2011. Vol.2, No.3, 252-263
Copyright © 2011 SciRes. DOI:10.4236/ce.2011.23034
Mediation in the Construction of Mathematical Knowledge: A
Case Study Using Dynamic Geometry
Gilmara Teixeira Barcelos1, Silvia Cristina Freitas Batista1,
Liliana Maria Passerino2
1Instituto Federal de Educação Ciência e Tecnologia Fluminense, Cam pos dos Goytacazes,
Rio de Janeiro, Brazil;
2Universidade Federal do Rio Grande do Sul, Porto Alegre, Rio Grande do Sul, Brazil.
Email: firstname.lastname@example.org, email@example.com. br, lili firstname.lastname@example.org
Received July 5th, 2011; revised July 20th, 2011; accepted July 26th, 2011.
According to the social-historical theory, interaction between individuals plays a major role in building the hu-
man being. It is through such interpersonal relationships that the individual’s psychological development takes
place. Therefore, in school education, interaction with teacher and with classmates is essential; in addition, their
mediation along the educational process is an extremely relevant factor t o achieve goals. Mediation also includes
use of tools and signs in the social context, enabling the development of superior psychological processes. This
research was carried out using the social-historical theory as theoretical background and considering that digital
technologies enable adequate spaces to perform investigation activities. It was performed with a group of stu-
dents Mathematics teachers and aimed at analyzing a learning situation about Triangle Similarity Cases using a
Dynamic Geometry software program. The importance of mediation (tools, teacher, classmates) in knowledge
building was evaluated by performing investigative activities aided by the software. Qualitative research,
through case study, was chosen and three data collection techniques were used: participant observation, ques-
tionnaire and semi-structured interview. Research-related tasks were divided into three stages: preparation; de-
velopment and analysis of collected data, relating them with the adopted theoretical background. Considering all
that was observed and collected through questionnaires and interview, it is possible to state that mediation
played an important role during software activities.
Keywords: Mediation, Social-Historical Theory, Mathematics, Dynamic Geometry
Many studies, using varied epistemological approaches, have
been developed with the aim of analyzing Mathematics learning
in schools (D’Ambrósio, 2001; Moysés, 2007; Bueno-Ravel &
Gueudet, 2009). Different forms of approaching mathematical
themes have been used (Artigue, 2002; Johnston-Wilder &
Pimm, 2004; Maschietto, 2008). In this sense, the role of inter-
action in knowledge building,1 either mathematical or not, has
been stressed (Laplane, 2000; Preti, 2002; Moysés, 2007). In-
teraction between individuals plays a major role in human de-
velopment (Vygotsky, 1978). It is through interpersonal rela-
tionships that the individual’s psychological development takes
A particular type of interaction is crucial in this study: me-
diation. This is extremely relevant in the development of edu-
cational processes, and the role of a mediator, encouraging
reflections through the help provided, is important for the stu-
dent’s autonomy and knowledge appropriation (Passerino et al.,
2008). Mediation takes place in the Zone of Proximal Deve-
lopment (ZPD), understood as a joint action structure in any
context in which there are participants with distinct levels of
responsibility and competence, who work together toward
problem solving (Cole, 1986). Mediation also includes use of
tools and signs in the social context, and the combination of
such mediating resources enable the development of superior
psychological processes (Vygotsky, 1978).
Two essential qualitative changes occur by using tools
(and/or signs). One is that use of external marks will become
internal mediation processes: this mechanism is called inter-
nalization process by Vygotsky (1978). The other change is that
symbolic systems are developed (language is the basic sym-
bolic system), which organize signs into complex and articulate
structures (Oliveira, 1993). Both internalization process and use
of symbolic systems are crucial to the development of superior
mental processes and show the importance of social relation-
ships between individuals in building psychological processes.
From the importance of social relationships results the impor-
tance given to the role of teachers and classmates (teacher and
classmate mediation) aiding the learning process.
Use of tools, associated with teacher and classmate media-
tion in investigative activities, may contribute to learning of
Mathematics. Mathematical investigations involve concepts,
procedures and representations, but the focus is on the conjec-
ture-test-demonstration sequence (Ponte et al., 2005). These
authors claim that, in Mathematics, “investigating is finding out
relationships between known or unknown mathematical ob-
jects, trying to identify their respective properties” (Ponte et al.
2005: p. 13).
1Studies in the educational area have shown the potential of mediation fo
learning and subject autonomy. Results can be seen in Coll (1990); Wertsch
(1998); Doise (1988), among others. In mathematical investigations, digital technologies can be
G. T. BARCELOS ET AL. 253
favorable resources. Such technologies highlight the role of
graphic language, relativizing the importance of calculation and
symbolic manipulation (Ponte et al., 2003). The graphic focus
fosters reflections and critical analyses, enabling less mechanic-
cal procedures than those purely algebraic. Digital technologies
are mediating tools that, whenever well used, can contribute to
creation of ZPD. Use of these technologies, however, should be
conscious and critical.
This paper describes a study performed with a group of un-
dergraduate students in Mathematics at Instituto Federal de
Educação, Ciência e Tecnologia Fluminense (IF Fluminense) -
Campos dos Goytacazes, RJ/Brazil. It aimed at verifying the
importance of mediation (tools, teacher and classmates) in
knowledge building at a situation of mathematical learning
about Triangle Similarity Cases using a dynamic geometry
software program. This article is structured in other five sec-
tions, in addition to this introduction. In section 2, two essential
themes on which to base the research are briefly discussed: the
social-historical theory and the importance of using digital
technologies in the teaching and learning process. Section 3
presents the objectives and methodology of this study. Section
4 describes the activities performed and their objectives. Sec-
tion 5 presents analysis and discussion of results in the light of
the adopted theoretical background. Finally, Section 6 brings
some considerations on the research and on obtained results.
This section focuses on the social-historical theory, mainly
considering the concepts of mediation, internalization and ZPD,
required for the study described herein. Next, the issue of using
digital technologies in the teaching and learning process is dis-
cussed, showing, according to the view of some authors, the
advantages such technologies may bring to education and par-
ticularly to Mathematics, if properly used.
