This document summarizes a case study that explored three teachers' levels of pedagogical content knowledge (PCK) regarding functions. The study found that the teachers had mediocre PCK levels, with more experienced teachers having better knowledge. However, all teachers lacked some conceptual knowledge of functions and focused more on procedures than conceptual understanding when teaching. The study suggested providing teachers with additional resources on mathematics concepts and ongoing training to improve their PCK and ability to effectively teach functions.

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A CASE STUDY OF TEACHERS’ PEDAGOGICAL CONTENT KNOWLEDGE OF
FUNCTIONS
Yusminah Mohd Yusof
Universiti Kebangsaan Malaysia
<ymy277@yahoo.com>
Effandi Zakaria
Universiti Kebangsaan Malaysia
<effendi@ukm.my>
Khalid Abdullah
Universiti Kebangsaan Malaysia
Abstract
This case study was conducted to explore, describe and determine the level of Pedagogical
Content Knowledge (PCK) of three school teachers, focusing on the topic of Functions. The
teachers‟ PCK levels were analyzed based on the techniques used in delivering the lessons.
This study was conducted using a qualitative approach where interviews based on teachers‟
answers to the vignette and observations were used as instruments for collecting data. These
data were analyzed using a similar approach by Ebert (1993). The findings showed that the
PCK of these three teachers were in mediocre levels with the more experienced teachers, A
and B had better level of PCK. Nonetheless, due to their lack of conceptual knowledge on
certain aspects, these teachers failed to deliver the related concepts of Functions accurately
and clearly in class. Alternatively, the students were taught the detailed procedures on how to
solve Mathematics problems with less emphasis on the conceptual understanding. The
textbook was seen as the main resource for teaching and organizing the lessons. The findings
suggested that teachers be adequately supplied with resource book on Mathematics consisting
of concepts and procedures to improve their knowledge, understanding and teaching of this
subject. Teachers should also be assisted consistently through in-service trainings and courses
to improve their PCK. This is to ensure the betterment of teachers‟ quality in teaching and
furthermore will help produce meaningful learning of Mathematics for students.
Introduction
As agents of mind innovation and a country‟s development, teachers are directly responsible for
performing the Integrated Curriculum of Secondary School (KBSM) and expected to materialize the aspiration
of National Education Philosophy. In relation to that, teachers need to equip themselves with the knowledge,
skills and positive attitude in line with the requirement of KBSM (Alimuddin 2008). Shulman (1987) has
developed a theoretical framework to conceptualize the required teachers‟ knowledge in performing effective
teaching and learning process. This theoretical framework is given the term Pedagogical Content Knowledge
(PCK). PCK is a combination of content knowledge, knowledge about the students and a variety of how content
knowledge is applied in a classroom‟s teaching and learning process (Hill et al. 2004).
Furthermore, experts in education claim that pedagogical content knowledge (PCK) is critical and it is
very important for a teacher to master it to be able to convey lesson content to students effectively (Ball et al.
2003; Hill et al. 2005; Lilia 1997; Shulman 1987). This is because PCK enables a teacher to predict
complications that may be faced by students and thus prepare themselves with methods, explanations including
useful and suitable analogies or representation /symbols in expressing certain lesson topics (Ball et al. 2001;
Shulman 1987).
Although the PCK construct has been in the education milieu for more than 20 years, it is still a useful
construct and idea in educational research as the understanding towards PCK allows a teacher to provide
knowledge, methods and objectives in preparing educator‟s capital or manpower and also teachers
professionalism enhancement course (Abell 2008). In a mathematics class specifically, teachers are responsible
to play the active role in the teaching and learning process until the desired change of behaviour among students
take place. To perform such task, each mathematics educator or teacher should acquire an in depth knowledge
of the mathematics lesson intended to be taught (Ma 1999) and also the pedagogical knowledge that suits the
level of the students taught (Ball et al. 2001). Moreover, Ball et al. (2001) emphasizes that the most essential
factor in determining the result of the learning process from the teaching strategy is how far the strategy used
could assist students in a meaningful lesson. Hence, the most important question is not just how much can a
teacher know about a form of knowledge but how a teacher uses what he knows to perform the teaching task by
providing results of the learning to the students effectively.
