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Are 9th grade students ready to engage in the theoretical
discursive process in geometry?
Yavuz Karpuz1 and Bülent Güven2
1) Recep Tayyip Erdogan University, Turkey.
2) Trabzon University, Turkey.
Date of publication: February 24th, 2022
Edition period: February 2022-June 2022
To cite this article: Karpuz, Y. & Güven, B. (2022). Are 9th grade students
ready to engage in the theoretical discursive process in geometry. REDIMAT
– Journal of Research in Mathematics Education, 11(1), 86-112. doi:
10.17583/redimat.3667
To link this article: http://dx.doi.org/10.17583/redimat.3667
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REDIMAT, Vol. 11 No. 1 February 2022 pp. 86-112
2022 Hipatia Press
ISSN: 2014-3567
DOI: 10.17583/redimat.3667
Are 9th Grade Students Ready to
Engage in the Theoretical Discursive
Process in Geometry?
Yavuz Karpuz
Recep Tayyip Erdogan University
Bülent Güven
Trabzon University
(Received: 29 July 2018; Accepted: 8 July 2021; Published: 24
February 2022)
Abstract
This study was conducted to examine whether newly enrolled 9th grade
students were ready to directly engage in the theoretical discursive process
from the perspective of Duval’s Cognitive Model. The sample of the study
was comprised of 51 newly enrolled 9th grade students between the ages of
14 and 15, who had not received any prior geometry instruction. These 51
students were posed two open-ended questions that would enable them to
make a transition between perceptual and discursive apprehension. According
to the findings obtained from the study, many of the students could not display
the necessary behaviors for theoretical discursive process. Students were
mostly unsuccessful in converting discursive information into perceptual
information, in writing discursive information based on perceptual
information, and making inferences based on discursive information. These
findings indicate that recent graduates of secondary school are not ready
enough to directly engage in theoretical discursive process and, thus, they
could experience difficulties in such high order skills as providing proof
requiring the theoretical discursive process.
Keywords: Duval’s cognitive model, theoretical discursive process,
geometrical figure apprehension.
REDIMAT, Vol. 11 No. 1 February 2022 pp. 86-112
2022 Hipatia Press
ISSN: 2014-3567
DOI: 10.17583/redimat.3667
¿Están Listos los Estudiantes de
Noveno Grado para Participar en el
Proceso de Discurso Teórico en
Geometría?
Yavuz Karpuz
Recep Tayyip Erdogan University
Bülent Güven
Karadeniz Technical University
(Recibido: 29 Julio 2018; Aceptado: 8 Julio 2021; Publicado: 24
February 2022)
Resumen
Este estudio se realizó para examinar si los estudiantes de noveno grado recién
matriculados estaban listos para participar directamente en el proceso
discursivo teórico desde la perspectiva del modelo cognitivo de Duval. La
muestra del estudio estuvo compuesta por 51 estudiantes de noveno grado
recién matriculados entre las edades de 14 y 15 años, que no habían recibido
ninguna instrucción previa en geometría. Muchos de los estudiantes no
pudieron mostrar los comportamientos necesarios para el proceso discursivo
teórico. Los estudiantes en su mayoría no lograron convertir información
discursiva en información perceptiva, escribir información discursiva basada
en información perceptiva y hacer inferencias basadas en información
discursiva. Estos hallazgos indican que los recién graduados de la escuela
secundaria no están lo suficientemente preparados para participar
directamente en el proceso discursivo teórico y, por lo tanto, podrían
experimentar dificultades en habilidades de orden tan alto como proporcionar
pruebas que requieren el proceso discursivo teórico.
_____________________________________________________________
Palabras clave: Modelo cognitivo de Duval, proceso discursivo teórico,
aprehensión de figuras geométricas.
Karpuz & Güven – Theoretical Discursive Process in Geometry
86
here is no doubt that using figures to solve problems in geometry is
highly beneficial because they provide an integrative presentation of
all the constituent relations of a geometrical situation (Duval, 1995).
Identifying geometrical properties based on figure constitutes the
most important step in a problem-solving process. However, that students
draw a conclusion in relation to a figure under the influence of its appearance
is one of the common situations that mathematics teachers often encounter.
For this reason, one of the points that needs to be focused on in mathematics
education is how such explanations encountered in learning environments as
“because it looked like a right angle” or “because it worked in another problem
case,” made under the influence of the appearance of the figure, can be
transformed to statements that are based on definitions, axioms and theorems
(Jones, 2000). According to Duval (1998), who names such kinds of
explanations as the theoretical discursive process, defines explanations based
on definitions, axioms and theorems as those made through deduction. Thus,
providing proof or logical deduction for geometrical properties is essentially
a process of constructing theoretical discursive process.
