ORIGINAL ARTICLE

Introducing students to geometric theorems: how the teachercan exploit the semiotic potential of a DGS

Maria Alessandra Mariotti

Accepted: 15 February 2013 / Published online: 3 March 2013

� FIZ Karlsruhe 2013

Abstract Since their appearance new technologies have

raised many expectations about their potential for inno-

vating teaching and learning practices; in particular any

didactical software, such as a Dynamic Geometry System

(DGS) or a Computer Algebra System (CAS), has been

considered an innovative element suited to enhance

mathematical learning and support teachers’ classroom

practice. This paper shows how the teacher can exploit the

potential of a DGS to overcome crucial difficulties in

moving from an intuitive to a deductive approach to

geometry. A specific intervention will be presented and

discussed through examples drawn from a long-term

teaching experiment carried out in the 9th and 10th grades

of a scientific high school. Focusing on an episode through

the lens of a semiotic analysis we will see how the tea-

cher’s intervention develops, exploiting the semiotic

potential offered by the DGS Cabri-Geometre. The semi-

otic lens highlights specific patterns in the teacher’s action

that make students’ personal meanings evolve towards the

mathematical meanings that are the objective of the

intervention.

1 Introduction

Introducing students to theoretical thinking is a key edu-

cational issue that teachers are asked to face at different

school levels and in relation to different mathematical

domains. The complexity of this issue as well the variety of

possible approaches has been discussed in the wide liter-

ature on this topic (Mariotti 2006, 2012a; Hanna and de

Villiers 2012). The content of this paper concerns this

specific educational issue and discusses how a particular

didactic intervention may be effective in overcoming cer-

tain difficulties related to enabling students to enter a

theoretical world. We will consider the crucial moment

when students are expected to move from intuitive geom-

etry to theoretical geometry, that is, geometry as a

deductive system. This didactic issue was the basis of a

long-term teaching experiment lasting for many years and

involving a number of teachers and classes. Outcomes of

such long-standing research work have been of different

natures, both theoretical and empirical.

A general trend in the Italian research context has been

that research work has been conceived as ‘‘research for

innovation’’ (Arzarello and Bartolini Bussi 1998), in which

action in the classroom is both a means and a result of the

evolution of research analysis. Teaching experiments of

this type allow one to generate at the same time possible

didactic sequences, a theoretical background that supports

them and a rich database of possible outcomes. Consis-

tently, one of the outcomes of the teaching experiments

carried out by our teams has been the development of a

theoretical framework according to a spiral process where

theoretical constructs emerged from results coming from

the classrooms and at the same time inspired the design of

new didactical interventions.

As far as our teaching experiments are concerned, such a

theoretical frame is centred on the seminal idea of semiotic

mediation introduced by Vygotsky (1978), which in the

following will be referred to as the Theory of Semiotic

Mediation (TSM) (Bartolini Bussi and Mariotti 2008).

After a short description of the TSM and according to the

theme of the Special Issue to which this paper belongs, I

will discuss a classroom-based intervention aimed at

addressing difficulties met by students in moving from an

M. A. Mariotti (&)

University of Siena, Siena, Italy

e-mail: mariotti.ale@unisi.it

123

ZDM Mathematics Education (2013) 45:441–452

DOI 10.1007/s11858-013-0495-5

intuitive to a deductive approach to geometry. The dis-

cussion will focus on the teacher, in particular on his/her

specific role in the teaching and learning intervention.

Within the frame of the TSM I will enlarge on a specific

phase of the intervention showing how the teacher’s

semiotic action may affect the development of personal

meanings towards the mathematical meaning that consti-

tutes the didactic objective of the intervention.

2 The theory of semiotic mediation: an overview

The idea of mediation has often been employed to refer to

the potentiality that the use of a specific artefact has in

fostering mathematical learning. However, the elements of

the mediation process, triggered by the use of an artefact in

the solution of a task and related to the particular mathe-

matical knowledge in focus, are not always made explicit

in all their complexity. Too often, teachers and researchers

apparently believe that the mathematical meaning evoked

by the use of the artefact is sufficiently transparent for

students not to require specific care to be mediated.

On the contrary, we recognize the crucial role of human

mediation in the teaching–learning process (Kozulin 2003,

p. 19) and the need for the teacher to perform specific

forms of mediation. The model of the mediation process

elaborated by the TSM is developed around two key ele-

ments: the notion of the semiotic potential of an artefact

and the notion of a didactic cycle.

2.1 The semiotic potential of an artefact

As far as the relationship between a certain artefact and

certain mathematical knowledge is concerned, one can

speak of evoked knowledge. As a matter of fact, for

experts—for instance, mathematicians—the use of an

artefact may evoke specific knowledge. For instance,

positional notation and the polynomial notation of numbers

may be evoked by an abacus and its use; similarly, a

Dynamic Geometry System may evoke the classic ‘‘ruler

and compasses geometry’’.

The notion of semiotic potential, introduced in the TSM,

aims at defining such an evocative power, stressing the

distinction between meanings emerging from the activity

with the artefact and the mathematical meanings evoked by

such activity.

The semiotic potential of an artefact consists in the

double relationship that occurs between an artefact and on

the one hand the personal meanings emerging from its use

to accomplish a task (instrumented activity), and on the

other hand the mathematical meanings evoked by its use

and recognizable as mathematics by an expert (Bartolini

Bussi and Mariotti 2008, p. 754).