Statement of the Problem
In performing the process of teaching and learning, teachers bring along with them the knowledge
components. Content knowledge, knowledge about the students and the various ways of using content
knowledge in a classroom‟s teaching and learning process indeed play a role (Hill et al. 2004). The integration
of these knowledge is recognized as Pedagogical Content Knowledge (PCK). Based on literature reviews, it is

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discovered that pedagogical content is an essential and critical element in determining a teacher‟s success in
handling the teaching and learning process and further produces effective teaching (Shulman 1986; Ball & Bass
2000; Hill et al. 2004). However, the trainings given by teacher training institutes are incapable of equipping
teachers with sufficient PCK skills. Lilia (1997) discovered that the educational trainings given by 4 teacher
training institutes provide limited opportunity for teachers to develop their PCK. Ratnadivel et al. (2002) next
summed up that only 2 out of 18 courses in mathematics major in Sultan Idris Teacher Training Institute gives
emphasis or attention to PCK. The educational trainings given by teacher training institutes are only a small part
of the real teaching and learning process and the teachers‟ PCK development is restricted to certain aspects only
(Noor Shah 2008).
Besides that, mathematics teachers‟ educational programme is said to be only focusing on the
acquiring of knowledge and the trainees‟ skills and not to train them to teach mathematics at school (Tengku
Zawawi 2005). Lesson on the pedagogy of teaching mathematics is centralized on the technical form of
teaching education only (Cheah 2001). Meanwhile, content knowledge and pedagogy are taught separately and
this is against the reality of teaching and learning whereby these knowledge function in integration or
assimilation (Tengku Zawawi 2005). Therefore, if teacher training could not provide a comprehensive
opportunity to teachers to develop their PCK, how do these teachers teach in schools? How do teachers use their
existing PCK while performing the mathematics teaching process? Thus, it is obligatory for us to acquire
understanding in performing teaching process centralized on PCK.
Conceptual Framework
This study is based on a conceptual framework that is partly adapted from a research conducted by
Ebert (1993), who studied about the Assessment of Secondary School Teachers‟ Pedagogical Content in the
topics of Functions and Graphs. The conceptual framework is as shown in Figure 1.
Figure 1. Conceptual Framework of Teachers‟ PCK on the topic of Functions.
Shulman (1987) stated that PCK means teachers‟ knowledge about what is taught and a teacher‟s
ability to adapt his existing content knowledge into a form feasible for students‟ understanding or
comprehension. In this study, PCK means a mathematics teacher‟s ability to interpret the content of the topic of
Functions based on teachers‟ content knowledge and pedagogical knowledge with the intention of providing
understanding to students. This conceptual framework observes PCK as the way to provide conceptual
explanation and procedure to students using suitable analogy, representation/ symbol or example and the way a
teacher stimulates learning (Ebert 1993).
Purpose of the Study
This study was designed to determine teachers‟ PCK in the topic of Functions. The topic was chosen
considering the Functions concept is the base for all mathematics fields (Spanier & Oldham 1987; Zaslavsky
1997). Among the vital concepts in Mathematics Curriculum starting from elementary to tertiary level is the
Functions concept in mathematics ( Dubinsky & Harel 1992). The Functions concept plays important role in our
lives as almost all life‟s significant processes can be summarized and simplified as a Functions process (Stein et
al. 1990; Zaslavsky 1997). In New Curriculum for Secondary School for Additional Mathematics, Functions
is an algebra component, which is the foundation for other topics. Meanwhile, students‟ grasp of the topic is a
necessity before they are exposed to a more complex topic (Additional Mathematics Syllabus Specification
2003).
In this study, the researcher would only focus on how the lesson‟s content is conveyed to students. The
PCK elements that would be observed in it involve the use of analogy, representation/symbols, examples,
PCK
Giving explanation – concept and procedure
Using Analogy
Using Representation/ Symbol
Giving Example
Teacher‟s method in stimulating teaching & learning
Enquiry technique
Activity performed
Assessment of students‟ understanding

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explanation or demonstration that are suitable in providing conceptual and procedural explanation or account
and the ways on how a teacher stimulates the teaching process (Ebert 1993).
Objectives of the study. This study is to explore and describe the elements of teacher‟s PCK on the
topic of Functions. It is also to determine teacher‟s level of PCK in the topic of Functions.