According to the approaches (Van Hiele model and Duval’s cognitive
model) that seek to explain geometrical reasoning students need certain
behaviors to engage in the theoretical discursive process (Jones, 1998). For
example, according to the Van Hiele model, in order for students to make
proof, they should know the properties of geometric figures and be able to
recognize the logical relationships among these properties (Fuys, Geddes and
Tischler, 1988; Güven, 2006; Mason, 1998). But it has limitations such as the
emphasis on sequential and hierarchical levels of geometry understanding,
Duval’s cognitive model is more attractive because it is concerned with
understanding the cognitive processes (Ramatlapama and Berger, 2018).
According to Duval (1998, 1995), who bases geometric reasoning on
cognitive processes, students can engage in the theoretical discursive process
only if they look geometrical figure mathematically. The mathematical way
of looking at figures in geometry requires that students can establish accurate
interactions between their perceptual and discursive apprehensions.
Generally, in school mathematics, particularly high school geometry
lessons are regarded as a transitional phase in making logical deductions and
providing proof (National Council of Teachers of Mathematics Standards
(NCTM, 2000; Sriraman, 2004). Thus, in the educational programs in Turkey,
T
REDIMAT, 11(1)
87
grade 9 (high school) geometry is regarded as a transition to the level of
deduction with the assumption that secondary school graduates know the
properties of geometric features and the logical relationships among these
features. Moreover, as a natural outcome of the education students receive,
they can look geometric figure mathematically and can establish accurate
interactions between their perceptual and discursive apprehensions when they
look at a figure. By means of the present study, to what extent this expectation
is realistic and the response to the following research question were
investigated: “Does the education provided to secondary school students
enable them mathematical way of looking at figures?” To this end, by making
use of Duval’s cognitive model, the current study aimed to explain whether or
not newly enrolled high school students (year 9) and those who had not yet
received any instruction in geometry were ready for theoretical discursive
process.
While reviewing the related literature, it is possible to encounter numerous
studies on students’ geometrical figure apprehension (Llinares and Clement,
2014; Michael, Gagatsis, Avgerinos, Kuzniak, 2011; Michael, 2013;
Torregrosa and Quesada, 2008). While the participants of some of these
studies were teacher candidates (Llinares and Clement, 2014; Torregrosa and
Quesada, 2008), in other studies, the participants were comprised of high
school students (Michael, Gagatsis, Avgerinos, Kuzniak, 2011; Michael,
2013). In studies on high school students, the structure of geometrical figure
apprehension of different grade level was examined. In these studies, it was
found that students’ figure apprehension generally developed as students
proceeded from one grade level to another, that students experienced
difficulties in questions related mostly to sequential and discursive
apprehension, and that the mistakes that students made in their responses to
questions were predominantly related to dominance of the perceptual
apprehension on the looking at the figure, when compared to the other
processes (Michael, Gagatsis, Avgerinos, Kuzniak, 2011; Michael, 2013).
These studies entail important findings related to high school students’ use of
figure apprehensions. However, while solving problems and making logical
deductions, students also need to establish relationships among these
processes (Duval, 1995, 1998). Hence, in addition to these studies, those
examining how students establish relationships among perceptual and
discursive apprehension are also needed. Because between perceptual and
Karpuz & Güven – Theoretical Discursive Process in Geometry
88
discursive apprehension is essential for engaging theoretical discursive
process.
Theoretical Framework
Duval’s Cognitive Model
Duval (1995) has sought to explain the types of processes involved when
looking at a geometric figure. Duval stated that these processes were made up
of four geometrical figure apprehension processes: perceptual apprehension,
discursive apprehension, sequential apprehension and operative apprehension
(Duval, 1995). According to Duval, each of these carries out different
functions, which enable the comprehension of mathematical relationships in
geometric figures, and solving problems very often requires an interaction
among these four processes. However, for an accurate establishment of this
interaction, these apprehensions should be developed separately (Duval,
1995).
Perceptual apprehension is the process which includes knowledge acquired
when one looks at a figure for the first time and is related to the structure
(external appearance) of the figure. It includes such processes as providing
information about the name and size of the figure and becoming aware of the
fundamental geometric elements (point, line segment, triangle, circle…) that
make up the figure. Moreover, identifying the subfigure also takes part in the
perceptual apprehension process. This apprehension is static and does not
enable one to recognize the relationships among the subfigures (Duval, 1995).
It is impossible to identify the mathematical properties of a geometric figure
merely through perceptual apprehension. For this to happen, some preliminary
information about the figure should be given. Based on the preliminary
information provided, establishing a relationship between a figure and
mathematical principles (definition, theorem, axiom, etc.) to draw a
conclusion is named as discursive apprehension (Duval, 1995, 1999; Michael,
2013).
In learning environments, the transformation of students’ discourses into
theoretical discursive processes can be attained with the replacement of
explanations derived from the appearance of the figure with conclusions
drawn based on definitions, axioms and theorems. According to Duval (1998),
REDIMAT, 11(1)
89
such a transformation is only possible by looking at a figure mathematically.
