EFFECTIVE TEACHERS’ PEDAGOGICAL CONTENT KNOWLEDGE IN TEACHING QUADRATIC FUNCTIONS IN MATHEMATICS

EFFECTIVE TEACHERS’ PEDAGOGICAL CONTENT KNOWLEDGE IN TEACHING QUADRATIC FUNCTIONS IN MATHEMATICS
EFFECTIVE TEACHERS’ PEDAGOGICAL CONTENT KNOWLEDGE
IN TEACHING QUADRATIC FUNCTIONS IN MATHEMATICS
BY
CHARLES DUZEPHI SIBUYI
Submitted in partial fulfilment of the requirements for the degree
M.Ed Assessment and Quality Assurance
Department of Science, Mathematics and Technology Education
Faculty of Education
University of Pretoria
PRETORIA
Supervisor: Prof: G.O.M Onwu
SEPTEMBER 2012
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© University of Pretoria
APPROVAL
This research work has been examined and is approved as meeting the required standards of
scholarship for partial fulfilment of the requirements for the degree of Master of Education at
the University of Pretoria.
_________________
_______________
Supervisor
Date
_________________
_______________
External Examiner
Date
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DEDICATION
I dedicate this dissertation to my father, Sibuyi Mabihana Willias, who passed on the 7th of
February 2009 to be with his Lord and Saviour Jesus Christ.
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ETHICAL CLEARANCE CERTIFICATE
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ACKNOWLEDGEMENTS
The successful completion of this study was purely not a one man effort, namely the
researcher, but was as a result of the support of other people. I owe to acknowledge this
support from the following:

My supervisor, Professor G.O.M Onwu for his guidance, support, patience and
encouragement throughout the study.

My colleague, Mr Mnisi Thabo, for the encouragement not to lose hope when there
were challenges during the course of this study.

My wife, Joyce Sibuyi, for her unflinching and loving support in more ways than one;
as well as her encouragement to pursue the study.

The two teachers and their learners for their willingness and cooperation in taking part
in this study.

The Mpumalanga Department of Education, for giving me permission to do this study
in their schools.

God Almighty for taking care of me on the road as I drove from home to the
University and back during contact times.
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TABLE OF CONTENTS
Title Page
i
Approval
ii
Dedication
iii
Ethical Clearance Certificate
iv
Declaration of Originality
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Acknowledgements
vi
Table of Contents
vii
List of Tables
x
List of Figures
xi
Abstract
xii
List of Terms
xii
CHAPTER 1: INTRODUCTION AND BACKGROUND
1
1.1 Introduction and Background
1
1.2 Background of the Study
2
1.3 The Purpose of the study
5
1.4 Problem statement
6
1.5 The Research Questions
6
1.6 Significance of the study
6
1.7 Overview of Chapters
7
1.8 Chapter Conclusion
8
CHAPTER 2: LITERATURE REVIEW
9
2.1 Introduction
9
2.2 What is Pedagogical content knowledge?
9
2.2.1 Components of Pedagogical content knowledge
11
2.2.2 Pedagogical content knowledge in the teaching of mathematics
13
2.2.2.1 Knowledge of subject matter and Mathematics teaching
15
2.2.2.2 Knowledge of Instructional strategies and Mathematics
Teaching
18
2.2.2.3 Knowledge of learners’ Conceptions , preconceptions and
Misconceptions in Mathematics teaching
21
2.2.3 Conceptual framework for the study
24
2.3 Chapter Conclusion
27
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CHAPTER 3: RESEARCH METHODOLOGY
28
3.1 Introduction
28
3.2 Research Design and Method
28
3.3 Research Site and Population
29
3.4 Sampling Procedure and Profile of participants
29
3.5 Data Collection Instruments
31
3.5.1 Classroom Observation Protocol
31
3.5.2 Teachers’ Pre-Lesson Interview Questions
32
3.5.3 Document Analysis: The Lesson Plan
34
3.5.4 Questionnaire on the Teachers’ Mathematics teaching knowledge
Development
34
3.6 Validation of Instruments
36
3.7 Preparing for the Study
36
3.8 Administration of the Main Study
36
3.9
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Data analysis
3.10 Trustworthiness of the study
38
3.11 Elimination of bias
39
3.12 Ethical considerations
39
3.13 Chapter Conclusion
41
CHAPTER 4: FINDINGS ON THE CASE STUDIES
42
4.1 Introduction
42
4.2 The Teachers’ Pedagogical Content Knowledge
42
4.2.1 Description of Classroom Observations for Teacher A
42
4.2.2 Pre-Lesson Interviews With Teacher A
48
4.2.3 Summary of the Pedagogical Content Knowledge of Teacher A
based on Pre-Lesson Interviews and Lesson Observations
50
4.2.4 Description of Classroom Observations for Teacher B
53
4.2.5 Pre-Lesson Interviews with Teacher B
57
4.2.6 Summary of the Pedagogical Content Knowledge of Teacher B
Based on Pre-Lesson Interviews and Lesson observations
58
4.2.7 Lesson Plan Analysis of the Participating Teachers
60
4.2.7.1 Lesson Plan Analysis of Teacher A
61
4.2.7.2 Lesson Plan Analysis of Teacher B
63
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4.2.8 Development of the Pedagogical Content Knowledge of the Teachers
67
4.2.8.1 Interviews on Development of the PCK of Teacher A
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4.2.8.1 Interviews on Development of the PCK of Teacher B
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4.9 Chapter Conclusion
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CHAPTER 5: DISCUSSION OF FINDINGS AND CONCLUSION
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5.1 Introduction
72
5.2 Discussion of Themes
72
5.2.1 Pedagogical Content Knowledge of the Participating Teachers
72
5.2.2 Development of the Pedagogical Content Knowledge of the Participating
Teachers
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5.3 Conclusion
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5.4 Limitations of the Study
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5.5 Recommendations
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References
Appendices:
Appendix A: Pre-Lesson Interview Questions
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Appendix B: Classroom Observation Protocol
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Appendix C: Guiding Question on Lesson Plan Analysis
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Appendix D: Interview Questions on PCK Development
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Appendix E: Letter to request permission from the Department of Education
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Appendix F: Letter of Approval to conduct research from the Department of
Education
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Appendix G: Letter to request permission to conduct research from the Principal 100
Appendix H: Letter to invite participants
101
Appendix I: Letter to request permission from parents of learners
102
Appendix J: Informed consent form for parents of learners
103
Appendix K: Excerpts of Teacher A’s Lesson Observation Description
104
Appendix L: Teacher A’s Pre-Lesson Interviews
109
Appendix M: Teacher A’s Lesson Plan Analysis
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Appendix N: Excerpts of Teacher B’s Lesson Observations
114
Appendix O: Teacher B’s Pre-Lesson Interviews
116
Appendix P: Teacher B’s Lesson Plan Analysis
118
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LIST OF TABLES
1. Table 2.1: Framework of PCK used by Bukova-Guzel (2010).
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2. Table 2.2: Framework of PCK elements used in the study.
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3. Table 3.1: Profile of the participating teachers in the study
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4. Table 3.2: Classroom observation protocol.
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5. Table 3.3: Pre-lessons Interview questions.
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6. Table 3.4: Lesson plan analysis guiding questions.
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7. Table 3.5: Pedagogical Content Knowledge development questionnaire
35
8. Table 4.1: Description of Classroom observations for Teacher A
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9. Table 4.2: Pre-Lesson Interview questions and responses of Teacher A
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10. Table 4.3: Description of Classroom observations for Teacher B
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11. Table 4.4: Pre-Lesson Interview Questions and Responses for Teacher B
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12. Table 4.5: Guide lines for Lesson Plan Analysis
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13. Table 4.6: How Teacher A May have developed his PCK
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14. Table 4.7: How Teacher B may have Developed his PCK
69
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LIST OF FIGURES
1. Figure 2.1: Schematic diagram of the three elements of PCK to be observed.
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ABSTRACT
This study investigated the pedagogical content knowledge supposedly held by two FET
mathematics teachers from Mpumalanga Province as they taught quadratic functions in grade
11 classes. The criterion for selecting the two teachers was that they had consistently
produced good results (overall pass rate of 80% or more) in the grade 12 mathematics
examinations of the National Senior Certificate for the past three years or more and thus, they
were classed as effective. The two teachers prepared and taught lessons on quadratic
functions in grade 11 whilst they were being observed. The study focused on teacher
knowledge base as exemplified in the teachers’ pedagogical content knowledge (PCK). Three
elements of PCK were investigated; namely; (i) knowledge of the subject matter; (ii)
knowledge of teaching strategies and (iii) knowledge of learners’ conceptions. Qualitative
research approach using the case study research method was used to collect qualitative data
on the pedagogical content knowledge of the two teachers through lesson observations,
lesson plan analysis and interviews. Analysis of the results suggests that the two teachers
have adequate subject matter knowledge but have limited knowledge on the aspects of
teaching strategies and knowledge of learners’ pre-conceptions and misconceptions on the
topics of quadratic functions that they taught. The study recommends that teachers be
exposed to workshops that deal specifically with the various topic specific teaching strategies
and knowledge of learners’ pre-conception and misconceptions on the topic of quadratic
functions.
Keywords: Pedagogical content knowledge, Knowledge of teaching strategies, Knowledge of
learners’ conceptions, knowledge of the subject matter, quadratic functions
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LIST OF TERMS
Effective teachers: In this study, effective teachers refer to those teachers who have
consistently produced good results, an average pass rate of 80% or more in the
National Senior Certificate grade 12 mathematics examinations for the past three
years or more as reflected in the district’s grade 12 performance statistics.
Pedagogical Content Knowledge (PCK): In this study pedagogical content knowledge
mean an amalgam of (i) specific content knowledge on quadratic functions, (ii)
knowledge of teaching strategies and application and (iii) prior knowledge of
learners’ conceptions that allow a teacher to transform specific content
knowledge in a more conceptually accessible version for the learners.
Subject matter knowledge: In this study subject matter knowledge (as displayed by the
teacher) mean the correct application of mathematical concepts, facts and
procedures, the reasons underlying mathematical procedures and the relationship
between mathematical concepts during classroom teaching of quadratic functions.
Knowledge of learners’ conceptions: Knowledge of learners’ conceptions in the study is
defined as the teachers’ awareness of learners’ prior knowledge which may
include pre-conceptions, misconceptions, learning difficulties and correct
conceptions they may have, which can be used by the teacher on the students
behalf during classroom teaching and lesson planning for effective teaching.
Knowledge of Pedagogy: In the study, knowledge of pedagogy refers to knowledge of
planning and organization of a mathematics lesson and teaching strategies for
effective teaching of the particular topic under investigation.
Knowledge of curriculum: In the study; knowledge of curriculum refers to knowledge about
learning goals for different grade levels for use to organise lesson planning and
classroom teaching.
Procedural knowledge: Procedural knowledge is regarded as knowledge of mathematical
rules, algorithms and procedures that a teacher uses to assist learners to learn how
to solve mathematical problems quickly and efficiently because it is to some
extent automated through drill work and practice.
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Conceptual Knowledge: In this study, conceptual knowledge refers to knowledge, as
displayed by the teacher, of the core concepts and principles and their
interrelations in the mathematics domain.
Misconceptions: In this study misconceptions refer to pieces of wrong knowledge that may
arise as result of learners’ prior experience and learning both inside and outside of
the classroom and effective mathematics teachers should have knowledge to
diagnose and eliminate such wrong knowledge.
Teaching Strategies: In this study, teaching strategies refers to methods used by teachers to
create learning environments and to specify the nature of the activities in which
the teacher and learners will be involved during the lesson to ensure that the
sequence or delivery of the lesson helps learners to understand the topic taught.
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CHAPTER 1
INTRODUCTION AND BACKGROUND
1.1 INTRODUCTION
Skills in mathematical reasoning are becoming even more important than ever before in the
workplace and everyday living, driven in part by emerging technologies and job demands.
To assure that learners in schools gain the indispensable mathematical reasoning required in
life, the new curriculum in South Africa, the National Curriculum Statement (NCS); calls for
educators to assure that their learners participate during mathematics lessons and express
their mathematical ideas. According to Brodie (2007:p.3), “getting learners to talk is seen as
important because it (i) shows that learners are attending to the lesson; (ii) allows learners to
express and clarify their own ideas; (iii) enables learners to share ideas with each other; and
(iv) provides teachers with information about what learners know and do not know and how
learners are thinking and trying to make sense of ideas. Teachers are encouraged to make
their mathematics lessons more learner-centred by encouraging learners to contribute to the
lesson.”
To achieve this kind of approach to teaching, schools need quality teachers who have the
appropriate knowledge about the art of teaching. Without doubt teachers are one of the most
powerful influences on students’ engagement with mathematics (Attard, 2011). Such
teachers, according to Tanner (2003), “create experiences that help students make sense of
the knowledge and skills being studied”. According to Turnuklu and Yesildere (2007:p.1),
“although a number of factors may influence the effective teaching of a particular subject,
teachers play an important role in that success”. Good teachers, Attard (2011) claims, can
achieve high and consistent levels of engagement and effective learning.
Contrary to
common belief in society that a teacher who knows a particular subject very well is best
suited to teach such a subject, research has shown that this belief is not necessarily true
(Shulman 1986, 1987; Hill, Rowan and Ball 2005; Etkina, 2010). Various researchers such as
Shulman (1986, 1987), Hill, Rowan and Ball (2005) and in particular, Etkina (2010:p.1)
emphasise that “teachers of a specific subject should possess special understandings and
abilities that integrate their knowledge of the content of the subject that they are teaching as
well as having knowledge of the learners who are learning the content”. Knowledge of the
learners includes; amongst other things; having knowledge of what pre-conceptions,
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misconceptions and difficulties that the learners might have about a topic to be taught. As
educators know, teaching is a complicated practice that requires an interweaving of many
aspects of specialised knowledge (Mishra and Koehler, 2006). Such specialised knowledge
includes knowledge of pedagogy; knowledge of the subject matter and knowledge of the
learners as explained above.
1.2
BACKGROUND OF THE STUDY
In South Africa, the general performance of learners in mathematics and Science in the
National Senior Certificate examinations was recorded as being poor for the period between
2008 and 2011 (Report on National Senior Certificate Examination, 2011). In terms of this
report, the percentage of learners in the whole country who managed to obtain a mark above
the 40% pass level in mathematics in the 2011 Grade 12 final examination is shown in the
table below.
YEAR
2008
2009
2010
2011
Total registered learners in Mathematics
300 008
290 407
263 034
224 635
% of learners who obtained passes above 40% pass level
29.9
29.4
30.9
30.1
The information in the table shows a decline in the number of learners taking mathematics at
school and also their low levels of performance. This trend of poor performance in
mathematics by South African learners can be traced back to earlier pre-democracy years of
South Africa (CDE, 2004). In a research report released by the Centre for Development and
Enterprise (CDE, 2004), educational planners and those involved in education in the country
are concerned about the poor grade 12 mathematics results and the quality of education in
mathematics and science offered in schools generally. Aside from the poor quality of
education, there is a growing concern in the country about the dwindling numbers of learners
leaving school with sufficiently good grades to enter mathematics and science-based courses
at tertiary institutions. In this same report (CDE, 2004), statistical data reveal that between
the period 1991 to 2003, enrolment in mathematics Higher Grade, a subject essential for entry
into many tertiary education science courses, plummeted from 53 631 to 39 159. In
Mpumalanga province for example, of the 359 schools that offered mathematics at higher
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Grade level in 2003, only 19 schools managed to produce one or more candidates who
obtained symbols in the 80% pass region. The CDE report (2004) goes on to conclude in
rather ominous terms that failure to improve mathematics and science education is probably
the most significant obstacle to African advancement in South Africa because this reality
undermines the country’s ambition for expanded economic growth, black empowerment and
community development. Based on this information, it would be of interest to know what
research would say about the possible causes of poor performance in mathematics by South
African learners.
A more recent report by CDE (2011) that investigated the quantity and quality of South
African school teachers indicates that South Africa is at or near the bottom of other
developing countries when ranked in terms of student performances in mathematics and
Science. In the same report, it is revealed that many of the existing teachers of mathematics
and Science are not teaching the subjects well and are also poorly managed. This report
points to poor teaching by teachers at schools as the main cause of poor performance in
Mathematics and Science.
Other researchers in South Africa (Howie, 2003; De Clercq, 2008; CDE, 2004 and 2011) also
identified various in-schools and out-of-school factors that impact on learner performance in
mathematics. De Clercq (2008), in her study of teacher quality, appraisal and development,
asserts that factors contributing to poor learner performance in mathematics in developing
countries (such as South Africa), include teacher quality, the socio-economic background of
learners and their communities, the context of schooling, poor school leadership and poor or
under-resourced school facilities. The CDE (2004 and 2011) research reports also point to
teacher quality, classroom environment and language of instruction as factors accounting for
the poor performances of learners in Mathematics and Science.
The language of learning and teaching may also contribute to South African learners’ poor
performance in mathematics (Howie 2003). Based on data gathered from TIMSS (Trends in
mathematics and Science Studies, 1999), a study was done Howie (2003) to assess whether
language and other background factors affected secondary school pupils’ (Grades 7, 8 and
12) performance in mathematics in South Africa. It was found that learners, whose
proficiency in English (as a medium of learning and teaching) was good, performed
significantly better in mathematics than learners who had a poor proficiency in English. In
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most rural and township schools of South Africa, English is used as a language of learning
and teaching although it is a second language for most learners (CDE, 2004).
The history of the South African education system may also be assumed to have had an
impact on the current situation of poor performance in Mathematics at schools. Adler and
Reed (2002), state that segregation, fragmentation, authoritarian and bureaucratic control of
the curriculum, institutions and governance, inefficiency and inequality have characterised
South African education for a long time. Each of these are said to have had a considerable
effect on the present performance levels in Mathematics and Science, particularly, in the rural
and township schools where the culture of teaching and learning is said to have virtually
collapsed. Furthermore, Bush (2003) posits that years of struggle against apartheid inevitably
affected schools, particularly those in the townships. Teachers formed teacher unions that
played a key role in the political struggle and because educators were frequently absent from
school to engage in protest activities, the culture of learning and teaching was not sustained.
Educator factors that have consistently been linked to poor performance in Mathematics and
Science include teachers’ knowledge of Mathematics and the skill of performing the teaching
task (Ingvarson, Beavis, Bishop, Peck and Elsworth, 2004; Baumert, Kunter, Blum, Brunner,
Voss, Jordan, Kusman, Kraus, Neubrand and Tsai, 2010). Hill, Rowan and Ball (2005) in
their study that explored whether and how teachers’ mathematical knowledge for teaching
contributes to gains in learners’ Mathematics achievement, found that teachers’ mathematical
knowledge was significantly related to learners’ achievement. The knowledge about teaching
and learning that teachers bring to the classroom has an impact on whether learners will
access the topics that teachers teach.
Still on teacher factors, Baumert et al (2010) posit that the pool of alternative mathematical
representations and explanations given by teachers to learners in the classroom are largely
dependent on the breadth and depth of the teachers conceptual understanding of the subject,
and that insufficient understanding of the mathematical content, limits the teachers’ capacity
to explain and represent that content to learners in a sense-making way. This is a deficit that
cannot be offset by pedagogical skills alone. Anecdotal evidence suggests that efforts of
teachers with limited conceptual understanding of the mathematics topics that they teach fall
short of providing students with powerful mathematical experiences.
4
Although the general performance of most learners in the National Senior Certificate
mathematics examination in most South African schools is poor, as already alluded to in the
previous paragraphs, there are, however, some schools that are consistently producing good
National Senior Certificate results (Grade 12) and high quality work (80% and above pass
level) in Mathematics (CDE, 2004; NSTF, 2007). Mathematics teachers in such schools may
be presumed to be doing something differently from what other teachers in less effective
schools are doing. Effective teachers in such good performing schools were the focus of this
study.
Influenced by the fact that research has pointed to pedagogical content knowledge of a
teacher as having an influence on how teachers make a lesson topic accessible to their
learners (Shulman, 1986), and the fact that research also points to poor teacher knowledge as
being a contributory factor to poor performance (CDE, 2011), the interest of this study
concerned investigating the pedagogical content knowledge held by two teachers who where
classed as effective, as they taught topics of quadratic functions in Grade 11.
The concept of quadratic functions was chosen for this study for several reasons. First, it
serves as an entry point to the study of polynomial functions in mathematics. Second,
according to Zaslavsky (1997), the functions concept is the foundation for all mathematics
fields. The third reason for selecting this concept is that it has many uses in career-related
professions such as business, engineering and science where the concept is used for
modelling ideal situations. In business it may be used to help in forecasting profit and loss.
The U-shape of a parabola is incorporated in science in the construction of structures such as
the parabolic reflectors of satellite dishes and car head lamps. Good insight into quadratic
functions will enable learners to deal with different types of functions such as trigonometric
functions, linear functions, exponential functions and logarithmic functions, leading to an
understanding of real-life uses of this concept. Moreover, several mathematics teachers in the
circuit where the study was conducted complain that learners are not performing well in this
topic during examinations. On the basis of this explanation, the concept of the quadratic
function was chosen for the study.
1.3
THE PURPOSE OF THE STUDY
The purpose of the study was to investigate the pedagogical content knowledge held by two
teachers who were classed as effective since their learners have consistently achieved good
5
passes (average pass rate of 80% and above) in Grade 12 mathematics National Curriculum
Statement examination in mathematics for the past three or more years.
Specifically, the study investigated what pedagogical content knowledge (PCK) these two
teachers considered to be effective have with regard to the teaching of quadratic functions.
The study also sought to determine how the teachers had acquired the PCK that they were
using.
1.4
PROBLEM STATEMENT
The problem of this study was to determine what teacher knowledge base two effective
mathematics teachers have and display in the context of teaching the topic on quadratic
functions in Grade 11 Mathematics classrooms and how they acquired it (PCK).
The following research questions are derived from the problem statement.
1.5
RESEARCH QUESTIONS

What pedagogical content knowledge do the two teachers display in teaching
quadratic functions in Grade 11?

