EXPLORING PROOF & REASONING INSTRUCTION IN LEBANON
LEBANESE UNIVERSITY
Faculty of Pedagogy
Deanery
Exploring the Instruction of Proof and Reasoning in Textbooks and by Lebanese
Secondary Teachers:
Grade 10 –Absolute Value
Proposal for a Thesis in the Field of Mathematics Education
Submitted by
Salam Hassan
In Partial Fulfilment of the Requirements for
a Research Masters in Mathematics Education Degree
Beirut
Date: October 14th, 2021
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EXPLORING PROOF & REASONING INSTRUCTION IN LEBANON
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Table of Contents
Table of Contents ........................................................................................................... 2
Chapter One: Introduction.............................................................................................. 4
1.1 Research Problem................................................................................................. 4
1.2 Rationale .............................................................................................................. 4
1.3 Significance .......................................................................................................... 5
1.4 Theoretical and Conceptual Frameworks............................................................. 6
The Whole Picture: Conceptual Framework .......................................................... 6
1.5 Purpose and Research Questions ......................................................................... 7
Chapter Two: Literature Review .................................................................................... 8
2.1 Thematic Review ................................................................................................. 8
Concluding Statement ............................................................................................ 9
2.2 Operational Definitions ........................................................................................ 9
Justification ............................................................................................................ 9
Proof ..................................................................................................................... 10
Proof-related Reasoning ....................................................................................... 10
Discourse .............................................................................................................. 10
Chapter Three: Methodology ....................................................................................... 11
3.1 Methodology Selected........................................................................................ 11
3.2 Research Sample ................................................................................................ 11
3.3 Instruments of Study .......................................................................................... 12
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Studying Textbooks ............................................................................................. 12
Teacher Questionnaire ......................................................................................... 13
Post-questionnaire Interview ................................................................................ 14
Studying Discourse .............................................................................................. 14
3.4 Validity of Instruments ...................................................................................... 14
3.5 Reliability of Instruments................................................................................... 14
3.6 Ethical Concerns ................................................................................................ 15
3.7 Limitations of the study ..................................................................................... 15
3.8 Procedures of the study ...................................................................................... 16
Chapter 4: Results ........................................................................................................ 17
4.1 Data Analysis Procedure .................................................................................... 17
Appendix A .................................................................................................................. 18
Codes for Mathematical Discourse .......................................................................... 18
Classroom Observation Discourse Form ................................................................. 19
Appendix B .................................................................................................................. 20
Section 1: Demographics ......................................................................................... 20
Section 2: Past proof-related reasoning work with students .................................... 21
Section 3: Proof and Reasoning in Absolute Value lesson of grade 10 ................... 22
References .................................................................................................................... 24
List of Figures .............................................................................................................. 29
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Chapter One: Introduction
1.1 Research Problem
Reasoning-and-proving has been recognized worldwide as a standard of teaching
mathematics in schools, and has been incorporated in the modern mathematical curricula with
strong emphasis (Hanna & de Villiers, 2008; Hanna et al., 2012; NCTM, 2000, 2009;
Stylianides & Stylianides, 2017). Nevertheless, students and even teachers face difficulty
with the study of proof (Almeida, 2001; Thompson, 1996; Stylianides, 2014). Furthermore,
many studies have conceded that the performance of students in this topic on the secondary
and the undergraduate levels is worrisome (e.g., Harel & Sowder, 2007; Hemmi, 2008;
Weber, 2001). Through my experience, as a secondary mathematics teacher, I have met
colleagues who lack the pedagogical and/or content knowledge to teach reasoning-and-proof
to students. They tend to skip this part, as if it is optional and unnecessary. As for students in
the secondary level, it is very common to see students struggling with the topic of
mathematics in general, and in the reasoning and proof component in particular.
1.2 Rationale
The fact that students usually encounter difficulty with reasoning and proof calls for a
change in the ways students learn to prove at schools. The change starts by understanding
how the three components: the curriculum, the teacher and students interact in classroom
settings incorporating mathematical reasoning discourse (Bieda, 2010).
