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International Journal of Mathematical
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The preference of visualization in
teaching and learning absolute value
Alper Cihan Konyalioğlu
a
, Zeki Aksu
a
& Esma Özge Şenel
a
a
Department of Mathematics Education, K.K. Education Faculty,
Atatürk University, 25240 Erzurum, Turkey
Version of record first published: 08 Nov 2011.
To cite this article: Alper Cihan Konyalioğlu, Zeki Aksu & Esma Özge Şenel (2012): The preference
of visualization in teaching and learning absolute value, International Journal of Mathematical
Education in Science and Technology, 43:5, 613-626
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2. International Journal of Mathematical Education in
Science and Technology, Vol. 43, No. 5, 15 July 2012, 613–626
The preference of visualization in teaching and learning
absolute value
Alper Cihan Konyalioğlu*, Zeki Aksu and Esma Özge S
° enel
Department of Mathematics Education, K.K. Education Faculty,
Atatürk University, 25240 Erzurum, Turkey
(Received 23 June 2011)
Visualization is mostly despised although it complements and –
sometimes – guides the analytical process. This study mainly investigates
teachers’ preferences concerning the use of the visualization method and
determines the extent to which they encourage their students to make use of
it within the problem-solving process. This study was conducted for the
ninth-grade students and their mathematics teacher in a social science
intensive public school in the city of Erzurum, Turkey. Utilizing case study
as the preferred method, data were collected through observations,
interviews and student evaluations. This study revealed that visualization
has a positive effect at the preliminary phases of teaching the absolute value
concept but generates a lack of stimulation during problem solving in
further phases of the instruction. This could be explained as a result of
current examination system which requires a habituation of the analytical
process in solving mathematical questions.
Keywords: mathematics education; visualization; visual representations;
absolute value
1. Introduction
Operations necessitating conceptual understanding within the experimental sciences
often cause faster perception and easier comprehension when compared to
mathematical operations and concepts due to both the laboratory setting and
external world, which provides visual representations of abstract concepts in the
experimental sciences.
Since mathematics is not a science based on experiments, students usually find it
harder to comprehend mathematical concepts and operations when compared with
experimental sciences. When this occurs, students try to learn the information which
they have failed to comprehend by memorization. As a result, students fail to
recognize the sensory aspects of mathematics and end up drawing away a belief
that doing mathematics means working with meaningless symbols [1]. For this
reason, measures have to be taken by teachers to eliminate such difficulties using
activities in the classroom that allow the students to make sense of mathematical
concepts.
*Corresponding author. Email: ackonyali@atauni.edu.tr
ISSN 0020–739X print/ISSN 1464–5211 online
ß 2012 Taylor & Francis
http://dx.doi.org/10.1080/0020739X.2011.633627
http://www.tandfonline.com
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3. Given the physical-world reflections of geometric concepts and shapes, as well as
certain concepts in mathematics, visualization can be thought as a phenomenon that
could introduce experimentality to mathematics to a certain extent. Expressing
abstract or algebraic statements as concrete objects or semi-concrete drawings
results, in a sense, in carrying abstract and algebraic statements into an experimental
environment. In other words, visual models could be seen as the reflections of
abstract concepts and symbols in the physical world. Ersoy et al. [2] argued that
mathematics is not an experimental (laboratory) science in a classical sense; yet,
shapes can usually be used for easier understanding and teaching of mathematical
subjects and problems where the main point is the expectation of students to produce
a text independent from shapes or drawings. Nevertheless, it should be remembered
that intuition mainly relies on drawings in producing such products. In a way, this is
a statement that visualization through shapes and drawings points out the
experimental aspect of mathematics.
Looking at it from this perspective, visualization could be defined as a bridge
built between ‘the world of experiments’ and ‘the world of thinking and reasoning’
[3]. Visualization presented in the form of drawings, shapes and concrete models and
representing the experimental aspect of mathematics, will, as suggested by
Hac|salihoğlu [4], ensure a stronger perception of operations and concepts as
drawings about abstract concepts lead to mental interpretation. Students can
effectively get the information and retain it by classifying, arranging and schema-
tizing the abstract concepts that they perceive as concrete or formal [3].