How did the teachers develop the pedagogical content knowledge that they use in
teaching quadratic functions?
1.6 SIGNIFICANCE OF THE STUDY
The study was undertaken because the issue of effective teaching of mathematics is of vital
importance to the provision of quality education. It is believed that the results of the study
will have beneficial application as it seeks to reveal the PCK held by the two participating
teachers with regard to the teaching of quadratic functions. Such knowledge and skills could
be used to develop in-service and pre-service teacher education programmes aimed at
improving the quality of teaching of mathematics specifically, the teaching of quadratic
functions. The teacher educational programmes developed would be of practical value, at the
same time making a significant contribution to general educational theory.
Pedagogical content knowledge (PCK) remains a vague form of knowledge which cannot be
easily isolated and studied separately from other teacher knowledge bases because it is
6
individualistic, is developed by the teacher and is unique to that teacher. However, it (PCK)
provides researchers with a starting point for collecting and analysing data regarding teacher
knowledge. It was felt that such knowledge would be useful to practising mathematics
teachers and could be used to lay a pathway to finding improved methods for teaching topics
in mathematics which learners may find difficult to understand. The scientific contribution of
the study is that it would contribute to closing the knowledge gap if any, between teacher
practices and their knowledge base which has an impact on effective teaching and the
achievement of learners with regard to the topic of quadratic functions.
1.7
OVERVIEW OF CHAPTERS
Chapter one provides an introduction to the study, the background, significance and purpose
of the study, the problem statement, the research questions and an overview of the content of
the chapters in the study.
The second chapter reviews and discusses some aspect of the literature needed to justify the
study. The findings and the research methodologies employed by other researchers in similar
studies were examined. Special attention was given to findings as well as methods used in
other studies that investigated the pedagogical content knowledge of teachers where the three
elements, knowledge of subject matter, knowledge of teaching strategies and knowledge of
learners’ conceptions were used as a framework that was adopted for this research. The
chapter concludes by discussing the conceptual framework designed for this study.
The third chapter outlines the study’s methodology. It also describes the population and the
procedures for sampling the two teachers for the case studies. The development of the
research instruments are described as well as their validation. The data analysis is also
explained. Ethical issues that were considered for this study are dealt with as well.
The fourth chapter of the study presents the findings. Each case study is narrated in terms of
the data obtained from the observations, document analysis and interview schedules.
The fifth and last chapter discusses the findings as reflected in the data obtained from the two
case studies using the framework developed in Chapter Two. Similarities and differences of
the case studies are highlighted by focusing on the three themes identified for the study:
knowledge of the subject matter, knowledge of teaching strategies and knowledge of
7
learners’ conceptions. As a conclusion to the study; recommendations for appropriate teacher
development and further research on the issue are offered.
1.8
CHAPTER CONCLUSION
This chapter discussed the background of the study which was mainly about the poor
performance in Mathematics by most Grade 12 learners in the South African education
system. Various possible factors causing poor performance were tabled. The purpose,
research questions, significance of the study which emphasised the need to investigate the
pedagogical content knowledge (PCK) held by the two case teachers were also discussed.
The chapter ends with an overview of the chapters contained in this report.
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CHAPTER 2
LITERATURE REVIEW
2.1
INTRODUCTION
In this chapter, the concept of Pedagogical content knowledge (PCK) as applied in the
teaching of mathematics is discussed. Each of the three elements of Pedagogical content
knowledge that had been used as framework for this study are explained. The elements are;
(i) knowledge of subject matter; (ii) knowledge of teaching strategies and (iii) knowledge of
learners’ conceptions are also explained. The conceptual framework for the study is also
presented in this chapter.
2.2 What is Pedagogical Content Knowledge?
In a study of the knowledge bases that teachers must possess to teach effectively, Shulman
(1987) identified pedagogical content knowledge (PCK) as one of the most important
knowledge bases that teachers should possess in order to teach effectively. He maintained
that having knowledge of the subject matter is not enough to teach it. Teachers need to
possess pedagogical content knowledge as well. This knowledge base, PCK, must; according
to Shulman (1987:p.8) include “knowledge of learners and their characteristics, knowledge of
educational contexts, knowledge of educational ends, purposes and values and their
philosophical and historical bases”. This has led researchers to now consider PCK as
important as the subject matter knowledge. According to Shulman (1987), PCK depends on a
teacher’s subject matter knowledge, knowledge of pedagogy and on how the teacher
transforms this knowledge into various forms that enable students in different learning
environments to understand the subject matter. He acknowledges that pedagogical content
knowledge is difficult to isolate and to measure.
Kwong, Joseph, Eric, Khoh, Gek and Eng (2007:p.28), indicate that “Shulman (1986; 1987)
has suggested that pedagogical content knowledge (PCK) forms a unique and distinct
knowledge domain of teacher cognition. PCK emphasises the manner in which teachers relate
their subject matter knowledge (what they know about what they teach) to their pedagogical
knowledge (what they know about teaching, how their learners’ learn and the learners’
conceptions) and how subject matter knowledge is part of the process of pedagogical
9
reasoning”. Shulman (1986: p.9) defined PCK as “the most useful forms of representation of
those ideas, the most powerful analogies, illustrations, examples, explanations, and
demonstrations – in a word, the ways of representing and formulating the subject that make it
comprehensible to others… . It also includes an understanding of what makes the learning of
specific concepts easy or difficult, the conceptions and preconceptions that students of
different ages and backgrounds bring with them to the learning environment”. Thus,
pedagogical content knowledge (PCK), according to Kwong et.al (2007:p.28) is viewed as
that “distinctive knowledge domain of teaching that differentiates the expert teacher in a
subject area from the subject expert”. Furthermore, as Kwong et al (2007:p.28) asserts;
“while general pedagogical knowledge can be generically applied to all teaching subjects,
much of PCK is specific to individual topics in subjects”. Darling-Hammond, 2000 indicate
that “an emerging consensus is that teachers’ knowledge of discipline-specific pedagogy is
critical to being able to present topics within a range of subjects in a manner that learners will
comprehend” while Kagan, 1992 and Reynolds, 1992 asserts that “studies have shown that
novice teachers often struggle to present concepts in a manner understandable to their
students because they have little or no PCK at their disposal”. From the above paragraph, it
can be seen that pedagogical content knowledge is an important knowledge base for teachers
to have in order to teach mathematics topics effectively.
Mishra and Koehler (2006:p.1027) see pedagogical content knowledge as “that knowledge
base which is concerned with the representation and formulation of concepts, pedagogical
techniques, and knowledge of what makes concepts difficult or easy to learn, and knowledge
of learners’ prior knowledge” This is the same view that Shulman, 1986 and De Jong, 1999
hold regarding pedagogical content knowledge. Mishra and Koehler (2006:p.1027), further
cite Rohaan, Taconis and Jochems, 2009 who see pedagogical content knowledge as that
knowledge which “involves knowledge of teaching strategies that incorporate appropriate
conceptual representations in order to address learner difficulties and misconceptions and to
foster meaningful understanding”. Carpenter, Fennema, Peterson and Carey (1988:p.386),
see pedagogical content knowledge as the knowledge, held by a teacher, which “includes
knowledge of the conceptual and procedural knowledge that students bring to the learning of
a topic, the misconceptions about the topic that they may have developed, and the stages of
understanding that they are likely to pass through in moving from a state of having little
understanding of the topic to mastery of it. It also includes knowledge of techniques for
assessing students' understanding and diagnosing their misconceptions, knowledge of
10
instructional strategies that can be used to enable students to connect what they are learning
to the knowledge they already possess, and knowledge of instructional strategies to eliminate
the misconceptions they may have developed”
From the above paragraphs, it can be seen that teachers need to have pedagogical content
knowledge in order to teach their subjects effectively. It is for this reason that the study had
interest in investigating the pedagogical content knowledge supposedly held by the two
participating teachers.
2.2.1 Components of pedagogical content knowledge
The components of pedagogical content knowledge, according to Shulman (1986), comprise
first, knowledge of the specific subject matter; second, knowledge of instructional strategies;
third, knowledge of learners’ conceptions; and fourth, an understanding of what makes the
learning of a specific topic difficult or easy for learners. Shulman’s (1986) fifth category of
teachers’ knowledge bases, curriculum knowledge, involves awareness of how topics are
arranged both within a school year and over a given longer period and ways of using
curriculum resources, such as textbooks, to organise a programme of study for students.
Pedagogical content knowledge, which is at the centre of this study, is an amalgam of a
teacher’s knowledge bases that Hagevik et al (2010) and Yusof and Zakaria (2010) say
includes:

knowledge of context, curriculum and assessment

knowledge of student learning

knowledge of instructional strategies and representations of Mathematics

Knowledge of student understanding about concepts in Mathematics.
De Miranda (2008:p.17) described pedagogical content knowledge as “the knowledge of
three knowledge bases coming together to inform teacher practice: namely, subject matter
content knowledge, pedagogical knowledge and knowledge of context. Subject matter
content knowledge is described as knowledge that is unique to mathematics teachers and
separates, for example, an engineering and technology teacher from an engineer”. Since
different versions of the definition of PCK exist, for the purpose of this study, Shulman’s
11
(1986) description of pedagogical content knowledge will be adopted as its theoretical
framework in which he sees it as an amalgam of a teacher’s knowledge base that includes:
1) Knowledge of the representation of subject matter for teaching
2) Knowledge of relevant instructional strategies
3) Knowledge of learners’ conceptions (preconceptions and misconceptions).
This study focused on these three elements mentioned in the previous paragraph and were
consciously integrated when observing how the two effective teachers displayed them when
teaching quadratic functions in their respective Grade 11 mathematics classes. The choice of
the three elements of PCK was influenced by the fact that they form the core of what
Shulman (1986) indicated as teachers’ PCK that would enable them (the teachers) to
transform the subject matter in such a way that their learners would be readily able to access
the content. First, the teacher needs to have a good grasp of the subject matter before being
able to transform it. Second, the teacher needs a teaching strategy to use to make the subject
accessible to the learners. Third, the teacher needs to have an idea of possible learners’
conceptions that the learners may have about the topic in order to prepare explanations that
will help to eliminate or reinforce the conceptions as is necessary. Furthermore, studying the
PCK elements that a teacher holds cannot be done in isolation of each of its other elements
since PCK is an amalgam within a teacher’s total knowledge base that is uniquely constructed
by the individual teacher. It is the intersection of knowledge of pedagogy, knowledge of
learners’ conceptions and subject matter knowledge (Mishra and Koehler, 2006), and is thus
individualistic.
These three elements of PCK in this study were also used by Winsor (2003) in a study
involving an investigation of pre-service mathematics teachers’ knowledge of functions
where he regarded them (the three elements of PCK) analogously as the legs of “a threelegged stool. The seat of the stool represents the PCK and each one of the legs represents
subject matter knowledge, knowledge of learners and knowledge of instructional strategies. It
is reasoned that the seat needs equal support from each leg while the legs need help from the
seat to stand firmly”. They are interdependent in relation to each other, and can thus not be
studied in isolation although each has its own characteristics and function.
12
2.2.2 Pedagogical content knowledge in the teaching of mathematics
In a study conducted by Yusof and Zakaria (2010) that explored and described the level of
pedagogic content knowledge of three teachers, focusing on the topic of functions at
secondary level mathematics, it was found that the teachers who participated in that study
lacked conceptual knowledge of topics on functions. Hence their lessons were inaccurate and
lacked substantial clarity for the learners. In turn this inhibited understanding. As was the
case with this study, these two scholars focused on how the lessons’ contents were conveyed
to learners. The PCK elements that were observed in that study involved the use of analogy,
representation using symbols, examples, explanation or demonstration that were suitable for
providing conceptual and procedural explanation or description, as well as noting the ways in
which the teachers stimulated the teaching process. Specifically, the particular study sought
to explore and describe the elements of teacher’s PCK and also to determine the teacher’s
level of PCK on the topic of functions. A qualitative research approach based on a case study
research design was used. The case study method used a combination of interviews,
classroom observation and document analysis to collect data on the three teachers’ PCK. In
order to describe the teachers’ PCK, the following guidelines, were used:
According to Yusof and Zakaria (2010:p.34), for “Level 1 PCK; at this level, the teacher
explains the wrong concept or a concept which is not quite clear” to the learners. The teacher
asks low level questions, the lesson is teacher-centred, and the teacher is unable to detect
learners’ difficulties with the topic. The teacher does not relate his lesson presentation to the
learners’ existing knowledge. For Level 2 PCK; at this level, the teacher explains the correct
mathematical concepts but gives the same type of examples to back his explanations, no
variety of examples are given. The teaching is seen to be teacher-centred. The teacher has an
awareness of learners’ difficulties but does not probe further through asking questions that
allow learners to speak out their ideas about the topic. Level 3 PCK: The teacher’s
explanations of concepts are more accurate and clear along with the incorporation of suitable
examples. Learners’ participation is seen to be positive through the provision of relevant
activities which provide both conceptual and procedural understanding of mathematics
principles that are being studied”
In the current study the researcher observed the type of PCK that the two participating
effective teachers used as they presented their lessons. The framework for observing the
13
teachers’ PCK was based on the PCK elements that were given in earlier paragraphs but no
classification of the PCK levels was done like what Yusof and Zakaria (2010) did.
The theme of Chick, Pham and Baker’s study (2006) involved teachers’ pedagogical content
knowledge as they taught the subtraction algorithm. What became clear was that the teachers
had a good lesson presentation but lacked knowledge of how to identify and correct students’
misconceptions. A qualitative research approach using the case study research design was
also used in that study. Data was collected via questionnaires, lesson observations and
interviews and the following framework, containing these three attributes, was used to
evaluate teachers’ PCK. First, if the teacher had knowledge of the subject matter, this was
evident when the teacher exhibited deep and thorough conceptual understanding of identified
aspects of the subtraction algorithm; identified critical mathematical components within the
concept of the subtraction algorithm that are fundamental for understanding and applying the
concept; and displayed skills for solving a problem. Second, if the teacher had knowledge of
instructional strategies and application, it would be evident if the teacher used appropriate
activities during the instruction phase; used real life examples; applied different instructional
strategies in the presentation if need be; and also used different representations in the
instruction. Third, if the teacher had knowledge about learners’ conceptions, evidence would
be obvious if the teacher showed interest in the learners’ prior knowledge; dealt with the
learners’ difficulties during the lesson; took care of possible learners’ misconceptions about
the topic during the lesson; and also had instruments to measure the level of learners’
learning of the topic. Although this researcher has a different topic, namely quadratic
equations, the same three components as those used by Chick et.al (2006) in their framework
were adopted for this study.
Bukova-Güzel (2010) investigated pre-service mathematics teachers’ pedagogical content
knowledge by using solid objects and found that the participating teachers did not pay
attention to possible student misconceptions. In the said study, data was collected through
semi-structured interviews, analysis of lesson plans prepared by the students and video
recordings of instructional applications. The framework for the analysis of PCK used in that
study (Table 2.1) uses knowledge of teaching strategies, knowledge of learners and the
curriculum. Two of these components, knowledge of teaching strategies and knowledge of
learners were incorporated in this study.
14
Knowledge of teaching
strategies and multiple
representations
 Using appropriate
activities in Instruction
 Using real life examples
and analogies in
instruction
 Utilising different
instructional strategies
in presentations
 Making use of different
representations in
instruction (graphics,
tables, formulas, etc.)
Knowledge of learner





Having knowledge of
students’ prior
knowledge
Using real life examples
and analogies in
instruction
Having knowledge of
the difficulties students
will face during
learning
Having knowledge of
possible student
misconceptions
Having knowledge of
student differences
Knowledge of curriculum


Being aware of the
elements of the
mathematics curriculum
(conception, purposes,
etc.)
Being aware of the
varieties of instructional
tools in the mathematics
curriculum and how to
use them

Being aware of the
instruments to measure
student learning and
how to use them
Table: 2.1: Bukova-Güzel’s framework for pedagogical content knowledge Source:
Bukova-Gṻzel (2010)
The remaining sections of this chapter will discuss each of the three identified elements of
PCK and the conceptual framework used in this study. It will draw attention to how other
researchers, in their respective studies, observed these elements as contributing to a teacher’s
PCK.
2.2.2.1
Knowledge of the Subject Matter and mathematics Teaching
One of the three elements of pedagogical content knowledge identified for this study is
subject matter knowledge. According to Turnuklu and Yesildere (2007), knowledge of
mathematics and knowledge of mathematical representations are related to content
knowledge. The teaching process of mathematics topics starts from the teacher’s
understanding of what must be taught and how it must to be taught to the learners. Such a
teaching process proceeds through a series of activities in which learners are given a series of
instructions and an opportunity to learn, although ultimately the learning itself remains the
learner’s responsibility (Shulman, 1987). If the teaching action has been effective, it should
end up with the learner having newly acquired comprehension (Shulman, 1987).
15
The study by Turnuklu and Yesildere (2007) that investigated pre-service primary teachers’
competency of pedagogical content knowledge in mathematics, found that, to teach
mathematics effectively, teachers ought to have a deep understanding of the mathematical
knowledge of the topics that they teach. Their findings indicated that there is a link between
knowledge of mathematics topics by the teacher and effective teaching of mathematics. They
argue that, if a teacher has good conceptual understanding of mathematics topics, the
influence on the quality of their instruction and the instructions used and provided would be
positive. Mishra and Koehler (2006) agree with this conclusion as they contend that teachers
who have a good understanding of the subject matter find different ways to represent it and
make it accessible to learners.
It was clear to Turnuklu and Yesildere (2007) that the pre-service primary teachers’
mathematical knowledge on the topics of fractions, decimal fractions and integers was
mediocre, thus they could not assist their learners with the misconceptions that their learners
displayed. These specific findings point to the fact that, for teachers to be able to present their
mathematics topics in a way that will be understood by their learners, and be able to identify
their learners’ problems, they must have good subject matter knowledge specific to the
topics. It is for this reason that this particular study has included subject matter knowledge as
one of the components to be addressed.
Subject matter knowledge by teachers of any subject is important in teaching as evidenced by
findings of Mishra and Koehler (2006) in a study focusing on developing a framework for
investigating teacher knowledge. They posit that subject matter knowledge is knowledge
about the actual subject matter that is to be taught and learned. Teachers need to know the
subject matter very well. The mathematics content to be covered in high school mathematics
is different from the mathematics in graduate computer science hence the purpose it serves
has to be considered too. Mishra and Koehler (2006:p.1026) posit that “teachers must know
and understand the mathematics that they teach, including knowledge of central facts,
concepts, theories and procedures within a given topic; knowledge of explanatory
frameworks that organise and connect ideas; and knowledge of the rules of evidence and
proof”. Though Turnuklu and Yesildere (2007) worked according to a framework that could
be used to assess teachers’ abilities to incorporate technological devices in teaching, they also
emphasised the need for teachers to have a good understanding of the topics that they intend
to teach, in order for them to select an appropriate technological device that could be used in
16
the teaching. The inclusion of subject matter knowledge as part of the framework used for
this study is thus further justified by these findings.
A study conducted by Ball (1990) investigating the mathematical understanding that
teachers-in-training bring to education, revealed that prospective teachers of mathematics in
both secondary and elementary schools have limited understanding of the mathematics when
teaching a lesson. The teachers’ knowledge of the subject matter appeared to be rule-bound
and thin. In the study, interviews and questionnaires were used to collect data from 252
prospective teachers. The interviews and questionnaires were designed to explore the
participants’ knowledge of mathematics and the teaching of it. In this study, interviews were
also used to collect data about how the teachers intended to handle their lessons.
Ryan and McCrae (2005:p.641) see subject matter knowledge as more than just the
knowledge of facts or concepts. To them, subject matter knowledge “requires knowledge of
both the substantive structure (facts and their organising principles) and the syntactic
structure (legitimacy principles for the rules) of a subject domain”. They further indicate that
teachers need to have a “good understanding of both the conceptual knowledge and the
procedural knowledge of mathematics to be able to provide learners with clear explanations”.
Their study was particularly interested in the conceptual and procedural knowledge that the
participants exhibited as they taught their learners. It is critical that there is a clear
understanding of what the two types of knowledge mean in mathematics.
To bring clarity about what conceptual and procedural knowledge is in mathematics,
reference is made to Schneider and Stern (2008:p.2) who “see conceptual knowledge as the
knowledge of the core concepts and principles and their interrelations in the mathematics
domain. It is knowledge that is rich in relationships. On the other hand, procedural
knowledge in mathematics allows learners to solve problems quickly and efficiently because
it is to some extent automated through drill work and practice. Procedural knowledge can
thus be viewed as rules and procedures for solving mathematics problems”. These two points
are accommodated in this research initiative that also investigated the two participating
teachers’ display of both conceptual knowledge and procedural knowledge when they were
presenting lessons on quadratic functions.
Zerpa, Kajander and Van Barneveld (2009:p.59) stress that “teachers need to have deep
conceptual understanding of the mathematics that they are teaching their learners and must be
17
able to illustrate why mathematical algorithms work, and how these algorithms may be used
to solve problems in real-life situations”. Cockburn (2008) posits that having a sound
understanding of mathematics is a crucial component of an effective teacher’s repertoire. A
good conceptual understanding of the topics will enable teachers to diagnose learners’
misconceptions and misunderstandings easily (Kiliḉ, 2011).
The relevance of teachers’ domain-specific knowledge for effectiveness in teaching has also
been emphasised repeatedly by Ball, Lubienski and Mewborn (2001). Following up on the
work of Shulman (1986) and Kraus, Neubrand, Blum and Baumert (2005), three subdimensions of pedagogic content knowledge that are specifically important to mathematics
teachers and that help to make the subject matter accessible to the learners, can be identified.
These are: (i) tasks given to learners, (ii) using learners existing conceptions and prior
knowledge and (iii) giving appropriate instructional support and guidance in the form of
explanations, analogies, illustrations and examples that will enable learners to master the
content.
Of prime importance during lesson observations in this study was how teachers displayed
their subject matter knowledge of topics on quadratic functions at the same time ensuring that
the learners were able to internalise the content during the lesson. The teachers’ subject
matter knowledge was assessed by checking: (i) the accuracy of mathematical facts; (ii)
flexibility of presenting explanations displayed by the teachers; (iii) sequential presentation
of facts;(iv) and hierarchical presentation of facts and also the (v) flow of ideas of presenting
the topics by the teacher during the presentation of the lesson. Pedagogical content
knowledge, like all other forms of knowledge is useful only when it is applied and inferred,
which is why this researcher wanted to observe its application and not measure it as was done
in the study by Hill et al (2004) where they developed a measure for a teacher’s mathematics
knowledge for teaching mathematics.
2.2.2.2
Knowledge of Instructional Strategies and Mathematics Teaching
According to Brodie (2007), the new curriculum that has been recently introduced into South
African schools calls for learners to participate in mathematics lessons and to express their
mathematical ideas. Teachers are encouraged to make their lessons more learner-centred by
encouraging learners to contribute to the lesson. The choice of the instructional strategy to be
used by the teacher is very important. Different lessons require different teaching methods.
18
According to Shulman (1987), the correct choice of such an instructional strategy does not
depend on the teachers’ knowledge of the subject matter only but also on the teacher’s
knowledge of the learners’ level of understanding. Since this research investigated the use of
instructional strategies during lessons, it is important to know what “good” teaching
strategies the teacher used in mathematics teaching.
Lim (2007), in his study of the characteristics of mathematics teaching in Shanghai, noted
that the success of a teacher in teaching a specific mathematics topic depends on the depth
and breadth of the individual teacher’s pedagogical content knowledge because, prior to the
commencement of a lesson, a mathematics teacher needs to (i) plan the lesson; (ii) choose a
teaching strategy; and (iii) select content that will suit the learners’ level of understanding.
These three activities are all assumed to be elements of PCK. Teachers with a sound
knowledge of the elements of PCK, always select teaching strategies that are appropriate for
the level of development of their mathematics learners. Cockburn (2008) asserts that,
although content knowledge is central to an educator’s effectiveness in teaching mathematics,
the method of teaching plays an equally important role if any learning is to take place. In the
case of this study the teaching strategies that the effective teachers used when they taught
quadratic functions was also investigated.
Tanner (2003) posits that good instructional strategies should: (i) actively engage the
learners; (ii) assist them in using their prior knowledge and skills to solve problems in
mathematics; (iii) motivate the learners to participate during the lesson; and also;(iv) create
an appropriate learning environment. According to Ingvarson, Beavis, Bishop, Peck and
Elsworth (2004) excellent teachers of mathematics are aware of a wide range of effective
teaching strategies and techniques for teaching and learning mathematics that promote the
learners’ enjoyment of the subject. Furthermore, such teachers usually choose teaching
strategies that tend to create the best learning experience for every learner. The PCK of
teachers according to De Miranda (2008:p.17) “involves knowing how to take advantage of
different teaching approaches that make a learning experience most appropriate for the
learners. This includes being flexible and adjusting instruction that takes into account various
learning styles, abilities and interests. Knowing how to best teach a concept so that the
learners will receive the best learning experience speaks to the essence of PCK. The different
teaching approaches employed will vary from teacher to teacher and in differing contexts, but
invariably will revolve around similar principles for each approach”.
19
Westwood (2004:p.79) asserts that “studies have indicated that although expert teachers
differ in their actual style of teaching and management, they all use instructional strategies
that
(i) maximize students’ time and engagement in learning tasks; and (ii) encourage
students’ active participation during lessons. In addition, (iii) they ensure that students
understand the work they are required to do; and, (iv) they set tasks and activities at the right
level to ensure high rates of success. Expert teachers also (v) create a positive and supportive
classroom environment; (vi) they are good managers of behaviour; and (vii) are skilled in
motivating learners to learn”. This study too, investigated how the teachers used their
teaching strategies to benefit the learners. The teaching strategy that the participating teachers
used during lesson presentation was investigated by checking the method used such as telling
method, group work and self discovery teaching method.
It is important to know what is meant by an effective instructional strategy. An effective
instructional strategy is one that triggers active learning by the learner (Eysink, de Jong,
Berthold, Kolloffel, Opfermann and Wouters, 2009). Active learning, according to Eysink, et
al. 2009), encompasses processes such as interpreting, exemplifying, classifying and
organising the content by the learner.
Baumert et al (2009), in their study involving teachers’ mathematical knowledge, cognitive
activation in the classroom and student progress’, mention three components of instructional
strategies that are crucial for initiating and sustaining insightful learning processes in
mathematics lessons. These three components are:

Cognitively challenging and well-structured learning opportunities

Learning support through monitoring of the learning process and individual feedback
and adaptive instruction