A key concept to understand the social-historical theory is
mediation. Mediation is the intervention process of an interme-
diate element in a relationship, which is no longer direct and
starts being mediated by this element (Vygotsky, 1978). As the
Marxist theory conceives the tool mediating man’s labor acti-
vity, Vygotsky (1978) conceives the notion of sign2 mediating
thought. Sign acts as a tool of psychological activity similarly
to the role of tool at work. However, Vygotsky (1978) stresses
that this analogy does not imply identity between similar con-
cepts and states that the fundamental difference between sign
and tool consists of the different forms by which they guide
human behavior. The tool’s function is to serve as the conduc-
tor of human influence on the object of activity, it is externally
oriented, it must lead to changes in objects. The sign, on the
other hand, changes nothing in the object of a psychological
operation. It is a means of internal activity aimed at mastering
oneself; the sign is internally oriented (Vygotsky, 1978).
Human activity can only be understood if we consider that
such tools and signs participate in it. This participation, which
is not auxiliary but central to the development of activity as it
acquires qualitatively different characteristics, is called media-
tion in the social-historical theory.
When signs are incorporated into the structure of cognitive
processes, as an essential means of orientation and domain in
psychic processes (Vygotsky, 1993) we are facing internaliza-
tion. Therefore, internalization is the process through which an
external operation is reconstructed and starts occurring inter-
nally (Vygotsky, 1978). Human beings, in their development
process, no longer need external marks and start using internal
signs (mental representations that replace objects in the real
world). Internalized signs represent objects, events or situa-
In addition to mediation through tools and signs, there is
human mediation (Kozulin, 2003). To Vygtosky (1978), the
main function of intentional action of one subject over another
in a child’s development is based on the idea of mediation.
Thus, education is seen as a social space for mediation in which
tools, signs and people mediate the development process (Kar-
pov, 2003). Mediated learning necessarily goes through active
participation of an adult or more experienced partner that se-
lects, modifies and interprets context conditions present in the
learning process of another subject (less experienced) (Gindis,
2003). According to Vygotsky (1978), functions in a child’s
development occur in two different levels: first in the social
level (between people, as interpsychological category) and then
in the individual level (child’s self, as intrapsychological cate-
gory). This is equally valid for voluntary attention, logical
memory and concept formation. In fact, all superior functions
are originated from real relations between individuals. Al-
though there are many forms of mediation, it has not yet been
possible to identify a pattern for them because they are con-
text-dependent (Kozulin, 2003).
Moysés (2007) clarifies that each psychic function that is in-
ternalized requires a mental restructuring as such function will
interact with other existing functions in the child’s mind.
However, this is not about overlapped layers, but coordination
between the new function and other existing functions.
As to the formal educational process, Moysés (2007), in his
research,3 verified that mediation through different resources
favors knowledge building by the student. A type of mediation
often present at school is that developed by another subject,
who creates learning situations in which tools and signs are
intentionally introduced in an action (Baquero 1996). Such
teacher mediation acts directly on each student’s ZPD. ZPD is
the distance between the actual developmental level, as deter-
mined by independent problem solving, and the level of poten-
tial development, as determined through problem solving under
adult guidance or in collaboration with more capable peers
(Vygotsky, 1978). The author states that a child’s state of men-
tal development can only be identified if the level of actual
development and ZPD are revealed.
In school education, the teacher is this more experienced sub-
ject that through guide-questions, examples and demonstrations,
performs interventions that can help the student do a task
(Moysés, 2007). It should be stressed that interaction between
students also causes interventions in their development. In
3Moysés (2007) reports a study in the area of Mathematics including
6th-grade students from a public school, based on the theory proposed by
Vygotsky and his followers. The researcher investigated possible applica-
tions of this theory for the educational process being studied.
2Examples of signs are language, counting systems, mnemonic techniques,
algebraic symbolic systems, among others (Moysés, 2007).
G. T. BARCELOS ET AL.
groups of students generally heterogeneous as to previous
knowledge, a more advanced student may contribute to the de-
velopment of others (Oliveira, 1993).
Moysés (2007) stresses that, although learning through de-
monstrations presupposes imitation, it should not be seen as a
copy, but as something involving constructive experimenting.
Such experimentation enables concept formation.
Concept formation was also studied by Vygotsky (1986), who
classified them as spontaneous and scientific and analyzed how
they interacted. Spontaneous concepts are those in which the
child learns in her daily life, originating from contact with ob-
jects, facts, phenomena, among others, of which she is fre-
quently unaware. Scientific concepts are systemized, hierar-
chized and intentionally transmitted, generally following a me-
thodology. These are essentially the concepts learnt at school.
Development of scientific concepts presupposes a conscious
and consented relation between knowledge subject and object.
Their formation is a mental operation that demands attention on
the issue, deriving essential aspects from it and inhibiting se-
condary concepts and aiming at wider generalizations through
synthesis (Vygotsky, 1986). There is then a process of analysis
and synthesis, abstraction and inhibition of given characteristics;
in addition, there is movement of thought within the pyramid of
concepts, constantly alternating from the particular to the ge-
neral, and vice-versa (Vygotsky, 1986).
Several topics dealt with in this section were observed in the
research and are commented in the analysis and discussion of
results (Section 5).
Use of Digital Technologies in Mathematical Learning
Digital technologies enable adequate spaces to perform in-
vestigation activities, in which the teacher observes, guides,
suggests questions and reflections, encourages socialization of
ideas and fosters critical analysis of results.
However, it should be highlighted that such technologies are
favorable tools to new educational practices, but alone they are
not the solution for educational problems. As supported by
Valentine and Soares (2005), change is not in technology itself,
but in new relations it stimulates. In this sense, it is crucial that:
1) there is a restructuring of the student’s and teacher’s role; 2)
the focus is on learning, not on teaching; 3) the teacher provides
interventions and guidance based on social and cognitive ob-
servations of students; 4) relations emerging from interaction
are considered, enabling learning to learn and development of
competences (Valentine & Soares, 2005).
In Mathematical learning, these technologies can collaborate
to content understanding since they facilitate things through use
of computational ability, graphic visualization, discovery and
confirmation of properties, possibility to run experiments with
data collection and problem modeling, speculations, among
others (Baldin, 2002). Particularly, use of online resources and
specificity of some of them should be the attention focus in
studies on Mathematical Education (Bueno-Ravel & Gueudet,
2009). According to the authors, this requires theoretical evolu-
tion that enables capturing the phenomena associated with the
teaching and learning process.
