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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)


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.


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.


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.


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.


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.



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 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


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.


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


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).



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 xyxb 12 +−

−= . She

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.

M: “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).


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

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

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, &

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.

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

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

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).


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.


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.


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


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.


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


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.


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

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.


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.


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.


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.


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


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.


References

Allen, R., Wallace, M., Cederberg, J., & Pearson, D. (1996). Teachers empowering teachers:

Vertically-integrated, inquiry-based geometry in school classrooms. Mathematics Teacher,

90, 254-255.

Arnon, I. & Dubinsky, E. (2001, February). Research on technology and other dangerous

pedagogical considerations. Paper presented at the Conference on Research on Technology

and the Teaching and Learning of Mathematics, University Park, PA.

Artigue, M. (1997). Le logiciel ‘Derive’ comme révélateur de phénomènes didactiques liés à

l'utilisation d'environnements informatiques pour l'apprentissage. Educational Studies in

Mathematics, 33, 133-169.

Balacheff, N. (1994). Didactique et intelligence artificielle, Recherches en Didactique des

Mathématiques 14(1/2), 9-42.

Brousseau, G. (1997). Theory of didactical situations in mathematics. Dordrecht: Kluwer

Academic Publishers.

Chevallard, Y. (1992). Intégration et viabilité des objets informatiques, le problème de

l'ingénierie didactique. In B. Cornu (Ed.), L'ordinateur pour enseigner les mathématiques,

(pp. 183-203). Paris: Presses Universitaires de France.

Cobb, P., Gravemeijer, K. P. E., Yackel, E., McClain, K., & Whitenack, J. (1997).

Mathematizing and symbolising: The emergence of chains of signification in one first-grade

classroom. In D. Kirshner & J. A. Whitson (Eds.), Situated cognition theory: Social,

semiotic, and neurological perspectives (pp. 151-233). Hillsdale, NJ: Lawrence Erlbaum

Associates.


Drijvers, P. (2002). Learning mathematics in a computer algebra environment: obstacles are

opportunities. Zentralblatt für Didaktik der Mathematik, 34(5), 221-228.

Drijvers, P. (2003). Learning algebra in a computer algebra environment. Design research on

the understanding of the concept of parameter. Doctoral dissertation. Utrecht, the

Netherlands: CD- press. Also available through www.fi.uu.nl/~pauld/dissertation .

Goldenberg, E. P. (1988). Mathematics, metaphors, and human factors: Mathematical, technical

and pedagogical challenges in the educational use of graphical representation of functions,

Journal of Mathematical Behavior, 7, 135-173.

Gravemeijer, K. (1999). How emergent models may foster the constitution of formal

mathematics. Mathematical Thinking and Learning, 1, 155-177.

Gravemeijer, K., Cobb, P., Bowers, J., & Whitenack, J. (2000). Symbolising, modeling and

instructional design. In P. Cobb, E. Yackel, & K. McClain (Eds.), Symbolising and

communicating in mathematics classrooms: Perspectives on discourse, tools, and

instructional design (pp. 225-273). Hillsdale, NJ: Lawrence Erlbaum Associates.

Guin, D., Delgoulet, J., & Salles, J. (2000). Formation aux TICE: concevoir un dispositif

d'enseignement autour d'un fichier rétroprojetable. Paper presented at the L'enseignement des

mathématiques dans les pays francophones, EM 2000, Grenoble.

Guin, D., Joab, M., & Trouche, L. (2003). SFoDEM (Suivi de Formation à Distance pour les

Enseignants de Mathématiques), bilan de la phase expérimentale. Montpellier: Institut de

Recherche sur l’Enseignement des Mathématiques, Université Montpellier II.

Guin, D., Ruthven, K., & Trouche, L. (2004). The didactical challenge of symbolic calculators:

turning a computational device into a mathematical instrument. Dordrecht: Kluwer



Guin, D., & Trouche, L. (1999). The complex process of converting tools into mathematical

instruments: The case of calculators. International Journal of Computers for Mathematical

Learning, 3, 195-227.

Guin, D., & Trouche, L. (2005, February). Distance training, a key mode to support teachers in

the integration of ICT? Towards collaborative conception of living pedagogical resources.

Paper presented at the Fourth Conference of the European Society for Research in

Mathematics Education,Sant Feliu de Guíxols, Spain.

Haspekian, M. (2003, March). Between arithmetic and algebra: a space for the spreadsheet?

Contribution to an instrumental approach. Tools and technologies in mathematical didactics.

Paper presented at the Third Conference of the European society for Research in Mathematics

Education, Italy. Retrieved, 24 March 2005, from

http://www.dm.unipi.it/~didattica/CERME3/proceedings/Groups/TG9/TG9_Haspekian_cerme3.pdf

Healy, L. (2002). Iterative design and comparison of learning systems for reflection in two

dimensions, Unpublished doctoral dissertation, University of London, .

Hollebrands, K., Laborde C., & Straesser, R. (2005). Technology and the learning of geometry

at the secondary level. In M. K. Heid & G. W. Blume (Eds.), Research on technology and the

teaching and learning of mathematics: Syntheses, cases, and perspectives. Volume 1:

Research syntheses. Greenwich, CT: Information Age Publishing, Inc.