Efficient classroom and time management.
From the discussion so far, what is being implied is that teachers need to select teaching
strategies that encourage discussion and justification of ideas in the content of the topic so as
to demonstrate mathematical understanding (Eysink, et. al 2009). Furthermore, teachers need
to support their learners through guided practice until they are independent, and confront
misconceptions that learners may have about a given topic in mathematics (Tanner, 2003;
Lim, 2007). According to Westhood (2004), the choice of a teaching strategy must also
20
encourage a disciplined learning environment that allows learners to listen to other learners’
inputs and encourages the sharing of mathematical ideas as they unfold from the lesson.
The researcher in this investigation observed the kind of instructional strategies that the two
effective teachers used in their mathematics lessons on quadratic functions. The intention was
to see whether the strategies suited the topics that were being taught during that specific
lesson, and whether the chosen teaching strategy was able to challenge the learners
cognitively; the examples used as well as providing the best learning opportunities for the
learners.
2.2.2.3
Knowledge of Learners’ Conceptions, Preconceptions and Misconceptions in
Mathematics Teaching.
According to Fennema and Franke (1992:148), “knowledge of learners is generally defined
as knowing about the characteristics (conceptions, pre-conceptions, misconceptions and
learning difficulties) of a certain group of learners, establishing a classroom environment and
planning instruction accordingly to meet the needs of these learners”. Smith, DiSessa and
Roschelle (1993) think that learners do not come to class as blank slates. They come to the
classroom with certain preconceptions about topics in mathematics.
As they learn
mathematics, the sense they make of what they are presented with can differ from what their
teachers expect, and teachers may also not be aware of the total experience that these learners
bring along to the class. From a constructivist view of learning, all learning involves the
interpretation of phenomena, situations and events including classroom instruction, through
the perspective of the learners’ existing knowledge.
A study by Kiliḉ (2011) that investigated pre-service secondary teachers’ knowledge of their
students revealed that having strong subject matter knowledge is essential to becoming a
good teacher, but it is not sufficient for effective teaching. The findings of the study revealed
that teachers should, in addition, “know how to teach a particular mathematical concept to
particular learners, how to represent a particular mathematical idea, how to respond to
learners’ questions, and what curriculum materials and tasks to use to engage students in a
new topic” Kiliḉ (2011: p.18) . Kiliḉ (2011:p.18) cite An, Kulm and Wu (2004) who assert
that since “PCK is perceived as knowledge of how to teach particular subject matter,
knowledge of subject matter and knowledge of pedagogy is not enough to achieve an
effective teaching practice without knowing the learners”. Kiliḉ (2011) used classroom
21
observations, structured interviews, questionnaires and journals as data collection
instruments. The results in the study by Kiliḉ (2011) showed that pre-service teachers have
insufficient knowledge of learners’ conceptions and “when the pre-service teachers were
given examples of learners’ errors and asked how to address them, the pre-service teachers
tended to repeat how to carry out the procedures or explained how to apply a rule or
mathematical fact to solve the problem” instead of explaining the correct concepts that would
help eliminate the learners’ errors. In this current study the two participating teachers’
knowledge of learners’ conceptions, knowledge of the subject matter and knowledge of
teaching strategies were investigated using interviews, observations and lesson plan analysis
as data collection instruments. Kiliḉ (2011) used the same data collection instruments.
As Tanner, Gene, Caro and Amy (2003) report about their study entitled ‘Instructional
strategies: how teachers teach matters’, learners were seen to have varying knowledge and
interest levels about the mathematics topic that their teachers brought to the classroom. Their
prior knowledge (preconceptions) about the topic can interfere with their new learning
experience. This could result in a concept being incorrectly understood by the learners. In this
research study, teachers’ lesson plans were analysed to check which concepts were to be
taught, what prior learning was required of the learners and what possible misconceptions if
any the teacher anticipated the learners might have about the topic being taught.
When teachers choose teaching strategies to help learners understand mathematics topics, it is
important to consider what prior conceptions about the topic their learners in the class have
acquired. As teachers transform the content to a more accessible form for the learners,
documented misconceptions about the topic or content must be considered (Tanner, Gene,
Caro and Amy, 2003). Knowledge of common mathematical errors and misconceptions of
learners can provide teachers with insight into the learners’ thinking as well as offering ideas
that would serve as a focus for teaching and learning (Ryan and McCrae, 2009). Teachers
need to do baseline assessment to find out what learners know about the topic. If teachers are
familiar with what learners know, they can help build bridges between the known and the
unknown, and this will help teachers to clarify misconceptions in order to assist learners in
comprehending new information (Ryan and McCrae, 2009).
In order to investigate the misconceptions learners could possibly have about the topics that
they were previously taught, it is important to know what misconceptions are.
Misconceptions are pieces of wrong knowledge that may arise from learners’ prior
22
experience and learning, both inside and outside of the classroom (Smith et al. 1993).
Teachers who have poor subject matter knowledge of mathematics may also contribute to the
development of such misconceptions in learners since misconceptions are faulty extensions
of productive prior learning (Smith et al. 1993). Reasoning based on misconceptions leads to
consistently wrong problem solutions (Kὄrner, 2005). Such misconceptions are usually
formed when the new lesson is not compatible with the learner’s existing conceptual prior
knowledge about the topic. Furthermore, because of their strength and flawed content,
misconceptions interfere with learning new concepts.
Ingvarson et al. (2004) indicate that excellent teachers of mathematics have rich knowledge
of how students learn mathematics. They have an understanding of current theories relevant
to the learning of mathematics. Such educators have knowledge of the mathematical
development of students including learning sequences, appropriate representations, models
and language. Effective teachers are aware of the common misconceptions that learners may
have regarding a particular topic and they structure their lessons in such a way so as to
confront the misconception that the learners may have. Such teachers encourage discussion
and justification of the ideas that learners may bring during a mathematics lesson (Ingvarson
et. al, 2004).
According to Smith et al. (1993), for classroom instruction to be successful in eliminating
misconceptions, teachers must present the correct concepts in clear opposition to the students'
faulty conceptions. The chosen instruction (if the teacher has knowledge of learners’
misconception) should “include demonstrations and activities that produce counter-evidence
and plausible conceptual alternatives to target misconceptions. The confrontation of ideas
through discussions in the classroom is then internalised by students as a psychological
process of competition that finally results in the replacement of the misconception”.
As already indicated, the key elements of Shulman’s (1986) conception of PCK are
knowledge of representations of subject matter on the one hand and the understanding of
specific learning difficulties and student conceptions on the other. Obviously, these elements
are intertwined and should be used in a flexible manner: It is assumed that the more
representations teachers have at their disposal and the better they recognise learning
difficulties, the more effectively they can deploy their PCK (Shulman 1987).
23
Teachers need to have good knowledge of possible difficulties that learners might experience
when a particular mathematics topic is presented. This will enable the teacher to prepare
possible explanations and examples that will enable the learners to access the content of the
topic that is being taught. Knowledge of such learning difficulties will also allow the teacher
to prepare possible prior content knowledge necessary for a particular topic that learners
could easily link to the new knowledge.
Learning difficulties may arise due to the language of instruction in mathematics. Teachers
need to choose a teaching strategy that allows them to use a language of teaching that the
learners are familiar with. Howie (2003), Kanyongo et al (2007), Maree and Erasmus (2007)
conducted studies on factors that affect the performance of learners in Mathematics in subSaharan Africa. Each of these studies revealed, among other factors, that learners whose
proficiency in the language of teaching and learning was good performed better than learners
whose proficiency in the language of teaching and learning was poor. Teachers who teach in
the second language of the learners must have that awareness. Learners with a language
problem
may
have
difficulty
in
understanding
written
mathematical
problems,
communicating mathematical ideas orally and reading text.
2.2.3
Conceptual Framework for the Study
The review of relevant literature has shown that to teach effectively, teachers need to have
both content knowledge and the pedagogy of the topics that they teach together with
knowledge of the learners’ conceptions (Deubel, 2009). The three elements of PCK, namely,
knowledge of the subject matter, knowledge of teaching strategies and knowledge of
learners’ conceptions in this research synchronise with the views and constructs of PCK used
by several researchers in this domain, like Shulman (1986), Hailm and Meerah (2002), De
Jong (2003), Ball, Lubienski and Mewborn (2005); De Jong et al., (2005) and Kraus,
Neubrand, Blum and Baumert (2005). Thus these three elements formed the core of the
conceptual framework that was used in the development of data collection instruments as
well as analysis of data for this study.
The proposed study intends to find answers to the following research questions:
1. What pedagogical content knowledge do effective teachers display with regard to
teaching quadratic functions in Grade 11?
24
2. How did the teachers develop the PCK that they use when teaching quadratic
functions?
The conceptual framework developed for this study was based on the three elements of
pedagogical content knowledge that have been discussed in conjunction with the literature
reviewed in this chapter. It is also presented in tabular format (Table 2.2) and indicates all the
expected sub-items within each component. The PCK elements are listed as a heading for
each column with the items that were observed for that particular component of PCK being
listed in each column. Such an approach was used by other researchers such as Bukova-Guzel
(2010) and Yusof and Zakaria (2010).
PCK
Knowledge of teaching
strategies
PCK
Knowledge of subject
Knowledge of
matter
learners’
conceptions
PCK
PCK
Figure 2.1: Schematic diagram showing the interdependence of the three PCK elements. PCK is at the
centre of the intersection of the three elements. Each element can stand independently but as an amalgam
they provide stability to the PCK construct .
25
a. Knowledge of subject matter
b. Knowledge of Teaching
c.
strategies
 Exhibits deep and thorough
conceptual and procedural
understanding of identified
aspects of quadratic
functions.(guided by checking
(i) correctness of mathematical
facts
(ii) flexibility of explanations
(iii) sequential representation of
facts
(iv) hierarchical presentation
(v) easy flow of ideas
 Identifies critical mathematical
 Using appropriate
conceptions
 Addresses learners’
activities in Instruction.
 Using real life examples
Knowledge of learners’
misconceptions.
 Displays expectations of possible
and analogies in
difficulties that learners may face
instruction
during learning and address such.
 Utilises different
 Discusses learners’ ways of
instructional strategies in
presentations
thinking about a concept.
 Being aware of the instruments
to measure student learning and
components within the concept of
how to use them.
quadratic functions that are
fundamental for understanding
and applying the concept.
 Displays skills for solving
problems in the area of quadratic
functions
Table 2.2: A framework of PCK Elements used in the study Source:
Bukova -Güzel (2010)
The PCK elements supposedly held by the two effective teachers in the study; were
investigated by checking the sub-items indicated in each column of Table 2.2 as the teachers
presented their lessons on quadratic functions to grade 11 learners.
26
2.3 CHAPTER CONCLUSION
In this chapter, the notion of pedagogical content knowledge was discussed with reference to
the work of other scholars. The three elements selected for this study, namely, knowledge of
the subject matter, knowledge of teaching strategies and knowledge of learners’ conceptions,
were also presented. The implication of each element for the teaching of mathematics was
also considered. The chapter ends with a presentation of the conceptual framework that was
used for both the collection and analysis of the data.
27
CHAPTER 3
RESEARCH METHODOLOGY
3.1
INTRODUCTION
In this chapter the research procedure that was used to investigate the research questions is
described with reference to the research design, method and sampling procedures for the
identification of the participating teachers, the research instruments and their validation, the
data gathering process and the administration of the main study.
3.2
RESEARCH DESIGN AND METHOD
A qualitative research approach which focused on case study method was used in this
investigation. In their respective studies, this method was also applied by Yusof and Zakaria
(2010), Chick, Pham and Baker (2005) as well as Bukova-Guzel (2010) whose focus too was
on teachers’ pedagogical content knowledge (PCK). In all these studies qualitative research
was fitting as the researchers endeavoured to describe an event in the social world from the
standpoint of the individuals who were part of the ongoing event (Sinkovic et al., 2008). So,
in this study, the research interest was to study the PCK held by the two participating teachers
as they taught topics on quadratic functions in grade 11 and qualitative research approach was
seen as being consistent with the intentions of the research questions.
The qualitative research approach in education, according to Mason (2006), enables the
researcher, after data analysis, to understand and explore the richness, depth, context and
complexity within which teachers in the research site operate. In this study the units of
analysis were the two teachers who were considered as being effective in the teaching of
mathematics based on their good performance (pass rate of 80% and above) in the Grade 12
external National Senior Certificate mathematics examination. The researcher collected data
regarding the teachers’ pedagogical content knowledge through direct observation, document
analysis and interviews. The case study method allowed the researcher to gain insight into the
type of pedagogical content knowledge the teachers were using as they taught. The advantage
of the case study method is that it allows the researcher to focus and gain insight on a specific
phenomenon but the disadvantage of this method is that findings cannot be generalized.
Lessons were observed using an observation schedule. Verbal responses to the interviews
28
with participants were transcribed in written form so as to serve as a backup as well as for
easy reference to what had transpired during each session devoted to data collection.
3.3
RESEARCH SITE AND POPULATION
The research site was located in a certain school circuit of the Mpumalanga Department of
Education in South Africa. The population for this study comprised all mathematics teachers
who came from schools that had obtained an average of 80% overall pass rate and an average
of 80% in the National Senior Certificate Grade 12 mathematics examinations for the past
three (or more) consecutive years.
As a point of entry to identify the two teachers who could be classified as “effective”,
statistical records of performance of schools from the Province’s district education office in
which this circuit fell, were used to compile a record of performance of schools in
mathematics. The Chief Education Specialist (CES) for further Education and Training (FET)
band in the district was consulted to supply this information. He supplied the researcher with
schools that were seen to be closer to the set criterion of 80% pass rate. Finding schools with
a pass rate of 80% was difficult as most schools that could be regarded as good performing
schools around the district had an average performance of 70% pass rate in mathematics and
the researcher had to settle for schools in this category. Of the 135 secondary schools in the
district; only eight schools had such an average pass rate level in mathematics during the past
three to four years. The eight schools formed the pool of schools from which the sample of
teachers could be drawn.
In the pool of eight schools; there were fourteen teachers who taught mathematics in grades
10 to 12 and they formed the population from which the sample could be drawn.
3.4
SAMPLING PROCEDURE AND SAMPLE
From the eight identified schools which satisfied the performance criterion of 70% pass rate
or more for the past three to four years, two strata of schools were formed, one comprising of
schools from rural settings and the other stratum of schools from an urban setting. A second
criterion was that teachers from these eight schools had to be teaching Grade 12 classes when
such success was achieved. Two schools; with 5 teachers; were from a rural setting while the
other six schools; with 9 teachers; were from an urban setting satisfied the two set criteria. A
third criterion was that the two participating teachers should be from schools in the same
29
circuit. One teacher was then purposively identified from each stratum and letters inviting
them to participate in the proposed study were sent to them. Two male teachers were found
from the same circuit, one from a rural school and the other one from an urban school. This
approach of selecting two teachers to do in-depth studies was employed by other researchers
such as Chick and Harris (2007), Randall (2008) and Li and Yu (2009) when they conducted
in-depth studies to explain teachers’ pedagogical content knowledge in their own studies.
The profiles of the two participating effective teachers are given in tabular form (Table 3.1)
to enable easy comparison of the similarities and differences between the two teachers.
ITEM
TEACHER A
TEACHER B
1.
Educational qualifications
Diploma in Education; ACE
Certificate in Education
(Mathematics)
2.
Current school location
B.Sc (Mathematics and
Statistics); ACE (Advanced
Certificate in Education
(Mathematics)
Township setting
3.
Gender
Male
Male
4.
Age
34
38
5.
Experience( at the time of the
research)
Grades teaching since
appointed.
Current studies (if any)
18
12
Grade 10 to 12
Grade 10 to 12
None
B.Ed (Educational
Management)
6.
7.
Deep rural
Table 3.1: Profile of the Participating Teachers
Thus, the participants are qualified mathematics teachers and are presumed to have sufficient
general subject matter knowledge to enable them to develop the specific content knowledge
for teaching quadratic functions in school mathematics as can be seen from their
qualifications. The general believe in society is that teachers who hold such qualifications,
with the relevant years of experience, should be able to develop adequate content knowledge
and PCK for teaching school mathematics.
30
3.5
DATA COLLECTION INSTRUMENTS
Data was collected via observation of the teachers’ lesson presentation, lesson plan analysis
and one-on-one interviews with each mathematics teacher to find answers to the research
questions (see Table 3.2 to 3.5 for format and scoring of each instrument). Chick and Harries
(2007) used observation schedule when they studied the pedagogical content knowledge of
teachers by observing the examples that teachers gave to learners on the topic of ratios.
Observations were also used by Kilic (2006) when he studied the components of pedagogical
content knowledge that pre-service teachers gain through the method course offered at a
certain university.
The format and how each instrument was used during data collection are now presented.
3.5.1
Classroom Observation Protocol
PCK ELEMENT TO
BE OBSERVED
EVIDENT WHEN THE
TEACHER….
a. Knowledge of the
subject matter
1. Exhibits deep and thorough conceptual
understanding of identified aspects of
functions.
2. Identifies critical mathematical
components within the concept of
functions that are fundamental for
understanding and applying that
concept.
3. Displays skills for solving problems in
the area of functions.
1. Uses appropriate activities in
Instruction
2. Uses real-life examples and
analogies in instruction
3. Utilises different instructional strategies
in presentations.
b. Knowledge of
Teaching strategies
c. Knowledge of
learners’ conceptions
1. Addresses learners’ misconceptions
2. Displays expectations of possible
difficulties learners may face during
learning and address such.
3. Discusses learners’ ways of thinking
about a concept.
4. Shows an awareness of the instruments
to measure student learning and how to
use them
Table 3.2: Observation protocol used in the study
Source: Adapted from Chick, Baker, Pham (2006)
31
OBSERVED PRACTICE
DISPLAYED
To answer the first research question which sought to find out what pedagogical content
knowledge (PCK) effective mathematics teachers display in the teaching of quadratic
functions, a lesson observation protocol based on Yusof and Zakaria (2010) model was used
to collect data on the two teachers’ presentation of lessons. Yusof and Zakaria (2010) as well
as Bukova-Guzel (2010) used the observation method to assess the teachers’ PCK on
mathematics. The following elements were assessed during lesson observations (see table
3.2):

Knowledge of the content of the topic in which the teacher is engaged, that is,
conceptual knowledge and procedural knowledge where the following were used to
assess this element: accuracy of mathematical facts presented; flexibility of
presentation; sequential representation of facts; flow of ideas and hierarchical
presentation of facts.

Knowledge of teaching strategies that enabled the teacher to present the lesson in a
way that was comprehensible to the learners for which the following guidelines were
used: organisation of the lesson; choice of examples; representations, use of
chalkboard and appropriate teaching strategies.

Knowledge of learners’ conceptions (misconceptions and pre-conceptions) about the
topic under discussion where the following were used to check the teacher’s
knowledge of this element: assessing learners’ understanding; identifying errors
learners made; addressing learners’ difficulties, and determining sources of such
difficulties; identification of misconceptions and elimination of them by probing
questions; and using appropriate tasks.
The observation protocol schedule assessed how teachers presented their lessons in order to
assist learners in comprehending the topic; how the teachers assessed their learners’ after the
lessons; the teaching strategies that they employed; and how the teachers dealt with
misconceptions and learner difficulty during lesson presentation. Table 3.2 depicts the
classroom observation protocol that was used during lesson observations.
3.5.2
Teachers Pre-lesson interview Questions
The teachers were interviewed before each lesson to find out more about their PCK. The
purpose of these interviews was to find out how the teacher had organised the lessons, the
32
teacher’s knowledge of key concepts to be taught, teaching strategies to be used, how the
teacher planned to assist learners with difficulties, assessment tasks, any expectations of
learners’ misconceptions that the teacher might have had and also to triangulate lesson
observation data. The questions that were used during the interviews are presented in Table
3.3 below.
ELEMENTS OF PCK
QUESTION RELATED TO PCK ELEMENT
RESPONSES
FOR THIS STUDY
a.
Knowledge of
1.
the subject
matter
What are the key concepts in the lesson that you
are about to teach?
2.
Draw a concept map illustrating the sequence
you will follow to teach these key concepts.
3.
Does the lesson involve any procedural
knowledge that the learners must know? If so,
what does the procedure involve?
b.
Knowledge of
1.
teaching
strategies and
Which teaching strategy will you employ to
ensure successful delivery of this lesson?
2.
Why did you choose such a strategy?
3.
In your selection of examples to be used in this
application
lesson, have you selected real-life examples?
c.
Knowledge of
learners’
1.
What is the goal/aim of your lesson?
2.
Which learners’ prior knowledge is regarded as
conceptions
important before the above key concepts can be
successfully taught to learners?
3.
What possible learner misconceptions do you
anticipate regarding this lesson?
4.
How will you assist learners who experience
difficulties with this lesson?
5.
Have you prepared an assessment instrument to
evaluate whether the goal of the lesson have
been achieved?
Table 3.3 Interview Questions: Source: Adapted from Chick, Baker, Pham (2006)
(Probing teachers’ pedagogical content knowledge: Lessons from the case of the subtraction algorithm)
33
3.5.3
Document Analysis: The lesson plan analysis
In addition to interviews and lesson observation, the teachers’ lesson plans were analysed
using a structured format (see Table 3.4). This was intended to check for other PCK attributes
that may not have been observed during lesson presentation and also not mentioned during
interviews. Such attributes include key concepts, teaching strategies and dealing with
misconceptions.
ELEMENT OF PCK
Knowledge of subject matter
Knowledge of teaching
strategies and application
CHECKED IN THE PREPARATION
OBSERVATIONS
1.
Are key concepts to be taught during
the lesson indicated in the preparation?
2.
Does the preparation indicate possible
mathematics procedures to be taught to
the learners?
3.
Does the lesson preparation reflect
accurate concepts associated with the
topic quadratic functions?
1.
Is the teaching strategy to be used
stated in the preparation?
2.
Are alternative teaching strategies to
be used during the lesson reflected in
the preparation?
3.
Are examples to be used during the
lesson indicated in the lesson
preparation?
1.
Does the preparation reflect possible
misconceptions that will be addressed
during the lesson?
2.
Does the preparation reflect the
required learners’ prior knowledge
before the start of the new topic?
3.
Are possible learners’ difficulties
reflected in the preparation?
4.
Is an assessment instrument indicated
in the preparation?
5.
Is the goal of the lesson clearly stated
in the preparation?
Knowledge of learners’
conceptions
Table 3.4: Lesson plan analysis: Guiding questions: Source:
Adapted from Chick, Baker,
Pham (2006): Probing teachers’ pedagogical content knowledge: Lessons from the case of the
subtraction algorithm).
34
3.5.4 Questionnaire on the Teacher’s Mathematics Teaching Knowledge Development
This questionnaire sought to find out how the two mathematics teachers acquired the
mathematics teaching knowledge that they have which helped them to teach effectively; by
producing good results. The questions that were used are tabulated below (Table 3.5).
Item
1
Question to be asked
Did you receive any special training as a mathematics teacher after your initial teacher training?
2
(a) Do you attend workshops that focus on teacher development?
(b) What have you gained from attending such workshops?
3
(a) Have you ever observed your colleagues when they were teaching a mathematics lesson?
(b) What did you learn from such an observation?
4
What are your qualifications as a mathematics teacher?
5
For how long have you been a mathematics teacher?
6
How often do you review the lessons that you have taught?
Table 3.5
Pedagogical content knowledge development questionnaire: Source:
Shulman (1987)
3.6
VALIDATION OF INSTRUMENTS
To ensure that the three data collection instruments (namely; observation schedule, interview
questions and lesson plan analysis) have content and face validity, three experts in the
mathematics department of a certain University were requested to scrutinise the instruments
in order to establish the validity of both the content and the format of each research
instrument. The experts worked independently of each other to scrutinize the instruments.
Furthermore, a pilot study was done specifically to test the observation protocol and the
consistency of the researcher’s observations in one school that was amongst the eight that
were regarded as having met the criteria for selecting an effective teacher. It was found that
the researcher could use the instruments with consistency. The instruments were then
modified and fine-tuned in consultation with the three experts. Language related errors were
noted on the instruments and changes were effected to improve their usefulness.
35
3.7 PREPARING FOR THE MAIN STUDY
Preparation for the research process involved each of the following steps:

Obtaining permission from the Mpumalanga Provincial Department of Education

Obtaining permission from the principals of selected schools

Validation of research instruments

Obtaining consent letters from the participating teachers

Obtaining consent letters from parents whose children would be taught by the
participants. Parents were assured that if they so wish, they could withdraw their
children from participating and that the lessons missed by their children would be
repeated by the teacher. Such an arrangement was also agreed upon with the two
participating teachers.