Research Design
The study used a qualitative approach based on the design of a case study in a certain school. The case study
was a research conducted intensively by the researcher on a social unit such as individual, a family, a village, a
school or a community. The method of the case study was an empirical enquiry, investigating a phenomenon in
the real context using various means to collect the data (Yin 1994). The qualitative approach of the case study
combines interview, observation and document analysis, which are required to administer or manage the data on
teachers‟ PCK.
As content knowledge is fundamental to PCK, questions and vignette on the topic of Functions were
used in this study to perceive teachers‟ content knowledge. Other than that, the instrument used could also
provide insights on the way teachers deliver understanding or comprehension to students. Meanwhile, verbal
interview was conducted on the research subjects upon the responses given towards the Functions questions and
vignette. In addition, observations on all the teachers in their classrooms were conducted throughout the
teaching of the Functions topic.
Participants and Location of the study
The research participants were all teachers who taught Form 4 Additional Mathematics in a school
with a qualification of at least professional certificate in Mathematics Education. There were 4 of them.
Nevertheless, there were only 3 teachers who fit the characteristics given in operational definition to become
research participants. All of them have been teaching Form 4 Additional Mathematics for more than a year and
acquired professional eligibility in Mathematics education.
Instrumentation and Data Collection
The instrument used was a set of vignettes on the topic of Functions and observation protocol. Next,
the data collection is illustrated as the following;
Vignette based on the topic of Functions. The study conducted is a qualitative research that involves
questions in the form of vignettes on Functions. Vignettes are questions based on a certain scenario. As a
research instrument, vignette is gaining constant use in qualitative research on teachers‟ knowledge (Aini 2001),
for instance as what was conducted by McDiarmid and Wilson (1991) who gave out questions based on a series
of teaching scenarios to gather more detailed information on teacher‟s knowledge and belief.
The vignettes used in this study are questions along with students‟ responses that consist of several
errors or mistakes that require teacher‟s capability or skill to identify and correct those errors. Ebert (1993)
stated that the use of vignette could also provide alternatives to evaluate content knowledge and teacher‟s PCK
by looking at teacher‟s structural responses and the kind of activity that they suggest to answer questions and
prevent confusion or uncertainty among students in learning the topic of Functions. The teachers were also
interviewed based on their responses in the vignette.
Observation protocol. Observation is used to gain a more accurate account on the teaching method
applied in teaching all units of the lesson on the chosen topic. The teaching aspect that would be observed is on
how teachers explain in providing comprehension to students and how they evaluate students‟ understanding.
Besides that, this method is conducted to observe the elements of content knowledge and teachers‟ pedagogical
knowledge in forming teachers‟ PCK. The observation applied is unstructured observation and non-participant
observation whereby the researcher was not involved in the teaching session but acted merely as an observer.
This is because the researcher planned to record the teaching session that took place in its natural state. The
researcher recorded and took notes of all that are observed throughout the participants‟ teaching process. This
method was also conducted according to the teachers‟ teaching schedule. The observation was done for each
teaching session on the sub topics of Functions. Whenever the researcher was unable to attend the class during
which a subtopic is taught, she would ask the participants to flashback or recapitulate the teaching session
conducted and the notes that had been taken. However, there was only one session that could not be observed or
administered by the researcher that is on the topic of Inverse Function conducted by participant C.
Data Analysis of Interview and Observation. The data analysis is done through questions and
responses of the participants towards the vignettes on the topic of Functions and the observation done is
recorded based on how the teachers provide explanation on the concepts and procedures to students using
suitable analogy, representation/symbols or examples and how teachers stimulate the teaching process.
Teachers’ pedagogical content knowledge (PCK). A teacher‟s PCK is formed when he conveys lesson
content for students‟ understanding. This element can be observed through the data collected based on teacher‟s
responses in the aforementioned vignettes and classroom observation. Through the data obtained, a teacher‟s

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approach in conveying lesson content to students is through having lectures and minimal discussion activities.
The themes are how the teacher provides conceptual explanation, the method of providing procedural
explanation related to the concept taught and how the teacher stimulates the teaching process.