To be able to look at a figure mathematically, accurate interactions should be
established between perceptual and discursive apprehensions. In such an
interaction, the perceptual information presented on a figure (the information
presented on the figure: point, line segment, angle, etc.) should be accurately
converted to discursive information (sentences or symbols showing the
mathematical relationships on a figure, such as the lengths of the two line
segments are equal or line segment AB is the angle bisector, etc.) or the given
discursive information should be converted accurately to perceptual
information. However, the result should be obtained based only on discursive
information (Duval, 1998). In this process, while perceptual apprehension
enables one to recognize the perceptual information on the figure (line
segment, angle, point, etc.), discursive apprehension enables one to gain
discursive information based on perceptual data, which in turn leads to the
construction of new information. Thus, discursive apprehension serves two
different functions: The first function is to establish a link between
mathematical principles and the geometric figure; that is, expressing visual
data utilizing mathematical principles, while the second function is to enable
the construction of new information by utilizing mathematical principles
(Llinares and Clemente, 2014). Evidently, to be able to look at a figure
mathematically, it is essential to display certain behaviors (e.g., converting
perceptual information into discursive information).
Based on Duval’s explanations regarding perceptual and discursive
perception and the transitions between them, the behaviors arising from the
interaction between discursive and perceptual apprehensions can be presented
as below (Table 1).
As can be seen in Table 1, to be able to engage in the theoretical discursive
process in geometry, the perceptual information presented on the figure
should be accurately converted to discursive information and vice versa by
establishing correct interactions between perceptual and discursive
apprehensions. In fact, to prevent the influence of the appearance of the figure,
conclusions should be drawn based merely on discursive information. As
these behaviors show what must be done to engage in the theoretical
discursive process, it is possible to consider these behaviors as criteria for
cognitive readiness in engaging in the theoretical process within Duval’s
Cognitive Model.
Karpuz & Güven – Theoretical Discursive Process in Geometry
90
Table 1
Interaction Between Discursive and Perceptual Apprehension (The Mathematical Way of Looking at a Figure) and
Sample Student Behaviors
Student Behavior
Sample Student Behavior
1st Behavior
Converts the given discursive information to
perceptual information.
(Transition from discursive to perceptual
apprehension)
2nd Behavior
Converts the given perceptual information to
discursive information.
(Transition from perceptual to discursive
apprehension)
REDIMAT, 11(1)
91
Table 1 (continue)
Interaction Between Discursive and Perceptual Apprehension (The Mathematical Way of Looking at a Figure) and
Sample Student Behaviors
Student Behavior
Sample Student Behavior
3rd Behavior
Draws a conclusion based on the discursive
information obtained from the figure.
(Acquiring new information using discursive
apprehension)
4th Behavior
Does not become influenced by the appearance of
a figure while drawing a conclusion.
(Drawing a conclusion based on perceptual
apprehension)
then DE AB If
(A probable conclusion that can be
drawn under the influence of the appearance of the
figure.)
Karpuz & Güven – Theoretical Discursive Process in Geometry
92
Method
This study aimed to reveal whether newly enrolled high school students who
had not yet received any instruction in geometry had cognitive readiness to
engage in theoretical discursive process. To this end, requirements for
theoretical discursive process were identified based on Duval’s Cognitive
Model (see Table 1) and two open-ended questions were prepared to measure
these requirements. These questions were administered to the students during
one of the convenient class hours. Subsequently, to observe the students’
mathematical behaviors and draw conclusions from these observations
regarding their cognitive processes (Goldin, 1997), approximately 10-to-15-
minute clinical interviews were held on a voluntary basis with three self-
expressive students selected from each category of results obtained from the
analyses of the answers to the open-ended questions (see Table 3). With the
permission of the students, their responses were recorded. In these interviews,
the students were asked to explain and justify their answers. To this end, the
students were posed the question, “Could you please explain and justify your
response?” In this way, the justifications underlying students’ written
responses were tried to be revealed.
Sample
The study was carried out in a state high school (Anatolian High School),
which admits its students through a centralized exam in Turkey and is at a
moderate level of success in its region with reference to student scores.
Excluding vocational schools, Anatolian High Schools are the most preferred
type of high school among those aiming to provide academic education to
students. One such school was chosen to implement the study in order to
address the common student level. In the selected high school, there were 75
students in year 9. These students were introduced formal proof for the first
time in the geometry course. The fact that students had not previously attended
a geometry course with instruction on formal proof in their high school was
paid attention to since the study aimed towards the readiness of students’
theoretical discursive process. For this reason, 24 students were not included
in the study as their teacher had started teaching the geometry units in their
class. Thus, the sample of the study was comprised of 51 fourteen-to-fifteen-
REDIMAT, 11(1)
93
year-old 9th grade students who had just enrolled in high school and had not
previously received a geometry course in their high school. These students
had learned triangles, polygons, geometric objects and transformation
geometry within the scope of sub-learning topics of geometry in secondary
school. At the end of secondary school, students are expected to explain the
features of geometric figures, logical relationships among them and as a
natural outcome of education accurately establish an interaction between their
perceptual and discursive apprehensions when they look at a figure. When
they become high school students, they are expected to systematically prove
the geometric relationships by theoretical discursive process as of year 9.