Agreeing how data collection through observation, interviews and document analysis,
all of which focused on each of the three elements of pedagogical content knowledge
would be done with the two participating teachers and their school managers.
The identity of the two participating teachers was protected and codes were allocated to
them. The code name of one of the teachers was Teacher A, and the other teacher’s code
name was Teacher B. In the research instruments, all data collected from the participating
teachers is referred to according to their code names, Teacher A or Teacher B. This is in
line with the principle that one of the cornerstones of research ethics is that respondents
should be offered the opportunity to have their identity hidden in a research report
(Oliver, 2003).
3.8 ADMINISTRATION OF THE MAIN STUDY
Before the data gathering process started, the researcher met with each of the participating
teachers to discuss the whole research process and to clarify any issue that they might have
been concerned about regarding the research process. The two teachers were anxious about
the fact that the June examinations were about to be written, which meant that the research
process could be interrupted for some days. They suggested that observations of the lessons
36
should take place on days when the teachers and learners were free from writing
examinations. The researcher agreed to the teachers’ input. The teachers also negotiated that
the lessons to be observed and key discussions on lesson plan analysis and pre-lesson
interviews should be based on lessons and topics that they (teachers) had already prepared
according to their interpretation of the syllabus. Furthermore, the participating teachers
agreed to repeat any lesson that may have been missed by any learner whose parent may have
not concerted the learner to participate in the study. Lesson observations were done during
school hours and during the mathematics period of the grade 11 class. Learners were given
letters of consent to be given to their parents to sign if they consented. Fortunately, all parents
of both set of schools signed the consent forms.
Data was collected over a period of two weeks from each of the two teachers due to the
interruptions of the half-yearly examinations. Five lessons from each teacher were observed.
The lessons were audio-taped; consent was obtained from the two teachers to do so. Prelesson interviews for each teacher were always done the day before the actual lesson
presentation. Each recorded interview was transcribed as written notes. Each teacher taught
only one grade 11 Mathematics class.
The teachers’ demographic profiles regarding how they may have developed their
mathematics teaching knowledge were collected through interviews on the day the last lesson
was observed. Once data was collected; it was ready for analysis.
3.9
DATA ANALYSIS
Data collected was analysed using the conceptual framework according to the following
categories:

Knowledge of the subject matter (checking for the teacher’s conceptual understanding
of the topic; display of skills in problem solving (procedural knowledge) ( Refer to
Appendix A, B and C)

Knowledge of teaching strategies (checking for use of appropriate activities; use of
real-life examples; and use of different teaching strategies) (Appendix A, B and C)

Knowledge of the learners’ conceptions (checking for the teacher’s ability to address
the learners’ misconceptions; expectation of possible learners’ difficulties; discussion
37
of learners’ ways of thinking; awareness of the instruments to measure student
learning) (Appendix A, B and C)

Teachers’ PCK development (finding out about the qualifications of the teacher;
teaching experience; workshop attendance; and peer observation during lessons)
The above stated categories and themes were investigated through the observation of lessons,
interviews with the teacher and lesson plan analysis.
The next section will discuss the trustworthiness of the study.
3.10
TRUSTWORTHINESS OF THE STUDY
According to Sinkovic et al (2008), trustworthiness in a qualitative study aims to support the
argument that the study’s findings are worthy of receiving attention. In order to establish
trustworthiness, credibility, dependability, transferability and confirmability need to be
established.
Credibility focuses on establishing a match between the constructed realities of the
participants and those represented by the researcher (Lincoln and Guba, 1989). To ensure
credibility in this study, the lessons that were observed were video-taped and this ensured that
the researcher could re-visit the lessons with ease to ensure that the reality that the researcher
had recorded was not a fabrication. The researcher ensured that there was accurate reflection
on the observations by cross-checking with the participants regarding what had been
experienced during the lesson. Field notes reflected what transpired during the lesson. Peer
debriefing was used to ensure that the items in the observation checklist did indeed relate to
aspects of pedagogical content knowledge.
Dependability deals with the consistency of research results obtained over time.
Dependability, according to Sinkovic et.al (2008) can be established by using different
methods of data collection and different times of collecting the data on the same research
problem. In this study, dependability was established by having prolonged and concentrated
engagement with the participants about the study, two to three weeks in this case. In addition,
pre-lesson interviews and also lesson plan analysis were used as evidence when collecting
data about the teachers’ PCK.
38
According to Rodwell and Byers (1997), confirmability can be established if the results can
be linked to the data itself. It speaks to data management and the analysis of the data itself. In
this study, confirmability was established by keeping the collected data that was used for
interpretation safely, so that any interested researcher could access the data for inspection. In
addition, an audit trail was done by independent critical readers whom the researcher had
asked to evaluate the methods used for the gathering of the data.
Transferability refers to the applicability of the findings to another setting (Lincoln and Guba,
1985). As this was a qualitative study and no substantive generalisations could be made, the
researcher gave thick description with enough detail of the findings so that readers could
decide on their own whether the results of the study would be transferable to their own
research contexts or not.
3.11
ELIMINATION OF BIAS
Bias occurs when interfering factors distort the truth or accuracy of the information. Bias can
be easily eliminated if the sources of such bias are known. Sources of interfering factors
could include the use of leading questions, incorrect recording of respondents’ answers and
situational factors such as discomfort or anxiety among participants.
All these contributing factors were carefully avoided starting from the construction of the
questionnaires and preparation of all the other research instruments. The interview
questionnaire did not contain leading questions and all the additional observations noted from
the participants were verified with the participants before being finally recorded for further
analysis. The respondents’ identity was not be revealed, which further assisted in eliminating
bias.
3.12
ETHICAL CONSIDERATIONS
Sometimes there is a sense of insecurity among human science practitioners that their
approach is not as objective as that of the general scientist who deals with measurable and
quantifiable phenomena. Bochner (2002) suggests that the human sciences are a little untidy
and show signs of inferiority, stating that ‘Traditionally we have worried much more about
how we are judged as “scientists” by other scientists than about whether our work is useful,
insightful, or meaningful – and to whom” (2002:259). Even other scientists who work with
quantitative information needs to bear the human aspect of their research in mind thus ethics
39
would play a pivotal part in these studies too. Wherever people are involved in studies the
ethical aspect raises its head. And, as we live in a constitutional democracy with a Bill of
Rights, as formalised in 1996, this is an important consideration.
According to Schurink, Schurink and Poggenpoel (1998), important ethical considerations
include:

Voluntary participation on the part of those requested to be part of the data gathering
process. Participants will be also informed that they can voluntarily leave the project
whenever they choose to do so, and this without penalty.

The participants will need to give their informed consent – this will mean that they
will be informed of what the research entails and of how they can participate. Their
superiors in the school hierarchy would also be included. This will include parents of
learners whom the participants teach. Lessons missed by learners who could be
withdrawn from the project would be repeated by the teacher. In this study all parents
of learners in schools that were selected consented that their children take part in the
research project.

Confidentiality and anonymity should be assured in the contract drawn up between
the researcher and the participants. For this study, the two participants were allocated
code names, Teacher A and B respectively.

Feedback regarding the results and findings of the research would need to be
contractually arranged and the agreement be effected over time as the project
progressed. For this project, as soon as the results have been certified as valid by the
ethics committee of the University, the two participating teachers will be informed of
the outcomes of the study.

The competency of researcher should be assured, as well as the scientific soundness
of project. For this researcher to have successfully defended the proposal, it reflects
adequate competency levels to be able to manage this research project.
Ethical considerations of this nature reflect that even in the teaching of Mathematics the
teachers participating in this Mpumalanga study have to be regarded with respect and dignity.
Consent was sought from all participants including the learners’ parents so they were aware
40
that such a study project was underway. All participants were free to choose to participate or
not. Code names were used to protect their identities. Interview transcripts and all
observational notes were kept in a locked, safe place to ensure that no one other than the
researcher could access the information.
3.13
CHAPTER CONCLUSION
In this chapter the research methodology was detailed with regard to the research procedure,
the research site, the population size, the sample, the data collection instruments, the
validation of the research instruments and the research and data gathering processes. Data
analysis was dealt with and the trustworthiness of the study was established. Attention was
drawn to the elimination of bias and ethical considerations.
41
CHAPTER 4
DATA ANALYSIS AND RESULTS OF THE CASE STUDIES
4.1
INTRODUCTION
This chapter reports the findings on the pedagogical content knowledge of each of the two
case teachers who are herein referred to as Teacher A and Teacher B. For each case study,
findings on classroom observations (by first describing the lesson observations), pre-lesson
interviews, and lesson plans analysis will be presented. The presentation will include an
analysis of how their presumed pedagogical content knowledge (PCK) was used in the
teaching of quadratic functions.
4.2. The Teachers’ Pedagogical Content Knowledge
This section presents the findings regarding the participating teachers’ PCK with reference to
the three elements identified for this study: namely, knowledge of the subject matter,
knowledge of teaching strategies and application and knowledge of learners’ conceptions.
Classroom lesson observations, pre-lesson interviews and lesson plan analysis were used to
collect data about the participants’ PCK on the topic of quadratic functions. A description of
the observations made during each teacher’s lessons presentation will be given and a link to
one of the elements of PCK will be done if such a link did exist. Furthermore, a summary of
the findings on pre-lesson interviews and lesson plan analysis on the three elements of PCK
will also be given. The next paragraph will describe the lesson observation of Teacher A.
4.2.1 Description of Classroom Observations for Teacher A
The purpose of lesson observation was to examine the interaction patterns at work in the
classroom for each of the teachers, namely how they used their content knowledge in
teaching a particular topic on quadratic functions. The instructional skills and strategies used
by the teachers, the ways in which they tried to identify learners’ preconceptions and learning
difficulties, and what they did to try to rectify these misconceptions, if any, were also
examined. The topic in which most lessons were observed were on quadratic functions (how
to draw the graph, how to sketch the graphs, determining the equation of a parabola given
three points).
42
LESSON OBSERVATION: TEACHER A
DESCRIPTION OF LESSON OBSERVATION (TEACHER A)
CATEGORISATION/THEMES
Condition of the classroom
Despite the large class size, the classroom
presented
There were 35 male learners and 20 female learners. Each learner had a desk,
a
highly
conducive
learning
environment.
a textbook and a chair though there were many learners which limited
movement between the rows of tables. Teacher A had a full view of all the
learners. A chalkboard, a teacher’s table and a duster were available in the
classroom. Teacher A had his chalks and textbook.. Wall charts of
Mathematics topic were hung on the walls
LESSON OBSERVATION:
Grade 11
CATEGORISATION/THEMES
Topic: How to draw the graph of a Quadratic function
Line 1: Teacher A, standing in front of the class, introduced the topic of the
Teacher A used content knowledge to present
day “Today’s lesson will be about drawing the graph of a parabola”;. He
the three variations of the equation using the
wrote the variations of a quadratic function on the chalk board, numbered as
lecture method. He did a review about the
2
shown here (i) y= ax , where, a≠0,
2
b=0 and c=0 (ii) y = ax + bx, a≠0, b≠0, c
approach of how to draw the graph of a straight
=0 and (iii) y = ax2 + bx + c and said; “These are the three variations of a
line and emphasises that it is different from the
2
quadratic equation. Note that “a” the coefficient of x can never be equal to
2
approach of drawing that of a quadratic
zero. The simplest form of the equation is y = x ”. He reminds learners of the
function. His question of asking learners
graph of a straight line y =mx + b which they had learnt in grade 10 and that it
whether they understand his explanation does
is different from the quadratic function graph as he writes the equation of a
not allow learners to express their views on the
straight line on the chalkboard. “In grade 10 you learnt how to draw the
given explanation.
graph of a straight line y= mx + b, and you needed to choose just three xvalues to substitute in the equation to get coordinates to draw the graph. For
a quadratic function, you need more x- values as you will see when I
demonstrate later” He ends his introduction with a question; “Do you
understand?”
Line 2: All the learners in the class responded to Teacher A’s question asked
Teacher A’s closed question elicits chorus
in line 1 above “ Yes”
answer from learners s without an opportunity
to explain their thinking.
Line 3: Teacher A then wrote the equation y = x2- 4 on the chalk board
Teacher A used a topic specific example to
followed by; “Now I will show you how to draw the graphs of such functions.
demonstrate how to draw the graph of a
You must pay attention as I demonstrate how it is done. We will start by the
parabola using a table of values. He engages
2
graph of y= x -4 as an example”. He then asked “In the given equation, what
2
is the coefficient of x ”
43
learners by asking them questions.
Line 4: A learner responds; “Two, Sir”. Teacher A does not accept the answer
Teacher A correctly rejects the learner’s answer
given and points at another learner to give the correct answer. He said; “No,
but does not probe further to allow the learner to
not correct! Any one else to help us?”
explain why he/she thinks the answer is “2”.
Line 5: Another learner responds to the question; “ The coefficient of x2 is
Learners are actively involved in the lesson as
one”
they participate in the development of the
lesson.
Line 6: Teacher A accepts the solution given by the learner; “ Yes, the
2
2
coefficient of x in the equation y = x – 4 is one”
Teacher A accepts the learner’s answer but does
not probe the learner to find out how he got the
correct answer. Teacher A uses his content
knowledge here.
Line 7: Teacher A explains further whilst he writes on the chalkboard that if
Teacher A uses content knowledge to provide an
the coefficient of x2 was zero the equation y = x2 -4 would become y = -4
explanation of why the coefficient of x2 is never
which is a constant function and no longer a quadratic function. That is the
equal to zero and he further gives a conceptual
2
2
reason a≠ 0 in the equation y= ax + bx + c. “If the coefficient of x was equal
2
reason why “a” must never be equal to zero in a
to zero, the equation y = x - 4 would become y= 4 in this case which is a
quadratic function. If “a” is equal to zero, the
constant function or just a linear function as you can see that the equation
quadratic function will become linear.
2
y = ax + bx + c becomes y = bx + c. Do you understand?” He wrote down
all these equations on the chalkboard for the learners to see.
Line 8: The entire class responds with a “Yes”
Once again the class responds as a whole-chorus
answer- without the opportunity of individuals
expressing
their
comprehension
of
the
2
coefficient of x in a quadratic function. The
teacher uses inefficient questioning technique to
probe learners. The question usually require
“yes” or “no” type of response (pedagogic
knowledge )
Line 9: Teacher A then proceeds with the lesson:; “Now, to draw the graph of
2
Teacher A demonstrates knowledge of content
y = x – 4 we choose, say, 9 values of x; which will be substituted in the given
on the topic of quadratic functions. He
equation to get the corresponding y- values and form coordinates written in
predominantly uses procedural knowledge to
the form (x;y)” The following x-values were chosen; -4;-3;-2;-1;0;1;2;3;4 and
provide explanations on how to draw the graph
2
were to be substituted in the equation y = x -4 to get y-values. Teacher A
drew a table as shown and explains “For each x-value, we are going to
calculate the corresponding y- value so that we have a set of coordinates
which we are going to plot on the graph paper that I will issue to you.
Remember that coordinates are a set of ordered number pairs of x and y
44
of a parabola.
written in the form (x;y)”
X
-4
-3
-2
-1
0
1
2
3
4
Y
(x;y)
Line 10: A boy learner asked Teacher A why he chose 9 x-values; “Sir, how
The learners are participating in Teacher A’s
many x-values must one choose?”
lesson as they ask questions about the number
of x-values that they can choose in order to
calculate the y values.
Line 11: Teacher A responds to the learner’s question “You can choose as
Teacher A gives learners a procedure of how to
many x- values as you like, the minimum being seven but in your choice, you
choose independent values of x in order to
must include zero and there must be equal number of positive and negative x-
calculate the corresponding y-values from a
values as you can also see from my choice. I chose 9 values just to help you
given quadratic function. The teacher’s content
see how it is done”. He re-wrote the set of numbers that he had chosen earlier
knowledge
and underlined each negative value and its corresponding positive value for
assisted him in providing such an explanation.
the learner to see what he meant by equal number of positive and negative
Teacher A demonstrates both procedural and
numbers. He further explained that for a quadratic function, they need to
conceptual knowledge approach for the question
choose more values because three x-values would not be sufficient to reveal
asked by the learner.
and
knowledge
of
procedures
the curvilinear nature of a quadratic function. “You need more x-values so that
you have enough coordinates to reveal the curvilinear nature of a quadratic
function.”
Line 12: Learner accepts the explanation without any further questioning
Teacher A gave an explanation that seemed to
which can be assumed that he was satisfied with the teacher’s explanation
have satisfied the learner but he does not probe
given.
the learner with a follow-up question to see if he
(the learner) has indeed understood the number
of x-values to choose and how they should be
arranged.
Line 13: Teacher A proceeds to show the learners how to substitute the chosen
2
Teacher
A
accurately
demonstrates
his
x-values in the equation y = x – 4 to get the corresponding y- value which he
procedural
called the independent values of y. He starts by substituting x= -4 and say;
corresponding y-values in order to plot the
“Now that we have the chosen x-values, we need to substitute these values in
points. He uses an algorithm to determine the
to the equation of the function to get the corresponding y-values and be able
corresponding y values for plotting.
to form coordinate points that will be plotted on the graph. So, for x = -4, we
get y= (-4)2 -4 = 16 -4 = 12 and the coordinates are (-4; 12)” The calculated
value and the corresponding coordinates were written in the table drawn in
45
knowledge
for
calculating
the
line 9.
Line 14: A girl learner questions the calculated value of y as 12 and thinks the
The learners are actively involved in the lesson
value should be -12 and say; “No sir, y must be equal to -12”
and
one
of
the
learners
exposes
preconception or misconception
regarding
multiplication
of
her
(line 14)
integers
and
exponents.
Line 15: Teacher A responds to the learner’s rejection of the calculated y-
Teacher A asks for an explanation from the
value by asking the learner a question; “Why do you think so?” as he moved
learner in order to get more insight into the
closer to the learner’s desk.
learner’s thinking; which is a good diagnostic
instructional strategy (pedagogic knowledge)
Line 16: The leaner explains her thinking that displays that she is not able to
The explanation demanded by the teacher of the
2
distinguish or differentiate between -4 and -4x2. She has difficulty perhaps
learner’s thinking and effective pedagogical
in manipulating exponential notations or does not have the knowledge that a
knowledge of the teacher helps to expose the
negative number times a negative number is positive and that ‘squared’ means
learner’s misunderstanding and difficulty with
multiply a number by itself: “ Because -4 squared is -8 and -8-4 = -12” as
exponential notations. At the same time it could
she said
be a language difficulty were “squared” means
multiply by 2 to her.
Line 17: Teacher A responds by correctly explaining that -42 = -4x-4 =16
2
2
Teacher A correctly explains to the learner the
“No, (-4) is like -4x-4 = 16. In general a = a x a. Do you understand?” The
correct concept. His knowledge of the subject
learner appeared not convinced but accepted the explanation as Teacher
matter assisted in giving the correct explanation
proceeded to substitute the next chosen value of x being -3 and said; “For x
and
2
also
how
to
deal
with
learners’
=-3, we have y = (-3) -4 = (-3x-3) – 4 = 9-4 = 5. Can you see that?” [Whilst
misconceptions but to eliminate the learner’s
looking at the girl learner who had shown a misconception about squared
doubts, Teacher A should have given the learner
numbers]
a similar problem for her to solve using the
newly acquired knowledge.
Line 18: Teacher A then allowed learners to do the rest of the calculations of
Teacher displayed good content knowledge on
the corresponding y-values from the remaining x-values on their own whilst
the topic and concepts associated with quadratic
offering support to the learners who were not confident in doing the
functions. He also has shown good knowledge
calculations. He had instructed them to raise their hands in case of difficulties.
of pedagogy by the way in which he supported
He said; “Now that you have an idea of how to calculate the coordinates, I
learners who experienced difficulties with
want you to continue to find the corresponding y-values using the given x-
certain concepts such as calculations involving
values on own. If you experience problems, raise your hand and I will come
exponential notations (line 16) and choosing x-
and assist you.” Learners were randomly pointed at to give the corresponding
values (line 10) during the lesson. Teacher A
y-value from a given x-value and ultimately a table as shown on the next page
was observed assisting them as he moved
was formed.
around (line 14).
46
X
-4
-3
-2
-1
0
1
2
3
4
Y
12
5
0
-3
-4
-3
0
5
12
(x;y)
(-4;12)
(-3;5)
(2;0)
(-1;-3)
(0;-4)
(1;-3)
(2;0)
(3;5)
(4;12)
Line 19: Teacher A goes on to explain how to choose a suitable scale to draw
Teacher
A
displays
knowledge
(content
the graph once coordinate points have been calculated. “Now that we have all
knowledge) of how to choose the appropriate
the required coordinate points, we can now plot the points on the graph
scale (content-specific procedural knowledge)
papers which I am now handing out to you. To plot the points you need an X-Y
to draw the graphs but displays poor questioning
plane which is drawn according to scale” He explained to learners how to
techniques (pedagogic knowledge) to probe
choose a suitable scale. “Use your ruler to draw the X-Y plane with zero at the
learners with regard to assessing whether his
point of intersection. Consider a certain number of squares on your graph to
learners have understood his explanation of
represent I unit of length in each of the axis, for example 4 small squares to
choosing a suitable scale.
represent 1 unit. Calibrate each axis to accommodate the values associated
with that axis. For example, your x-axis should accommodate values from -4
up to 4 units whilst your y-axis should accommodate values from -4 units up
to 12 units. Do you understand?”
Line 20: Class responds to the question in line 19 with a chorus; “Yes’
Teacher A did not ask any individually
directed questions to ensure that overall the
learners
have
a
common
or
shared
understanding of the concepts being taught.
A probing question directed to a specific
learner would have been appropriate.
Line 21: A girl Learner says; “Sir, I find it difficult to choose a suitable
Learners are actively engaged. A learner
scale” On hearing this comment, Teacher A moved closer to the
expresses difficulty in choosing a suitable
learner’s desk.
scale for drawing the graph (learner
difficulty in quadratic functions involving
graphs).
Line 22: Teacher A explains to the learner how to choose a scale using
Teacher A is responding to learner’s
the squares of the given graph paper. He said; “To decide on a scale,
difficulty on selecting a suitable scale. He
choose a number of small squares to represent one unit. As an
repeats the procedure (content specific
example, choose 5 small squares of your graph paper to represent one
procedural knowledge) on how to choose a
unit, then ten small squares will represent 2 units and so on” He
scale while demonstrating on the learner’s
concludes with a non-probing type of question; “Do you understand?”
graph paper that he had earlier issued to
them.
47
Line 23: Learner responds to Teacher A’s question and says; “yes”
The teacher once again did not follow up
with an example-a demonstration by the
learner to see if the learner was able to
choose a suitable scale for graphing. There
is a need for follow-up questions in order
assure learner comprehension.
Line 24: Teacher A in concluding the lesson outlines a procedure that learners
Teacher A uses more of his procedural
should follow when they have to draw the graph of a quadratic function “
knowledge and gives a home work to conclude
Take note that the following steps must be followed in the order shown on the
his post-activity lesson on how to draw the
chalkboard to draw the graph”; he said as he numbered the steps from 1 to 5
graph of quadratic functions. He uses more of a
(see appendix k). He gave learners a home work and said; “For your home
procedural knowledge
2
work, you will draw the graphs of the following functions: (i) y = x -9
(ii) y
2
= x -4x +4”
approach
which is
amenable to the nature of the concept or topic
taught. He also displays sound
conceptual
knowledge in explaining some aspects of the
topic
Table 4.1 Description of Classroom observation for Teacher A
Before summarising Teacher A’s observed PCK, an excerpt of the interviews held with
Teacher A before the lesson will be presented. This will then be used to triangulate the PCK
elements of Teacher A observed during lesson observations.
4.2.2 Pre-Lesson Interviews.
Pre-lesson interviews were done to collect data about the participating teachers’ PCK in
teaching quadratic functions in grade 11. The interview questions and responses (Table 4.3)
were used to collect data on each of the three elements of pedagogical content knowledge of
the two teachers. This was for the purposes of triangulation with the lesson observations
described earlier in the preceding section 4.2.1.
Pre-lesson Interviews with Teacher A
Question Posed
Response of Teacher A
Line 1: Researcher: What are the key concepts in the
The key concepts in this lesson are the x-axis, y-axis
lesson that you are about to teach?
dependent values, coordinates, dependent and
48
independent values
Line 2: Researcher: Draw a concept map illustrating
Well, I will just give you how the lesson will flow from
the sequence you will follow to teach these concepts.
one aspect to the other: formula of equation
values
substitution to get y
x-
choose scale
plot
the points on Cartesian plane
Line 3: Researcher: Does the lesson involve any
I want to show the learners a procedure that they
procedural knowledge?
would use to get coordinate points to be able to draw
graphs of a parabola
Line 4: Researcher: Which teaching strategy will be
The lecture method is appropriate for this lesson
employed to ensure successful delivery of the lesson?
because other methods such as group work would
need that I move around the learners’ desks and that
is not possible given the size of the class”
Line 5: Researcher: Why do you choose such a
The lecture method helps me to save time and it is
teaching strategy?
appropriate to be used given the large size of the
class.
There is no
room
for movement and
rearrangement of the sitting plan for learners to allow
for group work would waste valuable teaching time.
Line 6: Researcher: In your selection of examples for
No, there are no real life examples but I have selected
illustration of the topic or concept, have you selected
a question from the regional grade 11 final
real life examples?
examination paper 1 to be used as an example.
Line 7: Researcher: What is the goal/aim of your
The goal of this lesson is to draw (according to scale)
lesson?
the graph of a given parabola using a table of values.
Line 8: Researcher: Which learners’ prior knowledge
To draw the graph of a parabola, you must be able to
do you regard as important before the above topic can
substitute chosen values of x in to the equation of the
be successfully taught to learners?
function and be able to get coordinates. You also need
to be able to choose a suitable scale to label your axis
Line
9:
Researcher:
What
possible
learners’
I have no idea about the possible misconceptions that
misconceptions do you anticipate regarding this
the learners might have regarding this lesson, but
particular lesson?
should such a situation arise during lesson
presentation, I will deal with it in the classroom. I
mean whatever misunderstanding the learners might
bring to my attention during the lesson, I will assist
the learners.
49
Line 10: Researcher: How would you assist learners
I will give individual attention to the learners who are
who experience difficulties with regard to this
experiencing difficulties with the lessons or I may as
particular lesson (on any topic about quadratic
well refer such learners to the learners who have
function)?
shown good understanding of the topics or I will
repeat the lesson if the situation warranted that. To
repeat the lesson would depend on the number of
learners who need help.
Line 11: Researcher: Have you prepared an
Yes, I always have a class work or home work to
assessment instrument to evaluate whether the goal of
gauge the level of learning that may have taken place.
the lesson was achieved?
Line 12: Researcher: Thank you for your time, we
I thank you.
will meet during lesson presentation.
Table 4.2: Pre-Lesson Interview questions and Responses of Teacher A
4.2.3 Summary of the PCK of Teacher A based on Pre-Lesson Interviews and Lesson
Observations
A summary of Teacher A’s PCK will now be presented based on the lesson observation
description and the conducted interviews. Each of the three elements of Teacher A’s PCK
elements will be presented.

