Aldon Incident

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O RI G I N A L A RT I CL E

Handheld calculators between instrument and document

Gilles Aldon

Accepted: 2 August 2010 / Published online: 26 August 2010

FIZ Karlsruhe 2010

Abstract The new generations of handheld calculators

can be considered either as mathematical tools withopportunities for calculation and representation or as a part

of the teachers and students sets of resources. Framed by

the Theory of Didactical Situations and the documenta-

tional approach, we take advantage of a particular experi-

ment on introducing complex calculators in scientific

classes to investigate the position and the role of this

handheld technology both for students and teachers. The

results show how different functionalities can be shared

among teachers and students, but also how other func-

tionalities remain private and may even conflict with the

teachers intentions.

1 Introduction

An important part of the teachers activity inside the

classroom is to manage and to control the dynamics he/she

has launched and promoted with a view to construct

knowledge. In a particular context, when complex calcu-

lators belong to the classroom environment, the dynamics

run up against both the power and the difficulties of inte-

gration of the technology but also against the position and

the role teacher and students give to the technology. Fol-

lowing Chevallard (2007, p. 725), a didactic system is a

social arrangement S(X;Y;Q) in which X is a group of

persons studying question Q in order to build up some

answer A under the guidance or supervision of a team

Y. The didactic system interacts with the milieu, set of

didactic tools organized by Y to produce knowledge. The

aim of this paper is to understand the role that complex

calculators may play in the dynamics of knowledgeconstruction.

A metaphor of the dynamics of knowledge in the

classroom can be borrowed from the fractal geometry.

More precisely, the collage theorem (Barnsley, 1993) states

that starting from a given set, it is possible to find an

iterated function system (IFS) whose attractor is close to

the given set; to find this IFS, one must endeavor to find a

set of transformations, contraction mappings on a suitable

space within which the given set lies, such that the union,

or collage, of the images of the given set under the trans-

formations is near to the given set (Ibid., p. 95).

Launching the dynamics of this IFS gives the set as a fixed

point in the Hausdorff space. This theorem can be seen as a

metaphor of the knowledge dynamics in the students and

teachers joint action; outside the class, the teacher has

didactical intentions, and plans lessons searching and

organizing resources in order to arrange the students

milieu (Brousseau, 1997) and to foresee the knowledge

dynamics in the classroom. At the same time, students

organize their work through their own sets of resources,

copybooks textbooks, but also digital resources, computers

and handheld technologies. The teacher orchestrates

(Trouche, 2004) the mathematical situation within the

dynamic environment of the class (Fig. 1) such that the

targeted result of the learning process, viewed as a fixed

point, becomes close to the set of intentions (Fig. 2).

Teaching can then be seen as the management of an open

dynamic system (Rogalski, 2003); the characteristic of a

dynamic system is to have the possibility of a modifica-

tion under its own dynamic (Ibid., p. 361); the open

nature indicates a self-evaluation of the objects and of their

actions by the actors of the system. As in the collage the-

orem, some small changes in the definition of the IFS can

G. Aldon (&)

Institut National de Recherche Pedagogique, Lyon, France

e-mail: [email protected]

123

ZDM Mathematics Education (2010) 42:733745

DOI 10.1007/s11858-010-0275-4


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result in a very different fixed point, taking into account the

sensibility to the initial conditions, thought through thea priori analysis of the situation and transmitted through

the phase of devolution, where the teacher gives the

learning responsibility to students and accepts the conse-

quences of this transfer. But also, the decisions taken by the

teacher in the interactions with students, within the class-

room, can restore or disrupt the teachers didactical aim

and the trajectory of the dynamics can be completely

modified if only one of the transformations changes during

the process (Fig. 3).

In the leaves example, and starting from four transfor-

mations with contracting similarities acting on the blue tri-

angle (Fig. 1), the dynamic system converges to the fixed

point (Fig. 2); a modification of the dynamic during the

process changes the fixed point leading to the third drawing

(Fig. 3). In this process, the steps consist in finding the

adapted transformations and then to launch the dynamic and

to follow the trajectory paying close attention to ensure thatan incident does not disturb the predicted dynamic. In the

classroom framework we define an incident as a public event,

linked to the teaching but out of phase with the teachers

intentions (Roditi, 2001). In the case of the Fig. 3, only one

of the four transformations has been slightly modified.

The resources at any actors disposal are of different

natures and, following Jonnaert et al. (2004, p. 676), they

may have an internal origin (knowledge, know-how, etc.)

or an external origin, either social (resorting to external

expertise) or material (books, copybooks, tools, artifacts,

etc.). We consider teachers resources in a broad perspec-

tive, including the curriculum material but also every-thing likely to intervene in teachers documentation work:

discussions between teachers, orally or online; students

worksheets, etc. (Gueudet & Trouche, 2009, p. 200).