Data Collection Instrument
For theoretical discursive process, three fundamental behaviors need to be
realized: converting discursive information into perceptual information,
arriving at discursive information based on the perceptual information, and
arriving at logical conclusions based only on the discursive information
(Torregrosa and Quesada, 2008; Llinares and Clemente, 2014). Thus, whether
or not students were ready to engage in the theoretical discursive process were
tried to be revealed based on whether or not they displayed these behaviors.
Accordingly, two open-ended questions that could reveal each behavior were
prepared together with two experts holding a doctoral degree in the field of
math education. Subsequently, a pilot study was conducted at a high school to
determine, by consulting expert opinion, whether or not the questions could
reveal the behaviors expected of the students, and the questions were revised
to take their final shape. The questions that were prepared and the behaviors
that were aimed to be measured are as follows (see Table 2).
In the first question, a geometric figure with discursive information was
presented, and the students were asked to convert the discursive information
to perceptual information by using appropriate symbols. In the second
question, the students were given a figure with perceptual information (such
as perpendicularity) displaying certain mathematical relationships and were
asked to write the mathematical properties of the given geometric figure. The
figure given in the second question was designed to display with mathematical
properties such as “parallelism” and “perpendicularity”. For example, while
at first sight the figure seemed like a rectangle with all its angles being 90
Karpuz & Güven – Theoretical Discursive Process in Geometry
94
Table 2
Measurement Instruments
The Questions Prepared to Measure students’ competencies in the theoretical discursive
process
The Behaviors To Be Measured
Try to show the given information on the geometric figure by using symbols
• Converting discursive information to
perceptual information accurately.
Look at the geometric figures below and write their mathematical properties using
appropriate symbols and representations in the space provided on the right
• Converting the given perceptual information
to discursive information accurately.
• Arriving at a conclusion based on the
discursive information obtained through the
figure.
• Not being influenced by the appearance of
the figure while arriving at a conclusion.
REDIMAT, 11(1)
95
degrees, it is, in fact, in the most general sense, a trapezoid, and it cannot be
claimed that all its angles are 90 degrees.
Data Analysis
The qualitative data obtained from the responses given to the open-ended
questions and the interview constitute the data of the study. The data obtained
were analyzed in two phases. Initially the written responses to the open-ended
questions and subsequently the qualitative data obtained in the interviews
were analyzed. The written responses given to the open-ended questions were
analyzed by two researchers. Three previously formed categories were used
in data analysis (see Table 3). These categories were used to group students
according to the behaviors of accurately converting discursive information
into perceptual information, arriving at accurate discursive information based
on the perceptual information, and arriving at logical conclusions based only
on the discursive information. The first group of the categorization was
comprised of students who had perfectly displayed the behavior measured via
the two open-ended questions (see Table 2). The second group was made up
of students who had displayed some deficiencies in the behaviors measured.
Finally, the third group consisted of students who did not display the
behaviors at all. For example, while the written responses to the first question
was analyzed, the students who could convert all the given discursive
information to perceptual information completely were assigned to group 1,
while those students who had some deficiencies (such as not being able to
convert some discursive information into perceptual information), but could
convert some of the discursive information into perceptual information were
assigned to group 2 and finally those students who could not accurately
convert any of the discursive information into perceptual information were
assigned to group 3. In order to validate the categorization and to clarify the
number of students in each category, the data were analyzed by two
researchers separately and independent of each other. Subsequently, the
obtained data were compared and the responses of the students assigned to
different groups were re-examined. At the end of the examination, the
researchers arrived at a common conclusion and a complete agreement was
established. In this way, the number of students in each category, which both
researchers agreed upon, were revealed. The student numbers that were
Karpuz & Güven – Theoretical Discursive Process in Geometry
96
determined arranged into tables, which are presented in the findings section.
The categories identified and their explanations are as follows:
Table 3
The Categories Used to Group Students and Their Explanations
1st CATEGORY: Displays
the behavior accurately
2nd CATEGORY: Responding
accurately despite some deficiencies
in displaying the behavior
3rd CATEGORY:
Unable to display the
behavior
This category includes the
students who answered all the
questions accurately (by
writing the expected
responses).
This category includes the students
who provided accurate answers but
could not write some of the expected
answers.
This category includes
the students who
provided erroneous
answers or did not
respond to the
questions at all.
After the students’ written responses given to the open-ended questions
were categorized by the researchers, the qualitative data obtained from the
clinical interviews were analyzed. After the recorded interviews were
transcribed, the students’ oral responses given during the interviews and their
written responses to the open-ended questions were compared. Both the
consistent and conflicting aspects of the written and oral responses were tried
to be identified during the comparisons. When an inconsistency was observed
between a student’s written and oral response to the extent of impacting
his/her category, the students’ category identified for that particular question
was changed based on the data obtained from the interview. Thus, as a result
of the data analysis, one student who was placed in “Group 3- No response”
based on his/her written responses to the open-ended questions was moved
into “Group 3- Erroneous responses produced under the influence of the
appearance of the figure” according to the findings obtained from the
interview (see Table 6).