Knowledge of The subject matter of some aspects of quadratic functions
Teacher A demonstrated that he has the required content knowledge in the teaching of
quadratic functions topics during the actual lesson presentation as he was able to present
accurate mathematical facts on the topics on quadratic functions (ref. section 4.2.1). This
display of content knowledge is in line with what transpired during interviews when he was
able to outline the concepts he was about to teach on how to draw the graph of a parabola,
(ref. section 4.2.2, line 1). He said; “The key concepts in this lesson are the x-axis, y-axis
dependent values, coordinates, dependent and independent values” These key concepts were
explained during lesson presentation.
During interviews, he outlined a sequence on how he will teach the concepts of how to draw
the graph of a parabola using a table of values. The concept map that he drew can be
50
presumed to have played a pivotal role in assisting him to present sequential explanations on
the topic that he taught which helped learners to understand the concepts in the topics (ref.
section 4.2.2, line 2). In some instances, he emphasized on procedural knowledge approach to
his lesson on how to draw the graph of a parabola using a table of values where rules and
algorithms were emphasized; he said during interviews; “I want to show the learners a
procedure that they would use to get coordinate points to be able to draw graphs of a
parabola” (ref. section 4.2.1, line 24 and 4.2.2, line 3). In the lesson about how to draw the
graph of a parabola, he displayed skilful execution of the procedures of calculating the ycoordinate with no errors. He further skilfully explained to learners how to choose an
appropriate scale to be able to draw the graph using the calculated coordinate points where
both his procedural knowledge and conceptual knowledge were displayed (ref. section 4.2.1
line 11). The concept of drawing the graph of a parabola demands that the learner should
understand the curvilinear nature of the parabola; otherwise, learners are inclined to join
adjacent points of a parabola in a straight line which distorts the nature of a parabola.
Teacher A’s demographic profile (Table 3.1) further suggests that he should have adequate
knowledge about this topic since he holds a B.Sc degree in Mathematics and statistics and an
Advanced certificate in education in mathematics education. Furthermore, he has 18 years of
experience as a teacher which should have helped him develop knowledge of the subject
matter and specifically, the topics on quadratic functions.

Knowledge of Learners’ Misconceptions and Conceptions
Whilst his content knowledge appeared to be adequate, his knowledge of dealing with
learners’ conceptions and misconceptions appear to be limited. He seldom asked probing
questions to his learners during lessons on how to draw the graph of a parabola which could
have helped him to gain some insight into the learners’ misconceptions (ref. section 4.2.1,
line 16). Though he used oral probing questions during the lesson on the topic of how to draw
the graph of a parabola, most of the questions that he asked were of low order level, requiring
a “Yes” or “No” response which did not assist the teacher in eliciting learner thinking,
learning difficulty and misconceptions if any, and in enhancing teacher-learner discussion
(ref. section 4.2.1, lines 8 and 20). It was only in one instance where Teacher A asked a
learner to explain why that learner thought a particular answer to a given question was not the
correct one (ref. see section 4.2.1, line 15).During this interaction with the learner, it was
51
discovered that the learner had misconceptions and difficulties regarding exponential
notation, multiplication of negative numbers and language difficulty.
Teacher A did not appear to plan his lesson based on some anticipated learners’
preconceptions about the topic that he was about to teach on quadratic functions. He
acknowledged during interviews that he did not know the possible preconceptions or
misconceptions that his learners might have as he taught them how to draw the graph of a
parabola (ref. section 4.2.2, line 9). He said; “I have no idea about the possible
misconceptions that the learners might have regarding this lesson, but should such a
situation arise during lesson presentation, I will deal with it in the classroom. I mean
whatever misunderstanding the learners might bring to my attention during the lesson, I will
assist the learners” Misconceptions and learner difficulties were discovered when Teacher A
asked probing questions and allowed learners to do exercises on their own or learners
exposing their own misconceptions or difficulties themselves during the lesson activities (ref.
section 4.2.1, line 16).Based on the observations just given, Teacher A’s knowledge of
learners’ difficulties and misconceptions can be presumed to be limited on the topic of
quadratic functions. But he adequately addressed any difficulties his learners experienced.

Knowledge of Teaching Strategies
On the use of teaching strategies, he preferred using the telling method to deliver his lessons
whilst occasionally using oral probing questions to determine learners’ knowledge. Asked
during the interviews about which teaching method he would use, he said; “The lecture
method is appropriate for this lesson because other methods such as group work would need
that I move around the learners’ desks and that is not possible given the size of the class” He
combined the lecture method with discussion method where learners were guided on how to
draw the graph of a parabola.
He would explain a concept and then demonstrate how to use a specific procedure associated
with solving problems in quadratic functions (ref. see section 4.2.1, line 9). Learners were
actively involved in his lessons; some asked questions which the teacher tried to address. His
apparent deficiency was primarily in lack of follow up examples to ensure understanding on
the part of learners. Others were poor questioning technique such as asking questions not
directed to a specific learner, which resulted in chorus answers from the entire class He
however used some effective teaching strategies such as actively involving learners as he
52
delivered the lesson, handling of learners’ questions and effective use of the chalkboard in
drawing and explaining graphs. Quadratic functions demand that individual learners be able
to carry out certain operations and skills of graph construction and interpretation on their own
and teachers need to know that individual learners are able to do so, through proper inclusive
assessment procedures and questioning techniques. Furthermore, home work was seen by the
teacher as an extended opportunity for the learners to learn the concepts that he taught them
(ref. see section 4.2.1, line 24). Learners were given an opportunity to work on their own
which helped the teacher to diagnose learners’ misconceptions and difficulties and also
enhanced their learning opportunities on the topics. Overall the teacher’s knowledge of
teaching strategies was regarded as inadequate on the topic of quadratic functions.
4.2.4 Description of Classroom Observations for Teacher B
DESCRIPTION OF LESSON OBSERVATION TEACHER B (LESSON 1)
CATEGORISATION/THEMES
Condition of the classroom
There were 12 female learners and 23 male learners The classroom represented a
The classroom displayed a fairly
well organised learning environment with double desks neatly arranged in four
conducive learning environment
columns and six rows as viewed from the position where the teacher was standing.
though poorly resourced.
Though some window panes were missing in some of the windows, the classroom
was fairly warm. A chalkboard and a teacher’s table with a box of white chalks
and a duster were available. There was sufficient room for the teacher to move
around.
LESSON OBSERVATION: GRADE 11
CATEGORISATION/THEMES
Topic: How to draw the graph of a quadratic function
Line 1: Teacher B: Introduced the topic for the lesson about quadratic functions;
2
Introduces the topic by referring learners
“Last year, in grade 10 you drew graphs of y = ax , which represent the simplest
back to the prerequisite knowledge taught
form of a quadratic function and I want us to quickly review certain facts about
the previous year. Pedagogical Knowledge
this type of graph”. He proceeds and asks learners; “Can someone tell me the
of the teacher is helping to set the scene for
2
shape of the graph of y = ax ?” as Teacher B points to one of the learners for an
the new lesson as the teacher checks what
answer.
do the learners know and have that will help
facilitate access to the new knowledge.
53
Line 2: The learner responds correctly to the question; “ It is cup shaped or bell
The teacher engages learners through recall
shaped”
type questions and they appear to have the
required pre-requisite prior knowledge.
Line 3: Teacher B accepts the learner’s answer to the question and proceeds to ask
Teacher B continues to use recall type
another question “Yes the shape is not like that of a straight line graph. Now,
questions to assess learners’ pre-requisite
2
under which conditions is a graph of y = ax cup shaped and under which
knowledge and conceptions.(Pedagogy)
conditions is it bell shaped?” and points at another learner to give the answer.
Line 4: A Learner responds to the question correctly by explaining the conditions
The learners demonstrate that they still have
under which graphs are cup shaped or bell shaped. The learner said; “It will be
knowledge of facts about the graph of a
cup shaped when it has a minimum turning point and bell shaped when it has a
simple quadratic function from grade 10.
maximum turning point”
This diagnostic questioning might assist the
teacher in knowing that the learners are
aware that a quadratic function is not a
straight line graph at least.(Pedagogy)
Line 5: Teacher B appreciates the learner’s response to the question and continues
Teacher B acknowledges positively the
to clarify the conditions for a minimum turning point or a maximum turning point
learner’s response and proceeds to provide
of the parabola using the cup and bell shape concept. He said; “Well done! When
conceptual knowledge on the topic under
2
the coefficient of x is positive the graph has a minimum or is cup shaped and
2
discussion-. He has knowledge of the
when the coefficient of x is negative, the graph has a maximum and is bell
subject matter and has used the “cap” and
shaped” He then proceeded with the lesson of the day and said; “To draw the
“bell” concept to illustrate minimum and
graph of a parabola you need to choose 7 or 9 integer x-values which must
maximum turning points. These are real life
include zero”
examples (ref. line 4).
Line 6: Learner asks a question; “What do you mean by integer values?”
Learners are attentive to what teacher B is
saying during the lesson as they ask relevant
questions.
Line 7: “ They are whole numbers”; Teacher B explains to the learner what he
Teacher B displays knowledge of the subject
meant by integer values and gave an example of such set of chosen x-values
matter and checks for understanding of the
which are symmetrical about zero; namely; -4;-3;-2;-1;0;1;2;3;4. He concludes his
explanation from the learner through asking
explanation by asking the learner to choose his own set of x-values; “Can you
a question (Pedagogy).
choose your own x-values”
Line 8: The learner respond to teacher B’s question and say: “Yes, -6;-5;-4;-3;-2;-
The learner understands the teacher’s
1;0;1;2;3;4;5;6;”
explanations and can apply his newly
gained knowledge about how to choose xvalues.
54
Line 9: Teacher B acknowledges the learner’s choice of x-values and says; “That
2
Teacher B displays knowledge of content
is good.” He then chooses y = x -9 as the function to be used to illustrate how to
and pedagogy by the way he explains and
use a table of values to draw the graph of parabola. He draws a table as shown
handles learner question and answer and
below so that the corresponding values of y can be calculated.
how he chooses the appropriate example to
illustrate the topic of the day (Line 5, 6 and
X
-4
-3
-2
-1
0
1
2
3
4
8).
Y
(x;y)
Line 10: “For x = -4; y = (-4)2 -9 = 7”; he writes the sum on the chalkboard and
He involves learners in the calculation of
then tells the learners to also do their own calculations using the approach that he
the corresponding values of y using the
had just demonstrated with x= -4. After a while, all the x-values were substituted
equation of the function (pedagogy). He
and the corresponding y-values were obtained. The table was then completed as
applies his procedural knowledge frequently
punctuated conceptual knowledge (ref. Line
X
-4
-3
-2
-1
0
1
2
3
4
Y
7
0
-5
-8
-9
-8
-5
0
7
5) to provide explanations to the learners.
Learners were seen to be enthusiastic in
doing the calculations on their own. It can
(x;y)
(-4;7)
(-3; 0)
(2;-5)
(-1;-8)
(0;-9)
(1;-8)
(2-5)
(3; 0)
(4;7)
be presumed to mean that they made
meaning of what they were doing.
shown:
Line 11: Teacher B explains to the learners how to choose a suitable scale to draw
Teacher B explains to learners how to
the parabola on a graph paper: “Now that we have the coordinates, we need to
choose a scale which is an important part
choose a suitable scale to use on your graph paper. Choose 1unit: 10 mm. If 1
for the learners to be able to draw the graph.
unit: 10 mm, how many millimetres will represent 2 units; 3 units etc?” He further
He uses procedural knowledge approach to
says to the learners; “I hope you still remember what coordinates are?
deliver his explanation. The teacher has
appropriate subject matter knowledge and is
able to involve his learners (Pedagogy) as
he explains how to choose a scale.
Line 12: A learner responds to Teacher B’s question regarding coordinates and
The learners have the appropriate pre-
says; “ A set of ordered number pairs x and y”
requisite knowledge.
Line 12: Before Teacher B responds to the answer given on coordinates, another
The learners are actively involved in the
learner responds to the question on the scale; “2 units will be 20 mm and 3 units
lesson. Teacher B asks questions that
will be 30 mm”
engage learners in expressing their ideas
about
the
(Pedagogy).
55
topic
under
discussion
Line 13: Teacher B appreciates the learner’s response to the question and says;
Teacher
“Excellent”. He further gives learners information about how to use their chosen
rewarding them with appropriate words
scale; “You must calibrate both the X and Y axis with the same scale. In your
such
graph paper, one small square is approximately 2 mm so 5 small squares will be
proceeds to give a conceptual meaning of
equivalent to 10 mm” Learners were then told to draw their graphs on the graph
how to choose a scale using the graph
paper while Teacher B was moving around their desks to offer assistance.
papers and he further allows the learners to
as
B
encourages
“excellent”
learners
(pedagogy).
by
He
do the graphing themselves as he moves
around to offer support. A good display of
the knowledge of pedagogy.
Line 14: A learner informs the teacher of the difficulty she is experiencing in
The learners are not inhibited to expose
plotting the graph; “Sir; I cannot plot the point (-3; 0). Show me how it is done”
their difficulties to teacher B. They are
encouraged by the positive way in which he
handles their questions. Good display of
teaching strategy.
Line 15: Teacher B provides a conceptual way of identifying the coordinate point
Teacher B uses conceptual knowledge
on the graph. “I expected this type of difficulty from some of you. Most learners
approach to deal with learners’ topic-
are unable to plot the correct coordinate points because of failure to read the x
specific difficulty. The teacher anticipated
and y values correctly. X-values are read along the vertical lines whilst y-values
such difficulty in plotting correct coordinate
are read along the horizontal lines. As an example, the (-3; 0) is at x= -3 on the x-
points. He uses the chalk board to illustrate
axis since y=0 along the x-axis. The point (2; 5) is at the intersection of the point
where to put coordinate points on the X-Y
x=2 and y =5 see the sketch on the chalkboard.’ He explained while drawing the
plane. He displays sound mathematics
sketch below on the chalk board and also drawing horizontal and vertical dotted
knowledge on drawing the graph of a
lines to illustrate the exact position of the coordinates.
parabola and provides accurate and logical
explanations using both conceptual and
(-3;0)
Y
-3
procedural knowledge.
2
-5-
X
(2;-5)
Line 16: Learner accepts explanation; “ I understand, I will plot the other points”
The teacher gave a satisfactory explanation
and a solution to the learner’s difficulty.
Line 17: Teacher B indicates to his learners that he will give them assistance as he
Teacher B gave his learners home work;
moves around their tables and concludes this lesson with a home work. He said; “I
which serves to offer the learners an
will be moving around your desks to assist those who need assistance. For your
extended opportunity to learn how to draw
home work you will draw the graphs of the following:
2
(i) y =x -4
2
x -2x”. The lesson of the day was concluded.
(ii) y =
the graph of a parabola using a table of
values. A good display of pedagogy.
Table 4.3 Description of Classroom observations for Teacher B
56
4.2.5 Pre-lesson interviews with Teacher B
Pre-lesson interview excerpt of Teacher B will be indicated before a summary of his PCK as
observed in the lesson observations (section 4.2.1) is discussed as was done with Teacher A.
Question Posed
Response of Teacher A
Line 1: Researcher: What are the key concepts in the
The key concepts in this lesson are the x-axis, y-axis,
lesson that you are about to teach?
coordinates, and scale.
Line 2: Researcher: Draw a concept map illustrating
Well, my lesson is planned as shown
the sequence you will follow to teach these concepts.
Choose values of X
Y
substitute in equation
form coordinates points
choose
get
scale
coordinate points on graph paper
Line 3: Researcher: Does the lesson involve any
In a way, the lesson involves both conceptual
procedural knowledge?
knowledge and procedural knowledge. Knowing why
certain things are done the way they are done and
what must be done first before the next step.
Line 4: Researcher: Which teaching strategy will be
I will first explain to the learners how to draw this
employed to ensure successful delivery of the lesson?
graph then arrange the learners in groups of say five
to six learners and then demonstrate how to draw this
graph should be drawn using a table of values.
Line 5: Researcher: Why do you choose such a
Demonstration and lecture method will allow me to
teaching strategy?
give the learners a guided practice whilst group work
will enable them to learn from each other.
Line 6: Researcher: In your selection of examples for
Well, in a way there is real life examples if you
illustration of the topic or concept, have you selected
consider that the shapes of the quadratic functions can
real life examples?
be described in terms of either cup shaped or bell
shaped.
Line 7: Researcher: What is the goal/aim of your
The goal of this lesson is to draw (according to scale)
lesson?
the graph of a given parabola using a table of values.
Line 8: Researcher: Which learners’ prior knowledge
Knowledge of the shape of the simplest quadratic
do you regard as important before the above topic can
function y = ax2. This knowledge allows learners to
be successfully taught to learners?
form a good link between the expanded forms of the
57
quadratic function y = ax2 + bx + c and y = ax2
Line 10: Researcher: What possible learners’
Generally, most learners find it difficult to choose an
misconceptions do you anticipate regarding this
appropriate scale for a graph and secondly, they find
particular lesson?
it difficult to plot points correctly as they have a
problem in reading the values of the system of axis in
the X-Y plane.
Line 11: Researcher: How would you assist learners
I usually give guided practice first for all learners and
who experience difficulties with regard to this
having arranged them in groups helps me to move
particular lesson (on any topic about quadratic
around each group and offer explanations to
function)?
struggling groups. If all fails, I usually repeat the
lesson during extra lesson times.
Line 13: Researcher: Have you prepared an
Yes, I always have a class work or home work to
assessment instrument to evaluate whether the goal of
assess the level of learning that took place.
the lesson was achieved?
Line 14: Researcher: Thank you for your time, we
You are welcomed.
will meet during lesson presentation.
Table 4.4: Pre-Lesson Interviews and Responses of Teacher B
4.2.6 Summary of the PCK of Teacher B Based on Lesson Observations and Pre-lesson
Interviews
A summary of Teacher B’s PCK based on the description of the observed lessons and the prelesson interview is presented as follows.

Knowledge of The Subject Matter
Teacher B displayed adequate knowledge of the subject matter as he presented lessons on
quadratic functions in grade 11. Key concepts to be taught on the lesson about how to draw
the graph of a parabola were clearly articulated during pre-lesson interviews as he provided
them correctly without even referring to the text book. He said “The key concepts in this
lesson are the x-axis, y-axis, coordinates, coordinates and scale” He combined procedural
knowledge and conceptual knowledge in the presentation of his lessons to his learners (ref.
section 4.2.4, lines 5, 10 and 13). Teacher B displayed comprehensive knowledge, both
conceptual and procedural as well as flexibility in aspects of quadratic functions that he
58
taught. Conceptual knowledge approach was seen to be predominant in his presentation as he
emphasised and insisted on learners’ comprehension and understanding of the concepts rather
than mere routine application of rules and algorithms of the topics taught. He avoided that his
learners memorized procedures without any understanding. The explanations that he provided
during the lessons were seen to help his learners to access the topics that he taught under
quadratic functions. They were able to do exercises associated with the topic correctly (ref.
section 4.2.4, line 15). He showed flexibility in his explanations since he was able to provide
two different explanations on the same concept so as to make the topic under discussion
accessible to his learners (ref. section 4.2.5, line 11 and 13).
Furthermore, Teacher B has the required professional qualifications to teach quadratic
functions in grade 11 (Table 3.1) and has twelve years of experience as a mathematics
teacher.

Knowledge of Teaching Strategies
Teacher B demonstrated that he has insufficient knowledge of teaching strategies to teach
quadratic functions in grade 11. Though he engaged learners with questions that assisted
learners to express their mathematical thinking and knowledge (ref. see section 4.2.4, lines 4
and 8); it has to be mentioned that some of his questions were of low order level because they
were mainly recall type of questions (section 4.2.4, line 2); but these could be justified in
terms of accessing base knowledge before introducing new topic. Furthermore, as he asked
questions, he pointed at learners who raised their hands and mostly left out those seemed to
know the correct answers only and did not engage those who did not. It can only be presumed
that he focused on getting the correct answers from the learners who appeared to know the
answers. It would be good to get answers from any of the learners so as to pick up
misconceptions and conceptions that learners might have regarding the topic.
He mainly used the lecture method where he was observed as the main imparter of
information to learners to present his lessons. As he used this method, he would explain
concepts to the learners in a way that assisted the learners to access the topic that he was
presenting on quadratic functions. Group work was sometimes used by Teacher B in the
delivery of his lessons to assist learners who were seen to be having difficulties with the
topics. He also evaluated the learners’ prior knowledge on the topic through oral questioning
before presenting the new lesson (ref. section 4.2.4 line 1 and 3). He occasionally asked
59
learners questions that required learners to express their mathematical thinking or knowledge
on the topic. His encouraging words such as “well done” or “excellent” encouraged learners
to be involved as they were able to reveal their own misunderstanding without being coerced
or inhibited (ref. section 4.2.4, line 14). Teacher B ensured that all the learners felt confident
and safe enough to pose questions or answer them rightly or wrongly.
He used the chalkboard to the advantage of all learners as they were able to have a clear view
of what was written on the chalkboard. The examples he used were relevant to the topics on
quadratic functions. His main source of information was the school mathematics textbook
used by his learners as well. It encouraged learners to also use the book to their advantage in
knowing the page numbers where additional practice questions on each section on quadratic
functions can be referred if need be.

Knowledge of Learners’ Misconceptions and Conceptions
While his knowledge of the subject matter appeared to be adequate, his knowledge of
learners’ conceptions with regard to the topic seemed to be inadequate. There were few
instances where he came to class with an anticipation of the type of difficulties that his
learners might experience about a topic (ref. section 4.2.4, line 15). During the pre-lesson
interviews on the lesson about how to draw the graph of a parabola; he clearly articulated
possible learners’ difficulties and said; “Generally, most learners find it difficult to choose an
appropriate scale for a graph and secondly, they find it difficult to plot points correctly as
they have a problem in reading the values of the system of axis in the X-Y plane” There were
few instances of course; where Teacher B asked probing or follow-up questions during the
lesson, where he also discovered learners’ difficulties and addressed them confidently and
adequately(ref. section 4.2.4, line 5). These were rare.
4.2.7 Lesson Plan Analysis of the participating Teachers
This section focuses on the results of the analyses of the lesson plans prepared by the teachers
in teaching quadratic functions. They were analysed to get more information about the
teachers’ PCK in terms of knowledge of the subject matter; knowledge of teaching strategies
and knowledge of learners’ misconceptions and conceptions. Data about these three elements
were obtained from the lesson plan analysis using a set of guiding questions as shown in table
4.5 below. The two teachers used a common lesson plan template. Asked why they used a
60
common lesson plan template they said; “We received these templates from the curriculum
implementer of the district and all schools in the district are expected to use these templates
for mathematics preparations.”
ELEMENT OF PCK
a.
Knowledge of subject
matter
CHECKED IN THE PREPARATION



b.
Knowledge of teaching
strategies

Is the teaching strategy to be used stated in the
preparations?

Are alternative teaching strategies to be used during
the lesson reflected in the preparations?

Are examples to be used during the lesson indicated
in the lesson preparation?
Does the preparation reflect possible
misconceptions that will be addressed during the
lesson?
Does the preparation reflect the required learners’
prior knowledge before the start of the new topic?
Are possible learners’ difficulties reflected in the
preparations?
Is an assessment instrument indicated in the
preparations?
Is the goal of the lesson clearly stated in the
preparations?

c.
Knowledge of learners’
conceptions
Are key concepts to be taught during the lesson
indicated in the preparations?
Does the preparation indicate possible procedures
to be taught to the learners?
Does the lesson preparation reflect accurate
concepts and procedures associated with the topic
on quadratic functions?