Educational software programs, among other digital tech-
nologies, are tools that can collaborate to improvement of
learning environment quality. These allow exploration, visuali-
zation and experimentation of varied situations, of which some
are virtually impossible to be performed without their aid.
Compass and Ruler,4 a program used in this research, is a
free software program that has a quite open proposal, favoring
knowledge building. Its purpose is to work with dynamic geo-
metry, that is, its function is to allow creation of geometric
constructions that can be changed by moving one of the basic
points and enabling preservation of original properties. This
program has no specific content to be studied, but tools for
geometric constructions to be used, depending on the intended
objective. Some positive points of Compass and Ruler: 1) it is
easy to use; 2) it has a pleasant, didactic interface; 3) enables
simple to quite complex geometric constructions, depending on
the user’s ability and need; 4) it favors knowledge building; 5)
it stimulates creativity and questioning; 6) it enables interacti-
vity; 7) it offers resources that make Internet constructions
Computer programs focused on dynamic geometry are very
useful resources to study Geometry since they allow movement
of constructions, favoring visualization of a varied number of
situations that collaborate to understanding of the theme under
study. In addition, such resources enable relating empirical
explorations and formal tests, and can be used to highlight a
deductive argument in demonstrations (Guven, 2008).
Fainguelernt (1999) states that studying Geometry is ex-
tremely relevant for the development of spatial thinking. Ac-
cording to the author, thinking activated by visualization in-
volves intuition, perception and representation, which are es-
sential skills for world reading and not to have a distorted view
Conviction of the authors of this study regarding the impor-
tance of digital technologies for Mathematical Education, co-
herent with the ideas of the above-mentioned authors, is based
on research and experience in the classroom. This favorable
position in relation to digital technologies guided this entire
Research Configuration: Objectives and
The general objective of this study was to verify the impor-
tance of mediation by teacher, classmates and tools in know-
ledge building at a situation of mathematical learning about
Triangle Similarity using the Compass and Ruler program. Se-
condary to the above, the following objectives were also in-
cluded: 1) identify and analyze forms of teacher mediation
(individual and in group); 2) verify whether and how collabora-
tion between classmates in group activities took place; 3) iden-
tify forms of software use in problem solving; 4) verify whether
the activity performed using the software collaborated to the
understanding of involved concepts.
The sample included a group of students of Geometry II,
Mathematics Teacher Education Program, IF Fluminense
Campus Campos-Centro (RJ/Brasil) —2nd semester of 2008.
This course is taught by one of the authors and the group was
composed by 15 students; these aspects facilitated real-life
It was a qualitative research through case study. Preference
for a qualitative study was based on some of its basic charac-
4Created by René Grothmann (professor of the Catholic University o
Eichstätt, Germany), available at
G. T. BARCELOS ET AL. 255
teristics that, according to Bogdan and Biklen (1998), are hav-
ing natural environment as a direct source of data and the re-
searcher as its main instrument; collected data are predomi-
nantly descriptive; higher concern about the process than about
The advantage of using a case study lies on the possibility of
having deeper knowledge as it has no restrictions of comparing
the case study with other cases. Furthermore, in this type of
study, researchers have more time to adapt their tools, change
their approach to explore unpredictable elements, need some
details and create an understanding of the case that takes all that
into consideration (Laville & Dionne, 1997).
Data collection techniques were basically participant obser-
vation, questionnaire and semistructured interview. In partici-
pant observation the researcher is integrated and takes part in
group activities. A reflexive position toward observed subjects
is needed, making notes, recording and collecting data through
tools considered to be convenient during the investigation.
There is no restriction to the investigation neither an a priori
analysis structure; therefore, it is possible to have a wider view
of the situation and consider many aspects, without isolating
them from each other. However, it should be stressed that, dur-
ing participant observation it is not always possible to make
notes during the process, which demands discipline and good
memory by the observer (Laville & Dionne, 1997). Although
this technique has some inconveniences, such as those men-
tioned above, it is considered that its advantage in terms of
context understanding makes it an adequate technique for this
research. As a complement to participant observation, ques-
tionnaires as well as a semi-structured interview as an attempt
to obtain possible information that was missed in participant
observation, or even to confirm what was observed.
The questionnaire was used because, among other advan-
tages, it allows anonymity and does not expose subjects to the
influence of the interviewers’ opinions and personal aspects.
However, this technique prevents knowing the circumstances in
which it was completed (Gil, 1999), which explains use of the
interview. Using more than one data collection technique en-
ables a better triangulation of sources and methods (Yin, 2003).
Questionnaires were completed by students Mathematics
teachers in the classroom, in the presence of the researchers
during study stages. Efforts were made to ensure all questions
were clear, so that there were no misunderstandings.
The interview is wider than the questionnaire in terms of or-
ganization. The interviewer, who is not restricted by a docu-
ment, can better explain some questions throughout the inter-
view or rephrase them to favor understanding (Laville & Dionne,
1997). This study used a semistructured interview, which con-
sists of a series of open questions, made verbally according to a
predicted order, but with a possibility of inserting new ques-
tions depending on the interviewer’s needs (Laville & Dionne,
1997). The aim was to collect data that might not have been
asked by the questionnaires neither perceived during observa-
After choosing the methodology to be used, three study
stages were defined:
- planning of the learning situation, describing actions to
be developed and strategies to be used;
- creation of exercises to be performed without using the
software, with the aim to check whether students al-
ready knew the theme to be studied;
- creation of pedagogical activities about Triangle Simi-
larity using the Compass and Ruler program;
- creation of exercises to be performed without using the
software, with the aim to check understanding of the
theme to be studied;
- creation of questionnaires and questions for the semis-
Development: application of activities and questionnaires,
observation of the entire process, register of observations
Analysis of collected data, relating them with the adopted
Research results are presented next.
This section reports study development. Activities performed
and their objectives are described, but there is no deep analysis
on the varied aspects under observation as this will be discussed
in Section 5.