Findings
Converting the Given Discursive Information to Perceptual
Information
The first of the open-ended questions was asked to determine whether the
students could convert the given discursive information to perceptual
REDIMAT, 11(1)
97
information accurately. The distribution of the number of students according
to the categories identified based on the data obtained from the analysis is as
follows (Table 4).
Table 4
The Distribution of the Number of Students Based on Their Responses to the First
Question
As can be seen in Table 4, only two of the students were able to completely
convert the discursive information given about the figure to perceptual
information. On the other hand, even though 32 students in the 2nd category
could convert some of the discursive information to perceptual information
accurately, they could not convert some of the information at all. Finally, the
17 students in the 3rd category converted the discursive information to
perceptual information erroneously. When the number of students in each
group are taken into consideration, it can be claimed that the majority of the
students displayed the behavior insufficiently or erroneously. Some of the
student responses for each category are presented below (see Figure 1).
Figure 1. Some student responses from each of the three categories
As can be observed from the responses, the student in the 1st Category
converted all the given discursive information to perceptual information
accurately. While the student was doing so, s/he established different
Behavior
1st Category
2nd Category
3rd Category
Can convert the given discursive
information to perceptual information
accurately.
2
32
17
1st Category
2nd Category
3rd Category
Karpuz & Güven – Theoretical Discursive Process in Geometry
98
perceptual information for different angles and sides. However, when the
response of the student in the 3rd category was examined, it was observed that
the student established the same perceptual information for the different
discursive information (e.g. [AD] is a median and AB = AC ) given
and this caused the student to make mistakes in converting discursive
information to perceptual information.
It is obvious that students who make such mistakes will obtain erroneous
results when they use the perceptual information, they form on a geometric
figure to determine the figure’s mathematical properties. Thus, in the
interviews held with the students, when students were given the perceptual
information, they had created themselves (see Figure 2) and asked to explain
the mathematical properties of the figure, based on their perceptual
information, they arrived at wrong results. One excerpt from an interview with
a student is as follows:
Researcher: Please write the mathematical properties of the figure
by considering the symbols on the given figure. (The researcher
shows Figure 2.)
Student: Here side AC and side DC are shown to be equal...
Researcher: What else can be said?
Student: Side BD, side DC and side AB are also of equal length. And
there, in D, there are two equal angles. (The student points to angles
ADB and ADC.)
Figure 2. The perceptual information previously formed by the student
As can be observed in the interview excerpt, the student arrives at
erroneous results by giving the same perceptual information for different
verbal information. Thus, this shows that students are unsuccessful even in
REDIMAT, 11(1)
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the most fundamental behavior they need to display in solving geometric
problems.
When the response of the student in the second category is examined, it is
observed that s/he converted some of the given discursive information to
perceptual information accurately. However, s/he could not convert some of
the discursive information (that AD line segment is an angle bisector, and BC
and AC side lengths are equal) to perceptual information at all. In the
interviews held with the students, when students were asked why they could
not show some of the information on the figure, they stated that they did not
read all the information on the figure and that previously this had not caused
any problems. An excerpt of an interview held with one of these students is as
follows:
Researcher: You haven’t shown on the figure that the side is a
median.
Student: I didn’t read that; I didn’t see it.
Researcher: Why didn’t you read it?
Student: I don’t read all the information when solving problems. It
doesn’t cause any problems. The information is given on the figure
anyway.
As can be understood from the interview excerpt, the student has said that
s/he does not read some of the given information due to the problem-solving
habits he gained in geometry and thus could not show some of the information
on the figure. Furthermore, s/he states that this behavior has not previously
caused any problems for him/her.
Converting the Given Perceptual Information to Discursive
Information
The second of the open-ended questions was asked to measure two behaviors.
The first of these aimed to determine whether students could convert the given
perceptual information to discursive information. When the responses to the
open-ended questions were examined, it was found that students converted
perceptual information to discursive information either accurately or did not
convert them at all. That is, students were classified only according to the 1st
and 3rd categories. As there were no students who had not written incomplete
discursive information, no student was placed in the 2nd category. The
Karpuz & Güven – Theoretical Discursive Process in Geometry
100
distribution of student numbers across the categories defined is presented in
Table 5.
Table 5
The Distribution of the Number of Students Based on Their Responses to the
Second Question
Behavior
1st Category
2nd Category
3rd Category
Converts the given perceptual
information to discursive
information accurately
23
0 (zero)
28
As can be observed in Table 5, twenty-three students converted the given
perceptual information to discursive information accurately by using the
necessary symbols and representations to show the perpendicularity of the
angles on the figure. On the other hand, 28 students could not convert the
given perceptual information to discursive information. Consequently, this
behavior, which enables one to acquire the necessary discursive information
(hypotheses) to arrive at a conclusion without being influenced by the
appearance of the figure, was not displayed by most of the students (3rd
behavior, see Table 1). In addition, as the perceptual information given on
figures constitutes the data of a geometrical problem, it can also be asserted
that most of the students do not determine the given data in a problem. Some
student responses in relation to categories are shown in Figure 3.