A strong emphasis is put on the role of the textbook in affecting students’ conception
of proof, and teacher’s beliefs and attitudes toward teaching this concept (Stylianides, 2014;
Stylianides & Silver, 2004). However, the present literature has shown that textbooks are not
helping students to engage in justifying and reasoning (Bieda, 2010).
On the other hand, various research suggests that the cause of students' difficulties in
mathematical proof and reasoning is teachers' practice rather than textbooks (Bieda, 2010;
EXPLORING PROOF & REASONING INSTRUCTION IN LEBANON
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Stylianides, 2014). Interestingly, exemplary teaching practices have made reasoning and
justifying tasks accessible even to young students at the elementary level (Lampert, 1990;
Zack, 1997).
Stylianides and Silver (2004) hint at a deficiency in the analyses of the modern
curricula for reasoning and proof, and describe it as unavailable. In addition, there is not
much research about the extent and type of proof-related educational tasks in domains of
mathematics other than geometry (Johnson et al., 2010). With respect to evaluating practice
and implementation, Bieda (2010) states that there is a scarcity of available information about
the teaching of reasoning skills at schools. In Lebanon, there are several papers that relate to
mathematical reasoning education, but they either have different purposes or target different
classes (e.g. Al Masri, 2013; Nassar, 2010; Osta et al., 2017).
As a result, a further need for multi-level investigation of proof and reasoning
education is obvious. In this study, we intend to study textbooks, the teacher plan and
implementation of such reasoning tasks with students in classrooms.
1.3 Significance
Although our study will focus on one Algebra lesson only in one secondary class, it
should give a general view of the nature and extent of proof-related tasks in the Lebanese
Math Algebra curriculum. The results should provide an example of how teachers plan tasks
of this kind, and how they implement them in a classroom setting. Moreover, we expect that
our study may provide an explanation for some of the difficulties that students encounter in
the field of proof, and as a result, suggest possible solutions to this issue. As a result, we will
then have a research-based study that we can rely on if we want to adapt a change that
ensures Lebanese students are receiving and achieving the required skills as adopted by the
international vision and standards for mathematics education.
EXPLORING PROOF & REASONING INSTRUCTION IN LEBANON
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In our study, we are targeting multiple audiences. First, curriculum designers might
benefit from the results in order to develop the curriculum materials to support proof and
reasoning in Algebra. Furthermore, this thesis will be of help for student teacher trainers and
tertiary education course designers in the preparation of prospective teachers, to be aware of
current problems in practice and tackle this issue in their lesson planning and delivery.
Besides, educational stakeholders at the Ministry of Education, and educational supervisors at
schools might use the results of this dissertation, to design workshops to help in-practice
teachers in overcoming the present difficulties. Last but not least, we aim to help researchers
understand more the implication of the textbook, the planning and implementation of proofrelated tasks, on students’ cognitive level in the reasoning and justifying domain.
1.4 Theoretical and Conceptual Frameworks
This study is rooted in an integrated theoretical framework composed of three
theories: Mathematical tasks framework (Stein et al., 1996), Revised Bloom’s taxonomy
(Anderson et al., 2001), and 5 practices model for orchestrating productive mathematics
discussions (Stein et. al., 2008). The details of the three theories were not presented in this
proposal due to the word limit.
The Whole Picture: Conceptual Framework
To provide a better and clearer understanding of the framework for our study, we have
constructed the map of Figure 1 showing connections between different parts of the theories.
Figure 1
Conceptual Framework of this Study
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Note. Highlighted text represents areas that will be explored in the study.
1.5 Purpose and Research Questions
The purpose of this study is to relate the textbook material, the teachers' preparation
and the students' implementation and discussion of mathematical tasks to the extent and
nature of proof and reasoning task opportunities in the lesson Absolute Value for grade 10
students in Lebanon.
Please note that this study is a mixed approach study with a qualitative component,
which is the reason why no hypotheses are presented.