For any artefact the semiotic potential can be identified

through the analysis of its use in the solution of specific

tasks, an analysis that will involve at the same time a

cognitive, an epistemological and often a historic per-

spective (Bartolini Bussi 1996; Bartolini Bussi et al. 2005).

In the case of new technologies, the analysis of the

didactical functionalities outlined by the designer can be a

first basis for exploring the semiotic potential but new

paths can emerge and open unexpected directions of

analysis (Falcade et al. 2007). The specification of the

semiotic potential is to be considered an indispensable a

priori phase leading a teacher to design a teaching–learning

sequence centred on the use of a given artefact.

2.2 The didactic cycle

Once an artefact and its semiotic potential have been

identified, a key question arises: how may it happen that a

student relates practices with the artefact to mathematics?

We take a semiotic perspective, that is we recognize the

central role that signs1 have in the teaching–learning pro-

cess, both as product and as medium in the construction of

knowledge. Thus we can reformulate this question as fol-

lows: how may it happen that personal meanings arising in

the accomplishment of a task through the use of a certain

artefact may become mathematical meanings?

The TSM intends to provide a possible answer to this

question. Focusing on the production of signs, it offers a

model of the teaching–learning process centred on

describing semiotic processes and explaining how such

processes may regulate the meanings’ evolution and pro-

mote students’ mathematical learning. Each intervention of

the teacher aiming at fostering or inhibiting such evolution

can be considered a didactic action, that is, an expression of

his/her teaching intention (Mariotti and Maracci, 2010).

The structure of a teaching sequence may be outlined as

an iteration of didactic cycles (Bartolini Bussi and Mariotti

2008, p. 754). Each cycle consists of phases where dif-

ferent typologies of activities take place; each type of

activity contributes differently but complementarily to

develop the complex process of semiotic mediation and can

be classified in terms of its contribution to the semiotic

mediation process.

The teaching–learning process starts with activities

proposing to the students tasks to be accomplished using a

specific artefact. The phenomenon of meanings’ emergence

in relation to the use of the artefact is called unfolding of

1 When using the term ‘‘sign’’ we refer to the indissoluble

relationship between signified and signifier. In the stream of other

researchers (Radford 2003; Arzarello 2006) we developed the idea

that meaning originates in the intricate interplay of signs (Bartolini

Bussi and Mariotti 2008; for a thoughtful discussion see also Sfard

2000, p. 42ff).

442 M. A. Mariotti

123

the semiotic potential. The emergence is witnessed by the

appearance of specific signs—words, sketches, gestures,

…—referred to the artefact’s use. The unfolding of the

semiotic potential can be fostered by activities of individ-

ual production of signs. Students are involved individually

in different semiotic activities, most of them concerning

written production of texts. For instance, students may be

asked as homework to write individual reports on their own

experience, including their doubts and questions arising.

The design of these activities aims at promoting the

emergence of a rich set of signs (Mariotti and Maracci

2012) which, though related to the use of the artefact,

testify to a first detachment from the contingency of the

situated action and may offer the opportunity to create a

semantic link to mathematical signs. They are personal

because they are strictly linked to the specific situation in

which individuals experience the use of the artefact, and

for this same reason they are situated signs (Lave 1988).

Once personal signs have emerged, their evolution into the

mathematical signs has to be fostered by the teacher

through specific social activities designed to exploit the

semiotic potential of the artefact.

Collective discussions constitute the core of the semiotic

mediation process on which teaching and learning is based.

The whole class is engaged in discussion around a specific

topic that is explicitly articulated at the beginning. For

instance, after a problem-solving session the various solu-

tions can be presented to the class for discussion, and

students’ written texts may be selected by the teacher and

presented to be collectively analysed, commented on and

elaborated. Such collective discussions are to be considered

real mathematical discussions, in the sense that their main

characteristic is the cognitive dialectics, promoted by the

teacher, between different personal meanings and the

mathematical meaning that constitutes the didactic goal,

and that are related to specific mathematical signs

belonging to mathematics practice (Bartolini Bussi 1998).

Such an evolution of signs, from personal signs, stemming

from the activity with the artefact, towards mathematical

signs, is not expected to be either spontaneous or simple,

thus the guiding role of the teacher becomes crucial. The

main objective of a teacher’s action in a mathematical

discussion is that of fostering the move towards mathe-

matical signs, taking into account individual contributions

and exploiting the semiotic potentialities coming from the

use of the particular artefact.

The role played by the teacher is crucial at every step of

the didactic cycle. The teacher has to design tasks which

favour the unfolding of the semiotic potential of the arte-

fact, observe students’ activity with the artefact, collect and

analyse students’ written solutions and reports, and in

particular has to pay attention to the signs which emerge

from students’ texts. Then, taking into account this analysis

of students’ productions, the teacher has to design and

manage the classroom discussion in such a way as to foster

the evolution towards the desired mathematical signs.

In the following sections some examples will be pre-

sented focusing on this crucial phase of the didactic cycle,

and specifically on the role played by the teacher in fos-

tering the process of semiotic mediation. It will be clear

how much the teacher’s action is crucial but also how much

it has to change with respect to standard classroom routines

(Lagrange and Monaghan 2009).

The examples are drawn from the teaching experiment

concerning the unfolding and development of the semiotic

potential of the DGS Cabri-Geometre (Laborde and

Bellemain, 1995), with respect to the mathematical

meaning of a geometrical construction. However, before

discussing the examples I will give a general account of the

specific didactic scenario that constitutes their background.