Students also carry out their work with their own resources

of internal or external origins, everything likely to inter-

vene in students documentation work to follow and

extend the previous citation. Each resource, both for stu-

dents and teacher, must be viewed as a part of a wider set

of resources. In this context, the handheld plays a specific

role: an element of the sets of resources of teachers and

students and a technical object, an artifact, provided by

human activity and offered to mediate the teachers and

students mathematical activities of calculation and repre-

sentation. The double evolution of the role of this artifact

both in the system of resources of teachers and in the

system of resources of students has to be studied, to

understand the possible gap between the teachers inten-

tions and the students learning. These evolutions occur

mainly outside the classroom but manifest themselves

inside the classroom and participate in the complex rela-

tions between teachers and students.

Considering teachers activity as the management, the

orchestration of a mathematical situation in the dynamic

environment of the classroom in a particular context and

handheld calculators as a part of teachers and students

sets of resources, this paper tries to give elements of answer

to the questions:

What is the place that teachers and students give to the

handheld calculators inside and outside the classroom?

What are the handheld calculators properties and

functionalities shared inside the classroom through the

interactions between students and teacher?

Fig. 1 The four

transformations at the

beginning of the process

Fig. 2 The fixed point

of the IFS

Fig. 3 An incident during

the process

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2 The research setting

In order to address the above questions, we consider a

particular teaching experiment which gives an opportunity

to study the effect of a wide introduction of handheld

technology in a school. We took the opportunity of a

natural experiment in the sense that the schools and

classes contexts are not created by researchers but by theteam of mathematics teachers. In this team, one particular

teacher, called Jean in this text, plays a role of leader due

to his involvement in the French research team e-CoLab

(Aldon et al., 2008). During the school year 20082009,

Jeans high school participated in a project in which, with

the help of Texas Instruments, all the students of the

scientific classes had a TI-Nspire handheld calculator,

written as HHT in the following. Even if there is not a

particular observation of Jeans classes in this paper, his

role in the school and with the mathematics teachers team

sets him in a central position. The high school, field of our

observation, is ranked as average for French high schoolsby the French National Education Ministry;1 the mathe-

matics teachers are experienced teachers who have a low

rate of technological integration (Aldon & Sabra, 2009),

except for Jean who was the originator of the experiment,

and who turned out to be, during the year, the leader of the

mathematics teachers. Our choice to study the introduction

of this calculator is linked to its particularly new nature;

apart from the fact that it includes a computer algebra

system, a spreadsheet, a graphical and geometrical

environment, this calculator has the following specific

properties:

it exists as a handheld version of the TI-Nspire CAS

software for computers,

files can be organized into directories, each file consist-

ing of one or more activities of one ore more inter-

connected pages,

pages can be copied and transferred from one activity to

another,

the different applications can be connected. For exam-

ple, while animating geometrical objects in the graph-

ical environment, measurements can be stored in the

spreadsheet application, files can be transferred between

calculators and between calculators and computers.

A team of the INRP (Institut National de Recherche

Pedagogique) has monitored this experiment and has built

a global methodology which is described below, taking into

account the time in order to study the interactions between

teachers and students, the modifications of the teachers

systems of resources and the instrumental genesis (Guin &

Trouche, 2002) of both teachers and students.

3 Theoretical frameworks

The study focuses on the modifications brought by the

introduction of the handheld calculator TI-Nspire inteaching and learning. We assume that the calculator is both

a tool, allowing calculation and representation of mathe-

matical objects, and an element of students and teachers

sets of resources (Gueudet & Trouche, 2008a, b, 2009).

Considering the HHT as a digital resource, part of a set

of resources, we are particularly concerned with the hand-

helds properties of memorization, organization of knowl-

edge and communication (Pedauque, 2006). From the

different elements of the students sets of resources, we

mainly focus on the HHT leaning on the fact that it pro-

vides/supports the main functions required for documentary

production:

The two cognitive functions, memorization and

organization of ideas, seem to be the fundamental

basis for the documentary production. []

The function of creativity comprising enrichment due

to the domain of interest related to the document

surpasses that kind of organization just mentioned.

[]

The third and last constituting function of the docu-

mentary production is the transmission function.

(Ibidem, p. 3)

The HHT is designed to store and organize files, and can

be seen as an extension of the human memory in a per-

spective ofaugmented reality (AR). AR allows the user to

see the real world, with virtual objects superimposed upon

or composited with the real world. Therefore, AR supple-

ments reality, rather than completely replacing it. (Azuma,

1997, p. 2). The possibilities of structuring, memorizing and

organizing evoke a sharpening of ideas and highlight the

link between different problems. The two functions, mem-

orization and organization of ideas, are intertwined and

participate to remember and consolidate information in a

unique framework. The function of creativity is linked to

the semiotic context and expresses the relationships

between the manipulated objects and their different repre-

sentations (see, Multimodality in multi-representational

environments, Arzarello, Robutti in this issue). The

transmission function rests on both the possibility to share

and to appropriate and to modify files.