In particular, our study examines the following questions:
1. What opportunities for proof and reasoning as defined and measured by Thompson et
al. (2012) are designed in the mathematics textbooks for grade 10 students in the
lesson Absolute Value in Lebanon?
2. To what extent does the Lebanese grade 10 teacher select tasks for instruction that
promote students’ reasoning and proving skills in this lesson?
3. What relationship can be seen between teacher’s and students’ engagement in a
mathematical discourse in terms of the development of high-level reasoning skills of
Lebanese grade 10 students?
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Chapter Two: Literature Review
2.1 Thematic Review
Literature review will be divided into three themes relative to the three areas of the
study, and it will be concise in order to respect the word limit of the proposal.
The first theme is analyzing the textbooks. Three articles were reviewed, and they all
began from the problem of students having difficulty with proof (Bergwall, 2019; Bergwall &
Hemmi, 2017; Thompson et al., 2012). They used or built on the Thompson et al. framework
to analyze and code the narrative and student tasks sections of the textbooks. The results
agreed that there are few opportunities in the textbooks for students to encounter proofrelated reasoning especially in the student tasks parts. It is notable however that in Thompson
et al. study (2012), half of the properties in the expository sections were justified either
generally or specifically.
Second, we are targeting teacher preparation. Davidson et al.’s study (2019) started
from the difficulties elementary teachers encounter with reasoning. They focused on
preparing material that may help teachers with teaching and assessing reasoning after a short
professional development. The results were that teachers need assistance in this field, and this
assistance might be in the form of available teaching material. On the other hand,
Mwadzaangati (2019) initiated her study again from the difficulties of students with proof.
She aimed at the comparison between mathematical tasks as set up in the textbooks and their
implementation by teachers. She found that teachers practices have mostly lowered the high
level of cognition of tasks set up in the textbooks.
Finally, literature on mathematical discourse is reviewed. Legesse et al. (2020) have
originated their study from students having trouble in understanding the concepts and
procedures of mathematics. They tested the effect of discourse-based instruction on eleventh
graders’ performance, and got positive results that this type of instruction, if conducted
EXPLORING PROOF & REASONING INSTRUCTION IN LEBANON
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properly, would enhance students’ understanding. Alternately, Maoto et al. (2018) have
studied the learning of representation, reasoning and proof by ten high school math teachers,
then analyzed the classroom of one of them while teaching the same processes. The results
were that teachers hurry to reach the end process of learning (numerical and symbolic
representation) without paying careful attention to the discourse (multiple forms of
reasoning).
Concluding Statement
The factors influencing students’ difficulties with proof explored in the literature
suggest three themes: curriculum, teacher choices, and classroom discourse. These factors
call for the need to study textbooks, evaluate teacher preparation, and assess students’
engagement in classroom discourse. The reviewed articles provided the framework to study
textbooks, hinted at difficulties in teachers’ preparation, and raised the issues of classroom
discourse and its importance.
Our study will extend the analyses of textbooks to a new domain of Algebra. In
addition, it will explore how the content material is used and implemented in the classroom,
as suggested by Bergwall (2019) as a “necessary next step” (p. 18). On the other hand, this
study will explore the teacher’s choices and actions in terms of opportunities provided for the
development of students’ proof and reasoning skills. Needless to say, our study will be done
at a different site, locally in Lebanon, where a recorded research of the questions at hand is
unavailable.
2.2 Operational Definitions
Justification
As defined by Bergwall (2019), a justification is any rationale whose goal is to
convince the reader that a certain mathematical statement is correct.
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Proof
Thompson et al. (2012) used the terms proof and general argument synonymously.
Muca (2015) defines proof as “a chain of mathematical statements starting from a hypothesis
to a conclusion by means of logical deduction and proper uses of mathematical language and
symbolic notation, where each mathematical statement connects … two or more
mathematical ideas” (p. 21).