Table 4.5: Guiding for Lesson Plan Analysis
An analysis of each teacher’s lesson plan was done and findings for both teachers on each of
the three elements of pedagogical content knowledge presented.
4.2.7.1 Lesson Plan Analysis of Teacher A
KNOWLEDGE OF THE SUBJECT MATTER
Guiding question
Observation made
Categorization/Theme
a) Are key concepts on quadratic
Some of the key concepts on how
Teacher’s preparation reflects
functions to be taught indicated in
to draw the graph of a quadratic
knowledge of the subject matter
the lesson plan?
function were indicated in the
including concepts for the topic
61
lesson plan
b)
Does the preparation indicate
The preparation indicates that the
Procedures to be taught were
possible mathematical procedures
teacher
indicated but not all of them
for the topic to be taught?
calculate the y-values and complete
will
help
learners
to
a given table of values.
c) Does the lesson preparation
The lesson plan reflects accurate
Teacher A has good knowledge of
reflect accurate concepts associated
concepts associated with quadratic
the subject matter
with
functions and especially on how to
the
topic
on
quadratic
functions?
draw the graph of a parabola.
KNOWLEDGE OF TEACHING STRATEGIES
Guiding Question
Observation Made
a) Is the teaching strategy to be
The
used stated in the lesson plan?
teacher’s activities as well as the
demonstration methods to teach the
learners’ activities. The teacher
rules.
lesson
plan
Categorization/Theme
reflects
the
Teacher A used the lecture and
will first do an exposition of how a
graph of a quadratic function is
drawn. Learners’ will calculate the
y-values from the chosen x-values
b)
Are
alternative
teaching
The lesson plan does not have an
Teacher A prefers to use one
strategies to be used during the
alternative
teaching method in a lesson. Does
lesson reflected in the preparation?
indicated
teaching
strategy
not demonstrate sufficient in
flexibility in his teaching approach
Guiding Question
Observation Made
Categorization/Theme
c) Are examples to be used during
Examples to be used during the
Teacher A prepares his examples in
the lesson indicated in the lesson
lesson were indicated in the lesson
advance before the lesson. Does
preparation?
2
plan. For this lesson, y= x -4 was
not use real-life examples to make
used as the main example. No
concepts meaningful to learners.
everyday life examples
KNOWLEDGE OF LEARNERS’ CONCEPTIONS AND MISCONCEPTIONS
Guiding Question
Observations Made
62
Categorization/Theme
a) Does the preparation reflect
The preparation does not reflect
Teacher A does not show that he is
possible misconceptions that will
possible learners’ misconceptions
aware of possible misconceptions
be addressed during the lesson?
that will be addressed during the
to be addressed.
lesson.
b) Does the preparation reflect the
Yes, the required learners’ prior
Teacher A knows the required prior
required learners’ prior knowledge
knowledge was reflected in the
knowledge of learners before
required before the new topic?
lesson plan for the topics to be
teaching a new topic on quadratic
taught on quadratic functions.
functions.
learners’
No, possible learners’ difficulties
Teacher A has no knowledge about
in
were not indicated in the lesson
the possible difficulties in the topic
preparation?
plan.
on quadratic functions
d) Is an assessment instrument
Yes, a home work was given as an
Teacher A knows that learning
indicated in the preparations?
instrument
must be assessed after each lesson
c)
Are
difficulties
possible
reflected
the
to
assess
learning.
2
Expand y = x -9 was given as a
and uses past examination
home work.
questions to assess his learners.
e) Is the goal of the lesson clearly
The lesson plan did not reflect the
The goal or objective of a lesson
stated in the preparation?
goal of the lesson.
must always be stated for the sake
of the learners and teachers as well,
but this was not the case.
4.2.7.2 Lesson Plan Analysis of Teacher B
KNOWLEDGE OF THE SUBJECT MATTER
Guiding question
Observation made
Categorization/Theme
a) Are key concepts on quadratic
Key concepts on how to draw the
Teacher’s preparation reflects
functions to be taught indicated in
graph of a quadratic function were
knowledge of the subject matter
the lesson plan?
indicated in the lesson plan
including concepts for the topic
b)
The preparation mentions that the
Procedures to be taught indicated
possible mathematical procedures
teacher
in the plan.
for the topic to be taught?
calculate the y-values, plot the
Does the preparation indicate
will
help
learners
to
points and also completion of a
given table of values.
63
c) Does the lesson preparation
The lesson plan reflects accurate
Teacher B has good knowledge of
reflect accurate concepts associated
concepts associated with quadratic
the subject matter.
with
functions and especially how to
the
topic
on
quadratic
functions?
draw the graph of a parabola.
KNOWLEDGE OF TEACHING STRATEGIES
Guiding Question
Observation Made
a) Is the teaching strategy to be
The
used stated in the lesson plan?
teacher’s activities as well as the
of teaching. It may be due to the
learners’ activities. The teacher
fact that he wanted to use that
will first do an exposition of how a
method for the observed lessons.
lesson
plan
Categorization/Theme
reflects
the
Teacher B used the lecture method
graph of a quadratic function is
drawn. Learners will be grouped
and then calculate the y-values
whilst
the
teacher
will
offer
assistance
b)
Are
alternative
teaching
Group work is mentioned as an
Teacher B indicated only one
strategies to be used during the
alternative teaching strategy. For
teaching method in a lesson though
lesson reflected in the preparation?
what? Group work in itself is
he used group work as well. This
meaningless without a particular
might suggest that he only wanted
purpose
to use the lecture method only in
that lesson.
Guiding Question
Observation Made
Categorization/Theme
c) Are examples to be used during
Examples to be used during the
Teacher B did not reflect the
the lesson indicated in the lesson
lesson were not indicated in the
examples to be used on the lesson
preparation?
lesson plan but there was an
plan and no real life example was
2
example. For this lesson, y= x -9
used to enable concept
was used as the main example.
understanding.
KNOWLEDGE OF LEARNERS’ CONCEPTIONS AND MISCONCEPTIONS
Guiding Question
Observations Made
Categorization/Theme
a) Does the preparation reflect
The preparation does not reflect
Teacher B may not have had
possible misconceptions that will
possible learners’ misconceptions
misconceptions to be addressed or
be addressed during the lesson?
that will be addressed during the
is not used to the practice of
lesson.
including knowledge of learner
preconceptions in the lesson plan.
64
b) Does the preparation reflect the
Yes, the required learners’ prior
Teacher B knows the required prior
required learners’ prior knowledge
knowledge was reflected on the
knowledge of learners before this
required before the new topic?
lesson plan. Learners should be
lesson on how to draw the graph of
2
able to the graph of y = ax as
a parabola.
taught in grade 10 for what topic?
c) Are possible learners’
No, possible difficulties were not
Teacher B may have ignored
difficulties reflected in the
written on the lesson plan.
indicating possible difficulties in
preparation?
the topic. However during lesson
observations; he anticipated and
identified possible learners’
difficulties.
d) Is an assessment instrument
Yes, a home work was given as an
Teacher B knows that learning
indicated in the preparations?
instrument to assess learning on
must be assessed after each lesson
how to draw a graph of a parabola
and uses the appropriate
using a table of values. The
instrument. Past examination
following equations were given: y
questions were used as home to
2
2
= x -4 and y = x – 2x
assist learners to learn how to draw
the graph of a parabola.
e) Is the goal of the lesson clearly
The outcome was indicated that the
The goal of the lesson must be
stated in the preparation?
learners must be able to
given for each lesson to assist the
investigate, analyse, describe and
teacher and the learners to remain
represent the function.
focused.
Based on the presented lesson plan analysis, a joint summary of the PCK of the two
participating teachers; Teacher A and Teacher B is given focusing on each of the three
elements of pedagogical content knowledge as displayed in the lesson plans.

Knowledge of The subject Matter
On analysing the two participating teachers’ lesson plans (section 4.2.7.1 and 4.2.7.2) it was
found that both teachers’ lesson plans contained accurate information about the concepts on
the topics on quadratic functions that they were about to teach which may imply that they
have knowledge about the content of the topics they were about to teach. Both Teacher A and
Teacher B did not indicate possible rules and algorithms which could be used to solve
problems associated with the topics that they intended to teach they mentioned that there
were procedures to be taught to learners during the pre-lesson interviews (see section 4.2.5).
Lesson plans contained accurate information about the topics on quadratic functions.
65
The two teachers’ lesson plans displayed that they both have adequate content knowledge on
the topic of quadratic functions.

Knowledge of Teaching Strategies
The analysis of the lesson plans of Teacher A and Teacher B revealed that they both have
inadequate knowledge of teaching strategies to teach quadratic functions (section 4.2.1 and
4.2.4). They both relied heavily on the lecture method of teaching which puts them as the
main imparters of the information about the topics on quadratic functions to their learners.
The lecture method appeared to be adequate for presenting the topic on how to draw the
graph of a parabola; given that it is fairly new to most learners. Teacher B occasionally varied
the lecture method with group work as indicated in his lesson plan which was also observed
during lesson presentation. How the teacher would approach the lesson activities was
indicated on the lesson plan where the teacher’s activities as well as the learners’ activities
were indicated. Specifically, Teacher A indicated in his lesson plans that he would first
explain how a graph of a parabola is drawn and then guide learners to calculate y-values from
the chosen x-values. Teacher B’s teaching strategy was to first revise work that was done
previously, explain and demonstrate how to use a procedure to calculate for the y- values and
then organise learners in to groups and then monitor and assist learners in the various groups.
The two teachers’ knowledge of teaching strategies of teaching quadratic functions was seen
as inadequate since they both relied on one teaching method, namely; the lecture method, to
teach.

Knowledge of Learners’ Misconceptions and Conceptions
From the lesson plan analysis, it was noted that Teacher A did not reflect possible learners’
misconceptions on the topic on how to draw the graph of a parabola using a table of values
but he did indicate possible difficulties that his learners would experience. The difficulties
that he anticipated (as per lesson plan) his learners would experience were about choosing the
appropriate scale (appendix o). This somewhat contradicted what he said during lesson
interviews when he said; “I have no idea about the possible misconceptions that the learners
might have regarding this lesson, but should such a situation arise during lesson
presentation, I will deal with it in the classroom. I mean whatever misunderstanding the
learners might bring to my attention during the lesson, I will assist the learners” There was
neither indication of possible learners’ difficulties nor misconceptions on Teacher B’s lesson
66
plans, but during interviews as well as during lesson observations, learners’ difficulties were
anticipated, identified and addressed (section 4.2.4, line 15 and section 4.2.6). It can be
presumed that the inconsistency of indication of learners’ misconceptions and learners’
difficulties during interviews, lesson presentation and lesson plans, is as a result of the
teachers not used to being asked to provide such information in their lesson plans and
secondly not perhaps having adequate and systematic knowledge about learners’
preconceptions regarding the topic on quadratic functions.
4.2.8 Development of the Pedagogical Content Knowledge of the Teachers
The study further wanted to find answers to the question of how the teachers may have
developed the PCK that they were using in teaching quadratic functions. Interviews were held
with each teacher as depicted in the tables shown below (Table 4.6 and 4.7) which display
both the question posed and the responses to each question.
4.2.8.1 Interviews on Development of the Pedagogical Content Knowledge of Teacher A
QUESTION POSED
RESPONSE BY PARTICIPANT
Did you receive any special training as a
Yes, after my initial teacher training (a B.Sc degree, majoring
mathematics teacher after your initial teacher
in Mathematics and Statistics) in secondary teaching), I
training?
registered and passed an advanced certificate (ACE) in
Mathematics Education with the University of South Africa.
[Previous question continued]
This has helped me to master the content of most of the topics
such as functions and others that I am supposed to teach my
grade 10 to 12 learners.
Do you attend workshops that focus on teacher
Yes, in our district, curriculum implementers, who are
development?
specialists in mathematics teaching and employed at the district
level, always arrange content and method workshops in
mathematics that focus on how to teach certain topics such as
quadratic functions and also about the content of some topics.
Have you ever observed any of your colleagues
Yes I do. In most cases, during the IQMS (Integrated Quality
when they are teaching a mathematics lesson?
Management System) development circles, we are bound to
observe our peers to help in their assessment of the teaching
standards that the department has set for educators to display
as they teach. Doing such observations has helped to discover
some mistakes that my colleagues do when teaching. These
include teaching without a clearly stated goal, checking
learners’ prior knowledge and not giving feedback after an
67
exercise.
What have you learnt from such observation?
I have learnt certain techniques of how to develop a lesson and
how to conclude a lesson. I have also learnt how to deal with
classroom control especially when there are learners who
behave in an unacceptable way during lessons.
What are your qualifications as a mathematics
I have already told you that I hold a B.Sc degree specialising in
teacher?
Mathematics and Statistics and I also have an Advanced
Certificate in Education specialising in the teaching of
mathematics.
For how long have you been a mathematics
I have been teaching Mathematics for the past 18 years and
teacher?
mostly in grade 11 and 12 though I can also teach grade 10
classes.
How often do you review the lessons that you have
I always make a review of the lessons that I have taught so as to
taught?
check how well or badly I have presented the lesson. This helps
me to plan better for the next lesson and identify areas where I
need to improve my presentation.
Thank you for your time!
You are welcome!
Table 4.6: How Teacher A may have developed the PCK
From the responses as noted in table 4.5 above including table 3.1 in chapter 3, Teacher A
holds a B.Sc degree, majoring in Mathematics and Statistics and also holds an Advanced
Certificate in Education (ACE) specialising in the teaching of Mathematics. Such
qualifications may have helped shape Teacher A’s content knowledge of quadratic functions.
He has 18 years of experience as a mathematics teacher. Teacher A has continued to study for
an advanced certificate in Education (ACE) with a higher institution of learning in South
Africa. Such training may have helped shape the teacher’s content knowledge of mathematics
during the initial training as well as during further studies.
He has developed a practice of reviewing his lessons when after each lesson in order to
check how well or badly he has presented the lesson and this helps him to plan better for the
next lesson. The teacher attends workshops which focus on how certain mathematics topics
including how the topic on quadratic functions should be taught to learners. He sometimes
68
observes his colleagues as they present lessons to learners and has learnt techniques on how
to develop and conclude lessons and manage classrooms. In summary, initial teacher training,
further studies, attending workshops, review of his own lessons and peer observations have
helped shape Teacher A’s pedagogical content knowledge. Although he has eighteen years
of teaching mathematics experience he does not seem to have adequate learner knowledge on
conceptions or potential learning difficulties on the topics of quadratic function that he taught
grade 11 class (section 4.2.1). (Knowledge of these help the teacher to prepare suitable
explanations that will assist learners in the topics on quadratic functions-discussion.)
4.2.8.2 Interviews on Development of the Pedagogical Content Knowledge of Teacher B
QUESTION POSED (TEACHER B)
RESPONSE TO THE QUESTION
Did you receive any special training as a
Yes, I managed to obtain Mathematics I and II at the
mathematics teacher after your initial teacher
University of South Africa and also registered for an
training?
Advanced Certificate in Mathematics with the University
of Limpopo
Do you attend workshops that focused on teacher
Yes, there are workshops that are arranged by the region
development?
that are compulsory to be attended by all mathematics
teachers
What have you gained from attending such
One workshop that I attended focussed on content matter
workshops?
such as functions, data handling, transformation of
functions and probability
Have you ever observed your colleagues when
Yes,
with
other
colleagues
in
the
Mathematics
they were teaching a mathematics lesson?
Department, we have agreed to observe each other and
then do constructive criticism on how the colleague may
have presented the lesson.
What have you learnt from such observations?
I have learnt different teaching strategies that can be
incorporated in one lesson.
What are your qualifications as a mathematics
I hold a National Professional Diploma in Education,
teacher?
obtained at the University of North West. I also have an
advanced certificate in Education, specialising in
Mathematics Education. Currently, I have enrolled for a
B.Ed in Mathematics Education with the University of
North West
For how long have you been a mathematics
I have twelve (12) years teaching experience and as a
teacher?
mathematics teacher.
69
How often do you review the lessons that you have
I try to reflect on almost all the lessons that I teach.
taught?
Basically I check whether the approach I have used in
the previous lesson was the appropriate one and also
whether I cannot improve on the type of explanations that
I have given to the learners.
Table 4.7: How Teacher B may have developed the PCK.
From the above interview responses of Teacher B it can be noted that he received training as
a mathematics teacher during his initial teacher training and has also furthered his studies by
obtaining Mathematics I and II from a higher learning institution in South Africa. He also
attends in-service workshops. Most of the workshops that he has attended focused on
improving teacher pedagogic content knowledge in school mathematics regarding the new
areas or topics of functions and data handling. It is hardly surprising that Teacher B was able
to demonstrate adequate subject matter knowledge in teaching some aspects of quadratic
functions.
Furthermore, Teacher B does observe some of his colleagues as they teach. From such
observations it can be presumed that he learnt different teaching strategies that could be
employed when he teaches his learners as he claimed. His knowledge of teaching strategies
as observed when he presented lessons serves to confirm this claim. Teacher B holds a
diploma in professional education, specialising in the teaching of mathematics. He has 12
years experience as a teacher of mathematics at grades 10 to 12 and is currently studying for
a B.Ed in mathematics education. The teacher claims to always reflect on his previous lessons
so as to see how best to improve that lesson in future. This claim concurs with Shulman
(1987) where sources of PCK include teachers’ reflections on lessons taught.
70
4.9
CHAPTER CONCLUSION
In this chapter the results of the two case studies were presented. Findings on the two
teachers’ knowledge of the subject matter; knowledge of teaching strategies and knowledge
of learners’ conceptions on the topic of quadratic functions were presented. The findings
revealed that both teachers have adequate knowledge of the subject matter on the topic of
quadratic functions but insufficient knowledge of teaching strategies and knowledge about
learners’ misconceptions and learners’ difficulties on the topic of quadratic functions.
The two participating teachers have adequate professional qualifications to enable them to
teach quadratic functions effectively in grade 11. The teachers attend workshops which help
them to develop knowledge about the content on quadratic functions and teaching strategies.
The two teachers also observe their peers and do lesson reviews to help improve on their
teaching approaches.
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CHAPTER 5
DISCUSSION OF RESULTS
5.1
INTRODUCTION
This chapter aims to comment on the findings from the two case studies in relation to the
main research questions. Similarities and differences displayed by the two participating
teachers with regard to their teaching approaches and practices are highlighted in accordance
with the framework that guided the study.
Two main themes that were presumed to have an impact on effective teaching of quadratic
functions were identified. The first theme, pedagogical content knowledge, with three subthemes, endeavours to answer the first research question; “what pedagogical content
knowledge effective teachers display with regard to teaching quadratic functions in grade 11
classes?” This theme focused on the teaching and learning process in the classrooms and
how the participating teachers displayed knowledge of each of the three elements of
pedagogical content knowledge as they taught lessons on quadratic functions in their grade
11 classes.
The second theme sought to find answers to the second research question; “how did the
teacher develop the pedagogical content knowledge that they use?” The theme focused
mainly on how the participating teachers could have developed the knowledge on the use of
each of the three elements of pedagogical content knowledge, namely, knowledge of the
subject matter, knowledge of teaching strategies and knowledge of learners’ conceptions and
misconceptions on the topic of quadratic functions.
5.2 DISCUSSION OF THEMES
5.2.1 Pedagogical Content Knowledge of the Participating teachers
This main theme has been divided into three sub-themes; namely; knowledge of the subject
matter, knowledge of teaching strategies and knowledge of learners’ conceptions and
misconceptions. The discussion of findings on this theme, pedagogical content knowledge, is
based on how the teachers displayed the three elements of PCK during lesson observations,
lesson plan analysis and one-on-one interviews with the participating teachers.
72
Knowledge of the Subject Matter of Quadratic Functions
This sub-theme was aimed at finding out about the teachers’ knowledge on concepts of
quadratic functions, “facts and procedures, the reasons underlying procedures and the
relationships between concepts” in quadratic functions as well as how the teacher deployed
his knowledge during lesson presentation.
Based on the results from the three research instruments as triangulated, both teachers have
adequate knowledge of quadratic functions topics. When the two teachers taught their
respective grade 11 learners about how to draw the graph of a parabola using a table of
values, concepts such as dependent y- values, independent x-values and scale were accurately
explained by both teachers (ref. see section 4.2.1 and 4.2.4). Teacher A emphasised on the
use of procedural knowledge to calculate the y-values from a given set of x-values in order to
plot the graph.
On the other hand, Teacher B used both procedural knowledge and
conceptual knowledge (to a limited extent) to explain the concepts on how to draw the graph
of a parabola (ref. see section 4.2.4, line 15). The use of procedural knowledge by Teacher A
and the use of a combination of both procedural and conceptual knowledge by Teacher B
were seen to be effective by the researcher because learners were able to successfully do
problems associated with the concepts taught on their own.
While the use of procedural knowledge only by Teacher A was seen as effective (though
insight of concepts by learners may not be guaranteed) in solving problems in quadratic
functions, Star (2002) contends that the learning of procedures must be connected with
conceptual knowledge in order to foster the development of understanding of the concepts,
which is in line with what Teacher B was seen doing. Bosse' and Bahr (2008) indicate that if
teachers of mathematics apply the alliance of factual knowledge, procedural proficiency and
conceptual understanding, it provides a powerful way of learning quadratic functions by
learners. Furthermore; learners who are only taught procedures without understanding the
concepts, are often not sure when or how to use what they know and such learning is fragile
and ineffective.
The two teachers’ adequate knowledge of the subject matter on quadratic functions may be
linked to their academic and professional qualifications including experience. Teacher A
holds a degree in mathematics and has done an advanced professional course (ACE) in the
teaching of mathematics. He has eighteen years of experience as a mathematics teacher in
73
grades 10-12. Similarly, Teacher B holds a diploma in mathematics teaching and has done an
advanced professional course (ACE) in the teaching of mathematics. He has 12 years
experience in the teaching of mathematics in grades 10 to 12.
In summary, both teachers have adequate knowledge of quadratic functions topics. Teacher A
emphasises procedural knowledge to teach the topics whilst Teacher B incorporates both
procedural and conceptual knowledge (to a lesser extent though) in the teaching of quadratic
functions to bring effectiveness. The experience they have as teachers of grades 10-12,
professional and academic qualifications is presumed to have contributed to the development
of their subject matter knowledge on quadratic functions. The next paragraph will focus on
knowledge of teaching strategies of quadratic functions by the two teachers.
Knowledge of Teaching Strategies of Quadratic Functions
This particular element of pedagogical content knowledge focused on investigating the two
teachers’ knowledge of the following: (i)how to plan and teach lessons using a variety of
teaching strategies, (ii) how they engaged learners through questions and assessment
tasks,(iii) how they used the chalkboard during lesson presentation on topics of quadratic
functions.
Teacher A usually started his lessons by first reviewing the concepts which were dealt with
during the previous lessons by posing questions to the learners (ref. see section 4.2.1). The
questions were however mostly recall questions which required “yes” or “no” type of
response (ref. see section 4.2.1). It can be assumed that the aim of this (questioning) activity
was merely to check if the learners recall the concepts since there was no probing of the
learners.
Similarly, Teacher B reviewed lessons taught during the previous day and also did
corrections of exercises given the previous day. During such reviews and corrections,
Teacher B also engaged his learners through oral probing (ref. see section 4.2.4). The
difference with Teacher A regarding oral probing is that Teacher B sometimes asked his
learners questions that allowed them to express their mathematical thinking on the topic of
quadratic functions (ref. see section 4.2.4). Unfortunately, during such oral probing in the
lessons observed, he only pointed at learners who had raised their hands presumably because
he thought they knew the correct answers. This researcher thinks it would have been very
74
good had the teacher also tried to involve all the learners including those who did not raise
their hands just to get their thinking about the topic on quadratic functions under discussion
(own anecdote).
Both teachers used the telling or lecture method to present their lessons on quadratic
functions. According to Anthony and Walshaw (2009), when a teacher uses the telling
method, he or she is the main imparter of information while learners are passively listening.
The same authors (Anthony and Walshaw) indicate that effective teachers encourage
classroom exchanges in the form of carefully planned questions that encourage learners to
speak out their mathematical ideas about the concepts on quadratic functions. In such an
environment the teacher will be seen as guiding the learners on a topic about quadratic
functions whilst the learners themselves are the main contributors in the lesson. The two
teachers seldom used teaching methods that encouraged the learners to be the main speakers
(ref. see section 4.2.1 and 4.2.4); instead the teachers were the main imparters of information.
The over use of the telling method of teaching by the two teachers made their lessons on
quadratic functions to be teacher-centred. On rare occasions (especially in the lessons
observed), Teacher B sometimes allowed a learner to present a solution of a given exercise on
the chalkboard whilst the other learners were allowed to ask questions to the presenter.
The two teachers used the chalkboard very well during each lesson on quadratic functions.
Notes on the lesson were neatly developed on the chalkboard as the lesson progressed (ref.
see section 4.2.1 and 4.2.4). Examples that were used during the lessons; in particular on the
lesson about how to draw the graph of a parabola were written on the chalkboard for learners
to see how to draw the graph of a parabola. According to Anthony and Walshaw (2009),
effective teachers use tools and representation to bring about effectiveness in their teaching.
Class work and home work assignments were given at the end of each lesson by each of the
two participating teachers to assess the learners (ref. see 4.2.1 and 4.2.4).
In summary, the two teachers; Teacher A and Teacher B, used the telling method to present
most of their lessons on quadratic functions. Teacher A, usually asked recall type of questions
during lesson presentation whereas Teacher B sometimes posed questions that required his
learners to speak out their mathematical thinking regarding topics on quadratic functions. The
two teachers assessed their learners at the end of each lesson and gave them an additional
opportunity to learn the concepts through home work. The next paragraph will focus on
knowledge of learners’
75
Knowledge of Learners’ conceptions (misconceptions and pre-conceptions)
This element of pedagogical content knowledge was intended to find out the teachers’
knowledge of possible difficulties, errors, misconceptions and preconceptions that learners
might have on the topics of quadratic functions.
Based on the five observed lessons for each teacher, both teachers, Teacher A and Teacher B
assisted learners who had been identified as experiencing difficulties (through their failure of
correct application of concepts) regarding certain concepts such as exponential notation and
multiplication of integers and plotting of points on the graph respectively (ref. see 4.2.1 and
4.2.4). Errors, including learners’ difficulties; were usually discovered when the teachers
were doing reviews of lessons done the previous day. These errors and misconceptions were
also discovered by the two teachers when doing corrections of exercises given and also when
learners asked questions themselves.
The questioning technique of the two teachers was however not effective enough to reveal the
learners’ misconceptions (ref. Section 4.2.1 and 4.2.4). Teacher A did not ask probing
questions that would help expose the learners’ misconceptions about certain aspects of
quadratic functions during most of his lesson presentation (ref. see section 4.2.1). Most of the
questions that he asked during the lessons were mostly recall type of questions that required a
‘yes” or “no” response as already alluded to in earlier paragraphs. It was only in one instance
during a lesson about how to draw the graph of a parabola where Teacher A asked a learner a
probing question which resulted in revealing that a the learner had difficulties in dealing with
exponential notation and multiplication of integers (ref. see 4.2.1, line 15). This was the only
instance where the teacher asked a probing question which led to the discovery of the
learner’s difficulties through questioning. According to Kiliḉ (2011), “teachers need to
identify
learners’
misconceptions
and
difficulties
correctly
and
eliminate
such
misconceptions and difficulties by asking probing questions or using appropriate tasks.
Moreover, teachers need to be able to determine the source of students’ difficulties and errors
in order to correct them effectively”.
On the other hand, Teacher B sometimes asked his learners probing questions that allowed
them to express their mathematical thinking. This observation was made when the teacher
taught learners how to draw the graph of a parabola (ref. see 4.2.4, line 7). In that lesson he
also came to class with an idea of possible learners’ difficulties that his learners might
76
experience about how to draw the graph of a parabola (ref. see section 4.2.4, line 15). In this
case he had some prior knowledge of students’ potential learning difficulty and used this
knowledge on the learners’ behalf. This was one of the rare occasions where he came to class
with knowledge of possible learners’ difficulties on the topic of quadratic functions.
The two teachers’ knowledge of learners’ difficulties and misconceptions on the topic of
quadratic functions appears to be limited, though Teacher B is better than teacher A on this
aspect. They both neither engaged learners with probing questions with any measure of
consistency ,nor came to their respective classes with a clear set of learners’ difficulties and
misconceptions about the topic on quadratic functions. The two teachers’ failure to use
probing questions in order to diagnose misconceptions and their habitual failure to come to
class with a clear set of possible learners’ difficulties mean that they do not have sufficient
knowledge about this component of pedagogical content knowledge.
According to (An, Kulm, & Wu, 2004), “teachers who posses a strong knowledge base in
this domain know which mathematical concepts are difficult to grasp, which concepts
learners typically have misconceptions about and also know possible sources of their
learners’ errors”. Moreover they are aware of how to eliminate those difficulties, errors and
misconceptions.
In summary, the two teachers do oral probing to try to discover misconceptions that learners
may have regarding the topic on quadratic functions but their questioning technique is not
effective. The teachers mostly use recall type of questions though Teacher B, occasionally
asked his learners thought provoking questions. The two teachers were able to deal with
learners’ misconceptions and learners’ difficulties on the topics of quadratic functions once
discovered. They used the learners’ responses to identify the type of misconceptions that their
learners might have had. It has been noted that the two teachers’ knowledge of learners’
misconceptions and difficulties is however very limited; though comparatively, Teacher B
was better than Teacher A on this knowledge domain.
The next section will focus on how the teacher may have developed his pedagogical content
knowledge.
77
5.2.2 Development of the Pedagogical Content Knowledge of the Participating Teachers
This theme, which helps to answer the second research question, was intended to reveal how
the two teachers may have acquired the type of pedagogical content knowledge that they
were using by analysing the information gleaned from the interviews conducted (ref. see
sections 4.2.2 and 4.2.3).
Teacher A received further training on the teaching of mathematics after his initial teacher
training programme. His initial teacher training is a BSc degree, majoring in Mathematics
and Statistics and he also has an advanced certificate in Education, specialising in the
teaching of mathematics. Teacher B holds a National Professional Diploma in Education
obtained from a certain University. He obtained mathematics I and II at a University after his
initial teacher training programme. He further obtained an advanced certificate in
mathematics teaching. In terms of the Employment of Educators Act 1998 (South Africa),
both educators are relevantly qualified to teach mathematics at grades 10-12 level.
Their qualifications may be the reason why their content knowledge of the subject matter on
quadratic functions can be considered to be adequate. They both have a good grasp of the
various topics of quadratic functions.
The two teachers attend workshops arranged by their district. Most of the workshops that
Teacher A attended dealt with aspects of how to teach certain topics such as sequences and
analytical geometry. For Teacher B, most of the workshops that he attended dealt with
content matter such as functions, data handling and transformation of functions. It appears as
if the workshops that they have attended did not discuss learners’ conceptions and learners’
difficulties on the topic of quadratic functions. The two teachers appear to have limited
knowledge about knowledge of learners’ conceptions and misconceptions.
Regarding their teaching experience which is also regarded as a source of pedagogical
content knowledge (Shulman, 1987), Teacher A has 18 years experience as a mathematics
teacher whilst Teacher B has 12 years. Their experience in teaching the subject may have
contributed to their effectiveness in terms of getting good results in the grade 12 final
mathematics examination. Surprisingly, despite their experience in teaching the subject, their
knowledge of learners’ conceptions and misconceptions is low. Both teachers observe other
78
mathematics teachers teaching quadratic functions and other topics in the grade 11 and 12
Mathematics syllabi.
In summary, the two teachers are relevantly qualified to teach mathematics in grades 10-12
and have the required content knowledge to teach quadratic functions. They also have
appropriate experience in terms of long service but it is surprising that their knowledge of
learners’ misconceptions is limited though teacher B is better compared to Teacher A.
5.3 CONCLUSION
The study effectively investigated the pedagogical content knowledge held by two successful
teachers whose selection to participate was based on their learners’ performance in the Grade
12 Mathematics examination the past three years. Their average performance was in the
region of a 70% pass rate over the past three years. A qualitative research approach using the
case study was used. A framework to guide the study and assist in data collection was
developed. The framework that was developed focused on three elements of pedagogical
content knowledge as having influence on effective teaching of quadratic functions.
Observation protocols, interviews and lesson plan analysis were used to gather data about the
teachers’ approaches to teaching quadratic functions as a way of collecting the teachers’
pedagogical content knowledge. The data collected was triangulated via one-on-one
interviews with the teachers before each of five lesson presentations, during lesson
observation and through lesson plan analysis. In addition, the study attempted to find out how
the teachers had developed the pedagogical content knowledge that they used in the teaching
of quadratic functions.
The findings of the study
The findings of the study in relation to the research questions stated below are presented:
1. What pedagogical content knowledge do effective teachers display with regard
to teaching quadratic functions in grade 11?
2. How did the teachers develop the PCK that the y use when teaching quadratic
functions?
For the first research question, the two effective teachers were found to have adequate
knowledge of the subject content knowledge on quadratic functions. They presented logical
79
and accurate lessons to their learners which made the topics on quadratic functions accessible
to the learners. Exercises given to learners were usually taken from past examination question
papers by both teachers. Teacher A used procedural knowledge to present his lessons on
quadratic functions. Teacher B sometimes incorporated procedural knowledge and conceptual
knowledge (to a limited extent) to teach. Drill work on past examination question papers was
emphasised by both teachers. These practices may be presumed to have assisted the two
teachers to produce good results in grade 12 mathematics National Senior Certificate
examinations.
However, the study found that the teachers’ knowledge of instructional strategies was limited
though effective. Both teachers relied heavily on the lecture method. They used the lecture
method in all the lessons presented, though Teacher B would occasionally employ the use of
group work where learners were grouped in fours or fives to work on a solution of a given
exercise and the teacher would then choose one learner from a volunteering group to present
the solutions. No other teaching method was used by the teachers. Particularly noteworthy
was the absence of probing questions that would help determine learners’ preconceptions
about the topic on quadratic functions or to make a note of them. From the literature review
(ref. section 2.2.2.2), it was found that effective teaching strategies are able to maximize
learners’ time and engagement in learning tasks and encourage learners’ active participation
during lessons. The attributes mentioned above can only be realized if the teacher uses
instructional strategies that can initiate and sustain insightful learning processes in the
mathematics classroom lesson.
Findings on the teachers’ knowledge of learners’ conceptions and misconceptions on the
topics of quadratic functions revealed that, the two teachers have limited knowledge on this
knowledge domain. They were sometimes unable to detect the misconceptions which the
learners had by analysing their (learners) responses to questions.
These findings are
consistent with the findings of Prediger (2010), who in a study which focused on diagnostic
competencies of pre-service teachers on learners’ misconceptions and difficulties; found that
the teachers have difficulties in properly analysing their learners’ responses to diagnose
misconceptions. To resolve the problem of learner difficulties, such as the common issue of
poor arithmetic skills, asking probing questions could well reveal the actual source of the
cause of the learners’ misunderstanding, rather than the teachers merely reiterating the
procedures that they had taught. Most of the lessons were teacher-centred with questions
80
posed to the learners being recall type, closed questions requiring one word answers only.
The type of questions posed mostly did not allow learners to explain their thinking so that any
misconception could be noted and corrected. Learners’ solutions were mostly used as a
resource to deal with misconceptions. Both teachers selected exercises from past examination
question papers for their learners and the learners were drilled on these. This may suggest that
their success in producing good grade 12 results in the National Senior Certificate
mathematics examinations is based on the drill and practice strategy solely for the purpose of
passing the examination.
The teachers’ effectiveness as defined in this study could be linked to their experience as
mathematics teachers, their knowledge of the subject matter and their use of examinationrelated questions that allowed the learners to become fully acquainted with the form of
questions that would be found in the final examination. So, knowledge of the subject matter,
the use of procedural knowledge and drill work makes some teachers effective in terms of
producing grade 12 results. The findings of Star (2005) in his study that investigated the role
of procedural knowledge in mathematics learning partly support the results emerging from
this current study. He (Star) found that procedural knowledge applied with insight of the
concepts is a useful tool in mathematics learning by learners.
5.4
Limitation of the study
The limitations of this study need to be taken into account when considering the findings as
the researcher was aware of them from the outset and care was taken to accommodate them in
the best possible way.