These specific properties of memorization, organization,

creation and transmission are possibly used in different

domains of mediation, namely the private, the collec-

tive and the public domain. Crossing the functions and the

1http://indicateurs.education.gouv.fr/resultlyceeg.php?code=0340039H

&annee=2008.

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domains of mediation gives a two-dimensional grid allow-

ing analysis of the position of the HHT in the set of

resources of the different actors, in our case students

and teachers. Several studies (e.g., Artigue, 1997; Guin &

Trouche, 2002; Laborde et al., 2005) show that the process

of integrating (handheld) technology in the mathematics

classroom is a slow process in which, back and forth

between artifact and actors, the artifact progressivelytransforms into an instrument through the two comple-

mentary movements of instrumentalization and instrumenta-

tion; the subject shapes the artifact for her/his own

mathematical goals and the artifact modifies the subjects

activity. The instrument is the result of this instrumental

genesis (Rabardel & Pastre, 2005) and,at a given moment and

for a particularuse,consists of both the artifact and schemes. A

scheme is described by Vergnaud (1996) as an invariant

organization of activity for a given class of situations.

We introduce here a distinction between artifact and

resource to stress the different resources we consider: a

textbook, a piece of software, the HHT, an interactionbetween students and teacher, etc., are parts of the set of

resources of actors who build schemes of utilization for a

given class of situations. This construction constitutes what

Gueudet and Trouche (2009) call documentational genesis.

The documentational genesis has a dual nature: The

instrumentalization dimension conceptualizes the appro-

priation and reshaping processes [] The instrumentation

dimension conceptualizes the influence on the teachers

activity of the resources she draws on. (Ibidem, p. 205). In

this approach, we distinguish between resources and doc-

uments, a document being composed of resources and

schemes of utilization, the transformation of resources into

documents coming from this double movement of instru-

mentalization and instrumentation. At a given moment, a

document can be seen as the state of the process where

schemes of utilization are associated with a set of resources:

Thus the document is much more than a list of

exercises; it is saturated with the teachers experi-

ence, just as a word, for a given person, is saturated

with sense in a Vygotskian perspective (Vygotsky

1978). The formula we retain here is:

Document Resources Scheme of Utilization:

Ibidem; p: 205

We stress that the documentary productions properties

are, both for students and teachers, an important element in

the transformation of resources into documents through this

documentational genesis. Looking at the calculator with

its different potentialities, we consider it on the one hand

as an artifact with opportunities for calculation and

representation, and with the potential of becoming an

instrument for both teachers and students. On the other

hand, it can be seen as a digital resource, part of the

teachers and students sets of resources, with the potential

of becoming a document, i.e. a resource with schemes of

utilization.

The subjects studied by Gueudet and Trouche (2008a)

are the teachers, and we extend this to the students. We

consider that both teachers and students base their work on

resources and that these two geneses are not completelyindependent and sustain each other. The two processes are

built simultaneously, the resources given by the teacher to

the students becoming part of the students set of resources

in their process of learning and the way students react to

the documents contributes to the modification of the set

of resources of the teachers in their professional task of

teaching.

We place this study in the field of the didactics of

mathematics and we lean on the Theory of Didactical

Situations and more precisely, on the concept of milieu

defined by Brousseau (1997, p. 57):

The milieu is the system opposing the taught system

or, rather, the previously taught system.

One very important hypothesis of this theory is the idea

that knowledge is built by learners by adaptation with the

milieu through interactions:

The Theory of Didactic Situations assumes that

learning in a school situation is an adaptation to a

milieu. The task of the teacher is then to organize the

milieu in such a way that the adaptation will result

in the student developing the target knowledge.