The literature contains many studies on the role of visualization in mathematics.
Some mathematicians and mathematics instructors are in favour of using visuali-
zation in teaching mathematical concepts, while others oppose it. Researchers
supporting the visualization method believe that it will enhance comprehension, self-
confidence and creativity in mathematics education [5–12]. Similarly, some maintain
that visual thinking may be a strong alternative resource for students by bringing
new ways of thinking in mathematics and also underlining the importance of
visualization and visual reasoning within mathematics teaching [13–16].
Hac|salihoğlu [4] refers to drawings, one of the visualization forms, as a factor
that can help improve students’ thinking ability. Jencks and Peck [17] report that
using visual models in problem solving facilitates students’ comprehension and
creates solution-finding opportunities. They asserted that visualization also helps
learning particularly in the beginning phases in which the basic concepts are taught
to students. They also maintain that visualization also has a crucial part in enhancing
problem-solving skills by playing an active role in ensuring long-term recall. Jencks
and Peck [17] argue that after working with concrete models, students establish links
within the logic of problem solving and develop distinctive formal rules, a process in
which the teacher’s function is simply to find a good model for the problem. In one
of his studies on the role of visualization in general mathematics course, Tall [15]
argues that visualization is much more effective than conventional approaches in
strengthening students’ intuitions and facilitating learning. Konyal|oğlu [3] considers
visualization as a tool which serves to attract students’ attention by drawing
geometric concepts and models with varying effects to implicate the presence of
various mathematical systems and various spaces; to help individuals acquire the
habit of abstraction and thus improve their cognitive independence and productivity;
to ensure meaningful learning and retention of information. Arcavi [18], on the other
hand, mentions the power of visualization ‘to make the invisible visible’, stating that
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4. since mathematical concepts are abstract, it would be useful to make concepts visible
for effective learning through teaching–learning process. Alsina and Nelsen [19]
underline the fact that visualization has a key educational value as a tool with a
possible central role in the beginning stages of problem solving, defining a concept,
and presenting a proof.
Although there is much research demonstrating the importance of visualization
in teaching–learning process of mathematics, visualization has never been brought to
the foreground in this process. Eisenberg and Dreyfus [20] attribute this to the fact
that teachers usually prefer the analytical process in teaching (even though they use
visualization in their own work), since they believe that visualization cannot form a
proof and it is not easy to establish and understand visual models. Ersoy et al. [2]
argues that it is chiefly images that allow one to discern the proof types in
mathematics and the ways used to solve a problem, whereas Eroğlu [21] states that
while the role of images in implicating and understanding relationships is undeniable,
images can never be a part of proof on their own and can only implicate the accuracy
of a judgement.