Level 1 PCK. At this level, the PCK reveals that the teacher explains the wrong concept or that is not
quite clear. The examples given are limited to a certain form only. The procedures explained are integrated with
its conceptual understanding, whereas the teacher‟s question techniques, if existed, are of low level. The teacher
is the main medium and students‟ participation is very minimal throughout the teaching and learning process.
The teacher may not realize students‟ difficulties within the topic. There is no clear indication of the teacher
relating the lesson with students‟ existing knowledge.
Level 2 PCK. At this level, the teacher‟s explanation of concept is accurate but with limited variety of
examples. There are attempts to integrate conceptual understanding with the knowledge of performing
mathematics‟ procedures. However, the focus of the teaching remains on performing a systematic or organized
mathematics‟ procedures that must be adhered to by students. Activities that involve students are provided but
of limited participation. The teaching is teacher-centered. The teacher is aware of students‟ difficulties but
perhaps there is no attempt for further explanation through the teaching and learning process. The questions
forms are of no variety. The use of analogy, representation /symbols and demonstration may be observed at this
level.
Level 3 PCK. The teacher‟s explanations of concept are more accurate and clear along with suitable
examples. The examples given are of many variety including situational and application concepts. The
questions asked towards students are to stimulate their minds and comprehension. Students‟ participation is
active through activities focused towards providing conceptual and procedural understanding that are integrated,
creating conjectures and generalization of mathematics. The use of meaningful analogies, representations/
symbols and demonstrations occur at this level. Suggestions on the use of technological application are also
possible. Students‟ mistakes are seen as a way to observe the process of performing the correct procedures
besides evaluating students‟ understanding of the concept. The teaching is also considered as student-centered
and the development of conceptual understanding is accurate without neglecting procedural understanding and
knowledge.
Findings
The results of the study are presented in table form as well as descriptive analysis.
Background of Research Participants
Three teachers are involved as the research participants this study.
Table 1
Participants’ Background
Demographic Characteristics Teacher A Teacher B Teacher C
Gender Male Male Female
Academic Qualification Bachelor of Marine
Science
Bachelor of Finance Bachelor of
Education
(Mathematics)
Profesional Qualification KPLI (Mathematics &
Science)
KPLI (Mathematics) Bachelor of
Education
(Mathematics)
Teaching Experience (year) 7 4 2
Course attended Additional Mathematics
Enhancement Course
Contextual Learning
(Science and
Mathematics)
None
Mathematics and Additional
Mathematics SPM Result
2A, 6C 3B, Did not take
Additional
Mathematics
2A, 5C
As displayed in Table 1, the three research participants: Teacher A, Teacher B and Teacher C have
been teaching Additional Mathematics for 7, 4 and 2 years respectively. All the teachers have acquired
professional trainings in mathematics education whether in teacher training institute or in university. Teacher C
is a graduate of Bachelor of Education in Mathematics from a local university, whereas Teacher A and Teacher
B had received training to be a qualified mathematics teacher through the post-graduate teacher training course
(KPLI) for a year in a teacher training institute. Teacher A and Teacher C had taken the subject while in
secondary school whereas Teacher B did not have Additional Mathematics background. Their performance of

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2A and 3B in Malaysia Certificate of Education Mathematics is considered as good and excellent whereas
Teacher A and Teacher C‟s performance in Additional Mathematics is considered as average.
Teacher A and B each has had the opportunity to attend contextual teaching and learning enhancement
course respectively once within their working experience so far. All three of them have positive perceptions
towards Additional Mathematics and claimed that they enjoy teaching the subject. Nevertheless, perhaps due to
different educational backgrounds and the duration of working experience, there are differences among them
especially in terms of the knowledge of Functions concept.
Teacher A is the head panel of Additional Mathematics in the school, teaching the subject in 4 classes
(1 Form 4 class (16-year old students) and 3 Form 5 classes (17–year old students)). The teacher also had taught
Science and Mathematics for 5 years before being appointed as the head panel. The teacher is considered as
„relaxed‟ and poised while performing his teaching session.
Teacher B teaches two Additional Mathematics classes (one Form 4 class and one Form 5 class) and
two Science classes. This teacher does not have Additional Mathematics background throughout secondary
school. He is the school‟s head of cooperative unit. He frequently claimed that he has been constantly busy
since being appointed with the position. While the researcher was at the research location, he was seen occupied
with distributing necessity items to Form 4 students. This research participant also stated his lack of preparation
for the lesson in the third observation. He has a clear and distinct voice, good humour and provides systematic
solution with reason so that it is easier for students to memorize or understand it.