The case study was performed during three meetings (once a
week), each lasting 2.5 hours. There were 15 students in the
first meeting; 10 in the second, and 14 in the third. The study
purpose was presented to the students in the beginning of the
first meeting, when the importance of their committed partici-
pation was also stressed.
Initially, exercises were proposed to diagnose the knowledge
students already had on Triangle Similarity. These exercises
were collected by the researchers5 and analyzed between the
first and the second meeting.
Next, three activities were proposed using the Compass and
Ruler software.6 These are activities of geometrical construc-
tion that aim at enabling establishment of conjectures about
triangle similarity cases. Until the end of the first meeting, most
students had finished activity 1 of this set of activities. They
were asked not to solve the other activities at home, but in the
In the second meeting, the students did the activities using
the software. Established conjectures and developed construc-
tions were discussed and presented to the group. Conjectures,
recorded in the activity sheet, and files containing the answers,
were collected and analyzed between the second and the third
Also in the second meeting, the students completed a ques-
tionnaire relative to the activities solved by using the software,
as an attempt to collect data on: 1) the role of teacher and
classmate as mediators of the learning situation; 2) the impor-
tance of activities for learning the theme; 3) the importance of
using the software as a learning tool.
In the third meeting, researchers formally demonstrated the
first case of triangle similarity. Demonstration of the second
case was only commented. Conjectures raised and discussed in
the previous meeting were then generalized.
5It should be stressed that the researchers also played the role of teachers
througho ut th e study, th at is, t hey were n ot onl y obser ving the situ ation , but
also acting as mediators.
6All students had already used the software in other learning situations;
therefore, activities to learn its features were not necessary.
G. T. BARCELOS ET AL.
Initial exercises were handed back to the students participat-
ing in the study for analysis and discussion of what was initially
done. A reflection on initial resolutions was proposed in the
light of what was studied aided by the software and discussed
in group. Exercises 2 and 3 of the initial test were then remade.
Constructions made in the software were also commented based
on formalization of the studied theme.
Next, students, as a group, solved a list of contextualized
problems without aid from the software. They were collected
for further analysis. Students also completed a second ques-
tionnaire, which aimed at collecting data on: 1) the role of
teacher and classmate as mediators of the learning situation; 2)
the influence of software activities on exercise solving without
the software; 3) the importance of using geometrical pictures
(triangles) as a learning tool.
It should be stressed that, in all described stages, attitudes
and comments considered significant to achieve the study’s
objective were observed and recorded. Based on the analysis of
everything that was developed, a semi-structured interview was
performed with five students (selected after analysis of re-
sponses and attitudes), aiming to obtain further information.
This study design was based on the statement by Cole and
Scribner (1978: p. 13):
“To serve as effective means to study “the course of a pro-
cess development,” an experiment must provide as many op-
portunities as possible to have the experimental subject engaged
in the most varied activities that can be observed and not only
rigidly controlled. An effectively used technique by Vygotsky
with this purpose was introducing obstacles or difficulties in the
task to break routine problem-solving methods. [...] Another
method was providing alternative paths for problem solving,
inclu ding many type s of materials (cal led “externa l auxiliari es”
by Vygotsky), which might be used in different ways to meet
the test’s demands. [...] A third technique was confronting the
child with a task that exceeded her knowledge and abilities, as
an attempt to evidence the rudimentary beginning of new
The three techniques suggested in the above excerpt were
applied to the study case described in this research. Students
participated in varied activities, with distinct objectives and
methodologies (activities of diagnosis, knowledge building,
socialization of conjectures and application). In addition, they
were prepared considering different levels of difficulty. Soft-
ware features enabled different paths in activity solving, which
enriched the process. Although activities did not exceed much
of students’ ability, they approached new contents for most of
them, differently than those traditionally used in classroom.
This fact allowed observation of students’ first steps in terms of
Analysis and Discussion of Results
Five subsections were organized for analysis and discussion
of results. They correspond to the sequence stages of the study.
Resolution of Initial Exercises
As described in Section 4, in the beginning of the first meet-
ing three exercises were proposed with the aim of diagnosing
the knowledge already possessed by students about Triangle
Similarity, before performing the activities that would be pro-
Since one of the authors teaches this discipline, it was known
that by the time the study was carried out the theme Triangle
Similarity Cases had not been studied in the course. However,
such fact was no guarantee that it had not been studied in an-
other context. Thus, this stage was important as it allowed
verifying whether or not the theme was new to students. This is
in accordance with Vygotsky (1978), when he stated it is im-
portant to identify the level of actual development (starting
point) to act in ZPD.
Students were told that during exercise solving they should
base their responses. Furthermore, they knew they could leave
some or even all exercises unanswered, as long as they pro-
vided a justification for it.
All students tried to individually solve the exercises using the
definition of similarity (which had already been studied in the
course), but it was affirmed that triangles were similar. This
fact implied verification of such condition (similarity cases),
which was only performed by one participant.7 It is important
to stress that, although it was an individual activity, students
exchanged ideas during problem solving.
Some students left exercises unanswered or incomplete, but
in the analysis it was possible to perceive such fact resulted
from the difficulty in identifying homologous sides, and not
because they had no guarantee of similarity. Attempt of solving
the exercises “automatically” is something common in Mathe-
matics classes, as a consequence of traditional practices.
It is important to highlight that the imitation view differs from
that of Vygotsky (1978; 1986), for whom imitation is not a
copy of a model, but the reconstruction of what is observed in
the other’s action. Imitative activity is not a mechanical process,
but an aid to perform actions beyond one’s ability, which con-
tributes to development.
In summary, except for one student, all others were unaware
or could not remember the theme being addressed in this study
—Triangle Similarity Cases. They only knew the definition of
In the first research stage, it was possible to perceive the im-
portance of figures in problem solving. Such fact was perceived
by observing that students moved the exercise sheet or repro-
duced one of the triangles on the paper, drawing it in the same
position as the other, which was expected by the researchers
based on previous experience. In exercise three, for example,
similar triangles were in different positions, so one student
asked help for the researchers to draw the lower triangle in the
same position as the larger one. This fact portrays the difficulty
in identifying homologous sides and the importance of teacher
intervention in the development of activities. According to the
group’s teacher, when triangle similarity was defined, students
were told they did not have to draw triangles in the same posi-
tion. The teacher also reported it was stressed that it would be
enough to identify respectively congruent angles so that ho-
mologous sides could be identified. In summary, this idea was
not internalized; in addition, this fact showed that even adults
need mediation through tools (triangle drawing) to think and
solve. According to Vygotsky (1978), use of mediating tools
radically reorganizes superior psychological functions, such as
7During resolution of exercises, this student asked if he could use simi-
larity cases to justify his resolution. He also claimed to have previously
studied triangle similarity cases in high school. This means that the exercise
objective was achieved.