3 The teaching experiment

As said above, the research study started some years ago

and aimed at introducing students to theoretical thinking in

geometry. With the TSM approach the teaching–learning

process is expected to be very slow, thus we planned a

long-term experiment, consisting in implementing a spe-

cific teaching sequence lasting for a period of at least

2 years, corresponding to the 9th and 10th grades. At the

beginning, three regular classes at the upper secondary

school level were involved for 2 years (9th and 10th

grades), in different high schools, but since then for about

10 years a group of teachers has been following the project,

which of course has developed also in other directions.

Since the very beginning, the research study was carried

out in collaboration with the school teachers who imple-

mented the teaching sequence in the classroom. This means

that the sequence of activities was carefully designed and

repeatedly revised by the whole group, consisting of

teachers and researchers, but realized by the teachers dur-

ing their regular math classes as part of the regular cur-

riculum. The design of the sequence of activities was

consistent with the structure of the didactic cycle described

above, so that the sequence consisted of the iteration of

didactic cycles, each starting with proposing a task in the

Cabri environment. Taking a semiotic mediation perspec-

tive, the educational goal was that of exploiting the use of

Cabri as a tool of semiotic mediation for introducing stu-

dents to the idea of geometrical construction as a theoret-

ical solution of a construction problem.

Different kinds of data were collected: any kind of

students’ production (e.g. worksheets or written reports)

and traces of classroom activities were recorded and tran-

scribed. In the following, after a general description of the

Introducing students to geometric theorems 443

123

semiotic potential of the artefact, I shall give two examples

drawn from the transcripts of two collective discussions

and aiming at illustrating the role of the teacher in the

specific teaching intervention.

4 The semiotic potential of the artefact Cabri

with respect to the mathematical meaning

of geometrical construction

As said, the goal of the teaching sequence was to introduce

students to a theoretical perspective, thus, consistent with

this educational issue, the design of the teaching sequence

began with searching for a specific context to make it

possible (Mariotti 2000, 2012a). Our choice was inspired

by history and by the relationship, immediately evoked in

the mind of any mathematician, between a Cabri figure2

and a geometrical construction. Such a relationship can be

elaborated from the point of view of TSM from a historical,

an epistemological and a cognitive perspective.

Construction problems constitute the core of classic

Euclidean geometry, and the use of specific artefacts, i.e.

ruler and compasses, can be considered to be at the origin

of the set of axioms defining the theoretical system of

Euclid’s Elements. Accordingly, any geometrical con-

struction corresponds to a theorem, which means that there

is a proof that validates the construction procedure that

solves the corresponding construction problem. Thus in

classic Euclidean geometry the theoretical nature of a

geometrical construction is clearly stated, in spite of the

apparent practical objective, i.e. the accomplishment of a

drawing following the construction procedure.

Since the advent of DGS, geometrical constructions

have seen a revival. The use of virtual tools producing lines

and circles on the screen simulates the use of ruler and

compasses in classic geometry. The parallel between

geometry theory and construction activities in Cabri can be

developed further: a sub-set of the tools available in Cabri

can be related to its corresponding set of constructing tools

in Euclidean geometry and, consequently, it can be related

to the corresponding set of axioms (Laborde and Laborde

1991). The novelty of a DGS, with respect to drawings in

the ‘‘paper and pencil’’ classic medium, consists in the fact

that ‘‘screen drawings’’ can be acted upon using the drag-

ging modality. In dragging any basic point from which the

construction is initiated, the whole drawing is transformed;

however, all the properties defined by the constructing

procedure are maintained or, as we usually say, remain

‘‘invariant’’. Moreover, any property that is a consequence

of the constructed properties is also maintained. This

functioning is the cause of the perceptual permanence of a

figure, which is what makes the user recognize ‘‘the same

figure moving on the screen’’; at the same time such a

stability of the drawn figure in respect to dragging consti-

tutes the standard test of correctness for any construction

task in a DGS.

Let us elaborate more on the relationship between the

different ‘‘invariants’’ that may be perceived while the

drawing is moving under the effect of dragging a basic

point, and the theoretical status of the corresponding geo-

metrical properties within geometry theory. As said, the

invariant properties of any Cabri figure are related

according to a hierarchy: some of them are defined by the

commands used in the construction procedure, others are

consequences of them. Such a hierarchy of properties

corresponds to a relationship of logical dependence

between them, which can be expressed in the conditional

statement of a specific theorem. This correspondence,

between construction invariants and derived invariants,

allows us to establish a relationship between the perceptual

control by dragging in a DGS and the logical control by

theorems and definitions within the system of Euclidean

geometry (Holzl et al. 1994; Jones 2000; Mariotti 2000,

2012b).3

In summary, a twofold system of meanings can be

related to the use of specific tools available in the artefact

Cabri. On the one hand, the use of certain tools is related to

the solution of a construction problem resulting in the

appearance of a drawing on the screen and in the stability

of such a drawing by dragging; on the other hand, specific

DGS tools can evoke geometrical axioms and theorems of

classic Euclidean geometry, axioms and theorems that can

be used to prove the theorem that validates the solution of

the corresponding geometrical construction problem.

Moreover, if the use of basic tools may be referred to

axioms and definitions of a theory, the introduction of any

new tool, corresponding to the application of a specific

Macro construction, may be referred to the enlargement of

the theory through the addition of a new theorem. Adding

new tools to those already available corresponds to the

meta-theoretical operation of adding new theorems to the

theory.