1st Category
3rd Category
Figure 3. The responses of some of the students in relation to the specified
categories
As can be observed in Figure 3, the student in the 1st category converted
the given information of perpendicularity to discursive information
REDIMAT, 11(1)
101
accurately. However, the student in the 3rd category has directly written
results (such as it is a rectangle and its angles are 90 degrees) related to the
mathematical properties of the figure without writing the discursive
information. In the interviews held with some students in the 3rd category,
students stated that they did not see the need to convert perceptual information
to discursive information and that it was meaningless to write the relationships
they saw when they looked at the figure. An excerpt from an interview held
with one of the students is presented below:
Researcher: Why didn’t you state that angles A and D are 90
degrees?
Student: It’s already stated on the figure. Is there a need to rewrite
it?
Researcher: What do you think?
Student: I think it’s not necessary. It’s clearly seen in the figure, it’s
pointless to write it…
As can be seen in the interview script, the student finds it meaningless to
convert the perceptual information presented on the figure to discursive
information. For this reason, these students directly wrote explanations about
the mathematical properties of the figure.
Arriving at a Result Based on the Discursive Information
Obtained from the Figure
The second of the open-ended questions was asked to measure two behaviors.
The second behavior is to write correct mathematical properties of a figure
based on the perceptual information obtained from the figure. When the
answers were examined, it was found that only two of the students had written
accurate mathematical properties about the figure, while the others provided
erroneous answers or did not answer at all, thereby not providing any
explanation about the mathematical properties of the figure. Consequently, in
terms of this behavior, these students were grouped only in the 1st and 3rd
categories. Since those who provided erroneous responses or no response at
all had not displayed the behavior, they were placed in the 3rd category and it
is for this reason that the 3rd category was divided into two subcategories,
namely “3rd category- Erroneous responses produced under the influence of
the appearance of the figure” and “3rd category- No responses.” The
Karpuz & Güven – Theoretical Discursive Process in Geometry
102
distribution of the number of students according to categories is shown in
Table 6.
Table 6
The Distribution of Student Numbers Based on Their Responses to the Second
Question
Behavior
1st Category
3rd Category
• Arrives at a conclusion based
on the discursive information
obtained from the figure.
• Is not influenced by the
appearance of the figure while
making deductions.
Correct
responses
Erroneous responses
produced under the
influence of the
appearance of the
figure
No
responses
2
38
11
As can be seen in the table, most of the students are comprised of students
who gave wrong answers because they were influenced by the appearance of
the figure. All these students arrived at the conclusion that the figure was a
rectangle and thus wrote the properties of a rectangle. Unlike these students,
the students who wrote the mathematical properties correctly were those
students who based their response on discursive information without being
influenced by the appearance of the figure. On other hand, those students who
could not arrive at any conclusion only converted perceptual information to
discursive information. Some student responses in relation to categories are
presented in Figure 4.
Figure 4. Some student responses
REDIMAT, 11(1)
103
As can be observed in Figure 4, the student in the 1st category has stated
that angle A and angle D are 90 degrees, so it can definitely not be a rectangle
but could be a trapezoid. This indicates that the student determined the
mathematical properties of the figure based on preliminary information
(discursive information). In other words, the student first acquired certain
discursive information (e.g. that angles A and D are 90 degrees) and then
arrived at a conclusion based on this information.
As the discursive information s/he acquired was insufficient to consider
the figure as a rectangle, the student arrived at the conclusion that the figure
must be a trapezoid. Similar findings were also obtained during the interviews
held with the students. When this student was asked why s/he believed the
figure was not a rectangle, s/he stated that it would be insufficient to consider
the figure as a rectangle when only angles A and D are stated to be 90 degrees.
An excerpt from the interview held with the student is as follows:
Researcher: What can you say about the mathematical properties of
the given figure?
Student: Angles A and D are 90 degrees. But it doesn’t have to be a
rectangle.
Researcher: Why?
Student: Because the other angles (refers to angles B and C) don’t
have to be 90 degrees. They can change.
Researcher: What else can be said?
Student: What else… I think it can be a trapezoid, I mean if we turn
it like this (meaning rotating the figure to make side DC the base), it
will look more like a trapezoid.
Researcher: Why did you think it was a trapezoid?
Student: As these angles are 90 degrees (points to angles A and D),
these sides become parallel (sides AB and DC); that’s why it
becomes a trapezoid.