G. T. BARCELOS ET AL. 257
memory and attention. Other students also called the research-
ers to explain they “turned” the triangles. Six students did not
solve the third activity. Although there was enough time, they
were not able to do it neither justified lack of solving.
When the first student finished and handed in the exercise
sheet, another student immediately called him to ask for help.
This student helped solving some activities by showing,
through software recourse, the sequence of constructions he
used. This fact reinforces the importance of imitation, consid-
ered in a wide sense, as previously stressed,
Problem Solving Aided by the Compass and Ruler
In the second stage three activities were proposed to be
solved using the Compass and Ruler Software. These are ac-
tivities of geometrical construction that aim at enabling estab-
lishment of conjectures about triangle similarity cases, as de-
scribed in Section 4. Observation of attitudes and questions
during classes, and analysis of problem solving allowed re-
cording important situations to learn the theme, as described
Proposed activities are not traditional, as commonly found in
didactical books. As it can be seen in Figure 1, they are inves-
tigative activities. According to Ponte et al. (2005), investigat-
ing is a powerful form of building knowledge.
Students had much difficulty in performing what was re-
quested in the first activity because they misinterpreted the
headings, displaying a certain degree of functional illiteracy8 in
problem solving. Triangles were built using the definition, and
not the condition imposed in the heading. This problem was
mitigated in activity 2, although it was still present. In addition,
in activity 1 students had difficulty transposing angles, although
they had already performed such action in another course disci-
pline, but without using any software program. Therefore, there
was the need of mediation by the researchers, who used ques-
tions to remind students of the required sequence for angle
transposition. Furthermore, some students saw the construction
in their classmates’ monitor to try to discover how it had been
done. This attitude, as described earlier, helps performing ac-
tions beyond one’s own ability, which contributes to develop-
ment. Teacher’s mediation and use of software were essential to
achieve the objective in activity 1.
Figure 2 presents the requested construction in activity 1.
Gray lines indicate the drawing needed for construction, which
can be hided using software features. All drawings are shown to
analyze constructions. Another resource used was “Repeat con-
struction,” which enables visualization of the sequence per-
formed by the student. After analysis of constructions using the
features above, it was possible to observe that, except for one
student, all of them used angle transposition. The student that
did not use this path applied parallel lines, which is a more
efficient form than that used by the other students. This student
had already studied the theme, and therefore was at a more
advanced stage than the others. According to Vygotsky (1978),
each theme studied at school has a specific relation with student
development, and such relation varies as students move from
one stage to another.
Activity 1—Compass and ruler software.
AB/MN = 0.61
BC/NP = 0.61
AC/MP = 0.61
By the end of the first meeting, only one student had con-
cluded the second activity. The second meeting was then started
to continue problem solving. In activity 2 (Figure 3) it was
necessary to construct proportional segments. Students proved
to know what the process was (which was confirmed by oral
description). However, since the construction required many
circumferences, some students drew wrong segments. By mov-
ing the construction using software features, some students
realized there was something wrong, since they had understood
the activity objective and knew that, to have similar triangles,
angles should be congruent to those of the given triangle, which
did not occur. At that moment, some students did the construc-
tion again, others asked help for a classmate and others re-
quested help for the researchers. When solving activity 2, two
students said they found it very difficult to do the construction.
The students claimed to have done it three times, but they did
not give up, felt challenged and persisted until obtaining the
correct construction (in the fourth attempt). This was the acti-
vity students had more difficulty to perform. Such fact is attri-
buted to the drawing of proportional segments that would re-
quire construction of several circumferences, as shown in Fig-
According to Vygotsky (1978), a crucial aspect of learning is
that it creates ZPD, as explained in Section 2. To have an effect-
tive learning, many internal development processes should be
awakened. In addition, this author states that it occurs when the
8Functional illiterate is the person that, even knowing how to read and
write, has not reading, writing and calculating skills required to foster their
personal a nd professional development.
G. T. BARCELOS ET AL.
Construction - Activity 2.
student interacts with people in his environment and in coo-
peration with his partners. The attitude of both students and
also of the other students indicate that the activities enabled
creation of ZPD; in order to reach their objectives, it was ne-
cessary to develop internal processes, as well as an interaction
with researchers and with classmates.
Activity 3 was solved without any difficulty, since it in-
volved concepts used in the previous activities. Such fact sug-
gests that activities 1 and 2 were solved consciously, as they
had a positive influence on the following activity. As in both
previous activities, although each student was using a computer,
they discussed the entire process of problem solving. This con-
firms that development of superior functions derives from ac-
tual relations between individuals (Vygotsky, 1978). According
to this author, shared activity is important for cognitive deve-
lopment as it allows living in the external plane what will be
By the end of each activity, students generally called the re-
searchers to show their constructions. Questions such as “Is it
correct?” were answered with other questions, making them
debugging what they had done and identify the error in case it
occurred. Such attitude was more frequent during the first two
activities. In the third activity, most students reflected a little
more on what they were doing (students identified their errors
by themselves or aided by the classmate sitting next to them).
Discussion and Formalization of Conjectures
Established in Activities
The stage of discussion and formalization of established
conjectures was started after students concluded the activities.
The researchers coordinated this stage by requesting students to
orally describe construction and answer to item c of each acti-
vity. This item requested the student to write what was ob-
served based on the construction and on its investigation. When
a student described a different path from other students, he
should socialize the procedure and present his construction to
In activity 1, all students orally answered that if two triangles
have two orderly congruent angles, then they are similar. One
student described the steps of her construction using the soft-
ware feature “Repeat construction,” showing that she used the
angle transposition process to construct congruent angles.