Researchers maintain that relying too much on visualization may prevent
mathematical thinking due to the limiting effect of a single-case scenario represented
by an image [22–27]. Similarly, some researchers discuss an old prejudice against
visualization in mathematics which concerns the reasons behind the preferences:
assuming that mathematics should be exact, analytical, symbolic and algorithmic
[14,28–32]. Speaking on this issue, Barwise and Etchmendy [13] indicates, ‘Despite
the particular importance of visual images on cognitive activities of human beings,
visual representation remains as ‘a second class citizen’ both in theory and practical
applications of mathematics. Particularly, we are taught to look at the arguments
which make the use of representation forms of diagrams and graphics unavoidable
and we transfer this scorn to our students’ (p. 9). Eisenberg and Dreyfus [28] pointed
out about the use of the figures that this contempt is more common among
educationists, whereas it is not that common among professional mathematicians,
who use math as a tool in reality. Eisenberg and Dreyfus [28] explain the reason for
rejection of the figures as a radical perception of philosophic belief in which a figure-
based piece of evidence is neither strong nor valid, and denote the radical belief in the
non-visual aspect of mathematics: ‘if there is someone to be blamed for retaining this
mathematics aspect in academic society it is us’. According to Cunningham [33],
emergence of this anti-visualization prejudgement dates back to late nineteenth and
early twentieth centuries as the remarkable achievements in formal and symbolic
mathematics impelled the mathematicians towards wholly symbolic studies and thus
discredited the visualization approaches in mathematics. Tall [15], on the other hand,
suggests that the denial or disclaiming of visualization would mean the denial of the
roots of most mathematical thoughts, as visualization represents the fundamental
source of those ideas within the first periods of mathematic development. As the
result of his study, Vinner [32] reveals that there are mathematical prejudgements
against visualization in mathematics. Attitude analysis demonstrates that students
appreciate the visual approach more and understand the terms better once the
concepts and arguments are visually exhibited frequently. Nonetheless, in the final
exam, in which they were free to use any method to solve the problems, all the
students were observed attempting to solve the problem using analytic approach
(although none of them fully succeeded). It was also revealed by Vinner’s [32] study
that most students avoid using visualization approach in formal evaluation.
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5. Furthermore, they were not satisfied even after assuming encouragement in using the
visualization approach to mathematical explanations. Vinner [32] pointed out that
this perception was deeply engrained in the field with regard to his previous
experiences with mathematicians.
Lean and Clement [30] ascertained that students with higher analytic abilities
reached better results on classic course problems than the students with higher visual
intelligence. They also stated, regarding the results of this study, that visual thinking
was not encouraged and rewarded during courses or in many course books. In his
study, Stylianou [34] ascertained that analytic and visual are not two differently
functioning processes; on the contrary, their thinking style is complementary.
A visual image becomes an important factor for both interpreting the data
significantly and explaining analytic development of the solution [35]. Vinner and
Hershkowitz [36] observed learning deficiencies in students when the course subject
was taught as a mainly analytic definition or with limited visual examples. They also
put emphasis on the necessity of using visual representations for accurate conception
of the image. Eisenberg and Dreyfus [20] predicate that although they could easily
answer each problem with the visualization approach, students face considerable
trouble in their attempt to answer the questions using the classic algebraic approach
in problem solving and thus they failed to correctly answer the questions. However,
appropriate visual representation may help the students’ to make sense of symbolic
systems which they have to cope with in algebra lessons [9]. Ben-Chaim et al. [5], on
the other hand, interpret it this way: ‘. . .regarding our experiences we consider that
secondary school students as well as the pre-school teachers need experiences which
have concrete and semi-concrete representational approaches towards the problems
before they understand more abstract algebraic generalizations.’
It is not that easy to reach a definite conclusion on the use of visualization
method in mathematics education and more difficult to decide on which subject and
in which phase of the teaching process to use it. Yet, proper knowledge of teachers’
and students’ preferences on the use of the visualization method in addition to
deciding to what extent the teacher’s problem-solving style influences students in
absorbing mathematical concepts provides both with a more suitable and effective
education environment. In this respect, the purpose of this study is to analyse
teachers’ preference for visualization and encouragement to students to use
visualization in learning absolute value and to reveal to what extent the students
make use of this approach within the problem-solving process.
Experiencing many difficulties in the teaching and absolute value in schools [37]
and availability of foreknown numerical axis as a model in absolute value issues led
the researcher to choose this topic. This study intends to answer the following
questions: (1) is the visualization which is thought to support the analytic process in
either solving a problem or in understanding the topic being used by teachers and
students? and (2) do the teachers encourage students to use visualization?
2. Method
Case study method is the predominant method of experimentation for this study.
Case study is a holistic and in-depth investigation of a single case or fact instead of
following certain rules for investigating the limited number of variables. It is the way
of observing the events in the real environment, collecting and analysing the data in a
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6. systematic way and revealing the results. The result of this method shows why an
event happens that way and guides further researches to be done on the details which
are important to focus on. The case study method is more applicable for the analysis
of events in a real-world environment rather than testing or suggesting
hypothesis [38].