Thus, the semiotic potential of the artefact Cabri can

be summarized in the following list where some of the

Cabri’s features are related to specific mathematical

meanings:

2 The expression ‘‘Cabri figure’’ is meant to express both the image

produced on the screen and its dynamic behaviour preserving the

properties defined in its construction.

3 Actually, if we consider the whole set of tools available in a DGS,

including for instance ‘‘measure of an angle’’, ‘‘rotation of an angle’’

and the like, the set of possible constructions does not coincide with

that attainable only with ruler and compasses. For a full discussion see

Stylianides and Stylianides (2005).

444 M. A. Mariotti

123

• The dragging test can be related to the theoretical

validation of a geometric construction: a Cabri figure,

the solution of a construction task, will be stable by

dragging if the correctness of the procedure accom-

plished can be proved within the geometry theory.

• Specific tools can be related to specific elements of the

corresponding geometry theory: axioms, theorems … .

• Actions concerning the management of the Cabri’s

menu can be related to fundamental meta-theoretical

actions concerning the construction of a theory, such as

the introduction of a new theorem or a definition.

Thus the artefact Cabri may be considered a possible

tool of semiotic mediation as far as its use and its func-

tioning in solving specific construction tasks provides an

environment for phenomenological experiences referring to

the mathematical meanings of:

– geometrical theorems that validate specific geometrical

construction

– meta-theoretical actions related to the development of

the theory by adding new theorems.

5 Implementing a teaching sequence

5.1 The construction task

According to the previous analysis of the semiotic potential

of the artefact Cabri, the activities designed in the teaching

sequence were centred on a specific type of construction

task, composed of two requests, the former corresponding

to acting with the artefact, the latter corresponding to

producing a written text referring to such actions. Specif-

ically, students were asked:

• to produce a specific Cabri figure that should be stable

by dragging, and

• to write a description of the procedure used to obtain

the figure appearing on the screen and produce a

validation of its stability by dragging.

The request to produce a written text includes both

describing and commenting on the procedure carried out.

The request to validate the solution made sense with

respect to the Cabri environment, that is, with respect to the

need to explain and gain insight into the reason why the

figure on the screen could pass the dragging test.

Consistent with the fundamental hypothesis on semiotic

mediation, solving such a construction task aimed at

making personal meanings emerge, meanings that could be

related to the use of specific Cabri tools but that also had

the potential to be related to the mathematical meaning of

geometrical construction, and specifically meanings related

to the geometrical axioms or theorems supporting the

validity of the construction procedure with respect to a

theory (for a more detailed discussion on the construction

task, see Mariotti 2000, 2012b).

Tasks were designed to be accomplished in pairs, which

meant sharing the management of the computer, mainly

sharing the use of the mouse. Thus, we expected an intense

exchange between the two students and consequently a

production of varied—belonging to different systems of

representation—signs used in communicating about the

solution of the task. At the same time, the explicit request

to produce a written text aimed to pair activities with the

artefact with activities of individual production of signs.

5.2 The mathematics notebook

Besides the production of texts during the solution of a

construction task, the teaching sequence foresaw the pro-

duction of other texts that students were asked to produce

individually after a specific request. For the most part these

would be verbal texts, but they could also be drawings or

diagrams. Among others, there are two main types of text

request that play a key role in the development of the

semiotic mediation process. One is asking each student to

write a personal report on classroom activities, the other is

asking each student to write his/her personal mathematics

notebook (in Italian, ‘‘quaderno di classe’’).

In this text production, the signs introduced in the pre-

vious work and related to the use of the artefact are

expected to be reinvested, thus it is expected to find both

personal signs and some mathematical signs related to

them. Writing one’s own personal mathematics notebook is

a semiotic activity that has the objective of fostering the

evolution of meanings, as well as the development of a

semiotic net relating previous knowledge with new

knowledge emerging from the activity with the artefact (for

a discussion of some examples of students’ reports, see

Mariotti 2001). Such a semiotic net should constitute the

base that could be related to the mathematical meanings

that are the educational goals.

Concerning the development of a theoretical perspective,

a very specific role is played by writing the mathematics

notebook. Each student is asked to write her/his personal

notebook where any mathematical result, discussed and

socially accepted in the collective discussions, will be

reported. The activity of writing helps students in a process

of de-contextualization, transforming the realization of a

construction in Cabri into the description of a procedure and

its justification, thus fostering the passage from the realiza-

tion of the Cabri figure to the conception of the corre-

sponding geometrical theorem. The content of each personal

notebook is expected to report on what has been elaborated

Introducing students to geometric theorems 445

123

and shared in the collective discussions; in particular, the

personal notebook contains the sequence of the axioms,

definitions and theorems of the shared geometry theory as

has been developed up to that moment. Any notebook may

be considered a representation of the culture and the history

of the class, both at the personal and at the social level. In

the notebook, the elements of the theory are fixed into an

ordered sequence, so that both the elements and their logical

relationships are represented. It is a temporary result of the

individual and the collective endeavour towards the social

construction of mathematical meanings, and at the same

time it is the point of departure for further evolution. For this

reason the notebook is a semiotic object that plays a key role

in the evolution of signs within classroom activities.