Hoyles, C., & Noss, R. (2003). What can digital technologies take from and bring to research in

mathematics education? In A. J. Bishop, M.A. Clements, C. Keitel, J. Kilpatrick, & F. Leung

(Eds.), Second international handbook of mathematics education (Vol 1, pp. 323-349).

Dordrecht: Kluwer Academic Publishers.


Laborde, C. (1999). Vers un usage banalisé de Cabri-Géomètre en classe de seconde: analyse des

facteurs de l'intégration. In D. Guin (Ed.), Calculatrices symboliques et géométriques dans

l'enseignement des mathématiques, colloque francophone européen (pp. 79-94). Montpellier:

Institut de Recherche sur l’Enseignement des Mathématiques, Université Montpellier II.

Lagrange, J. B., Artigue, M., Laborde, C., & Trouche, L. (2003). Technology and mathematics

education: A multidimensional study of the evolution of research and innovation. In A.J.

Bishop, M.A. Clements, C. Keitel, J. Kilpatrick, & F. K. S. Leung (Eds.), Second

international handbook of mathematics education (Vol. 1, pp. 239-271). Dordrecht: Kluwer


Lajoie, S. P. (1993). Cognitive tools for enhancing learning. In S. P. Lajoje & S. J. Derry (Eds.),

Computers as cognitive tools (pp. 261-289). Hillsdale, NJ: Erlbaum.

Meira, L. (1995). The microevolution of mathematical representations in children's activity.

Cognition and Instruction, 13, 269-313.

Nemirovsky, R. C. (1994). On ways of symbolizing: The case of Laura and the velocity sign.

Journal of Mathematical Behavior, 13, 389-422.

Rabardel, P. (2001). Instrument mediated activity in situations. In A. Blandford, J.

Vanderdonckt, & P. Gray (Eds.), People and computers XV - Interactions without frontiers

(pp. 17-30). Berlin: Springer-Verlag.

Rabardel, P. (2002). People and technology - a cognitive approach to contemporary instruments.

Retrieved 1 March, 2005 from: http://ergoserv.psy.univ-paris8.fr.

Rabardel, P., & Samurçay, R. (2001, March). From artifact to instrument-mediated learning:New

challenges to research on learning. International symposium organized by the Center for


Activity Theory and Developmental Work Research, Helsinki. Retrieved 1 March, 2005 from

http://ergoserv.psy.univ-paris8.fr/Site/default.asp?Act_group=1

Roth, W., & Tobin, K. (1997). Cascades of inscriptions and the re-presentation of nature: how

numbers, tables, graphs, and money come to re-present a rolling ball. International Journal of

Science Education, 19, 1075-1091.

Sarama, J., & Clements, D. H. (2005). Linking research and software development. In M. K.

Heid & G. W. Blume (Eds.), Research on technology and the teaching and learning of

mathematics: Syntheses, cases, and perspectives. Volume 2: Cases and perspectives.

Greenwich, CT: Information Age Publishing, Inc.

Trouche, L. (2004). Managing complexity of human/machine interactions in computerized

learning environments: Guiding student's command process through instrumental

orchestrations International Journal of Computers for Mathematical Learning, 9, 281-307.

Vivet, M. (1991). Usage des tuteurs intelligents: prise en compte du contexte, rôle du maître. In

M. Baron, R. Gras, & J. F. Nicaud, , (Eds.), Deuxièmes journées Environnements

informatiques d'apprentissage avec l'ordinateur (pp. 239-246). Cachan: Ecole Nationale

Supérieure.

Vergnaud, G. (1996). Au fond de l’apprentissage, la conceptualisation . In R. Noirfalise & M. J.

Perrin (Eds.), Actes de l’école d’été de didactique des mathématiques (pp. 174-185).

Clermont-Ferrand, France: Institut de Recherche sur l’Enseignement des Mathématiques,

Université de Clermont-Ferrand II.

http://ergoserv.psy.univ-paris8.fr/Site/default.asp?Act_group=1


Verillon, P., & Rabardel, P. (1995). Cognition and artefacts: A contribution to the study of

thought in relation to instrumented activity, European Journal of Psychology of Education,

10, 77-103.

Vygotsky, L. S. (1930/1985). La méthode instrumentale en psychology. In B. Schneuwly & J. P.

Bronckart (Eds.), Vygotsky auhourd’hui (pp. 39-47). Neufchâtel: Delachaux et Niestlé.

Wenger, E. (1998). Communities of practice. New York: Cambridge University Press.

Zbiek, R. M., & Hollebrands, K. F. (2005). A research-informed view of the process of

incorporating mathematics technology into classroom practice by inservice and prospective

teachers. In M. K. Heid & G. W. Blume (Eds.), Research on technology and the teaching and

learning of mathematics: Syntheses, cases, and perspectives. Volume 1: Research syntheses.

Greenwich, CT: Information Age Publishing, Inc.

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