Proof-related Reasoning
This term is also expressed as “reasoning-and-proving”. As its name suggests, it refers
to proof-related aspects of reasoning, and comprises making mathematical generalizations
and providing support to mathematical claims (Stylianides, 2009).
Discourse
To define mathematical discourse, we will refer to the view of Moschkovich (2007):
I view Discourse practices as dialectically cognitive and social. On the one hand,
mathematical Discourse practices are social, cultural, and discursive because they
arise from communities and mark membership in Discourse communities. On the
other hand, they are also cognitive, because they involve thinking, signs, tools, and
meanings. (p. 25)
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Chapter Three: Methodology
In this chapter, we will present the proposed research methodology of the study, along
with the sample, instruments of study, their validity and reliability, in addition to ethical
considerations, limitations and procedures of the study.
3.1 Methodology Selected
Problems in education do not come in an exclusive quantitative or qualitative
appearance, but rather they are usually a blend of both. This is the reason why it is widely
known that a mixed methods approach yields a better understanding of a research problem by
providing multiple views and different study methods (Cohen et al., 2018; Creswell, 2012;
Creswell & Creswell, 2018; Creswell & Plano Clark, 2017; Hesse-Biber, 2010; Leavy, 2017).
In this study, we will start by studying textbooks quantitatively according to Thompson et al.
(2012). We will then conduct a questionnaire for teachers followed by some interviews.
Finally, we will be analyzing qualitatively classroom discourse while delivering the lesson
Absolute Value in grade 10. In order to tackle the three different aspects of our problem, a
sequential mixed study approach is essential.
3.2 Research Sample
Since we aim to study textbooks, teachers and discourse, our sample will be divided
into three groups.
Concerning the textbooks, we will study the national textbook “Building Up
Mathematics” by CRDP for grade 10 which is used in public schools in Lebanon, and the
Collection: Puissance Al-Ahlia textbook “Mathematics FSG” which is widely used in private
Lebanese schools.
As for the teachers, the sample will be 50 to 100 teachers for the questionnaire
answering voluntarily on a Google form. Interviews will be conducted via online virtual
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meetings based on the results of the questionnaire with teachers who are willing to do the
interview.
Finally, to study discourse, we will be analyzing sound recorded periods in grade 10
classes of two schools, one public and another private, chosen by convenience. This last
method of choice, although may not be representative, will be the only possible choice since
recording classroom sessions needs a lot of approvals from teachers and administrators, and it
is not easy to get these consents given the present circumstances in Lebanon.
3.3 Instruments of Study
Studying Textbooks
In the study of textbooks, Thompson et al. (2012) framework will be adopted to code
the narrative sections and the exercises. A copy of this framework is provided in figures 2
and 3 below.
Figure 2
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Figure 3
Teacher Questionnaire
The teacher questionnaire is composed of three sections: Demographics, Information
about past proof-related reasoning work with students, and Information about incorporating
reasoning and proving in the lesson Absolute Value. A copy of this questionnaire is provided
in Appendix B.
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Post-questionnaire Interview
The interview questions will be built based on the teachers’ responses to the
questionnaire to be more meaningful and informative.
Studying Discourse
The transcripts of discourse, obtained from audio-recorded sessions, will be studied
using the revised OMLI Classroom Observation Protocol (Appendix A). This protocol was
adapted from a dissertation by Cayton (2012).
3.4 Validity of Instruments
Several measures were and will be taken to ensure the validity of the instruments in
the study. First, the framework for coding textbooks was taken from Thompson et al. (2012)
which is a valid source, and was built on in several subsequent studies (Bergwall & Hemmi,
2017; Bieda et al., 2014; Davis et al., 2014; Zhang & Qi, 2019). The questionnaire was
formed based on the literature review and the research question RQ 2. It will be piloted on a
sample of teachers other than the sample of study. It will also be reviewed by three math
experts before implementation. The interview questions will be based on the results of the
questionnaire, and will also be triangulated before use. The revised OMLI Classroom
Observation Protocol, the instrument used to study discourse, is also adapted from a valid
source (Cayton, 2012).