The crucial phase in the evolution of meanings character-

izing the semiotic mediation process occurs in a mathematical

discussion (Bartolini Bussi 1996, 1998): social interaction

around some mathematical content organized by the teacher

and involving the whole class. Maintaining a difficult balance

between free students’ interventions and the achievement of

her didactic aims, the teacher has to orchestrate social inter-

actions in the classroom, fostering the development of a dis-

course referring to the activities carried out with the artefact,

but also explicitly relating such a discourse to the specific

mathematical meanings that are the goal of the didactic

intervention. In the following, I will present examples of a

teacher’s orchestration of a collective discussion, where

besides the standard strategies that are used by the teacher to

manage discussions in whichever context, specific strategies

are performer-related to the use of specific Cabri tools. In the

first example I shall analyse the critical starting point of the

activities on Cabri figures, to show how the teacher initiates

the development of the meaning of geometrical construction

during a mathematical discussion and exploits the semiotic

potential offered by Cabri.

6 The first example: the germ of the meaning

of geometrical construction

As said before, the theoretical nature of a geometrical

construction consists in the fact that each action carried out

with a tool corresponds to the validity of specific properties

of the obtained figure, thus the correctness of a construction

is given by the correctness of the whole sequence of

actions, that is, the construction procedure. In the evolution

from a practical to a theoretical perspective, the key point

is the move from validating the correctness of a specific

drawing to validating the correctness of the sequence of

actions that led to it. In other words, the key point is the

shift of focus from the product to the procedure that pro-

duced it; this shift of focus is not spontaneous and is dif-

ficult to be achieved (Schoenfeld 1985; Mariotti 1996).

Acting in Cabri may foster this shift, providing a context

within which the request to evaluate a procedure rather

than a product becomes meaningful. Drawing a specific

geometric figure, for instance a square, may be carried out

in Cabri through different kinds of construction procedures:

some maintain and others do not maintain certain geo-

metric properties when dragged. As soon as dragging is

accepted as a validating tool, a meaningful problem arises:

why are some constructions stable and others not?

Answering such a question requires a focus on the con-

struction procedure rather than its product. Though acting in

the Cabri context may foster the shift from evaluating the

product to evaluating the procedure, nevertheless, the con-

text itself is not sufficient and the intervention of the teacher

becomes determinant to accomplish such a shift.

In the next sections, I will present the analysis of a

collective discussion where the role of the teacher’s inter-

vention is clearly highlighted.

6.1 The scenario of the activity

The experiment concerns a 9th grade class of a scientific

high school (Liceo Scientifico), with 19 out of the 23 pupils

of the class participating in the first cycle of activities and

specifically in the collective discussion. The first part of the

didactic cycle takes place in the computer laboratory;

pupils sit in pairs at the computer and have been allowed to

explore the software for about half an hour to get a first

acquaintance with the Cabri environment. Then the fol-

lowing task is presented:

Construct a line segment on the screen. Construct a

square which has this line segment as one of its sides. Write

down a description of your procedure and of your reasoning.

The task requires both acting with the artefact Cabri and

reporting on this action. According to the didactic cycle, a

semiotic process is initialized and concerns the interpretation

and the use of signs related to the idea of construction and the

use of the artefact. At this initial point, the term ‘‘construct’’ is

used on purpose because of its ambiguity: it is a term used in

common language and to some extent in school language; it is

commonly related to drawing a figure, yet it is not expected to

have a theoretical meaning. Thus different interpretations of

the task, and accordingly different solutions, are expected,

together with a verbalization of the procedure.

At the end of the lab session, the Cabri figures are saved

in the working folders, and the written productions of the

students are collected. With the aim of organizing the fol-

lowing collective discussion the teacher analyses the dif-

ferent solutions and classifies them according to whether

the Cabri figures have been obtained by using tools refer-

ring to geometrical properties and/or referring to perceptual

control. Three different types of procedure are identified:

the first type consists of drawing four segments arranged in

446 M. A. Mariotti

123

a square ‘‘by eye’’; the second type consists of geometri-

cally correct procedures obtained by using circles, or per-

pendicular and/or parallel tools; and the third type consists

of partially correct procedures, that is, part of the square is

realized by using Cabri tools and part of the square is

realized through arranging the image ‘‘by eye’’.

6.2 The evolution of the meaning of construction

The following day the teacher opens the discussion by

suggesting analysing the solution given by Group 1

(Giovanni and Fabio).

The discussion takes place in the computer lab, with the

computers connected in a local net so that the teacher can

manage and control the students’ screens. Thus the Cabri

figure obtained by Group 1 is sent and displayed on all the

screens (Fig. 1a), but it can be dragged only from the

teacher’s computer. The figure was obtained by drawing

four consecutive segments and arranging them in a square

by eye. Students are requested to evaluate the constructed

figure and explain why it is correct or not. All the pupils

agree on the fact that according to properties of a square,

the correctness of the answer can be evaluated by mea-

suring sides and angles of the figure.

The main elements arising from the discussion as shared

criteria of evaluation are the use of measure and the precision

related to such use. In spite of the attempts of the teacher to

direct pupils towards a process of generalization, the debate,

lasting for a good while, does not change the focus that

remains on the drawn figure and on the need to measure

angles and sides, bearing witness to an empirical attitude to

checking the correctness of the solution. The discussion is

interrupted by the end of the lesson and resumed the sub-

sequent day when one of the pupils summarizes the main

points of the previous discussion. At this point the teacher

opens again the figure proposed by Group 1 (Giovanni and

Fabio) and begins to drag one of the vertices (Fig. 1b).