(Sierpinska, 1999, lecture 3, p. 4)

Actors act on the milieu receive information from it and

modify it; students construct their knowledge by inter-

actions with the milieu and teachers have to organize this

milieu to facilitate the encounter between knowledge and

students but also to find information when students are

confronted with the milieu and modify it by their reactions,

such that the produced knowledge becomes an institutional

knowledge through the phase of institutionalization, when

knowledge becomes autonomous from the conditions of

its emergence. With Brousseau (Ibid.), we assume that a

situation is characterized within an institution, by a set of

relations and roles of one or more actors (student, teacher,

etc.) with a milieu, aiming at the transformation of this

milieu, in accordance with a project. The HHT can be

considered as an element of the milieu of the students in a

didactic situation as well as an element of the milieu of the

teacher in a situation of lessons construction. The degrees

of instrumentation and instrumentalization determine the

position of the HHT in the milieu and, as an echo effect,

the position in the milieu gives information on both

instrumentalization and instrumentation: HHT can be part

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of the material milieu (Margolinas, 2004), the HHT is an

artifact with unexplored possibilities or of the objective

milieu, the HHT is the place of experiments allowing

understanding of the functionalities of the machine and

interaction with the inside data or, finally of the milieu of

reference, the HHT allows experiments giving information

on mathematical objects but also linking problems through

the mathematical objects and the solving processes. On theother hand, the HHT can be seen as an element of the

teachers milieu when, in a situation of construction of

lessons, he or she builds and organizes the students milieu.

The observation of the HHT positions evolution in the

students milieu is an important indication of the instru-

mental genesis, and gives information on the relationships

between the teachers intentions and the actual learning

activity of students.

Combining these three previously described approaches,

the instrumental genesis framework gives information

about the way mathematical concepts are taught and rep-

resented; the documentational genesis shows how theactors invest the documentary properties of the HHT and

the notion of milieu gives information about the place of

the HHT both in the teachers construction situation and in

the students learning situations and allows analysis of

these situations.

4 Methodology

The different frameworks concern processes linked toge-

ther and built in the long range, evolving in different

dynamics, say the dynamic of the professional develop-

ment, the dynamic of the interactive processes of the dif-

ferent geneses and the dynamic of knowledge construction.

The purpose of the methodology is then to catch and to

follow these dynamics looking more at the trajectory than

at the final result. In this context, we construct our meth-

odology (Aldon & Sabra, 2009) to observe the intersection

and the perturbation in the dynamics created by the

encounters of the teachers and students documentational

geneses; the methodology leans on the following choices

and effective devices.

In this study, we have chosen to passively follow,

without intervention from the researchers, the experiment

in the last class of the high school (Lycee Terminale S,

scientific class, 18-year-old students). The choice of this

class level comes from the practical examination2 that

students have to take at the end of the year. We thought that

this examination would be an attractor of the dynamics,

particularly related to the use of the HHT.

We also wanted to observe an ordinary class and this

justifies our position of passive observer. We have chosen a

particular teacher (Marie, in the following) who agreed to

our presence in her class and who agreed to fill in a log

book (Gueudet & Trouche, 2008a) and also to be inter-

viewed. This teacher has a long teaching experience but ashort technological one. The topics as well as the didactical

and mathematical responsibility of the lessons have been

left to the teacher. These choices were done because we

wanted:

to observe an ordinary classroom in the sense that

the responsibility of the teaching lies with the teacher,

to focus on the uses of the technology in the class

without being distracted by mathematics teaching

difficulties.

The TI-Nspire calculator, as previously said, has a

functionality which allows users to structure its contentinto folders, files, problems and pages. Leaning on this

potentiality, we decided to follow twice, the work of four

students in each of Jeans and Maries classes, the evolu-

tion of the content of their calculators in order to follow the

students documentational genesis through the internal

organization of the HHT. We decided to ask Marie to

choose four student representatives of different types: with

technical skills or not, with good mathematics results or

not; we assumed that this sample would provide interesting

information about students behavior. In order to somewhat

enlarge the panel and to obtain elements of comparison, we

also asked Jean to choose four students in his class with the

same profiles; in total, we obtained the contents of eight

calculators from two different classes. The contents of the

small number of calculators allow us to understand some

behavior and to formulate hypotheses that have to be

confirmed by a future study.

It was foreseen that these contents would give us

information about instrumentation, the evolution of the use

of the HHT being part of the students individual set of

resources (school books, handbooks, web sites, etc.), linked

with the collective system in the classroom (students files,

blackboard, etc.) and with the teachers sets of resources.

We gathered the content of the students handhelds; the

students sent us the content of their calculator half-monthly

from December to June; these data allow us to follow a part

of their documentational genesis and give information on

the HHTs use outside the classroom.

We organized observations in the classroom: on three

occasions we observed 4-h mathematics lessons, which

were divided into 2 h of laboratory classes (equipped with

computers) and 2 h with the whole class in a regular

classroom. In these lessons, students use the HHT to solve

2 During the school year 20082009, an experiment of a practical

examination was carried out by the ministry of education, and all

students ofthe last scientificclasses have to take a practicalexamination

at the end of the year: http://eduscol.education.fr/cid47793/epreuve-

pratique-de-mathematiques-du-baccalaureat-serie-s.html.