3.5 Reliability of Instruments
Reliability of an instrument reflects the consistency of results it produces when
applied several times at different times (Creswell, 2012). To ensure reliability of the
instruments of the study, multiple measures will be taken. First, there will be a test-retest
reliability by administering the questionnaire to the same population of teachers twice. In
addition, different judges will analyse the results (data) obtained by the instruments. As for
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the quantitative data, reliability, especially the internal consistency, will be calculated using
the Cronbach’s alpha measure.
3.6 Ethical Concerns
Ethics will be guaranteed in this study through multiple procedures. As for the
questionnaire, it will be filled voluntarily by teachers. Moreover, the interview will be
conducted with teachers who offer to do it after telling them that their information will be
kept safe, and no personal identifiers will be connected to their responses. As for recording
sessions in two classes, the teachers of these classes will do this willingly after getting verbal
consent from the administration and the students, and after assuring them that the school’s
name and the identity of the students will remain anonymous. The research purposes and
procedures of the study will also be communicated to the participants in advance.
3.7 Limitations of the study
Of course, it would be inconvenient to study every grade 10 mathematics textbook
that might be used in every school in Lebanon. That’s why we chose the two most frequently
used textbooks, so our results on textbook analysis will cover most but not all what Lebanese
students are exposed to in their school education at the first secondary level regarding proof
in the lesson Absolute Value.
In order to get more teachers to answer the questionnaire and have a more
representative sample, many subjective detailed questions were removed from it. These
questions will be postponed to the interview part which will be done with volunteering
teachers by convenience, which might be another limitation of the study.
Finally, the two grade 10 classes where the sessions will be recorded will be chosen
by convenience, and they constitute a very small sample with no specific characteristics that
the researcher might want to have or control during the study. In addition, the researcher
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won’t be able to observe the classes on site, but rather he will ask for recordings of the
sessions, which might elicit a new limitation.
3.8 Procedures of the study
Our study will be divided into three parts done at different times.
First, we will study the textbooks by coding the narrative and the application sections
of the lesson using the coding already presented in this chapter.
As for the teachers, we will be sending the questionnaire via any available electronic
means. Interviews will be conducted via online virtual meetings based on the results of the
questionnaire with teachers who are willing to do the interview.
Finally, to study discourse, we will be analyzing sound recorded periods in grade 10
classes of two schools. Sessions to be observed will be audio-recorded. The focus of the
study will be on teacher to students discourse and vice versa. For this, a sound recorder will
be placed next to the teacher, which should capture this discourse clearly enough. Portions of
the audio which include the sought for discourse will be transcribed. To analyse the
transcripts, the revised OMLI Classroom Observation Protocol (Appendix A) will be used.
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Chapter 4: Results
4.1 Data Analysis Procedure
Data will be dealt with carefully to ensure privacy and confidentiality. Quantitative
data obtained from textbooks coding and the questionnaire will be analysed using SPSS
software. Analysis of the data will be done through descriptive and inferential statistics.
Qualitative data analysis will be content-based, and interpretation will be triangulated to get
reliable results.
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Appendix A
Codes for Mathematical Discourse
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EXPLORING PROOF & REASONING INSTRUCTION IN LEBANON
Classroom Observation Discourse Form
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Appendix B
This teacher questionnaire consists of four sections. Kindly, answer all the questions.
Section 1: Demographics
Q1: Based on its location, which of the following best describes your school?
. urban
. suburban
. rural
Q2: Is your school public or private?
. public
. private
Q3: Based on the grade levels it contains, your school is:
. secondary only (Grades 10 to 12)
. primary and secondary (Grades 1 to 12)
Q4: How many students are there in your school?
___________
Q5: How many years have you been teaching secondary classes?
. 0 to 5
. 6 to 10
. 11 to 15
. 16 to 20
. more than 20
Q6: How many years have you been teaching grade 10?