Excerpt #1

Transcript Analysis

13 Chorus: That is a quadrilateral

… with different sides.

The solution was obtained using

perceptive adaptation, thus after

the dragging the figure is

deformed14 Marco: with four different

sides …15 I: Is it a square??

16 Chorus: No, no, no Students immediately recognize

that the figure is no more a

square

17 I: OK. Let’s consider another

solution … I’ll show you what

has been done by Dario and

Mario …

The teacher asks pupils’ opinions

and proposes the solution from

Group 3 (Dario and Mario)

The intervention of the teacher aims at introducing the

dragging function and its effect on a figure in order to

destabilize students’ evaluation of the solution based on the

produced drawing. Everybody agrees that the figure is no

more a square. Immediately after, the teacher proposes

another solution. From its appearance it becomes clear that

Dario and Mario used some of the Cabri tools (the Circle

tool and the Perpendicular tool) to obtain the new figure.

Fig. 1 a What appears on the screen. b What appears on the screen after dragging

Introducing students to geometric theorems 447

123

Excerpt # 2

Transcript Analysis

21 I: Well, I’d like to know youropinion about the constructionof Dario and Mario

22 Marco: They did a circle, thentwo perpendicular lines …

Marco identifies some of thetools used by his classmates

23 I: Do you know what theystarted from?

The teacher asks for the first step

24 Michele: We can use thecommand ‘‘history’’

Michele proposes to use theHistory tool

25 I: Let’s do it. They took asegment, then they … Theydrew a line perpendicular to thesegment, then the circle … inyour opinion, what is it for?What is the use of it?

The teacher activates the Historytool and while the softwaredisplays the constructionprocedure step by step sheinterprets each step as an actionof the users

SILENCE The students are puzzled and donot react

Is there a logic in doing so, or didthey do it just because they feltlike to draw a perpendicularline … a circle … Alex, tell us…

The teacher insists

26 Alex: The measure of thesegment is equal to the measurereported by the circle on theperpendicular line

Alex proposes an explanation

27 I: You mean that the circle isused to assure two equalconsecutive segments, the firstone and that on theperpendicular line … and theperpendicular …

The teacher reformulates whatAlex said

28 Chorus: is used … to obtain… an angle of 90�

The students follow the model offormulation proposed by theteacher

29 I: I know that the square hasan angle of 90� and four equalsides or three equal angles …then let’s see if it is true … let’sgo on. Intersection between lineand circle. They (Dario andMario) determined theintersection point between theline and the circle … why didthey need that point?

The teacher initiatesestablishment of a relationshipbetween the properties of asquare and what could havebeen done by Dario and Mario.She invites the students tointerpret what they did makingexplicit the scope of their action

30 Chiara: The intersection pointbetween the line and thesegment …

Chiara follows the invitation ofthe teacher

31 I: And what should you drawfrom there ?

The teacher invites the studentsto guess what could have beendone

32 Chiara: A segment,perpendicular to the line

Chiara and her classmates acceptthe game

33 I: What else??

34 Chorus: Parallel to thesegment …

35 I: Let’s see what did theydo …

Finally, the teacher proposeschecking what had been done

Following Marco’s suggestion the teacher executes the

command History, describing each step of the construction

as an action of the users. In this way she evokes the idea

that behind the drawn figure there is a sequence of

actions. Thus, the meaning of construction as a procedure

may emerge, according to the teacher’s aim to foster the

shift from focusing on the drawing to focusing on the

procedure.

At a certain point, she interrupts the description and

starts asking the pupils to reflect and try to detect the

‘‘motivations’’ for the actions that are displayed by the

History tool. This intervention (23) aims to provoke a new

shift from the procedure to the justification of the proce-

dure itself. The silence that follows shows that pupils do

not immediately understand the teacher’s question. How-

ever, after the teacher’s prompt (25), Alex (26) expresses

the relationship between two of the segments according to

the series of commands previously executed, and the tea-

cher (27) reformulates the statement of Alex in terms of

motivations: ‘‘You mean that the circle is used for assuring

two equal consecutive segments ….’’ The chorus appro-

priates the expression ‘‘… is used for …’’, introduced by

the teacher, and continues in terms of motivation. This kind

of intentional semiotic game that we call an interpretation

game shows its effectiveness: the shift from the description

of the procedure to the motivation of the procedure occurs

through the appropriation of the teacher’s words.

The discussion continues, developing the analysis star-

ted by the teacher of other construction procedures: each

time, the pupils are asked to foresee the next step, provide

motivation for it and then compare it with the step recorded

by the History tool. This exchange between teacher and

pupils is what we have called a prediction game. It follows

the analysis of a particular procedure, but at the same time

allows one to propose choices different from those already

made.

It is important to remark that the History tool provides

the basis for the analysis, but it is not sufficient to

accomplish the shift from the operations to the intentions:

the software only shows the steps’ sequence, whilst

through the activation of the interpretation game, rein-

forced by the prediction game, the teacher may introduce

the point of view of the geometrical reasons for each of

the actions performed. In other words, the intervention of

the teacher aims at making pupils figure out how their

classmates performed the drawing and why they did that;

the students are invited to imagine themselves acting

with Cabri tools following a similar logic. In order to

guarantee that the final product will always result in a

square the construction has to be accomplished according

to the geometrical properties that characterize it; at this

point, the relationship between stability by dragging and

geometrical solution of a construction task emerges, and

it is possible to share (Excerpt #3) the stability by

dragging as the criterion for accepting a solution for a

construction task.