2.1. Sampling
This study was conducted in the Social Science Intensive State School in the Erzurum
province of Turkey. At the time this research was being conducted, this high school
admitted students scoring between 395 minimum and 444 maximum out of 500 full
score from High School Entrance Exam. The school had approximately 200
students. There were two ninth grade classes in this school and two different teachers
were teaching in each of these classes. The math teacher of the class with whom this
study was conducted had 10 years of experience by that time. Since the teacher of the
other class had not taught absolute value topic yet, the study was conducted in only
one class which had 22 students with an almost equal gender distribution. With
regard to the purpose of this study which is to ascertain whether students use the
technique the teacher uses in the class in problem-solving process, a moderate level
school was chosen.
2.2. Data collection instruments
One of the important strategies used in assuring reliability and validity in a
qualitative research is variation. Affirming the data obtained through observation
with interview or affirming the results reached by an interview with observation
could set a good example of variation. Denzin (1970) suggests that the basic principle
in variation is to collect data from different individuals in different environments
with different methods and thus to prevent misunderstandings or prejudgements that
could emerge from the results. Furthermore, variation could serve to evaluate and
give meaning to the results of the research from different dimensions. Thereby,
readers could get a better view of the reliability and validity of the research
results [39].
For this reason, the data constituting the basis of this study were obtained
through three different data collection instruments: observation, interview and the
achievement test. The main reason for using these three instruments together is to
make up the deficiencies of each other and to put more distinct data forward. In
order to confirm teacher’s use of visualization during the lesson, observation method
was used. The achievement test was applied to determine students’ use of the
visualization method within problem-solving process. Interviews were conducted in
order to confirm the data obtained through the first two instruments.
2.2.1. Observation
Overall, 6 h throughout the term which the teacher was teaching, the absolute value
topics were observed. A semi-structured observation was performed. The observer in
the class watched the lessons’ purpose being to confirm teacher’s use of the
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7. visualization method during the class, and to monitor students’ reactions to their
teacher’s approach especially in problem-solving process.
2.2.2. Achievement test
After the observation, students were asked six open-ended questions, four of which
were from ninth grade mathematics course book [40] and two from university
(student) selection exam. Questions are given in the annexes. Out of 22 students, 20
of them, 11 male and 9 female, took the exam. The test consisted of questions
regarding the definition of absolute value term and its geometrical interpretation.
The main purpose of this exam was to see students’ use of the visualization approach
in the problem-solving process in cases where the teacher uses the same approach.
Students were informed before the exam that the scores they got from the exam
would not be subjected to any formal evaluation and were given 1 class hour to
answer the questions. The exam papers were evaluated by the researchers.
2.2.3. Interview
In accordance with the test applied and the observation performed, interviews were
conducted with three female students and one male. Also, additional interviews were
conducted with the class teacher and two instructors who were experts in the field in
order to ask for their experience on the topic. In those interviews, each was asked
different questions by the researchers after the literature review The main purpose of
the interviews with students was to see the effectiveness of visualization within the
process of topic comprehension, the effect of the teacher’s preference of using
visualization on students and students’ use of this approach in the problem-solving
process. The main purpose of the interview with the class teacher was to get her
opinion about the necessity of the use of this method and the reasons obliging the
students to use this method in problem-solving process. The purpose of conducting
interviews with the instructors, on the other hand, upon the interviews held with the
instructors of the university, was to clarify their aspects regarding the necessity of
using this method in teaching and the visualization method in general. Interviews
were recorded and then transcribed.
2.3. Data analysis
After transcription, interviews and observations were analysed with the code-
category-theme system. The achievement exam papers of the students were analysed
and all transcripts and obtained scores were evaluated by the researchers. Different
methods were applied to ensure the reliability and validity of the research. First, all
three researchers collaboratively acted in each phase of research design, composition
of research statement and hypothesis, and data collection and analysis of the study
from the beginning to the end. Consensuses were reached in cases of disagreement.