As can be observed in the interview excerpt, the student has used the given
discursive information (that angles A and D are 90 degrees) without being
influenced by the appearance of the figure and arrived at a conclusion in
relation to the mathematical properties of the figure. That is, s/he has
determined the hypotheses and arrived at a conclusion based on this
information. However, most of the students wrote the mathematical properties
of the figure under the influence of the appearance of the figure. As can be
understood from Figure 4, the student (in the 3rd category- wrong answers)
draws the conclusion that the figure is a rectangle and states that all the angles
Karpuz & Güven – Theoretical Discursive Process in Geometry
104
are 90 degrees. As can be seen, the student arrives at a conclusion under the
influence of the appearance of the figure and then writes mathematical
properties suitable to this conclusion. Similar findings were also yielded by
the interviews. When the student was asked during these interviews how s/he
had arrived at that conclusion, it was understood that s/he was influenced by
the appearance of the figure and expressed mathematical properties that were
in agreement to this conclusion.
An excerpt from an interview with one of the students is presented below:
Researcher: What can you say about the mathematical properties of
the given figure?
Student: The figure is a rectangle. All its angles are 90 degrees...
Researcher: Why did you arrive at the conclusion that it is a
rectangle?
Student: Because all its angles are 90 degrees...
Researcher: How did you understand that all its angles are 90
degrees?
Student: Because it is a regular quadrilateral. (Here the student
refers to the appearance of the figure.)
As can be seen in the interview excerpt, the student arrives at a conclusion
by being influenced by the appearance of the figure and s/he determines the
mathematical properties of the figure based on this conclusion. In other words,
s/he states that the figure is a rectangle without referring to any mathematical
reason and then states the properties of a rectangle. This shows that the student
initially states a conclusion and then writes hypotheses to explain the
conclusion.
In the 3rd category, there are not only students who wrote erroneous
mathematical properties but also students who converted perceptual
information to discursive information but did not write any mathematical
properties based on this information (3rd category-no response). However, it
was understood during the interviews that although a student had converted
perceptual information to discursive information in his/her written response
but had not written a mathematical property based on this information, s/he
had actually arrived at a conclusion by being influenced by the appearance of
the figure and when asked questions about the mathematical properties of the
figure, s/he drew a conclusion based on the appearance of the figure. For this
reason, this student was placed into “3rd category- Erroneous responses
produced under the influence of the appearance of the figure.”
REDIMAT, 11(1)
105
Following is an excerpt from an interview with one of the students:
Researcher: What can you say about the mathematical properties of
the given figure?
Student: Angles A and D are 90 degrees. Side BA is perpendicular
to side AD, and side CD is perpendicular to side AD.
Researcher: What else can be said?
Student: Nothing else is given.
Researcher: Then let me reword the question... What can be said
about angles B and C?
Student: Angles B and C (the student thinks for a while) ... 90
degrees.
Researcher: What can be said about the given figure?
Student: Well it’s a rectangle… That is what’s given.
Researcher: Why is it a rectangle?
Student: (referring to the appearance of the figure) Because its sides
are perpendicular.
Researcher: Why didn’t you write this information?
Student: I understood the question as “write the mathematical
properties presented on the figure.”
Researcher: How do you think the question should have been
worded?
Student: I think it should have been “Write the properties that are
not presented on the figure.”
As can be observed, even though the student found the perceptual
information on the figure sufficient and did not write any other information,
when asked questions about the figure, s/he gave responses based on the
appearance of the figure. When the student was asked why s/he had not written
the mathematical properties she expressed during the interview, s/he said that
s/he had misunderstood the question and thought that s/he had to write only
the mathematical properties presented on the figure. Moreover, the student,
who thought that converting perceptual information to discursive information
meant writing the mathematical properties presented on the figure, believed
that the question should be reworded as “Write the mathematical properties
not presented on the figure.” This indicates that the student regards the
conversion of perceptual information into discursive information not as an
essential hypothesis formation process to draw a conclusion, but a process to
determine the mathematical properties of the given figure.
Karpuz & Güven – Theoretical Discursive Process in Geometry
106
Discussion and Conclusion
By means of the current study, whether students at the very beginning of their
high school education possess cognitive readiness to pass on to the theoretical
discursive process has been investigated. To this end, some of the behaviors
needed for the theoretical discursive process were identified by utilizing
Duval’s Cognitive Model, and these were used to explain students’ cognitive
readiness. Based on the findings obtained in the study, it can be asserted that
most of the students could not display the behaviors that were essential for the
theoretical discursive process. The students were generally unsuccessful in
converting discursive information to perceptual information, writing
discursive information based on perceptual information and making
deductions based on discursive information. This indicates that 9th grade
students in Turkey contrary to assumption do not have readiness in directly
engaging in the theoretical discursive process and, thus, will be unsuccessful
in higher order skills such as providing proof, which necessitates the
theoretical discursive process. In fact, in numerous studies conducted on high
school students, students’ proof-writing ability was found to be very low
(Healy and Hoyles, 1998; McCrone and Martin, 2004; Senk, 1985).