448 M. A. Mariotti

123

Excerpt # 3

Transcript Analysis

55 I: […] What is better, a square

which is always a square, also

when it is dragged, or a square

which can become whatever

else?

The teacher focuses on the

invariance of the figure in

relation to the use of dragging

and its effect

56 Chorus: A square which

always remains a square

Pupils seem to accept the

dragging test as validation

6.3 Towards a first shared meaning of ‘‘construction’’

In order to elaborate further on the relationship between the

stability of a Cabri figure and the geometrical solution of a

construction problem, the teacher proposes the analysis of

other solutions using the History tool; the pupils meet

difficulties in autonomously performing the prediction

game and there is often the need for the teacher to inter-

vene and solicit responses. The pupils go through and

analyse the links created by the different tools used in the

constructions. The discussion ends (Excerpt #4) with the

final evaluation of the different types of solution analysed,

and eventually the term ‘‘geometrical construction’’

appears (186) with a meaning clearly related to the stability

by dragging and the use of related properties: the solution

of a construction task must be a figure which cannot be

messed up by dragging (183).

Excerpt #4

Transcript Analysis

180 I: Then, there is a hierarchy:

a bad drawing, a slightly better

one, and that which is correct

… what would you say?

Request to evaluate the different

solutions. The hierarchy is

implicitly based on the idea of

stability by dragging

181 Daniele: The ‘‘first’’ is that

(Dario and Mario)

182 I: Why is it so? What is the

motivation?

Request to make explicit the

reason

183 Chorus: Because it is

impossible to mess it up

Explicit reference to the dragging

test

184 I: Why is it impossible to

deform it? How was it

constructed? What did they

used?

Request to refer to the

geometrical properties

characterizing the figure

185 Chorus: All in function … The relation between the

properties is recognized186 Fabio: They used a

geometrical construction

From now on pupils agree on the acceptance of a

solution in terms of the dragging test, but it is also clear

that it is possible to explain why a solution is acceptable

and that this can be done by referring to the geometrical

properties realized by the use of the different commands.

This is just the first approach to the construction prob-

lem, and the facilities offered by the artefact Cabri allowed

the geometrical meaning of construction to emerge. As a

matter of fact, the episode presented above, and the semi-

otic games highlighted, provide a good example of a

mediation intervention that can be accomplished by the

teacher: exploiting the semiotic potential of Cabri, she

guides pupils to shift the focus from the empirical level of

the drawing produced to the construction procedure and its

geometrical motivations, as they emerge from the solution

of the construction task accomplished through the use of

specific Cabri tools.

As the teaching sequence goes on, the meaning of the

term construction evolves and enriches. This evolution

comes from the elaboration of a geometrical theory based on

the use of specific construction tools. The choice of an

appropriate set of Cabri tools to be added to an empty menu

brings out the set of construction axioms that constitute the

first core of the geometry theory within which students have

to produce the validation of their construction procedures.

Then, as long as new constructions are produced, the cor-

responding theorems are validated and added to the theory,

while the corresponding Cabri tools are added to the avail-

able menu (Mariotti 2000, 2001). It is against this back-

ground that the following example has to be interpreted.

7 Example of collective discussion on the notebook

The second example concerns a collective discussion that

took place after some didactic cycles and is aimed at

revising students’ personal notebooks.

Among the activities asking students to personally

elaborate mathematical texts, the writing of the mathe-

matics notebook has a very specific role. As a matter of

fact, the notebook is a very special kind of text: it belongs

to the sphere of the individual, but it also makes sense with

respect to the social sphere, primarily to the community of

the class but also, through the link of the teacher, to the

community of mathematicians. Thus the collective dis-

cussions that are centred on revisiting the notebooks have a

special function in the teaching–learning process (Cerulli

and Mariotti 2003). Taking into account the specific goal of

the teaching–learning sequence concerning students’

introduction to a theoretical perspective, the discussion on

the mathematics notebook has the objective of elaborating

on mathematical meanings related to the content of the

specific geometry theory, but also to meta-theoretical ideas

such as the specific status of axioms or of theorems. On the

one hand the discussion aims at rooting the meaning of the

specific axioms and theorems referring to the different

Introducing students to geometric theorems 449

123

Cabri tools and their different use in the construction task;

but on the other hand the discussion aims at de-contextu-

alizing such meanings, fostering the elaboration of the

mathematical meaning of ‘‘theory’’.

As said, the episode we are interested in concerns a

collective discussion launched with the aim of revising

students’ personal notebooks, and starts when the discus-

sion has already progressed towards an agreement about

what has to be taken as the start of the theory. At a certain

point the teacher asks students to recall the specific con-

struction task that led to the formulation of the first axiom.

Excerpt #5

Transcript Analysis

66. Teacher: Do you remember

the first activity that we did …that from three segments not

always …

The teacher invites the students

to recall the activity with Cabri

67. BAZZ: You can construct atriangle…

A first contribution refers to the

construction problem that

originates the first axiom

68. Teacher: What does it [thenotebook] say then?

The reference to the ‘‘authority’’

of the notebook intends to

stress the meaning of the theory

as a reference frame for the

community

69. MOR: axiom 1 triangular

inequality… given three

segments it is possible toconstruct a triangle which has

these segments sides, if the sum

of the minor segments is larger

or equal to the largest segment

The formulation of the axiom

still maintains the reference to

the construction problem that

originated it

70. Teacher: It gives me a rule to

construct …MOR: the triangles

… Teacher: with the segments

that I have already defined …

The intervention of the teacher is

intertwined with that of the

student

71. MOR: This is an axiom, not a

theorem…The student emphasizes the

theoretical status of the

statement

72. Teacher: Why is it an axiom?

Could we give a proof of this?