Second, the data obtained were shared with field experts in the same faculty. Thus,
the pre-determined codes representing the categories in the study were confirmed by
the experts.
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8. 3. Findings
Regarding the interviews conducted with the class math teacher and the
instructors, their opinions on visualization in general were recorded, in addition to
their opinions on the use and importance of visualization in mathematics teaching
(is visualization approach applicable for each course subject, what are the
advantages and restrictions of visualization from students’ aspects?) and the
applicability of visualization approach in absolute value subject. As the result of
those interviews, the offered considerations of the teacher and the instructors are
shown in Figure 1.
In the interview conducted, the teacher described visualization as expressions
which build concrete conceptions in an individual’s brain. She asserted that the
visualization within mathematics teaching clarified the problem in student’s mind
and thus made the solution of the problem much easier. She then indicated that she
did not think that visualization is advantageous or applicable in each subject and
that she could not agree with that idea;
The class teacher: ‘When you draw a function graphic, for example, a set is
demonstrated within a simple curve and the student can easily discuss it with you.
However, when it gets more complicated, it causes confusions in students.’
Considerations regarding the conception of visualization and
use of the method
According to the teacher According to the instructors
Visualization method,
should build concrete
conceptions in an
individual’s brain
This method should
especially be used in the
introduction part of the
course subject
It could not be benefited
from in each course subject
It should certainly be used
in absolute value subject
Visualization method,
should make the apparent
things visible and
embody the abstract things
This method should
especiallybe used at the
introduction part of the
course subject
It cannot be applicable in
each course subject
It is applicable for being
used in absolute value
subject
Figure 1. Considerations regarding the conception of visualization and the use of the method
in mathematic education and especially in absolute value subject from the aspects of the
teacher and the students.
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9. In the interviews conducted, in parallel with the teacher, the instructors described
the visualization as making the apparent visible embodying abstract subjects. They
also indicated that visualization was not applicable in all mathematical subjects and
in every stage of each subject. However, they emphasized that visualization ‘should
be applied in especially introduction parts of some subjects.’
The class teacher specified the absolute value as a difficult subject for the
students. She suggested that visualization should certainly be used while teaching
absolute value and be brought up while discussing the numeric axis. In addition to
this, she also suggested that visualization method be used especially at the
introduction part of the absolute value subject. Simply put, the instructors regarded
the absolute value subject as one of the topics in which visualization should be
used.
The majority of the students used the analytic method in order to reach solutions
to the problems asked in the applied achievement test. It was also seen that they did
not make use of the visual expressions (the numeric axis and coordinate plane, etc.)
the teacher used in solving the problems and they answered most of the questions
which they intended to solve incorrectly with analytic methods. The reasons why the
visualization method was not used in problem solving by students and the class
teacher’s theories are given in Figure 2. In the interviews conducted with the
students, they were asked to offer their thoughts on the absolute value subject and
about the visualization method the teacher used at the time of instruction. Moreover,
they were asked why they did not use visualization method in answering the
questions in the achievement test.
Below is the summary of the interviews held with four students.
Student #1 suggested that absolute value expressed an entity that he understood
better when the teacher taught it using visualization method. Figures were more
effective to the student than the verbal expressions when it came to apprehension of
the subject:
Student #1: ‘Well, absolute value subject was far more complicated to me at first till this
year. This year, I have a better understanding of it. It might have something to do with
those figures as well . . . You know. On top of that, there is the teacher’s giving the
examples in the class from our daily life in teaching a subject that has made us
understand the subjects better. You see, it is very different . . .’
Referring to ignoring the use of visualization in problem solving, he asserted that
visualization was more effective in the conventionalizing phase; it was a waste of time
because the present examination system compelled the students not to use the
visualization system.