. 0 to 5
. 6 to 10
. 11 to 15
. 16 to 20
. more than 20
Q7: Have you attended any professional development programs in teaching proof/
absolute value in the last 5 years?
. Yes, proof only.
. Yes, absolute value only
. Yes, both
. No
Q8: How many classes (sections) do you teach?
.1
.2
.3
.4
.5
Q9: How many preparations (different grades) do you have?
.1
.2
.3
.4
.5
Q10: How many total students do you teach at school?
___________
Q11: How many grade ten students do you teach? In how many sections?
EXPLORING PROOF & REASONING INSTRUCTION IN LEBANON
___________
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___________
Q12: How many other teachers are teaching grade 10 at your school?
.0
.1
.2
.3
.4
.5
Q13: Do you have flexibility in selecting what you will teach to students regarding the
explanation, exercises and problems?
. Yes
. No, it is imposed by the math coordinator/Head of Department
. Other:…
Section 2: Past proof-related reasoning work with students
Q1: How often do you involve your students with proof and reasoning activities in
your secondary classes?
. Rarely
. Sometimes
. Frequently
. Usually
Q2: Do you explain to your students the concept and methods of proof?
. Yes
. No
Q3: If the answer to the previous question is yes, what types of proof-related
reasoning do you usually teach and encourage? (Check all that apply)
. Make a conjecture
an argument
. Investigate a conjecture
. Counterexample
. Develop an argument
. Correct or identify a mistake
. Evaluate
. Other:…
Q4: If the answer to Q2 is No, what is the main reason?
. Students’ level
. Coordinator’s decision
proof . Limited amount of time
. Personal discomfort and difficulty with
. Another: ….
Q6: How do you implement proof in class?
. Direct teacher instruction
. Pair/ Group work
social talk dealing with mathematical concepts)
. Mathematical Discourse (purposive
. Other:
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Section 3: Proof and Reasoning in Absolute Value lesson of grade 10
Q1: Which book for grade 10 mathematics do you use in your school?
. National Textbook (CRDP) . Collection: Puissance Textbook (Dar Al Ahlia) .
Other: ……
Q2: Do you use proof-related reasoning in the narrative section (text/ explanation) of
the lesson Absolute Value in grade 10?
. Yes
. No
Q3: If your answer to the previous question is yes, what source do you use for the
proof part of the explanation?
. The class textbook
. Other: ….
. The internet
Q4: If your answer to Q2 in this section is yes, how do you implement such part in
class?
. Direct teacher instruction
. Pair/ Group work
. Mathematical Discourse (purposive
social talk dealing with mathematical concepts)
Q5: If your answer to Q2 in this section is no, what is the main reason for not
incorporating proof in the explanation of this lesson?
. Students’ level
proof
. Coordinator’s decision
. Limited time
. Personal discomfort and difficulty with
. Another: ….
Q6: Do you use proof-related reasoning in the exercises/problems section of the
lesson Absolute Value in grade 10?
. Yes
. No
Q7: If your answer to the previous question is yes, what source do you use for these
exercises/problems?
. The class textbook
. The internet
. Other: ….
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Q8: If your answer to Q6 in this section is yes, how do you implement such part in
class?
. Direct teacher instruction
. Pair/ Group work
. Mathematical Discourse (purposive
social talk dealing with mathematical concepts)
Q9: If your answer to Q6 in this section is no, what is the main reason for not
incorporating proof in the exercises/problems section of this lesson?
. Students’ level
proof
. Coordinator’s decision
. Limited time
. Personal discomfort and difficulty with
. Another: ….
Q10: If you were given the choice to incorporate proof in the exercises/problems part
of the lesson, which type of proving exercises would you choose?
. Proving equalities
. Proving inequalities . Another:….
Section 4: Possibility of Future Collaboration
Would you like to conduct a follow-up interview with the researcher about this topic
in the future?
. Yes
. No
If your answer is yes, kindly provide your contact information to communicate with
you. __________________________________________________________
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List of Figures
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