123
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exercises or problems. During these observations, we

mainly focused on interactions between students and

teacher and on incidents related to the mathematical or

technical content of the lesson, distinguishing by the nature

of the answer brought by the teacher, either mathematical

or technical. These observations inform us on the instru-

mental genesis of both teacher and students in relation with

the didactical performance (Drijvers et al., 2010) of theteacher, but also about the position of the HHT in the

milieu and the HHTs properties and functionalities shared

inside the classroom.

Thirdly, the log books written by Marie and interviews

with her completed these previously mentioned data; we

asked Marie to fill in a small journal in the 2 weeks before

the observations in class; the goal of this log book is to

position the observation into the whole-year teaching pro-

cess and to catch the memorable facts related to the HHT,

from the viewpoint of the teacher, that occurred in class

during the two previous weeks; we mainly asked questions

about the role the teacher attributed to the HHT and herfeeling about the students use. Still in the perspective of

positioning the observation in the process, we interviewed

the teacher just before and just after the lesson to obtain her

intention and her first lesson analysis on the spot. We

complete the description of the teachers landscape with

interviews and informal discussions with the other mathe-

matics teachers of the school.

Finally, we asked students to fill in a questionnaire at the

beginning and at the end of the year. The aim of the first

questionnaire is to assess the students attitudes towards

mathematics and technology and their first opinions on the

use of the HHT, particularly outside the classroom. The

second questionnaire focuses more on the use of the HHT

in connection with mathematics learning; even if we do not

discuss the results of these questionnaires in this paper,

they give a clear landscape of the students in this particular

school.3 We took advantage of the experimental mathe-

matics examination to interview students both on the role

of technology in the particular context of the examination

and the position of the handheld in the mathematics land-

scape during the year.

5 Some results

5.1 Class observations

In the two following paragraphs we select information

from the collected data in order to show the HHTs position

in the classroom and we illustrate our purpose using first a

particular observation in Maries classroom and second the

contents of the students TI-Nspires which give us infor-

mation about the place students pay attention to the HHT

outside the classroom. We articulate these observations

with others data collection, as for example, the interviews

of students and teachers, the content of the teachers log

books and the observation of Jeans students during the

practical exam focusing more particularly on the propertiesof the HHT that appear important in the relations between

students and teachers. In Maries class, students work on

the task, completely described below, of finding the num-

ber of solutions of the equation ln(x) = kx2 in x as a

function of the parameter k. From this observation we are

going to analyze the position of the HHT in the milieu and

to link this position with the main documents functions in

a collective domain of mediation. During this observation,

the classroom is organized in such a way that students

could work in pairs and a student, called sherpa student

(Trouche, 2004) worked, doing what the teacher requested

and projecting his calculators screen through an overheadprojector.

As already said, the position of the HHT in the milieu is

a teachers decision as it can be illustrated by the wording

of the problem; in the first question, the teacher put

the HHT in the students material milieu: Using your

TI-Nspire. In the next paragraph, we are going to

describe the position of the HHT in the milieu through the

interactions between students and teacher, following a

particular and important aspect of this problem, that is to

say, the role of the calculator in the understanding of the

parameter (Drijvers, 2003) (Fig. 4).

During the classroom observation the dialogues between

teacher and students were recorded and we pay particular

attention to the interactions between teacher and students

related to the functionalities of the HHT. After having

given the wording sheet to the students, the teacher read

and commented on the problem and the following dialogue

took place.

1. Teacher: Then you have to find the number of

solutions as a function of the value of the parameter

k. What can we do?

2. Student: We draw the curve

3. T: Yes which one? []4. S: We can choose values.

5. T: Yes, we can choose values, I dont know, for

example, 1x2, 2x2, 3x2 What about k? Which

constraints on k?

6. S: x is positive

7. T: Yes x is strictly positive, but k?

8. S: It is real

9. T: It is real, hence you must allow negative values.

Lets go3

It will be possible to have a complete discussion about these

questionnaires in Aldon & Sabra (2009).

738 G. Aldon

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The HHT is part of the milieu of the students, since the

wording asks them to use it to answer the first questions. It

is then obvious how to obtain the answer (line 2), but it isimportant to notice that this student speaks of the curve

and not the curves; the HHT is still an artifact of the

material milieu, with graphing possibilities but not yet

linked with the mathematical situation or linked with

the mathematical situation that students do not actually

understand. So, the teacher tries to make a link between the

mathematical situation and the graphing possibilities (line

5) giving particular values to the parameter. She modifies

the milieu and the students task: instead of experimenting

with the generalization aspect of the parameter, the given

task becomes: draw the curves of the functions x ? x2,

x ? 2x2, However, line 6, we can observe that the stu-dent confuses the parameter and the variable, which seems

to be solved by the dialogue in lines 8 and 9. The intention

of the teacher is to illustrate graphically the number of

solutions depending on a parameter as she explained to us

in the interview just before the lesson. The students,

however, get hold of the situation with their own knowl-

edge and add another issue which is the understanding of

the letters in the mathematical expression as shown in the

next excerpt:

Number of solutions of an equation

A real number kis given.