Student #2 indicated that the absolute value implied optimism in everyday life in
the sense of every negative occasion’s turning out to be positive in the end.
Furthermore, he also emphasized that they understood the absolute value subject
better once the teacher tried to teach the concepts with a given visual meaning.
He admitted that teacher’s use of visualization approach in the comprehension phase
was more effective for him and he found the approach interesting.
Student #2: ‘The teacher explained to us, well, demonstrated the numeric axis, and told
us the distance to the adjacent numbers. Here, for instance, this example stuck us and
she even demonstrated it to me, followed by some students nearby. In other words, she
made the students comprehend the subject this way and we saw that for the first time
this year, it was amazing.’
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10. He attributed his not using this method in the achievement exam simply to their
being compelled to briefly and concisely reach the answers.
Student #2: ‘Her using such methods as visualization for having us comprehend the
subject is really favorable but because we live in a system which directly correspond to
university (student) selection exam, I cannot think of using this method. Since we, as
students, do direct problem solving (question analysis or so) teacher’s efforts to have us
comprehend the subject is quite useful. But it remains in class, you know. To tell the
truth, teacher’s having us comprehend the subject is very useful from the aspect of
transferring it to the real life; however, it feels like a fantasy in the exams.’
Student #3 said that absolute value evoked optimism to him. He specified that all
necessary terms were easily remembered since they became more concrete due to the
visualization method and thus learnt easier. As the reason for not using this method
in the exam, he determined that this method did not immediately come to mind, and
said he had not noticed which method he used in solving the problems during the
exam, although he understood the subject better with the visualization method:
Student #3: ‘It, for sure, sticks in our minds since it is visual, you know. Well,
I demonstrated in the numeric axis and the coordinate plane . . .’
Considerations on the use of visualization in
achievement test
According to
students
According to the
class teacher
Result’s being more important than
the process
University (student) selection exam
system
The waste of time
Inapplicability of the method in
problem solving
Using the term “what’s the result….?”
consistently asking questions to the
students”
Readiness level of the students
Teachers’ deficiencies
Students’ attitudes
Figure 2. Students’ considerations over not using the visualization method solving a problem
in the achievement test although this method was used by the teacher in instruction period.
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11. Here, the student perceived the numeric axis and the coordinate plane as
comprehensible and apparent in assisting in understanding of the subject.
Student #4 said that absolute value evoked optimism and distance from his view
point. He also mentioned that learning is more permanent with the figures and the
graphics regarding the visualization. He added that he did not need to use this
method in some questions because he wanted to briefly and concisely reach the
conclusion in others.
The class teacher, on the other hand, claimed that the very first reason why the
students did not use the visualization approach was that their study is result-focused
and they do not push their limits incoming up with different ways of solving and
viewing the problem. Here, she also pointed out that the problem was not only with
the students but also with the teachers. The emphasis put by the teacher on the
phrase ‘what is the result . . .?’ in asking questions in the exams parallels the students’
focusing on the result in working out the problem.
4. Discussion and conclusion
Contrary to teacher’s suggestion to use the visualization method (especially in the
introduction part of the absolute value subject), students were determined to abstain
from using it in problem solving. Although the teacher used visualization method as
a supportive process for the analytic approach – especially in the introduction part of
the absolute value subject – she did not prefer using it in following sections and, in
addition, did not encourage the students to use it. However, the use of
visualization method and encouraging students accordingly could help students
look at the problems in different ways and develop different thinking styles for
problem solving.
The main reason why students gave no answers to some questions came up with
so many wrong answers and overall, failed in the achievement test, was their
preference for using analytic process first instead of demonstrating the expression in
the question on the numeric axis. As a result, they could not put the correct symbols
between the terms. This is because they could not reflect their knowledge upon the
analytic process in problem solving because they had learnt absolute value by visual
methods during the class. However, those questions could be answered in an easier
way using visualization method, and his assessment corresponds to the study of
Eisenberg and Dreyfus [20]. Considering a mathematic problem, an analytic
solution simply offers a syntactic interpretation, whereas a visually enriched solution
would play a more complementary role. This kind of solution supports the analytic
results.