Among other students, only one claimed to have used parallel
lines to construct congruent angles, but at the end he estab-
lished the same conjecture. When this activity was developed,
the resolution process used by this student was not predicted.
This fact is warned by Borba and Penteado (2005), who
stressed that when using digital technological resources, a
teacher should be able to deal with unpredictable situations. It
is worth stressing that there was yet another way (using the
feature “Fixed amplitude angle” to construct congruent angles).
In activity 2, only one student associated the condition im-
posed in the activity with the definition of triangle similarity;
all others said that only sides are proportional and that angles
are congruent, which did not prevent reaching the activity ob-
jective as responses are equivalent. One student described the
steps of construction using the software feature “Repeat con-
struction.” At that moment, this student said his construction
was not the first he had done; in his opinion, his first attempt
was too complex, so he decided to do the activity again to sim-
plify it. This statement describes the path followed to solve the
activity, stressing how the student thought. Such fact highlights
the importance of discussing and presenting results in investi-
gation activities, as supported by Ponte et al. (2005). After all,
without this moment the student would not have the opportu-
nity to express himself, and consequently process analysis
would be incomplete. All other students claimed to have used
the same construction process.
In activity 3, analysis of oral responses showed that students
realized imposed conditions for construction ensured triangle
similarity. This fact is attributed to the comments made in ac-
tivities 1 and 2. This reinforces the idea that, when performing
a given activity, representations are being formed and their
richness allows going beyond a mere description and/or memo-
rization of the theme being studied, as supported by Moysés
(2007). Since the construction process in activity 3 was analo-
gous to that of activities 1 and 2, description of constructions
was not requested. Analysis was performed based on construc-
tion files collected by the researchers.
After discussion was over, one of the researchers asked if all
three exercises proposed before the software activity could be
solved without knowing about similarity cases. Students were
also asked about what led them to construct proportions using
measurements of triangle sides. Students did not give immedi-
ate responses; one student said it was due to Thales’ theorem.
The researcher then explained that this theorem is not sufficient
to support resolution of the activity. Statements showed that
students (except for one, who was already aware of the theme)
constructed the proportion merely because they were studying
similar triangles, which have proportional sides. That is, for
most students any two triangles were similar. However, these
students reported that, after solving the activities using the
software, they realized that the first exercises were solved
without any background. The second meeting eliminated these
The third and last meeting started with a review of the stages
performed in previous meetings. Next, one of the researchers
demonstrated the first case of similarity on the board. Demon-
stration of the second case, as it was similar to the first, was
conducted by the students, who listed the demonstration steps.
Demonstration of the third case was given as homewo rk.
G. T. BARCELOS ET AL. 259
After similarity cases were formalized, the researchers com-
mented on constructions performed in the software (whose files
were collected and analyzed between the second and the third
meeting), based on the analysis and considering the demonstra-
tions. It was not possible through file analysis to understand
one student’s construction, so she was asked to explain the
procedure and its stages. This is another situation that high-
lights the importance of discussing and presenting results in
Next, students were asked to do exercises 2 and 3 again. Ex-
ercise 2 was solved along with the researchers (in a dialogical
manner), and students had to do exercise 3 by themselves.
Similarity cases were easily identified. Response accuracy was
much higher than in the first resolution (diagnostic stage). Posi-
tive results are attributed to the sequence of activities, which
included use of the software (tool), intervention by mediators,
classmate collaboration, discussions and socialization of pro-
cedures used in previous stages.
Resolution of Final Exercises
According to Vygotsky (1978), the level of actual develop-
ment is the level of an individual’s mental functions that were
established as a result of completed development cycles. With
the aim of analyzing the level of actual development of partici-
pants in relation to the theme under investigation, six exercises
were proposed about what was conjectured in the software-
aided activities and further demonstrated. Figure 4 shows three
of these exercises.
At this study stage, students were distributed into small
groups, chosen by them. However, the students that missed the
second meeting (four students) were grouped in such way to
ensure there were students who attended all meetings. Exercises
were problems that required domain of the content being stu-
died to be solved. Students were asked to make a full register of
exercise solving to enrich the analysis. During exercise solving,
students were quite interested and more rigorous in establishing
the resolution (by providing a background for each resolution
stage). They constantly requested presence of the researchers to
check their answers.
It is important to stress that there was a dialogue between
group members and between different groups to answer ques-
tions and check their responses. Students compared resolutions
and sometimes concluded that although different, paths were
correct. One student’s statement reflects what was described, as
well as shows that the theme was understood: “I went from the
small to the large triangle and you did the opposite, but the
result is the same.” In this stage it was possible to experience
situations in which learning of concepts had its origins in social
practices (Vygotsky, 1978).
When one of the students, who had missed the second meet-
ing, was asked if she was being able to solve the exercises, she
answered that “I can manage it with your support and helped
by my classmates.” This statement reinforces the idea that in
groups of students, generally heterogeneous as to previous
knowledge, more advanced students may contribute to the de-
velopment of others (Oliveira, 1993).
In exercise 4, as well as in others, students easily identified
the similarity case, but the difficulty in calculating one of the
sides was remarkable. Students had to use the Pythagorean
theorem, which had already been used in other exercises, but
most of them could not identify such need. Since this difficulty
remained for a certain time, one of the researchers intervened
making questions that gave the idea of using that theorem. As
stated by Moysés (2007), guide questions, examples and de-
monstrations are important intervention forms that can help
students perform an activity.
The two last exercises had no figures, a situation that had
been previously planned. Importance of figures for problem
solving was manifested in students’ statements. During the
activity, two students of different groups said the following:
“My problem is when there is no figure.” “Is my figure is cor-
rect?” These statements were not addressed to anyone in par-
ticular, which in the social-historical theory is called “interior
discourse,” which is a discourse focused on thought, addressed
to the subject himself, with the aim of helping the individual in
his psychological operations. Language as a tool of thinking
The importance of figures is stressed by Moysés (2007: p.