In the theoretical discursive process, it is important for students to convert
discursive information to perceptual information and vice versa. However,
most of the students in this study were unsuccessful in displaying this basic
level behavior of converting the given discursive information to perceptual
information. According to Duval (1998), the most important reason
underlying this is that in learning environments importance is attached to
increase in knowledge, while cognitive and perceptual processes are
neglected. Thus, the teaching of concepts should not be the sole focus in
primary and secondary school education. Behaviors essential for theoretical
discursive process should also be given place in learning environments
because mathematics is not a branch of science consisting of concepts and
mathematical results found by some people, but a way of thinking (Cuoco and
Goldenberg, 1996).
Another behavior that is essential for engagement in theoretical discursive
process is to be able to make deductions based on the discursive information
derived from the figure. However, the findings obtained indicate that most
students are influenced by the appearance of figures when making deductions.
REDIMAT, 11(1)
107
Similar findings were revealed in numerous other studies (Michael, 2013;
Ubuz, 1999). While Ubuz (1999) attributed it to the fact that students were not
at the necessary Van Hiele level, Duval (1995, 1998) attributed it to the
dominance of perceptual apprehension and its influence on the discursive
processes. According to Duval (1995, 1998), drawing conclusions in relation
to figures should begin with discursive apprehension. If discursive
apprehension does not dominate the reasoning process, perceptual
apprehension becomes dominant and impacts students’ reasoning processes.
Even though he utilizes different concepts in his explanations, Fischbein
(1993), like Duval (1998), believes that when discursive information does not
dominate the reasoning process, the appearance of figures will affect the
conclusions (Fischbein, 1993; Fischbein and Nachlieli,1998). On the other
hand, Harel and Sowder (1998) attribute students’ behavior of making
deductions based on the figures’ appearance to their being within the
Perceptual Proof Scheme. According to Harel and Sowder (1998), students
within this scheme apprehend figures as static and cannot take into
consideration the different conditions of figures.
When both the written responses and the interview oral responses of
students who were influenced by the appearance of the figures were examined,
it was found that students initially made claims by being influenced by the
appearance of the figure and then put forward hypotheses to explain their
conclusions. When the students’ reasoning processes were examined, it can
be claimed that most students could not reason deductively and, hence, arrived
at conclusions under the influence of the appearance of the figures. In
deductive reasoning, the process begins with hypotheses, and conclusions are
essentially derived from these hypotheses (Özlem, 1994). Thus, similar results
were reported in a study by Healy and Hoyles (1998), which examined
students’ proof-writing ability and their opinions about the role of proof. In
this study, it was found that most students could not reason deductively while
they arrived at a conclusion. Furthermore, in studies where students’ being
influenced by the appearance of figures is attributed to their not being at the
necessary Van Hiele level (Ubuz, 1999) actually implicitly emphasize
students’ weaknesses in the deductive reasoning process. The rationale is that
a person who proceeds to the upper levels in Van Hiele should not be
influenced by the appearance of figures when drawing conclusions, which is
naturally a reference to the deductive reasoning process characteristic of the
upper levels.
Karpuz & Güven – Theoretical Discursive Process in Geometry
108
When the other reasoning processes (inductive and abduction) other than
deductive reasoning are considered, the only type of reasoning that starts with
a conclusion is the abduction reasoning process. The course of an abduction
process that seeks to reach possible explanations that would validate a
conclusion to be resulting from observation is conclusion-rule-hypothesis and
the hypotheses arrived at are not absolute explanations (Meyer, 2010). Thus,
it can be stated that the abduction thinking process was dominant in most of
the students who participated in the study. All these results should not lead to
the general conclusion that the abduction process always leads students to
make wrong conclusions. However, it can be concluded that when it is
considered that an abduction process begins with an observation, and when
students do not have the necessary instruments for them to make sufficient
observations, using the abduction reasoning process can lead to wrong
conclusions.
The most important instrument for students to make a sufficient
observation is a dynamic geometry software because by means of the
software, geometric figures can easily be made on the computer screen. Such
properties as angle, side, perimeter and area can be measured, and the
geometric figures made with certain associations can be moved around on the
screen. Consequently, all the measured properties of figures also change
dynamically (Güven and Karataş, 2009). For this reason, it can be said that
learning environments in which dynamic geometry software is used are more
conducive to the abduction reasoning process.
Based on the results of the present study, teachers are recommended to
determine how students interactively use their perceptual and discursive
apprehension before starting the geometry course and integrate into their
lesson activities that would accurately develop the relationship between these
two processes. Activities that specifically enable students to understand that
the appearance of figures is deceptive should be integrated into the lessons.
Moreover, the approach to discover the properties or identify the common
properties of geometric figures as of primary education should, over time,
leave their place to a process directed by definitions, axioms or theorems.
REDIMAT, 11(1)
109
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Yavuz Karpuz is Lecturer in the Vocational School of Technical
Studies, at the Recep Tayyip Erdogan University, Turkey.
Bülent Güven is professor in the Faculty of Education, at the Trabzon
University, Turkey.
Contact Address: Direct correspondence concerning this article,
should be addressed to the author.
Email: yavuz.karpuz@erdogan.edu.tr