We are interested in the number of solutions of the equation (E) : ln(x)=kx forx

strictly positive.

1) Using your TI-Nspire:

a) Conjecture, related to the values ofk, the number of solutions of the

equation (E).

Call the examiner in order to validate your conjecture

b) Ifk>0, find graphically an approximate value ofksuch that the equation

(E) has only one solution.

Call the examiner in order to validate your conjecture

2) Prove that, for k0, prove the conjecture formulated in 1-a) and validated your

proof by examiner.

b) Find the exact value of the real k for which the equation (E) has a

unique solution and calculate this solution.

Technical hints

How to use a cursor to modify the value of a parameter k

In a page Graphs & Geometry, create a slider menu 1: Actions A:

Insert Slider

Enter the name of the parameter (k) type kand Enter

kreplace the name v1 given

by default

Modify the domain [minimum, maximum] Ctrl, menu 1:

for kand eventually the step size Settings... and fill in the

different sections; Slider may

be horizontal or vertical or

minimized.

Use the slider to modify k Grab the slider and drag it

(Ctrl, hand) and move it.

If the slider is minimized,

click on the arrow

Fig. 4 Wording of the problem


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T: No, be careful, there are not two variables. k is a

parameter, but it is fixed, OK?

P: Yes, but the derivatives function

T: Only x varies.

P: But, to find the sign of the derivative function, we

T: Answer the question 2!

P: Then, its less than zero

T: k is less than zero, yes!

Another student says at the same time: I have under-

stood nothing! showing that the experiments with dif-

ferent curves and different values of the parameter are not

sufficient to link the technical ability and the mathematical

understanding. This misunderstanding between the pro-

tagonists is well illustrated by the following dialogue by

four students S1, S2, S3, S4 and the teacher T; students,

following the hints in the students sheet, have built a slider

and experiment with different values of k such that the

curves of the two functions x ? lnx and x ? kx2 have a

unique intersection point:

S1: Zero point two

T: Then, for zero point two, what happens?

S1: From zero to zero point two, there is only one value?

T: Then, when k is greater than zero point two, what do

you say?

S2: No solution

S1: Its about zero point two

T: OK! Approximately, uh! Well, perhaps you can find

better. What will you take between zero and zero point

two? Then, it seems that for k greater than zero point

two, the curve is above. Then, change the settings For

example, for zero point one, how many solutions?

S1: Zero point one nine

T: Then your schoolmate says

S3: No, no, zero point one eight!

Sherpa: Look, Miss for zero point one nine, the curves

do not touch themselves

S3: No

S4: Yes!

T: Ah, you dont agree

This long excerpt, which ran on among the students for

five more minutes, shows that the students completely

investigate a situation with the HHT without linking the

experiment and the theoretical aspect of the mathematical

situation. They move the slider, change the values of k and

observe the screen, using the zoom to enlarge the window,

and experiment with the HHT, or in the words of Brous-

seau, play the game against the objective milieu. The HHT

gives results (the objective milieu reacts), allowing the

students to answer the question more and more precisely.

However, they do not investigate the predicted learning

situation and stay without encounters with the knowledge

that the teachers planned to teach. There is no incident

during this phase, in the sense that the dialogue took place

without visible blockade, and the teacher has no possibil-

ities of understanding on the spot the new game played by

students who seem to understand the role of the parameter,

since they obtain the awaited conjecture.

To summarize this paragraph, it appears that:

the teacher organizes the situation to benefit from the

HHTs functionality of dynamic representation in order

to make students understand and act on a mathematical

situation, assuming that the underlying mathematical

concepts are available for the whole class;

in the meantime, students use the creativity property of

the HHT in a private domain of mediation to investi-

gate a different situation in which the targeted knowl-

edge is absent.

These observations give some elements of answer rela-

ted to the place students and teacher give to the calculator

inside and outside the classroom. As a next step, it isinteresting to go into detail and to cross the function and

the domain of mediation of the HHT in the students set of

resources.

5.2 Handheld devices as part of the students sets

of resources

We lean on the analysis of the contents of the students

HHT to draw a parallel between the main functions of a

documentary production and the actual students organi-

zation of their HHT in a documentational genesis. We lean

on the hypothesis that through the content of the HHT, we

can approach the private part of the HHT; we cross our

information with the private and collective utilizations in

the classroom.