The study dealt with only two cases; hence generalisation would need to be handled
with discretion.

Investigating teachers’ pedagogical content knowledge is challenging as it reveals
itself in many places and ways, such as, in teachers’ planning, classroom interactions,
explanations, mathematical competency, and so on and a study of only one
environment could lead to a limited perspective emerging.

The study was restricted to one circuit only due to funding and time constraints.
81
5.5
Recommendations
The study has revealed that though the teachers are effective in terms of producing good
mathematics results in Grade 12, their knowledge of teaching strategies and that of knowing
their learners’ conceptions and misconceptions on the topic of quadratic functions; is
inadequate. The teachers’ use of procedural knowledge and drill work contributes to their
learners achieving good results in mathematics. To improve the two teachers’ knowledge of
learners misconceptions and conceptions on the topics of quadratic functions, they should use
the learners’ responses as a resource of revealing the learners’ thoughts about a specific topic
on quadratic functions. They need to ask questions that require learners to explain their ideas
about a topic on quadratic functions. In addition, the way the teachers prepare their lessons
needs to reflect possible learners’ difficulties that would need to be addressed during the
lesson. Possible learners’ difficulties can be obtained from the learners’ responses during oral
probing and also from the learners’ solutions of exercises given. Furthermore, the teachers’
lesson plans should also indicate all key concepts, prior knowledge of learners, the teaching
strategy to be used as well as possible misconceptions that the teacher thinks the learners
might experience. The teachers need to be exposed to workshops that not only deal with the
content knowledge of quadratic functions only but also focuses on knowledge domains of
learners’ misconceptions and knowledge of teaching strategies that could be used to
effectively teach quadratic functions.
Recommendation for further studies
Based on the findings of this study, it has been noted that some successful teachers do not
necessarily know or practise all the identified elements of pedagogical content knowledge. A
good knowledge of the subject matter and use of drill work can bring success in teaching
based on the teaching approach that teachers use. Teachers teach mathematics based on their
understanding and belief of what they think mathematics is and how it should be taught. So, a
study to investigate whether there is a relationship between the pedagogical content
knowledge held by a teacher on a specific topic in mathematics and the teacher’s
mathematical beliefs is necessary.
82
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92
APPENDIX A : Pre-Lesson Interview Questions
ELEMENTS
OF
PCK
QUESTION RELATED TO PCK ELEMENT
RESPONSES
FOR THIS STUDY
a.
Knowledge
the
of
1.
subject
matter
2.
.
3.
What are the key concepts in the
………………………………
lesson that you are about to teach?
…………………………………
Draw a concept map illustrating the
…………………………………
sequence you will follow to teach
…….…………………………..
these key concepts.
….……………………………
any
…………………………………
the
…………………………………
learners must know? If so, what
…………………………………
does the procedure involve?
………………………………
Which teaching strategy will you
………………………………………….
employ
…………………………………………...
Does
the
procedural
b.
Knowledge
lesson
involve
knowledge
that
of
teaching
1.
strategies
to
ensure
successful
delivery of this lesson?
2.
………………………………
Why did you choose such a
strategy?
3.
In your selection of examples to be
used in this lesson, have you
selected real-life examples?
4.
Knowledge
learners’
conceptions
of
1.
2.
3.
4.
5.
What is the goal/aim of your
lesson?
Which learners’ prior knowledge is
regarded as important before the
above key concepts can be
successfully taught to learners?
…………………………………………
…………………………………………
…………………………………………
……………………………
………………………………………….
What
possible
learner
misconceptions do you anticipate
regarding this lesson?
…………………………………………...
How will you assist learners who
experience difficulties with this
lesson?
…………………………………………..
Have you prepared an assessment
instrument to evaluate whether the
goal of the lesson have been
achieved?
………………………………………….
…………………………………………..
………………………………………….
………………………………………….
Adapted from Chick, Baker, Pham (2006): Probing teachers’ pedagogical content knowledge: Lessons from the case
of the subtraction algorithm)
93
APPENDIX B : Classroom Observation Protocol
Lesson number:-__________
Topic__________________
Duration of period:_____
1. What pedagogical content knowledge successful teachers display with regard to the teaching of
quadratic functions?
EVIDENT WHEN THE TEACHER….
OBSERVED PRACTICE DISPLAYED
Knowledge of the
1.
thorough
1…………………………………………..........................
subject matter
conceptual understanding of identified
……………………………………………………………
aspects of functions.
……………………………………………………………
PCK ELEMENT TO BE
OBSERVED
b.
2.
Exhibits
deep
Identifies
components
functions
critical
within
that
understanding
and
mathematical
the
concept
are fundamental
and
applying
of
……………………………………………………………
for
……………………………………………………………
that
……………………………………………………………
concept.
……………………………………………………………
3. Displays skills for solving problems in
3 ………………………………………………………….
………………………………………………………….
the area of functions.
b. Knowledge of Teaching
strategies
c. Knowledge of learners’
conceptions
2…………………………………………..........................
1. Uses appropriate activities in
Instruction
1……………………………………………………………
2. Uses real life examples and
analogies in instruction
2.……………………………………………………………
3. Utilizes different instructional
strategies in presentations.
3. ……………………………………………………………
1. Addresses learners’ misconceptions
1…………………………………………………………..
2. Displays expectations of possible
difficulties learners may face during
learning and address such.
……………………………………………………………
……………………………………………………………
…………………………………………..........................
……………………………………………………………
2……………………………………………………………
…………………………………………………………….
3. Discusses learners’ ways of thinking
about a concept.
3……………………………………………………………
4. Shows an awareness of the
instruments to measure student
learning and how to use them
4………………………………...........................................
……………………………………………………………
…………………………………………………………….
Framework for observing Pedagogical Content Knowledge of teachers (Based on Chick, Baker, Pham (2006): Probing
teachers’ pedagogical content knowledge: Lessons from the case of the subtraction algorithm)
94
APPENDIX C: Guiding Questions on Lesson Plan Analysis
ELEMENT OF PCK
CHECKED IN THE
PREPARATION
a.
Knowledge of subject
matter
1. Are key concepts to be taught
during the lesson indicated in the
preparations?
2. Does the preparation indicate
possible procedures to be taught to
the learners?
3. Does the lesson preparation
reflect accurate concepts and
procedures associated with the
topic on quadratic functions?
b.
Knowledge of teaching
strategies
1. Is the teaching strategy to be
used stated in the preparations?
2. Are alternative teaching
strategies to be used during the
lesson reflected in the
preparations?
c. Knowledge of
learners’ conceptions
3. Are examples to be used during
the lesson indicated in the lesson
preparation?
1. Does the preparation reflect
possible misconceptions that will be
addressed during the lesson?
2. Does the preparation reflect the
required learners’ prior knowledge
before the start of the new topic?
3. Are possible learners’ difficulties
reflected in the preparations?
4. Is an assessment instrument
indicated in the preparations?
5. Is the goal of the lesson clearly
stated in the preparations?
95
OBSERVATIONS MADE
APPENDIX D: Interview questions on Pedagogical Content Knowledge Development of the Participating
Teachers
Item
Question to be asked
1
(a) Did you receive any special training as a mathematics teacher after your initial teacher training?
2
(a) Do you attend workshops that focus on teacher development?
(b) What have you gained from attending such workshops?
3
(a) Have you ever observed your colleague when he/she is teaching a mathematics lesson?
(b) What did you learn from such an observation?
4
What are your qualifications as a mathematics teacher?
5
For how long have you been a mathematics teacher?
6
How often do you review the lessons that you have taught?
Adapted from Shulman (1987): Knowledge and Teaching: Foundations of the new reform.
96
APPENDIX E: Letter to request for Permission to conduct a Research
Enq: Sibuyi C.D
P.O Box 336
Cell: 082 499 8277
THULAMAHASHE
E-mail: [email protected]
1365
18th February 2011
The Head of Department
Department of Education
Private Bag X11341
NELSPRUIT
1200
Sir/Madam
REQUEST FOR PERMISSION TO CONDUCT RESEARCH IN SOME OF YOUR SCHOOLS:
MYSELF
The above matter bears reference;
1.
I, Charles Duzephi Sibuyi, hereby request to conduct a research in some of your schools that will meet
the requirements of the sampling technique that will be used in the study.
2.
I am currently registered with the University of Pretoria, as an M.Ed (Assessment and Quality
Assurance) student.
3.
The title of my research study is: Investigating Successful Teachers’ Pedagogical Content
Knowledge in teaching Quadratic Functions in School Mathematics and I have successfully
defended it.
4.
The study will use a qualitative research design using a case study method where data will be collected
from selected teachers.
5.
Hoping for a favourable response to my request.
Yours Faithfully
__________________
Sibuyi Charles Duzephi
97
APPENDIX F: Permission from The Provincial Department of Education
98
99
APPENDIX G: Letter to the Principals
Enq: Sibuyi C.D
P.O Box 336
Cell: 082 499 8277
THULAMAHASHE
1365
………………………..
The Principal
………………………………………………………………..
………………………………………………………………..
………………………………………………………………..
Sir/Madam
REQUEST FOR PERMISSION TO CONDUCT A RESEARCH IN YOUR SCHOOL: MYSELF
The above matter refers;
1.
I, Charles Duzephi Sibuyi, hereby request to conduct a research in your school.
2.
I am currently registered with the University of Pretoria as an M.Ed student.
3.
The title of my study is: Investigating Successful teachers’ pedagogical content knowledge when
teaching quadratic functions in school mathematics.
4.
The research questions are :
1.
What pedagogical content knowledge do successful teachers display when they teach quadratic
functions in grade 11?
2.
5.
How did the teachers develop the pedagogical content knowledge that they use?
Your grade 11 mathematics teacher has been identified as being one of the successful teachers in the
district based on his/her previous grade 12 mathematics results in the National Senior Certificate for the
past three years.
6.
Your teacher will be expected to prepare and teach five lessons based on quadratic functions. Each
lesson will be observed and video taped. Two interviews, one before the start of lesson observations
and the other one after all the lessons have been observed, will be conducted with your teacher. The
teacher’s lesson plans will also be analysed.
7.
There will be no financial incentives for participating in the research but findings will be made known
to your teacher. The teacher may withdraw at anytime that he/she feels like and the data collected
before withdrawal will not be used any further.
8.
The data collection instruments as well as consent forms to participate are herein attached for your
attention.
Thanking you in advance.
Yours Faithfully
____________________
Sibuyi Charles Duzephi
100
APPENDIX H: Letter to Invite Participants
Enq: Sibuyi C.D
P.O Box 336
Cell: 082 499 8277
THULAMAHASHE
1365
………………………….
Mr/Ms/Dr/Hon.……………………………………….
…………………………………………………………………
Sir/ Madam
INVITATION TO PARTICIPATE IN A RESEARCH PROJECT: YOURSELF
The above matter refers;
1.
I, Charles Duzephi Sibuyi, hereby invite you to be a participant in a research to be conducted in your
school.
2.
I am currently enrolled for an M.Ed degree with the University of Pretoria and this research project is a
pre-requisite for me to be able to fulfil the requirements of the mentioned degree.
3.
The title for my research is: Investigating Successful teachers’ pedagogical content knowledge
when teaching quadratic functions in school mathematics.
4.
The research questions are :
1.
What pedagogical content knowledge do successful teachers display when they teach quadratic
functions in grade 11?
2.
5.
How did the teachers develop the pedagogical content knowledge that they use?
As a grade 11 mathematics teacher, you have been selected to participate in this research based on your
previous good performance in grade 12 mathematics National Senior Certificate examination results
for the past three years.
6.
You will be expected to prepare and teach five lessons based on quadratic functions. All your lessons
will be observed and will also be video taped. Two Interviews will be conducted with you, one before
the start of lesson observations and the other, after the last observed lesson. Furthermore, all your
lesson plans for the five lessons will be analysed.
7.
Kindly note that learners who may withdraw from the project, will have to be taught the lesson
that they have missed due to their withdrawal.
8.
Neither risks nor health hazards are anticipated during your participation in this project.
9.
There will be no financial benefit for participating but the findings of the study will be made known to
you. More over, you are free to withdraw from participation at any time you feel like and the data
collected from you before your withdrawal will not be used any further.
10. Data collection instruments and a consent form are herein attached for your attention.
Yours Faithfully
_________________________
Sibuyi Charles Duzephi
101
APPENDIX I: Letter to request for permission of Learner Participation from Parents
Enq: Sibuyi C.D
P.O Box 336
Cell: 082 499 8277
THULAMAHASHE
1365
To :
………………………………………………………………..
………………………………………………………………..
Sir/Madam
REQUEST FOR PERMISSION TO HAVE YOUR CHILD AS PART OF THE GROUP OF LEARNERS WHO
WILL BE TAUGHT DURING OBSERVATIONS OF THEIR MATHEMATICS TEACHER
The above matter refers;
1.
I,
Charles
Duzephi
Sibuyi,
hereby
request
for
your
permission
to
include
your
child………………………………………………….(full names) to be part of the group that will be taught by their
mathematics teacher when he/she is observed by me ( a student researcher ) as he teaches lessons on quadratic functions at
school.
2. I am currently registered with the University of Pretoria as an M.Ed student.
3. The title of my study is: Investigating Successful teachers’ pedagogical content knowledge when teaching quadratic
functions in school mathematics.
4. The research questions are :
1. What pedagogical content knowledge do successful teachers display when they teach quadratic functions in
grade 11?
2. How did the teachers develop the pedagogical content knowledge that they use?
5. The grade 11 mathematics teacher in the school where your child is attending has been identified as one of the successful
teachers in the district based on his/her previous grade 12 mathematics results in the National Senior Certificate for the past
three years and this research is based on such teachers.
6. The teacher will be expected to prepare and teach five lessons based on quadratic functions. Each lesson will be observed
and video taped. Two interviews, one before the start of lesson observations and the other one after all the lessons have been
observed, will be conducted with your teacher. The teacher’s lesson plans will also be analysed.
7. There will be no financial incentives for participating in the research but findings will be made known to your child’s
teacher. You may withdraw your child from participating at anytime that you feel like and any data collected, which is
directly linked to your child before the withdrawal of your child will not be used any further. In case your child is not part of
the project, lessons missed by your child will be re-done by the teacher.
8. Consent forms to allow for voluntary participation of your child are herein attached for your attention.
Thanking you in advance.
Yours Faithfully
____________________
Sibuyi Charles Duzephi
102
APPENDIX J: Informed consent form
(Form for parents to allow their children to participate in the research project)
(Must be signed by the parent of each learner research participant who accepts to participate, and be
returned with the letter of acceptance of the invitation to participate in the research)
1
Title of research project: Investigating Successful Teachers’ pedagogical Content Knowledge when
teaching Quadratic Functions.
2
I,…………………………………………………, (full names) hereby voluntarily grant my permission
for my child ……………………………………………………..(full names) who is in grade 11
at………………………………………………………….(full names of school) to participate as a
learner when the mathematics teacher will be observed as part of data collection in the project as
explained to me in the letter of invitation to participate in a research project
by……………………..……………………………………………………………………………..
3
The purpose, research procedures, objectives, possible safety and health implications have been
explained to me and I understand them.
4
I understand my right to choose whether my child can participate in the project or not and that the
information obtained will be handled confidentially.
5
I am aware that the results of the investigation may be used for the purposes of publication.
6
I am aware that it is within my rights to withdraw my child from participation in the project at any time
I may feel like.
7
I am aware that in case I withdraw my child from the project, the subject teacher will teach my child
the lesson that he/she may have missed.
(Upon signature of this form, the parent will be provided with a copy).
Signed: _________________________
Date: _______________
Witness: _________________________
Date: _______________
Researcher: ______________________
Date: _______________
103
APPENDIX K: TEACHER A LESSON OBSERVATION DESCRIPTION
Excerpt 1: How to draw the graph of a Quadratic Function (Parabola)
[In each of the excerpts, names of learners that appear are not their real names. Pseudonyms
have been used to protect their identities]
Line 1: Teacher A: “Today’s lesson will be about drawing the graph of a quadratic
equation function. Such a function is of the form y = a x2 + b x + c or y = a x2 + b x or y = x2.
These are the three forms of the equation. Note that “a” the coefficient of x2; can never be
equal to zero. The simplest form of the equation is y = x2. Do you understand? [The three
forms of the equation were written on the chalkboard]
Line 6: Learners: “Yes” […In a Chorus]
Line 7: Teacher A: “Now I will show you how to draw the graphs of such functions. You
must pay attention as I present how it is done. We will start by the graph of y = x2 – 4 as an
example. In the given equation, what is the coefficient of x2? Raise your hand if you know the
answer. A..h..h!![Humming and looking around the class]Yes Lufuno”.
Line 11: Lufuno: “Two Sir”.
Line 12: Teacher A: “No! Any one else to help, Zenani, give us the solution.”
Line 13: Zenani: “The coefficient of x2 is one”.
Line 14: Teacher A: ‘Yes, the coefficient of x2 in the equation y = x2 – 4 is one. Now, to
draw the graph of the given quadratic function, we need to choose, say, nine values of x and
substitute each one of them in the equation y = x2 – 4 to get the corresponding value of y and
then write the values of x and y in coordinate form; that is (x; y).We are going to draw a
table of values and the coordinate as shown on the chalkboard. Lets choose the following xvalues: -4;-3;-2;-2;-1;0;1;2;3;4 and we call the x-axis the independent variable. The y-axis
will form the dependent variable since y-values depend on the x-values.”
Line 21: Mpendulo [A learner in the class]: “Sir, but why did you choose these x-values?
Can I choose other x- values besides the ones that you have chosen?”
Line 23: Teacher A: ‘You have asked me two questions neh? For your first question, I have
decided to choose these values and for your second question; YES you can choose other
104
values but there must be equal number of negative numbers and positive numbers including
zero. Do you understand?”
Line 27: Mpendulo: “Yes”
Line 28: Teacher A: “Now that we have the values of x, we need to substitute these values in
to the equation y = x2 – 4 to get the corresponding values of y and be able to form
coordinates that will be plotted on the graph. So, for x=-4; y = (-4)2 – 4 = 16 – 4 = 12”
Line 31: Zenani [A learner in the class, interjects]: “No Sir, y must be -12”
Line 32: Teacher A: ‘Why do you think so?”
Line 33: Zenani: “Because -4 squared is -8 and -8-4 = -12”
Line 35: Teacher A: “No, -4 squared is 16 and 16 – 4 = 12”
Line 36: Zenani: “I don’t understand this”.
Line 37: Teacher A: “You see Zenani; -4 squared is like -4x-4 = 16. Do you understand?”
Line 38: Zenani:” Ok”. [Showing doubts]
Line 39: Teacher A: “Let’s proceed. For x = -3; y = (-3)2 – 4 = 5. So the coordinates are (3; 5)”
[Teacher A calculated the corresponding y-values using the chosen x-values and the table
below was developed]
x-values
-4
-3
-2
-1
0
1
2
3
4
y-values
12
5
0
-3
-4
-3
0
5
12
(x;y)
(4;12)
(-3;5)
(-2; 0 )
(-1;-3)
(0;-4)
(1;-3)
(2; 0)
(3;5)
(4;12)
Line 46: Teacher A: “Now that we have all the coordinate points for our chosen x-values,
we can now plot the points on the graph paper that I have given you. To plot the points, you
need to first choose a scale that will be used to represent the x and y values. The graph of the
105
function must be legible enough but not too big for your chosen scale. To decide on the scale,
choose a number of small squares to represent one unit. You can choose, say, 5 small squares
to represent one unit. So, five squares represent 1 unit then ten small squares, represent 2
units etc. Do you understand?” [The graph shown on section 4.2.21 was drawn]
Line 53: Class: “Yes!” [In a chorus]
Line 54: Learner: “Sir, I find it difficult to choose a scale”
Line 55: Teacher A: “Ok, as I said before, decide on the number of small squares in your
graph paper which will make one unit. As an example, 5 small squares may form 1 unit, so 10
small squares will be 2 units and so forth. Do you see that?”
Line 58: Learner: “Yes”
Line 59: Teacher A: “For your home work, you are to draw graphs of the following
functions using a table of values: (i) y = x2 – 9 (ii) –x2 +2x -3 (iii) –x2 + 2x + 8. Take note