75): “[…] by establishing a relation between a given situation
in- volving calculation and a representation—either formed by
different or richer mental images, or through diagrams, schemes,
more evocative verbal descriptions, gestures, simulations—
contextualized thinking favors articulation of variables and
contributed to the success of the mathematical problem solving
In exercise 5, the difficulty of not having a figure was over-
come by exchange of ideas between group members; therefore,
identification of two pairs of similar triangles was correctly
performed based on similarity cases. This fact is attributed to
the set of activities performed throughout the meetings. To end
this exercise, students had to construct two proportions and
later solve a system. Some groups had difficulty, so one of the
researchers organized the conclusion on the board through an
active participation of students. This exercise was considered
very difficult, but all groups came close to the final answer
before the intervention.
In exercise 6, two students had difficulty interpreting “in-
scribed circumference.” One student drew a circumscribed
circumference and another drew a tangent in only two sides.
Another problem was the drawing of a right angle in the
smaller triangle (from the center to the tangential point), which
generated errors in identification of homologous sides. The
G. T. BARCELOS ET AL.
situations above influenced exercise solving. For this reason,
the researchers provided the required support to the groups,
enabling them to successfully solve the activity. Changing a
person’s performance through interference of another is
stressed in the social-historical theory (Oliveira, 1993; Baquero,
1996), as it represents a moment of development (students
cannot do it by themselves). In addition, such change is a result
of social interaction in the process of building superior psycho-
logical functions, an important aspect of the social-historical
theory (Oliveira, 1993).
Students actively participated in exercise solving. Group in-
teraction was very strong, and the richness of interpersonal
exchanges contributed to problem solving.
It was possible to perceive that mathematical investigations
involve concept, procedures and representations, in addition to
the importance of the conjecture-test-demonstration sequence in
learning (Ponte et al., 2005), as described in the introduction.
Analysis of Question naires and In t erv i e w
Students had to complete two questionnaires during the study.
The first was completed soon after solving activities using the
Compass and Ruler software, and the second was completed by
the end of the last meeting, after solving the exercises in group.
The students that missed the second meeting did not answer all
questions of the second questionnaire.
To facilitate analysis of questionnaire answers, students were
numbered (Student 1, Student 2, and so on), and the numbers
were maintained in both analysis.
Analysis of Questionnaire 1
Ten students completed the first questionnaire. When asked
about the importance of the teacher as mediator in problem
solving using the software, 80% of the students considered it
“very important” and the remaining 20% found it “important.”
The comments made by two students that considered the
teacher’s mediation as “very important” stand out. They stress
the role of teachers as advisers, enabling development of acti-
vity and knowledge building.
The teacher’s participation was crucial for problem solving,
because when there were construction mistakes she helped me
find where the error was so I could go on with the construction
The teacher is essential for knowledge building. It was not
different here. She served as a bridge so we could reach the
proposed objectives (student 9).
The statements above show that students realized the impor-
tance of mediation that is not directive and authoritative, as
supported by the social-historical theory (Oliveira, 1993).
As to importance of classmates for problem solving using the
software, 50% of the students considered it “important” and the
remaining found it “very important.” Analysis of statements
showed the importance of interaction in the learning process, as
stressed by the social-historical theory. Two comments conside-
red significant for the context of this study should be stressed:
My classmates also helped a lot in the construction, because
together we could exchange information and realize where and
why we were making a mistake (student 4).
Interaction between students to exchange ideas is important
for a better development of the student (student 9).
When asked about the level of the activities solved using the
software, most students (60%) considered it “moderate” and the
remaining classified it as “difficult.” These rates are attributed
to the investigative nature of the activities. Although the theme
is simple (it is part of the curriculum in elementary and high
school), students are not generally used to solving investigative
activities. Some justifications stand out (the first refers to a
student that considered it “hard,” and the second is from a stu-
dent that considered it “moderate”).
Solving it was difficult because we needed a lot of help from
classmates and from the teacher. Without such help, I would
take a long time to finish the exercise (student 4).
The activities were developed to make us think, do and un-
derstand the class’s objective (student 8).
Although activities were not considered easy, it was a satis-
faction to perceive that software (tool) use to learn the theme
was considered “very important” by 90% of the students and
“important” for the remaining 10%. These rates reinforce the
importance of using tools during an individual’s development,
in agreement with the social-historical theory. In addition, the
statements below indicate that students are aware of the advan-
tages of using the Compass and Ruler (and similar) software to
study Geometry. This is an important fact, given that they are
students Mathematics teache rs.
Because the student has the opportunity to find out the con-
tent through experiences that confirm the theoretical part of the
content (student 3).
The software gave us a better perception of the activity con-
clusions. Without it, I think it would have been very difficult to
perceive them. Moreover, it makes the class more dynamic and
interesting (student 4).
Using the software we can, for example, move points and see
that cases are valid for all other triangles, meeting the same
conditions (student 6).
Widening the focus, the following question was about the
importance of using the software in Mathematics learning in
general. All students considered such use as “important.” Their
statements demonstrate that students are aware of the possibili-
ties generated by digital technologies: computational ability,
graphic visualization, discovery and confirmation of properties,
possibility to run experiments with data collection and problem
modeling, speculations, among others, as supported by Baldin
(2002), described in second section.
The software is a tool to help students understand some sub-
jects they have difficulty visualizing (student 1).
Because students interact with the content, making learning
more pleasant (student 3).
When students do and observe what they are doing, they
have a better understanding and learn the theory (student 5).
As to discussion and formalization of activities, 70% of stu-
dents considered it “very important,” while the remaining found
it “important.” Analysis of comments shows the importance of
interaction and idea exchange enabled by such actions.
Discussion is a way of aggregating new knowledge through
your classmates (student 1).
By exchanging ideas people learn more and better (student
These discussions show many forms of solving the same
question (student 9).
Analysis of Questionnaire 2
After resolution of final exercises (without using the soft-
ware), 14 students completed the second questionnaire.
G. T. BARCELOS ET AL. 261
Most students reported having little difficulty solving the fi-
nal exercises (Chart 1), which can be considered a quite satis-
factory result. Three students did not answer this question be-
cause they missed one of the meetings. It is important to stress
that students highlighted the importance of software activities
to solve the final exercises, which reinforces the value of this
tool in learning.
After the software activity it was easy to visualize similarity
cases (student 1).
By constructing similarity cases in the software it was easier
to remember what was constructed and associate it with the
activities (student 6).