5.2.1 A notebook allowing the memorization of knowledge

From the viewpoint of the students, the handheld may be a

means of data storage, which they perceive as helpful:

Its reassuring in the perspective of the exam to have

proofs stored, because we have to know very many

proofs and its also possible to verify our calcula-tions (Interview, 19th May)

The storage functionality of the HHT embodies the

students organization of knowledge. Figure 5 shows the

folder structure of the HHT of this student.

Figure 6a and b shows the content of the folder Maths

lessons,4 and gives insight into the whole knowledge

4 The folder Maths oblig cours (Fig. 4).

740 G. Aldon

123


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organization, which is in line with the teachers lesson

plan.

Depending on the teachers conceptions, the properties

of memorization and organization of ideas of the HHT are

rejected, ignored or in the contrary promoted; it is inter-

esting to observe that the handheld calculators with a

strong organization shown in Figs. 4 and 5 come from the

Jeans class, Jean being a teacher using technology often in

the classroom as well as personally. The students calcu-

lators of Maries class are not, or not very organized and

the data storage remains hidden and private, because she

does not take into account the HHTs organization of data

in her teaching. For example, to the question do we save

our work? asked by students at the end of a lesson,

Maries answer was: As you like it, its your problem.

In an interview, Marie announced.

Fig. 5 General organization of

a students calculator,

December 2009

Fig. 6 a The evolution of the content of a particular folder, January 2009.b The evolution of the content of a particular folder, June 2009


123


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Sometimes its difficult because they (the students) do

not know their lessons. They do not learn because they

believe that they have everything inside their calculators.

5.2.2 An artifact for the personal mathematical work

including a property of creation

Looking at the contents of the calculators, it appears thatvery often students use the HHT as a draft, using a file, or a

set of files to do the current calculations; as an example, the

content of this calculator is particularly significant (Fig. 7).

Only two folders were built by this student, the first one

containing elements of the lessons and the second one,

shown above, containing temporary products, draft files

(In French, draft is brouillon and the names of the files

are draft 7, 8, , r, s, x). Looking at the previous contents

of this HHT, we can also see that the first files were

deleted; we can assume that they were of no more use or no

longer readable, because they were directly linked to a

particular context. The HHT is seen in this case as a directcreation tool bringing immediate feedback to a given

question in a personal domain of mediation. The dimension

of organization is not used in the continuity and the

resource cannot become a document of reference for this

student. It is interesting to link this calculators content of a

student having good results in mathematics and good skills

in using technology and the teachers use of technology

in the classroom: more or less, Marie disconnected

the experiment on the calculator and the mathematical

knowledge as we can observe in these excerpts:

The first excerpt takes place at the end of a lesson;

a student asks:

S: Do we save our work?

T: You save if you want, but tomorrow we do the

theoretical part.

The second excerpt takes place when students just

discovered a conjecture using the HHT:

T: Well, now, we are trying to actually prove it, uh!

Then you are going to answer the theoretical questions.

The teacher does not view the HHT as part of the public

part of students set of resources and does not promote

documentational genesis on the HHT and consequently,

students do not borrow this memorization property or

investigate it privately, for instance in this spreadsheet

(Fig. 8), where all the derivative functions were typed in

manually and not calculated using the formal algebraic

features of the calculator (Fig. 9).

It is also interesting to observe the content of the

Maries HHT which is very well organized, with specific

folders for each of her classes as shown in Fig. 8. In the

class observations, it often appears that Marie does not

want to interfere with the organization of students HHT, as

already mentioned. It appears clearly that Marie gives to

this property a private status which, consequently, is not

Fig. 7 The content of a folder of another HHT, Marieclass, May 2009

Fig. 8 A spreadsheet table of derivative functions

742 G. Aldon

123


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shared inside the classroom: the memorization and orga-nization of ideas properties remain private both for teacher

and students.

The potential of creation of the HHT can be illustrated

by a creation of a function or a program allowing students

to answer a particular question but also giving them the

opportunity to summarize knowledge. It can sometimes be

very simple, as for example giving quotient and remainder

of an Euclidean division, and sometimes very elaborate, as

for example a function solving a Diophantine equation of

type ax ? by = c, a program that the student has shared

(or wants to share) publicly as shown by the second line of

the program, a commentary in the form of copyright(Fig. 10).