Meanwhile, Teacher C has started teaching Additional Mathematics in the school for about one year.
She also teaches 2 Additional Mathematics classes, which are one Form Four class and one Form 5 class and
the other 3 classes are for mathematics subject. The teacher seemed a bit passive probably due to the lack of
experience being this is only her second year of teaching. She was seen to have less communication with the
students and did not give out many questions. All the three participants expressed that they enjoy teaching
Additional Mathematics that is compulsory for all students under the technical stream in the school.
Pedagogical Content Knowledge. Overall, the participants‟ PCK level can be categorized as at Level 1
and 2 with minimum difference. Examples on the aspects of domain, co-domain, image, object and range were
not given accurately. The teacher also did not explain explicitly that equation is also a form of representation
/symbols for Functions and this reveals the teacher‟s inability to identify that linear equation is a Function. The
examples given were also limited to examination form without any application form question. The elements of
teachers‟ PCK were summarised in Table 2 below
Table 2
Teacher’s Pedagogical Content Knowledge
No. TEACHER‟S EXPLANATION Teacher A Teacher B Teacher C
CONCEPT EXPLANATION
1 Introduction to Functions
Relating it to students‟ life √ √ √
Provide relationship aspects √ √ √
Give relationship example in the form of graphs, diagram
and organized pairs. √ √ √
Give numerical, situational and trigonometric function
examples
√ √ √
Give definition of Functions in general √ √ √
Explain domain, codomain, range, image, object √ √ √
Provide inaccurate list of image, object, domain,
codomain and range
√ √ √
2 Composite Function
Give definition of Composite Function √ √ √
Explain the definition and concept with charts in general √ √ √
Use Analogy √
3 Inverse Function
Give definition √ √ √
Give examples of diagram that display inversion √ √ √
Use analogy √

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TEACHER‟S PROCEDURAL EXPLANATION
1 Find image or object
Give examples of questions and the solutions √ √ √
Emphasize image concept and object in explanation √ √
2 Composite Function
Gives examples of questions and the solutions √ √ √
Use analogy √
Provide solutions and their steps √ √ √
Provide solutions with more systematic steps √
3 Inverse Function
Give examples of questions and the solutions √ √ √
Use analogy √
Emphasize the need to maintain using one „a‟ variable
only due to students‟ difficulties. √
Discuss questions based on non-linear stimulus √ √
Integrate concept and procedure √ √
Perform Organized solutions and their steps √
TEACHERS‟ METHODS OF STIMULATING THE LESSON
1 Question Techniques
Low level question; Knowledge stage „what‟ √ √ √
Analysis stage „why‟ √ √
2 Enhancement Exercises
Examination form questions √ √ √
3 Evaluation of students‟ understanding
Monitoring and providing guidance √ √ √
TEACHERS‟ METHODS OF STIMULATING THE LESSON
Request students to demonstrate on blackboard √ √
Guide students while answering on the blackboard √
Provide explanation on students‟ answers √ √
Solve the question √
The result of this section is to answer the research question that is “what is the teacher‟s level of PCK
in performing the teaching and learning process in the topic of Functions?” PCK is defined as how a teacher
provides explanation on the lesson content and how a teacher stimulates the lesson to assist in providing
understanding to students.
Teacher’s Methods of Explanation. Based on the details in Table 2, it is understood that the knowledge
in the topic of Functions conveyed by the teacher to the students is based on what has been prepared in the
textbook. The teachers felt that the lesson content arranged in the textbook is simplified and suit the level of
their students. There was no discrepancy in the teachers‟ explanation with what has been prepared.
The teachers also did not realize that their knowledge on domain, co-domain, image, range and object
is not quite strong and hence gave students the wrong information regarding those aspects. Meanwhile, the
explanation on the concept of Composite Function was just to provide a diagram for Composite Function and
later give explanation on the procedures involved. The belief that Mathematics must be mastered with the
ability to perform its procedures justifies the approach taken by the teachers in the teaching and learning process
conducted in classrooms. Providing intensive and repetitive exercises has become the practice of these teachers.