The teacher endorses MOR’s

intervention and asks for the

approval of the classmates

73. Chorus: No…74. BERN: I could make a

drawing with three segments …but it agrees with the other

three…

The students come back to

referring to the practical

drawing, to generalizing to all

the possible occurrences

75. MOR: but they are infinite…76. BAZZ: But how can we

explain that… I mean, prove

that it is true…

The status of the axiom, as a

statement to be accepted

without proof, is restated with

the interesting allusion to the

impossible cases77. MAS: The axiom includes all

the cases, also of the segments

with which we can’t construct

the triangle…

This short excerpt clearly shows the twofold perspective

from which the discussion is orchestrated by the teacher. On

the one hand (67) the content of Axiom 1 is recalled and its

meaning related to the construction rules that emerged from

the experience in Cabri; on the other hand the distinction

between axioms and theorems is raised after the intervention

of MOR (71). While the teacher suggests explaining it in

terms of proof, students elaborate on the relationship between

drawing and proving, making clear that drawing cannot be

used to explain … to prove that it is true (76). Such a dis-

tinction comes again later when the teacher prompts the

students to explain why the construction of an angle bisector

leads to introduction of the first theorem of the shared theory.

Excerpt #6

Transcript Analysis

139. Teacher: Thus let us behave

as … if this is a theorem then

there should be a premise (in

Italian, ipotesi) and a

conclusion (in Italian, tesi).

Which is our premise, in your

opinion?

The teacher refers to the theorem

that validates the construction

of an angle bisector, and asks

for identifying the premise and

the conclusion

140. STEF: The triangle… Different proposals are made,

some partially correct, as that

of BERN141. BERN: The angle from

which we started

142. Teacher: Then …143. Chorus: The line …144. Teacher: The initial data …

those from which I have started

… [those] are my premise …then what else do I know that is

true? …

The intervention of the teacher

refers again to the activity with

the artefact where the meaning

of premise has a counterpart in

the initial data of the

construction

145. Chorus: The construction

Though very short, this episode shows how the discus-

sion may develop, relating meanings emerging from the

construction experience to mathematical meanings related

to the notion of theorem, such as those of premise and

conclusion, or to the distinction between axioms and

theorems.

As a final remark, I present some evidence of the efficacy

of the collective discussion in the elaboration of personal

meanings into mathematical meanings. Analysing students’

personal reflections—written in their reports on this collec-

tive discussion—we can find clear traces of the complex

work of texture developed by the teacher to lead students to

grasp mathematical meanings without losing the original

meaning rooted in their experience with the artefact. Con-

sider the following excerpt of Cris’s writing report.

450 M. A. Mariotti

123

In Cris’s words we clearly find a trace of the discussion

dealing with the relationship between theorems, conceived

as the unity of a statement and proof, and constructions;

there also appears the idea of an order in the sequence of

the theorems, corresponding to the development of the

theory. Similar traces of the discussion can be found in

the following two examples—the last excerpt is perhaps

the most impressive, showing how far the elaboration of

mathematical meanings of theory can progress.

8 Conclusions

As said above, this paper gives an example of a teacher’s

intervention designed to face a specific educational issue

and the related didactic difficulties.

The general framework of TSM provides a teaching–

learning model inspiring the teacher’s action and leading

classroom practice. A semiotic lens allowed our analysis to

highlight specific semiotic strategies to accomplish the

educational goals. The TSM postulates that an artefact—in

this case the DGS Cabri—could be exploited by the teacher

to make students develop genuine mathematical meanings

through purposefully organized classroom activities. The

intervention was designed as a sequence of individual and

collective activities centred on the development of semiotic

processes. Though each type of activity plays an essential

role and contributes to the process of semiotic mediation,

social interactions orchestrated by the teacher in true

mathematics discussions play a key role. The distance

between personal meanings, rooted in the phenomenology

of the activities performed with the artefact, and mathe-

matical meanings evoked by such experience, can be

bridged only through a careful and purposeful semiotic

action of the teacher that, as discussed in this paper, can

take the form of specific semiotic games. The use of the

term semiotic game, firstly introduced in Mariotti and

Bartolini Bussi (1998), is consistent with the use of the

same term in other studies (Arzarello and Paola 2007) and

as clearly pointed out by these authors it may appear as a

practice ‘‘rooted in the craft knowledge of the teacher, and

most of times is pursued unconsciously by her/him’’

(p. 18). Identifying and making such practices explicit

contributes to outlining a possible model of the teacher’s

intervention in the development of the semiotic mediation

process. Hence, the findings presented in this paper indi-

cate a promising research direction for the study of

teachers’ action in classroom activities—the identification

and the explicit description of a new class of semiotic

games that may constitute a powerful corpus of semiotic

strategies that teachers can appropriate and exploit to foster

semiotic mediation processes related to the use of specific

artefacts. Such a semiotic lens used in describing and

modelling teachers’ intervention in the classroom consti-

tutes a promising innovation avenue to be explored.

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