It is interesting to cross this observation with the actual

behavior of this student (Julien is a Jeans student) during

the final exam: having to study a recursive sequence, he

alternated calculation by hand and with the spreadsheet,

verifying his results using different modes of calculation

and different representations (numerical, formal and

graphical). The creation property of the HHT was natu-

ralized such that it was an effective tool to solve a math-

ematical problem. The interview with the same student

confirms that this property of the HHT was worked in

class:

Yes, in math lessons, for derivative functions, inte-

grals, and so on [] we really use it (the HHT)

during math lesson. (Interview, 19th of May)

5.2.3 An artifact for communication

The HHT allows for the transmission of information

between students and between teacher and students, and

thus has a communication function. This possibility is very

differently used by teachers and the habits condition the

position of the students use from private to collective or

public. For example, in our observation, the teacher uses a

particular class organization: students work with their own

HHT in a face to face disposition of the classroom. In this

configuration, the Sherpas HHT, working under the con-

trol of the teacher offers a collective view of a part of the

students document and the organization facilitates the

collective communication from an individual question; in

the next excerpt the students search for the intersectionpoint of two curves:

Sherpa: Madam, here

T: Yes, we can see nothing much.

S: Here, the curve is here and after there is no more

curve!

T: There is no curve?

Another student: They are at the same place, you dont

see it.

S: Yes but you must see the intersection.

AS: Yes, but where is your other curve?

The dialogue begins with an individual interrogation andis continued by a dialogue in the class about the graphical

possibilities of the calculator and the links with the math-

ematical problem of intersection of two curves. The func-

tion of communication of the document intervenes with the

teachers pedagogical choices and the HHT appears to be a

catalyst for this communication.

On the other hand, the Sherpas HHT appears to be a

vector of communication between the teacher and the

whole class:

T: Yes! You delete v1 you write k, enter, OK! Then, you

have the curve of 5x2, yes. Well! Now would you like to

change the value? You reach the slider, yes with the

small hand and you move it. Yes, well done. (To the

class) Do you see what happens? The curve is modified.

You dont have to write the formula.

S: Madam, why is it always positive?

T: Speaking to the whole class: Then, why is it always

positive? (Showing the screen of the calculator) There is

something written here on the slider, yes here, you see,

the value is between 0 and 10. Then it is possible to set

that! Look at your file.

Fig. 9 The files organization of Maries HHT

Fig. 10 A program with Juliens copyright


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The teacher uses the discussion with one student to

transmit information to the whole class. In that case the

Sherpas HHT appears to be a generic HHT and the tea-

cher regulates the march of work of the whole class

through the dialogue with this student. In other words, the

teacher transforms the private domain of mediation into a

collective one. It is also important to notice that the tea-

cher does not institutionalize the technological approach,keeping the communication tool in a collective but not

public domain. On the other hand, a personal work may

become public as shown in the example of Juliens pro-

gram (Fig. 9) which is the result of a private creation but

built on a public communication (and intended for a public

communication):

Yes, this program, I wrote it starting from a program

somebody else did. (Interview, 19th of May)

These three different examples illustrate different aspects

of the HHT property of communication crossed with the

domains of mediation in which the teacher and studentsdocumentational geneses follow different dynamics.

To summarize this section, it appears that the docu-

mentational activity of students and teacher with respect to

memorization, organization of knowledge, creativity-

encouragement, and communication participate in the

construction of knowledge dynamics. The instrumental

approach stresses the transformation of an artifact into an

instrument, the HHT becoming an instrument of calcula-

tion, representation and creativity. The documentational

approach adds memorization, communication and organi-

zation of ideas properties. The transformation of the

resource (the HHT with its intrinsic properties) into a

document (the HHT with schemes of utilization) takes into

account the different domains of mediation and creates

distinct and sometimes discrepant documents. These doc-

uments are part of the milieu and modify the dynamics of

the construction of knowledge.

6 Conclusion

The introduction of this paper starts with the metaphor of

the collage theorem describing the way to find an iterated

map system knowing the knowledge fixed point we want to

reach. Following Rogalski (2003), the joint teacherstudent

activity can be seen as an open dynamic system. From a

didactical point of view, the handheld calculator is part of

the didactical contract. The position of the machine as an

element of the system of resources of both teachers and

students makes the negotiation of this contract extra com-

plex. The availability of the HHT evokes different kinds of

tensions:

tensions between the memorization properties and the

teachers conceptions of the students resources;

tensions between the property of creativity and the

teachers intentions;

tensions between the communication property and the

teachers pedagogical organization.

The observations show that the role given to the HHT ina private domain of mediation by students and teacher is not

shared; a consequence is that the documentational geneses

become distinct and separated processes for teacher and

students. These processes are confronted with each other

only in a collective domain and concern mainly the property

of the creation. The communication and cognitive proper-

ties (memorization and organization of ideas) seem to

remain private but are important parts of the documenta-

tional genesis. It is surely a strong difference between the

instrumental approach and the documentational approach to

consider the HHT not only as an artifact becoming an

instrument for mathematical purpose but also a resourcebecoming a document for teaching and learning purpose.

New perspectives of research can extend this study, in

particular, trying to explain and describe these tensions and

understand how it is possible to lean upon the different

documentary properties to enhance teaching and learning.

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