The reason why students did not prefer visualization in problem solving is not
simply by prejudgement as Vinner [32] stated, but stems from the present university
(student) selection system in Turkey, whereby the test strategy is the desire to reach
the result in the fastest way possible. Students, therefore, perceive this method in
problem solving as a waste of time in the exams. However, the students who used
analytic process in order not to waste time answered most of the questions
incorrectly. The present examination system and its requirements compel students to
work in a way where the result becomes more important than the method. If students
had used the visualization method in supporting the analytic process, they would
have reached the right answers.
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12. It was determined that teacher’s use of visualization method at the beginning of
the class attracted students’ attention to the subject and relieved classroom boredom.
Interviews held with related participants also revealed that visualization caused
students to develop positive attitudes towards the concepts. In other words, the
teacher’s use of visualization method turned the passive behaviours of the students at
the beginning of the class into a tendency to become more active and increased the
participation of the students. Consequently, the class was successful in participating
in the instruction process and the students reached higher levels of motivation.
This finding verifies Vinner [32]’s findings.
The usefulness and necessity of using visualization for each subject within the
field of mathematics cannot be taken for granted. Studies beforehand would be
necessary for determining what kind of visualization method would be used in which
course subject or in which part of any subject (e.g. this method is useful in the
lecturing and problem-solving portions of teaching process). Furthermore, students
found the visual model easy despite the negative reviews of the complexity of the
model in current literature, a scenario which, whether or not it discredits the current
view of the visualization method, warrants further study.
References
[1] A.B. Oaks, Writing to learn mathematics: Why do we need it and how can it
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Appendix 1. Interview questions and the achievement test
Interview questions (students)
Questions
(1) What is absolute value? What does absolute value mean to you? How could you
explain us?
(2) Studying the absolute value subject, upon which approach of the teacher did you
consider yourself having a better understanding?
(3) What could you say about geometric interpretation of absolute value?
(4) Alternative: Assuming that you are asked to interpret the absolute value other than
arithmetical operations, what would you say over correlating it to the geometric
elements such as numeric axis or tables?
(5) What is the first way of solution that comes to your mind when you are asked a
question of absolute value? Alternative: What is the first approach or strategy you use
in solving a problem?
(6) What could you say on teacher’s instructing the subject with figures from the aspect of
teacher’s taking the advantage of? Probe: such as getting easy, concretizing, better and
more comprehensive.
(7) Well, why didn’t you use this approach in problem solving?
Interview questions (the teacher)
Questions
(1) What comes to your mind in general regarding visualization?
(2) What do you think of visualization in mathematics? What kind of conveniences would
be provided with the use of visualization in mathematics education?
(3) How is absolute value subject perceived by the high school students? How do you
make use of visualization technique in having the students apprehend the topic?
(4) What are the advantages of using this approach from students’ aspect?
(5) Can you give any example about the use of this approach in absolute value subject? It
can be one of the ways you use.
(6) What kind of contributions do you consider the visualization approach can make to
the field of mathematics in modern education system?
International Journal of Mathematical Education in Science and Technology 625
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15. The achievement test
Questions
(1) Find the solution set of the equation given by j5 jx þ 1jj ¼ 3.
(2) Write down the inequality given by 16 5 x 5 38 using absolute value.
(3) If 3 5 x 5 7, then jx 8j þ jx 2j ¼ ?
(4) Find the sum of elements of the solution set of the equation given by jx 7j ¼ 2007.
(5) Find the sum of elements of the solution set of the equation given by jx 4j þ jxj ¼ 8.
(6) If x ¼ j
ffiffiffi
5
p
3j, y ¼ jx 5j and z ¼ j y 2j, then z¼?
626 A.C. Konyalioğlu et al.
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