Furthermore, based on the teachers‟ limited knowledge in the application of Functions, thus the
questions and whatever examples given by them do not involve Functions application. The conceptual
explanation among these teachers does not differ much even though Teacher A has a more stable content
knowledge than the other two research participants. This results supports the statement by Nik Azis and Ng
(1991) that claimed, although an in-depth content knowledge in mathematics is required, it is not enough as a
prerequisite to teaching mathematics. This is because all three participants rely very much on the textbook for
lesson content as well as organizing and performing the lesson content. Therefore, the teaching and activities
performed in the classroom are not much different among these teachers.
Teacher B also has his own strengths in providing clear and systematic explanation that did make it
easier for his students to understand the lesson specifically in performing the Mathematics procedures involved.

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The teacher also has attempted to use analogy even though it was only to provide procedural comprehension.
Besides that, his traits such as having a clear voice, cheerful and using the blackboard systematically and neatly
are perceived by the researcher as the teacher‟s individual strengths while performing lessons in the classroom.
This result is in contrast with the study conducted by Noorhashikim (2002) that discovered experienced teachers
have better PCK level than those with less working experience. This may be due to the lack of differences
resulting from the 7 years and 4 years of experience between Teacher A and Teacher B respectively.
Teacher’s Methods of Stimulating the Lesson. Moreover, students‟ active involvement in mathematics
activity is limited because teachers were also seen as less stimulating in the lesson without having high level
questions. The students only participated in providing answers to exercises in the classroom. They were not
given the chance to explain or justify how an answer is obtained.
The exercises questions were taken from textbook and also examination questions. The application
form questions provided in the textbook were not given to the students even though the questions could provide
students with the opportunity to explore Functions better and face more challenges. This is because the teachers
are of the opinion that the questions are not based on examination format and are more difficult for students to
solve.
Besides that, the teachers‟ knowledge towards students‟ conceptual understanding is also limited.
Although the teachers realized students‟ difficulty in learning certain topics, they did not take any step to
explain further or provide better clarification for the students. For instance, Teacher B realized the difficulty that
the students‟ may face regarding the concept of variable in learning Functions, but the teacher did not attempt to
explain further. The teacher just chose to limit the use of one variable that is „a‟ in solving related questions.
This is against the importance of variable concept as being something fundamental in the topic of Functions
(Leinhardt et al. 1990). Meanwhile, students‟ mistake is considered as arithmetic error that later provides
options for teachers to just explain to them on the correct mathematical procedures without asking students to
give reasons towards the procedures performed as a way to observe the process that has taken place.
Summary
When someone has chosen a career of teaching mathematics, the passion towards the field or subject
exists inherently. Therefore, a teacher must use this advantage and always look forward to explore a better
understanding of a certain mathematical concept and not only to teach students to pass in the examination. The
passion towards mathematics must be conveyed to students and this can only be done if the teacher himself sees
Mathematics as an interesting subject to be explored and appreciated through more dynamic activities that
maximize students‟ participation. This is possible if the teacher possesses firm conceptual knowledge and could
perceive its relation with the subject or reality of life that is linked to mathematics along with related
pedagogical knowledge.
Teachers must be open and willing to accept changes in the reformation of education towards suitable
technological application for the teaching and learning process, based on research results that have proven the
success of applying technology effectively in the teaching and learning of mathematics. Certain departments
such as the Ministry of Education and Centre of Curriculum Development could never have changed the
attitude and perception of teachers except the teachers themselves believe that dynamic education, technology,
and lesson content will go through changes within time. Teachers must be prepared to change and adapt their
teaching methods starting with shifting their paradigm for the sake of developing a functional generation
effectively in the future.
A good pedagogical content knowledge within teachers would not be developed easily without the
effort of certain departments and also the initiative required within each teacher specifically Additional
Mathematics teachers. Indeed a change could be expedited if the individuals involved, teachers generally, are
ready to change to a better, more progressive stage. With the knowledge that is being enhanced and renewed
consistently, teachers‟ understanding on the subject and its topics that need to be conveyed to students will be
more established and later capable of fulfilling this adage: Those who can, do. Those who understand, teach
(Shulman 1986: 14).
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