax and
knowing the difference between an expression and an equation.
3. Realizing that an equation is solved with respect to an unknown, being able to identify the
unknown in the parameterized problem situation and not forget to add it at the end of the
command Solve (equation, unknown).
4. Being able to accept the result, particularly when it is an expression, as a solution, to
interpret it with some “symbol sense,” and to relate it to graphical representations.
Of course, the observation reveals no more than one student’s behavior. Meanwhile, it
may clarify how the instrumental approach, and the identification of instrumented action schemes
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in particular, may help us to see the complexity of tool use, and to understand the obstacles and
opportunities that emerge during the instrumental genesis.
Discussion
We conclude this section on the instrumental approach of the “soloist” learning process.
For what does the instrumental approach to learning mathematics using technological tools
allow? It allows for an analysis of the learning process in technological environments of
increasing complexity, and takes into account the non-trivial character of using computerized
environments. Furthermore, it stresses the subtle relationship between machine technique and
mathematical insight, and provides a conceptual framework for investigating the development of
schemes, in which both aspects are included. This is helpful for designing student activities, for
observing the interaction between students and the computer algebra environment, for
interpreting it and for understanding what works well and what does not. The seemingly technical
obstacles that students encounter while working in the computer algebra environment often have
conceptual components, and the instrumental framework helps one to be conscious of this and to
turn such obstacles into opportunities for learning.
Essentially, the instrumental approach reflects “old” ideas, already reflected in the notions
of tool use by Vygotsky and others. Its originality lies in the fact that these ideas have been
appropriated for the case of using technology in mathematics education mainly by French
researchers (Artigue, Laborde, Lagrange, Guin, Trouche) during the most recent decade.
How about the articulation of the instrumental approach with other theoretical
perspectives? For example, many current research studies on mathematics education focus on
semiotics, symbolizing, modeling, and tool use (e.g., Cobb, Gravemeijer, Yackel, McClain, &
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Whitenack, 1997; Gravemeijer, 1999; Gravemeijer, Cobb, Bowers, & Whitenack, 2000; Meira,
1995; Nemirovsky, 1994; Roth & Tobin, 1997). These approaches stress the dialectic relation
between symbolizing and development of meaning in increasing levels of formalism. This
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.
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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
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
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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
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
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).
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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
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
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.
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
20
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 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
Running head: Artifacts to Instruments
21
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 sherpastudent’s calculator which remains visible on the big screen) and sometimes
calculators are 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 sherpastudent 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?
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
In the frame of this configuration, teachers and students play new roles. The sherpastudent 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 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 sherpastudent 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
Running head: Artifacts to Instruments
23
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

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.
Running head: Artifacts to Instruments
24
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.
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
Running head: Artifacts to Instruments
25
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
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.
Running head: Artifacts to Instruments
26
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.
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:
Running head: Artifacts to Instruments

27
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.
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.
Running head: Artifacts to Instruments
28
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.
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
Running head: Artifacts to Instruments
29
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
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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