With the support of the software and studying similarity
cases, it was easier to solve the exercises (student 8).
Also in the resolution of final exercises, most students con-
sidered teacher’s and classmate’s mediation as “very impor-
tant” and “important.” This result is similar to that obtained in
data tabulation of the previous questionnaire. It is curious that,
in the resolution of final exercises, classmate’s mediation was
considered “very important” by a higher number of students
than teacher’s mediation (as opposed to software activities).
This is attributed to the fact that the exercises were solved in
groups and to the activities previously performed. Without
being requested, one student, when commenting on question 2,
compared the teacher's mediation in both learning situations:
The teacher’s mediation was important, but the number of
times we needed her support was much lower than that of the
previous activity (student 4).
With regard to classmate’s mediation, some statements
showing the importance of exchanging information during ex-
ercise solving stand out (the word exchange was present in five
out of 14 comments):
Students’ collaboration was very important because through
them we could have a better visualization of what was right or
wrong. Through information exchange we got to the result
more easily (student 2).
It is of great help to develop thinking, and information ex-
change accelerates the development of questions (student 9).
The fourth question (open) asked whether, when solving the
exercises after software activities, any association had been
Resolution of final exercises.
made with the activities solved with the support of the software.
Four students did not answer because they missed one of the
meetings. All others answered affirmatively, stressing the im-
portance of visualization enabled by the software, which influ-
enced resolution of final exercises.
Only two students answered negatively to the question on
having more difficulty solving the exercises that had no figure
(tool); their justification is that they can easily represent the
figure based on the heading information. These students are at a
more advanced developmental level because they have already
internalized the theme being studied. The need of figures was
justified in the students’ comments, of which the following
Yes, because it is easier to visualize the drawing, and some-
times we make mistakes because the drawing is incorrect (stu-
Yes, because we are never sure about the drawing, and a
wrong drawing leads to a misunderstanding of the exercise
Yes, when you have ready figures it is easier to visualize tri-
angle similarities (student 13).
In addition to questionnaires, five students were interviewed.
As mentioned earlier, a semi-structured interview was chosen
because it allows insertion of new questions according to the
In general, it was possible to perceive that all students con-
sider mediation by another person as something important in
the teaching and learning process, either a teacher or a class-
mate. However, there is a certain preference for the mediation
of a classmate. Of the five interviewees, only one preferred a
teacher’s mediation, claiming he has more knowledge on the
subject and, therefore, his guidance and questions lead to a
correct solution more objectively; three said they feel more
comfortable with a classmate and ask help to the teacher only
when they still have a question; one claimed he has no prefer-
ence, but stressed that it depends on the teacher (he said some
teachers are not open to questions).
They all considered that the activities performed with the
Compass and Ruler software enabled occurrence of ZPD. One
of the interviewees stated there were activities she could only
do if helped by someone (a classmate or a researcher), but this
made her learn, because nobody did it for her (she emphasized
that, although helped, she did the activities by herself). Her
statement is very interesting since it is obvious she is aware that
doing an activity supported by someone is different from
someone doing it for her, and here lies the essence of ZPD.
A comment by the first interviewee led to the formulation of
a new question. This student said she really needs to visualize
the figure when solving the type of question proposed in the
study. Figures are essential to her, but she stressed they should
be included in the question since she usually makes a mistake
when drawing them according to the heading information. After
this comment, all the other interviewees were asked about the
importance of the figure and whether they find it difficult to
draw it based on the information given. All the others affirmed
the figure is important for problem solving, but they comple-
mented by saying that, even without the figure, they can men-
tally visualize it based on the information given and then re-
G. T. BARCELOS ET AL.
produce it on paper. The first interviewee is in an internalize-
tion process that differs from the others. She is also very de-
pendent on the external mediation of figures as tools; in fact,
she has not internalized them as signs.
In general, the data collected in questionnaires and interview
showed the importance of using tools associated with investiga-
tive activities as well as of teacher’s and classmate’s mediation
in resolution of activities.
This section reviews some important issues about the study.
Although the entire process has already been detailed in previ-
ous sections, an important aspect should be stressed: acknow-
ledgment of the importance of social-historical theory for the
teaching and learning process of Mathematics. Such acknow-
ledgment originated from the perception, in practice, of cohe-
rence between theoretical background and students’ actions and
Throughout the study, shared activity was emphasized. Its
presence sometimes made creation of ZPD evident. Through
the mediation of classmates and/or researchers, many students
were able to perform tasks they could not do by themselves,
clearly showing cognitive development. Considering all that
was observed and collected through questionnaires and inter-
view, the importance of mediation during activities performed
using the software is unquestionable. This was the focus of this
study—to analyze the importance of mediation, which was
It should also be stressed the importance of establishing in-
creasing levels of difficulties and of presenting counter-ques-
tions (questions asked based on students’ answers), which
stimulate thinking. Such behavior was adapted during all re-
search activities; it was possible to observe that it enabled stu-
dents to consciously progress in terms of what was being stu-
died through analysis and reflection. However, stressing this
behavior only reinforces the previous idea about the fundamen-
tal importance of a well coordinated mediation by the teacher.
The technique of participant observation was very important
in the entire process, since it allowed researchers to register
observations, generally reflecting on what was occurring. Some
perceptions were only possible due to an integration between
researchers and students. In addition, the questionnaires and the
semi-structured interview complemented the observed data,
ensuring a deeper analysis.
The fact that all the activities performed demands time that is
often not available for teachers (both in terms of implementing
the planning with students and in relation to the time required
to prepare investigation activities) was certainly not ignored.
However, an experiment such as this reported in the study at
least leads to a reflection on the teaching practice of many
teachers. How many misjudge that they teaching when they are
actually only transmitting information? It should be stressed
that the focus here is not on teachers without commitment, but
on dedicated people who take their work seriously, although
sometimes mistakenly. Thus, it is worth reflecting if spending
some time preparing activities that aim at knowledge building
is not advantageous after all.
Finally, the students who participated in the study belong to a
group of students Mathematics teachers. Therefore, they had
the opportunity of having a quite interesting experience not
only in terms of learning the mathematical theme, but also in
relation to their future teaching practices. Readings are funda-
mental, but experiences are generally more remarkable—they
speak for themselves.
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