that the following steps must be followed in the order shown to be able to draw the graph of a
quadratic function:
1. Know the equation of the function
2. Choose seven or nine values of x, with equal number of negative values and
positives and must include a zero. [They are called the independent values. So
the x-axis is the independent axis]
3. Substitute each value in to the equation of the function and get the
corresponding value of y and write in coordinate form (x;y).
4. Decide on a suitable scale to draw the x and y values of your Cartesian plane.
5. Plot the points and then join them. Your graph is then drawn. Do not forget to
label the axis as well as your graph by writing the equation of the function.”
Line 72: Teacher A: “Make sure that you do your home work to gain confidence in drawing
these graphs”. [The lesson was concluded by giving learners an exercise as a home work as
the bell rung indicating the beginning of the next period]
106
APPENDIX K (Continued)
TOPIC: How to draw Sketch graphs of a Parabola
Line 1: Teacher A: “We have seen how we can draw a graph of a parabola using a table of
values. All of you will have noted that it involves a lot of calculation for the corresponding
values of y after having chosen the seven or nine values of x. Do you still remember how it is
done?”
Line 5: Learners: “Yes!!!” [In a Chorus]
Line 6: Teacher A: “Today, I want to show you a much shorter method of drawing the
graph but this time it is a sketch graph which is drawn not according to a scale. You do not
need a scale to draw the sketch graph of a parabola. You only approximate the positions of
the points”.
Line 10: Njabulo: [A learner in the class]: “How are we going to do that?”
Line 11: Teacher A: “Be patient please. To draw a rough sketch of a parabola you need the
following points :(i) the roots of the equation or the x-intercepts which we obtain when y=0;
that is; when the function value is zero. You also need the (ii) y-intercept; which is obtained
when x=0. The third point that is important is (iii) the axis of symmetry or the line of
symmetry which we obtain by using the formula x=-b/2a. Lastly you need the (iv) the vertex
or the turning point which is obtained when x=-b/2a. The value of –b/2a is substituted in to
the main equation to get the y-value. Sometimes you can use the formula y = -(b2- 4 a c)/4a to
get the y-value. In other words, (-b/2a; - [b2- 4 a c]/4a) are the coordinates of the turning
point”.[ Teacher A wrote the four points on the chalkboard as he spoke]
Line 20: Learner: “Sir, can you show an example of how to do this? It seems there are so
many formulae to be used in order to draw the sketch graph and it looks difficult”.
Line 22: Teacher A: “Ok, I was just about to illustrate that with an example. Please pay
attention! Given y = x2 -2x – 8, draw a rough sketch of this function. You do not have to fear;
it is easy to do. Just watch. Four important steps must be followed in order to draw a sketch
graph of a parabola. You do not have to do the steps in the same sequence as I have
numbered them.”
107
(i) “We proceed to get the x-intercepts which are obtained when y=0; that is; x2- 2x 8=0 and we factorise the trinomial which becomes (x-4)(x-2)=0. This imply that
x=4 or x=-2. Written in coordinate points; the x-intercepts for this graph are
(4;0) and (-2;0). Do you understand?”
(ii) “The y-intercept is obtained when x=0; so for y=x2 -2x -8; when x=0 we have y=022(0) -8 = -8. The coordinates of the y-intercepts are (0;-8). Do you see that
class?”
(iii) “For the axis of symmetry or line of symmetry x=-b/2a. By the way, who can tell me
what the coefficients of x and x2 are in the given equation respectively?”
Line 36: Learner: “The coefficient of x is 2 and the coefficient of x2 is 1”.
Line 37: Another Learner: “No, the coefficient of x is -2 and that of x2 is 1”
Line 38: Teacher A: “Yes the coefficient of x is equal to the value of “b” which is -2 and the
coefficient of x2 is equal to “a” which is 1”.
(iv) “For x= -b/2a, imply that x=-(-2)/2(1) =1. So y = (1)2 -2(1) -8 = -9 hence the vertex
of the parabola will be (1;-9). Any question?”
Line 41: Class: “No question”.
Line 42: Teacher A: “I will now draw the rough sketch, using the coordinate points obtained
from the four important steps that I have shown you.”
[The teacher drew the rough sketch as shown in section 4.2.1.1(c), figure 4.2 and concluded
his lesson by giving learners an exercise and homework]
108
Appendix L:
TEACHER A
PRE-LESSON INTERVIEWS
TOPIC: HOW TO DRAW THE GRAPH OF A PARABOLA
1.
KNOWLEDGE OF THE SUBJECT MATTER
Researcher: What are the key concepts in the lesson that you are about to teach?
Teacher A: The key concepts in this lesson are the x-axis, y-axis, dependent values, and independent values
Researcher: Draw a concept map illustrating the sequence you will follow to teach these concepts.
Teacher A: Well, I will just give you how the lesson will flow from one aspect to the other. In this case it will
follow the sequence as drawn:
Formula for equation
choose values of X
substitute in equation
get Y
choose scale
plot
points on graph paper
Researcher: Does the lesson involve any procedural knowledge?
Teacher A: I want to show the learners a procedure that they would use to get coordinate points to be able to
draw graphs of a parabola
2.
KNOWLEDGE OF TEACHING STRATEGIES
Researcher: Which teaching strategy will be employed to ensure successful delivery of the lesson?
Teacher A: The lecture method is appropriate for this lesson because other methods such as group work would
need that I move around the learners’ desks and that is not possible given the size of the class”
Researcher: Why do you choose such a teaching strategy?
Teacher A: The lecture method helps me to save time and it is appropriate to be used given the large size of the
class. There is no room for movement and rearrangement of the sitting plan for learners to allow for group
work would waste valuable teaching time.
Researcher: In your selection of examples for illustration of the topic or concept, have you selected real life
examples?
Teacher A: No, I do not have real life examples but I have taken two questions from the Regional grade 11 final
examination papers 1 to be used as examples.
109
KNOWLEDGE OF LEARNERS’ MISCONCEPTIONS
3.
Researcher: What is the goal/aim of your lesson?
Teacher A: The goal of this lesson is to draw (according to scale) the graph of a given parabola using a table
of values.
Researcher: Which learners’ prior knowledge do you regard as important before the above topic can be
successfully taught to learners?
Teacher A: To draw the graph of a parabola, you must be able to substitute chosen values of x in to the
equation of the function and be able to get coordinates. You also need to be able to choose a suitable scale to
label your axis.
Researcher: What possible learners’ misconceptions do you anticipate regarding this particular lesson?
Teacher A: I have no idea about the possible misconceptions that the learners might have regarding this lesson,
but should such a situation arise during lesson presentation, I will deal with it in the classroom. I mean
whatever misunderstanding the learners might bring to my attention during the lesson, I will assist the learners.
Researcher: How would you assist learners who experience difficulties with regard to this particular lesson (on
any topic about quadratic function)?
Teacher A: I will give individual attention to the learners who are experiencing difficulties with the lessons or I
may as well refer such learners to the learners who have shown good understanding of the topics or I
will repeat the lesson if the situation warranted that. To repeat the lesson would depend on the number
of learners who need help.
Researcher: Have you prepared an assessment instrument to evaluate whether the goal of the lesson was
achieved?
Teacher A: Yes, I always have a class work or home work to gauge the level of learning that may have taken
place.
Researcher: Thank you for your time, we will meet during lesson presentation.
110
APPENDIX M: TEACHER A’S LESSON PLAN ANALYSIS
TOPIC: HOW TO DRAW THE GRAPH OF A PARABOLA
---------------------------------------------------------------------------------------------------------------KNOWLEDGE OF THE SUBJECT MATTER
Guiding question
Observation made
Categorization/Theme
a) Are key concepts on quadratic
Some of the key concepts on how
Teacher’s preparation reflects
functions to be taught indicated in
to draw the graph of a quadratic
knowledge of the subject matter
the lesson plan?
function were indicated in the
including concepts for the topic
lesson plan
b)
Does the preparation indicate
The preparation only mentions that
Procedures to be taught not
possible mathematical procedures
the teacher will help learners to
planned in advance
for the topic to be taught?
calculate the y-values and complete
a given table of values.
c) Does the lesson preparation
The lesson plan reflects accurate
Teacher A has good knowledge of
reflect accurate concepts associated
concepts associated with quadratic
the subject matter
with
functions and especially how to
the
topic
on
quadratic
functions?
draw the graph of a parabola.
KNOWLEDGE OF TEACHING STRATEGIES
Guiding Question
Observation Made
a) Is the teaching strategy to be
The
used stated in the lesson plan?
teacher’s activities as well as the
lesson
plan
Categorization/Theme
reflects
the
Teacher A used the lecture and
demonstrate method of teaching
learners’ activities. The teacher
will first do an exposition of how a
graph of a quadratic function is
drawn. Learners’ will calculate the
y-values from the chosen x-values
b)
Are
alternative
teaching
The lesson plan does not have an
Teacher A prefers to use one
strategies to be used during the
alternative
teaching method in a lesson
lesson reflected in the preparation?
indicated
teaching
APPENDIX N (continued)
111
strategy
Guiding Question
Observation Made
Categorization/Theme
c) Are examples to be used during
Example to be used during the
Teacher A prepares his examples in
the lesson indicated in the lesson
lesson was indicated in the lesson
advance before the lesson.
2
preparation?
plan. For this lesson, y= x -4 was
used as the main example.
KNOWLEDGE OF LEARNERS’ CONCEPTIONS AND MISCONCEPTIONS
Guiding Question
Observations Made
Categorization/Theme
a) Does the preparation reflect
The preparation does not reflect
Teacher A may not be aware of
possible misconceptions that will
possible learners’ misconceptions
possible misconceptions to be
be addressed during the lesson?
that will be addressed during the
addressed.
lesson.
b) Does the preparation reflect the
Yes, the required learners’ prior
Teacher A knows the required prior
required learners’ prior knowledge
knowledge was reflected on the
knowledge of learners before this
required before the new topic?
lesson plan. Learners should be
lesson.
able to calculate y-values given the
x-values
c)
Are
difficulties
possible
reflected
learners’
Yes, possible learners’ difficulties
Teacher A has some knowledge
in
were indicated in the lesson plan.
about the possible difficulties in the
Possible
topic.
the
preparation?
difficulties
included
learners having difficulty
with
choosing an appropriate scale.
d) Is an assessment instrument
Yes, a home work was given as an
Teacher A knows that learning
indicated in the preparations?
instrument to assess learning. y =
must be assessed after each lesson
2
x -9 was given as a home work.
and uses the appropriate
instrument.
e) Is the goal of the lesson clearly
The lesson plan did not reflect the
The teacher may not be attaching
stated in the preparation?
goal of the lesson.
any value in writing the goal of the
lesson in his lesson plans.
112
APPENDIX M (Continued)
113
APPENDIX N: Excerpts of Teacher B’s Lesson Observation Description
Topic: How to draw graphs of the form y = ax2
Line 1: Teacher B: Last year, in grade 10, you drew graphs of y= ax 2, which represents the simplest
form of a quadratic function. This year in grade 11, you are expected to draw the graphs of y = ax 2 +
bx and y = ax2 + bx + c. Can someone tell me the shape of the graph of y = ax2?
Line 4: Learner: It is cup shaped or bell shaped
Line 5: Teacher B: Correct. All graphs of quadratic functions are either cup shaped or bell shaped.
When doe s the graph has a cup shape and when does it have a bell shape?
Line 7: Learner: When the coefficient of x2 is positive it is cup shaped and when the coefficient of x2
is negative it is bell shaped.
Line 9: Teacher B: Well done! When the coefficient of x2 is positive we say the graph has a
minimum or cup shaped whilst when the coefficient of x2 is negative the graph has a maximum or bell
shaped. Now, to draw the graph of a parabola, you need to choose x-values which are symmetrical
about zero.
Line 13: Learner: What do you mean being symmetrical about zero?
Line 14: Teacher B: It is when you have equal number of negative values and positive values with
zero at the centre of the two set of numbers, for example -4;-3;-2;-1;0;1;2;3;4. Can you choose your
own numbers using the example given?
Line 17: Learner: Yes, -6;-5;-4;-3;-2;-1;0;1;2;3;4;5;6
Line 18: Teacher B: That is good. You do not have to choose such a lot of x-values. To illustrate the
use of a table of values; suppose we have to draw the graph of y = x2 – 9 and we choose our x-values
to be -4;-3;-2;-1;0;1;2;3;4. We need to substitute our x-values in to the equation y = x2 -9 in order to
get the y-values and form coordinate points. Let’s draw a table as shown on the chalkboard, and then
calculate the corresponding value of y. For x= -4; y = (-4)2 -9= 16 -9=7 [Teacher B calculated the
corresponding values of y together with the learners]
X
-4
-3
-2
-1
0
1
2
3
4
Y
7
0
-5
-8
-9
-8
-5
0
7
(x;y)
(-4;7)
(-3;0)
(-2;-5)
(-1;-8)
(0;-9)
(1;-8)
(2;-5)
(3;0)
(4;7)
Line 27: Teacher B: Now that we have the values of x and y, we are now in a position to draw the
graph of the parabola. We need to choose a suitable scale to use on our graph papers. As an example
choose 1 unit: 10mm and then calibrate your x and y axis. If 1 unit: 10mm, how many millimetres will
represent 2 units; 3 units etc?
Line 31: Learner: 2 units will be 20 mm and 3 units will be 30 mm
Line 32: Teacher B: Excellent! More over, you must calibrate both the x and y axis with the same
scale. In your graph paper, one small square is approximately 2 mm, so 5 small square units will be
equivalent to 10 mm.
114
Line 35: Learner: Sir, I cannot plot the point (-3;0). Show me how I can do it.
Line 36: Teacher B: Common problems experienced by learners in drawing graphs is their failure to
read the x and y values. Take note that the x- values are read in a vertical way whilst the y-values are
read along the horizontal direction. For (2;-5), the coordinate is the point of intersection of the x and
y values. For the coordinate point (-3;0), the point is right at x= -3 since y=0 along the x-axis and x=0
along the y-axis. See diagrams on the chalkboard.
(-3;0)
y
-3
2
-5-
x
(2;-5)
Do you see how you should plot the points?
Line 42: Learners: Yes!
Line 43: Teacher B: I will be moving around your desks to assist those who need assistance. For
your home work you will do the following: (i) y = x2- 4 (ii) y = x2-2x
[Lesson was concluded in this way]
115
Appendix O: TEACHER B’S PRE-LESSON INTERVIEWS
TOPIC: HOW TO DRAW THE GRAPH OF A PARABOLA
1.
KNOWLEDGE OF THE SUBJECT MATTER
Researcher: What are the key concepts in the lesson that you are about to teach?
Teacher B: The key concepts in this lesson are the x-axis, y-axis, coordinates, and scale.
Researcher: Draw a concept map illustrating the sequence you will follow to teach these concepts.
Teacher B: Well, my lesson is planned as shown
Choose values of X
substitute in equation
get Y
form coordinates points
choose scale
coordinate points on graph paper
Researcher: Does the lesson involve any procedural knowledge?
Teacher B: In a way, the lesson involves both conceptual knowledge and procedure knowledge. Knowing why
certain things are done the way they are done and what must be done first before the next step.
2.
KNOWLEDGE OF TEACHING STRATEGIES
Researcher: Which teaching strategy will be employed to ensure successful delivery of the lesson?
Teacher B:I will first explain to the learners how to draw this graph then arrange the learners in groups of say
five to six learners and then demonstrate how to draw this graph should be drawn using a table of values.
Researcher: Why do you choose such a teaching strategy?
Teacher B: Demonstration and lecture method will allow me to give the learners a guided practice whilst group
work will enable them to learn from each other.
Researcher: In your selection of examples for illustration of the topic or concept, have you selected real life
examples?
Teacher B: Well, in a way there is real life examples if you consider that the shapes of the quadratic functions
can be described in terms of either cup shaped or bell shaped.
3.
KNOWLEDGE OF LEARNERS’ MISCONCEPTIONS
Researcher: What is the goal/aim of your lesson?
Teacher B: The goal of this lesson is to draw (according to scale) the graph of a given parabola using a table
of values.
116
APPENDIX O (Continued)
Researcher: Which learners’ prior knowledge do you regard as important before the above topic can be
successfully taught to learners?
Teacher B: Knowledge of the shape of the simplest quadratic function y = ax 2. This knowledge allows learners
to form a good link between the expanded forms of the quadratic function y = ax 2 + bx + c and y = ax2
Researcher: What possible learners’ misconceptions do you anticipate regarding this particular lesson?
Teacher B: Generally, most learners find it difficult to choose an appropriate scale for a graph and secondly,
they find it difficult to plot points correctly as they have a problem in reading the values of the system of axis in
the X-Y plane.
Researcher: How would you assist learners who experience difficulties with regard to this particular lesson (on
any topic about quadratic function)?
Teacher B:I usually give guided practice first for all learners and having arranged them in groups helps me to
move around each group and offer explanations to struggling groups. If all fails, I usually repeat the
lesson during extra lesson times.
Researcher: Have you prepared an assessment instrument to evaluate whether the goal of the lesson was
achieved?
Teacher B: Yes, I always have a class work or home work to assess the level of learning that took place.
Researcher: Thank you for your time, we will meet during lesson presentation.
117
APPENDIX P: TEACHER B’S LESSON PLAN ANALYSIS
TOPIC: HOW TO DRAW THE GRAPH OF A PARABOLA
---------------------------------------------------------------------------------------------------------------KNOWLEDGE OF THE SUBJECT MATTER
Guiding question
Observation made
Categorization/Theme
a) Are key concepts on quadratic
Key concepts on how to draw the
Teacher’s preparation reflects
functions to be taught indicated in
graph of a quadratic function were
knowledge of the subject matter
the lesson plan?
indicated in the lesson plan
including concepts for the topic
b)
The preparation mentions that the
Procedures to be taught not
possible mathematical procedures
teacher
planned in advance
for the topic to be taught?
calculate the y-values, plot the
Does the preparation indicate
will
help
learners
to
points and also completion of a
given table of values.
c) Does the lesson preparation
The lesson plan reflects accurate
Teacher B has good knowledge of
reflect accurate concepts associated
concepts associated with quadratic
the subject matter
with
functions and especially how to
the
topic
on
quadratic
functions?
draw the graph of a parabola.
KNOWLEDGE OF TEACHING STRATEGIES
Guiding Question
Observation Made
a) Is the teaching strategy to be
The
used stated in the lesson plan?
teacher’s activities as well as the
demonstrate method of teaching
learners’ activities. The teacher
and also grouped his learners
lesson
plan
Categorization/Theme
reflects
the
Teacher B used the lecture and
will first do an exposition of how a
graph of a quadratic function is
drawn. Learners will be grouped
and then calculate the y-values
whilst
the
teacher
will
offer
assistance
b)
Are
alternative
teaching
strategies to be used during the
Group work is mentioned as an
Teacher B indicated only one
alternative teaching strategy.
teaching method in a lesson though
lesson reflected in the preparation?
he used group work as well.
APPENDIX Q(Continued)
118
Guiding Question
Observation Made
Categorization/Theme
c) Are examples to be used during
Examples to be used during the
Teacher B did not reflect the
the lesson indicated in the lesson
lesson were not indicated in the
examples to be used on the lesson
preparation?
lesson plan but there was an
plan.
2
example. For this lesson, y= x -9
was used as the main example.
KNOWLEDGE OF LEARNERS’ CONCEPTIONS AND MISCONCEPTIONS
Guiding Question
Observations Made
Categorization/Theme
a) Does the preparation reflect
The preparation does not reflect
Teacher B may not be aware of
possible misconceptions that will
possible learners’ misconceptions
possible misconceptions to be
be addressed during the lesson?
that will be addressed during the
addressed.
lesson.
b) Does the preparation reflect the
Yes, the required learners’ prior
Teacher B knows the required prior
required learners’ prior knowledge
knowledge was reflected on the
knowledge of learners before this
required before the new topic?
lesson plan. Learners should be
lesson.
2
able to the graph of y = ax as
taught in grade 10
c) Are possible learners’
No, possible difficulties were not
Teacher B may have ignored
difficulties reflected in the
written on the lesson plan.
indicating possible difficulties in
preparation?
the topic because during
observations; he identified possible
learners’ difficulties.
d) Is an assessment instrument
Yes, a home work was given as an
Teacher B knows that learning
indicated in the preparations?
instrument to assess learning. y =
must be assessed after each lesson
2
2
x -4 and y = x – 2x was given as a
and uses the appropriate
home work.
instrument.
e) Is the goal of the lesson clearly
The outcome was indicated that the
The teacher understands that
stated in the preparation?
learners must be able to
setting the goal of the lesson; helps
investigate, analyse, describe and
him to remain focused on the goal
represent the function.
as he teaches.
119
APPENDIX P (Continued)
120
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